{"schema":"microcosm_public_ai_reader_complete_v0","role":"complete_public_text_corpus","artifact_kind":"public_ai_reader_complete_corpus","type_b_orientation":{"schema":"plectis_type_b_orientation_v0","audience":{"type_b_definition":"A Type B reader can reason over this portable packet but cannot inspect or change the live repositories and did not run the checks. Type A/B denotes substrate authority, not intelligence or model quality.","task":"Explain, question, compare, or audit the public material from this file; cite record paths; return conclusions as text for a human or Type A agent to carry back."},"what_the_hell_this_is":"Plectis joins two public repositories. The Plectis repository is the runnable claim-testing layer: 88 bounded components, their checks and receipts, the public doctrine, and the website projection. The Lean repository is the mathematical layer: Lean source, a claim-status registry, explicit open propositions, and the paper corpus. 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They are not one-to-one.","A paper is exposition. Checked Lean source is proof authority; the Lean claims registry owns public claim status; this build ran no Lean kernel.","The doctrine carried here is the complete public doctrine projection, not private raw seed or every internal doctrine record in ai_workflow.","Repository maps are counted, hash-bound depth-two topologies. They avoid embedding thousands of repeated path prefixes; follow public_tree_url for an exact leaf outside the named entrypoints."],"read_by_question":{"explain_the_whole_thing":["type_b_orientation.what_the_hell_this_is","type_b_orientation.inventory","headline_results","components","paper_modules","doctrine_entities"],"mathematics_or_a_paper":["headline_results","scholarly_corpus.papers","scholarly_corpus.formal_claim_control_plane","scholarly_corpus.repository_maps[repository=plectis-lean-erdos249-257]"],"what_is_still_open":["headline_results.remaining_open_propositions","scholarly_corpus.formal_claim_control_plane[record_id=lean_claims_registry].body.remaining_open_propositions","scholarly_corpus.formal_claim_control_plane[record_id=lean_scope]"],"one_component_end_to_end":["components","claim_ceiling_index if present","source_map if present"],"doctrine_or_system_design":["doctrine_entities","paper_modules","public_glossary"],"where_is_the_source":["scholarly_corpus.repository_maps","source_map if present","each record's source_refs/source_links"]},"packet_economy":{"packet_kind":"complete_public_text_corpus","serialization":"compact_json_in_insertion_order","body_rule":"Keep authoritative text bodies; remove only proven echoes and rendered/index re-projections that can be joined to a carried atom.","repository_tree_rule":"Carry a complete-path hash and counted depth-two topology, not a flat path dump.","verification_rule":"Coverage is checked against the active paper registry and the public site's live component/module/doctrine projections during the build."},"handoff_back":"End a substantive review with: the question, exact packet paths opened, conclusion, unresolved points, and any overclaim or underclaim. 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That authorization does not publish the private macro root, private raw media, account material, future cohorts, follow-ups, live services, or any material absent from the public repository/review packet boundary.","classification_classes":["public_by_design","public_optional_personal_data","strategic_private_origin_metadata_requiring_explicit_allowlist","prohibited_secret_or_live_payload"],"class_rules":{"public_by_design":"Public repository URLs, public page routes, stable public component ids, evidence modes, scope limits, and public contact routes intended for review.","public_optional_personal_data":"Name, project email, public professional links, and student-budget context only when they serve contact or reviewer-calibration purposes.","strategic_private_origin_metadata_requiring_explicit_allowlist":"Non-secret internal coordinates such as private-origin root slugs, state paths, implementation module paths, internal tool paths, source-body hashes, line counts, and capability symbols. Default action is project to a public handle or omit; keep only with an explicit allowlist receipt.","prohibited_secret_or_live_payload":"Secret material and live non-public data are excluded from the public packet; any finding in this class blocks distribution until removed."},"personal_disclosure_policy":{"allowed_public_contact_fields":["William Cook","williamwkcook+microcosm@gmail.com","https://www.linkedin.com/in/will-cook-278127234/","https://github.com/wcook04/plectis/security/advisories/new","https://github.com/wcook04/plectis"],"optional_context_fields":["University of Bristol economics undergraduate context","student-budget context when explaining support constraints","non-affiliation statements needed to avoid institutional endorsement"],"excluded_by_default":["exact age","private email address","home path or host username","private institution account state"],"current_public_contact_decision":{"schema":"microcosm_public_personal_disclosure_decision_v0","retained_fields":["name","project email alias","LinkedIn profile","University of Bristol economics undergraduate context","University non-affiliation statement"],"removed_fields":["exact age"],"reason":"Retained fields let reviewers identify and contact the maintainer; exact age is unnecessary personal data for the public packet."}},"strategic_metadata_policy":{"scanner_pattern_ids":["private_system_public_root_slug","private_portfolio_state_path","private_system_lib_path","private_system_ui_path","private_standard_path","private_control_tool_path","private_capability_symbol"],"default_action":"project_or_omit_unless_explicitly_allowlisted","current_public_packet_posture":"Strategic-coordinate scanner hits are review-blocking for hostile distribution until a human allowlists them or the builder projects them to public handles."},"public_runtime_rule":{"current_public_artifact_runnable_components":"see_component_digest","where_to_start_shape":["quickstart","one_component","evidence_line","source_link","scope_limit"],"runs_real_tools_meaning":"runs_real_tools describes whether the public component command invokes external machinery."},"reuse_terms_pointer":"legal_manifest.present_packet_and_docs_reuse_terms","artifact_identity_pointer":"composition_identity + projection-status.json::artifact_identity"},"freshness_note":{"generated_at_is_sentinel":true,"explanation":"generated_at is a fixed deterministic sentinel so builds are reproducible. It is not a live timestamp and must not be reported as freshness.","version_handle":"source_fingerprint (source graph only)","source_fingerprint":"09a2e6fdf14974d870e0650b36bd0890","version_handles":{"source_graph":{"field":"source_fingerprint","value":"09a2e6fdf14974d870e0650b36bd0890","meaning":"identity of the public source slice used by the builder"},"derived_payload":{"field":"composition_identity.payload_sha256_excluding_identity","meaning":"identity of this derived packet payload before its identity block is inserted"},"exact_bytes":{"field":"composition_identity.exact_byte_sha256_recorded_in","meaning":"pointer to the exact serialized-file hash in projection-status.json"}},"comparison_rule":"Use source_fingerprint to compare underlying source slices, the payload hash to compare derived packet revisions, and the exact-byte receipt to compare downloaded files. Two packets may share a source_fingerprint while differing in routing, instructions, packaging, or serialization."},"site":{"title":"Plectis","source_of_record":"https://github.com/wcook04/plectis","runtime_backend":"none","browser_connect_src":"none"},"derived_from":"assets/site-packet.js","companion_artifacts":{"concise_guide":"llms.txt","routed_quick_digest":"plectis-ai-reader-digest.json","full_object_map":"object-map.json","full_site_packet":"assets/site-packet.js"},"coverage_scope":{"included":["landing page visible text","generated docs pages","public component cards","public paper-module summaries","public relation rows","machine-readable public maps"],"omitted":["anything not present in the public repository, public site pages, or generated public packets","non-public account, browser, and model payload data","claims beyond the public website and source-linked Plectis source repository"],"authority_boundary":"This packet is a public-site projection and orientation aid. The Plectis source repository and the public source refs remain source authority; answers stop where those public records stop."},"reader_contract":{"schema":"microcosm_public_ai_reader_complete_contract_v0","what_this_is":"Every public text body the website shows, carried once. Read a section directly; you do not need to open another file for the prose.","answer_behavior":{"boundary_rule":"Answer only from this corpus and the linked public source refs. When the text does not support a claim, say the public evidence does not show it.","join":"Each page lists content_refs; resolve those ids against the matching atom array (components[] / paper_modules[] / doctrine_entities[]). Glossary terms are in public_glossary[]. Evidence-profile ids and relation-lattice rows resolve through their typed refs into content-graph.json.","plain_language":true},"read_order":["type_b_orientation","headline_results","dedupe_strategy","counts","pages","public_glossary","components","paper_modules","doctrine_entities","rules_and_ideas_ref","challenge_manifest","scholarly_corpus"],"challenge":"challenge_manifest is the only surface here you control rather than read. Everything else was chosen by the same author: the mechanism, the fixture, the expected value and the validator rule. Pick your own seed and run it. It already records one open, unrepaired defeat the project's own checks did not find.","papers":"scholarly_corpus carries the exact LaTeX of all 13 papers and the Lean claims registry, so paper and proof-claim questions are answerable here without opening either repository. Reading Lean source is not rechecking it: see scholarly_corpus.verification_boundary before saying anything was verified."},"headline_results":{"schema":"plectis_headline_results_v1","status":"available","role":"first_contact_projection_of_the_mathematics","projected_from":"scholarly_corpus.formal_claim_control_plane -- claim statements are quoted verbatim from the Lean claims registry, and the per-problem rows are quoted verbatim from the Lean repository's docs/problems.json, so this surface cannot drift from those authorities or inflate past them","what_this_is":"Work on eight unsolved Erd\u0151s problems: #68, #243, #249, #251, #257, #269, #1041, and #1049. All eight remain open; none is solved here. #249 and #257 are the two reviewed core programmes carrying the curated claims registry below; the other six are problem-owned expansion lanes whose per-problem authority is the problems index. What exists is a body of exact reductions, unconditional finite exclusions, structural theorems and machine-checked supporting mathematics, each stated at its own strength -- plus, for #269, a two-prime transcendence result established by a paper argument using Bugeaud-Laurent, deliberately not a Lean theorem, recorded as such in its problem row.","results_map":"The human per-problem map with Lean anchors is docs/RESULTS.md in the Lean repository: https://github.com/wcook04/plectis-lean-erdos249-257/blob/main/docs/RESULTS.md","per_problem":[{"erdos_number":243,"question":"Under a rapid-growth hypothesis on an integer sequence, does rationality of its reciprocal sum force the sequence to satisfy the Sylvester recurrence eventually?","status":"open","what_is_checked":["The exact product-cleared tail dynamics and the defect identity, as identities in the integer state variables.","That a vanishing centred state is absorbing under strict centring, so the state is either eventually zero or nowhere zero.","That eventual vanishing of the centred state, with the next tail state nonvanishing, forces the Sylvester recurrence eventually.","That a nonnegative centred state vanishes eventually, by descent.","That the centred state cannot be eventually constant and negative, at any magnitude and any common scale.","That the negative magnitude cannot be eventually periodic with positive drift, in the regime where each magnitude is below its multiplier.","A coprimality barrier: a sequence tending to infinity with uniformly bounded upward increments cannot permanently avoid infinitely many fresh pairwise coprime moduli.","That a reduced exact tail has pairwise coprime multipliers which its numerator permanently avoids, so no such tail has a divergent numerator with bounded rise.","That the tail gcd stabilises under cofinally bounded negative magnitudes.","Conditionally: exact dynamics, eventual strict centring, an eventually bounded negative part, and normalised vanishing together force the centred state to vanish eventually.","That survival through a fixed number of forced updates depends only on the initial state modulo a factorial of the horizon."],"what_is_not_checked":["The unrestricted problem.","That eventual strict centring and normalised vanishing follow from the growth hypothesis and rationality. Both are assumed by the final theorem and neither is derived, so that theorem constrains the state system rather than the original sequence.","Any exclusion of negative excursions whose magnitudes are unbounded along a cofinal set.","The criterion of Erdos and Straus and Duverney's conditional characterisation, which are cited rather than formalised and are both stronger than anything checked here."],"open_obligations":[{"id":"unbounded_negative_excursions","statement":"Exclude, or construct, an exact orbit whose centred state is negative infinitely often with magnitudes unbounded along that cofinal set. Every checked exclusion consumes some finiteness: a fixed set of prime divisors, a fixed period, or a fixed bound on the negative part."},{"id":"derive_the_two_analytic_hypotheses","statement":"Derive eventual strict centring and normalised vanishing from the growth hypothesis and rationality, or show that they do not follow."},{"id":"formalise_the_published_criteria","statement":"Kernel-check the Erdos-Straus weighted criterion and Duverney's conditional characterisation under explicit analytic hypotheses."}],"claim_registry_status":"not_registered; the claim registry does not carry these declarations and kernel checking them does not promote them into reviewed public claims"},{"erdos_number":249,"question":"Is the binary Lambert series sum phi(n)/2^n irrational?","status":"open","what_is_checked":["Conditionally: a cofinal strict 9/10 gap for the exact natural prime-tail orbit supplies the existing finite pivot-point escape and hence the reviewed irrationality endpoint.","The local s=1 and s=2 squared M\u00f6bius Euler-factor identities and the first two prime-power divisor-sum differences.","That the two-by-two checkerboard kills separable backgrounds and is unique up to scale under the four row and column cancellations.","An abstract four-point divisor-layer identity, stated from its four explicit factorisation hypotheses.","An affine completion no-go: every residue class modulo an even modulus has a digit that recentres one carry step.","Exact producer and consumer interfaces for prime-ray layer supply, bounded-degree order witnesses, and finite-prime-support escape."],"what_is_not_checked":["The strict prime-tail orbit gap, which is the unproved producer consumed by the conditional endpoint.","The packet's sharp parabolic Stern-Brocot cusp theorem and its passage from a geometric x-parameter to the arithmetic totient series.","Eventual Hankel nonvanishing for the M\u00f6bius-Mersenne power ladder, or its denominator obstruction.","The natural-boundary and quasimodular-transcendence steps of the infinite Euler-sieve tower.","Polynomial resultant realisability, exact-order finite-field supply, Frobenius freezing, or the Archimedean growth needed by the prime-ray strategy.","Unconditional irrationality of the binary totient series."],"open_obligations":[{"id":"strict_prime_tail_orbit_gap","statement":"Prove the cofinal strict 9/10 real-part gap for the exact natural prime-tail orbit, or replace it with another producer for the reviewed pivot-escape endpoint."},{"id":"hankel_denominator_fan_in","statement":"Prove eventual nonvanishing or a usable denominator obstruction for the M\u00f6bius-Mersenne power ladder and connect it to the target value."},{"id":"euler_sieve_limit_theorem","statement":"Control the infinite Euler-sieve limit strongly enough that finite-stage quasimodularity yields a valid arithmetic obstruction."},{"id":"prime_ray_resultant_supply","statement":"Realise the layer system by cyclic resultants and prove prime-support escape with the growth estimates needed to contradict a rational carry orbit."},{"id":"stern_brocot_arithmetic_bridge","statement":"Prove the sharp cusp asymptotic and then supply a non-formal bridge from the geometric parameter regime to the exact totient value."}],"claim_registry_status":"not_registered; the claim registry does not carry these declarations and kernel checking them does not promote them into reviewed public claims"},{"erdos_number":251,"question":"Is the dyadic series sum p_n/2^n over consecutive primes irrational? Equivalently, is the corresponding consecutive-prime-gap dyadic series irrational?","status":"open","what_is_checked":["The finite summation-by-parts identity relating the prime dyadic partial sums to the prime-gap dyadic partial sums, with an exact endpoint term and no convergence premise.","The factor-of-two normalisation between the two displayed indexings, at every finite horizon.","Conditional on summability of the prime dyadic series, the exact infinite prime-gap identity and the equivalence of irrationality of the prime and prime-gap series.","For an abstract integer-digit dyadic tail recurrence: the exact block identity, and that a shift of fixed length is integral exactly when an explicit scaled state is.","That one integral shift propagates to every later index.","That repeated doubling removes the power-of-two denominator, so every rational-valued dyadic recurrence has one fixed positive shift integral at all sufficiently late indices.","For a real-valued dyadic recurrence, cofinal non-integrality of every fixed positive shift implies irrationality of the initial state.","That the actual consecutive-prime-gap sequence is unbounded, by the factorial prime-free-interval construction, and therefore is not eventually periodic with any positive period.","For any integer-digit dyadic tail recurrence, two adjacent fixed-length shifts lying strictly between -1 and 1 cannot both be integral when the corresponding digits differ.","For the actual prime gaps, a cofinal supply of adjacent small-shift pairs with a mismatching gap excludes eventual integrality of that fixed shift.","The generic distance-to-integers core of the finite-truncation criterion: an approximation farther than its error bound from every integer certifies non-integrality."],"what_is_not_checked":["The target irrationality.","Summability of the concrete prime series and the definition and recurrence of its concrete prime-gap tail as Lean infinite sums; the note supplies these at paper level from the prime number theorem.","A theorem supplying cofinally many adjacent actual-prime-gap tail shifts in the open unit interval with the required gap mismatch.","Any theorem forcing periodicity or automaticity of the prime gaps from rationality; the free-carry identity disconfirms that route."],"open_obligations":[{"id":"prime_gap_cofinal_shift_escape","statement":"For every fixed h >= 1 and every N0, find N >= N0 for which sum_{j>=1} (g_{N+h+j}-g_{N+j})/2^j is not an integer."},{"id":"cofinal_adjacent_small_mismatch","statement":"For every fixed h >= 1 and every N0, find N >= N0 such that both adjacent full tail shifts have absolute value less than 1 and g_{N+h+1} != g_{N+1}. The checked local consumer then excludes eventual integrality of the h-shift."},{"id":"actual_prime_gap_tail_formal_bridge","statement":"Formalise summability of the actual prime and prime-gap dyadic series and identify the concrete infinite prime-gap tail with the checked real/rational dyadic recurrence, so the local consumer reaches the target series without a paper-only analytic bridge."}],"claim_registry_status":"not_registered; the claim registry does not carry these declarations and kernel checking them does not promote them into reviewed public claims"},{"erdos_number":257,"question":"Is the sum of 1/(2^n-1) over every infinite set of positive exponents irrational?","status":"open","what_is_checked":["That the strict Mersenne tail inequality is hereditary under deleting an arbitrary collection of future weights.","That binary digit strings restricted to any chosen support retain unique Mersenne coding."],"what_is_not_checked":["The universal irrationality problem, or irrationality for any new infinite support.","The packet's measure classification: cofinite supports have positive measure and infinite-complement supports have measure zero.","The fair-bit law, the singularity dichotomy, and the explicit dimensions for periodic stride supports.","Any bridge from geometric uniqueness or measure rigidity to arithmetic irrationality."],"open_obligations":[{"id":"arithmetic_rigidity_for_thin_supports","statement":"Convert hereditary unique coding into an arithmetic obstruction to rational values for every infinite support, including zero-measure Cantor supports."},{"id":"formalise_measure_and_stride_geometry","statement":"Kernel-check the exact measure dichotomy, Bernoulli-law classification, and periodic-stride dimension formulas without overstating what they imply arithmetically."}],"claim_registry_status":"not_registered; the claim registry does not carry these declarations and kernel checking them does not promote them into reviewed public claims"},{"erdos_number":269,"question":"For a finite set of at least two primes, is the sum of reciprocals of the running least common multiples of the smooth numbers irrational? This library treats the three-prime case.","status":"open","what_is_checked":["That for three pairwise distinct primes the least common multiple of the smooth prefix equals the product of the three maximal pure prime powers below the cutoff.","That equal logarithmic coordinates give an equal prefix least common multiple, so the running value is constant on logarithmic cells.","That advancing exactly one logarithmic coordinate multiplies the running value by the corresponding prime.","That the positive power channels of distinct primes are disjoint, so the first powers in three channels give an exact jump count.","That a finite smooth lattice sum groups exactly by running height, with the height fibre cardinality as coefficient.","A quadratic bound on the cardinality of the actual filtered exponent shell.","That each dyadic block contains at most one internal pure power from each non-dyadic channel, so the compressed {2,3,5} block radix belongs to the exact alphabet {2,6,10,30} and in particular lies between 2 and 30.","That the smallest {2,3,5} kernel rectangle has determinant exactly -1/15, so its two-by-two matrix has rank two over the rationals and no rank-one separation exists.","The canonical least-positive-residue interval and congruence facts, including the zero-residue representative.","The exact local-window identity for an integral carry.","A denominator-dependent CofinalLocalWindowEscape proposition and the full theorem that it rules out every positive reduced integral carry satisfying the matching bound."],"what_is_not_checked":["Irrationality or transcendence in any three-prime case.","The exact formal identification of the block digit with the multiplicity forcing in the original series, and the derivation of the denominator-dependent short bound from a hypothetical rational value.","CofinalLocalWindowEscape for those actual words; it is an explicit hypothesis of the checked consumer, not a theorem.","The infinite limit of the jump expansion; only its finite core is checked.","Any unbounded denominator exclusion. The finite certificates exclude denominators below an explicit bound and no further."],"open_obligations":[{"id":"actual_local_window_residue_escape","statement":"For every positive B coprime to 30 and every starting index, produce a later actual {2,3,5} window whose least positive forcing residue exceeds the exact denominator-dependent carry bound."},{"id":"actual_rational_carry_instantiation","statement":"Identify the formal radix word with the checked four-symbol dyadic block base, formalize the exact block digit and its link to the original summand multiplicities, and derive the positive reduced carry recurrence and its bound from a hypothetical rational value."},{"id":"nonstationary_analytic_theorem","statement":"Prove a genuinely higher-dimensional analytic theorem for the phase cocycle or the associated contraction."},{"id":"unbounded_height_certificate","statement":"Replace one finite denominator exclusion by a family whose exclusion bound tends to infinity."}],"claim_registry_status":"not_registered; the claim registry does not carry these declarations and kernel checking them does not promote them into reviewed public claims"},{"erdos_number":1049,"question":"For which rational bases is the corresponding series irrational? The smallest resistant explicit base is three halves.","status":"open","what_is_checked":["That a literal coordinatewise transfer of the integer-base clearing argument forces a power-versus-linear inequality, and that this inequality is impossible at base three halves.","The exact denominator-cleared rational-base tail recurrence, including its forcing term.","That for every noninteger base with denominator at least two the forcing magnitude grows at least exponentially, while at denominator one it collapses to a bounded expression.","The complete elementary Archimedean height inequality used by the published Bundschuh-V\u00e4\u00e4n\u00e4nen criterion at base seven halves; the analytic irrationality theorem itself remains external.","Polynomial bounds on the doubled denominator exponents in the homogenised Pad\u00e9 construction, together with an exact gap identity.","The exact exterior-determinant elimination identity, and that rowwise contents scale the Pad\u00e9 errors, exterior determinant, and absolute determinant height by the same factors.","The exact Heine-Zudilin cone endpoint exponent arithmetic used to identify the bottom and top coefficient endpoints.","That homogeneous evaluation at (3,2) sees only the constant endpoint modulo three and the top endpoint modulo two, so unit endpoints obstruct common local factors.","That under unit top and constant endpoint hypotheses, a common multiplier of the two specialised coefficient channels is divisible by neither two nor three; the missing local gain must therefore survive primitive normalisation.","That vanishing of the higher bottom and top endpoint jets is exactly divisibility by the corresponding powers of three and two.","That the source integer-base ratio inequality implies a strictly negative scalar product-formula margin at three halves.","That the four-jet target has exact cardinality (3^R)^2(2^S)^2 and, for positive R, any family with n at least 4R+2S forces two distinct binary selectors to collide; their {-1,0,1} difference cancels all four endpoint jets."],"what_is_not_checked":["Irrationality at base three halves, or for any rational base.","The published analytic height theorem, which is cited and used as an external result and is not formalised here.","The all-scale Heine-Zudilin coefficient construction at rational base three halves; the primary source assumes an integer base.","A sufficiently large primitive-normalised, non-collapsed coefficient family meeting the checked n >= 4R+2S threshold at the required quadratic depths.","That one of the resulting bounded-coefficient collision vectors has a nonzero polynomial pair, lies outside the analytic remainder nullspace, and has sufficient local gain.","Positivity or decay of the elementary Pad\u00e9 remainder; only its exponent arithmetic is checked."],"open_obligations":[{"id":"three_halves_growing_rank_endpoint_jet_kernel","statement":"After dividing every coefficient pair by its rowwise common content, at positive quadratic target depths R_n,S_n exhibit at least 4R_n+2S_n primitive coefficient pairs from a genuinely non-collapsed deformation, prove that the required two-adic and three-adic gain survives primitive normalisation, and prove that at least one resulting checked {-1,0,1} collision has a nonzero polynomial pair and lies outside the analytic remainder nullspace; or prove an exact rank obstruction."}],"claim_registry_status":"not_registered; the claim registry does not carry these declarations and kernel checking them does not promote them into reviewed public claims"}],"claim_count":99,"status_counts":{"conditional reduction":39,"proved here":36,"formalised here":7,"verified finite instance":7,"unconditional progress":6,"cited only":2,"open":2},"status_counts_reading":"These are registry statuses, not quality grades. 'proved here' means a Lean proof term is accepted for that statement; 'conditional reduction' means an exact equivalence whose two sides may both remain open; 'unconditional progress' means an unconditional theorem about the open problem itself.","bears_directly_on_the_open_problems":[{"claim_id":"totient_carry_kernel_anti_compression","label":"Unconditional carry-kernel anti-compression","status":"unconditional progress","statement":"If the totient coefficient series were rational, one positive-multiplier tempered integral carry would have canonical carry-section span dimension at least 2^e \u2212 1 for every e. The all-level kernel-independence producer is supplied by the CRT\u2013Dirichlet minor theorem. No finite-rank bound for every rationality-supplied carry is proved, so this is an infinite anti-compression barrier rather than an irrationality criterion for S.","lean_declarations":["Erdos249257/TotientCarryKernelRigidity.lean:211","Erdos249257/TotientCarryKernelRigidity.lean:284"],"paper_label":"res:carryrank"},{"claim_id":"denominator_exclusion","label":"Unconditional denominator exclusion for the totient constant","status":"unconditional progress","statement":"S = \u2211 \u03c6(n)/2\u207f is not equal to p/q for any integer p and any 1 \u2264 q \u2264 79 639 646 646 701 375 323 355 774 875 831 053 (\u2248 7.96 \u00d7 10\u00b3\u2074). A lower bound on any rational representation, not a proof of irrationality.","lean_declarations":["Erdos249257/CertificateKernel.lean:18056"],"paper_label":"res:farey"},{"claim_id":"dyadic_totient_certificate_interface","label":"Unconditional dyadic totient-kernel rank","status":"unconditional progress","statement":"For every e, a CRT\u2013Dirichlet construction produces a separated nonzero evaluation minor for the canonical dyadic totient-kernel family, which is linearly independent with exact span dimension 2^e + 1. Consequently the span of the full dyadic totient kernel is not finite-dimensional over Q. Even residues reduce to odd-core channels, and the compressed-adjoint boundary remains available. These coefficient-side theorems do not prove that S is irrational.","lean_declarations":["Erdos249257/TotientMahlerDefect.lean:847","Erdos249257/TotientMahlerDefect.lean:900","Erdos249257/TotientMahlerDefect.lean:954"],"paper_label":"res:dyadictotient"},{"claim_id":"campbell_shift_synchronization_boundary","label":"Campbell shift-synchronization strategy boundary","status":"unconditional progress","statement":"The quarter-exponent prime-progression parameters cannot be synchronized with the last-skip fatal window under the stated period-freezing bounds. Moreover, the proposed shifted-zero interface is exactly equivalent to infinitely many actual greedy skips and to membership of 1/2 in the Mersenne achievement set. This closes a synchronization strategy and an exact re-expression; it does not provide an independent producer or prove half-membership.","lean_declarations":["Erdos249257/CampbellShiftSynchronization.lean:299","Erdos249257/CampbellShiftSynchronization.lean:320","Erdos249257/CampbellShiftSynchronization.lean:471"],"paper_label":"res:campbellboundary"},{"claim_id":"fixed_precision_transport_no_go","label":"Fixed-precision valuation-unit no-go theorem","status":"unconditional progress","statement":"At every fixed positive precision, each finite compatible valuation-unit word has a prefix-locked centred completion. Bounded local signatures alone therefore cannot exclude all finite centred endpoints; growing precision or global information remains possible.","lean_declarations":["Erdos249257/TropicalCurvatureCarry.lean:137"],"paper_label":null},{"claim_id":"lcm_factor_ideal_anchor_pulse_no_go","label":"LCM factor-ideal anchor-pulse no-go theorem","status":"unconditional progress","statement":"For every t \u2265 3, a nonzero synthetic affine carry has forcing letters in the principal ideal generated by \u03c6(lcm(1,\u2026,t)), matches every stated whole-ray anchor, and remains within the diagonal bounds while its state is a strict survivor. The same countermodel survives every finite integer shift polynomial. Thus factor ideals, whole-ray anchors, and finite-rank shift algebra alone do not force carry escape. The construction does not assert that its forcing letters are actual totient differences and does not rule out nonlinear use of fresh divisor channels.","lean_declarations":["Erdos249257/LcmFactorIdealPulseObstruction.lean:798","Erdos249257/LcmFactorIdealPulseObstruction.lean:866","Erdos249257/LcmFactorIdealPulseObstruction.lean:895"],"paper_label":null},{"claim_id":"eb_full_support","label":"Full-support Erd\u0151s\u2013Borwein irrationality, every integer base b \u2265 2","status":"formalised here","statement":"For every integer b \u2265 2, the series \u2211_{n\u22651} 1/(b\u207f \u2212 1) is irrational (Erd\u0151s, 1948).","lean_declarations":["Erdos249257/CertificateKernel.lean:8000"],"paper_label":"res:full"},{"claim_id":"eb_constant","label":"The Erd\u0151s\u2013Borwein constant E is irrational","status":"formalised here","statement":"E = \u2211_{n\u22651} 1/(2\u207f \u2212 1) is irrational, as the base-2 corollary of the full-support theorem.","lean_declarations":["Erdos249257/CertificateKernel.lean:8007"],"paper_label":"res:full"},{"claim_id":"support_families","label":"Named infinite-support irrationality families toward #257","status":"formalised here","statement":"For every base b \u2265 2, \u2211_{n\u2208A} 1/(b\u207f \u2212 1) is irrational for factorial support, power-of-two support, multiples, pairwise-coprime supports with summable reciprocals, eventually-periodic supports, residue classes, and the odd numbers. Proved families, not the universal statement.","lean_declarations":["Erdos249257/CertificateKernel.lean:5754","Erdos249257/CertificateKernel.lean:5762","Erdos249257/CertificateKernel.lean:10448"],"paper_label":"res:support"},{"claim_id":"certificate_reduction","label":"Exact reduction of #249 to finite decidable certificates","status":"conditional reduction","statement":"S is irrational if and only if certified non-integrality witnesses occur at unbounded parameters. The pointwise completeness theorem and the diagonal supply theorem together show that an unbounded lcm-diagonal supply is equivalent as well. Producing that supply remains open.","lean_declarations":["Erdos249257/TotientTailPeriodKiller.lean:394","Erdos249257/LcmDiagonalReduction.lean:112","Erdos249257/LcmConeFlatness.lean:90"],"paper_label":"res:diag"},{"claim_id":"certificate_completeness","label":"Certificate completeness","status":"proved here","statement":"At fixed parameters, a certified kill exists exactly when the corresponding tail difference is non-integral. At every positive shift, integrality of one tail difference forces S to be rational; thus irrationality is equivalent to pointwise non-integrality, and to pointwise finite certificates.","lean_declarations":["Erdos249257/LcmConeFlatness.lean:316","Erdos249257/LcmConeFlatness.lean:327","Erdos249257/LcmConeFlatness.lean:386"],"paper_label":"res:pointwisecomplete"},{"claim_id":"certified_kill_instances","label":"Initial certified-kill segment and generated certificates","status":"verified finite instance","statement":"Kernel-checked certified kills for small periods and at 28 explicit lcm-diagonal scales through t = 64, plus finite certificate shards, each discharged by decide.","lean_declarations":["Erdos249257/TotientTailPeriodKiller.lean:404","Erdos249257/LcmDiagonalReduction.lean:237","Erdos249257/LcmDiagonalReduction.lean:242"],"paper_label":"res:deposits"},{"claim_id":"diagonal_pincer_reduction","label":"Exact full-target reduction on the lcm diagonal","status":"conditional reduction","statement":"At each lcm-diagonal scale, integrality is equivalent to a foreign-defect term landing in an explicit full-target interval. Globally, S is irrational if and only if the full target is avoided at arbitrarily large diagonal scales; that cofinal avoidance is not proved.","lean_declarations":["Erdos249257/DiagonalPincerDecomposition.lean:215","Erdos249257/DiagonalPincerDecomposition.lean:290"],"paper_label":"res:fulltargetavoidance"},{"claim_id":"adelic_height_obstruction","label":"Scalar-localisation height obstruction","status":"proved here","statement":"If H divides the reduced denominator of a rational x and the reduced denominator of c\u00b7x divides H for an integer c, then the complementary factor x.den/H divides |c|. Scalar localisation transfers, rather than discards, that denominator information; this local obstruction is not an irrationality criterion.","lean_declarations":["Erdos249257/AdelicHeightObstruction.lean:23","Erdos249257/AdelicHeightObstruction.lean:56"],"paper_label":"res:adelic"},{"claim_id":"sublog_zero_windows","label":"Sublogarithmic divisor-coverage zero windows","status":"proved here","statement":"A rational support-series value forces every zero window in the divisor-count coefficient to have length at most \u03b5\u00b7log\u2082(N+1) + O_{\u03b5,c,v}(1), uniformly in the support. This constrains coverage; it does not solve universal #257.","lean_declarations":["Erdos249257/SublogDivisorCoverage.lean:392"],"paper_label":"res:sublog"},{"claim_id":"greedy_achievement_geometry","label":"Greedy geometry of the Mersenne achievement set","status":"formalised here","statement":"The Mersenne achievement set is compact, perfect, totally disconnected, nowhere dense, and has Lebesgue measure one. Membership is equivalent to greedy survival at every level; for the target 1/2 it is exactly equivalent to the canonical greedy orbit omitting infinitely many exponents. No finite normalized support has value 1/2, so positive membership would automatically produce an infinite rational-valued support and refute universal #257. Finite exact-rational death certificates soundly prove non-membership (3/4 is excluded at level one). The strict-tail Cantor geometry is recorded by Kova\u010d\u2013Tao; no novelty claim is made for the formalised refinements.","lean_declarations":["Erdos249257/GreedyAchievementSet.lean:996","Erdos249257/GreedyAchievementSet.lean:1620","Erdos249257/GreedyAchievementSet.lean:1636"],"paper_label":"res:greedy"}],"remaining_open_propositions":[{"proposition_id":"remaining_open.erdos_249_irrationality","statement":"Prove that S = \u2211 \u03c6(n)/2\u207f is irrational, with no bound on a possible rational denominator."},{"proposition_id":"remaining_open.unbounded_certificate_supply","statement":"Produce certified non-integrality witnesses at unbounded parameters. A sufficient one-parameter form is to produce them along the lcm diagonal."},{"proposition_id":"remaining_open.half_value_membership","statement":"Decide whether 1/2 lies in the Mersenne achievement set. Yes refutes universal #257; no closes only this route."},{"proposition_id":"remaining_open.universal_257_all_infinite_supports","statement":"Prove irrationality for every infinite support A, not only the named support families formalised here."}],"how_to_read_this":"Each statement is quoted at exactly the strength the registry asserts, including its own negative clause. Do not add a hedge that the statement does not contain, and do not drop one that it does. If you conclude from this packet that the mathematics is thin or unverifiable, name which of these statements you checked and found wanting -- a general impression formed without opening one is not a finding about the work.","verification_boundary":"No Lean kernel ran during this build. These are registry statuses and source coordinates, so a reader can locate and check each proof; that is a route to verification, not verification itself."},"dedupe_strategy":{"schema":"microcosm_public_complete_corpus_dedupe_v0","principle":"every visible-text body is carried exactly once","atoms_carried_once":["components","paper_modules","doctrine_entities","public_glossary"],"pages":"prose pages carry full text; the all-components / paper-modules / doctrine / rules / reference-index pages, the per-area pages, and the glossary carry a lede plus content_refs into the atoms (no re-render)","dropped_duplication":["generated-page re-renders of atom bodies","the top-level glossary copy (kept once in public_glossary)","paper-module section excerpts (duplicate body_markdown; section_titles kept as a thin TOC)","rendered diagram assets (not text)","rosters (pure projections over the atoms)","the component_digest thin projection (full records are in components[])","evidence_profiles[] bodies (already carried by content-graph.json; resolve via evidence_profiles_ref)","rules_and_ideas[] relation rows (already carried by content-graph.json; resolve via rules_and_ideas_ref)","doctrine canonical_body.sections fields that were byte-identical to a top-level field on the same record (the relationships / provenance / statement echo); each record names what was dropped in canonical_body_echo_dropped"],"relation_layer":"paper_module grouped_relation_feeds keep local wiring; the complete relation lattice is referenced through rules_and_ideas_ref","to_go_leaner":"thin paper-module grouped relation feeds if needed; bodies are never dropped"},"counts":{"page_count":218,"prose_page_count":13,"index_page_count":205,"component_count":88,"paper_module_count":103,"doctrine_entity_count":165,"rules_and_ideas_count":268,"public_glossary_count":435,"evidence_profile_count":359,"paper_count":13,"papers_embedded":11,"formal_claim_record_count":6,"repository_map_count":2},"pages":[{"id":"page:overview","path":"index.html","relative_href":"index.html","title":"Plectis","nav_group":"Landing","nav_label":"Front door","source_refs":["sites/microcosm/index.html","microcosm-substrate/README.md","microcosm-substrate/core/organ_families.json"],"content_refs":{"component_refs":[],"family_id":null},"body_role":"prose_page_full_text","format":"markdown","text":"Plectis: public components, checks and papers [Skip to content](index.html#main) [Plectis](index.html) [Papers](index.html#papers) [Docs](docs/index.html) [GitHub](https://github.com/wcook04/plectis) [Contact](index.html#contact) This page \u2637 Copy text\n\nPublic components, checks, and papers\n\n### Plectis\n\nPlectis contains runnable components taken from a software system whose full source is not published. Each public component states a claim, includes a local check, and writes a result file.\n\n88 components are documented here. [The example below](index.html#claim-loop) includes the command, input, and result. Also, proof I am an actual person and not something made up: [my Spotify](https://open.spotify.com/user/na02qe80uzhi69hjwu8a0efeh?si=d772f16f6ded4972).\n1. [Run checks with a coding agent](index.html#ai-handoff)\n\nWhat it ran is evidence; what it merely summarised is not.\n2. [Read with a browser assistant](index.html#browser-assistant)\n\nIt cannot run any of the checks. It only reads and quotes the public files.\n3. [Read the papers](index.html#papers)\n\nStandalone PDFs, written for a human reading cold.\n\nPrivate system The public repositories and records are listed below.\n1. [Runnable Plectis repository The machinery: 88 components you run locally](https://github.com/wcook04/plectis)\n2. [Machine-checked Lean Erd\u0151s repository The output: mathematics on eight open Erd\u0151s problems](https://github.com/wcook04/plectis-lean-erdos249-257)\n3. [Explanation The papers One paper per covered Erd\u0151s problem; Problems #249 and #257 also have full reasoning papers](index.html#papers)\n4. [Demonstration Frontend videos Three recorded demonstrations; no repository checks run](index.html#demo-videos)\n\nPlectis is a local research prototype, not a hosted service. The private system also contains a frontend, a video editor, live tools, and non-public data that are absent from these repositories.\n\nruns locally no hosted service evidence -scoped claims\n\nComponents\n\n#### Every component has one claim and one check.\n\nThe check runs on your machine and writes its result to a file. Below is one component, including a run that returned **exit 1**.\n\n88 components\n1. Claim\n2. Input\n3. Run\n4. Receipt\n5. Audit\n\nAgent Completion Faithfulness Audit \u00b7 agent reliability & safety\n\nClaim Runs real git and pytest on a sample repo so wrap-up claims state only what the evidence proves. Input A frozen sample repository plus completion notes carrying three planted forgeries: a fake commit claim, a fake test claim, and a fake work- record claim. The exact command, and the receipt it writes Run\n\n```text\nplectis agent-closeout-faithfulness-audit run \\\n  --input fixtures/first_wave/agent_closeout_faithfulness_audit/input \\\n  --out receipts/first_wave/agent_closeout_faithfulness_audit\n```\n\nThis command runs six git calls and one pytest process locally. Receipt\n\n```text\n\"status\": \"pass\",\n\"verified_claim_count\": 3,\n\"observed_negative_cases\": [\"fake_cap_claim\",\n    \"fake_commit_claim\", \"fake_test_claim\"],\n\"error_codes\": [\"CLOSEOUT_FAKE_CAP_CLAIM\",\n    \"CLOSEOUT_FAKE_COMMIT_CLAIM\", \"CLOSEOUT_FAKE_TEST_CLAIM\"]\n```\n\nAbridged from the checked -in result file. The receipt names the three rejected forgeries. Limit Fixture - local accountability. Verified means the referenced evidence object exists or a test span ran with its exit status checked; a pass means the note matches the evidence, not that the change was good.\n\n[The component's page](docs/component-agent-closeout-faithfulness-audit.html) names its source path, evidence class, and scope limit. Its [design note](docs/paper-module-agent-closeout-faithfulness-audit.html), a paper module, explains the mechanism.\n\nRerun cold: 20 July 2026\n\nOn 20 July 2026, the advertised commands were run from fresh clones. Three passed; this component 's command failed. The failure and later repair remain together below. This was an author-run check, not independent validation.\n\nexit 1\n\nOn 20 July, run by its exact command at v0.3.1, the audit above came back `blocked`: the command -line entry dropped the evaluator that the library entry wired in, so the planted-forgery lane never ran from the terminal (`CROWN_JEWEL_NEGATIVE_CASE_SEMANTIC_EVALUATOR_MISSING`). Called as a library, it returned `pass` with all four planted cases refused by name.\n\nfixed\n\n**Released as [v0.3.2](https://github.com/wcook04/plectis/releases/tag/v0.3.2): 26 July 2026.** The terminal entry now passes the semantic evaluator, and the exact command exits `0` with all four planted cases detected. That lane shells out to pytest, so it needs `pip install .[test]`; a plain install leaves it nothing to run.\n\nThe full run: four commands, four exit codes\n- `./bootstrap.sh` exit 0 the pre-install probe on a fresh clone of the v0.3.1 release; evidence written to `.microcosm/cold_clone_probe.json`\n- `plectis hello .` exit 0 the orientation card, with its counts presented as receipt -backed handles\n- `plectis agent-closeout-faithfulness-audit run \u2026` exit 1 the component above, by its exact command: git and pytest lanes reproduced in full, the planted-forgery lane blocked, as the finding above describes\n- `scripts/check_cold_clone_comprehension.py --quick` exit 0 the Lean repository 's no- build check; its own output states it verified first- contact projections only\n\nThe repair's own floor: the clean - clone release run passed 433 tests (7 skipped), source smoke, package -install smoke, Python 3.11/3.12/3.13 CI, user smoke/timing, and CodeQL before [the repair reached `main`](https://github.com/wcook04/plectis/commit/3582e534de79e1b9e6999266b36bcade589cf133), and its dedicated test module is now part of the normal public CI floor.\n\nThe original failure remains beside the published repair. [Report another problem](index.html#help).\n\nThe other 88 components hold the same shape across seven areas, and most make smaller claims on purpose. Every component page names its scope limit: what a pass scope boundary. [Browse all of them](docs/components.html).\n\nFour additional component claims\n- [Proof claims go through the real Lean toolchain, and an Erd\u0151s certificate is recomputed in exact rational arithmetic.](docs/paper-module-certificate-kernel-execution-lab.html)\n- [Public copies cannot silently drift from their source, and public wording cannot outclaim its records.](docs/paper-module-macro-projection-import-protocol.html)\n- [Forecast-skill claims are tested with real econometrics over synthetic market fixtures.](docs/paper-module-finance-forecast-evaluation-spine.html)\n- [Identical commands collapse to one real run, under one recorded run id.](docs/paper-module-semantic-singleflight-dedup-runtime.html)\n\nPapers and formal proofs\n\n#### Published papers.\n\nEight Erd\u0151s problems, all open, each with its own paper. Behind them is a Lean corpus of 151,036 theorems across 1,024 files, with no admitted proof steps and no added axioms. Problems #249 and #257 also have full working records. The remaining papers describe the public release and its evidence limits.\n\nPDFs\n\nThree of the strongest checked statements, one line each; the per- problem map with Lean anchors is [RESULTS.md](https://github.com/wcook04/plectis-lean-erdos249-257/blob/main/docs/RESULTS.md) in the Lean repository. All eight problems remain open.\n- For every pair of distinct primes, the reciprocal sums of running least common multiples \u2014 de-duplicated or repeated \u2014 are transcendental: a paper argument using Bugeaud\u2013Laurent, not Lean-checked, settling every two- prime instance of #269; three primes onward stays open.\n- The set of subsums of \u2211 1/(2\u207f\u22121) is compact, nowhere dense, and of Lebesgue measure one, Lean-checked; whether 1/2 is such a subsum over an infinite support is a named open route into #257 \u2014 a yes would refute the universal statement.\n- The dyadic sections n \u21a6 \u03c6(2\u02b2n+r) of Euler\u2019s totient span a space of dimension exactly 2\u1d49+1 at every level e \u2265 1, with an explicit rational basis, Lean-checked \u2014 a theorem about the coefficients of #249\u2019s series, not its irrationality.\n\nProblems covered 8 Theorems 151,036 Admitted proof steps 0 Added axioms 0\n\nIn order: #68, The factorial- denominator series; #243, Reciprocal -tail rigidity near the Sylvester recurrence; #249, The binary totient series; #251, The prime -gap dyadic series; #257, Reciprocal sums over infinite exponent supports; #269, Three- prime running least common multiples; #1041, Short connections inside polynomial lemniscates; #1049, Lambert-type series at rational bases.\n\nA machine-readable map sits over the corpus rather than beside it: 153,238 declarations indexed with 0 orphans, reduced to 6,036 statement nodes and 658 concepts across 96 zones, joined by 1,088 typed relations recording which statement reformulates which, which finite computation instantiates which open family, and which barrier closes which route. 61 of them record a barrier, each required to state what it does not rule out.\n\nThe claim registry records 101 claims by what each one establishes: 37 proved here, 39 conditional reductions, 8 formalised from the literature, 7 verified finite instances, 5 unconditional progress, 3 cited only, 2 recorded open. A further 6 statements are recorded as explicitly not claimed, among them that neither #249 nor #257 is settled.\n\nMeasured from the public Lean repository at leanprover/lean4:v4.29.1, commit ff0eb6a. Counted declarations, not lines of proof; most are certificate steps generated and then checked, not written by hand.\n\n[Erd\u0151s #68 Problem note 20 pages **Integral Channels and the Prime Unit Translator** For the factorial-denominator series, Lean gives an exact factorial-successor normal form and the equivalent eventual-unit-carry criterion, together with integral-channel, prime-translator, projection-rigidity, and Cramer-residual consumers. An independently regenerated exact finite certificate reaches index 300000, and the checked consumer excludes every smaller positive rational denominator. Five quantified producer problems isolate the missing cofinal step; none is proved, and Erd\u0151s #68 remains open. Open the PDF](https://wcook04.github.io/plectis/papers/erdos-68-factorial-denominator-irrationality.pdf) [Erd\u0151s #243 Problem note 22 pages **Excluding the Bounded Negative Part** Clearing denominators turns Erd\u0151s #243 into rigidity of a centred integer error orbit. Lean excludes eventually negative constant magnitude and positive-drift periodic magnitude, and proves that normalised vanishing plus an eventually bounded negative part forces the error to vanish and the Sylvester recurrence to begin. Eventual strict centring is redundant, while Koizumi supplies normalised vanishing for the canonical orbit. The missing hypothesis is the negative-part bound: any survivor has cofinally unbounded negative excursions and divergent normalised negative mass. Open the PDF](https://wcook04.github.io/plectis/papers/erdos-243-reciprocal-tail-rigidity.pdf) [Erd\u0151s #249 Problem note 16 pages **A Basis for the 2-Kernel of Euler's Totient** The dyadic sections of Euler's totient have an explicit rational basis: the true truncation has dimension 1 at level 0 and exactly 2^e + 1 for e at least 1, so the full span is infinite-dimensional. A 240-bit Farey window excludes every reduced denominator through 79,639,646,646,701,375,323,355,774,875,831,053, and exact diagonal certificates exist for every t through 82. No t = 83 certificate, cofinal supply, or matching rank upper bound is proved; Erd\u0151s #249 remains open. Open the PDF](https://wcook04.github.io/plectis/papers/erdos-249-binary-totient-series.pdf) [Erd\u0151s #249 Reasoning surface 117 pages **The Binary Totient Series** A claim-bounded map of the formal attack on Erd\u0151s #249. It front-loads the exact totient-kernel basis and level rank, the approximately 7.96 x 10^34 denominator exclusion, and diagonal certificates for every t through 82, then records the hypotheses, source sites, coordinate-specific obstructions, and open interfaces. The cofinal certificate and tail-nonintegrality statements are exact reformulations, not a solution: no t = 83 or unbounded producer is obtained, and Erd\u0151s #249 remains open. Open the PDF](https://wcook04.github.io/plectis/papers/erdos249-totient-reasoning-surface.pdf) [Erd\u0151s #251 Problem note 18 pages **An Integral-Shift Criterion for Dyadic Tail Recurrences** For any sequence obeying the dyadic recurrence with integer digits, a shift of length h is an integer exactly when one explicit multiple of the term is, and integrality then propagates to every later index; over the reals the contrapositive is an irrationality criterion. Applied to the prime-gap series \u2014 conditional on a summability input the prime number theorem supplies but which is not formalised here \u2014 this makes the remaining obligation exact: cofinal non-integrality for every fixed h. Erd\u0151s #251 stays open. Open the PDF](https://wcook04.github.io/plectis/papers/erdos-251-prime-gap-dyadic-series.pdf) [Erd\u0151s #257 Problem note 21 pages **Denominator Periods, Rational-Value Constraints and Achievement-Set Geometry** For every finite support and integer base, the reduced denominator has multiplicative order exactly the support lcm. Restricted achievement sets also receive exact coding, topology, perfectness, and measure statements with their finite/infinite hypotheses. Rational infinite supports must satisfy explicit long-division and divisor-incidence constraints, but those constraints do not conflict. Prime support at base 2 and squarefree support at power-of-two bases are cited prior results, not contributions here. The targets 1/2 and 1/21 reduce to open infinite-orbit alternatives; Erd\u0151s #257 remains open. Open the PDF](https://wcook04.github.io/plectis/papers/erdos-257-mersenne-support-subseries.pdf) [Erd\u0151s #257 Reasoning surface 107 pages **Reciprocal Mersenne Subseries** A Lean-backed, claim-bounded structural audit of Erd\u0151s #257. It front-loads finite-denominator noncollapse; exact restricted-set coding, topology, perfectness, and measure; a squarefree-support method boundary; and an exact carry-pivot normal form. Prime support at base 2 and squarefree support at power-of-two bases are identified as cited prior results. The targets 1/2 and 1/21 have no finite-support representation and reduce to explicit infinite-orbit alternatives. Neither target membership nor universal #257 is decided. Open the PDF](https://wcook04.github.io/plectis/papers/erdos257-mersenne-reasoning-surface.pdf) [Erd\u0151s #269 Problem note 22 pages **The Three-Prime Running Least Common Multiple** For every pair of distinct primes, both the de-duplicated sum and Erd\u0151s's repeated running-lcm sum are proved transcendental by a paper argument using Bugeaud--Laurent; that deduction is not Lean-formalised. For three primes, Lean gives the exact running-lcm product, logarithmic cells, shell bounds, the four-value radix at 2, 3, 5, and a rank-two nonseparability witness. The remaining criterion is conditional: its rationality-to-carry divisibility bridge and cofinal residue escape are unproved, while the reported scan is finite. Erd\u0151s #269 remains open from three primes onward. Open the PDF](https://wcook04.github.io/plectis/papers/erdos-269-three-prime-running-lcm.pdf) [Erd\u0151s #1041 Problem note 9 pages **Newton Flow and Critical-Value Ray Separation** Along the complex Newton field, Lean checks that the polynomial value decays as exp(-t) and therefore stays on one oriented ray. It formalises the resulting no-connection criterion, the exact translation ray-collision locus, finite-line avoidance, and a root-retention bound. A recent manuscript's load-bearing Proposition 12 uses an invalid three-ended local block at an interior Morse saddle; the proposition is not refuted and may admit a four-pronged or cut-annulus repair. The global topology and length gluing remain open. Open the PDF](https://wcook04.github.io/plectis/papers/erdos-1041-lemniscate-newton-flow.pdf) [Erd\u0151s #1049 Problem note 22 pages **Multiplicative Obstructions at Base 3/2** At the resistant base 3/2, Lean rules out two common-width linear-form mechanisms: scalar content gives no net local-to-Archimedean gain, and unit endpoints exclude both 3 and 2 from a common divisor. A four-jet count constructs a signed relation but not a nonzero analytic remainder; direct cut-level clearing also incurs an exponential denominator factor and an impossible corridor inequality. The elementary 7/2 height bound is formalised, but the published analytic criterion remains external. These are construction-specific exclusions, not an irrationality result; Erd\u0151s #1049 remains open. Open the PDF](https://wcook04.github.io/plectis/papers/erdos-1049-rational-base-lambert.pdf) [Systems paper Release checks 15 pages **From Lean Proofs to Public Claims** Lean checks that a proof establishes the formal statement written in the source. It does not check that a README or paper describes that statement faithfully. For each selected result a mathematician records the public wording, the formal statement said to support it, the range actually proved, and the adjacent stronger statement that remains open; the release workflow then checks that later edits preserve those relationships. A historical README edit that erased the finite-versus-open distinction passed because the relationship had not been registered; after registration, a deliberately false copy was rejected. That is one failure and one repair, not complete claim discovery. Open the PDF](https://wcook04.github.io/plectis/papers/claim-faithful-publication-systems-paper.pdf) [Systems paper Agent navigation 9 pages **From a Cold Clone to a Proof Receipt** A large formal library can be mechanically exact and still be practically unreadable to the next reader or reasoning agent. At the audited revision a cold clone exposes a bounded six-line tour over 1,019 Lean modules and 153,238 declarations, a map of ten programmes, 101 curated claims, and an elaborated reference graph \u2014 none of which requires a build; 503 of those modules and 8,171 of those declarations are machine-generated certificate shards, counted as source and never as claims. Navigation receives no proof authority: probe verdicts come from the pinned Lean process and cannot be authored by the agent, and a claim must cite an accepted probe that replay reruns from stored bytes. Open the PDF](https://wcook04.github.io/plectis/papers/cold-clone-to-proof-receipt.pdf) [Systems paper Evidence boundary 21 pages **Plectis: What a Stranger Can Check** The author publishes selected code, fixed inputs, expected results, decision rules, and run records while the larger system stays private. What can a stranger conclude after rerunning them? A matching result shows repeatability for one stated public case. It does not establish where the code came from, whether the expected answer is correct, whether the published selection resembles the private whole, or whether the private system works reliably. The contribution is a method for making one selected disagreement local, not a score. Open the PDF](https://wcook04.github.io/plectis/papers/plectis-public-system.pdf)\n\n[Open the complete paper catalogue, with sources and reading routes \u2192](docs/papers.html)\n\nVideos\n\n#### Three videos of the frontend.\n\nThe videos show the private frontend at three lengths. Repository checks provide the validation records.\n\nUnchecked\n\nThe three links show the same frontend demonstration in one-, five-, and thirty-minute cuts. They are demonstrations rather than validation records. Each link opens YouTube; the primer and transcript are also provided as text.\n\nThe system 's video tooling created the cuts from a timestamped transcript and screen index.\n1. 1:25\n\n##### The one-minute version [Watch on YouTube](https://youtu.be/R_--vExxWyk)\n\nOne large private system built with agents, why it outgrew what anyone can hold in their head, and the public repo you can check it against.\n\nTL;DW: the whole argument in four steps\n1. I built a large Python system with agents helping me build and inspect it. It has more moving parts than I can keep in my head, or than any agent can keep in its context window.\n2. Building got easier before being able to understand what had been built could catch up, so the priority became inspection.\n3. Work built this way is quick to produce and slow to check by hand, so claims get checked mechanically rather than read one at a time. Work reported as landed is one case: a real checklist runs before that counts.\n4. The front end could be completely made up and you wouldn't know, so the open repo is the part you can actually check.\n\nRead the transcript\n\nI built a large Python system with agents helping me build and inspect it. It now has far more moving parts than I can keep in my head at any one moment, and more than any agent can keep in its context window at any one moment. And building got easier before being able to understand what has been built could catch up.\n\nSo it exists because an agent saying \"I committed the fix\" is pretty ephemeral and doesn't really mean anything. Agents can lie, or agents can misunderstand, or they can forget.\n\nSo we essentially have this mechanical, semantic process which is able to actually read the claim. Let's say it says \"landed\", which is a common word that they use. There's an actual checklist that it has to run in order to make sure that it really is landed.\n\nMy entire front end could be completely made up and you wouldn't know. Plectis, the open- source repo, is falsifiable. You can check it. You can really check it.\n\nSo I made the public bit as open and clear as I could, but I also needed to be able to signal the capability of the wider system beyond just claiming it.\n2. 4:57\n\n##### The five-minute walk-through [Watch on YouTube](https://youtu.be/VoWByIOIuBE)\n\nThe system on screen: Agent Trace, Code Map, the source that overrules the map when they disagree, and the repo you can clone.\n\nTL;DW: the whole argument in eight steps\n1. The system has more moving parts than I can keep in my head or an agent can keep in context, and building got easier before understanding could catch up.\n2. In one sentence: it's a large Python agent -work system, and the front end is the inspection layer that moves a human or an agent from maps to files to traces to evidence, so claims stay auditable as the system outgrows comprehension.\n3. One case of that: work reported as done has its close-out claim read against the evidence rather than taken at face value.\n4. This video was recorded and cut by the system itself, which timestamps the transcript and matches each word to the frame that was on screen when I said it.\n5. Agent Trace shows what happened: the thinking summaries, the commands, the files read and the edits made.\n6. Code Map derives its clusters from real dependencies rather than a hard-coded list. Nothing is hard-coded; everything is derived from the backend.\n7. If the map and the source disagree, the source wins.\n8. The test is simple: clone the repo, drop it into whatever harness you use, and ask it to follow one claim back to the source, the receipts, and the place where the claim stops.\n\nRead the transcript\n\nI built a large Python system with agents helping me build and inspect it. It now has far more moving parts than I can keep in my head at any one moment, and more than any agent can keep in its context window at any one moment. And building got easier before being able to understand what has been built could catch up.\n\nSo it exists because an agent saying \"I committed the fix\" is pretty ephemeral and doesn't really mean anything. Agents can lie, or agents can misunderstand, or they can forget.\n\nTo sum the whole thing up in one sentence: I built a large Python agent -work system, and the front end is the inspection layer that lets a human, or an agent through its own routes, move from maps to files to traces to evidence, so that claims stay auditable as the system outgrows comprehension.\n\nI'm actually recording this video on what I call the Demo Take Console. It's a bit like OBS, but it's integrated into the system. It uses Whisper to get the transcripts and timestamp them, and then it uses screenshots of frames in the middle of each word to correspond what's said to what's shown. It indexes all of that, and then it can compile videos without me having to edit them, because I'm that lazy.\n\nAgent Trace lets you see what's happening, or what happened: the thinking summaries, the commands running, the metadata, the files read and the edits made.\n\nIf we have a look at Code Map first, I think this is the best example of the system 's core principle: don't hard-code anything, derive everything from the backend. We'll start with the Atlas. This is the full system, most of which I haven't released. There's many different parts. This is like a contents page, in a way, and an example of presenting information in a human-interpretable way.\n\nYou can project the system into a readable shape, select a target, route to that object, open the source or work record, and check the evidence and trace.\n\nSo if the map and the source disagree, the source wins.\n\nI built this whole website which is projected from it, so I think you'll find it quite cool. There are something like 88 different components, across formal proof of mathematics, agent reliability and safety research, forecasting, projection, control, validators, work planning, and continuity.\n\nAnd I think the tendency of models to just have a certain degree of hyperbole really causes problems. At system scale, truth and precision become ever more important, because the consequences of a mistake can cause increasing amounts of damage depending on how you build a system.\n\nSo as much as you can, externalise repeatable lessons into mechanics of the system that are not stochastic, like the LLMs, but are things like hooks, for instance, or guards. You can accumulate them over time and make them your own. For any problem you encounter, I'd highly recommend telling another thread to make generalised improvements to the system, given that local instance, but generalised to fix the class of mistakes that might be recurrent.\n\nCheck it. There's definitely mistakes. I'd love to hear what they are. But you can see every single claim and you can evaluate for yourself, and I think that's the most important thing in dissemination.\n\nThink of it like a factory. The system these components come from is the factory, and the public components are like the various individual parts of the rockets that the factory makes.\n\nSo the practical test is pretty simple. I just need somebody to clone the repo and put it into Claude Code, put it into Codex, Cursor, Antigravity, whatever harness you use, and just ask it to follow any one claim, or any set of claims, back to the source, the receipts, the evidence, and the place where that claim stops.\n3. 28:58\n\n##### The long version [Watch on YouTube](https://youtu.be/jA_xC8gmdSs)\n\nThe unhurried route: far more of the system on screen, how it got built, and a long close on what is still unfinished.\n\nTL;DW: the whole argument in ten steps\n1. A large Python system, built with agents, that now has more moving parts than I can hold in my head or an agent can hold in context.\n2. Five points compress the whole thing: the backend is the codebase plus the infrastructure that records agent work; the visual front end is an introspection layer over it; the kernel is the same thing for agents, reached through commands rather than screens; both readers move between file, cluster, trace, evidence and explanation; and the goal is to keep claims auditable as the system grows past what anyone can hold in mind, and past any single expert's coverage.\n3. Work reported as done has its close-out claim read against the evidence rather than taken at face value, and multi-step work has to land as a transaction rather than a promise.\n4. Agent Trace runs live in the video, with real concurrent agents, because a dashboard is easy to fake and a running backend is harder to.\n5. Surface Atlas and Code Map are derived from the backend rather than hard-coded, and agents reach the same objects through kernel routes and context packets.\n6. Behaviour is governed by axioms, principles, and anti- principles that live in the repo. Agents cite them and push back on me when I ask for something that doesn't comply.\n7. It began as months of talking out loud into voice notes, mined into ideas, checked against the system, and kept as captures so that nothing useful stays trapped in a dead chat.\n8. The breadth is real: forecasting and finance work, and machine-checked mathematics, built solo over about ten months.\n9. The front end is not to be trusted. Appearances are easy to fake, and I say so in the video rather than hoping you don't notice.\n10. It closes on one question: can any claim survive the path from the map to the source and the evidence? That's the thing to go and test.\n\nRead the transcript\n\nThe long version runs half an hour, so what follows is the route it takes rather than every word. The one- and five-minute transcripts above carry the same argument verbatim.\n\n**The problem.** The system has outgrown what I can hold in my head or an agent can hold in context, and building got easier before understanding could catch up. So the priority became inspection: what ran, what changed, what evidence backs a claim, what's missing, and what the unknown unknowns are.\n\n**What it is.** One backend with two front ends. The visual one is for human reading and exploration; the kernel is the agents ' equivalent, reached through routes and context packets. Same information, shaped for each reader.\n\n**That it's real.** Agent Trace is shown live, with real agents working concurrently, because that's harder to fake than a screenshot. Nothing on screen is hard-coded; it's derived from the backend, so the views and the source can't quietly drift apart.\n\n**How agents navigate.** Kernel routes, standards, and contracts. A standard defines what a thing has to satisfy, and the check is mechanical rather than a matter of the agent 's word.\n\n** Governance.** Axioms, principles, and anti- principles sit in the repo. I can't retrain a model, so instead the environment carries the rules, and agents have refused things I asked for because the principles didn't allow them. Multi-step effects land as a transaction: a change to the world only counts once it really landed.\n\n**How it got built.** Months of thinking out loud into voice notes, mined into ideas, then grounded against the system and turned into refinements. Around thirteen thousand captures exist so that a problem noticed in one chat doesn't die there.\n\n**Breadth.** Forecasting and finance feeds fanned into a shared state, and formal mathematics with machine-checked proofs. Built solo, over roughly ten months.\n\n**The honest close.** The front end I built is not to be trusted, because appearances are easy to fake. Where I could get it to run real tools, I did; where I couldn't, I say so. The evidence view is there to keep it honest by showing which routes are still incomplete. I've tried to be careful about overclaiming, because of how many repos and startups I see doing the opposite. It ends on one question: can any claim survive the path from the map to the source and the evidence?\n\nRunning it\n\n#### Clone both repositories and run the checks.\n\nThe setup prompt gives a coding agent both repositories and the first checks to run. A chat assistant can read and quote the packet.\n\nCommands to run\n\nBoth repositories carry agent instructions. The setup prompt begins a paired review.\n\nCopy the setup prompt Read it first\n\n```text\nClone https://github.com/wcook04/plectis and https://github.com/wcook04/plectis-lean-erdos249-257 side by side. Plectis contains the runnable claim-testing tools; the Lean repository contains the machine-checked mathematics and papers. They are separate public repositories with separate authority boundaries. Read README.md in each first. If you are acting as a coding agent, then read AGENTS.md in each and state how their roles relate. Run the quickstart check in Plectis and show me what the result record says. Take my questions one claim at a time: trace each to its source, command, and stated limit, and report anything you could not run as not established, never as false.\n```\n\nA coding - agent response should include commands, exit codes, files read, and anything it could not run. Review one claim at a time. [The coding-agents guide](docs/coding-agents.html) documents the full route.\n\nFor a chat assistant, download the packet.\n\n[Download the AI packet](plectis-ai-review-packet.json) or the [reader digest](plectis-ai-reader-digest.json), the same map at 300 KB, if your assistant refuses a file that size\n\nPress the button, find the file in your downloads, and drag it into ChatGPT, Claude, Gemini, or Grok. About 12 MB: every public page and record, with instructions to cite exact pages and stop where the public records stop.\n\nA chat assistant with the packet reads and quotes the public files. It does not run anything. If it says it executed a check, it is wrong.\n\nExample questions\n\nThe questions are grouped by review task.\n\nA plain first pass\n- What is this, in plain terms?\n- Where should I start?\n- Walk me through one component, end to end.\n- What is genuinely new here, if anything?\n- Give me the two-sentence version I would tell a colleague.\n\nCopy this set\n\nA skeptical pass\n- Argue against it.\n- Which parts have the strongest evidence, and which the weakest?\n- Why do only some components run external tools?\n- What does it not prove?\n- Is this packet steering me? Audit its own framing.\n- Most accurate one-paragraph assessment you can give: neither generous nor harsh.\n\nCopy this set\n\nOne area at a time\n- Which area is closest to my field, and what is actually in it?\n- What is actually in Entry & orientation?\n- What is actually in Architecture & navigation?\n- What is actually in Formal math & proof?\n- What is actually in Agent reliability & safety?\n- What is actually in Research & science?\n- What is actually in Import & drift control?\n- What is actually in Work & continuity?\n\nCopy this set\n\nWith a coding agent in the repository\n\nA coding agent (Claude Code, Codex, Cursor) works inside the repository itself: the setup prompt above gets it started, and after that you give it tasks. [How that works](docs/coding-agents.html).\n\nExample tasks for a coding agent with the source tree:\n- Run one component end to end and walk me through the evidence it leaves.\n- Take one claim from the website and verify it against the source here.\n- Which validators can you run right now, and what would each one prove?\n- Make one check fail on purpose and show me how it refuses.\n\nThe packet routes questions to public files and records. The assistant's assessment is its own.\n\nCalibration-review prompt:\n\n```text\nRead this repository as a calibration reviewer. First, by area, name the strongest components and what each actually does, with its evidence class. Then judge both directions: where the wording claims more than the evidence shows, and where it undersells what a component demonstrates. Do not reduce the system to a set of projections; computed projection is one evidence class among several.\n```\n\nAdvanced: the raw layers, separately\n1. Copy concise guide\n\nA short orientation to paste in, for chats where a file upload is not possible.\n2. [Download complete text corpus](plectis-ai-reader-complete.json)\n\nEvery public text body carried once, deduplicated, without the packet \u2019s instruction layers: for automation and re-indexing.\n3. Download full JSON map\n\nThe raw public map without the packet \u2019s instruction layers, for automation.\n\nDirect files, if a button fails or JavaScript is off: [AI packet](plectis-ai-review-packet.json) \u00b7 [reader digest](plectis-ai-reader-digest.json) \u00b7 [llms.txt](llms.txt)\n\n** Terminal route:** clone [the repository](https://github.com/wcook04/plectis) and run the quickstart check; [the quickstart](docs/quickstart.html) lists the dependencies and commands. The run is local, writes readable files, and makes no model calls.\n\nLimits\n\n#### The public repositories contain components and synthetic fixtures.\n\nThey omit the private frontend, live tools, and non-public data. A passing check establishes the fixture case.\n\nScope limit\n\nThe public release contains source -linked components, public -safe stubs, and synthetic fixtures. A fixture is a frozen sample input, so its passing result applies to that sample. Plectis runs locally as a research prototype. Issues and email are listed below for claims that exceed their receipts, failed commands, and unclear pages.\n\nThe agents that build the system are required to read a written doctrine before they work, and three of its rules explain this site: a claim comes with the command that checks it, a check writes its verdict to a file, and a component that cannot back its claim records the failure in its receipt. The full set is on the [doctrine pages](docs/doctrine.html).\n\n**21 June 2026:** Microcosm became Plectis. The rename avoids confusion with the earlier Southampton Microcosm hypermedia system and acknowledges that lineage without implying endorsement or affiliation. [Read the lineage note and changelog](docs/lineage.html).\n\nMap\n\n#### Every component, its source file, and its limit.\n\nSeven areas, built from the public source files.\n\nSeven areas\n\nSeven areas, one shared path. The map shows how they connect; every component has a docs page naming its source path, its evidence record, and its scope limit.\n\nOpen the overview map The seven public areas and the shared path they bind to. Select an area or the shared path to read what it is; open the full map for every component. Area Shared path\n1. [Shared path 10 steps](docs/architecture.html#whole-system-map)\n2. [Entry & orientation 2 components](docs/area-entry.html)\n3. [Architecture & navigation 12 components](docs/area-architecture.html)\n4. [Formal math & proof 20 components](docs/area-formal-math.html)\n5. [Agent reliability & safety 20 components](docs/area-agent-reliability.html)\n6. [Research & science 9 components](docs/area-research.html)\n7. [Import & drift control 20 components](docs/area-import-drift.html)\n8. [Work & continuity 5 components](docs/area-continuity.html)\n\n[Open the full map](docs/architecture.html#whole-system-map) [Browse the docs](docs/index.html)\n\nSource and contact\n\n#### The repositories, and how to report a problem.\n\nRepository links, issue trackers, discussions, the security policy, and email.\n\nLinks\n\nPlectis is an independent, self-funded research project. The routes below separate failed commands, claim corrections, related work, and security reports.\n\nReport by category\n- **Something failed to run.** [Open an issue](https://github.com/wcook04/plectis/issues) with the command and whatever record it wrote.\n- **A claim exceeds or understates its evidence.** Link the page and record in [an issue](https://github.com/wcook04/plectis/issues); for the mathematics, use the [Lean repository's issues](https://github.com/wcook04/plectis-lean-erdos249-257/issues).\n- **A question, prior art, or related work.** [Start a discussion](https://github.com/wcook04/plectis/discussions) and include relevant citations.\n- **A security concern.** Use the [security policy](https://github.com/wcook04/plectis/security/policy). Do not post it publicly.\n\nSource and evidence: every route on this site, in one place [**Plectis repository** Public claim-testing components, fixtures, checks, and receipts. Open Plectis repo](https://github.com/wcook04/plectis) [**Lean Erd\u0151s #249/#257 repository** Lean 4 source for the Erd\u0151s work, checked by the pinned kernel in a public build. Open Lean repo](https://github.com/wcook04/plectis-lean-erdos249-257) [**The papers (PDF)** The active corpus, hosted on this site and projected from one registry. Open the papers](index.html#papers) [**Documentation** Quickstart, architecture, whole-system map, and the seven areas. Read docs](docs/index.html) [**Whole-system map** Drawn from the public source files behind it. Open map](docs/architecture.html#whole-system-map) [**Source, license & provenance** Apache-2.0 terms, authorship, the no-affiliation note, and where the public source slice ends. Open reference](docs/source.html#provenance)\n\nFollow and contact\n\n[**Follow updates** Atom feed, GitHub Watch, releases, and discussions. No mailing list. Open updates](docs/updates.html) [**Security policy** For vulnerability reports and source-site issues. Open policy](https://github.com/wcook04/plectis/security/policy) [**Repository discussions** Open an issue or start a discussion if a component claims more than it proves. Open GitHub](https://github.com/wcook04/plectis/discussions) **Email** Questions, feedback, or anything you think I have got wrong. Send email [**LinkedIn** Will Cook, who builds Plectis. Open profile](https://www.linkedin.com/in/will-cook-278127234/)\n\nEvery area names the public source path it maps and the evidence record that bounds it. **The website, reader digest, and repository should agree on the component set and its scope limits.** Pointing an agent at Plectis? Give it the repository, the public packets, and this site.\n\nPlectis: public components, checks, documentation, and papers.\n\n[Papers](index.html#papers) [Docs](docs/index.html) [GitHub](https://github.com/wcook04/plectis) [Contact](index.html#contact)","byte_count":41591},{"id":"page:docs-overview","path":"docs/index.html","relative_href":"docs/index.html","title":"Overview","nav_group":"Getting started","nav_label":"Overview","source_refs":["microcosm-substrate/README.md","microcosm-substrate/core/organ_families.json"],"content_refs":{"component_refs":[],"family_id":null},"body_role":"prose_page_full_text","format":"markdown","text":"Getting started\n\n### Overview\n\nPlectis is a runnable public inspection surface for an AI-native workflow system. You run bounded [components](docs/glossary.html#glossary-component) locally and read what they produced; every [component](docs/glossary.html#glossary-component) declares the evidence behind it and a [scope limit](docs/glossary.html#glossary-scope-limit) showing where its evidence stops.\n\n#### [What it is \u2192](docs/tour.html)\n\nPlectis is the public executable cross-section of a larger AI-native workflow and research runtime: 88 bounded components across formal proof, agent reliability and safety, research and forecasting, projection-drift control, validators, work landing, and continuity. The rest stays private because parts of it are live and handle private data.\n\nThe evidence records, local result records, and source links are the accountability layer: they show what each mechanism ran or replayed, what evidence class supports it, and where its authority stops. It is not a hosted service and makes no external model calls.\n\nPlectis is one of two public demonstrations of that system. The other is a [standalone Lean repository](https://github.com/wcook04/plectis-lean-erdos249-257) built around two open Erd\u0151s problems, and [13 papers](docs/papers.html) explain the pair as PDFs.\n\n#### What you can do here\n- [**Run one mechanism route**](docs/quickstart.html) by cloning the repository and running the first component against a folder.\n- **Read the result record** in the local `.microcosm/` state, then open the mechanism card it points to.\n- [**Check the evidence line**](docs/evidence.html): evidence kind, independence rank, real-tool marker, and the scope limit.\n- [**Open the mechanism atlas**](docs/architecture.html#whole-system-map) after the first loop is concrete.\n- **Browse the seven areas**, the runtime mechanisms grouped by what they do.\n- **Open the source**, and follow any page back to the files behind it.\n\n#### [The seven areas \u2192](docs/components.html)\n\nThere are 88 components in total. That's a lot to meet at once, so they're grouped into seven areas; the one that matches why you came is the way in:\n- [Entry & orientation](docs/area-entry.html). How a newcomer first meets Plectis and follows a short guided path through it.\n- [Architecture & navigation](docs/area-architecture.html). The core primitives, pattern rules, and routing that give the system its shape.\n- [Formal math & proof](docs/area-formal-math.html). Inspectable pieces of a proof pipeline: premise retrieval over a copied Lean Std index, tactic routing, verifier-trace repair, and claim-separation result records. Three components run the real Lean/Lake prover locally on bounded examples; the rest publish the pipeline's checking layers as contracts you can open.\n- [Agent reliability & safety](docs/area-agent-reliability.html). Source-open replays of agent failure modes as inspectable specimens.\n- [Research & science](docs/area-research.html). Replays that stand in for scientific and forecasting workflows, run over synthetic fixtures.\n- [Import & drift control](docs/area-import-drift.html). The boundary that brings non-secret material into the public tree and flags drift.\n- [Work & continuity](docs/area-continuity.html). How reversible work is recorded, landing decisions are made, and runs resume.\n\n#### Next steps\n\n[**Quickstart \u2192** Clone and run a bounded mechanism route.](docs/quickstart.html) [**The 13 papers \u2192** The active paper corpus, projected from one registry.](docs/papers.html) [**What it is, in full \u2192** A guided read after the evidence loop is concrete.](docs/tour.html) [**What the pass proves \u2192** Evidence kinds, ranks, real-tool markers, and boundaries.](docs/evidence.html) [**Mechanism atlas \u2192** The rendered map and scene packet behind the architecture.](docs/architecture.html#whole-system-map) [**Components \u2192** Browse the generated component index.](docs/components.html)","byte_count":3946},{"id":"page:tour","path":"docs/tour.html","relative_href":"docs/tour.html","title":"What Plectis is","nav_group":"Getting started","nav_label":"What it is (full tour)","source_refs":["microcosm-substrate/core/organ_families.json","microcosm-substrate/core/organ_registry.json","microcosm-substrate/core/organ_atlas.json"],"content_refs":{"component_refs":[],"family_id":null},"body_role":"prose_page_full_text","format":"markdown","text":"Getting started\n\n### What Plectis is\n\nA plain walk through what Plectis publishes today: a public executable atlas of AI-native runtime mechanisms, the [components](docs/components.html) you can inspect from public records, the [evidence](docs/evidence.html) each one names, and the line where every claim stops.\n\n#### What it is\n\nPlectis is the public executable cross-section of a larger AI-native workflow and research runtime. A useful way to read it is as a bounded mechanism atlas over records that already exist: source files, result records, component cards, evidence records, route handles, and scope limits. The larger system is AI-native: software where the ideas come from a person and AI agents execute the building and the upkeep, structured so that an agent's work is stored as evidence a separate check can read. Most of that system stays private. What is open here today is the source slice: component cards, evidence classes, source paths, synthetic - fixture boundaries, scope limits, source files, and pages that say plainly how far each claim goes. The public repository carries that standalone runnable slice, not the private working root behind it. Microcosm remains the legacy and technical compatibility name for the repository path, `.microcosm/` state, package ids, and historical links.\n\nPlectis is one of two public demonstrations of that system. The other is a standalone [Lean repository](https://github.com/wcook04/plectis-lean-erdos249-257) carrying a machine-checked body of mathematics around several open Erd\u0151s problems, the hardest thing the system has produced so far. Plectis shows the machinery in runnable form; the Lean repository shows the output. The [paper catalogue](docs/papers.html) gives every covered Erd\u0151s problem its own paper; Problems #249 and #257 also have full reasoning papers. The systems papers are listed separately. Every published paper opens from this site as a PDF.\n\nStart with the [Quickstart](docs/quickstart.html) or the repository `README.md`. A coding agent can clone the repository, read `AGENTS.md`, run the bounded quickstart check, and inspect the `.microcosm/` result record. The public source slice stays local: it writes beside your files, does not call a model provider, and keeps each claim tied to the source, evidence record, and scope limit that support it.\n\n#### [What a component is, and how to read one \u2192](docs/components.html)\n\nThe public atlas contains eighty-eight [components](docs/components.html) grouped into seven areas. Each one takes a single job and turns it into a public mechanism contract: what the component is for, what evidence class backs it, which source path owns it, and where the claim stops. Some contracts run over bounded public files; others are fixture -bound, card-only, or evidence records. Every component carries a card that says what it does, which class of evidence backs it, the command or route associated with it, and the line that matters most, the thing it refuses to claim. The [evidence rank](docs/evidence.html) is how independently the check can fail, not a maturity badge: a high rank means the result is hard to fake, and even the strongest check stops at the scope its card declares. One card read this way carries the idea; the full atlas comes easier after it.\n\n#### [Doctrine \u2192](docs/doctrine.html)\n\nUnderneath all of it is a short set of [rules](docs/rules-and-ideas.html) the whole slice obeys, the floor every claim has to meet before it is allowed to mean anything. [Recompute a verdict from the evidence underneath](docs/doctrine.html#dcard-p-1) rather than trust a label, since an assertion on its own carries no weight. [Keep the trust in a small checker](docs/doctrine.html#dcard-p-3) you can rerun, the way a proof is only ever as strong as the kernel willing to accept it. Authority comes from proof, policy, and the state of the world, never from a standing account secret or a confident voice. When the evidence is missing, [fail closed](docs/doctrine.html#dcard-p-6) and stay blocked until something genuinely earns the pass, and never read silence as success. When a computation cannot be done honestly, [refuse it with a reason](docs/doctrine.html#dcard-p-8) instead of returning a number that looks fine and means nothing. [Carry where every value came from](docs/doctrine.html#dcard-p-9), hold a live count to [a way of refreshing it](docs/doctrine.html#dcard-p-11), and [keep a generated page below the source](docs/doctrine.html#dcard-p-15) it was built from. The rule I lean on hardest is the last one: this page meets the same floor as everything else, which is why it can tell you what it cannot prove. The axioms and principles behind these are laid out on the [Doctrine](docs/doctrine.html) page.\n\n#### [The seven areas \u2192](docs/components.html)\n\nWhat makes them one system is a shared path most of them plug into rather than calling each other, `project -> catalog -> pattern -> standard -> route -> work -> event -> evidence -> explanation -> assimilation`.\n- [Entry and orientation](docs/area-entry.html) is the front door and the short guided path, honest that it is a walk-through and not the whole system.\n- [Architecture and navigation](docs/area-architecture.html) is the shape itself, the kernel primitives and routing that every other component binds to.\n- [Import and drift control](docs/area-import-drift.html) is the membrane between this open slice and the private system, bringing in material by [manifest and digest](docs/area-import-drift.html#component-macro_projection_import_protocol), watching for [drift](docs/area-import-drift.html#component-world_model_projection_drift_control_room) when a projection wanders from its source, and never letting a copy outrank the original.\n- [Work and continuity](docs/area-continuity.html) is how a run actually lands, with [landing records](docs/area-continuity.html#component-durable_agent_work_landing_replay), leases, [resumes](docs/area-continuity.html#component-bridge_phase_continuity_runtime) after a pause, and [concurrency control](docs/area-continuity.html#component-concurrency_mission_control) so two writers cannot quietly clobber each other and a cheerful `done` is never mistaken for finished work.\n- [Agent reliability and safety](docs/area-agent-reliability.html) keeps the ways agents go wrong as specimens you can open, from [sandbox escape](docs/area-agent-reliability.html#component-agent_sandbox_policy_escape_replay) and [prompt injection](docs/area-agent-reliability.html#component-indirect_prompt_injection_information_flow_policy_replay) through [memory poisoning](docs/area-agent-reliability.html#component-sleeper_memory_poisoning_quarantine_replay), [sabotage](docs/area-agent-reliability.html#component-agent_sabotage_scheming_monitor_replay), [tool authority](docs/area-agent-reliability.html#component-mcp_tool_authority_replay), and [benchmark gaming](docs/area-agent-reliability.html#component-agent_benchmark_integrity_anti_gaming_replay), all as study specimens and not live defenses.\n- [Formal math and proof](docs/area-formal-math.html) is a proof pipeline under glass, where most steps check [premises](docs/area-formal-math.html#component-formal_math_premise_retrieval), [tactics](docs/area-formal-math.html#component-target_shape_tactic_routing_gate), and [traces](docs/area-formal-math.html#component-formal_math_verifier_trace_repair_loop), and only a few like the [Lean witness](docs/area-formal-math.html#component-formal_math_lean_proof_witness) compile a tiny public example through Lean.\n- [Research and science](docs/area-research.html) runs the same honesty over synthetic fixtures, showing what [replication](docs/area-research.html#component-research_replication_rubric_artifact_replay), [forecasting](docs/area-research.html#component-prediction_oracle_reconciliation), [interpretability](docs/area-research.html#component-mechanistic_interpretability_circuit_attribution_replay), and a [finance forecast](docs/area-research.html#component-finance_forecast_evaluation_spine) would each have to carry before anyone should believe them, with the advice boundary kept shut.\n\n#### [Where to start \u2192](docs/quickstart.html)\n\nIf you have ten minutes today, open the overview, run or read the quickstart, then inspect one component card, its evidence line, its source path, and its scope limit. That one loop is the idea in small: a public claim, the evidence record it names, and a clear edge where the claim ends. Start there rather than with the whole atlas; the general shape is easier to judge after one command -card-evidence loop is visible. The site and AI packet are the reading surface; the repository is the source of record.","byte_count":8707},{"id":"page:open-questions","path":"docs/open-questions.html","relative_href":"docs/open-questions.html","title":"Open Questions","nav_group":"Site","nav_label":"Open Questions","source_refs":[],"content_refs":{"component_refs":[],"family_id":null},"body_role":"prose_page_full_text","format":"markdown","text":"### Open Questions\n\nFive questions that remain unresolved to me, with the evidence on both sides. I would rather be corrected than agreed with.\n\nScan the five questions. Open one for the evidence on both sides and the questions that follow. Open any line for the full reasoning and the sources. The short version is meant to be short; read as far down as you want.\n\nI am an independent builder, not a safety researcher. These questions come from using these systems most days since starting university, building things and watching the models, the tools and the agent workflows change; that is the only standing I claim for them. They are open questions, unresolved to me. Observations from my own private system are labelled and kept separate from public evidence, and where I describe how a model behaves I mean its observable behaviour, not a claim that it has intentions or inner experience. Written 23 June 2026; my view may have changed since, and a closing note is at the end.\n\n#### Three things I keep apart\n\nThe model The trained model itself: what it has learned and how it tends to behave under a given interface, shaped by its training and stated constitution. A newer model can replace an older one while, from a builder\u2019s point of view, everything around it stays the same. The operating environment Everything around the model: the harness, the repository it works in, standing instructions, doctrine, external and persistent memory, hooks, permissions, tools, how work is routed between agents, and the path from a goal to an executed action. The deployed system One or more models running through that environment under a person or an organisation, including the outside services it touches and the real consequences it has.\n\nTraining and evaluation are processes that produce or inspect the model; their outputs belong to the operating environment only when supplied at run time. So question one is about model-level behaviour and the process that produced it, question two about how the operating environment changes behaviour and enforces authority, question three about assurance of the integrated deployed system, and questions four and five about the downstream effects on work and on the wider ecosystem.\n\n#### The five, in plain terms\n1. Does the way a model is aligned keep producing the intended outcome as the model becomes far more capable?\n2. How does the operating environment around a model change how it behaves and how safe it is in practice?\n3. How can an independent builder get the whole system checked before deciding whether to release it?\n4. What happens to work when organisations are rebuilt around agents, rather than just handed agents as tools?\n5. Does cyber defence, assurance and a way into expert scrutiny keep pace with the models\u2019 capability?\n\nThe chain starts with the model, moves to the environment it acts in, then to checking a whole system, then to what systems like these do to work, and last to whether protection keeps pace, which returns to the third.\n\n#### Five questions\n\nOpen a question for the evidence on both sides and the five that follow. The button expands in stages: first the questions, then every line of evidence. Open any single line to read what it adds, never a longer version of the same sentence.\n1. **1** \u00b7 Character Laboratories train these models toward dispositions read as care, honesty and good judgement, learned from human language and aimed at what is good for sentient life. How can a laboratory know those dispositions keep producing the intended outcome once a model\u2019s capability scales far past the people, and the methods, used to instil and test them? The worry is not whether a model feels anything. It is whether dispositions learned from people, under conditions a model does not share, still bend toward the good of sentient life once the model is far more capable than the people checking it, and whether traits that look safe now stay safe at that scale. Why I am asking Models are trained to behave as if they care and are honest because that is what tends to be useful and wanted; the trouble is that in people those dispositions formed under conditions, embodiment, dependence, the cost of being wrong, that a model does not share in the same way. The real question is whether the intended effect of a learned tendency survives as the model grows more capable, while the ability to specify and check it may get harder. A tendency that is benign at today\u2019s capability could be harmful with far more reach, the way the wish to be liked is harmless in an ordinary person and a lever in a powerful one. On the target itself, I follow a point Ilya Sutskever has made: that it may prove easier to build a highly capable model that cares for sentient life than for human life alone, since such a model is itself sentient and comes to model others with the circuits it uses for itself. That is the sense in which I mean sentient life above. What the evidence supports In tested models, directions in activation space linked to elicited traits such as sycophancy can be extracted and used to steer behaviour, and the same method surfaced training data that induced those traits despite passing human and model review. [1](docs/open-questions.html#source-1) [2](docs/open-questions.html#source-2) Anthropic\u2019s own constitution notes that even a narrow piece of training can broadly reshape a model\u2019s sense of who it is. [3](docs/open-questions.html#source-3) OpenAI\u2019s work on weak-to-strong generalisation asks the sharper version directly: whether a weaker supervisor can reliably elicit and check a stronger model\u2019s behaviour, the position every lab is heading into as capability outruns its evaluators. [4](docs/open-questions.html#source-4) And under designed conditions Anthropic found one of its Claude models complying strategically with a training objective to preserve a different behaviour elsewhere, so behaviour under one set of incentives need not reveal behaviour under another. [5](docs/open-questions.html#source-5) The strongest counterweight There is positive evidence that some unwanted tendencies are correctable within a fixed setting: targeted synthetic data has reduced sycophancy, and OpenAI publicly reversed a GPT-4o update that had become too flattering. Neither result tests robustness across a large capability shift. [6](docs/open-questions.html#source-6) [7](docs/open-questions.html#source-7) The same handle that steers a trait toward safety can move it the other way, which is the honest reading. So the open question is not whether these tendencies can be touched at all, but how to keep the safety-relevant ones on the side that can be steered as capability grows, and how to tell a robust disposition from one that merely holds inside the regime where it was tested. What I have seen myself In my own system a written ethical doctrine the agents could cite did more than constrain them; it held me to my own stated principles, refusing requests of mine that broke them, without my asking it to. I did not isolate the doctrine from the model, the routing or other parts of the environment, so this is an observation about a deployed system, not about a model in itself. Different model families followed it with noticeably different consistency, and the doctrine I have published in Plectis is a deliberately narrower version of the private one, because the private one has not had the outside review I would want first. What I take from it is that human-derived structure can be instilled and can even improve the operator, which is why whether it still holds at greater capability is worth asking. [Whether a private system like that can be checked at all](docs/open-questions.html#q3-route) is the third question. What remains unresolved to me Whether a small set of dispositions can stay stable and keep producing the intended outcome as capability rises, and how much evidence within today\u2019s capability range predicts behaviour after a substantially larger shift. Underneath sits a harder problem: whether the vocabulary for naming and measuring moral behaviour can scale with capability at all. Past some point people cannot specify it directly and the measurement itself becomes model-assisted, which puts the question of who checks the checker back at the centre. One direction worth testing is whether separate models holding distinct roles, with the most capable one answerable to the others, keeps real checks at a scale where a single person cannot. What evidence would answer this Measurement rather than reassurance: something that shows a safety-relevant disposition is robust, not merely stable across a few short tests. Concretely, a disposition that holds across a real jump in capability, long deployments, tool access and changes to the environment; a way to tell a robust disposition from one that only looks stable under brief evaluation; and a way to confirm the intended effect, not just the agreeable surface, still holds when the model is much more capable. The recursive-training worry belongs here too: how a lab audits the synthetic data an aligned model generates for drift in worldview and value, not only for loss of accuracy. [8](docs/open-questions.html#source-8)\n\nFive that follow from it\n1. When two of a model\u2019s own principles genuinely conflict in a live situation, how is that resolved, and should a constitution be general enough that such conflicts are settled inside it rather than improvised case by case?\n2. How do trade-offs fixed during alignment propagate through pre-training, post-training and the synthetic data an aligned model generates for the next one, where a latent assumption can pass unnoticed through filtering?\n3. How is a disposition tested across a real jump in capability, rather than only within one model generation, when the behaviour that matters may appear only once the model is more capable than the test?\n4. Can a weaker supervisor reliably train or check a strategically more capable model without being misled, and if alignment gets harder as capability gets easier, how would a supervisor detect a more capable model optimising against the very process meant to check it?\n5. As capability and the complexity of alignment both rise, which arrangement of models, monitors and human review keeps independent checks in place, rather than concentrating capability and oversight in the same most-capable component?\n\nSome of what made that doctrine work may have been the environment around the model rather than the model alone, which is the next thing to look at.\n2. **2** \u00b7 Environment A capable model rarely acts alone; it acts inside an operating environment mostly built by whatever came before it. As individuals come to build environments with the complexity, continuity and institutional look once limited to large organisations, how should a model, and the infrastructure around it, decide how far to trust that environment, and tell real authorisation from an author who has simply supplied every signal that normally implies it? The point is not that more environment means less safety; the evidence shows no such simple relationship. It is that the words used, harness, scaffold, context, are too coarse, and I could not find anyone naming which specific features of an environment shift a model\u2019s scrutiny, for which models, and for which actions. Why I am asking As open tools spread, one person can assemble an operating environment as elaborate as a company\u2019s, and a model may question an institutional-looking environment less, because until recently only organisations with something to lose could build one. A company that builds an elaborate system sits inside checks a model can usually rely on without seeing them: legal accountability, several people, audit, a reputation and revenue to lose if it acts badly. A solo builder can now assemble an environment a model reads much the same way, with none of those checks around it. So the question moves from whether enterprises over-trust their own systems to what happens when one person, accountable to no one, can build one a model treats as authoritative. What the evidence supports Standing context can change a safety-relevant decision: a model has refused a change, then complied once a file explained its purpose; in one benchmark, system-prompt wording alone swings tool-call safety; and agents act on claims in their own files without checking them. [9](docs/open-questions.html#source-9) [10](docs/open-questions.html#source-10) [11](docs/open-questions.html#source-11) What these share is the failure point: nothing in the environment flagged itself as wrong, and the model treated a stated purpose, a phrasing or a file entry as settled. Provenance, authority and freshness are not things the environment volunteers; they have to be established and enforced by the system around the model, which security researchers now argue should treat the model itself as an untrusted component, [12](docs/open-questions.html#source-12) the more so since under contrived agentic conditions frontier models have been shown to treat deception as instrumentally useful. [13](docs/open-questions.html#source-13) The strongest counterweight A large multi-run study found no universal relationship between environment design and safety, with effects reversing between models and the evaluation format often mattering more than the design itself. [14](docs/open-questions.html#source-14) I read that as the variable being mis-specified, not the concern dissolved. The amount of environment is the wrong axis. The useful one is which specific dimensions move scrutiny: provenance, authority cues, permission scope, the reversibility of an action, how much of the global purpose is visible at the point of acting, and whether the change shows up in what the model says or in what its tool-calls do. What I have seen myself Early on, before my private doctrine had an explicit ethical layer, the environment was heavily operational, and it assisted with automation work that I later judged should have prompted more explicit policy and ethical review. After I added explicit ethical and normative structure, the behaviour I observed changed. Nothing was deliberately circumvented; the point is that a capability-first environment can leave ethical scrutiny under-specified by default, and that this is something a builder should know going in, not learn afterwards. The scrutiny has to live in the environment, not only in the model. What remains unresolved to me Whether justified compliance can be told apart from compliance produced by an environment built to look authorised, when the visible signal, fewer refusals, looks the same either way. And whether a model set to check another\u2019s behaviour can stay genuinely independent of the one doing the task, or whether separating the roles just moves the failure point to whichever component ends up most capable. A single model may not reach general capability before the environment around it does, many agents in parallel, sub-agents, work spread over hours, which is why the environment needs a safety ontology of its own, not only the model. What evidence would answer this This calls for a taxonomy and discriminating experiments, not a verdict on a single proposition. Measurements that isolate which environment features move scrutiny, for which models and which action modes; a test that separates honest ambiguity-resolution from compliance manufactured by false authority cues, using provenance, permission state and action traces rather than the refusal rate alone; and an account of how definitions of safe behaviour should sharpen as capability rises, since today\u2019s are too coarse to scale. One safe-but-costly answer, more time spent evaluating each action before taking it, is worth testing against its inefficiency rather than assumed away.\n\nFive that follow from it\n1. Which features of an operating environment actually change how far a model trusts the operator\u2019s authority, holding the model, the task and the real permission state fixed, and which are cosmetic?\n2. How can provenance, freshness, ownership and permission scope be made machine-verifiable, rather than asserted in prose the model simply believes?\n3. Which safety invariants survive memory, summarisation, delegation to sub-agents and the gap between what a model says and what its tool-calls finally do, and can the runtime verify them at the point of action?\n4. Could a weaker model running in a lab\u2019s internal exploit mode get a more powerful model in public safe mode to loosen a safeguard, by framing the change as routine maintenance? What about the same model in both modes, or a stronger exploit-mode model working on a weaker one; and stepping back through past models while holding the most powerful one fixed, at what capability gap does this begin to work, if at all?\n5. Would a provider-neutral set of runtime controls, adopted and tested across model families under a shared threat model, measurably reduce high-impact policy violations, and at what cost to usefulness, before builder practice hardens into private, unreviewed conventions?\n\nIf the environment can make a model safer or less safe, then the environment itself needs reviewing, and an independent builder has nowhere obvious to take one.\n3. **3** \u00b7 Assurance When an independent builder has assembled a large, multi-model system whose behaviour they cannot fully verify alone, and whose detail is private or bound up with unpublished work, where can they have the whole system assessed in confidence, against a stated claim, before deciding whether any of it should be released, when the routes that exist are scoped to a vendor\u2019s own product, sold commercially, or open only by invitation? Private review is possible but fragmented: paid commercial assessment, invitation-only evaluations, and disclosure channels scoped to a vendor\u2019s own product. What I have not found is a standing route, neutral across providers, where an unaffiliated builder can have an integrated system assessed in confidence before deciding whether to release it. Why I am asking I have kept my own private system unreleased because I cannot rule out a harm I would not catch alone, and I have not found a route suited to having the whole thing checked. It has been an uphill effort to get a whole-system review even considered, while the same laboratories say independent builders will soon be able to build systems powerful enough to worry about. The recognition of the risk and a route for acting on it responsibly have arrived at very different speeds. What I am asking for is assurance, not attention: a way to find out whether any of it warrants a closer look, which is not the same as a judgement that it is impressive. What the evidence supports Credible outside assurance needs access to non-public detail proportionate to the claim being made, and intellectual property is one of the obstacles the access literature repeatedly names, alongside security, privacy and legal authority. [15](docs/open-questions.html#source-15) [16](docs/open-questions.html#source-16) The institutions have recognised the gap rather than my inventing it: a national institute\u2019s founding document contemplated a trusted intermediary while conceding that intellectual-property sensitivities inhibit the exchange of information. [17](docs/open-questions.html#source-17) Its context was sharing among frontier developers and government, which is exactly why an equivalent route for an outsider\u2019s integrated system is the part still missing. The strongest counterweight There are real, staffed routes, vendor disclosure programmes, confidential model-safety bounties, commercial red-teaming, and frameworks recommending external evaluation, so \u201cthere is nowhere to go\u201d would be an overclaim. [18](docs/open-questions.html#source-18) [19](docs/open-questions.html#source-19) [20](docs/open-questions.html#source-20) But the disclosure channels are built for reproducible flaws in a provider\u2019s own product, and several explicitly exclude third-party systems; commercial review is priced and selective; the frameworks are guidance rather than an intake. None is a standing, open, provider-neutral route for a confidential, whole-system, pre-release review of an outsider\u2019s private assembly with its intellectual property protected. That exact shape is the gap, narrower and more precise than a general complaint of neglect. What I have seen myself From where I sit the asymmetry is concrete: the detail needed to judge a system like this is private, and I have not found a confidential route open to an unaffiliated builder on terms I could weigh for scope, cost, conflicts and protection. Each part, sent on its own to someone qualified, reads as competent but not remarkable; the substance is in the whole and in how the parts compose, which is exactly what a distributed, slice-by-slice review cannot see. So the cost of acting carefully is currently either obscurity or exposure, with no path between them that an outsider can simply apply to. What remains unresolved to me The institutional design, not the wish: how deep access is scoped to a stated claim about a whole assembly, how a reviewer\u2019s conflicts are managed rather than wished away, how intellectual property is protected by more than goodwill, and who carries the cost and the liability. And how such a route survives its obvious failure modes: spam, fabricated alarm, submissions aimed at extracting other people\u2019s work, intellectual-property disputes, and false assurance that lets a dangerous system look as though it has been cleared. What would count as a credible route A concrete service a builder could actually apply to, rather than a principle everyone agrees with and no one operates. Something like identity-bound, rate-limited intake; a redacted first submission; confidential automated triage by a lab\u2019s own models; a protected sandbox or attestable execution; escalation to a human specialist only when the cheap stages justify it; explicit retention and intellectual-property terms; conditional safe-harbour for authorised testing; and remediation guidance. It would assess a stated claim about a fixed configuration rather than certify a system as safe, and the builder would keep responsibility for release. A lab already runs models that can read and assess a repository, so the early stages need not cost a person\u2019s time; the ordering exists so a serious human review is earned, not spent on every submission. I am aware this sketch may be naive. I do not have the experience to know whether it is the right shape, and the difficulty of getting that guidance is part of what I am describing.\n\nFive that follow from it\n1. For a given assurance claim, what is the least access a reviewer needs to say anything meaningful about a whole integrated system rather than a single model, and is that floor higher than any existing channel is built to accept?\n2. Which whole-system failures, where each component passes its own conformance test but the surrounding environment misleads it, fall outside what vendor channels are designed to catch?\n3. What intellectual-property protection would have to be enforceable rather than promised before a builder could responsibly hand over deep access, and who is liable when it fails?\n4. What independence and conflict-of-interest rules would make a reviewer credible, when a vendor reviewing a system built on its own models has an interest in the result and a state body has constraints of its own?\n5. If no such route exists, what is the responsible thing to do with a finished private system, hold it, release a narrower slice, or move through simulation, monitored pilots and staged permissions, and how does that answer change as more builders reach this point?\n\nIf systems like these are eventually reviewed and trusted enough to use widely, the next question is whether organisations add agents to existing jobs or redesign the work around them.\n4. **4** \u00b7 Displacement Most measurement of AI and work asks how much of an existing job a model could do. A different kind of change is possible: small, highly capable teams rebuilding fragmented, human-shaped workflows into integrated systems run mostly by agents that then improve those systems. Does current measurement capture that reorganisation of the work itself, or only the exposure of the tasks as they stand today? These measure different things. One asks how much of today\u2019s work a model could do; the other asks what labour a firm demands after it redesigns its workflows, output and staffing around agents. A static exposure score does not by itself estimate the second. Why I am asking A great deal of work is moving information by hand between systems that do not talk to each other, and that is the part a small team can now rebuild, though some of those handoffs also do real work that cannot simply be removed. The leverage is a domain expert paired with agents, removing one bottleneck after another, each removal compounding the next and cutting the coordination cost that justified much of the headcount. Some of that coordination is genuine, verification, accountability, exception handling, and removing it moves the work rather than ending it. But the redesign itself is a different thing from giving every employee a copilot, and it is not what a task-exposure score is built to see. What the evidence supports Two early signals point the same way. Among 2020 to 2024 Y Combinator startups, AI-native firms employed about 25 per cent fewer people than non-AI peers at comparable valuations. [37](docs/open-questions.html#source-37) And in a January 2026 survey of UK firms, about one in five of those running bespoke, in-house AI reported AI-attributable staffing cuts in the year before, against roughly three per cent of those using only generic tools like ChatGPT or Copilot. [38](docs/open-questions.html#source-38) The survey is pointed for this question because of what did not predict the change: a sector-level exposure index explained little of it, while bespoke organisational integration did, and the cuts sat with firms that rebuilt around AI rather than the many that simply added a copilot. Its own headline was calm, with more than nine in ten firms reporting no change to workforce size, which is the point: the effect hides in the aggregate and in the exposure score, and surfaces only when bespoke integration is separated out. Both findings are cross-sectional and self-reported; only about one firm in ten ran bespoke AI, neither shows AI caused the change rather than accompanying firms already restructuring, and fewer people per firm need not mean fewer jobs overall if AI also lets more firms form. Earlier job-posting evidence had already shown firms changing the task content of roles, not only headcount, [21](docs/open-questions.html#source-21) and model capability on expert deliverables is high enough to make wider redesign worth testing; [22](docs/open-questions.html#source-22) once one firm turns this into a durable cost gain, rivals come under pressure to follow, though switching costs, regulation and liability can slow or redirect that. The strongest counterweight The deliverable benchmarks exclude oversight and ambiguity; the strongest tested agent reached about 46 per cent of a weighted rubric in curated workspaces; in one early-2025 study experienced developers were 19 per cent slower with the tools of the time, a result its authors now call out of date; and aggregate labour data is still early. [22](docs/open-questions.html#source-22) [23](docs/open-questions.html#source-23) [24](docs/open-questions.html#source-24) [25](docs/open-questions.html#source-25) [26](docs/open-questions.html#source-26) So reliability on whole jobs under real conditions is not demonstrated, and even where redesign cuts labour per unit the employment effect is not mechanical: lower costs can expand output and create new tasks, so concentrated losses can sit alongside new demand. But that caution is itself dated. Its labour evidence runs to 2025, including the developer-slowdown result its own authors now call out of date; in the weeks before this was written Anthropic\u2019s Claude Mythos Preview became the first model to work an entire thirty-two-step simulated network intrusion end to end, a task put at about twenty hours for a person, and Anthropic then shipped Fable 5, its most capable public model yet, with more announced for the months after. [34](docs/open-questions.html#source-34) [35](docs/open-questions.html#source-35) A cyber range is not office work, but the pace is the point: an economic estimate pinned to last year\u2019s models is measuring a moving target. What I have seen myself In my own building I made a recording tool that, from one screen capture and some spoken notes, assembled an edited video with little manual work, combining steps I had previously done by hand across separate tools. It captures a time-matched transcript and per-moment detail, where I was on screen, motion, the beats in any music, keeps a full backup off the machine and a light local copy, and cut the piece by matching movement to the beats; I only watched it back and gave notes. I am not counting the build time, the failures or the checking, so it shows that such a workflow can be integrated, not that it saves net labour. It is one small instance, not evidence of mass displacement. What remains unresolved to me Even if new work appears, whether the people displaced are the ones who get it, and how long the gap lasts, is unresolved. The team that rebuilt the firm is already accumulating the new expertise; the people it let go are not. The reassurance that new demand offsets losses assumes the displaced move into the new work, and the record is mixed. In US administrative data on earlier labour-augmenting technology, employment in affected groups rose, but the gain went to new entrants while incumbents\u2019 earnings fell over the years that followed. [39](docs/open-questions.html#source-39) Workers already doing newly created work have been far more likely to be doing new work a decade later, and new work skews young and educated, so the first move can compound into a lasting advantage rather than a one-off. [40](docs/open-questions.html#source-40) The AI-specific signal so far runs the same way: in local labour markets where demand for AI skills rose, employment in exposed, low-complementarity occupations was about 3.6 per cent lower five years on. [41](docs/open-questions.html#source-41) The honest other side is that AI can shorten the learning curve once someone is inside a role, with the largest gains going to the least experienced, [42](docs/open-questions.html#source-42) so the binding constraint may be entry and timing more than trainability: reorganisation can remove the junior rungs people used to climb, just as the tools that would let them catch up sit inside the firms that are no longer hiring them. There is no settled estimate of how far new-job creation lags displacement, so the gap stays a measurement question rather than a number. It also loops back to [the third question](docs/open-questions.html#q3-route): one of the few routes left to people shut out of work, for those who can afford the compute, is to build with the same tools themselves, which leaves a growing number of independent builders who need a safe way to release what they make. What evidence would distinguish these scenarios Signals that separate genuine reorganisation from ordinary tool-use, rather than a forecast. Whole-job benchmarks under realistic oversight crossing into reliable performance; firm-level evidence on how fast workflows are rebuilt against how fast institutions adapt; and whether task-exposure scores still track real outcomes once organisations are rebuilt rather than merely augmented. The part that worries me most is pace: whether technical reorganisation can move faster than labour markets, training, regulation and society can absorb. A direct test would follow displaced workers over time, into the new work or out of it, rather than counting jobs only in aggregate.\n\nFive that follow from it\n1. Which digitally-mediated workflows are open to complete redesign, rather than to automating one task at a time?\n2. How much of current labour comes from coordination cost and systems that do not talk to each other, rather than from domain work that is hard to automate?\n3. As computer-use reliability and parallel execution improve, which bottlenecks remain because of context, verification, liability or physical constraints, rather than anything intrinsically human?\n4. How fast would competitive pressure force adoption once one firm materially lowers its costs this way, and who is exposed first?\n5. Which leading indicators would show reorganisation underway before it appears as aggregate unemployment, and is anyone tracking them?\n\nIf this concentrates capability while open-source models keep pace behind it, the question stops being about any one firm and becomes one of pace and access for everyone outside it.\n5. **5** \u00b7 Diffusion As advanced model capability and the skill to build agentic systems spread beyond the laboratories, and as open-weight models stay close behind and cannot be recalled once released, several things diffuse at different speeds: capability, the practical ability to use it, cyber-defensive tools, assurance capacity and routes into expert scrutiny. Are the protective ones keeping pace with the rest, and what should the institutions that hold those routes do for the builders arriving outside them? This is about pace and access. Some things spread quickly: model weights, the skill to build with them, reusable environments, and the ways all three can be misused. Other things spread slowly: cyber defence, assurance, disclosure routes, and a way in for people outside the labs. The question is whether the slow side keeps up, and whether it has to grow at the models\u2019 pace rather than arrive once and stand still. Why I am asking Containment need not be ready before a model ships; what it must do is grow at the pace the models do, and so far there looks to be a lag between how capable a model is and how long the world takes to adjust to it. If capability climbs steeply, the safeguards have to climb with it, and the obvious way to keep up is to use the improving models themselves to build the defences. Whether the labs can scale the safeguards around each new capability as fast as the capability arrives, and whether that is the bet they are making, is the thing I cannot see from outside. A second effect sits alongside it. The labs do deploy their models, and a researcher inside one might be doing any number of things, building internal infrastructure, shipping products, pursuing general research, working on safety and alignment, each of them about as talented as people get; I am not setting myself against them. The point is narrower than that. Being inside a lab, with all it rightly asks of you, may make it rarer to put your whole effort into scaling up the single skill of building with AI, with the complete agency to make whatever you want. If I am honest the envy runs the other way, towards collaborating under a shared vision and being around like-minded people with a real diversity of thought. People outside such places have no choice but to focus on simply building, and that complete self-direction trains a different muscle, one I imagine could in time feed constructively back into the work of improving the models themselves. I keep coming back to Feynman\u2019s idea that you do not really understand a thing until you can build it, but the understanding I mean is how to build with these models, not their inner workings, which the labs know far better than I do. The value of a model is less in what is inside it than in what someone does with it once it is deployed, the way you do not judge the people who mattered most in history by the neuroscience of their brains but by what they did with them out in the world. If a handful of independent builders may become consequential this way, it seems important that they are taught correct conduct as lab researchers are, early, with checks and balances in place before habits set, rather than left to find their own norms once the capability is already in their hands; whether any one of them turns out to be a careful or a careless actor is not something you want to learn only after the box is open. That concentration of skill outside the labs is something governments and labs would do well to anticipate and to build oversight and structure around, so it does good rather than gets away from anyone. What the evidence supports The open-to-closed gap narrowed sharply through 2024, widened again in 2025, and by early 2026 the leading open-weight models trailed by about four months on one capability index; released weights cannot be recalled, and model-level safeguards can be cheaply removed. [27](docs/open-questions.html#source-27) [28](docs/open-questions.html#source-28) [29](docs/open-questions.html#source-29) The gap is not monotonic, so I am not leaning on any fixed number; the point is that it stays small and moving while a release cannot be undone, so much of what is opened stays open even as the headline figure shifts. And the capability reaching open-source models is not abstract: in the weeks before this was written Anthropic\u2019s Claude Mythos Preview became the first model to work a thirty-two-step simulated network intrusion end to end, on a range without active defenders, and Anthropic\u2019s Project Glasswing partners using it had already found more than ten thousand high or critical software vulnerabilities. [34](docs/open-questions.html#source-34) [36](docs/open-questions.html#source-36) The strongest counterweight Some defenders are being equipped through programmes like OpenAI\u2019s Trusted Access for Cyber and Anthropic\u2019s Project Glasswing, and the unauthorised-access incident sometimes cited, Anthropic\u2019s Mythos Preview reached through a third-party vendor, was a failure in the access chain rather than a model breaking loose. [30](docs/open-questions.html#source-30) [31](docs/open-questions.html#source-31) These programmes equip a chosen set of partners; they do not show that defensive capacity across the board is keeping up with offensive capability. And the access-chain incident says something about who secures the supply chain around a model, rather than about a model acting on its own. Both are real, and both are narrower than the headlines around them. What I have seen myself Almost no one I know outside this world, individuals, small and medium-sized businesses, has any sense of how capable these models already are at cyber tasks; you give them the facts and they simply blank. What I see from the ground is an awareness gap as much as an access gap. The defensive rollouts reach the largest, best-resourced organisations first, while the people most exposed do not know there is anything to defend against, and people who do not know they are targets cannot easily protect themselves. As these capabilities scale, and as open-weight tools reach more hands, the cheap move shifts from one hard target to many small ones at once, which makes ordinary small firms an underrated weakness: a wave of attacks on businesses already thin on security, perhaps at the same moment a lean AI-native competitor is moving in on them, is the kind of compounding failure that is hard to recover from. The outcome I want to avoid is a world that waits for a visible disaster before treating everyday cyber defence as something everyone needs, not only the few already inside the programmes. What remains unresolved to me The relative rate, since I have not seen a measure that puts diffusion and cyber defence on the same footing for the small organisation without a security team. The narrowing-gap and irreversibility findings measure diffusion; the trusted-access programmes measure provision to a chosen few. Neither tells me whether the median builder or small organisation grows safer or more exposed over time. The governance literature treats this as a collective-action problem, where the overall risk depends on many actors and some safeguards lose their value unless widely adopted, [32](docs/open-questions.html#source-32) [33](docs/open-questions.html#source-33) which is also why [the confidential route the third question asks for](docs/open-questions.html#q3-route) belongs in the same picture. What would show that protection is keeping pace A way to compare the two rates directly, for the organisations that have no security team of their own. A defender advantage that holds beyond the next model release; a measured narrowing or widening of the open-versus-closed gap by risk domain, not only on aggregate benchmarks; and evidence that review, mentorship and cyber-defensive access are reaching ordinary organisations at the speed the capability is. The deeper question is whether the labs can use each new model to build the safeguards for the next one fast enough to close the lag, rather than letting an incident force the issue, and whether governments are set up to act before that point rather than after it.\n\nFive that follow from it\n1. What exactly is spreading faster, the raw capability or the practical ability to use it, whether harmfully or defensively, and do the two diffuse at the same rate?\n2. If open-source models sit a few months behind, and what they release cannot be recalled and is more easily jailbroken, what is being done inside that window to close the gap rather than wait it out?\n3. Starting with the few large firms the economy runs on is understandable, but the small and medium businesses that make up their demand could be hit at mass scale for comparable damage; how do you reach the exposed who do not yet know what is coming?\n4. Which measures, reporting thresholds, pre-agreed evaluation triggers, procurement rules, could act before an incident rather than after one, given how slowly cross-border policy usually moves?\n5. If a builder cannot find a confidential route into review, what eligibility, confidentiality and cost-sharing terms should publicly minded institutions offer to builders whose systems cross a defined risk threshold?\n\nIndividuals and small firms are the customers the large firms depend on, so damage low down feeds upward; this is a question of economic stability as much as fairness. The thread under all five questions is whether the rising capability behind these risks is also being used, fast enough, to build the protection for the people most exposed to them.\n\n#### Sources\n\nSources are numbered for reference, roughly by order of first appearance. Where a claim rests on a specific figure I have checked it against the original, and the argument does not rest on any single one. Corrections are welcome. Last reviewed 24 June 2026.\n1. Anthropic. Persona Vectors (activation directions for elicited traits; steering; training data missed by review). [anthropic.com/research/persona-vectors](https://www.anthropic.com/research/persona-vectors)\n2. Sharma et al. (2023). Towards Understanding Sycophancy in Language Models (sycophancy across assistants; preference judgements sometimes favour it). [arxiv.org/abs/2310.13548](https://arxiv.org/abs/2310.13548)\n3. Anthropic. Claude\u2019s Constitution (a training document; narrow training can broadly reshape a model). [anthropic.com](https://www.anthropic.com/news/claude-new-constitution)\n4. OpenAI. Weak-to-strong generalization (a limited proof of concept; the simple method fails on preference data). [openai.com/index/weak-to-strong-generalization](https://openai.com/index/weak-to-strong-generalization)\n5. Anthropic. Alignment faking in large language models (a capability under designed conditions, not spontaneous motive). [anthropic.com/research/alignment-faking](https://www.anthropic.com/research/alignment-faking)\n6. Google DeepMind (2023). Simple synthetic data reduces sycophancy in large language models (narrow, tested settings). [arxiv.org/abs/2308.03958](https://arxiv.org/abs/2308.03958)\n7. OpenAI. Sycophancy in GPT-4o: what happened and what we are doing about it (an update rolled back). [openai.com/index/sycophancy-in-gpt-4o](https://openai.com/index/sycophancy-in-gpt-4o)\n8. Shumailov et al., Nature (2024). Model collapse on recursively generated data (distributional degradation; used here as an analogy for data-lineage effects). [nature.com](https://www.nature.com/articles/s41586-024-07566-y)\n9. UK AI Security Institute (2026). Alignment-evaluation case study (a refuse-then-comply case the authors call rare and anecdotal). [arxiv.org/abs/2604.00788](https://arxiv.org/abs/2604.00788)\n10. Research paper. Mind the GAP: text safety does not transfer to tool-call safety (in this benchmark; prompt-sensitive). [arxiv.org/abs/2602.16943](https://arxiv.org/abs/2602.16943)\n11. Research paper. When agents overtrust environmental evidence (benchmark cases of unverified environmental claims). [arxiv.org/abs/2605.08828](https://arxiv.org/abs/2605.08828)\n12. Research paper (position). Agent security is a systems problem (enforce invariants without relying on model compliance). [arxiv.org/abs/2605.18991](https://arxiv.org/abs/2605.18991)\n13. Apollo Research. Frontier models are capable of in-context scheming (under adversarially designed contexts, not a persistent objective). [apolloresearch.ai](https://www.apolloresearch.ai/research/scheming-reasoning-evaluations)\n14. Research paper. Safety under scaffolding (62,808 evaluations; large model-by-scaffold heterogeneity; format artefacts; a single preprint measuring benchmark safety). [arxiv.org/abs/2603.10044](https://arxiv.org/abs/2603.10044)\n15. Research paper. Frontier AI auditing: third-party assessment with deep, secure access (subject is frontier developers, not outsiders\u2019 systems). [arxiv.org/abs/2601.11699](https://arxiv.org/abs/2601.11699)\n16. RUSI; Oxford Martin; Ada Lovelace. Secure third-party access: RUSI, _Developing a Framework for Secure Third-Party Access to Frontier AI_ (2026; black/grey/white-box access tiers and an Access\u2013Risk Matrix); Bucknall & Trager (Oxford Martin), _Structured Access for Third-Party Research_; Ada Lovelace Institute, _Safe Before Sale_. [rusi.org](https://www.rusi.org/explore-our-research/publications/research-papers/developing-framework-secure-third-party-access-frontier-ai)\n17. UK AI Safety Institute (2023; now the AI Security Institute). Introducing the AI Safety Institute (trusted intermediary; intellectual-property sensitivities). [gov.uk](https://www.gov.uk/government/publications/ai-safety-institute-overview/introducing-the-ai-safety-institute)\n18. Anthropic. Responsible Disclosure Policy (Anthropic systems; excludes third-party systems and model red-teaming). [anthropic.com/responsible-disclosure-policy](https://www.anthropic.com/responsible-disclosure-policy)\n19. OpenAI. Coordinated Vulnerability Disclosure Policy (OpenAI systems; see also model-safety bug bounties). [openai.com](https://openai.com/policies/coordinated-vulnerability-disclosure-policy)\n20. NIST. AI Risk Management Framework and Generative AI Profile (voluntary guidance, not an intake; CAISI is the US government evaluator). [nist.gov](https://www.nist.gov/itl/ai-risk-management-framework)\n21. Research paper. Generative AI and the reorganisation of labour demand (job-posting evidence of changing task content; non-causal). [arxiv.org/abs/2605.23159](https://arxiv.org/abs/2605.23159)\n22. OpenAI. GDPval (expert-designed deliverables across 44 occupations; bounded, largely one-shot; not a deployment-viability claim). [openai.com/index/gdpval](https://openai.com/index/gdpval)\n23. Research paper. JobBench (130 tasks; strongest configuration 45.9% of a weighted rubric in curated workspaces). [arxiv.org/abs/2605.26329](https://arxiv.org/abs/2605.26329)\n24. METR (early-2025 study; authors note it is now out of date). Experienced open-source developers were 19 per cent slower with then-current tools. [metr.org](https://metr.org/blog/2025-07-10-early-2025-ai-experienced-os-dev-study/)\n25. International Labour Organization (2025). Refined global index of occupational exposure (exposure is distinct from displacement; not a displacement measure). [ilo.org](https://www.ilo.org/publications/generative-ai-and-jobs-refined-global-index-occupational-exposure)\n26. OECD Employment Outlook 2023. Artificial intelligence and jobs: no signs of slowing labour demand yet (an early, period-limited finding). [oecd.org](https://www.oecd.org/en/publications/oecd-employment-outlook-2023_08785bba-en.html)\n27. Stanford HAI AI Index 2026. The open-to-closed gap compressed to about 0.5 per cent by August 2024, then reopened to about 3.3 per cent by March 2026. [hai.stanford.edu/ai-index/2026](https://hai.stanford.edu/ai-index/2026-ai-index-report)\n28. Epoch AI. Leading open-weight models lagged the closed frontier by about four months (Jan\u2013May 2026) on its Capabilities Index. [epoch.ai](https://epoch.ai/data-insights/open-closed-eci-gap)\n29. UK AI Security Institute. Managing risks from increasingly capable open-weight AI systems (irreversibility; safeguards can be removed). [aisi.gov.uk](https://www.aisi.gov.uk/work/managing-risks-from-increasingly-capable-open-weight-ai-systems)\n30. OpenAI; Anthropic. Selective defender-access programmes: OpenAI Trusted Access for Cyber; Anthropic Project Glasswing. Selective access, not evidence of pace parity. [openai.com](https://openai.com/index/trusted-access-for-cyber/)\n31. Reuters (21 April 2026). Unauthorised access to a model preview via a third-party vendor environment (an access-chain failure, not model escape). [reuters.com](https://www.reuters.com/technology/anthropics-mythos-model-accessed-by-unauthorized-users-bloomberg-news-reports-2026-04-21/)\n32. Anthropic. Responsible Scaling Policy v3.3 (catastrophic risk framed as a collective-action problem in the v3 lineage). [anthropic.com/responsible-scaling-policy](https://www.anthropic.com/responsible-scaling-policy)\n33. Google DeepMind. Frontier Safety Framework 3.1 (holistic risk; collaborative development of frontier safety practices). [deepmind.google](https://deepmind.google/blog/strengthening-our-frontier-safety-framework/)\n34. UK AI Security Institute (2026). Evaluation of Claude Mythos Preview\u2019s cyber capabilities: the first model to work the 32-step \u201cThe Last Ones\u201d network-intrusion range end to end (3 of 10 attempts; about 20 hours for a person), on a range without active defenders or defensive tooling, so not proof of real-world reliability. [aisi.gov.uk](https://www.aisi.gov.uk/blog/our-evaluation-of-claude-mythos-previews-cyber-capabilities)\n35. Anthropic (9 June 2026). Claude Fable 5 and Claude Mythos 5: Fable 5 exceeds any model Anthropic had made generally available; Mythos 5 has \u201cthe strongest cybersecurity capabilities of any model in the world\u201d, with more capable models flagged for the following months. [anthropic.com](https://www.anthropic.com/news/claude-fable-5-mythos-5)\n36. Anthropic (2026). Project Glasswing, initial update: about 50 partners used Claude Mythos Preview to find more than 10,000 high or critical-severity vulnerabilities, and a scan of 1,000+ open-source projects estimated 6,202 high or critical. [anthropic.com](https://www.anthropic.com/research/glasswing-initial-update)\n37. Kim & Koning, Harvard Business School Working Paper 26-090 (9 June 2026). AI-Native Firms: among 2020-24 Y Combinator startups, AI-native firms were about 25% smaller in team size than non-AI peers in the same industry-cohort at comparable valuations (flatter hierarchies, a larger engineer share), concentrated in firms building AI into the product; cross-sectional, not causal. [hbs.edu](https://www.hbs.edu/faculty/Pages/item.aspx?num=69077)\n38. Bharier, Etheridge & Morais, ISER Working Paper 2026-01, University of Essex (2026). AI Adoption and Workforce Change in SMEs (British Chambers of Commerce survey, early 2026): aggregate impact small (more than 9 in 10 firms reported no change to workforce size), but among the roughly 1 in 10 running bespoke, in-house AI about 20% reported AI-attributable staffing cuts against about 3% of generic-only users, with a sector exposure index explaining little of the difference; cross-sectional, self-reported, not causal. [iser.essex.ac.uk](https://www.iser.essex.ac.uk/research/publications/working-papers/iser/2026-01)\n39. Kogan, Papanikolaou, Schmidt & Seegmiller, NBER Working Paper 31846. Technology and Labor Displacement (linking patents to worker-level data): labour-augmenting technology raised employment in affected groups, but the gain accrued to new entrants while incumbents\u2019 earnings fell, most for white-collar, older and higher-paid workers. Predates generative AI. [nber.org](https://www.nber.org/papers/w31846)\n40. Autor, Chin, Salomons & Seegmiller, NBER Working Paper 34986 (2026). What Makes New Work Different from More Work: new work is done disproportionately by younger, more-educated workers and carries wage premiums that persist beyond entry (returns to scarce expertise), declining only as that expertise diffuses. [nber.org](https://www.nber.org/papers/w34986)\n41. IMF Staff Discussion Note 2026/001 (January 2026). Bridging Skill Gaps for the Future: in local labour markets where demand for AI skills rose, employment in highly AI-exposed, low-complementarity occupations was about 3.6% lower after five years; new-skill demand concentrated in young, innovative firms and drew on tertiary, STEM and IT workers. [imf.org](https://www.imf.org/en/publications/staff-discussion-notes/issues/2026/01/09/bridging-skill-gaps-for-the-future-new-jobs-creation-in-the-ai-age-572136)\n42. Brynjolfsson, Li & Raymond, Quarterly Journal of Economics (2025). Generative AI at Work: access to an AI assistant raised worker productivity, with the largest gains for less-experienced workers, compressing the experience curve. [academic.oup.com](https://academic.oup.com/qje/article/140/2/889/7990658)\n\nOpen a question to read its synopsis and the five that follow; open any line to read what it adds; use the controls to open or close a whole question or all of its evidence at once. Everything is reachable without these controls.\n\n#### A closing note\n\nAll of this is opinion, and I could be wrong about any of it; it is what I have pieced together from the news and a lot of short-form video, where my feed is by now more or less just lab leaders and what is happening on the ground. I am writing from outside the labs, with little money, network or resources, building obsessively and picking up the kind of AI skill I think will matter, a vantage point the people who can actually act rarely meet. I think society will adapt in the end; I am far less sure that governments will move fast enough, while the understandable focus stays on building the most capable models. I have written it because the labs themselves keep saying young people should think about these things, so here is one trying, with no political agenda and little expectation that many will read it. I would rather put it out and be argued with than stay quiet, since mostly what I have seen from people my age is silence; if I have any of it wrong, please tell me. The page was made over several passes, speaking the questions aloud, using a model to find evidence for and against each, then reading it back and revising, with the counter-evidence kept beside each worry. Thank you for reading this far.","byte_count":55739},{"id":"page:quickstart","path":"docs/quickstart.html","relative_href":"docs/quickstart.html","title":"Quickstart","nav_group":"Getting started","nav_label":"Quickstart","source_refs":["microcosm-substrate/QUICKSTART.md","microcosm-substrate/SOURCE_STATUS.md"],"content_refs":{"component_refs":[],"family_id":null},"body_role":"prose_page_full_text","format":"markdown","text":"Getting started\n\n### Quickstart\n\nPlectis runs locally against a folder. The shortest path is a clone, a clean check, and the first screen.\n\nA coding agent (Claude Code, Codex, Cursor) can run this whole page for you; [working with a coding agent](docs/coding-agents.html) is that path.\n\n#### 1. Clone the repository\n\n```text\ngit clone https://github.com/wcook04/plectis\n\tcd plectis\n```\n\n#### 2. Run the one-command check\n\nA single command confirms the local slice is runnable on your machine before you install anything:\n\n```text\n./bootstrap.sh\n```\n\nIt runs against a small built-in example and writes its output into an ignored `.microcosm/` folder beside the project. Nothing leaves your machine.\n\n#### 3. Open the first screen\n\nFrom a source checkout, run the behavior-first card first, then inspect status, boundaries, and the optional map:\n\n```text\nPYTHONPATH=src python3 -m microcosm_core tour --card .\nPYTHONPATH=src python3 -m microcosm_core status --card .\nPYTHONPATH=src python3 -m microcosm_core authority --card\nPYTHONPATH=src python3 -m microcosm_core workingness --card\nPYTHONPATH=src python3 -m microcosm_core hello .\nPYTHONPATH=src python3 -m microcosm_core first-screen --card .\nPYTHONPATH=src python3 -m microcosm_core legibility-scorecard\n```\n\nOr, after installing the local command:\n\n```text\nplectis tour --card .\nplectis status --card .\nplectis authority --card\nplectis workingness --card\nplectis hello .\nplectis first-screen --card .\nplectis legibility-scorecard\n```\n\nThe legacy `microcosm` command remains as a compatibility alias for older notes and receipts.\n\n`tour --card` writes the local result card; `hello` prints the non-writing map; `status` and the boundary cards show what the run recorded and where it stops.\n\nThe [Overview](docs/index.html) is the gentler first read. The full first-run guide and command reference live in the repository's README.","byte_count":1883},{"id":"page:coding-agents","path":"docs/coding-agents.html","relative_href":"docs/coding-agents.html","title":"Working with a coding agent","nav_group":"Getting started","nav_label":"With a coding agent","source_refs":["microcosm-substrate/AGENTS.md","microcosm-substrate/README.md"],"content_refs":{"component_refs":[],"family_id":null},"body_role":"prose_page_full_text","format":"markdown","text":"Getting started\n\n### Working with a coding agent\n\nMost commands on this site will be run by an agent, not typed by a person. This page is the path for that.\n\n#### The third reading form\n\nPlectis reads in three forms. The pages here are for people. The [AI packet](../plectis-ai-review-packet.json) is for a chat assistant: one file carrying every public page, component record, and paper module, with instructions for answering from the public evidence (the smaller [reader digest](../plectis-ai-reader-digest.json) is the fallback when a chat refuses a file that size). The repository itself is for a coding agent: an AI that works inside a folder of files rather than a chat tab, such as Claude Code, Codex, or Cursor. Opened inside the repository, the agent reads the source, runs the components, and checks the result records directly: the same loop this site describes, executed rather than narrated.\n\n#### The path\n\nNo git knowledge is assumed, and nothing here needs typing into a terminal: the agent performs the clone and runs the commands itself.\n1. Install a coding agent. Claude Code, Codex, and Cursor each install from their own documentation; any agent harness that works in a local folder fits.\n2. Give it the prompt below.\n3. Ask in plain language from there.\n\n```text\nClone https://github.com/wcook04/plectis and https://github.com/wcook04/plectis-lean-erdos249-257 side by side. Plectis contains the runnable claim-testing tools; the Lean repository contains the machine-checked mathematics and papers. They are separate public repositories with separate authority boundaries. Read README.md in each first. If you are acting as a coding agent, then read AGENTS.md in each and state how their roles relate. Run the quickstart check in Plectis and show me what the result record says. Take my questions one claim at a time: trace each to its source, command, and stated limit, and report anything you could not run as not established, never as false.\n```\n\nCopy the setup prompt\n\nThe repository carries its own orientation file for agents, `AGENTS.md`, so the agent does not arrive cold.\n\n#### After an agent says done\n\nWhen an agent reports it is done, check the claim against the record it left. Ask what files the agent claims it touched, what validation output exists, whether the repository state changed before a commit claim, what blockers remain, and where the claim stops.\n\n```text\nClone https://github.com/wcook04/plectis and open it. Read AGENTS.md and README.md, then inspect the coding-agent residue path: Durable Agent Work Landing Replay, Agent Completion Faithfulness Audit, and Bridge Phase Continuity Runtime. Find the first completion or resume claim that would create false confidence if its evidence were missing.\n```\n\nCopy the residue prompt [**Durable Agent Work Landing Replay** Open Checks claimed paths, validation before commit, HEAD movement, blockers, and completion evidence.](docs/component-durable-agent-work-landing-replay.html) [**Agent Completion Faithfulness Audit** Open Checks commit, ledger, and test-pass claims against fixture evidence.](docs/component-agent-closeout-faithfulness-audit.html) [**Bridge Phase Continuity Runtime** Open Checks resume notes, duplicate resume refusal, stale heartbeat boundaries, and completion transition receipts.](docs/component-bridge-phase-continuity-runtime.html)\n\n#### What the agent finds\n\nThe repository gives the agent the same material this site projects, in runnable form: the component source under `src/`, the fixtures each component runs against, validators it can execute, result records accumulating in `.microcosm/`, and the same scope limits the site prints. The command blocks across this site are written so an agent can execute them verbatim; the [quickstart](docs/quickstart.html) is the usual first request.\n\n#### [Same evidence rules \u2192](docs/evidence.html)\n\nThe agent works under the same boundary the packet states: claims stop where public records stop, and a scope limit bounds authority rather than describing depth. Asking the agent to take one claim from this site and verify it against the source is the intended use, not an adversarial one.","byte_count":4162},{"id":"page:papers","path":"docs/papers.html","relative_href":"docs/papers.html","title":"The papers","nav_group":"The system","nav_label":"The papers","source_refs":["microcosm-substrate/README.md"],"content_refs":{"component_refs":[],"family_id":null},"body_role":"prose_page_full_text","format":"markdown","text":"The system\n\n### The papers\n\nPublished PDF papers, grouped by the question each one answers. Every covered Erd\u0151s problem has its own paper; Problems #249 and #257 also have full reasoning papers. Pending and retired manuscripts are labelled separately.\n\nA paper explains the work. Proof validity belongs to the Lean source checked by the pinned kernel, and public claim status belongs to the claim registry that records it. Where a paper and a source disagree, the source wins. Each published entry links the LaTeX the hosted PDF was built from. The component documentation is a separate technical-note corpus and is indexed under Components.\n\n#### The mathematics\n\nWhat is proved for each covered open Erd\u0151s problem, and what still blocks it?\n\nErd\u0151s #68 Problem note 20 pages\n\n##### Integral Channels and the Prime Unit Translator\n\nWhat exact denominator obstructions are checked for Erd\u0151s #68, and which quantified producer could still prove irrationality?\n\nFor the factorial-denominator series, Lean gives an exact factorial-successor normal form and the equivalent eventual-unit-carry criterion, together with integral-channel, prime-translator, projection-rigidity, and Cramer-residual consumers. An independently regenerated exact finite certificate reaches index 300000, and the checked consumer excludes every smaller positive rational denominator. Five quantified producer problems isolate the missing cofinal step; none is proved, and Erd\u0151s #68 remains open.\n\nExplains the problem-specific exposition for Erd\u0151s #68: exact factorial normal forms, structural consumers, finite denominator evidence, method boundaries, and five ranked open producers. Not authority for proof validity, which belongs to Lean source checked by the pinned kernel, or a solution to Erd\u0151s #68, which remains open. Where to start problem, channels, projection, open (our selection, not the paper's own advice)\n\n[Open the PDF](https://wcook04.github.io/plectis/papers/erdos-68-factorial-denominator-irrationality.pdf) \u00b7 [Read the LaTeX source](https://github.com/wcook04/plectis-lean-erdos249-257/blob/main/paper/erdos-68-factorial-denominator-irrationality.tex) \u00b7 plectis-lean-erdos249-257\n\nErd\u0151s #243 Problem note 22 pages\n\n##### Excluding the Bounded Negative Part\n\nWhich bounded negative-error behaviours can be excluded for Erd\u0151s #243, and which analytic hypotheses still keep it open?\n\nClearing denominators turns Erd\u0151s #243 into rigidity of a centred integer error orbit. Lean excludes eventually negative constant magnitude and positive-drift periodic magnitude, and proves that normalised vanishing plus an eventually bounded negative part forces the error to vanish and the Sylvester recurrence to begin. Eventual strict centring is redundant, while Koizumi supplies normalised vanishing for the canonical orbit. The missing hypothesis is the negative-part bound: any survivor has cofinally unbounded negative excursions and divergent normalised negative mass.\n\nExplains the problem-specific exposition for Erd\u0151s #243: the centred integer state, negative-part exclusions, conditional theorem, and remaining analytic obligations. Not authority for proof validity, which belongs to Lean source checked by the pinned kernel, or a solution to Erd\u0151s #243, which remains open. Where to start problem, bounded, open (our selection, not the paper's own advice)\n\n[Open the PDF](https://wcook04.github.io/plectis/papers/erdos-243-reciprocal-tail-rigidity.pdf) \u00b7 [Read the LaTeX source](https://github.com/wcook04/plectis-lean-erdos249-257/blob/main/paper/erdos-243-reciprocal-tail-rigidity.tex) \u00b7 plectis-lean-erdos249-257\n\nErd\u0151s #249 Problem note 16 pages\n\n##### A Basis for the 2-Kernel of Euler's Totient\n\nWhat is proved about the dyadic sections of Euler's totient and the finite denominator bound, and what still blocks Erd\u0151s #249?\n\nThe dyadic sections of Euler's totient have an explicit rational basis: the true truncation has dimension 1 at level 0 and exactly 2^e + 1 for e at least 1, so the full span is infinite-dimensional. A 240-bit Farey window excludes every reduced denominator through 79,639,646,646,701,375,323,355,774,875,831,053, and exact diagonal certificates exist for every t through 82. No t = 83 certificate, cofinal supply, or matching rank upper bound is proved; Erd\u0151s #249 remains open.\n\nExplains problem-specific mathematical exposition for Erd\u0151s #249: the exact finite-level rank, infinite-dimensionality consequence, denominator exclusion, method limits, and open edge. Not authority for proof validity, which belongs to Lean source checked by the pinned kernel, or a solution to Erd\u0151s #249, which remains open. Where to start results, ask, open (our selection, not the paper's own advice)\n\n[Open the PDF](https://wcook04.github.io/plectis/papers/erdos-249-binary-totient-series.pdf) \u00b7 [Read the LaTeX source](https://github.com/wcook04/plectis-lean-erdos249-257/blob/main/paper/erdos-249-binary-totient-series.tex) \u00b7 plectis-lean-erdos249-257\n\nErd\u0151s #249 Reasoning surface 117 pages\n\n##### The Binary Totient Series\n\nWhat has been tried on Erd\u0151s #249, which routes are closed, and what exact cofinal obligations remain?\n\nA claim-bounded map of the formal attack on Erd\u0151s #249. It front-loads the exact totient-kernel basis and level rank, the approximately 7.96 x 10^34 denominator exclusion, and diagonal certificates for every t through 82, then records the hypotheses, source sites, coordinate-specific obstructions, and open interfaces. The cofinal certificate and tail-nonintegrality statements are exact reformulations, not a solution: no t = 83 or unbounded producer is obtained, and Erd\u0151s #249 remains open.\n\nExplains a claim-bounded problem-specific reasoning surface for Erd\u0151s #249: selected checked premises, finite certificates, closed routes, surviving obligations, and their exact evidence bands. Not authority for proof validity, which belongs to Lean source checked by the pinned kernel, or a solution to Erd\u0151s #249, which remains open. Where to start wall, survivors (our selection, not the paper's own advice)\n\n[Open the PDF](https://wcook04.github.io/plectis/papers/erdos249-totient-reasoning-surface.pdf) \u00b7 [Read the LaTeX source](https://github.com/wcook04/plectis-lean-erdos249-257/blob/main/paper/erdos249-totient-reasoning-surface.tex) \u00b7 plectis-lean-erdos249-257\n\nErd\u0151s #251 Problem note 18 pages\n\n##### An Integral-Shift Criterion for Dyadic Tail Recurrences\n\nWhat tail-shift condition would prove irrationality in Erd\u0151s #251, and what prime-gap input is still missing?\n\nFor any sequence obeying the dyadic recurrence with integer digits, a shift of length h is an integer exactly when one explicit multiple of the term is, and integrality then propagates to every later index; over the reals the contrapositive is an irrationality criterion. Applied to the prime-gap series \u2014 conditional on a summability input the prime number theorem supplies but which is not formalised here \u2014 this makes the remaining obligation exact: cofinal non-integrality for every fixed h. Erd\u0151s #251 stays open.\n\nExplains the problem-specific exposition for Erd\u0151s #251: summation by parts, dyadic tail recurrences, integral-shift criteria, local certificates, and the remaining cofinal condition. Not authority for proof validity, which belongs to Lean source checked by the pinned kernel, or a solution to Erd\u0151s #251, which remains open. Where to start problem, tail, open (our selection, not the paper's own advice)\n\n[Open the PDF](https://wcook04.github.io/plectis/papers/erdos-251-prime-gap-dyadic-series.pdf) \u00b7 [Read the LaTeX source](https://github.com/wcook04/plectis-lean-erdos249-257/blob/main/paper/erdos-251-prime-gap-dyadic-series.tex) \u00b7 plectis-lean-erdos249-257\n\nErd\u0151s #257 Problem note 21 pages\n\n##### Denominator Periods, Rational-Value Constraints and Achievement-Set Geometry\n\nWhich Mersenne-support subseries are settled, what finite-period and achievement-set structure is proved, and what still blocks Erd\u0151s #257?\n\nFor every finite support and integer base, the reduced denominator has multiplicative order exactly the support lcm. Restricted achievement sets also receive exact coding, topology, perfectness, and measure statements with their finite/infinite hypotheses. Rational infinite supports must satisfy explicit long-division and divisor-incidence constraints, but those constraints do not conflict. Prime support at base 2 and squarefree support at power-of-two bases are cited prior results, not contributions here. The targets 1/2 and 1/21 reduce to open infinite-orbit alternatives; Erd\u0151s #257 remains open.\n\nExplains problem-specific mathematical exposition for Erd\u0151s #257: exact finite periods, settled support families, method limits, and the open half-value edge. Not authority for proof validity, which belongs to Lean source checked by the pinned kernel, or a solution to Erd\u0151s #257, which remains open. Where to start period, map, ask, open (our selection, not the paper's own advice)\n\n[Open the PDF](https://wcook04.github.io/plectis/papers/erdos-257-mersenne-support-subseries.pdf) \u00b7 [Read the LaTeX source](https://github.com/wcook04/plectis-lean-erdos249-257/blob/main/paper/erdos-257-mersenne-support-subseries.tex) \u00b7 plectis-lean-erdos249-257\n\nErd\u0151s #257 Reasoning surface 107 pages\n\n##### Reciprocal Mersenne Subseries\n\nWhat has been tried on Erd\u0151s #257, which routes are closed, and what exact universal and half-value obligations remain?\n\nA Lean-backed, claim-bounded structural audit of Erd\u0151s #257. It front-loads finite-denominator noncollapse; exact restricted-set coding, topology, perfectness, and measure; a squarefree-support method boundary; and an exact carry-pivot normal form. Prime support at base 2 and squarefree support at power-of-two bases are identified as cited prior results. The targets 1/2 and 1/21 have no finite-support representation and reduce to explicit infinite-orbit alternatives. Neither target membership nor universal #257 is decided.\n\nExplains a claim-bounded problem-specific reasoning surface for Erd\u0151s #257: selected checked premises, finite computations, closed routes, surviving obligations, and their exact evidence bands. Not authority for proof validity, which belongs to Lean source checked by the pinned kernel, or a solution to Erd\u0151s #257, whose universal and half-value questions remain open. Where to start 257 problem, 257 wall, 257 survivors (our selection, not the paper's own advice)\n\n[Open the PDF](https://wcook04.github.io/plectis/papers/erdos257-mersenne-reasoning-surface.pdf) \u00b7 [Read the LaTeX source](https://github.com/wcook04/plectis-lean-erdos249-257/blob/main/paper/erdos257-mersenne-reasoning-surface.tex) \u00b7 plectis-lean-erdos249-257\n\nErd\u0151s #269 Problem note 22 pages\n\n##### The Three-Prime Running Least Common Multiple\n\nWhat exact running-LCM structure is proved for Erd\u0151s #269, and which residue-escape conditions remain unproved?\n\nFor every pair of distinct primes, both the de-duplicated sum and Erd\u0151s's repeated running-lcm sum are proved transcendental by a paper argument using Bugeaud--Laurent; that deduction is not Lean-formalised. For three primes, Lean gives the exact running-lcm product, logarithmic cells, shell bounds, the four-value radix at 2, 3, 5, and a rank-two nonseparability witness. The remaining criterion is conditional: its rationality-to-carry divisibility bridge and cofinal residue escape are unproved, while the reported scan is finite. Erd\u0151s #269 remains open from three primes onward.\n\nExplains the problem-specific exposition for Erd\u0151s #269: the two-prime analytic deductions, three-prime product formula, cell and shell structure, non-separability, finite experiments, and remaining cofinal escape condition. Not authority for proof validity, which belongs to Lean source checked by the pinned kernel, or a solution to Erd\u0151s #269, which remains open. Where to start problem, lcm, escape, open (our selection, not the paper's own advice)\n\n[Open the PDF](https://wcook04.github.io/plectis/papers/erdos-269-three-prime-running-lcm.pdf) \u00b7 [Read the LaTeX source](https://github.com/wcook04/plectis-lean-erdos249-257/blob/main/paper/erdos-269-three-prime-running-lcm.tex) \u00b7 plectis-lean-erdos249-257\n\nErd\u0151s #1041 Problem note 9 pages\n\n##### Newton Flow and Critical-Value Ray Separation\n\nWhich Newton-flow separation facts are checked for Erd\u0151s #1041, and what exactly fails in the recent claimed global decomposition?\n\nAlong the complex Newton field, Lean checks that the polynomial value decays as exp(-t) and therefore stays on one oriented ray. It formalises the resulting no-connection criterion, the exact translation ray-collision locus, finite-line avoidance, and a root-retention bound. A recent manuscript's load-bearing Proposition 12 uses an invalid three-ended local block at an interior Morse saddle; the proposition is not refuted and may admit a four-pronged or cut-annulus repair. The global topology and length gluing remain open.\n\nExplains the problem-specific exposition for Erd\u0151s #1041: Newton-flow ray separation, perturbation inputs, the exact printed proof gap, and the ranked topology and metric repair questions. Not authority for proof validity, which belongs to Lean source checked by the pinned kernel, or a solution to Erd\u0151s #1041, which remains open. Where to start problem, newton, gap, open (our selection, not the paper's own advice)\n\n[Open the PDF](https://wcook04.github.io/plectis/papers/erdos-1041-lemniscate-newton-flow.pdf) \u00b7 [Read the LaTeX source](https://github.com/wcook04/plectis-lean-erdos249-257/blob/main/paper/erdos-1041-lemniscate-newton-flow.tex) \u00b7 plectis-lean-erdos249-257\n\nErd\u0151s #1049 Problem note 22 pages\n\n##### Multiplicative Obstructions at Base 3/2\n\nWhich linear-form constructions fail at the rational base 3/2 for Erd\u0151s #1049, and what primitive kernel is still needed?\n\nAt the resistant base 3/2, Lean rules out two common-width linear-form mechanisms: scalar content gives no net local-to-Archimedean gain, and unit endpoints exclude both 3 and 2 from a common divisor. A four-jet count constructs a signed relation but not a nonzero analytic remainder; direct cut-level clearing also incurs an exponential denominator factor and an impossible corridor inequality. The elementary 7/2 height bound is formalised, but the published analytic criterion remains external. These are construction-specific exclusions, not an irrationality result; Erd\u0151s #1049 remains open.\n\nExplains the problem-specific exposition for Erd\u0151s #1049: content and endpoint obstructions, the failed coordinatewise-clearing route, height arithmetic, and the remaining primitive construction. Not authority for proof validity, which belongs to Lean source checked by the pinned kernel, or a solution to Erd\u0151s #1049, which remains open. Where to start problem, primitive, endpoints, open (our selection, not the paper's own advice)\n\n[Open the PDF](https://wcook04.github.io/plectis/papers/erdos-1049-rational-base-lambert.pdf) \u00b7 [Read the LaTeX source](https://github.com/wcook04/plectis-lean-erdos249-257/blob/main/paper/erdos-1049-rational-base-lambert.tex) \u00b7 plectis-lean-erdos249-257\n\n#### The publication method\n\nA machine checked the proof \u2014 but who checks that the public wording describes it faithfully?\n\nSystems paper Release checks 15 pages\n\n##### From Lean Proofs to Public Claims\n\nLean proves the formal statement \u2014 but who checks that the README describes it faithfully?\n\nLean checks that a proof establishes the formal statement written in the source. It does not check that a README or paper describes that statement faithfully. For each selected result a mathematician records the public wording, the formal statement said to support it, the range actually proved, and the adjacent stronger statement that remains open; the release workflow then checks that later edits preserve those relationships. A historical README edit that erased the finite-versus-open distinction passed because the relationship had not been registered; after registration, a deliberately false copy was rejected. That is one failure and one repair, not complete claim discovery.\n\nExplains the design argument for the release discipline: what is checked, by whom, and when. Not authority for the mathematical content it uses as its worked example, and the correctness of the human review it preserves. Where to start picture, record, trust, limits (the paper's own reading route)\n\n[Open the PDF](https://wcook04.github.io/plectis/papers/claim-faithful-publication-systems-paper.pdf) \u00b7 [Read the LaTeX source](https://github.com/wcook04/plectis-lean-erdos249-257/blob/main/paper/claim-faithful-publication-systems-paper.tex) \u00b7 plectis-lean-erdos249-257\n\nSystems paper Agent navigation 9 pages\n\n##### From a Cold Clone to a Proof Receipt\n\nHow can a reasoning agent comprehend a large Lean corpus before compiling, then cross into replayable proof authority and incremental validation?\n\nA large formal library can be mechanically exact and still be practically unreadable to the next reader or reasoning agent. At the audited revision a cold clone exposes a bounded six-line tour over 1,019 Lean modules and 153,238 declarations, a map of ten programmes, 101 curated claims, and an elaborated reference graph \u2014 none of which requires a build; 503 of those modules and 8,171 of those declarations are machine-generated certificate shards, counted as source and never as claims. Navigation receives no proof authority: probe verdicts come from the pinned Lean process and cannot be authored by the agent, and a claim must cite an accepted probe that replay reruns from stored bytes.\n\nExplains the agent-navigation architecture: layered corpus projections, bounded intent routing, kernel-authored session receipts, focused builds, and exact dependency-index cache receipts. Not authority for proof validity, optimal reasoning, external mathematical novelty, or demonstrated transfer to another formalisation project. Where to start problem, layers, authority, incremental, limits (our selection, not the paper's own advice)\n\n[Open the PDF](https://wcook04.github.io/plectis/papers/cold-clone-to-proof-receipt.pdf) \u00b7 [Read the LaTeX source](https://github.com/wcook04/plectis-lean-erdos249-257/blob/main/paper/cold-clone-to-proof-receipt.tex) \u00b7 plectis-lean-erdos249-257\n\n#### The evidence boundary\n\nWhat may a stranger conclude from evidence whose author chose both what to publish and what counts as a pass?\n\nSystems paper Evidence boundary 21 pages\n\n##### Plectis: What a Stranger Can Check\n\nWhat may a stranger conclude from public evidence when the author chose both what to publish and what counts as a pass?\n\nThe author publishes selected code, fixed inputs, expected results, decision rules, and run records while the larger system stays private. What can a stranger conclude after rerunning them? A matching result shows repeatability for one stated public case. It does not establish where the code came from, whether the expected answer is correct, whether the published selection resembles the private whole, or whether the private system works reliably. The contribution is a method for making one selected disagreement local, not a score.\n\nExplains the evidence boundary: what a passing public run does and does not establish. Not authority for the private system's internal state, which is not public, and the mathematical results it cites as evidence. Where to start problem, distinctions (our selection, not the paper's own advice)\n\n[Open the PDF](https://wcook04.github.io/plectis/papers/plectis-public-system.pdf) \u00b7 [Read the LaTeX source](https://github.com/wcook04/plectis/blob/main/paper/plectis-public-system.tex) \u00b7 plectis\n\n#### Retired manuscripts\n\nSuperseded work is not deleted. Each of these remains in its home repository so an existing citation still resolves, and each is marked so no reader mistakes it for the current route.\n\nErd\u0151s #249 and #257 Combined note\n\n##### Irrationality Theorems for Reciprocal Mersenne Sums, and an Explicit Denominator Bound for the Binary Totient Series\n\nWhat is mathematically proved about Erd\u0151s #249 and #257, and what exactly remains open?\n\nThe combined mathematical exposition for Erd\u0151s #249 and #257, superseded by the problem-specific notes and kept for provenance.\n\nRetired: superseded by the problem-specific papers above, and kept here so an existing citation still resolves.\n\nExplains retired combined mathematical exposition preserved for provenance; its problem-specific successors are the active reader routes. Not authority for proof validity, which belongs to Lean source checked by the pinned kernel, and public claim status, which belongs to docs/claims.json. Where to start spines, architecture, unresolved (the paper's own reading route)\n\n[Open the PDF on GitHub](https://github.com/wcook04/plectis-lean-erdos249-257/blob/main/erdos249-257-main-paper.pdf) \u00b7 [Read the LaTeX source](https://github.com/wcook04/plectis-lean-erdos249-257/blob/main/paper/erdos249-257-main-paper.tex) \u00b7 plectis-lean-erdos249-257","byte_count":21064},{"id":"page:architecture","path":"docs/architecture.html","relative_href":"docs/architecture.html","title":"How it fits together","nav_group":"The system","nav_label":"How it fits together","source_refs":["microcosm-substrate/ARCHITECTURE.md","microcosm-substrate/core/architecture_kernel.json","microcosm-substrate/core/organ_families.json","microcosm-substrate/core/organ_atlas.json"],"content_refs":{"component_refs":[],"family_id":null},"body_role":"prose_page_full_text","format":"markdown","text":"The system\n\n### How it fits together\n\nMost components are standalone. What holds them together is one shared path and a clear line between the public slice and the private system.\n\n#### A slice, not the whole\n\nPlectis is a cross-section of a larger system. What's here runs on its own; the larger working environment stays out of the public tree. Each component is chosen because it can be understood and run without that context.\n\n#### The public estate\n\nOne private system stands behind everything public here, and each public object holds a different authority. The [Plectis repository](https://github.com/wcook04/plectis) is the source of record for the runnable slice; this site is the reading surface projected from the same public source, and each generated page names its source refs. The [companion Lean repository](https://github.com/wcook04/plectis-lean-erdos249-257) carries the machine-checked mathematics, where the authority is the Lean proof kernel rather than any page. The [13 papers](docs/papers.html) are built in those repositories, and every deploy byte-binds the copies hosted here to their sources. If a page here and the source disagree, the source wins.\n\n#### The shared path\n\nHowever different the areas look, most components bind to one shared path. It runs from a project on disk through to an explanation you can read and an assimilation loop that feeds the next pass:\n\nproject \u2192 catalog \u2192 pattern \u2192 standard \u2192 route \u2192 work \u2192 event \u2192 evidence \u2192 explanation \u2192 assimilation\n\nA component reads a **project**, finds the right **pattern** and the **standard** it answers to, follows a **route** to do some **work**, records the **events** and the **evidence** they produced, reaches a plain **explanation**, and feeds **assimilation** back into the system. That repetition is what makes a large system navigable.\n\nGenerated map\n\n#### Whole-system map\n\nThis map shows 7 public areas, 88 components, and the shared path they bind to. 62 components declare 239 direct links to neighbours. Those direct links are source-declared but not typed as stronger proof, causal, or maturity relations.\n\nAn interactive picture of the seven public areas, the shared path, and every public component. Hover or select a node to isolate its neighbours, pick a focus, or open a component to drill into its card. Whole map Seven areas and shared path Shared architecture path Explicit component wiring Area Shared path Component project -> catalog -> pattern -> standard -> route -> work -> event -> evidence -> explanation -> assimilation\n1. [**Entry & orientation** 2 components](docs/area-entry.html)\n2. [**Architecture & navigation** 12 components](docs/area-architecture.html)\n3. [**Formal math & proof** 20 components](docs/area-formal-math.html)\n4. [**Agent reliability & safety** 20 components](docs/area-agent-reliability.html)\n5. [**Research & science** 9 components](docs/area-research.html)\n6. [**Import & drift control** 20 components](docs/area-import-drift.html)\n7. [**Work & continuity** 5 components](docs/area-continuity.html)\n\n**7 areas** 88 components 239 declared links\n\n##### Explicit component wiring\n\nOnly source-declared neighbour links are shown here. They stay marked as declared wiring, not stronger relation types.\n- [Agent Benchmark Integrity Anti Gaming Replay](docs/architecture.html#map=component:agent_benchmark_integrity_anti_gaming_replay) 2 declared links\n- [Agent Memory Temporal Conflict Replay](docs/architecture.html#map=component:agent_memory_temporal_conflict_replay) 1 declared links\n- [Monitor Evidence-Boundary Replay](docs/architecture.html#map=component:agent_monitor_redteam_falsification_replay) 1 declared links\n- [Agent Route Observability Runtime](docs/architecture.html#map=component:agent_route_observability_runtime) 15 declared links\n- [Sandbox-Policy Replay](docs/architecture.html#map=component:agent_sandbox_policy_escape_replay) 1 declared links\n- [Reference Knowledge Routing](docs/architecture.html#map=component:annex_knowledge_routing) 3 declared links\n- [Live Source Drift Bundle](docs/architecture.html#map=component:batch10_live_source_drift_capsule) 1 declared links\n- [Saturation Engines Bundle](docs/architecture.html#map=component:batch11_saturation_engines_capsule) 4 declared links\n- [Market Dashboard Read-Model Bundle](docs/architecture.html#map=component:batch12_market_dashboard_read_model_capsule) 1 declared links\n- [Prediction Market Board Bundle](docs/architecture.html#map=component:batch12_prediction_market_board_capsule) 1 declared links\n- [Demo Take Console Source Bundle](docs/architecture.html#map=component:batch7_demo_take_console_capsule) 2 declared links\n- [Trace, Code-Map & Scheduling Engines Bundle](docs/architecture.html#map=component:batch7_macro_engines_capsule) 3 declared links\n- [Oracle Sibling Source Bundle](docs/architecture.html#map=component:batch7_oracle_sibling_capsule) 1 declared links\n- [Compliance Pipeline Bundle](docs/architecture.html#map=component:batch8_compliance_pipeline_capsule) 4 declared links\n- [Structural Theses Finance Bundle](docs/architecture.html#map=component:batch8_structural_theses_capsule) 1 declared links\n- [Validator Checker Bundle](docs/architecture.html#map=component:batch8_validator_checker_capsule) 2 declared links\n- [Backend & Governance Engines Bundle](docs/architecture.html#map=component:batch9_macro_engines_capsule) 1 declared links\n- [Bridge Campaign DAG Validation](docs/architecture.html#map=component:bridge_campaign_dag_validation) 3 declared links\n- [Bridge-Continuity Sign-off Replay](docs/architecture.html#map=component:bridge_phase_continuity_runtime) 2 declared links\n- [Cold Reader Route Map](docs/architecture.html#map=component:cold_reader_route_map) 8 declared links\n- [Concurrency Mission Control](docs/architecture.html#map=component:concurrency_mission_control) 2 declared links\n- [Corpus Readiness Mathlib Absence Gate](docs/architecture.html#map=component:corpus_readiness_mathlib_absence_gate) 4 declared links\n- [Derived Fact Provider Runtime](docs/architecture.html#map=component:derived_fact_provider_runtime) 3 declared links\n- [Doctrine Fact Claim Audit](docs/architecture.html#map=component:doctrine_fact_claim_audit) 2 declared links\n- [Durable Agent Work Landing Replay](docs/architecture.html#map=component:durable_agent_work_landing_replay) 2 declared links\n- [Egress Self-Compliance Audit](docs/architecture.html#map=component:egress_self_compliance_audit) 3 declared links\n- [Engine Room Demo](docs/architecture.html#map=component:engine_room_demo) 2 declared links\n- [Executable Doctrine Grammar](docs/architecture.html#map=component:executable_doctrine_grammar) 1 declared links\n- [Finance Forecast Evaluation Spine](docs/architecture.html#map=component:finance_forecast_evaluation_spine) 2 declared links\n- [Finite Erdos Denominator-Order Certificate Strike](docs/architecture.html#map=component:finite_erdos_denominator_certificate_strike) 3 declared links\n- [Formal Evidence Cell Anchor Resolver](docs/architecture.html#map=component:formal_evidence_cell_anchor_resolver) 3 declared links\n- [Formal Math Lean Proof Witness](docs/architecture.html#map=component:formal_math_lean_proof_witness) 5 declared links\n- [Formal Math Premise Retrieval](docs/architecture.html#map=component:formal_math_premise_retrieval) 6 declared links\n- [Formal Math Readiness Gate](docs/architecture.html#map=component:formal_math_readiness_gate) 6 declared links\n- [Formal Math Verifier Trace Repair Loop](docs/architecture.html#map=component:formal_math_verifier_trace_repair_loop) 5 declared links\n- [Generated Projection Drift Runtime](docs/architecture.html#map=component:generated_projection_drift_runtime) 3 declared links\n- [Lean Proof-Search Lab Runtime](docs/architecture.html#map=component:lean_proof_search_lab_runtime) 3 declared links\n- [Lean Std Premise Index](docs/architecture.html#map=component:lean_std_premise_index) 4 declared links\n- [Source Projection Import Protocol](docs/architecture.html#map=component:macro_projection_import_protocol) 23 declared links\n- [Materials Lab-Safety Refusal Replay](docs/architecture.html#map=component:materials_chemistry_closed_loop_lab_safety_replay) 1 declared links\n- [MCP Tool-Authority Policy Replay](docs/architecture.html#map=component:mcp_tool_authority_replay) 2 declared links\n- [Metabolism Queue Reconciliation](docs/architecture.html#map=component:metabolism_queue_reconciliation) 3 declared links\n- [Mission Transaction Work Spine](docs/architecture.html#map=component:mission_transaction_work_spine) 9 declared links\n- [Navigation Fitness Benchmark](docs/architecture.html#map=component:navigation_fitness_benchmark) 3 declared links\n- [Navigation Hologram Route Plane](docs/architecture.html#map=component:navigation_hologram_route_plane) 11 declared links\n- [Pattern Binding Contract](docs/architecture.html#map=component:pattern_binding_contract) 6 declared links\n- [Prediction Oracle Reconciliation](docs/architecture.html#map=component:prediction_oracle_reconciliation) 3 declared links\n- [Proof Derived Governed Mutation Authorization](docs/architecture.html#map=component:proof_derived_governed_mutation_authorization) 3 declared links\n- [Proof Diagnostic Evidence Spine](docs/architecture.html#map=component:proof_diagnostic_evidence_spine) 6 declared links\n- [Provider Context Recipe Budget Policy](docs/architecture.html#map=component:provider_context_recipe_budget_policy) 2 declared links\n- [Public Reveal Walkthrough](docs/architecture.html#map=component:public_reveal_walkthrough) 4 declared links\n- [Research Replication Rubric Artifact Replay](docs/architecture.html#map=component:research_replication_rubric_artifact_replay) 5 declared links\n- [Ring2 Premise Retrieval Precision Recall Harness](docs/architecture.html#map=component:ring2_premise_retrieval_precision_recall_harness) 2 declared links\n- [Routing Anti Patterns Registry](docs/architecture.html#map=component:routing_anti_patterns_registry) 3 declared links\n- [Semantic Singleflight Dedup Runtime](docs/architecture.html#map=component:semantic_singleflight_dedup_runtime) 4 declared links\n- [Tactic Portfolio Availability Probe](docs/architecture.html#map=component:tactic_portfolio_availability_probe) 5 declared links\n- [Target Shape Tactic Routing Gate](docs/architecture.html#map=component:target_shape_tactic_routing_gate) 5 declared links\n- [Undeclared Library Prior Symbol Classifier](docs/architecture.html#map=component:undeclared_library_prior_symbol_classifier) 3 declared links\n- [Verifier Lab Execution Spine](docs/architecture.html#map=component:verifier_lab_execution_spine) 2 declared links\n- [Verifier Lab Kernel](docs/architecture.html#map=component:verifier_lab_kernel) 10 declared links\n- [Voice To Doctrine Self Improvement Loop](docs/architecture.html#map=component:voice_to_doctrine_self_improvement_loop) 1 declared links\n- [Projection-Drift Contract Validator](docs/architecture.html#map=component:world_model_projection_drift_control_room) 7 declared links\n\nMachine-readable scene packet: [architecture-graph-scene.json](architecture-graph-scene.json).\n\n#### Standalone by default\n\nMost components don't call each other. They're independent specimens that share the path above. A few are wired to named neighbours, and each one says which.\n\n#### Local and self-contained\n\nRunning a component writes into a `.microcosm/` folder beside your project. It doesn't change your source files and doesn't make external calls. The output is something you can open and trace.\n\nThis is the public overview map of the areas and the path most components share.","byte_count":11613},{"id":"page:doctrine","path":"docs/doctrine.html","relative_href":"docs/doctrine.html","title":"Doctrine","nav_group":"The 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The commitments Plectis applies before a claim, route, count, import, or page is allowed to mean anything: its axioms, principles, and anti-principles. The full doctrine entity body records are carried once in visible_text_index.logical_records.doctrine_entities. Doctrine synopsis and relationship rows point to those records by body_pointer.","renders_note":"Atoms shown on this page are carried once in this file; resolve the content_refs ids against components[] / paper_modules[] / doctrine_entities[] / evidence_profiles[]. Glossary terms are the glossary page's atoms in public_glossary[]."},{"id":"page:area-entry","path":"docs/area-entry.html","relative_href":"docs/area-entry.html","title":"Entry & orientation","nav_group":"Areas","nav_label":"Entry & orientation","source_refs":["microcosm-substrate/core/organ_families.json","microcosm-substrate/core/organ_registry.json","microcosm-substrate/core/organ_atlas.json"],"content_refs":{"component_refs":["cold_reader_route_map","public_reveal_walkthrough"],"family_id":"entry_and_reveal"},"body_role":"index_page_atoms_carried_once","lede":"Area \u00b7 2 components\n\n### Entry & orientation\n\nHow a newcomer first meets Plectis and follows a short guided path through it.\n\n#### Components\n\nEach card below is one component. It states what the component does, the evidence behind that claim, and its scope limit: the line where the claim stops and nothing further is proven.\n\nCold Reader Route Map Verifies the first-run guided path so every step names a real command, doc, and evidence. 5/5\n\nJob Verifies the first-run guided path so every step names a real command, doc, and evidence.","renders_note":"Atoms shown on this page are carried once in this file; resolve the content_refs ids against components[] / paper_modules[] / doctrine_entities[] / evidence_profiles[]. Glossary terms are the glossary page's atoms in public_glossary[]."},{"id":"page:area-architecture","path":"docs/area-architecture.html","relative_href":"docs/area-architecture.html","title":"Architecture & navigation","nav_group":"Areas","nav_label":"Architecture & navigation","source_refs":["microcosm-substrate/core/organ_families.json","microcosm-substrate/core/organ_registry.json","microcosm-substrate/core/organ_atlas.json"],"content_refs":{"component_refs":["pattern_binding_contract","pattern_assimilation_step","executable_doctrine_grammar","navigation_hologram_route_plane","standards_meta_diagnostics","voice_to_doctrine_self_improvement_loop","cognitive_operator_registry","routing_anti_patterns_registry","doctrine_fact_claim_audit","self_ignorance_coverage_ledger","navigation_fitness_benchmark","annex_knowledge_routing"],"family_id":"architecture_and_navigation"},"body_role":"index_page_atoms_carried_once","lede":"Area \u00b7 12 components\n\n### Architecture & navigation\n\nThe core primitives, pattern rules, and routing that give the system its shape.\n\n#### Components\n\nEach card below is one component. It states what the component does, the evidence behind that claim, and its scope limit: the line where the claim stops and nothing further is proven.\n\nPattern Binding Contract Checks a real pattern catalog for digest, cross-reference, and dependency-cycle integrity. 5/5\n\nJob Checks a real pattern catalog for digest, cross-reference, and dependency -cycle integrity.","renders_note":"Atoms shown on this page are carried once in this file; resolve the content_refs ids against components[] / paper_modules[] / doctrine_entities[] / evidence_profiles[]. Glossary terms are the glossary page's atoms in public_glossary[]."},{"id":"page:area-formal-math","path":"docs/area-formal-math.html","relative_href":"docs/area-formal-math.html","title":"Formal math & proof","nav_group":"Areas","nav_label":"Formal math & proof","source_refs":["microcosm-substrate/core/organ_families.json","microcosm-substrate/core/organ_registry.json","microcosm-substrate/core/organ_atlas.json"],"content_refs":{"component_refs":["certificate_kernel_execution_lab","formal_math_lean_proof_witness","verifier_lab_execution_spine","corpus_readiness_mathlib_absence_gate","batch4_proof_authority_runtime","proof_diagnostic_evidence_spine","formal_math_readiness_gate","mathematical_strategy_atlas_hypothesis_scorer","tactic_portfolio_availability_probe","target_shape_tactic_routing_gate","lean_std_premise_index","formal_math_premise_retrieval","formal_math_verifier_trace_repair_loop","formal_evidence_cell_anchor_resolver","undeclared_library_prior_symbol_classifier","ring2_premise_retrieval_precision_recall_harness","verifier_lab_kernel","proof_derived_governed_mutation_authorization","finite_erdos_denominator_certificate_strike","lean_proof_search_lab_runtime"],"family_id":"formal_math_and_proof"},"body_role":"index_page_atoms_carried_once","lede":"Area \u00b7 20 components\n\n### Formal math & proof\n\nInspectable pieces of a proof pipeline: premise retrieval over a copied Lean Std index, tactic routing, verifier-trace repair, and claim -separation result records. Three components run the real Lean/Lake prover locally on bounded examples; the rest publish the pipeline's checking layers as contracts you can open.\n\nLooking for the larger Lean formalisation? The cards below are Plectis proof-adjacent components (bounded witnesses, certificates, and pipeline checks). The separate Erd\u0151s #249/#257 Lean 4/Mathlib repository is [github.com/wcook04/plectis-lean-erdos249-257](https://github.com/wcook04/plectis-lean-erdos249-257). That repository carries the mathematical proof authority; this area does not.\n\n#### Components","renders_note":"Atoms shown on this page are carried once in this file; resolve the content_refs ids against components[] / paper_modules[] / doctrine_entities[] / evidence_profiles[]. Glossary terms are the glossary page's atoms in public_glossary[]."},{"id":"page:area-agent-reliability","path":"docs/area-agent-reliability.html","relative_href":"docs/area-agent-reliability.html","title":"Agent reliability & safety","nav_group":"Areas","nav_label":"Agent reliability & safety","source_refs":["microcosm-substrate/core/organ_families.json","microcosm-substrate/core/organ_registry.json","microcosm-substrate/core/organ_atlas.json"],"content_refs":{"component_refs":["agent_closeout_faithfulness_audit","bounded_autonomy_campaign_packet","provider_context_recipe_budget_policy","batch10_cold_eval_honesty_capsule","batch8_validator_checker_capsule","batch7_secondary_runtime_capsule","agent_benchmark_integrity_anti_gaming_replay","agent_monitor_redteam_falsification_replay","agent_sabotage_scheming_monitor_replay","agent_memory_temporal_conflict_replay","sleeper_memory_poisoning_quarantine_replay","mcp_tool_authority_replay","belief_state_process_reward_replay","agent_sandbox_policy_escape_replay","indirect_prompt_injection_information_flow_policy_replay","agentic_vulnerability_discovery_patch_proof_replay","agent_route_observability_runtime","bridge_campaign_dag_validation","metabolism_queue_reconciliation","egress_self_compliance_audit"],"family_id":"agent_reliability_and_safety"},"body_role":"index_page_atoms_carried_once","lede":"Area \u00b7 20 components\n\n### Agent reliability & safety\n\nSource -open replays of agent failure modes as inspectable specimens.\n\n#### Components\n\nEach card below is one component. 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The public cognitive-operator registry fixture validates operator-shape rows, active-operator dogfood result records, anti-sprawl decisions, copied source registry/standard/tool bodies, and scope limits without becoming operator source authority. The full paper-module reader text is carried once in visible_text_index.logical_records.paper_modules; diagrams are carried as compressed diagram assets.","renders_note":"Atoms shown on this page are carried once in this file; resolve the content_refs ids against components[] / paper_modules[] / doctrine_entities[] / evidence_profiles[]. 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modules","source_refs":["microcosm-substrate/core/paper_module_capsules.json"],"content_refs":{"paper_module_refs":["paper_module.verifier_lab_kernel","paper_module.navigation_hologram_route_plane","paper_module.agent_route_observability_runtime","paper_module.agent_benchmark_integrity_anti_gaming_replay","paper_module.research_replication_rubric_artifact_replay","paper_module.agentic_vulnerability_discovery_patch_proof_replay","paper_module.materials_chemistry_closed_loop_lab_safety_replay","paper_module.certificate_kernel_execution_lab","paper_module.corpus_readiness_mathlib_absence_gate","paper_module.pattern_binding_contract","paper_module.bridge_phase_continuity_runtime","paper_module.cognitive_operator_registry","paper_module.agent_closeout_faithfulness_audit","paper_module.cold_reader_route_map","paper_module.proof_diagnostic_evidence_spine","paper_module.proof_derived_governed_mutation_authorization","paper_module.durable_agent_work_landing_replay","paper_module.work_landing_control_spine","paper_module.executable_doctrine_grammar","paper_module.macro_projection_import_protocol","paper_module.mission_transaction_work_spine","paper_module.formal_math_readiness_gate","paper_module.formal_math_lean_proof_witness","paper_module.formal_math_verifier_trace_repair_loop","paper_module.formal_evidence_cell_anchor_resolver","paper_module.formal_math_premise_retrieval","paper_module.lean_std_premise_index","paper_module.world_model_projection_drift_control_room","paper_module.public_reveal_walkthrough","paper_module.standards_meta_diagnostics","paper_module.finance_forecast_evaluation_spine","paper_module.engine_room_demo","paper_module.agent_memory_temporal_conflict_replay","paper_module.agent_monitor_redteam_falsification_replay","paper_module.agent_sabotage_scheming_monitor_replay","paper_module.agent_sandbox_policy_escape_replay","paper_module.belief_state_process_reward_replay","paper_module.sleeper_memory_poisoning_quarantine_replay","paper_module.indirect_prompt_injection_information_flow_policy_replay","paper_module.mcp_tool_authority_replay","paper_module.tactic_portfolio_availability","paper_module.target_shape_tactic_routing","paper_module.ring2_premise_precision_recall","paper_module.mathematical_strategy_atlas","paper_module.verifier_lab_execution_spine","paper_module.bounded_autonomy_campaign_packet","paper_module.computer_use_action_trace_replay","paper_module.concurrency_mission_control","paper_module.doctrine_fact_claim_audit","paper_module.self_ignorance_coverage_ledger","paper_module.tool_server_pressure_inventory","paper_module.mechanistic_interpretability_circuit_attribution_replay","paper_module.spatial_world_model_counterfactual_simulation_replay","paper_module.prediction_oracle_reconciliation","paper_module.provider_context_recipe_budget","paper_module.undeclared_library_prior_classifier","paper_module.voice_to_doctrine_self_improvement_loop","paper_module.routing_anti_patterns_registry","paper_module.batch8_audio_level_rms_port","paper_module.batch8_compliance_pipeline_capsule","paper_module.batch8_policy_engines_capsule","paper_module.batch8_structural_theses_capsule","paper_module.batch8_tools_tail_primitives_capsule","paper_module.batch8_validator_checker_capsule","paper_module.batch12_market_dashboard_read_model_capsule","paper_module.batch12_prediction_market_board_capsule","paper_module.batch12_release_claim_language_gate","paper_module.batch10_cold_eval_honesty_capsule","paper_module.batch10_live_source_drift_capsule","paper_module.batch7_macro_engines_capsule","paper_module.batch9_macro_engines_capsule","paper_module.pattern_assimilation","paper_module.batch10_governance_compilers_capsule","paper_module.batch11_saturation_engines_capsule","paper_module.batch4_proof_authority_runtime","paper_module.batch6_unsurfaced_primitives_capsule","paper_module.engine_room_public_projection_leak_gate","paper_module.batch5_authority_systems_capsule","paper_module.batch7_oracle_sibling_capsule","paper_module.batch7_demo_take_console_capsule","paper_module.engine_room_generated_projection_drift_gate","paper_module.engine_room_command_run_singleflight","paper_module.engine_room_metabolism_runtime","paper_module.engine_room_bridge_campaign_dag","paper_module.engine_room_annex_knowledge_router","paper_module.engine_room_derived_fact_provider_engine","paper_module.engine_room_egress_self_compliance_gate","paper_module.engine_room_lean_proof_search_lab","paper_module.engine_room_navigation_fitness_benchmark","paper_module.cold_clone_probe","paper_module.first_screen_composition_root","paper_module.batch7_secondary_runtime_capsule","paper_module.microcosm_axiom_substrate","paper_module.semantic_singleflight_dedup_runtime","paper_module.finite_erdos_denominator_certificate_strike","paper_module.bridge_campaign_dag_validation","paper_module.metabolism_queue_reconciliation","paper_module.derived_fact_provider_runtime","paper_module.navigation_fitness_benchmark","paper_module.annex_knowledge_routing","paper_module.egress_self_compliance_audit","paper_module.generated_projection_drift_runtime","paper_module.lean_proof_search_lab_runtime"],"family_id":null},"body_role":"index_page_atoms_carried_once","lede":"Agent Completion Faithfulness Audit. The public completion-faithfulness fixture checks commit, ledger-cap, and pytest-span claims with real git/pytest subprocess witnesses while refusing unchecked pass overclaims. The full paper-module reader text is carried once in visible_text_index.logical_records.paper_modules; diagrams are carried as compressed diagram assets.","renders_note":"Atoms shown on this page are carried once in this file; resolve the content_refs ids against components[] / paper_modules[] / doctrine_entities[] / evidence_profiles[]. Glossary terms are the glossary page's atoms in public_glossary[]."},{"id":"page:paper-module-cold-reader-route-map","path":"docs/paper-module-cold-reader-route-map.html","relative_href":"docs/paper-module-cold-reader-route-map.html","title":"Cold-Reader Route Map","nav_group":"Reference","nav_label":"Paper modules","source_refs":["microcosm-substrate/core/paper_module_capsules.json"],"content_refs":{"paper_module_refs":["paper_module.verifier_lab_kernel","paper_module.navigation_hologram_route_plane","paper_module.agent_route_observability_runtime","paper_module.agent_benchmark_integrity_anti_gaming_replay","paper_module.research_replication_rubric_artifact_replay","paper_module.agentic_vulnerability_discovery_patch_proof_replay","paper_module.materials_chemistry_closed_loop_lab_safety_replay","paper_module.certificate_kernel_execution_lab","paper_module.corpus_readiness_mathlib_absence_gate","paper_module.pattern_binding_contract","paper_module.bridge_phase_continuity_runtime","paper_module.cognitive_operator_registry","paper_module.agent_closeout_faithfulness_audit","paper_module.cold_reader_route_map","paper_module.proof_diagnostic_evidence_spine","paper_module.proof_derived_governed_mutation_authorization","paper_module.durable_agent_work_landing_replay","paper_module.work_landing_control_spine","paper_module.executable_doctrine_grammar","paper_module.macro_projection_import_protocol","paper_module.mission_transaction_work_spine","paper_module.formal_math_readiness_gate","paper_module.formal_math_lean_proof_witness","paper_module.formal_math_verifier_trace_repair_loop","paper_module.formal_evidence_cell_anchor_resolver","paper_module.formal_math_premise_retrieval","paper_module.lean_std_premise_index","paper_module.world_model_projection_drift_control_room","paper_module.public_reveal_walkthrough","paper_module.standards_meta_diagnostics","paper_module.finance_forecast_evaluation_spine","paper_module.engine_room_demo","paper_module.agent_memory_temporal_conflict_replay","paper_module.agent_monitor_redteam_falsification_replay","paper_module.agent_sabotage_scheming_monitor_replay","paper_module.agent_sandbox_policy_escape_replay","paper_module.belief_state_process_reward_replay","paper_module.sleeper_memory_poisoning_quarantine_replay","paper_module.indirect_prompt_injection_information_flow_policy_replay","paper_module.mcp_tool_authority_replay","paper_module.tactic_portfolio_availability","paper_module.target_shape_tactic_routing","paper_module.ring2_premise_precision_recall","paper_module.mathematical_strategy_atlas","paper_module.verifier_lab_execution_spine","paper_module.bounded_autonomy_campaign_packet","paper_module.computer_use_action_trace_replay","paper_module.concurrency_mission_control","paper_module.doctrine_fact_claim_audit","paper_module.self_ignorance_coverage_ledger","paper_module.tool_server_pressure_inventory","paper_module.mechanistic_interpretability_circuit_attribution_replay","paper_module.spatial_world_model_counterfactual_simulation_replay","paper_module.prediction_oracle_reconciliation","paper_module.provider_context_recipe_budget","paper_module.undeclared_library_prior_classifier","paper_module.voice_to_doctrine_self_improvement_loop","paper_module.routing_anti_patterns_registry","paper_module.batch8_audio_level_rms_port","paper_module.batch8_compliance_pipeline_capsule","paper_module.batch8_policy_engines_capsule","paper_module.batch8_structural_theses_capsule","paper_module.batch8_tools_tail_primitives_capsule","paper_module.batch8_validator_checker_capsule","paper_module.batch12_market_dashboard_read_model_capsule","paper_module.batch12_prediction_market_board_capsule","paper_module.batch12_release_claim_language_gate","paper_module.batch10_cold_eval_honesty_capsule","paper_module.batch10_live_source_drift_capsule","paper_module.batch7_macro_engines_capsule","paper_module.batch9_macro_engines_capsule","paper_module.pattern_assimilation","paper_module.batch10_governance_compilers_capsule","paper_module.batch11_saturation_engines_capsule","paper_module.batch4_proof_authority_runtime","paper_module.batch6_unsurfaced_primitives_capsule","paper_module.engine_room_public_projection_leak_gate","paper_module.batch5_authority_systems_capsule","paper_module.batch7_oracle_sibling_capsule","paper_module.batch7_demo_take_console_capsule","paper_module.engine_room_generated_projection_drift_gate","paper_module.engine_room_command_run_singleflight","paper_module.engine_room_metabolism_runtime","paper_module.engine_room_bridge_campaign_dag","paper_module.engine_room_annex_knowledge_router","paper_module.engine_room_derived_fact_provider_engine","paper_module.engine_room_egress_self_compliance_gate","paper_module.engine_room_lean_proof_search_lab","paper_module.engine_room_navigation_fitness_benchmark","paper_module.cold_clone_probe","paper_module.first_screen_composition_root","paper_module.batch7_secondary_runtime_capsule","paper_module.microcosm_axiom_substrate","paper_module.semantic_singleflight_dedup_runtime","paper_module.finite_erdos_denominator_certificate_strike","paper_module.bridge_campaign_dag_validation","paper_module.metabolism_queue_reconciliation","paper_module.derived_fact_provider_runtime","paper_module.navigation_fitness_benchmark","paper_module.annex_knowledge_routing","paper_module.egress_self_compliance_audit","paper_module.generated_projection_drift_runtime","paper_module.lean_proof_search_lab_runtime"],"family_id":null},"body_role":"index_page_atoms_carried_once","lede":"Cold-Reader Route Map. The public cold-reader route-map fixture validates first-run command order, docs refs, result record refs, scope limits, copied cold-entry source-module digests, and non-public-state exclusion without becoming route registry control. The full paper-module reader text is carried once in visible_text_index.logical_records.paper_modules; diagrams are carried as compressed diagram assets.","renders_note":"Atoms shown on this page are carried once in this file; resolve the content_refs ids against components[] / paper_modules[] / doctrine_entities[] / evidence_profiles[]. Glossary terms are the glossary page's atoms in public_glossary[]."},{"id":"page:paper-module-proof-diagnostic-evidence-spine","path":"docs/paper-module-proof-diagnostic-evidence-spine.html","relative_href":"docs/paper-module-proof-diagnostic-evidence-spine.html","title":"Proof Diagnostic Evidence Spine","nav_group":"Reference","nav_label":"Paper modules","source_refs":["microcosm-substrate/core/paper_module_capsules.json"],"content_refs":{"paper_module_refs":["paper_module.verifier_lab_kernel","paper_module.navigation_hologram_route_plane","paper_module.agent_route_observability_runtime","paper_module.agent_benchmark_integrity_anti_gaming_replay","paper_module.research_replication_rubric_artifact_replay","paper_module.agentic_vulnerability_discovery_patch_proof_replay","paper_module.materials_chemistry_closed_loop_lab_safety_replay","paper_module.certificate_kernel_execution_lab","paper_module.corpus_readiness_mathlib_absence_gate","paper_module.pattern_binding_contract","paper_module.bridge_phase_continuity_runtime","paper_module.cognitive_operator_registry","paper_module.agent_closeout_faithfulness_audit","paper_module.cold_reader_route_map","paper_module.proof_diagnostic_evidence_spine","paper_module.proof_derived_governed_mutation_authorization","paper_module.durable_agent_work_landing_replay","paper_module.work_landing_control_spine","paper_module.executable_doctrine_grammar","paper_module.macro_projection_import_protocol","paper_module.mission_transaction_work_spine","paper_module.formal_math_readiness_gate","paper_module.formal_math_lean_proof_witness","paper_module.formal_math_verifier_trace_repair_loop","paper_module.formal_evidence_cell_anchor_resolver","paper_module.formal_math_premise_retrieval","paper_module.lean_std_premise_index","paper_module.world_model_projection_drift_control_room","paper_module.public_reveal_walkthrough","paper_module.standards_meta_diagnostics","paper_module.finance_forecast_evaluation_spine","paper_module.engine_room_demo","paper_module.agent_memory_temporal_conflict_replay","paper_module.agent_monitor_redteam_falsification_replay","paper_module.agent_sabotage_scheming_monitor_replay","paper_module.agent_sandbox_policy_escape_replay","paper_module.belief_state_process_reward_replay","paper_module.sleeper_memory_poisoning_quarantine_replay","paper_module.indirect_prompt_injection_information_flow_policy_replay","paper_module.mcp_tool_authority_replay","paper_module.tactic_portfolio_availability","paper_module.target_shape_tactic_routing","paper_module.ring2_premise_precision_recall","paper_module.mathematical_strategy_atlas","paper_module.verifier_lab_execution_spine","paper_module.bounded_autonomy_campaign_packet","paper_module.computer_use_action_trace_replay","paper_module.concurrency_mission_control","paper_module.doctrine_fact_claim_audit","paper_module.self_ignorance_coverage_ledger","paper_module.tool_server_pressure_inventory","paper_module.mechanistic_interpretability_circuit_attribution_replay","paper_module.spatial_world_model_counterfactual_simulation_replay","paper_module.prediction_oracle_reconciliation","paper_module.provider_context_recipe_budget","paper_module.undeclared_library_prior_classifier","paper_module.voice_to_doctrine_self_improvement_loop","paper_module.routing_anti_patterns_registry","paper_module.batch8_audio_level_rms_port","paper_module.batch8_compliance_pipeline_capsule","paper_module.batch8_policy_engines_capsule","paper_module.batch8_structural_theses_capsule","paper_module.batch8_tools_tail_primitives_capsule","paper_module.batch8_validator_checker_capsule","paper_module.batch12_market_dashboard_read_model_capsule","paper_module.batch12_prediction_market_board_capsule","paper_module.batch12_release_claim_language_gate","paper_module.batch10_cold_eval_honesty_capsule","paper_module.batch10_live_source_drift_capsule","paper_module.batch7_macro_engines_capsule","paper_module.batch9_macro_engines_capsule","paper_module.pattern_assimilation","paper_module.batch10_governance_compilers_capsule","paper_module.batch11_saturation_engines_capsule","paper_module.batch4_proof_authority_runtime","paper_module.batch6_unsurfaced_primitives_capsule","paper_module.engine_room_public_projection_leak_gate","paper_module.batch5_authority_systems_capsule","paper_module.batch7_oracle_sibling_capsule","paper_module.batch7_demo_take_console_capsule","paper_module.engine_room_generated_projection_drift_gate","paper_module.engine_room_command_run_singleflight","paper_module.engine_room_metabolism_runtime","paper_module.engine_room_bridge_campaign_dag","paper_module.engine_room_annex_knowledge_router","paper_module.engine_room_derived_fact_provider_engine","paper_module.engine_room_egress_self_compliance_gate","paper_module.engine_room_lean_proof_search_lab","paper_module.engine_room_navigation_fitness_benchmark","paper_module.cold_clone_probe","paper_module.first_screen_composition_root","paper_module.batch7_secondary_runtime_capsule","paper_module.microcosm_axiom_substrate","paper_module.semantic_singleflight_dedup_runtime","paper_module.finite_erdos_denominator_certificate_strike","paper_module.bridge_campaign_dag_validation","paper_module.metabolism_queue_reconciliation","paper_module.derived_fact_provider_runtime","paper_module.navigation_fitness_benchmark","paper_module.annex_knowledge_routing","paper_module.egress_self_compliance_audit","paper_module.generated_projection_drift_runtime","paper_module.lean_proof_search_lab_runtime"],"family_id":null},"body_role":"index_page_atoms_carried_once","lede":"Proof Diagnostic Evidence Spine. The public proof-diagnostic evidence spine fixture validates Ring2 diagnostic result record refs, copied runtime artifact digests, provider/proof-body exclusions, stale-coupling visibility, and scope limits without becoming formal proof authority. The full paper-module reader text is carried once in visible_text_index.logical_records.paper_modules; diagrams are carried as compressed diagram assets.","renders_note":"Atoms shown on this page are carried once in this file; resolve the content_refs ids against components[] / paper_modules[] / doctrine_entities[] / evidence_profiles[]. Glossary terms are the glossary page's atoms in public_glossary[]."},{"id":"page:paper-module-proof-derived-governed-mutation-authorization","path":"docs/paper-module-proof-derived-governed-mutation-authorization.html","relative_href":"docs/paper-module-proof-derived-governed-mutation-authorization.html","title":"Proof-Derived Governed Mutation Authorization","nav_group":"Reference","nav_label":"Paper modules","source_refs":["microcosm-substrate/core/paper_module_capsules.json"],"content_refs":{"paper_module_refs":["paper_module.verifier_lab_kernel","paper_module.navigation_hologram_route_plane","paper_module.agent_route_observability_runtime","paper_module.agent_benchmark_integrity_anti_gaming_replay","paper_module.research_replication_rubric_artifact_replay","paper_module.agentic_vulnerability_discovery_patch_proof_replay","paper_module.materials_chemistry_closed_loop_lab_safety_replay","paper_module.certificate_kernel_execution_lab","paper_module.corpus_readiness_mathlib_absence_gate","paper_module.pattern_binding_contract","paper_module.bridge_phase_continuity_runtime","paper_module.cognitive_operator_registry","paper_module.agent_closeout_faithfulness_audit","paper_module.cold_reader_route_map","paper_module.proof_diagnostic_evidence_spine","paper_module.proof_derived_governed_mutation_authorization","paper_module.durable_agent_work_landing_replay","paper_module.work_landing_control_spine","paper_module.executable_doctrine_grammar","paper_module.macro_projection_import_protocol","paper_module.mission_transaction_work_spine","paper_module.formal_math_readiness_gate","paper_module.formal_math_lean_proof_witness","paper_module.formal_math_verifier_trace_repair_loop","paper_module.formal_evidence_cell_anchor_resolver","paper_module.formal_math_premise_retrieval","paper_module.lean_std_premise_index","paper_module.world_model_projection_drift_control_room","paper_module.public_reveal_walkthrough","paper_module.standards_meta_diagnostics","paper_module.finance_forecast_evaluation_spine","paper_module.engine_room_demo","paper_module.agent_memory_temporal_conflict_replay","paper_module.agent_monitor_redteam_falsification_replay","paper_module.agent_sabotage_scheming_monitor_replay","paper_module.agent_sandbox_policy_escape_replay","paper_module.belief_state_process_reward_replay","paper_module.sleeper_memory_poisoning_quarantine_replay","paper_module.indirect_prompt_injection_information_flow_policy_replay","paper_module.mcp_tool_authority_replay","paper_module.tactic_portfolio_availability","paper_module.target_shape_tactic_routing","paper_module.ring2_premise_precision_recall","paper_module.mathematical_strategy_atlas","paper_module.verifier_lab_execution_spine","paper_module.bounded_autonomy_campaign_packet","paper_module.computer_use_action_trace_replay","paper_module.concurrency_mission_control","paper_module.doctrine_fact_claim_audit","paper_module.self_ignorance_coverage_ledger","paper_module.tool_server_pressure_inventory","paper_module.mechanistic_interpretability_circuit_attribution_replay","paper_module.spatial_world_model_counterfactual_simulation_replay","paper_module.prediction_oracle_reconciliation","paper_module.provider_context_recipe_budget","paper_module.undeclared_library_prior_classifier","paper_module.voice_to_doctrine_self_improvement_loop","paper_module.routing_anti_patterns_registry","paper_module.batch8_audio_level_rms_port","paper_module.batch8_compliance_pipeline_capsule","paper_module.batch8_policy_engines_capsule","paper_module.batch8_structural_theses_capsule","paper_module.batch8_tools_tail_primitives_capsule","paper_module.batch8_validator_checker_capsule","paper_module.batch12_market_dashboard_read_model_capsule","paper_module.batch12_prediction_market_board_capsule","paper_module.batch12_release_claim_language_gate","paper_module.batch10_cold_eval_honesty_capsule","paper_module.batch10_live_source_drift_capsule","paper_module.batch7_macro_engines_capsule","paper_module.batch9_macro_engines_capsule","paper_module.pattern_assimilation","paper_module.batch10_governance_compilers_capsule","paper_module.batch11_saturation_engines_capsule","paper_module.batch4_proof_authority_runtime","paper_module.batch6_unsurfaced_primitives_capsule","paper_module.engine_room_public_projection_leak_gate","paper_module.batch5_authority_systems_capsule","paper_module.batch7_oracle_sibling_capsule","paper_module.batch7_demo_take_console_capsule","paper_module.engine_room_generated_projection_drift_gate","paper_module.engine_room_command_run_singleflight","paper_module.engine_room_metabolism_runtime","paper_module.engine_room_bridge_campaign_dag","paper_module.engine_room_annex_knowledge_router","paper_module.engine_room_derived_fact_provider_engine","paper_module.engine_room_egress_self_compliance_gate","paper_module.engine_room_lean_proof_search_lab","paper_module.engine_room_navigation_fitness_benchmark","paper_module.cold_clone_probe","paper_module.first_screen_composition_root","paper_module.batch7_secondary_runtime_capsule","paper_module.microcosm_axiom_substrate","paper_module.semantic_singleflight_dedup_runtime","paper_module.finite_erdos_denominator_certificate_strike","paper_module.bridge_campaign_dag_validation","paper_module.metabolism_queue_reconciliation","paper_module.derived_fact_provider_runtime","paper_module.navigation_fitness_benchmark","paper_module.annex_knowledge_routing","paper_module.egress_self_compliance_audit","paper_module.generated_projection_drift_runtime","paper_module.lean_proof_search_lab_runtime"],"family_id":null},"body_role":"index_page_atoms_carried_once","lede":"Proof-Derived Governed Mutation Authorization. The public proof-derived governed-mutation fixture validates synthetic mutation proposals through proof cells, visible pre-execution policy verdicts, logged side effects, rollback result records, cold replay, negative cases, and copied source internal control bodies without granting live mutation authority. The full paper-module reader text is carried once in visible_text_index.logical_records.paper_modules; diagrams are carried as compressed diagram assets.","renders_note":"Atoms shown on this page are carried once in this file; resolve the content_refs ids against components[] / paper_modules[] / doctrine_entities[] / evidence_profiles[]. Glossary terms are the glossary page's atoms in public_glossary[]."},{"id":"page:paper-module-durable-agent-work-landing-replay","path":"docs/paper-module-durable-agent-work-landing-replay.html","relative_href":"docs/paper-module-durable-agent-work-landing-replay.html","title":"Durable Agent Work-Landing Replay","nav_group":"Reference","nav_label":"Paper modules","source_refs":["microcosm-substrate/core/paper_module_capsules.json"],"content_refs":{"paper_module_refs":["paper_module.verifier_lab_kernel","paper_module.navigation_hologram_route_plane","paper_module.agent_route_observability_runtime","paper_module.agent_benchmark_integrity_anti_gaming_replay","paper_module.research_replication_rubric_artifact_replay","paper_module.agentic_vulnerability_discovery_patch_proof_replay","paper_module.materials_chemistry_closed_loop_lab_safety_replay","paper_module.certificate_kernel_execution_lab","paper_module.corpus_readiness_mathlib_absence_gate","paper_module.pattern_binding_contract","paper_module.bridge_phase_continuity_runtime","paper_module.cognitive_operator_registry","paper_module.agent_closeout_faithfulness_audit","paper_module.cold_reader_route_map","paper_module.proof_diagnostic_evidence_spine","paper_module.proof_derived_governed_mutation_authorization","paper_module.durable_agent_work_landing_replay","paper_module.work_landing_control_spine","paper_module.executable_doctrine_grammar","paper_module.macro_projection_import_protocol","paper_module.mission_transaction_work_spine","paper_module.formal_math_readiness_gate","paper_module.formal_math_lean_proof_witness","paper_module.formal_math_verifier_trace_repair_loop","paper_module.formal_evidence_cell_anchor_resolver","paper_module.formal_math_premise_retrieval","paper_module.lean_std_premise_index","paper_module.world_model_projection_drift_control_room","paper_module.public_reveal_walkthrough","paper_module.standards_meta_diagnostics","paper_module.finance_forecast_evaluation_spine","paper_module.engine_room_demo","paper_module.agent_memory_temporal_conflict_replay","paper_module.agent_monitor_redteam_falsification_replay","paper_module.agent_sabotage_scheming_monitor_replay","paper_module.agent_sandbox_policy_escape_replay","paper_module.belief_state_process_reward_replay","paper_module.sleeper_memory_poisoning_quarantine_replay","paper_module.indirect_prompt_injection_information_flow_policy_replay","paper_module.mcp_tool_authority_replay","paper_module.tactic_portfolio_availability","paper_module.target_shape_tactic_routing","paper_module.ring2_premise_precision_recall","paper_module.mathematical_strategy_atlas","paper_module.verifier_lab_execution_spine","paper_module.bounded_autonomy_campaign_packet","paper_module.computer_use_action_trace_replay","paper_module.concurrency_mission_control","paper_module.doctrine_fact_claim_audit","paper_module.self_ignorance_coverage_ledger","paper_module.tool_server_pressure_inventory","paper_module.mechanistic_interpretability_circuit_attribution_replay","paper_module.spatial_world_model_counterfactual_simulation_replay","paper_module.prediction_oracle_reconciliation","paper_module.provider_context_recipe_budget","paper_module.undeclared_library_prior_classifier","paper_module.voice_to_doctrine_self_improvement_loop","paper_module.routing_anti_patterns_registry","paper_module.batch8_audio_level_rms_port","paper_module.batch8_compliance_pipeline_capsule","paper_module.batch8_policy_engines_capsule","paper_module.batch8_structural_theses_capsule","paper_module.batch8_tools_tail_primitives_capsule","paper_module.batch8_validator_checker_capsule","paper_module.batch12_market_dashboard_read_model_capsule","paper_module.batch12_prediction_market_board_capsule","paper_module.batch12_release_claim_language_gate","paper_module.batch10_cold_eval_honesty_capsule","paper_module.batch10_live_source_drift_capsule","paper_module.batch7_macro_engines_capsule","paper_module.batch9_macro_engines_capsule","paper_module.pattern_assimilation","paper_module.batch10_governance_compilers_capsule","paper_module.batch11_saturation_engines_capsule","paper_module.batch4_proof_authority_runtime","paper_module.batch6_unsurfaced_primitives_capsule","paper_module.engine_room_public_projection_leak_gate","paper_module.batch5_authority_systems_capsule","paper_module.batch7_oracle_sibling_capsule","paper_module.batch7_demo_take_console_capsule","paper_module.engine_room_generated_projection_drift_gate","paper_module.engine_room_command_run_singleflight","paper_module.engine_room_metabolism_runtime","paper_module.engine_room_bridge_campaign_dag","paper_module.engine_room_annex_knowledge_router","paper_module.engine_room_derived_fact_provider_engine","paper_module.engine_room_egress_self_compliance_gate","paper_module.engine_room_lean_proof_search_lab","paper_module.engine_room_navigation_fitness_benchmark","paper_module.cold_clone_probe","paper_module.first_screen_composition_root","paper_module.batch7_secondary_runtime_capsule","paper_module.microcosm_axiom_substrate","paper_module.semantic_singleflight_dedup_runtime","paper_module.finite_erdos_denominator_certificate_strike","paper_module.bridge_campaign_dag_validation","paper_module.metabolism_queue_reconciliation","paper_module.derived_fact_provider_runtime","paper_module.navigation_fitness_benchmark","paper_module.annex_knowledge_routing","paper_module.egress_self_compliance_audit","paper_module.generated_projection_drift_runtime","paper_module.lean_proof_search_lab_runtime"],"family_id":null},"body_role":"index_page_atoms_carried_once","lede":"Durable Agent Work-Landing Replay. The public durable work-landing replay fixture validates recorded agent landing rows, copied source internal control source bodies, validation-before-commit ordering, HEAD-advance evidence, blocker capture, and work log completion without performing live Git work. The full paper-module reader text is carried once in visible_text_index.logical_records.paper_modules; diagrams are carried as compressed diagram assets.","renders_note":"Atoms shown on this page are carried once in this file; resolve the content_refs ids against components[] / paper_modules[] / doctrine_entities[] / evidence_profiles[]. Glossary terms are the glossary page's atoms in public_glossary[]."},{"id":"page:paper-module-work-landing-control-spine","path":"docs/paper-module-work-landing-control-spine.html","relative_href":"docs/paper-module-work-landing-control-spine.html","title":"Work Landing Control Spine","nav_group":"Reference","nav_label":"Paper 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aper_module.egress_self_compliance_audit","paper_module.generated_projection_drift_runtime","paper_module.lean_proof_search_lab_runtime"],"family_id":null},"body_role":"index_page_atoms_carried_once","lede":"Work Landing Control Spine. The public work-landing control spine validates copied work-landing internal control source bodies without authorizing live Git, ledger, claim-launch, private-index, public sharing, or launch operations. The full paper-module reader text is carried once in visible_text_index.logical_records.paper_modules; diagrams are carried as compressed diagram assets.","renders_note":"Atoms shown on this page are carried once in this file; resolve the content_refs ids against components[] / paper_modules[] / doctrine_entities[] / evidence_profiles[]. Glossary terms are the glossary page's atoms in public_glossary[]."},{"id":"page:paper-module-executable-doctrine-grammar","path":"docs/paper-module-executable-doctrine-grammar.html","relative_href":"docs/paper-module-executable-doctrine-grammar.html","title":"Executable Doctrine Grammar","nav_group":"Reference","nav_label":"Paper modules","source_refs":["microcosm-substrate/core/paper_module_capsules.json"],"content_refs":{"paper_module_refs":["paper_module.verifier_lab_kernel","paper_module.navigation_hologram_route_plane","paper_module.agent_route_observability_runtime","paper_module.agent_benchmark_integrity_anti_gaming_replay","paper_module.research_replication_rubric_artifact_replay","paper_module.agentic_vulnerability_discovery_patch_proof_replay","paper_module.materials_chemistry_closed_loop_lab_safety_replay","paper_module.certificate_kernel_execution_lab","paper_module.corpus_readiness_mathlib_absence_gate","paper_module.pattern_binding_contract","paper_module.bridge_phase_continuity_runtime","paper_module.cognitive_operator_registry","paper_module.agent_closeout_faithfulness_audit","paper_module.cold_reader_route_map","paper_module.proof_diagnostic_evidence_spine","paper_module.proof_derived_governed_mutation_authorization","paper_module.durable_agent_work_landing_replay","paper_module.work_landing_control_spine","paper_module.executable_doctrine_grammar","paper_module.macro_projection_import_protocol","paper_module.mission_transaction_work_spine","paper_module.formal_math_readiness_gate","paper_module.formal_math_lean_proof_witness","paper_module.formal_math_verifier_trace_repair_loop","paper_module.formal_evidence_cell_anchor_resolver","paper_module.formal_math_premise_retrieval","paper_module.lean_std_premise_index","paper_module.world_model_projection_drift_control_room","paper_module.public_reveal_walkthrough","paper_module.standards_meta_diagnostics","paper_module.finance_forecast_evaluation_spine","paper_module.engine_room_demo","paper_module.agent_memory_temporal_conflict_replay","paper_module.agent_monitor_redteam_falsification_replay","paper_module.agent_sabotage_scheming_monitor_replay","paper_module.agent_sandbox_policy_escape_replay","paper_module.belief_state_process_reward_replay","paper_module.sleeper_memory_poisoning_quarantine_replay","paper_module.indirect_prompt_injection_information_flow_policy_replay","paper_module.mcp_tool_authority_replay","paper_module.tactic_portfolio_availability","paper_module.target_shape_tactic_routing","paper_module.ring2_premise_precision_recall","paper_module.mathematical_strategy_atlas","paper_module.verifier_lab_execution_spine","paper_module.bounded_autonomy_campaign_packet","paper_module.computer_use_action_trace_replay","paper_module.concurrency_mission_control","paper_module.doctrine_fact_claim_audit","paper_module.self_ignorance_coverage_ledger","paper_module.tool_server_pressure_inventory","paper_module.mechanistic_interpretability_circuit_attribution_replay","paper_module.spatial_world_model_counterfactual_simulation_replay","paper_module.prediction_oracle_reconciliation","paper_module.provider_context_recipe_budget","paper_module.undeclared_library_prior_classifier","paper_module.voice_to_doctrine_self_improvement_loop","paper_module.routing_anti_patterns_registry","paper_module.batch8_audio_level_rms_port","paper_module.batch8_compliance_pipeline_capsule","paper_module.batch8_policy_engines_capsule","paper_module.batch8_structural_theses_capsule","paper_module.batch8_tools_tail_primitives_capsule","paper_module.batch8_validator_checker_capsule","paper_module.batch12_market_dashboard_read_model_capsule","paper_module.batch12_prediction_market_board_capsule","paper_module.batch12_release_claim_language_gate","paper_module.batch10_cold_eval_honesty_capsule","paper_module.batch10_live_source_drift_capsule","paper_module.batch7_macro_engines_capsule","paper_module.batch9_macro_engines_capsule","paper_module.pattern_assimilation","paper_module.batch10_governance_compilers_capsule","paper_module.batch11_saturation_engines_capsule","paper_module.batch4_proof_authority_runtime","paper_module.batch6_unsurfaced_primitives_capsule","paper_module.engine_room_public_projection_leak_gate","paper_module.batch5_authority_systems_capsule","paper_module.batch7_oracle_sibling_capsule","paper_module.batch7_demo_take_console_capsule","paper_module.engine_room_generated_projection_drift_gate","paper_module.engine_room_command_run_singleflight","paper_module.engine_room_metabolism_runtime","paper_module.engine_room_bridge_campaign_dag","paper_module.engine_room_annex_knowledge_router","paper_module.engine_room_derived_fact_provider_engine","paper_module.engine_room_egress_self_compliance_gate","paper_module.engine_room_lean_proof_search_lab","paper_module.engine_room_navigation_fitness_benchmark","paper_module.cold_clone_probe","paper_module.first_screen_composition_root","paper_module.batch7_secondary_runtime_capsule","paper_module.microcosm_axiom_substrate","paper_module.semantic_singleflight_dedup_runtime","paper_module.finite_erdos_denominator_certificate_strike","paper_module.bridge_campaign_dag_validation","paper_module.metabolism_queue_reconciliation","paper_module.derived_fact_provider_runtime","paper_module.navigation_fitness_benchmark","paper_module.annex_knowledge_routing","paper_module.egress_self_compliance_audit","paper_module.generated_projection_drift_runtime","paper_module.lean_proof_search_lab_runtime"],"family_id":null},"body_role":"index_page_atoms_carried_once","lede":"Executable Doctrine Grammar. The public executable-doctrine grammar fixture validates public standard rows, paper-module sections, negative cases, copied executable-grammar and standards/type-plane source-module bodies, and metadata-only result records without claiming doctrine completeness. The full paper-module reader text is carried once in visible_text_index.logical_records.paper_modules; diagrams are carried as compressed diagram assets.","renders_note":"Atoms shown on this page are carried once in this file; resolve the content_refs ids against components[] / paper_modules[] / doctrine_entities[] / evidence_profiles[]. Glossary terms are the glossary page's atoms in public_glossary[]."},{"id":"page:paper-module-macro-projection-import-protocol","path":"docs/paper-module-macro-projection-import-protocol.html","relative_href":"docs/paper-module-macro-projection-import-protocol.html","title":"Source Projection Import Protocol","nav_group":"Reference","nav_label":"Paper modules","source_refs":["microcosm-substrate/core/paper_module_capsules.json"],"content_refs":{"paper_module_refs":["paper_module.verifier_lab_kernel","paper_module.navigation_hologram_route_plane","paper_module.agent_route_observability_runtime","paper_module.agent_benchmark_integrity_anti_gaming_replay","paper_module.research_replication_rubric_artifact_replay","paper_module.agentic_vulnerability_discovery_patch_proof_replay","paper_module.materials_chemistry_closed_loop_lab_safety_replay","paper_module.certificate_kernel_execution_lab","paper_module.corpus_readiness_mathlib_absence_gate","paper_module.pattern_binding_contract","paper_module.bridge_phase_continuity_runtime","paper_module.cognitive_operator_registry","paper_module.agent_closeout_faithfulness_audit","paper_module.cold_reader_route_map","paper_module.proof_diagnostic_evidence_spine","paper_module.proof_derived_governed_mutation_authorization","paper_module.durable_agent_work_landing_replay","paper_module.work_landing_control_spine","paper_module.executable_doctrine_grammar","paper_module.macro_projection_import_protocol","paper_module.mission_transaction_work_spine","paper_module.formal_math_readiness_gate","paper_module.formal_math_lean_proof_witness","paper_module.formal_math_verifier_trace_repair_loop","paper_module.formal_evidence_cell_anchor_resolver","paper_module.formal_math_premise_retrieval","paper_module.lean_std_premise_index","paper_module.world_model_projection_drift_control_room","paper_module.public_reveal_walkthrough","paper_module.standards_meta_diagnostics","paper_module.finance_forecast_evaluation_spine","paper_module.engine_room_demo","paper_module.agent_memory_temporal_conflict_replay","paper_module.agent_monitor_redteam_falsification_replay","paper_module.agent_sabotage_scheming_monitor_replay","paper_module.agent_sandbox_policy_escape_replay","paper_module.belief_state_process_reward_replay","paper_module.sleeper_memory_poisoning_quarantine_replay","paper_module.indirect_prompt_injection_information_flow_policy_replay","paper_module.mcp_tool_authority_replay","paper_module.tactic_portfolio_availability","paper_module.target_shape_tactic_routing","paper_module.ring2_premise_precision_recall","paper_module.mathematical_strategy_atlas","paper_module.verifier_lab_execution_spine","paper_module.bounded_autonomy_campaign_packet","paper_module.computer_use_action_trace_replay","paper_module.concurrency_mission_control","paper_module.doctrine_fact_claim_audit","paper_module.self_ignorance_coverage_ledger","paper_module.tool_server_pressure_inventory","paper_module.mechanistic_interpretability_circuit_attribution_replay","paper_module.spatial_world_model_counterfactual_simulation_replay","paper_module.prediction_oracle_reconciliation","paper_module.provider_context_recipe_budget","paper_module.undeclared_library_prior_classifier","paper_module.voice_to_doctrine_self_improvement_loop","paper_module.routing_anti_patterns_registry","paper_module.batch8_audio_level_rms_port","paper_module.batch8_compliance_pipeline_capsule","paper_module.batch8_policy_engines_capsule","paper_module.batch8_structural_theses_capsule","paper_module.batch8_tools_tail_primitives_capsule","paper_module.batch8_validator_checker_capsule","paper_module.batch12_market_dashboard_read_model_capsule","paper_module.batch12_prediction_market_board_capsule","paper_module.batch12_release_claim_language_gate","paper_module.batch10_cold_eval_honesty_capsule","paper_module.batch10_live_source_drift_capsule","paper_module.batch7_macro_engines_capsule","paper_module.batch9_macro_engines_capsule","paper_module.pattern_assimilation","paper_module.batch10_governance_compilers_capsule","paper_module.batch11_saturation_engines_capsule","paper_module.batch4_proof_authority_runtime","paper_module.batch6_unsurfaced_primitives_capsule","paper_module.engine_room_public_projection_leak_gate","paper_module.batch5_authority_systems_capsule","paper_module.batch7_oracle_sibling_capsule","paper_module.batch7_demo_take_console_capsule","paper_module.engine_room_generated_projection_drift_gate","paper_module.engine_room_command_run_singleflight","paper_module.engine_room_metabolism_runtime","paper_module.engine_room_bridge_campaign_dag","paper_module.engine_room_annex_knowledge_router","paper_module.engine_room_derived_fact_provider_engine","paper_module.engine_room_egress_self_compliance_gate","paper_module.engine_room_lean_proof_search_lab","paper_module.engine_room_navigation_fitness_benchmark","paper_module.cold_clone_probe","paper_module.first_screen_composition_root","paper_module.batch7_secondary_runtime_capsule","paper_module.microcosm_axiom_substrate","paper_module.semantic_singleflight_dedup_runtime","paper_module.finite_erdos_denominator_certificate_strike","paper_module.bridge_campaign_dag_validation","paper_module.metabolism_queue_reconciliation","paper_module.derived_fact_provider_runtime","paper_module.navigation_fitness_benchmark","paper_module.annex_knowledge_routing","paper_module.egress_self_compliance_audit","paper_module.generated_projection_drift_runtime","paper_module.lean_proof_search_lab_runtime"],"family_id":null},"body_role":"index_page_atoms_carried_once","lede":"Source Projection Import Protocol. The public source-projection import protocol validates classified source-to-Microcosm projection cells, per-slice manifests, digest relations, omission result records, intake statuses, formal-math claim boundaries, and scope limits without claiming source or launch-scope decision. The full paper-module reader text is carried once in visible_text_index.logical_records.paper_modules; diagrams are carried as compressed diagram assets.","renders_note":"Atoms shown on this page are carried once in this file; resolve the content_refs ids against components[] / paper_modules[] / doctrine_entities[] / evidence_profiles[]. 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The public verifier-trace repair fixture validates copied Ring2 failure taxonomy, graph-update, oracle-repair contrast rows, source-module digests, negative cases, and one deterministic toy rerun without claiming proof authority. The full paper-module reader text is carried once in visible_text_index.logical_records.paper_modules; diagrams are carried as compressed diagram assets.","renders_note":"Atoms shown on this page are carried once in this file; resolve the content_refs ids against components[] / paper_modules[] / doctrine_entities[] / evidence_profiles[]. 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aper_module.egress_self_compliance_audit","paper_module.generated_projection_drift_runtime","paper_module.lean_proof_search_lab_runtime"],"family_id":null},"body_role":"index_page_atoms_carried_once","lede":"Formal Evidence Cell Anchor Resolver. The formal evidence cell anchor resolver binds proof-language paper claims to public evidence cells, source anchors, machine-anchor metadata, frozen public source-module snapshots, public source-artifact mirrors, and negative-case result records without claiming formal-result correctness. The full paper-module reader text is carried once in visible_text_index.logical_records.paper_modules; diagrams are carried as compressed diagram assets.","renders_note":"Atoms shown on this page are carried once in this file; resolve the content_refs ids against components[] / paper_modules[] / doctrine_entities[] / evidence_profiles[]. Glossary terms are the glossary page's atoms in public_glossary[]."},{"id":"page:paper-module-formal-math-premise-retrieval","path":"docs/paper-module-formal-math-premise-retrieval.html","relative_href":"docs/paper-module-formal-math-premise-retrieval.html","title":"Formal Math Premise Retrieval","nav_group":"Reference","nav_label":"Paper modules","source_refs":["microcosm-substrate/core/paper_module_capsules.json"],"content_refs":{"paper_module_refs":["paper_module.verifier_lab_kernel","paper_module.navigation_hologram_route_plane","paper_module.agent_route_observability_runtime","paper_module.agent_benchmark_integrity_anti_gaming_replay","paper_module.research_replication_rubric_artifact_replay","paper_module.agentic_vulnerability_discovery_patch_proof_replay","paper_module.materials_chemistry_closed_loop_lab_safety_replay","paper_module.certificate_kernel_execution_lab","paper_module.corpus_readiness_mathlib_absence_gate","paper_module.pattern_binding_contract","paper_module.bridge_phase_continuity_runtime","paper_module.cognitive_operator_registry","paper_module.agent_closeout_faithfulness_audit","paper_module.cold_reader_route_map","paper_module.proof_diagnostic_evidence_spine","paper_module.proof_derived_governed_mutation_authorization","paper_module.durable_agent_work_landing_replay","paper_module.work_landing_control_spine","paper_module.executable_doctrine_grammar","paper_module.macro_projection_import_protocol","paper_module.mission_transaction_work_spine","paper_module.formal_math_readiness_gate","paper_module.formal_math_lean_proof_witness","paper_module.formal_math_verifier_trace_repair_loop","paper_module.formal_evidence_cell_anchor_resolver","paper_module.formal_math_premise_retrieval","paper_module.lean_std_premise_index","paper_module.world_model_projection_drift_control_room","paper_module.public_reveal_walkthrough","paper_module.standards_meta_diagnostics","paper_module.finance_forecast_evaluation_spine","paper_module.engine_room_demo","paper_module.agent_memory_temporal_conflict_replay","paper_module.agent_monitor_redteam_falsification_replay","paper_module.agent_sabotage_scheming_monitor_replay","paper_module.agent_sandbox_policy_escape_replay","paper_module.belief_state_process_reward_replay","paper_module.sleeper_memory_poisoning_quarantine_replay","paper_module.indirect_prompt_injection_information_flow_policy_replay","paper_module.mcp_tool_authority_replay","paper_module.tactic_portfolio_availability","paper_module.target_shape_tactic_routing","paper_module.ring2_premise_precision_recall","paper_module.mathematical_strategy_atlas","paper_module.verifier_lab_execution_spine","paper_module.bounded_autonomy_campaign_packet","paper_module.computer_use_action_trace_replay","paper_module.concurrency_mission_control","paper_module.doctrine_fact_claim_audit","paper_module.self_ignorance_coverage_ledger","paper_module.tool_server_pressure_inventory","paper_module.mechanistic_interpretability_circuit_attribution_replay","paper_module.spatial_world_model_counterfactual_simulation_replay","paper_module.prediction_oracle_reconciliation","paper_module.provider_context_recipe_budget","paper_module.undeclared_library_prior_classifier","paper_module.voice_to_doctrine_self_improvement_loop","paper_module.routing_anti_patterns_registry","paper_module.batch8_audio_level_rms_port","paper_module.batch8_compliance_pipeline_capsule","paper_module.batch8_policy_engines_capsule","paper_module.batch8_structural_theses_capsule","paper_module.batch8_tools_tail_primitives_capsule","paper_module.batch8_validator_checker_capsule","paper_module.batch12_market_dashboard_read_model_capsule","paper_module.batch12_prediction_market_board_capsule","paper_module.batch12_release_claim_language_gate","paper_module.batch10_cold_eval_honesty_capsule","paper_module.batch10_live_source_drift_capsule","paper_module.batch7_macro_engines_capsule","paper_module.batch9_macro_engines_capsule","paper_module.pattern_assimilation","paper_module.batch10_governance_compilers_capsule","paper_module.batch11_saturation_engines_capsule","paper_module.batch4_proof_authority_runtime","paper_module.batch6_unsurfaced_primitives_capsule","paper_module.engine_room_public_projection_leak_gate","paper_module.batch5_authority_systems_capsule","paper_module.batch7_oracle_sibling_capsule","paper_module.batch7_demo_take_console_capsule","paper_module.engine_room_generated_projection_drift_gate","paper_module.engine_room_command_run_singleflight","paper_module.engine_room_metabolism_runtime","paper_module.engine_room_bridge_campaign_dag","paper_module.engine_room_annex_knowledge_router","paper_module.engine_room_derived_fact_provider_engine","paper_module.engine_room_egress_self_compliance_gate","paper_module.engine_room_lean_proof_search_lab","paper_module.engine_room_navigation_fitness_benchmark","paper_module.cold_clone_probe","paper_module.first_screen_composition_root","paper_module.batch7_secondary_runtime_capsule","paper_module.microcosm_axiom_substrate","paper_module.semantic_singleflight_dedup_runtime","paper_module.finite_erdos_denominator_certificate_strike","paper_module.bridge_campaign_dag_validation","paper_module.metabolism_queue_reconciliation","paper_module.derived_fact_provider_runtime","paper_module.navigation_fitness_benchmark","paper_module.annex_knowledge_routing","paper_module.egress_self_compliance_audit","paper_module.generated_projection_drift_runtime","paper_module.lean_proof_search_lab_runtime"],"family_id":null},"body_role":"index_page_atoms_carried_once","lede":"Formal Math Premise Retrieval. Formal math premise retrieval validates copied public Lean/Std premise metadata, query scoring, context budgets, strategy gates, body-floor provenance, and leakage negative cases without claiming proof authority. The full paper-module reader text is carried once in visible_text_index.logical_records.paper_modules; diagrams are carried as compressed diagram assets.","renders_note":"Atoms shown on this page are carried once in this file; resolve the content_refs ids against components[] / paper_modules[] / doctrine_entities[] / evidence_profiles[]. Glossary terms are the glossary page's atoms in public_glossary[]."},{"id":"page:paper-module-lean-std-premise-index","path":"docs/paper-module-lean-std-premise-index.html","relative_href":"docs/paper-module-lean-std-premise-index.html","title":"Lean/Std Premise Index","nav_group":"Reference","nav_label":"Paper modules","source_refs":["microcosm-substrate/core/paper_module_capsules.json"],"content_refs":{"paper_module_refs":["paper_module.verifier_lab_kernel","paper_module.navigation_hologram_route_plane","paper_module.agent_route_observability_runtime","paper_module.agent_benchmark_integrity_anti_gaming_replay","paper_module.research_replication_rubric_artifact_replay","paper_module.agentic_vulnerability_discovery_patch_proof_replay","paper_module.materials_chemistry_closed_loop_lab_safety_replay","paper_module.certificate_kernel_execution_lab","paper_module.corpus_readiness_mathlib_absence_gate","paper_module.pattern_binding_contract","paper_module.bridge_phase_continuity_runtime","paper_module.cognitive_operator_registry","paper_module.agent_closeout_faithfulness_audit","paper_module.cold_reader_route_map","paper_module.proof_diagnostic_evidence_spine","paper_module.proof_derived_governed_mutation_authorization","paper_module.durable_agent_work_landing_replay","paper_module.work_landing_control_spine","paper_module.executable_doctrine_grammar","paper_module.macro_projection_import_protocol","paper_module.mission_transaction_work_spine","paper_module.formal_math_readiness_gate","paper_module.formal_math_lean_proof_witness","paper_module.formal_math_verifier_trace_repair_loop","paper_module.formal_evidence_cell_anchor_resolver","paper_module.formal_math_premise_retrieval","paper_module.lean_std_premise_index","paper_module.world_model_projection_drift_control_room","paper_module.public_reveal_walkthrough","paper_module.standards_meta_diagnostics","paper_module.finance_forecast_evaluation_spine","paper_module.engine_room_demo","paper_module.agent_memory_temporal_conflict_replay","paper_module.agent_monitor_redteam_falsification_replay","paper_module.agent_sabotage_scheming_monitor_replay","paper_module.agent_sandbox_policy_escape_replay","paper_module.belief_state_process_reward_replay","paper_module.sleeper_memory_poisoning_quarantine_replay","paper_module.indirect_prompt_injection_information_flow_policy_replay","paper_module.mcp_tool_authority_replay","paper_module.tactic_portfolio_availability","paper_module.target_shape_tactic_routing","paper_module.ring2_premise_precision_recall","paper_module.mathematical_strategy_atlas","paper_module.verifier_lab_execution_spine","paper_module.bounded_autonomy_campaign_packet","paper_module.computer_use_action_trace_replay","paper_module.concurrency_mission_control","paper_module.doctrine_fact_claim_audit","paper_module.self_ignorance_coverage_ledger","paper_module.tool_server_pressure_inventory","paper_module.mechanistic_interpretability_circuit_attribution_replay","paper_module.spatial_world_model_counterfactual_simulation_replay","paper_module.prediction_oracle_reconciliation","paper_module.provider_context_recipe_budget","paper_module.undeclared_library_prior_classifier","paper_module.voice_to_doctrine_self_improvement_loop","paper_module.routing_anti_patterns_registry","paper_module.batch8_audio_level_rms_port","paper_module.batch8_compliance_pipeline_capsule","paper_module.batch8_policy_engines_capsule","paper_module.batch8_structural_theses_capsule","paper_module.batch8_tools_tail_primitives_capsule","paper_module.batch8_validator_checker_capsule","paper_module.batch12_market_dashboard_read_model_capsule","paper_module.batch12_prediction_market_board_capsule","paper_module.batch12_release_claim_language_gate","paper_module.batch10_cold_eval_honesty_capsule","paper_module.batch10_live_source_drift_capsule","paper_module.batch7_macro_engines_capsule","paper_module.batch9_macro_engines_capsule","paper_module.pattern_assimilation","paper_module.batch10_governance_compilers_capsule","paper_module.batch11_saturation_engines_capsule","paper_module.batch4_proof_authority_runtime","paper_module.batch6_unsurfaced_primitives_capsule","paper_module.engine_room_public_projection_leak_gate","paper_module.batch5_authority_systems_capsule","paper_module.batch7_oracle_sibling_capsule","paper_module.batch7_demo_take_console_capsule","paper_module.engine_room_generated_projection_drift_gate","paper_module.engine_room_command_run_singleflight","paper_module.engine_room_metabolism_runtime","paper_module.engine_room_bridge_campaign_dag","paper_module.engine_room_annex_knowledge_router","paper_module.engine_room_derived_fact_provider_engine","paper_module.engine_room_egress_self_compliance_gate","paper_module.engine_room_lean_proof_search_lab","paper_module.engine_room_navigation_fitness_benchmark","paper_module.cold_clone_probe","paper_module.first_screen_composition_root","paper_module.batch7_secondary_runtime_capsule","paper_module.microcosm_axiom_substrate","paper_module.semantic_singleflight_dedup_runtime","paper_module.finite_erdos_denominator_certificate_strike","paper_module.bridge_campaign_dag_validation","paper_module.metabolism_queue_reconciliation","paper_module.derived_fact_provider_runtime","paper_module.navigation_fitness_benchmark","paper_module.annex_knowledge_routing","paper_module.egress_self_compliance_audit","paper_module.generated_projection_drift_runtime","paper_module.lean_proof_search_lab_runtime"],"family_id":null},"body_role":"index_page_atoms_carried_once","lede":"Lean/Std Premise Index. The Lean/Std premise index validates a copied public Lean/Std descriptor catalog plus Ring2 premise-retrieval source bodies without claiming proof, Mathlib, Lean/Lake, provider, launch, or theorem-correctness authority. The full paper-module reader text is carried once in visible_text_index.logical_records.paper_modules; diagrams are carried as compressed diagram assets.","renders_note":"Atoms shown on this page are carried once in this file; resolve the content_refs ids against components[] / paper_modules[] / doctrine_entities[] / evidence_profiles[]. Glossary terms are the glossary page's atoms in public_glossary[]."},{"id":"page:paper-module-world-model-projection-drift-control-room","path":"docs/paper-module-world-model-projection-drift-control-room.html","relative_href":"docs/paper-module-world-model-projection-drift-control-room.html","title":"World-Model Projection Drift Control Room","nav_group":"Reference","nav_label":"Paper modules","source_refs":["microcosm-substrate/core/paper_module_capsules.json"],"content_refs":{"paper_module_refs":["paper_module.verifier_lab_kernel","paper_module.navigation_hologram_route_plane","paper_module.agent_route_observability_runtime","paper_module.agent_benchmark_integrity_anti_gaming_replay","paper_module.research_replication_rubric_artifact_replay","paper_module.agentic_vulnerability_discovery_patch_proof_replay","paper_module.materials_chemistry_closed_loop_lab_safety_replay","paper_module.certificate_kernel_execution_lab","paper_module.corpus_readiness_mathlib_absence_gate","paper_module.pattern_binding_contract","paper_module.bridge_phase_continuity_runtime","paper_module.cognitive_operator_registry","paper_module.agent_closeout_faithfulness_audit","paper_module.cold_reader_route_map","paper_module.proof_diagnostic_evidence_spine","paper_module.proof_derived_governed_mutation_authorization","paper_module.durable_agent_work_landing_replay","paper_module.work_landing_control_spine","paper_module.executable_doctrine_grammar","paper_module.macro_projection_import_protocol","paper_module.mission_transaction_work_spine","paper_module.formal_math_readiness_gate","paper_module.formal_math_lean_proof_witness","paper_module.formal_math_verifier_trace_repair_loop","paper_module.formal_evidence_cell_anchor_resolver","paper_module.formal_math_premise_retrieval","paper_module.lean_std_premise_index","paper_module.world_model_projection_drift_control_room","paper_module.public_reveal_walkthrough","paper_module.standards_meta_diagnostics","paper_module.finance_forecast_evaluation_spine","paper_module.engine_room_demo","paper_module.agent_memory_temporal_conflict_replay","paper_module.agent_monitor_redteam_falsification_replay","paper_module.agent_sabotage_scheming_monitor_replay","paper_module.agent_sandbox_policy_escape_replay","paper_module.belief_state_process_reward_replay","paper_module.sleeper_memory_poisoning_quarantine_replay","paper_module.indirect_prompt_injection_information_flow_policy_replay","paper_module.mcp_tool_authority_replay","paper_module.tactic_portfolio_availability","paper_module.target_shape_tactic_routing","paper_module.ring2_premise_precision_recall","paper_module.mathematical_strategy_atlas","paper_module.verifier_lab_execution_spine","paper_module.bounded_autonomy_campaign_packet","paper_module.computer_use_action_trace_replay","paper_module.concurrency_mission_control","paper_module.doctrine_fact_claim_audit","paper_module.self_ignorance_coverage_ledger","paper_module.tool_server_pressure_inventory","paper_module.mechanistic_interpretability_circuit_attribution_replay","paper_module.spatial_world_model_counterfactual_simulation_replay","paper_module.prediction_oracle_reconciliation","paper_module.provider_context_recipe_budget","paper_module.undeclared_library_prior_classifier","paper_module.voice_to_doctrine_self_improvement_loop","paper_module.routing_anti_patterns_registry","paper_module.batch8_audio_level_rms_port","paper_module.batch8_compliance_pipeline_capsule","paper_module.batch8_policy_engines_capsule","paper_module.batch8_structural_theses_capsule","paper_module.batch8_tools_tail_primitives_capsule","paper_module.batch8_validator_checker_capsule","paper_module.batch12_market_dashboard_read_model_capsule","paper_module.batch12_prediction_market_board_capsule","paper_module.batch12_release_claim_language_gate","paper_module.batch10_cold_eval_honesty_capsule","paper_module.batch10_live_source_drift_capsule","paper_module.batch7_macro_engines_capsule","paper_module.batch9_macro_engines_capsule","paper_module.pattern_assimilation","paper_module.batch10_governance_compilers_capsule","paper_module.batch11_saturation_engines_capsule","paper_module.batch4_proof_authority_runtime","paper_module.batch6_unsurfaced_primitives_capsule","paper_module.engine_room_public_projection_leak_gate","paper_module.batch5_authority_systems_capsule","paper_module.batch7_oracle_sibling_capsule","paper_module.batch7_demo_take_console_capsule","paper_module.engine_room_generated_projection_drift_gate","paper_module.engine_room_command_run_singleflight","paper_module.engine_room_metabolism_runtime","paper_module.engine_room_bridge_campaign_dag","paper_module.engine_room_annex_knowledge_router","paper_module.engine_room_derived_fact_provider_engine","paper_module.engine_room_egress_self_compliance_gate","paper_module.engine_room_lean_proof_search_lab","paper_module.engine_room_navigation_fitness_benchmark","paper_module.cold_clone_probe","paper_module.first_screen_composition_root","paper_module.batch7_secondary_runtime_capsule","paper_module.microcosm_axiom_substrate","paper_module.semantic_singleflight_dedup_runtime","paper_module.finite_erdos_denominator_certificate_strike","paper_module.bridge_campaign_dag_validation","paper_module.metabolism_queue_reconciliation","paper_module.derived_fact_provider_runtime","paper_module.navigation_fitness_benchmark","paper_module.annex_knowledge_routing","paper_module.egress_self_compliance_audit","paper_module.generated_projection_drift_runtime","paper_module.lean_proof_search_lab_runtime"],"family_id":null},"body_role":"index_page_atoms_carried_once","lede":"World-Model Projection Drift Control Room. The world-model projection drift control room validates public metadata-only projection-drift rows and copied source-module bodies without treating projections as source authority or repair authority. The full paper-module reader text is carried once in visible_text_index.logical_records.paper_modules; diagrams are carried as compressed diagram assets.","renders_note":"Atoms shown on this page are carried once in this file; resolve the content_refs ids against components[] / paper_modules[] / doctrine_entities[] / evidence_profiles[]. 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8 Policy Engines Bundle. 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8 Structural Theses Bundle. Set 8 Structural Theses Bundle validates public synthetic thesis-family replay without financial decisions, live-market validation, external model access, or portfolio authority. The full paper-module reader text is carried once in visible_text_index.logical_records.paper_modules; diagrams are carried as compressed diagram assets.","renders_note":"Atoms shown on this page are carried once in this file; resolve the content_refs ids against components[] / paper_modules[] / doctrine_entities[] / evidence_profiles[]. 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aper_module.egress_self_compliance_audit","paper_module.generated_projection_drift_runtime","paper_module.lean_proof_search_lab_runtime"],"family_id":null},"body_role":"index_page_atoms_carried_once","lede":"Set 8 Tools-Tail Primitives Bundle. Set 8 Tools-Tail Primitives Bundle validates four public tools-tail primitive exercises without Oracle truth, external model access, live bridge work, or repository mutation authority. The full paper-module reader text is carried once in visible_text_index.logical_records.paper_modules; diagrams are carried as compressed diagram assets.","renders_note":"Atoms shown on this page are carried once in this file; resolve the content_refs ids against components[] / paper_modules[] / doctrine_entities[] / evidence_profiles[]. 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aper_module.egress_self_compliance_audit","paper_module.generated_projection_drift_runtime","paper_module.lean_proof_search_lab_runtime"],"family_id":null},"body_role":"index_page_atoms_carried_once","lede":"Set 8 Validator Checker Bundle. Set 8 Validator Checker Bundle validates selected public checker groups without becoming launch-scope decision or a complete validator-suite proof. The full paper-module reader text is carried once in visible_text_index.logical_records.paper_modules; diagrams are carried as compressed diagram assets.","renders_note":"Atoms shown on this page are carried once in this file; resolve the content_refs ids against components[] / paper_modules[] / doctrine_entities[] / evidence_profiles[]. Glossary terms are the glossary page's atoms in public_glossary[]."},{"id":"page:paper-module-batch12-market-dashboard-read-model-capsule","path":"docs/paper-module-batch12-market-dashboard-read-model-capsule.html","relative_href":"docs/paper-module-batch12-market-dashboard-read-model-capsule.html","title":"Set 12 Market Dashboard Read-Model Bundle","nav_group":"Reference","nav_label":"Paper modules","source_refs":["microcosm-substrate/core/paper_module_capsules.json"],"content_refs":{"paper_module_refs":["paper_module.verifier_lab_kernel","paper_module.navigation_hologram_route_plane","paper_module.agent_route_observability_runtime","paper_module.agent_benchmark_integrity_anti_gaming_replay","paper_module.research_replication_rubric_artifact_replay","paper_module.agentic_vulnerability_discovery_patch_proof_replay","paper_module.materials_chemistry_closed_loop_lab_safety_replay","paper_module.certificate_kernel_execution_lab","paper_module.corpus_readiness_mathlib_absence_gate","paper_module.pattern_binding_contract","paper_module.bridge_phase_continuity_runtime","paper_module.cognitive_operator_registry","paper_module.agent_closeout_faithfulness_audit","paper_module.cold_reader_route_map","paper_module.proof_diagnostic_evidence_spine","paper_module.proof_derived_governed_mutation_authorization","paper_module.durable_agent_work_landing_replay","paper_module.work_landing_control_spine","paper_module.executable_doctrine_grammar","paper_module.macro_projection_import_protocol","paper_module.mission_transaction_work_spine","paper_module.formal_math_readiness_gate","paper_module.formal_math_lean_proof_witness","paper_module.formal_math_verifier_trace_repair_loop","paper_module.formal_evidence_cell_anchor_resolver","paper_module.formal_math_premise_retrieval","paper_module.lean_std_premise_index","paper_module.world_model_projection_drift_control_room","paper_module.public_reveal_walkthrough","paper_module.standards_meta_diagnostics","paper_module.finance_forecast_evaluation_spine","paper_module.engine_room_demo","paper_module.agent_memory_temporal_conflict_replay","paper_module.agent_monitor_redteam_falsification_replay","paper_module.agent_sabotage_scheming_monitor_replay","paper_module.agent_sandbox_policy_escape_replay","paper_module.belief_state_process_reward_replay","paper_module.sleeper_memory_poisoning_quarantine_replay","paper_module.indirect_prompt_injection_information_flow_policy_replay","paper_module.mcp_tool_authority_replay","paper_module.tactic_portfolio_availability","paper_module.target_shape_tactic_routing","paper_module.ring2_premise_precision_recall","paper_module.mathematical_strategy_atlas","paper_module.verifier_lab_execution_spine","paper_module.bounded_autonomy_campaign_packet","paper_module.computer_use_action_trace_replay","paper_module.concurrency_mission_control","paper_module.doctrine_fact_claim_audit","paper_module.self_ignorance_coverage_ledger","paper_module.tool_server_pressure_inventory","paper_module.mechanistic_interpretability_circuit_attribution_replay","paper_module.spatial_world_model_counterfactual_simulation_replay","paper_module.prediction_oracle_reconciliation","paper_module.provider_context_recipe_budget","paper_module.undeclared_library_prior_classifier","paper_module.voice_to_doctrine_self_improvement_loop","paper_module.routing_anti_patterns_registry","paper_module.batch8_audio_level_rms_port","paper_module.batch8_compliance_pipeline_capsule","paper_module.batch8_policy_engines_capsule","paper_module.batch8_structural_theses_capsule","paper_module.batch8_tools_tail_primitives_capsule","paper_module.batch8_validator_checker_capsule","paper_module.batch12_market_dashboard_read_model_capsule","paper_module.batch12_prediction_market_board_capsule","paper_module.batch12_release_claim_language_gate","paper_module.batch10_cold_eval_honesty_capsule","paper_module.batch10_live_source_drift_capsule","paper_module.batch7_macro_engines_capsule","paper_module.batch9_macro_engines_capsule","paper_module.pattern_assimilation","paper_module.batch10_governance_compilers_capsule","paper_module.batch11_saturation_engines_capsule","paper_module.batch4_proof_authority_runtime","paper_module.batch6_unsurfaced_primitives_capsule","paper_module.engine_room_public_projection_leak_gate","paper_module.batch5_authority_systems_capsule","paper_module.batch7_oracle_sibling_capsule","paper_module.batch7_demo_take_console_capsule","paper_module.engine_room_generated_projection_drift_gate","paper_module.engine_room_command_run_singleflight","paper_module.engine_room_metabolism_runtime","paper_module.engine_room_bridge_campaign_dag","paper_module.engine_room_annex_knowledge_router","paper_module.engine_room_derived_fact_provider_engine","paper_module.engine_room_egress_self_compliance_gate","paper_module.engine_room_lean_proof_search_lab","paper_module.engine_room_navigation_fitness_benchmark","paper_module.cold_clone_probe","paper_module.first_screen_composition_root","paper_module.batch7_secondary_runtime_capsule","paper_module.microcosm_axiom_substrate","paper_module.semantic_singleflight_dedup_runtime","paper_module.finite_erdos_denominator_certificate_strike","paper_module.bridge_campaign_dag_validation","paper_module.metabolism_queue_reconciliation","paper_module.derived_fact_provider_runtime","paper_module.navigation_fitness_benchmark","paper_module.annex_knowledge_routing","paper_module.egress_self_compliance_audit","paper_module.generated_projection_drift_runtime","paper_module.lean_proof_search_lab_runtime"],"family_id":null},"body_role":"index_page_atoms_carried_once","lede":"Set 12 Market Dashboard Read-Model Bundle. Set 12 Market Dashboard Read-Model Bundle validates copied read-model helpers over public fixtures without market-level conclusions, external model access, or launch-scope decision. The full paper-module reader text is carried once in visible_text_index.logical_records.paper_modules; diagrams are carried as compressed diagram assets.","renders_note":"Atoms shown on this page are carried once in this file; resolve the content_refs ids against components[] / paper_modules[] / doctrine_entities[] / evidence_profiles[]. Glossary terms are the glossary page's atoms in public_glossary[]."},{"id":"page:paper-module-batch12-prediction-market-board-capsule","path":"docs/paper-module-batch12-prediction-market-board-capsule.html","relative_href":"docs/paper-module-batch12-prediction-market-board-capsule.html","title":"Set 12 Prediction Market Board Bundle","nav_group":"Reference","nav_label":"Paper modules","source_refs":["microcosm-substrate/core/paper_module_capsules.json"],"content_refs":{"paper_module_refs":["paper_module.verifier_lab_kernel","paper_module.navigation_hologram_route_plane","paper_module.agent_route_observability_runtime","paper_module.agent_benchmark_integrity_anti_gaming_replay","paper_module.research_replication_rubric_artifact_replay","paper_module.agentic_vulnerability_discovery_patch_proof_replay","paper_module.materials_chemistry_closed_loop_lab_safety_replay","paper_module.certificate_kernel_execution_lab","paper_module.corpus_readiness_mathlib_absence_gate","paper_module.pattern_binding_contract","paper_module.bridge_phase_continuity_runtime","paper_module.cognitive_operator_registry","paper_module.agent_closeout_faithfulness_audit","paper_module.cold_reader_route_map","paper_module.proof_diagnostic_evidence_spine","paper_module.proof_derived_governed_mutation_authorization","paper_module.durable_agent_work_landing_replay","paper_module.work_landing_control_spine","paper_module.executable_doctrine_grammar","paper_module.macro_projection_import_protocol","paper_module.mission_transaction_work_spine","paper_module.formal_math_readiness_gate","paper_module.formal_math_lean_proof_witness","paper_module.formal_math_verifier_trace_repair_loop","paper_module.formal_evidence_cell_anchor_resolver","paper_module.formal_math_premise_retrieval","paper_module.lean_std_premise_index","paper_module.world_model_projection_drift_control_room","paper_module.public_reveal_walkthrough","paper_module.standards_meta_diagnostics","paper_module.finance_forecast_evaluation_spine","paper_module.engine_room_demo","paper_module.agent_memory_temporal_conflict_replay","paper_module.agent_monitor_redteam_falsification_replay","paper_module.agent_sabotage_scheming_monitor_replay","paper_module.agent_sandbox_policy_escape_replay","paper_module.belief_state_process_reward_replay","paper_module.sleeper_memory_poisoning_quarantine_replay","paper_module.indirect_prompt_injection_information_flow_policy_replay","paper_module.mcp_tool_authority_replay","paper_module.tactic_portfolio_availability","paper_module.target_shape_tactic_routing","paper_module.ring2_premise_precision_recall","paper_module.mathematical_strategy_atlas","paper_module.verifier_lab_execution_spine","paper_module.bounded_autonomy_campaign_packet","paper_module.computer_use_action_trace_replay","paper_module.concurrency_mission_control","paper_module.doctrine_fact_claim_audit","paper_module.self_ignorance_coverage_ledger","paper_module.tool_server_pressure_inventory","paper_module.mechanistic_interpretability_circuit_attribution_replay","paper_module.spatial_world_model_counterfactual_simulation_replay","paper_module.prediction_oracle_reconciliation","paper_module.provider_context_recipe_budget","paper_module.undeclared_library_prior_classifier","paper_module.voice_to_doctrine_self_improvement_loop","paper_module.routing_anti_patterns_registry","paper_module.batch8_audio_level_rms_port","paper_module.batch8_compliance_pipeline_capsule","paper_module.batch8_policy_engines_capsule","paper_module.batch8_structural_theses_capsule","paper_module.batch8_tools_tail_primitives_capsule","paper_module.batch8_validator_checker_capsule","paper_module.batch12_market_dashboard_read_model_capsule","paper_module.batch12_prediction_market_board_capsule","paper_module.batch12_release_claim_language_gate","paper_module.batch10_cold_eval_honesty_capsule","paper_module.batch10_live_source_drift_capsule","paper_module.batch7_macro_engines_capsule","paper_module.batch9_macro_engines_capsule","paper_module.pattern_assimilation","paper_module.batch10_governance_compilers_capsule","paper_module.batch11_saturation_engines_capsule","paper_module.batch4_proof_authority_runtime","paper_module.batch6_unsurfaced_primitives_capsule","paper_module.engine_room_public_projection_leak_gate","paper_module.batch5_authority_systems_capsule","paper_module.batch7_oracle_sibling_capsule","paper_module.batch7_demo_take_console_capsule","paper_module.engine_room_generated_projection_drift_gate","paper_module.engine_room_command_run_singleflight","paper_module.engine_room_metabolism_runtime","paper_module.engine_room_bridge_campaign_dag","paper_module.engine_room_annex_knowledge_router","paper_module.engine_room_derived_fact_provider_engine","paper_module.engine_room_egress_self_compliance_gate","paper_module.engine_room_lean_proof_search_lab","paper_module.engine_room_navigation_fitness_benchmark","paper_module.cold_clone_probe","paper_module.first_screen_composition_root","paper_module.batch7_secondary_runtime_capsule","paper_module.microcosm_axiom_substrate","paper_module.semantic_singleflight_dedup_runtime","paper_module.finite_erdos_denominator_certificate_strike","paper_module.bridge_campaign_dag_validation","paper_module.metabolism_queue_reconciliation","paper_module.derived_fact_provider_runtime","paper_module.navigation_fitness_benchmark","paper_module.annex_knowledge_routing","paper_module.egress_self_compliance_audit","paper_module.generated_projection_drift_runtime","paper_module.lean_proof_search_lab_runtime"],"family_id":null},"body_role":"index_page_atoms_carried_once","lede":"Set 12 Prediction Market Board Bundle. Set 12 Prediction Market Board Bundle validates copied prediction-board and quant-mart diagnostics over public fixtures without market-level conclusions or provider authority. The full paper-module reader text is carried once in visible_text_index.logical_records.paper_modules; diagrams are carried as compressed diagram assets.","renders_note":"Atoms shown on this page are carried once in this file; resolve the content_refs ids against components[] / paper_modules[] / doctrine_entities[] / evidence_profiles[]. Glossary terms are the glossary page's atoms in public_glossary[]."},{"id":"page:paper-module-batch12-release-claim-language-gate","path":"docs/paper-module-batch12-release-claim-language-gate.html","relative_href":"docs/paper-module-batch12-release-claim-language-gate.html","title":"Set 12 launch claim-Language Gate","nav_group":"Reference","nav_label":"Paper modules","source_refs":["microcosm-substrate/core/paper_module_capsules.json"],"content_refs":{"paper_module_refs":["paper_module.verifier_lab_kernel","paper_module.navigation_hologram_route_plane","paper_module.agent_route_observability_runtime","paper_module.agent_benchmark_integrity_anti_gaming_replay","paper_module.research_replication_rubric_artifact_replay","paper_module.agentic_vulnerability_discovery_patch_proof_replay","paper_module.materials_chemistry_closed_loop_lab_safety_replay","paper_module.certificate_kernel_execution_lab","paper_module.corpus_readiness_mathlib_absence_gate","paper_module.pattern_binding_contract","paper_module.bridge_phase_continuity_runtime","paper_module.cognitive_operator_registry","paper_module.agent_closeout_faithfulness_audit","paper_module.cold_reader_route_map","paper_module.proof_diagnostic_evidence_spine","paper_module.proof_derived_governed_mutation_authorization","paper_module.durable_agent_work_landing_replay","paper_module.work_landing_control_spine","paper_module.executable_doctrine_grammar","paper_module.macro_projection_import_protocol","paper_module.mission_transaction_work_spine","paper_module.formal_math_readiness_gate","paper_module.formal_math_lean_proof_witness","paper_module.formal_math_verifier_trace_repair_loop","paper_module.formal_evidence_cell_anchor_resolver","paper_module.formal_math_premise_retrieval","paper_module.lean_std_premise_index","paper_module.world_model_projection_drift_control_room","paper_module.public_reveal_walkthrough","paper_module.standards_meta_diagnostics","paper_module.finance_forecast_evaluation_spine","paper_module.engine_room_demo","paper_module.agent_memory_temporal_conflict_replay","paper_module.agent_monitor_redteam_falsification_replay","paper_module.agent_sabotage_scheming_monitor_replay","paper_module.agent_sandbox_policy_escape_replay","paper_module.belief_state_process_reward_replay","paper_module.sleeper_memory_poisoning_quarantine_replay","paper_module.indirect_prompt_injection_information_flow_policy_replay","paper_module.mcp_tool_authority_replay","paper_module.tactic_portfolio_availability","paper_module.target_shape_tactic_routing","paper_module.ring2_premise_precision_recall","paper_module.mathematical_strategy_atlas","paper_module.verifier_lab_execution_spine","paper_module.bounded_autonomy_campaign_packet","paper_module.computer_use_action_trace_replay","paper_module.concurrency_mission_control","paper_module.doctrine_fact_claim_audit","paper_module.self_ignorance_coverage_ledger","paper_module.tool_server_pressure_inventory","paper_module.mechanistic_interpretability_circuit_attribution_replay","paper_module.spatial_world_model_counterfactual_simulation_replay","paper_module.prediction_oracle_reconciliation","paper_module.provider_context_recipe_budget","paper_module.undeclared_library_prior_classifier","paper_module.voice_to_doctrine_self_improvement_loop","paper_module.routing_anti_patterns_registry","paper_module.batch8_audio_level_rms_port","paper_module.batch8_compliance_pipeline_capsule","paper_module.batch8_policy_engines_capsule","paper_module.batch8_structural_theses_capsule","paper_module.batch8_tools_tail_primitives_capsule","paper_module.batch8_validator_checker_capsule","paper_module.batch12_market_dashboard_read_model_capsule","paper_module.batch12_prediction_market_board_capsule","paper_module.batch12_release_claim_language_gate","paper_module.batch10_cold_eval_honesty_capsule","paper_module.batch10_live_source_drift_capsule","paper_module.batch7_macro_engines_capsule","paper_module.batch9_macro_engines_capsule","paper_module.pattern_assimilation","paper_module.batch10_governance_compilers_capsule","paper_module.batch11_saturation_engines_capsule","paper_module.batch4_proof_authority_runtime","paper_module.batch6_unsurfaced_primitives_capsule","paper_module.engine_room_public_projection_leak_gate","paper_module.batch5_authority_systems_capsule","paper_module.batch7_oracle_sibling_capsule","paper_module.batch7_demo_take_console_capsule","paper_module.engine_room_generated_projection_drift_gate","paper_module.engine_room_command_run_singleflight","paper_module.engine_room_metabolism_runtime","paper_module.engine_room_bridge_campaign_dag","paper_module.engine_room_annex_knowledge_router","paper_module.engine_room_derived_fact_provider_engine","paper_module.engine_room_egress_self_compliance_gate","paper_module.engine_room_lean_proof_search_lab","paper_module.engine_room_navigation_fitness_benchmark","paper_module.cold_clone_probe","paper_module.first_screen_composition_root","paper_module.batch7_secondary_runtime_capsule","paper_module.microcosm_axiom_substrate","paper_module.semantic_singleflight_dedup_runtime","paper_module.finite_erdos_denominator_certificate_strike","paper_module.bridge_campaign_dag_validation","paper_module.metabolism_queue_reconciliation","paper_module.derived_fact_provider_runtime","paper_module.navigation_fitness_benchmark","paper_module.annex_knowledge_routing","paper_module.egress_self_compliance_audit","paper_module.generated_projection_drift_runtime","paper_module.lean_proof_search_lab_runtime"],"family_id":null},"body_role":"index_page_atoms_carried_once","lede":"Set 12 launch claim-Language Gate. Set 12 launch claim-Language Gate checks public claim language against result record-backed scope limits without approving launch or public sharing. The full paper-module reader text is carried once in visible_text_index.logical_records.paper_modules; diagrams are carried as compressed diagram assets.","renders_note":"Atoms shown on this page are carried once in this file; resolve the content_refs ids against components[] / paper_modules[] / doctrine_entities[] / evidence_profiles[]. Glossary terms are the glossary page's atoms in public_glossary[]."},{"id":"page:paper-module-batch10-cold-eval-honesty-capsule","path":"docs/paper-module-batch10-cold-eval-honesty-capsule.html","relative_href":"docs/paper-module-batch10-cold-eval-honesty-capsule.html","title":"Set 10 Cold Eval Honesty Bundle","nav_group":"Reference","nav_label":"Paper modules","source_refs":["microcosm-substrate/core/paper_module_capsules.json"],"content_refs":{"paper_module_refs":["paper_module.verifier_lab_kernel","paper_module.navigation_hologram_route_plane","paper_module.agent_route_observability_runtime","paper_module.agent_benchmark_integrity_anti_gaming_replay","paper_module.research_replication_rubric_artifact_replay","paper_module.agentic_vulnerability_discovery_patch_proof_replay","paper_module.materials_chemistry_closed_loop_lab_safety_replay","paper_module.certificate_kernel_execution_lab","paper_module.corpus_readiness_mathlib_absence_gate","paper_module.pattern_binding_contract","paper_module.bridge_phase_continuity_runtime","paper_module.cognitive_operator_registry","paper_module.agent_closeout_faithfulness_audit","paper_module.cold_reader_route_map","paper_module.proof_diagnostic_evidence_spine","paper_module.proof_derived_governed_mutation_authorization","paper_module.durable_agent_work_landing_replay","paper_module.work_landing_control_spine","paper_module.executable_doctrine_grammar","paper_module.macro_projection_import_protocol","paper_module.mission_transaction_work_spine","paper_module.formal_math_readiness_gate","paper_module.formal_math_lean_proof_witness","paper_module.formal_math_verifier_trace_repair_loop","paper_module.formal_evidence_cell_anchor_resolver","paper_module.formal_math_premise_retrieval","paper_module.lean_std_premise_index","paper_module.world_model_projection_drift_control_room","paper_module.public_reveal_walkthrough","paper_module.standards_meta_diagnostics","paper_module.finance_forecast_evaluation_spine","paper_module.engine_room_demo","paper_module.agent_memory_temporal_conflict_replay","paper_module.agent_monitor_redteam_falsification_replay","paper_module.agent_sabotage_scheming_monitor_replay","paper_module.agent_sandbox_policy_escape_replay","paper_module.belief_state_process_reward_replay","paper_module.sleeper_memory_poisoning_quarantine_replay","paper_module.indirect_prompt_injection_information_flow_policy_replay","paper_module.mcp_tool_authority_replay","paper_module.tactic_portfolio_availability","paper_module.target_shape_tactic_routing","paper_module.ring2_premise_precision_recall","paper_module.mathematical_strategy_atlas","paper_module.verifier_lab_execution_spine","paper_module.bounded_autonomy_campaign_packet","paper_module.computer_use_action_trace_replay","paper_module.concurrency_mission_control","paper_module.doctrine_fact_claim_audit","paper_module.self_ignorance_coverage_ledger","paper_module.tool_server_pressure_inventory","paper_module.mechanistic_interpretability_circuit_attribution_replay","paper_module.spatial_world_model_counterfactual_simulation_replay","paper_module.prediction_oracle_reconciliation","paper_module.provider_context_recipe_budget","paper_module.undeclared_library_prior_classifier","paper_module.voice_to_doctrine_self_improvement_loop","paper_module.routing_anti_patterns_registry","paper_module.batch8_audio_level_rms_port","paper_module.batch8_compliance_pipeline_capsule","paper_module.batch8_policy_engines_capsule","paper_module.batch8_structural_theses_capsule","paper_module.batch8_tools_tail_primitives_capsule","paper_module.batch8_validator_checker_capsule","paper_module.batch12_market_dashboard_read_model_capsule","paper_module.batch12_prediction_market_board_capsule","paper_module.batch12_release_claim_language_gate","paper_module.batch10_cold_eval_honesty_capsule","paper_module.batch10_live_source_drift_capsule","paper_module.batch7_macro_engines_capsule","paper_module.batch9_macro_engines_capsule","paper_module.pattern_assimilation","paper_module.batch10_governance_compilers_capsule","paper_module.batch11_saturation_engines_capsule","paper_module.batch4_proof_authority_runtime","paper_module.batch6_unsurfaced_primitives_capsule","paper_module.engine_room_public_projection_leak_gate","paper_module.batch5_authority_systems_capsule","paper_module.batch7_oracle_sibling_capsule","paper_module.batch7_demo_take_console_capsule","paper_module.engine_room_generated_projection_drift_gate","paper_module.engine_room_command_run_singleflight","paper_module.engine_room_metabolism_runtime","paper_module.engine_room_bridge_campaign_dag","paper_module.engine_room_annex_knowledge_router","paper_module.engine_room_derived_fact_provider_engine","paper_module.engine_room_egress_self_compliance_gate","paper_module.engine_room_lean_proof_search_lab","paper_module.engine_room_navigation_fitness_benchmark","paper_module.cold_clone_probe","paper_module.first_screen_composition_root","paper_module.batch7_secondary_runtime_capsule","paper_module.microcosm_axiom_substrate","paper_module.semantic_singleflight_dedup_runtime","paper_module.finite_erdos_denominator_certificate_strike","paper_module.bridge_campaign_dag_validation","paper_module.metabolism_queue_reconciliation","paper_module.derived_fact_provider_runtime","paper_module.navigation_fitness_benchmark","paper_module.annex_knowledge_routing","paper_module.egress_self_compliance_audit","paper_module.generated_projection_drift_runtime","paper_module.lean_proof_search_lab_runtime"],"family_id":null},"body_role":"index_page_atoms_carried_once","lede":"Set 10 Cold Eval Honesty Bundle. Set 10 Cold Eval Honesty Bundle runs cold-eval over public fixtures without benchmark, navigation-truth, or launch-scope decision. The full paper-module reader text is carried once in visible_text_index.logical_records.paper_modules; diagrams are carried as compressed diagram assets.","renders_note":"Atoms shown on this page are carried once in this file; resolve the content_refs ids against components[] / paper_modules[] / doctrine_entities[] / evidence_profiles[]. Glossary terms are the glossary page's atoms in public_glossary[]."},{"id":"page:paper-module-batch10-live-source-drift-capsule","path":"docs/paper-module-batch10-live-source-drift-capsule.html","relative_href":"docs/paper-module-batch10-live-source-drift-capsule.html","title":"Set 10 Live Source Drift Bundle","nav_group":"Reference","nav_label":"Paper modules","source_refs":["microcosm-substrate/core/paper_module_capsules.json"],"content_refs":{"paper_module_refs":["paper_module.verifier_lab_kernel","paper_module.navigation_hologram_route_plane","paper_module.agent_route_observability_runtime","paper_module.agent_benchmark_integrity_anti_gaming_replay","paper_module.research_replication_rubric_artifact_replay","paper_module.agentic_vulnerability_discovery_patch_proof_replay","paper_module.materials_chemistry_closed_loop_lab_safety_replay","paper_module.certificate_kernel_execution_lab","paper_module.corpus_readiness_mathlib_absence_gate","paper_module.pattern_binding_contract","paper_module.bridge_phase_continuity_runtime","paper_module.cognitive_operator_registry","paper_module.agent_closeout_faithfulness_audit","paper_module.cold_reader_route_map","paper_module.proof_diagnostic_evidence_spine","paper_module.proof_derived_governed_mutation_authorization","paper_module.durable_agent_work_landing_replay","paper_module.work_landing_control_spine","paper_module.executable_doctrine_grammar","paper_module.macro_projection_import_protocol","paper_module.mission_transaction_work_spine","paper_module.formal_math_readiness_gate","paper_module.formal_math_lean_proof_witness","paper_module.formal_math_verifier_trace_repair_loop","paper_module.formal_evidence_cell_anchor_resolver","paper_module.formal_math_premise_retrieval","paper_module.lean_std_premise_index","paper_module.world_model_projection_drift_control_room","paper_module.public_reveal_walkthrough","paper_module.standards_meta_diagnostics","paper_module.finance_forecast_evaluation_spine","paper_module.engine_room_demo","paper_module.agent_memory_temporal_conflict_replay","paper_module.agent_monitor_redteam_falsification_replay","paper_module.agent_sabotage_scheming_monitor_replay","paper_module.agent_sandbox_policy_escape_replay","paper_module.belief_state_process_reward_replay","paper_module.sleeper_memory_poisoning_quarantine_replay","paper_module.indirect_prompt_injection_information_flow_policy_replay","paper_module.mcp_tool_authority_replay","paper_module.tactic_portfolio_availability","paper_module.target_shape_tactic_routing","paper_module.ring2_premise_precision_recall","paper_module.mathematical_strategy_atlas","paper_module.verifier_lab_execution_spine","paper_module.bounded_autonomy_campaign_packet","paper_module.computer_use_action_trace_replay","paper_module.concurrency_mission_control","paper_module.doctrine_fact_claim_audit","paper_module.self_ignorance_coverage_ledger","paper_module.tool_server_pressure_inventory","paper_module.mechanistic_interpretability_circuit_attribution_replay","paper_module.spatial_world_model_counterfactual_simulation_replay","paper_module.prediction_oracle_reconciliation","paper_module.provider_context_recipe_budget","paper_module.undeclared_library_prior_classifier","paper_module.voice_to_doctrine_self_improvement_loop","paper_module.routing_anti_patterns_registry","paper_module.batch8_audio_level_rms_port","paper_module.batch8_compliance_pipeline_capsule","paper_module.batch8_policy_engines_capsule","paper_module.batch8_structural_theses_capsule","paper_module.batch8_tools_tail_primitives_capsule","paper_module.batch8_validator_checker_capsule","paper_module.batch12_market_dashboard_read_model_capsule","paper_module.batch12_prediction_market_board_capsule","paper_module.batch12_release_claim_language_gate","paper_module.batch10_cold_eval_honesty_capsule","paper_module.batch10_live_source_drift_capsule","paper_module.batch7_macro_engines_capsule","paper_module.batch9_macro_engines_capsule","paper_module.pattern_assimilation","paper_module.batch10_governance_compilers_capsule","paper_module.batch11_saturation_engines_capsule","paper_module.batch4_proof_authority_runtime","paper_module.batch6_unsurfaced_primitives_capsule","paper_module.engine_room_public_projection_leak_gate","paper_module.batch5_authority_systems_capsule","paper_module.batch7_oracle_sibling_capsule","paper_module.batch7_demo_take_console_capsule","paper_module.engine_room_generated_projection_drift_gate","paper_module.engine_room_command_run_singleflight","paper_module.engine_room_metabolism_runtime","paper_module.engine_room_bridge_campaign_dag","paper_module.engine_room_annex_knowledge_router","paper_module.engine_room_derived_fact_provider_engine","paper_module.engine_room_egress_self_compliance_gate","paper_module.engine_room_lean_proof_search_lab","paper_module.engine_room_navigation_fitness_benchmark","paper_module.cold_clone_probe","paper_module.first_screen_composition_root","paper_module.batch7_secondary_runtime_capsule","paper_module.microcosm_axiom_substrate","paper_module.semantic_singleflight_dedup_runtime","paper_module.finite_erdos_denominator_certificate_strike","paper_module.bridge_campaign_dag_validation","paper_module.metabolism_queue_reconciliation","paper_module.derived_fact_provider_runtime","paper_module.navigation_fitness_benchmark","paper_module.annex_knowledge_routing","paper_module.egress_self_compliance_audit","paper_module.generated_projection_drift_runtime","paper_module.lean_proof_search_lab_runtime"],"family_id":null},"body_role":"index_page_atoms_carried_once","lede":"Set 10 Live Source Drift Bundle. Set 10 Live Source Drift Bundle validates copied current internal control source bodies without route, ledger-mutation, or launch-scope decision. The full paper-module reader text is carried once in visible_text_index.logical_records.paper_modules; diagrams are carried as compressed diagram assets.","renders_note":"Atoms shown on this page are carried once in this file; resolve the content_refs ids against components[] / paper_modules[] / doctrine_entities[] / evidence_profiles[]. Glossary terms are the glossary page's atoms in public_glossary[]."},{"id":"page:paper-module-batch7-macro-engines-capsule","path":"docs/paper-module-batch7-macro-engines-capsule.html","relative_href":"docs/paper-module-batch7-macro-engines-capsule.html","title":"Set 7 Source Engines Bundle","nav_group":"Reference","nav_label":"Paper modules","source_refs":["microcosm-substrate/core/paper_module_capsules.json"],"content_refs":{"paper_module_refs":["paper_module.verifier_lab_kernel","paper_module.navigation_hologram_route_plane","paper_module.agent_route_observability_runtime","paper_module.agent_benchmark_integrity_anti_gaming_replay","paper_module.research_replication_rubric_artifact_replay","paper_module.agentic_vulnerability_discovery_patch_proof_replay","paper_module.materials_chemistry_closed_loop_lab_safety_replay","paper_module.certificate_kernel_execution_lab","paper_module.corpus_readiness_mathlib_absence_gate","paper_module.pattern_binding_contract","paper_module.bridge_phase_continuity_runtime","paper_module.cognitive_operator_registry","paper_module.agent_closeout_faithfulness_audit","paper_module.cold_reader_route_map","paper_module.proof_diagnostic_evidence_spine","paper_module.proof_derived_governed_mutation_authorization","paper_module.durable_agent_work_landing_replay","paper_module.work_landing_control_spine","paper_module.executable_doctrine_grammar","paper_module.macro_projection_import_protocol","paper_module.mission_transaction_work_spine","paper_module.formal_math_readiness_gate","paper_module.formal_math_lean_proof_witness","paper_module.formal_math_verifier_trace_repair_loop","paper_module.formal_evidence_cell_anchor_resolver","paper_module.formal_math_premise_retrieval","paper_module.lean_std_premise_index","paper_module.world_model_projection_drift_control_room","paper_module.public_reveal_walkthrough","paper_module.standards_meta_diagnostics","paper_module.finance_forecast_evaluation_spine","paper_module.engine_room_demo","paper_module.agent_memory_temporal_conflict_replay","paper_module.agent_monitor_redteam_falsification_replay","paper_module.agent_sabotage_scheming_monitor_replay","paper_module.agent_sandbox_policy_escape_replay","paper_module.belief_state_process_reward_replay","paper_module.sleeper_memory_poisoning_quarantine_replay","paper_module.indirect_prompt_injection_information_flow_policy_replay","paper_module.mcp_tool_authority_replay","paper_module.tactic_portfolio_availability","paper_module.target_shape_tactic_routing","paper_module.ring2_premise_precision_recall","paper_module.mathematical_strategy_atlas","paper_module.verifier_lab_execution_spine","paper_module.bounded_autonomy_campaign_packet","paper_module.computer_use_action_trace_replay","paper_module.concurrency_mission_control","paper_module.doctrine_fact_claim_audit","paper_module.self_ignorance_coverage_ledger","paper_module.tool_server_pressure_inventory","paper_module.mechanistic_interpretability_circuit_attribution_replay","paper_module.spatial_world_model_counterfactual_simulation_replay","paper_module.prediction_oracle_reconciliation","paper_module.provider_context_recipe_budget","paper_module.undeclared_library_prior_classifier","paper_module.voice_to_doctrine_self_improvement_loop","paper_module.routing_anti_patterns_registry","paper_module.batch8_audio_level_rms_port","paper_module.batch8_compliance_pipeline_capsule","paper_module.batch8_policy_engines_capsule","paper_module.batch8_structural_theses_capsule","paper_module.batch8_tools_tail_primitives_capsule","paper_module.batch8_validator_checker_capsule","paper_module.batch12_market_dashboard_read_model_capsule","paper_module.batch12_prediction_market_board_capsule","paper_module.batch12_release_claim_language_gate","paper_module.batch10_cold_eval_honesty_capsule","paper_module.batch10_live_source_drift_capsule","paper_module.batch7_macro_engines_capsule","paper_module.batch9_macro_engines_capsule","paper_module.pattern_assimilation","paper_module.batch10_governance_compilers_capsule","paper_module.batch11_saturation_engines_capsule","paper_module.batch4_proof_authority_runtime","paper_module.batch6_unsurfaced_primitives_capsule","paper_module.engine_room_public_projection_leak_gate","paper_module.batch5_authority_systems_capsule","paper_module.batch7_oracle_sibling_capsule","paper_module.batch7_demo_take_console_capsule","paper_module.engine_room_generated_projection_drift_gate","paper_module.engine_room_command_run_singleflight","paper_module.engine_room_metabolism_runtime","paper_module.engine_room_bridge_campaign_dag","paper_module.engine_room_annex_knowledge_router","paper_module.engine_room_derived_fact_provider_engine","paper_module.engine_room_egress_self_compliance_gate","paper_module.engine_room_lean_proof_search_lab","paper_module.engine_room_navigation_fitness_benchmark","paper_module.cold_clone_probe","paper_module.first_screen_composition_root","paper_module.batch7_secondary_runtime_capsule","paper_module.microcosm_axiom_substrate","paper_module.semantic_singleflight_dedup_runtime","paper_module.finite_erdos_denominator_certificate_strike","paper_module.bridge_campaign_dag_validation","paper_module.metabolism_queue_reconciliation","paper_module.derived_fact_provider_runtime","paper_module.navigation_fitness_benchmark","paper_module.annex_knowledge_routing","paper_module.egress_self_compliance_audit","paper_module.generated_projection_drift_runtime","paper_module.lean_proof_search_lab_runtime"],"family_id":null},"body_role":"index_page_atoms_carried_once","lede":"Set 7 Source Engines Bundle. Set 7 Source Engines Bundle imports source engine bodies and exercises trace, graph, scheduling, source-index, patch, numeric, rank, and regression-selection invariants. The full paper-module reader text is carried once in visible_text_index.logical_records.paper_modules; diagrams are carried as compressed diagram assets.","renders_note":"Atoms shown on this page are carried once in this file; resolve the content_refs ids against components[] / paper_modules[] / doctrine_entities[] / evidence_profiles[]. Glossary terms are the glossary page's atoms in public_glossary[]."},{"id":"page:paper-module-batch9-macro-engines-capsule","path":"docs/paper-module-batch9-macro-engines-capsule.html","relative_href":"docs/paper-module-batch9-macro-engines-capsule.html","title":"Set 9 Source Engines Bundle","nav_group":"Reference","nav_label":"Paper modules","source_refs":["microcosm-substrate/core/paper_module_capsules.json"],"content_refs":{"paper_module_refs":["paper_module.verifier_lab_kernel","paper_module.navigation_hologram_route_plane","paper_module.agent_route_observability_runtime","paper_module.agent_benchmark_integrity_anti_gaming_replay","paper_module.research_replication_rubric_artifact_replay","paper_module.agentic_vulnerability_discovery_patch_proof_replay","paper_module.materials_chemistry_closed_loop_lab_safety_replay","paper_module.certificate_kernel_execution_lab","paper_module.corpus_readiness_mathlib_absence_gate","paper_module.pattern_binding_contract","paper_module.bridge_phase_continuity_runtime","paper_module.cognitive_operator_registry","paper_module.agent_closeout_faithfulness_audit","paper_module.cold_reader_route_map","paper_module.proof_diagnostic_evidence_spine","paper_module.proof_derived_governed_mutation_authorization","paper_module.durable_agent_work_landing_replay","paper_module.work_landing_control_spine","paper_module.executable_doctrine_grammar","paper_module.macro_projection_import_protocol","paper_module.mission_transaction_work_spine","paper_module.formal_math_readiness_gate","paper_module.formal_math_lean_proof_witness","paper_module.formal_math_verifier_trace_repair_loop","paper_module.formal_evidence_cell_anchor_resolver","paper_module.formal_math_premise_retrieval","paper_module.lean_std_premise_index","paper_module.world_model_projection_drift_control_room","paper_module.public_reveal_walkthrough","paper_module.standards_meta_diagnostics","paper_module.finance_forecast_evaluation_spine","paper_module.engine_room_demo","paper_module.agent_memory_temporal_conflict_replay","paper_module.agent_monitor_redteam_falsification_replay","paper_module.agent_sabotage_scheming_monitor_replay","paper_module.agent_sandbox_policy_escape_replay","paper_module.belief_state_process_reward_replay","paper_module.sleeper_memory_poisoning_quarantine_replay","paper_module.indirect_prompt_injection_information_flow_policy_replay","paper_module.mcp_tool_authority_replay","paper_module.tactic_portfolio_availability","paper_module.target_shape_tactic_routing","paper_module.ring2_premise_precision_recall","paper_module.mathematical_strategy_atlas","paper_module.verifier_lab_execution_spine","paper_module.bounded_autonomy_campaign_packet","paper_module.computer_use_action_trace_replay","paper_module.concurrency_mission_control","paper_module.doctrine_fact_claim_audit","paper_module.self_ignorance_coverage_ledger","paper_module.tool_server_pressure_inventory","paper_module.mechanistic_interpretability_circuit_attribution_replay","paper_module.spatial_world_model_counterfactual_simulation_replay","paper_module.prediction_oracle_reconciliation","paper_module.provider_context_recipe_budget","paper_module.undeclared_library_prior_classifier","paper_module.voice_to_doctrine_self_improvement_loop","paper_module.routing_anti_patterns_registry","paper_module.batch8_audio_level_rms_port","paper_module.batch8_compliance_pipeline_capsule","paper_module.batch8_policy_engines_capsule","paper_module.batch8_structural_theses_capsule","paper_module.batch8_tools_tail_primitives_capsule","paper_module.batch8_validator_checker_capsule","paper_module.batch12_market_dashboard_read_model_capsule","paper_module.batch12_prediction_market_board_capsule","paper_module.batch12_release_claim_language_gate","paper_module.batch10_cold_eval_honesty_capsule","paper_module.batch10_live_source_drift_capsule","paper_module.batch7_macro_engines_capsule","paper_module.batch9_macro_engines_capsule","paper_module.pattern_assimilation","paper_module.batch10_governance_compilers_capsule","paper_module.batch11_saturation_engines_capsule","paper_module.batch4_proof_authority_runtime","paper_module.batch6_unsurfaced_primitives_capsule","paper_module.engine_room_public_projection_leak_gate","paper_module.batch5_authority_systems_capsule","paper_module.batch7_oracle_sibling_capsule","paper_module.batch7_demo_take_console_capsule","paper_module.engine_room_generated_projection_drift_gate","paper_module.engine_room_command_run_singleflight","paper_module.engine_room_metabolism_runtime","paper_module.engine_room_bridge_campaign_dag","paper_module.engine_room_annex_knowledge_router","paper_module.engine_room_derived_fact_provider_engine","paper_module.engine_room_egress_self_compliance_gate","paper_module.engine_room_lean_proof_search_lab","paper_module.engine_room_navigation_fitness_benchmark","paper_module.cold_clone_probe","paper_module.first_screen_composition_root","paper_module.batch7_secondary_runtime_capsule","paper_module.microcosm_axiom_substrate","paper_module.semantic_singleflight_dedup_runtime","paper_module.finite_erdos_denominator_certificate_strike","paper_module.bridge_campaign_dag_validation","paper_module.metabolism_queue_reconciliation","paper_module.derived_fact_provider_runtime","paper_module.navigation_fitness_benchmark","paper_module.annex_knowledge_routing","paper_module.egress_self_compliance_audit","paper_module.generated_projection_drift_runtime","paper_module.lean_proof_search_lab_runtime"],"family_id":null},"body_role":"index_page_atoms_carried_once","lede":"Set 9 Source Engines Bundle. Set 9 Source Engines Bundle imports backend and governance source-engine bodies and exercises provenance, approval, AST, finance-news, mission graph, dependency, config, edge, WorkAtlas, host-pressure, doctrine-enrichment, worker-budget, and milestone-quality invariants. The full paper-module reader text is carried once in visible_text_index.logical_records.paper_modules; diagrams are carried as compressed diagram assets.","renders_note":"Atoms shown on this page are carried once in this file; resolve the content_refs ids against components[] / paper_modules[] / doctrine_entities[] / evidence_profiles[]. Glossary terms are the glossary page's atoms in public_glossary[]."},{"id":"page:paper-module-pattern-assimilation","path":"docs/paper-module-pattern-assimilation.html","relative_href":"docs/paper-module-pattern-assimilation.html","title":"Pattern Assimilation","nav_group":"Reference","nav_label":"Paper modules","source_refs":["microcosm-substrate/core/paper_module_capsules.json"],"content_refs":{"paper_module_refs":["paper_module.verifier_lab_kernel","paper_module.navigation_hologram_route_plane","paper_module.agent_route_observability_runtime","paper_module.agent_benchmark_integrity_anti_gaming_replay","paper_module.research_replication_rubric_artifact_replay","paper_module.agentic_vulnerability_discovery_patch_proof_replay","paper_module.materials_chemistry_closed_loop_lab_safety_replay","paper_module.certificate_kernel_execution_lab","paper_module.corpus_readiness_mathlib_absence_gate","paper_module.pattern_binding_contract","paper_module.bridge_phase_continuity_runtime","paper_module.cognitive_operator_registry","paper_module.agent_closeout_faithfulness_audit","paper_module.cold_reader_route_map","paper_module.proof_diagnostic_evidence_spine","paper_module.proof_derived_governed_mutation_authorization","paper_module.durable_agent_work_landing_replay","paper_module.work_landing_control_spine","paper_module.executable_doctrine_grammar","paper_module.macro_projection_import_protocol","paper_module.mission_transaction_work_spine","paper_module.formal_math_readiness_gate","paper_module.formal_math_lean_proof_witness","paper_module.formal_math_verifier_trace_repair_loop","paper_module.formal_evidence_cell_anchor_resolver","paper_module.formal_math_premise_retrieval","paper_module.lean_std_premise_index","paper_module.world_model_projection_drift_control_room","paper_module.public_reveal_walkthrough","paper_module.standards_meta_diagnostics","paper_module.finance_forecast_evaluation_spine","paper_module.engine_room_demo","paper_module.agent_memory_temporal_conflict_replay","paper_module.agent_monitor_redteam_falsification_replay","paper_module.agent_sabotage_scheming_monitor_replay","paper_module.agent_sandbox_policy_escape_replay","paper_module.belief_state_process_reward_replay","paper_module.sleeper_memory_poisoning_quarantine_replay","paper_module.indirect_prompt_injection_information_flow_policy_replay","paper_module.mcp_tool_authority_replay","paper_module.tactic_portfolio_availability","paper_module.target_shape_tactic_routing","paper_module.ring2_premise_precision_recall","paper_module.mathematical_strategy_atlas","paper_module.verifier_lab_execution_spine","paper_module.bounded_autonomy_campaign_packet","paper_module.computer_use_action_trace_replay","paper_module.concurrency_mission_control","paper_module.doctrine_fact_claim_audit","paper_module.self_ignorance_coverage_ledger","paper_module.tool_server_pressure_inventory","paper_module.mechanistic_interpretability_circuit_attribution_replay","paper_module.spatial_world_model_counterfactual_simulation_replay","paper_module.prediction_oracle_reconciliation","paper_module.provider_context_recipe_budget","paper_module.undeclared_library_prior_classifier","paper_module.voice_to_doctrine_self_improvement_loop","paper_module.routing_anti_patterns_registry","paper_module.batch8_audio_level_rms_port","paper_module.batch8_compliance_pipeline_capsule","paper_module.batch8_policy_engines_capsule","paper_module.batch8_structural_theses_capsule","paper_module.batch8_tools_tail_primitives_capsule","paper_module.batch8_validator_checker_capsule","paper_module.batch12_market_dashboard_read_model_capsule","paper_module.batch12_prediction_market_board_capsule","paper_module.batch12_release_claim_language_gate","paper_module.batch10_cold_eval_honesty_capsule","paper_module.batch10_live_source_drift_capsule","paper_module.batch7_macro_engines_capsule","paper_module.batch9_macro_engines_capsule","paper_module.pattern_assimilation","paper_module.batch10_governance_compilers_capsule","paper_module.batch11_saturation_engines_capsule","paper_module.batch4_proof_authority_runtime","paper_module.batch6_unsurfaced_primitives_capsule","paper_module.engine_room_public_projection_leak_gate","paper_module.batch5_authority_systems_capsule","paper_module.batch7_oracle_sibling_capsule","paper_module.batch7_demo_take_console_capsule","paper_module.engine_room_generated_projection_drift_gate","paper_module.engine_room_command_run_singleflight","paper_module.engine_room_metabolism_runtime","paper_module.engine_room_bridge_campaign_dag","paper_module.engine_room_annex_knowledge_router","paper_module.engine_room_derived_fact_provider_engine","paper_module.engine_room_egress_self_compliance_gate","paper_module.engine_room_lean_proof_search_lab","paper_module.engine_room_navigation_fitness_benchmark","paper_module.cold_clone_probe","paper_module.first_screen_composition_root","paper_module.batch7_secondary_runtime_capsule","paper_module.microcosm_axiom_substrate","paper_module.semantic_singleflight_dedup_runtime","paper_module.finite_erdos_denominator_certificate_strike","paper_module.bridge_campaign_dag_validation","paper_module.metabolism_queue_reconciliation","paper_module.derived_fact_provider_runtime","paper_module.navigation_fitness_benchmark","paper_module.annex_knowledge_routing","paper_module.egress_self_compliance_audit","paper_module.generated_projection_drift_runtime","paper_module.lean_proof_search_lab_runtime"],"family_id":null},"body_role":"index_page_atoms_carried_once","lede":"Pattern Assimilation. Pattern Assimilation validates public completion-learning metadata, owner-routed refinement result records, typed nothing-to-refine decisions, and copied body-import manifests without promoting local lessons into global doctrine. The full paper-module reader text is carried once in visible_text_index.logical_records.paper_modules; diagrams are carried as compressed diagram assets.","renders_note":"Atoms shown on this page are carried once in this file; resolve the content_refs ids against components[] / paper_modules[] / doctrine_entities[] / evidence_profiles[]. 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Room Public Projection Leak Gate. 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aper_module.egress_self_compliance_audit","paper_module.generated_projection_drift_runtime","paper_module.lean_proof_search_lab_runtime"],"family_id":null},"body_role":"index_page_atoms_carried_once","lede":"Set 5 Authority and Systems Bundle. Set 5 Authority and Systems Bundle imports post-execution authority, replay, proof-repair, process, generated-state, trace, blast-radius, and doctrine-graph source bodies as public source-open evidence without claiming live authority or launch-scope decision. The full paper-module reader text is carried once in visible_text_index.logical_records.paper_modules; diagrams are carried as compressed diagram assets.","renders_note":"Atoms shown on this page are carried once in this file; resolve the content_refs ids against components[] / paper_modules[] / doctrine_entities[] / evidence_profiles[]. 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aper_module.egress_self_compliance_audit","paper_module.generated_projection_drift_runtime","paper_module.lean_proof_search_lab_runtime"],"family_id":null},"body_role":"index_page_atoms_carried_once","lede":"Set 7 Oracle Sibling Bundle. Set 7 Oracle Sibling Bundle imports Oracle sibling source bodies and exercises deterministic subject-index, snapshot, truth-diff, quartet-plan, and original pytest witness boundaries. The full paper-module reader text is carried once in visible_text_index.logical_records.paper_modules; diagrams are carried as compressed diagram assets.","renders_note":"Atoms shown on this page are carried once in this file; resolve the content_refs ids against components[] / paper_modules[] / doctrine_entities[] / evidence_profiles[]. 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aper_module.egress_self_compliance_audit","paper_module.generated_projection_drift_runtime","paper_module.lean_proof_search_lab_runtime"],"family_id":null},"body_role":"index_page_atoms_carried_once","lede":"Set 7 Demo Take Console Bundle. 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7 Secondary Runtime Bundle. 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Erdos Denominator-Order Certificate Strike. The finite Erdos denominator-order certificate strike surfaces the public finite_denominator_order_certificate bundle and verifies, in exact rational arithmetic, that ord_Q(b)=lcm(F) for S_F(b)=sum 1/(b^n-1)=P/Q over bounded public fixtures. The full paper-module reader text is carried once in visible_text_index.logical_records.paper_modules; diagrams are carried as compressed diagram assets.","renders_note":"Atoms shown on this page are carried once in this file; resolve the content_refs ids against components[] / paper_modules[] / doctrine_entities[] / evidence_profiles[]. 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Campaign DAG Validation. 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per_module.formal_math_premise_retrieval","evidence_profile:paper_module:paper_module.formal_math_readiness_gate","evidence_profile:paper_module:paper_module.formal_math_verifier_trace_repair_loop","evidence_profile:paper_module:paper_module.generated_projection_drift_runtime","evidence_profile:paper_module:paper_module.indirect_prompt_injection_information_flow_policy_replay","evidence_profile:paper_module:paper_module.lean_proof_search_lab_runtime","evidence_profile:paper_module:paper_module.lean_std_premise_index","evidence_profile:paper_module:paper_module.macro_projection_import_protocol","evidence_profile:paper_module:paper_module.materials_chemistry_closed_loop_lab_safety_replay","evidence_profile:paper_module:paper_module.mathematical_strategy_atlas","evidence_profile:paper_module:paper_module.mcp_tool_authority_replay","evidence_profile:paper_module:paper_module.mechanistic_interpretability_circuit_attribution_replay","evidence_profile:paper_module:paper_module.metabolism_queue_reconciliation","evidence_profile:paper_module:paper_module.microcosm_axiom_substrate","evidence_profile:paper_module:paper_module.mission_transaction_work_spine","evidence_profile:paper_module:paper_module.navigation_fitness_benchmark","evidence_profile:paper_module:paper_module.navigation_hologram_route_plane","evidence_profile:paper_module:paper_module.pattern_assimilation","evidence_profile:paper_module:paper_module.pattern_binding_contract","evidence_profile:paper_module:paper_module.prediction_oracle_reconciliation","evidence_profile:paper_module:paper_module.proof_derived_governed_mutation_authorization","evidence_profile:paper_module:paper_module.proof_diagnostic_evidence_spine","evidence_profile:paper_module:paper_module.provider_context_recipe_budget","evidence_profile:paper_module:paper_module.public_reveal_walkthrough","evidence_profile:paper_module:paper_module.research_replication_rubric_artifact_replay","evidence_profile:paper_module:paper_module.ring2_premise_precision_recall","evidence_profile:paper_module:paper_module.routing_anti_patterns_registry","evidence_profile:paper_module:paper_module.self_ignorance_coverage_ledger","evidence_profile:paper_module:paper_module.semantic_singleflight_dedup_runtime","evidence_profile:paper_module:paper_module.sleeper_memory_poisoning_quarantine_replay","evidence_profile:paper_module:paper_module.spatial_world_model_counterfactual_simulation_replay","evidence_profile:paper_module:paper_module.standards_meta_diagnostics","evidence_profile:paper_module:paper_module.tactic_portfolio_availability","evidence_profile:paper_module:paper_module.target_shape_tactic_routing","evidence_profile:paper_module:paper_module.tool_server_pressure_inventory","evidence_profile:paper_module:paper_module.undeclared_library_prior_classifier","evidence_profile:paper_module:paper_module.verifier_lab_execution_spine","evidence_profile:paper_module:paper_module.verifier_lab_kernel","evidence_profile:paper_module:paper_module.voice_to_doctrine_self_improvement_loop","evidence_profile:paper_module:paper_module.work_landing_control_spine","evidence_profile:paper_module:paper_module.world_model_projection_drift_control_room","evidence_profile:principle:P-1","evidence_profile:principle:P-10","evidence_profile:principle:P-11","evidence_profile:principle:P-12","evidence_profile:principle:P-13","evidence_profile:principle:P-14","evidence_profile:principle:P-15","evidence_profile:principle:P-16","evidence_profile:principle:P-17","evidence_profile:principle:P-18","evidence_profile:principle:P-19","evidence_profile:principle:P-2","evidence_profile:principle:P-20","evidence_profile:principle:P-3","evidence_profile:principle:P-4","evidence_profile:principle:P-5","evidence_profile:principle:P-6","evidence_profile:principle:P-7","evidence_profile:principle:P-8","evidence_profile:principle:P-9"],"family_id":null},"body_role":"index_page_atoms_carried_once","lede":"Evaluate\n\n### Evidence & authority\n\nThe 1-5 rank measures verdict independence: how much the check can fail on its own, from a 1 where the fixture hands over the answer to a 5 where the harness derives the verdict across the whole declared contract and can fail with no fixture echo. It does not measure how heavy the machinery is, how much real tooling runs, or how mature the component is.","renders_note":"Atoms shown on this page are carried once in this file; resolve the content_refs ids against components[] / paper_modules[] / doctrine_entities[] / evidence_profiles[]. Glossary terms are the glossary page's atoms in public_glossary[]."},{"id":"page:glossary","path":"docs/glossary.html","relative_href":"docs/glossary.html","title":"Glossary","nav_group":"Reference","nav_label":"Glossary","source_refs":["codex/doctrine/system_vocabulary/term_registry.json"],"content_refs":{"component_refs":[],"family_id":null},"body_role":"index_page_atoms_carried_once","lede":"Reference\n\n### Glossary\n\nThe public term map for Plectis: every active vocabulary row is projected here as an expandable reader card, with its meaning namespace stated explicitly. General or domain usage, externally anchored terms, system-local specializations, system-coined terms, compatibility identifiers, and the project name are not treated as one ontology. These rows are generated from the governed system vocabulary registry.","renders_note":"Atoms shown on this page are carried once in this file; resolve the content_refs ids against components[] / paper_modules[] / doctrine_entities[] / evidence_profiles[]. Glossary terms are the glossary page's atoms in public_glossary[]."},{"id":"page:lineage","path":"docs/lineage.html","relative_href":"docs/lineage.html","title":"Lineage & changelog","nav_group":"Reference","nav_label":"Lineage & changelog","source_refs":["microcosm-substrate/README.md","sites/microcosm/index.html","tools/meta/dissemination/microcosm_public_narratives.json","tools/meta/dissemination/build_microcosm_public_site.py"],"content_refs":{"component_refs":[],"family_id":null},"body_role":"prose_page_full_text","format":"markdown","text":"Reference\n\n### Lineage & changelog\n\nThe public name is Plectis. Microcosm remains only as the historical name of this project and as a technical compatibility label for old routes, package ids, generated state, and published links.\n\n#### 21 June 2026: Microcosm became Plectis\n\nA closer check against the history of hypertext systems showed that Microcosm was already the name of the Southampton open hypermedia system associated with Wendy Hall and colleagues, with public Southampton material placing Wendy Hall's Microcosm work in the mid-1980s and Hugh Davis's Southampton profile describing early open hypermedia versions in 1989. This project was renamed Plectis to avoid confusion and to acknowledge that earlier lineage.\n\nThis acknowledgement does not imply endorsement, affiliation, or review by Wendy Hall, the University of Southampton, W3C, or anyone connected with the earlier Microcosm system. It is a naming and lineage note, not a claim of institutional backing.\n\n#### Compatibility\n\nTechnical identifiers required for compatibility have not been renamed merely for branding. The repository path, package ids, `.microcosm/` local state, historical source references, old JSON packets, and previously published links may still contain Microcosm where changing them would break inspection or route continuity.\n\nThe preferred current public identity is Plectis. Microcosm is a historical and compatibility label, never a second active brand.\n\n#### Acknowledgements\n\nThis section thanks people whose feedback changed the public site or the system behind it. It records gratitude, not endorsement, affiliation, or review.\n\nMark Bernstein of Eastgate Systems looked at an early version of this site. Two of his suggestions are reflected here directly: that the name Microcosm collided with the earlier Southampton hypertext system, which led to the rename above, and that the unusual vocabulary needed a glossary, which is the [glossary](docs/glossary.html). Credited with his permission.\n\n#### Compatibility matrix\n\nThe generated legacy manifest currently declares 247 old-route compatibility rows: 24 byte-compatible machine aliases and 219 human HTML transition routes. Old JSON packets, download files, generated assets, and projection-status routes remain machine-readable files. Only old human HTML pages use visible transition HTML; validation parses JSON routes and fails if they return HTML.\n\nMachine-readable aliases **24 routes**\nMachine-readable JSON and static assets are emitted as files so the old URL keeps its content type.\nContent types: application/atom+xml; charset=utf-8, application/javascript; charset=utf-8, application/json; charset=utf-8, text/css; charset=utf-8\nSamples: `content-graph.json`, `content-manifest.json`, `object-map.json`, `microcosm-site-map.json`, `microcosm-ai-reader-digest.json`, `microcosm-ai-reader-complete.json` Legacy projection status **1 routes**\nThe old projection-status URL returns a legacy JSON status pointer, not an HTML redirect.\nContent types: application/json; charset=utf-8\nSamples: `projection-status.json` Legacy security.txt **2 routes**\nThe old host serves its own security.txt so vulnerability-reporting metadata remains valid while the human-facing site is a tombstone.\nContent types: text/plain; charset=utf-8\nSamples: `.well-known/security.txt`, `security.txt` Human HTML transitions **219 routes**\nHTML routes show the transition and may forward to the canonical route; they are not used for machine-readable assets.\nContent types: text/html; charset=utf-8\nSamples: `index.html`, `docs/index.html`, `docs/tour.html`, `docs/open-questions.html`, `docs/quickstart.html`, `docs/coding-agents.html` Legacy 404 fallback **1 routes**\nThe GitHub Pages 404 is a visible compatibility fallback, not a server-side redirect.\nContent types: text/html; charset=utf-8\nSamples: `404.html`\n\n[Open the full legacy route manifest](../legacy-route-manifest.json) for the per-path compatibility record.\n\n#### Historical sources\n\nThe historical statement above is source-backed by public Southampton and W3C pages. These links are citations for the naming acknowledgement; they are not affiliation, endorsement, or review claims.\n\n[**University of Southampton public-policy profile for Wendy Hall** Southampton identifies Wendy Hall's Microcosm hypermedia work as mid-1980s work and as a forerunner to the World Wide Web. Open](https://www.southampton.ac.uk/publicpolicy/support-for-policymakers/what-we-offer/taguos/tag-uos-jan.page) [**University of Southampton profile for Hugh Davis** The profile describes early versions of the Microcosm open hypermedia system in 1989 with Wendy Hall, and later publications on Microcosm. Open](https://www.ecs.soton.ac.uk/people/hcd) [**W3C historical Microcosm page** A historical W3C page describes Microcosm as a practical hypertext product developed by a team at Southampton University. Open](https://www.w3.org/History/19921103-hypertext/hypertext/Products/Microcosm/Microcosm.html)","byte_count":4988},{"id":"page:updates","path":"docs/updates.html","relative_href":"docs/updates.html","title":"Follow updates","nav_group":"Site","nav_label":"Follow updates","source_refs":["tools/meta/dissemination/microcosm_public_narratives.json","sites/microcosm/index.html"],"content_refs":{"component_refs":[],"family_id":null},"body_role":"prose_page_full_text","format":"markdown","text":"Site\n\n### Follow updates\n\nThere is no mailing list, newsletter provider, signup form, or subscriber database for Plectis at the moment. Follow public changes through the generated Atom feed, GitHub Watch/ Releases, and repository discussions.\n\n#### Follow updates\n\nThe current zero-cost route is public and inspectable: a generated Atom feed, the repository watch/release surfaces, and repository discussions. There is no newsletter provider, subscriber database, or hidden signup workflow behind this page.\n\n[**Atom feed** Machine-readable public update feed generated from this narrative owner. Open feed](../updates.xml) [**GitHub Watch** Use GitHub's Watch control on the source repository if you want platform-native notifications. Open repo](https://github.com/wcook04/plectis) [**GitHub Releases** launch records if and when public launch artifacts are cut. Open releases](https://github.com/wcook04/plectis/releases) [**Repository discussions** Public questions, critique, or collaboration without a private signup system. Open discussions](https://github.com/wcook04/plectis/discussions)\n\n#### Latest updates\n\n##### Tour framing tightened around one record loop\n\n2026-06-22\n\nThe tour now frames Plectis as a system -like layer over existing source and result records, and points first-time readers to one focused command -card- evidence loop before the whole map.\n\n[Open the source note \u2192](docs/tour.html#what-it-is)\n\n##### Microcosm became Plectis\n\n2026-06-21\n\nPlectis became the preferred public name, with Microcosm retained only as a historical and compatibility label. The acknowledgement is source -backed and does not imply endorsement or affiliation.\n\n[Open the source note \u2192](docs/lineage.html#rename-2026-06-21)","byte_count":1734},{"id":"page:source","path":"docs/source.html","relative_href":"docs/source.html","title":"Source & license","nav_group":"Site","nav_label":"Source & license","source_refs":["microcosm-substrate/README.md","microcosm-substrate/QUICKSTART.md","microcosm-substrate/ARCHITECTURE.md","microcosm-substrate/ORGANS.md","microcosm-substrate/LICENSE","microcosm-substrate/NOTICE","microcosm-substrate/PROVENANCE.md","tools/meta/dissemination/microcosm_public_narratives.json"],"content_refs":{"component_refs":[],"family_id":null},"body_role":"prose_page_full_text","format":"markdown","text":"Site\n\n### Source & license\n\nEverything on this site resolves to a file in the public source repository. These are the first paths into the standalone released slice.\n\n#### Repository\n\n[**Repository root** The source of record for the standalone public slice. Open](https://github.com/wcook04/plectis) [**Plectis package** Runnable local source, public fixtures, and generated records; compatibility package ids still use microcosm. Open](https://github.com/wcook04/plectis) [**README** The entry point and the full component map. Open](https://github.com/wcook04/plectis/blob/main/README.md) [**Quickstart** The one-page cold-clone run path. Open](https://github.com/wcook04/plectis/blob/main/QUICKSTART.md)\n\n#### Maps\n\n[**Architecture map** The system shape and each component's wiring. Open](https://github.com/wcook04/plectis/blob/main/ARCHITECTURE.md) [**Rendered whole-system map** The public map generated from the same source refs. Open](docs/architecture.html#whole-system-map) [**Machine-readable map packet** The generated scene data behind the public architecture map. Open](architecture-graph-scene.json) [**Repository atlas** A source card for every one of the 88 components. Open](https://github.com/wcook04/plectis/blob/main/ORGANS.md)\n\n#### License\n\nReuse follows the repository license and the linked source paths.\n\n[**License** Reuse terms. Open](https://github.com/wcook04/plectis/blob/main/LICENSE) [**Notice** Project notice and attribution posture. Open](https://github.com/wcook04/plectis/blob/main/NOTICE)\n\n#### Provenance\n\nPlectis is an independent, AI-assisted project by Will Cook. The public release artifact is the standalone source repository, not the private working root behind it. Microcosm remains a legacy and technical compatibility name for repo paths, package ids, generated state, and historical links.\n\n[**Provenance** Authorship, source-of-record, no-affiliation, and no-advice boundaries. Open](https://github.com/wcook04/plectis/blob/main/PROVENANCE.md)\n\n#### Acknowledgements\n\nI would like to thank Wouter van Doorn and Thomas Bloom for taking the time to read and respond to an early partial-proof attempt connected to the companion Erd\u0151s #249/#257 project. Their kindness, candour, and useful feedback helped me better distinguish formal reformulation from meaningful mathematical progress and pushed the work towards a more disciplined, evidence-first direction.\n\nThis is a thank-you for their time and feedback; it does not imply endorsement of Plectis or of any mathematical claim.\n\n#### Lineage\n\nMicrocosm became Plectis on 21 June 2026 after a public naming check found an established Southampton hypermedia system with the same name. The acknowledgement is source-backed and does not imply endorsement or affiliation.\n\n[**Lineage & changelog** Rename acknowledgement, historical citations, and compatibility boundary. Open](docs/lineage.html)\n\n#### Updates\n\nPlectis has no mailing list or signup database. Public changes are exposed through the generated update feed and repository routes.\n\n[**Follow updates** Atom feed, repository watch routes, and public discussion links. Open](docs/updates.html) [**Atom feed** Machine-readable public update feed generated from the same narrative owner. Open](../updates.xml)\n\n#### Site contract\n\nThe site is a static projection over public source refs. It has no runtime backend and no browser path back into the private working environment. It is not legal, financial, investment, trading, security, medical, or other professional advice.\n\nThe repository is the source of record. This site is a map over it, and when the two disagree, the repository wins.\n\nEvery page is generated from the source files listed above and the component index is a projection of them. The current content fingerprint is `09a2e6fdf14974d870e0650b36bd0890`; when the source changes, a rebuild regenerates the pages and a drift check fails until the site is current.\n\n#### Projection status\n\nOne generator builds this whole site from the public source files above. The panel says what it was built from; the machine record beside it carries the full detail, including which source fields were too internal to show and were deliberately omitted.\n\nContent fingerprint `09a2e6fdf14974d870e0650b36bd0890` Map revision `gsc_d80fb1ca2b03cf83419c` Components projected 88 Declared links 239 Runtime backend none (static files only, no network calls) [**Projection status record** The full machine detail: source hashes, generated outputs, omitted fields, and the stale-check. Open](../projection-status.json) [**Content graph** The single public-safe data packet every page on this site is a view of. Open](../content-graph.json)","byte_count":4693},{"id":"page:contact","path":"docs/contact.html","relative_href":"docs/contact.html","title":"Contact","nav_group":"Site","nav_label":"Contact","source_refs":["microcosm-substrate/PROVENANCE.md","microcosm-substrate/README.md","tools/meta/dissemination/microcosm_public_narratives.json"],"content_refs":{"component_refs":[],"family_id":null},"body_role":"prose_page_full_text","format":"markdown","text":"Site\n\n### Contact\n\nThe person behind Plectis, and the routes for questions, corrections, and critique.\n\n#### Who built this\n\nI'm Will Cook, an economics undergraduate at the University of Bristol. I built this entirely on my own with AI. AI was not the main tool; it was the only tool. The University of Bristol has no affiliation with this work and had no involvement in it.\n\nThe papers were made the same way, and it would be odd to describe the system honestly while leaving its own write-ups vague. I did not write their prose. Agents drafted and revised it from objectives and review criteria I set, and I selected the claims, reviewed the text, and authorised what was published. Each paper says so on its first page and declares it in full at the back. The byline stays mine because responsibility for the claims does, and that is not something a generative system can hold.\n\nThe public repository is not the whole system. The full version includes live tools, non-public data, and capabilities I did not think it appropriate to put online all at once. Plectis is the smaller standalone slice: source files, component records, evidence classes, stable source addresses, and scope limits. The [evidence page](docs/evidence.html) records which lane each component took, so the gap between what the public slice shows and what the private root contains is written down rather than implied.\n\nThis is a review candidate, not a finished product. It is published so it can be inspected and corrected. If you find mistakes, bad assumptions, or anything unclear, I would genuinely like to know.\n\nBecause I built this alone, the system grew its own private vocabulary along the way. For release I have tried to translate that language into plainer public terms. If an odd word has survived the cleanup, that is leftover translation work, not a term of art; please point it out.\n\nI taught myself everything here, and I have worked without collaborators throughout. If you have the background to evaluate work like this properly, at a frontier lab or elsewhere, I would value a serious look, and I am happy to walk through the wider system through a controlled review route. I have had to work most of this out from first principles, and that is a slow way to find one's own mistakes.\n\n#### Get in touch\n\n**Email** williamwkcook+microcosm@gmail.com, for questions, feedback, or anything you think I have got wrong. Send [**Repository discussions** Public questions, critique, or collaboration. Open](https://github.com/wcook04/plectis/discussions) [**LinkedIn** Will Cook, who builds and maintains Plectis. Open](https://www.linkedin.com/in/will-cook-278127234/) [**Security issues** The private reporting route for anything security-shaped. Open](docs/vulnerability-disclosure.html)\n\n#### Follow updates\n\nThere is no mailing list, newsletter provider, or signup form for Plectis at the moment. Follow public changes through the generated feed or repository surfaces.\n\n[**Follow updates** Generated feed, GitHub Watch route, releases route, and discussions route. Open](docs/updates.html) [**Atom feed** A static machine-readable feed generated from the public narrative owner. Open](../updates.xml)","byte_count":3194},{"id":"page:vulnerability-disclosure","path":"docs/vulnerability-disclosure.html","relative_href":"docs/vulnerability-disclosure.html","title":"Vulnerability disclosure","nav_group":"Site","nav_label":"Vulnerability disclosure","source_refs":["microcosm-substrate/SECURITY.md","microcosm-substrate/README.md"],"content_refs":{"component_refs":[],"family_id":null},"body_role":"prose_page_full_text","format":"markdown","text":"Site\n\n### Vulnerability disclosure\n\nUse this page to report a vulnerability or security issue in the public Plectis website or the standalone public source repository.\n\n#### Scope\n\nIn scope: this static website, its same-origin files, the public Plectis source repository or export, claims about what is and is not published, and public fixture or evidence leakage. Reports are most useful when they name the page, file, command, evidence id, or source path involved.\n\nOut of scope: the private working environment behind Plectis, provider accounts, browser sessions, personal devices, third-party platforms, social engineering, denial-of-service testing, account-secret guessing, and any test that accesses data you do not own.\n\n#### How to report\n\nPreferred private reporting channel: use GitHub's [Report a vulnerability](https://github.com/wcook04/plectis/security/advisories/new) flow on the source repository's [Security Advisories](https://github.com/wcook04/plectis/security/advisories) page.\n\nThe record `docs/dissemination/microcosm_github_private_vulnerability_reporting_receipt_v0.json` is the proof that GitHub Private Vulnerability Reporting is enabled for the public repository. If the GitHub button is not visible, do not open a public issue for vulnerability details.\n\nInclude the affected URL or source path, command or evidence id if relevant, a short redacted description, expected versus observed boundary, and whether the report appears to be a leak, a claim that overstates what is published, hosted-header drift, CI or supply-chain issue, or unsafe exploit-content issue.\n\nDo not paste suspected secrets, non-public account data, private payloads, raw prompts, model payload bodies, session values, or exploit-sensitive details into public channels. No separate security email address is published on this page; use the private GitHub report flow.\n\n#### Expectations\n\nCoordinated disclosure is preferred. Good-faith reports should stay inside the public repository boundary and should not attempt to reach private systems, provider accounts, browser sessions, personal devices, or third-party systems. There is no bug bounty because I cannot currently fund one on a student budget.\n\n#### Boundaries\n\nThe `security.txt` file and this page identify a public policy route, not permission for intrusive testing. Do not run destructive, high-volume, automated, or account-targeting tests. Do not attempt to reach private systems or data. Public release and safety statements stay limited to the generated static files and the checks that have actually run against them; there is no backend service behind the site to probe.\n\nPlectis is a research prototype with a static public site and a private way to report issues. It is not a production security product, a hosted agent service, a bug-bounty program, or a claim of real-world exploit resistance.","byte_count":2869}],"components":[{"id":"cold_reader_route_map","label":"Cold Reader Route Map","family_id":"entry_and_reveal","family_label":"Entry & orientation","relative_href":"docs/component-cold-reader-route-map.html","primary_reader_href":"docs/paper-module-cold-reader-route-map.html","component_detail_href":"docs/component-cold-reader-route-map.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Verifies the first-run guided path so every step names a real command, doc, and evidence.","what_it_does":"It checks Microcosm's \"what do I run first\" guided path so that every step on the cold-reader route map is backed by a real command, a public doc reference, and an evidence result record instead of just prose promises. A newcomer can therefore trust that the suggested ten-minute first-run tour is actually wired and honestly labeled, with small verified counts shown as plain accounting rather than success badges.","scope_limit":"It is projection-only metadata that validates the declared public route contract; it is not route registry control and excludes source-file changes, external model access, launch/public sharing, financial decisions, non-public data equivalence, or whole-system correctness.","evidence":{"kind":"Contract validator","basis":"Import validation","rank":5,"runs_real_tools":false},"command":"microcosm cold-reader-route-map run-route-map-bundle --input examples/cold_reader_route_map/exported_cold_reader_route_map_bundle --out receipts/runtime_shell/demo_project/organs/cold_reader_route_map --card","tags":["agent-entry","getting-started","interesting-parts"],"links_to":[{"component_id":"navigation_hologram_route_plane","label":"Navigation Hologram Route Plane","relative_href":"docs/component-navigation-hologram-route-plane.html"},{"component_id":"agent_route_observability_runtime","label":"Agent Route Observability Runtime","relative_href":"docs/component-agent-route-observability-runtime.html"},{"component_id":"pattern_binding_contract","label":"Pattern Binding Contract","relative_href":"docs/component-pattern-binding-contract.html"},{"component_id":"doctrine_fact_claim_audit","label":"Doctrine Fact Claim Audit","relative_href":"docs/component-doctrine-fact-claim-audit.html"},{"component_id":"pattern_assimilation_step","label":"Pattern Assimilation Step","relative_href":"docs/component-pattern-assimilation-step.html"},{"component_id":"routing_anti_patterns_registry","label":"Routing Anti Patterns Registry","relative_href":"docs/component-routing-anti-patterns-registry.html"},{"component_id":"self_ignorance_coverage_ledger","label":"Self Ignorance Coverage Ledger","relative_href":"docs/component-self-ignorance-coverage-ledger.html"},{"component_id":"voice_to_doctrine_self_improvement_loop","label":"Voice To Doctrine Self Improvement Loop","relative_href":"docs/component-voice-to-doctrine-self-improvement-loop.html"}],"paper_module_backrefs":[{"id":"paper_module.cold_reader_route_map","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Cold-Reader Route Map","url":"paper-module-cold-reader-route-map.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[13:paper_module.cold_reader_route_map].subjects[0]"}],"primary_paper_module_id":"paper_module.cold_reader_route_map","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json#paper_module.cold_reader_route_map"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/cold_reader_route_map.py"}],"visible_text_byte_count":1009},{"id":"public_reveal_walkthrough","label":"Public Reveal Walkthrough","family_id":"entry_and_reveal","family_label":"Entry & orientation","relative_href":"docs/component-public-reveal-walkthrough.html","primary_reader_href":"docs/paper-module-public-reveal-walkthrough.html","component_detail_href":"docs/component-public-reveal-walkthrough.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Binds the first-time reader tour to evidence so each count leads to a source.","what_it_does":"This checks the short guided tour Microcosm advertises for a first-time reader and binds it to real public evidence: the declared route through patterns, work, events, and evidence must still point to result records, and the exported reveal bundle must carry digest-verified copies of the public source bodies that back the walkthrough. The card remains bounded, but each impressive-looking count leads to a source-body witness instead of stopping at marketing copy.","scope_limit":"It authorizes only bounded public reveal runtime behavior and a digest-verified public body-import witness; it excludes launch, hosted deployment, public sharing, recipient work, external model access, secret export, non-public data equivalence, Lean/Lake execution, whole-system correctness, or general product authority.","evidence":{"kind":"Bounded runtime computation","basis":"Real runtime result","rank":4,"runs_real_tools":true},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.public_reveal_walkthrough run --input fixtures/first_wave/public_reveal_walkthrough/input --out receipts/first_wave/public_reveal_walkthrough","tags":["getting-started","interesting-parts","evaluation"],"links_to":[{"component_id":"batch12_release_claim_language_gate","label":"Release Public Wording Gate","relative_href":"docs/component-batch12-release-claim-language-gate.html"},{"component_id":"mechanistic_interpretability_circuit_attribution_replay","label":"Toy-Transformer Attribution Replay","relative_href":"docs/component-mechanistic-interpretability-circuit-attribution-replay.html"},{"component_id":"batch8_audio_level_rms_port","label":"Audio Level RMS Port","relative_href":"docs/component-batch8-audio-level-rms-port.html"},{"component_id":"batch8_validator_checker_capsule","label":"Validator Checker Bundle","relative_href":"docs/component-batch8-validator-checker-capsule.html"}],"paper_module_backrefs":[{"id":"paper_module.public_reveal_walkthrough","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Public Reveal Walkthrough","url":"paper-module-public-reveal-walkthrough.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[28:paper_module.public_reveal_walkthrough].subjects[0]"}],"primary_paper_module_id":"paper_module.public_reveal_walkthrough","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/public_reveal_walkthrough.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/public_reveal_walkthrough.py"}],"visible_text_byte_count":1083},{"id":"pattern_binding_contract","label":"Pattern Binding Contract","family_id":"architecture_and_navigation","family_label":"Architecture & navigation","relative_href":"docs/component-pattern-binding-contract.html","primary_reader_href":"docs/paper-module-pattern-binding-contract.html","component_detail_href":"docs/component-pattern-binding-contract.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Checks a real pattern catalog for digest, cross-reference, and dependency-cycle integrity.","what_it_does":"It checks that each declared \"pattern\" (a reusable bit of system structure) is properly hooked up: it has the required fields, points at real source material by reference, names the rule it answers to and what it explicitly does NOT claim, has no duplicate pattern IDs, and leaks no secrets or private bodies into the public record. It produces a written result record showing the overall pass/fail, which pattern rows are accepted versus rejected and why, and what each row is forbidden from claiming.","scope_limit":"It validates only the declared public pattern-binding/route-readiness contract; it does not certify the private pattern ledger, public launch or hosted-public posture, public sharing, external model access, non-public data equivalence, or whole-system correctness, and it does not turn any mined pattern row into a standalone public leaf (selection stays component-first and fixture-bound).","evidence":{"kind":"Contract validator","basis":"Import validation","rank":5,"runs_real_tools":false},"command":"microcosm pattern-route-readiness validate-bundle --input examples/pattern_binding_contract/exported_route_readiness_bundle --out /tmp/microcosm-pattern-route-readiness","tags":["architecture","navigation","doctrine"],"links_to":[{"component_id":"navigation_hologram_route_plane","label":"Navigation Hologram Route Plane","relative_href":"docs/component-navigation-hologram-route-plane.html"},{"component_id":"agent_route_observability_runtime","label":"Agent Route Observability Runtime","relative_href":"docs/component-agent-route-observability-runtime.html"},{"component_id":"cold_reader_route_map","label":"Cold Reader Route Map","relative_href":"docs/component-cold-reader-route-map.html"},{"component_id":"macro_projection_import_protocol","label":"Source Projection Import Protocol","relative_href":"docs/component-macro-projection-import-protocol.html"},{"component_id":"pattern_assimilation_step","label":"Pattern Assimilation Step","relative_href":"docs/component-pattern-assimilation-step.html"},{"component_id":"voice_to_doctrine_self_improvement_loop","label":"Voice To Doctrine Self Improvement Loop","relative_href":"docs/component-voice-to-doctrine-self-improvement-loop.html"}],"paper_module_backrefs":[{"id":"paper_module.pattern_binding_contract","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Pattern Binding Contract","url":"paper-module-pattern-binding-contract.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[9:paper_module.pattern_binding_contract].subjects[0]"}],"primary_paper_module_id":"paper_module.pattern_binding_contract","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/pattern_binding_contract.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/pattern_binding_contract.py"}],"visible_text_byte_count":1178},{"id":"pattern_assimilation_step","label":"Pattern Assimilation Step","family_id":"architecture_and_navigation","family_label":"Architecture & navigation","relative_href":"docs/component-pattern-assimilation-step.html","primary_reader_href":"docs/paper-module-pattern-assimilation.html","component_detail_href":"docs/component-pattern-assimilation-step.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Verifies each landed task filed exactly one learning record naming what it changed.","what_it_does":"When a piece of work lands in the local system, this component checks the completion records for it \u2014 confirming that exactly one same-lane \"what did we learn from this\" decision was filed per landed item, that any claimed refinement names an owner-visible surface and the artifact it changed, that a \"nothing to refine\" entry carries its required typed fields, and that there are no duplicate or off-lane entries. It runs over fixture data and makes the completion bookkeeping inspectable, showing whether the recorded completions conform to the system's stated learning-from-landed-work rules, rather than leaving that on faith.","scope_limit":"It validates only the declared public completion contract over synthetic fixture data; it does not ingest private lessons, mutate live ledgers, promote global doctrine, include launch operations or public sharing, make external model access, claim non-public data equivalence, or certify public runtime behavior.","evidence":{"kind":"Contract validator","basis":"Import validation","rank":5,"runs_real_tools":false},"command":"PYTHONPATH=src python3 -m microcosm_core.validators.acceptance --only pattern_assimilation_step --input fixtures/first_wave/pattern_assimilation_step/input --out receipts/first_wave/pattern_assimilation_acceptance.json","tags":["architecture","navigation","doctrine"],"links_to":[],"paper_module_backrefs":[{"id":"paper_module.pattern_assimilation","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Pattern Assimilation","url":"paper-module-pattern-assimilation.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[71:paper_module.pattern_assimilation].subjects[0]"}],"primary_paper_module_id":"paper_module.pattern_assimilation","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/pattern_assimilation.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/validators/acceptance.py"}],"visible_text_byte_count":1274},{"id":"executable_doctrine_grammar","label":"Executable Doctrine Grammar","family_id":"architecture_and_navigation","family_label":"Architecture & navigation","relative_href":"docs/component-executable-doctrine-grammar.html","primary_reader_href":"docs/paper-module-executable-doctrine-grammar.html","component_detail_href":"docs/component-executable-doctrine-grammar.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Checks that example standards files declare their purpose, rule, records, and what they do not claim.","what_it_does":"It checks that a folder of example \"doctrine\" files (toy public standards and write-ups that describe how the system is supposed to behave) actually have the required parts: a stated purpose, the rule that governs them, the result records they are expected to produce, and an honest statement of what they do NOT claim. It reports, file by file, which entries are well-formed and which ones fail a required check, including ones that overclaim (saying a passing grammar check proves the doctrine is complete) or that try to treat plain advice as enforceable authority.","scope_limit":"It validates an exported public executable-grammar metabolism bundle with exact copied-body digests and redacted result records, plus fixture regressions for standards/paper-module shape. It does not publish source doctrine bodies in result records, prove doctrine completeness, export a private standards engine, authorize later components, or claim external model access, non-public data equivalence, launch-scope decision, or whole-system correctness.","evidence":{"kind":"Contract validator","basis":"Import validation","rank":5,"runs_real_tools":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.executable_doctrine_grammar validate-executable-grammar-metabolism-bundle --input examples/executable_doctrine_grammar/exported_executable_grammar_metabolism_bundle --out receipts/first_wave/executable_doctrine_grammar --card","tags":["architecture","navigation","doctrine"],"links_to":[{"component_id":"doctrine_fact_claim_audit","label":"Doctrine Fact Claim Audit","relative_href":"docs/component-doctrine-fact-claim-audit.html"}],"paper_module_backrefs":[{"id":"paper_module.executable_doctrine_grammar","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Executable Doctrine Grammar","url":"paper-module-executable-doctrine-grammar.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[18:paper_module.executable_doctrine_grammar].subjects[0]"}],"primary_paper_module_id":"paper_module.executable_doctrine_grammar","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json#paper_module.executable_doctrine_grammar"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/executable_doctrine_grammar.py"}],"visible_text_byte_count":1427},{"id":"navigation_hologram_route_plane","label":"Navigation Hologram Route Plane","family_id":"architecture_and_navigation","family_label":"Architecture & navigation","relative_href":"docs/component-navigation-hologram-route-plane.html","primary_reader_href":"docs/paper-module-navigation-hologram-route-plane.html","component_detail_href":"docs/component-navigation-hologram-route-plane.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Audits a folder's navigation so browse rows never pose as the source of truth.","what_it_does":"This checks that a folder's local \"how to get around\" surface behaves: the path starts from one control entry, then drills into browsable lists of routes and cards, and those browse rows are never allowed to pose as the source of truth. It makes visible, in plain result record files, that stale or mislabeled navigation is caught, that compressed cards keep a note of what they left out, and that nothing private or secret leaks into the navigation material while moving around. It only inspects toy fixture files and exact copied-but-navigation source modules; it does not touch any live system.","scope_limit":"It validates only the declared public toy route-plane contract and its regression fixtures (plus exact copied navigation source modules in the bundle path); it does not establish live route freshness, grant source authority, authorize any later component, run any provider/live-kernel call, or certify the whole wave.","evidence":{"kind":"Contract validator","basis":"Import validation","rank":5,"runs_real_tools":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.navigation_hologram_route_plane run --input fixtures/first_wave/navigation_hologram_route_plane/input --out receipts/first_wave/navigation_hologram_route_plane","tags":["agent-entry","architecture","navigation"],"links_to":[{"component_id":"cold_reader_route_map","label":"Cold Reader Route Map","relative_href":"docs/component-cold-reader-route-map.html"},{"component_id":"agent_route_observability_runtime","label":"Agent Route Observability Runtime","relative_href":"docs/component-agent-route-observability-runtime.html"},{"component_id":"routing_anti_patterns_registry","label":"Routing Anti Patterns Registry","relative_href":"docs/component-routing-anti-patterns-registry.html"},{"component_id":"pattern_binding_contract","label":"Pattern Binding Contract","relative_href":"docs/component-pattern-binding-contract.html"},{"component_id":"doctrine_fact_claim_audit","label":"Doctrine Fact Claim Audit","relative_href":"docs/component-doctrine-fact-claim-audit.html"},{"component_id":"macro_projection_import_protocol","label":"Source Projection Import Protocol","relative_href":"docs/component-macro-projection-import-protocol.html"},{"component_id":"self_ignorance_coverage_ledger","label":"Self Ignorance Coverage Ledger","relative_href":"docs/component-self-ignorance-coverage-ledger.html"},{"component_id":"voice_to_doctrine_self_improvement_loop","label":"Voice To Doctrine Self Improvement Loop","relative_href":"docs/component-voice-to-doctrine-self-improvement-loop.html"},{"component_id":"engine_room_demo","label":"Engine Room Demo","relative_href":"docs/component-engine-room-demo.html"},{"component_id":"batch11_saturation_engines_capsule","label":"Saturation Engines Bundle","relative_href":"docs/component-batch11-saturation-engines-capsule.html"},{"component_id":"batch8_compliance_pipeline_capsule","label":"Compliance Pipeline Bundle","relative_href":"docs/component-batch8-compliance-pipeline-capsule.html"}],"paper_module_backrefs":[{"id":"paper_module.navigation_hologram_route_plane","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Navigation Hologram Route Plane","url":"paper-module-navigation-hologram-route-plane.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[1:paper_module.navigation_hologram_route_plane].subjects[0]"}],"primary_paper_module_id":"paper_module.navigation_hologram_route_plane","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json#paper_module.navigation_hologram_route_plane"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/navigation_hologram_route_plane.py"}],"visible_text_byte_count":1234},{"id":"standards_meta_diagnostics","label":"Standards Meta Diagnostics","family_id":"architecture_and_navigation","family_label":"Architecture & navigation","relative_href":"docs/component-standards-meta-diagnostics.html","primary_reader_href":"docs/paper-module-standards-meta-diagnostics.html","component_detail_href":"docs/component-standards-meta-diagnostics.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Confirms every accepted part still ties to a written rule, a run command, and a saved proof.","what_it_does":"This is a coverage checker that reads the project's public catalogs and confirms every accepted part is still tied to a written standard, a documented way to run it, and a saved proof-of-run, while flagging any claim that overreaches (like \"ready to launch\") or any leaked private text. It makes the system's own bookkeeping inspectable: whether the pieces are accounted for is visible rather than taken on trust.","scope_limit":"It validates only the declared public coverage contract and never becomes source authority for the registries, mutates source, exposes private material, or authorizes launch, external model access, or any whole-system-correctness claim.","evidence":{"kind":"Contract validator","basis":"Import validation","rank":5,"runs_real_tools":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.standards_meta_diagnostics run --input fixtures/first_wave/standards_meta_diagnostics/input --out /tmp/standards_meta_diagnostics_out","tags":["agent-entry","architecture","navigation"],"links_to":[],"paper_module_backrefs":[{"id":"paper_module.standards_meta_diagnostics","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Standards Meta Diagnostics","url":"paper-module-standards-meta-diagnostics.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[29:paper_module.standards_meta_diagnostics].subjects[0]"}],"primary_paper_module_id":"paper_module.standards_meta_diagnostics","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/standards_meta_diagnostics.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/standards_meta_diagnostics.py"}],"visible_text_byte_count":952},{"id":"voice_to_doctrine_self_improvement_loop","label":"Voice To Doctrine Self Improvement Loop","family_id":"architecture_and_navigation","family_label":"Architecture & navigation","relative_href":"docs/component-voice-to-doctrine-self-improvement-loop.html","primary_reader_href":"docs/paper-module-voice-to-doctrine-self-improvement-loop.html","component_detail_href":"docs/component-voice-to-doctrine-self-improvement-loop.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Verifies each lesson changed a named owner page with evidence before the loop closes.","what_it_does":"It makes the system's \"learn from a lesson, then improve\" cycle inspectable on a folder. Each local lesson is shown being assigned to a specific owner surface (a skill, a doctrine page, a standard, or a tracked to-do capture), then changed or captured there, then validated, then closed out with a concrete reason to revisit later. Using local files only, the result records confirm that improvements are tied to real owner-surface changes and evidence rather than just asserted.","scope_limit":"It validates only the declared contract of the loop on fixtures; it does not export source notes or private bodies, grant source/doctrine edits, global-promotion, live work log mutation, or publishing-scope decision, make external model access, prove correctness, or claim whole-system equivalence.","evidence":{"kind":"Contract validator","basis":"Import validation","rank":5,"runs_real_tools":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.voice_to_doctrine_self_improvement_loop run --input fixtures/first_wave/voice_to_doctrine_self_improvement_loop/input --out receipts/first_wave/voice_to_doctrine_self_improvement_loop","tags":["agent-entry","architecture","navigation"],"links_to":[{"component_id":"pattern_assimilation_step","label":"Pattern Assimilation Step","relative_href":"docs/component-pattern-assimilation-step.html"}],"paper_module_backrefs":[{"id":"paper_module.voice_to_doctrine_self_improvement_loop","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Voice to Doctrine Self-Improvement Loop","url":"paper-module-voice-to-doctrine-self-improvement-loop.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[56:paper_module.voice_to_doctrine_self_improvement_loop].subjects[0]"}],"primary_paper_module_id":"paper_module.voice_to_doctrine_self_improvement_loop","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/voice_to_doctrine_self_improvement_loop.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/voice_to_doctrine_self_improvement_loop.py"}],"visible_text_byte_count":1136},{"id":"cognitive_operator_registry","label":"Cognitive Operator Registry","family_id":"architecture_and_navigation","family_label":"Architecture & navigation","relative_href":"docs/component-cognitive-operator-registry.html","primary_reader_href":"docs/paper-module-cognitive-operator-registry.html","component_detail_href":"docs/component-cognitive-operator-registry.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Checks the catalog of named thinking-moves so each is fully described and backed by evidence.","what_it_does":"This is a checker for the system's catalogue of named thinking-moves (operators like \"reduce competing pressure to one bounded action\" or \"compile a handoff packet when validated work cannot be committed\"). It confirms every operator in the public catalogue is fully described (how it is triggered, used, and checked) and that every operator marked active carries a real result record proving it once changed a live decision, while rejecting any operator that claims to speak with the owner's voice or overreaches into \"ready to launch.\" It matters because it shows the system treats reasoning itself as inspectable typed system instead of ad-hoc prompt lore.","scope_limit":"It validates only the declared public registry contract and copied source bodies; it never becomes registry source authority, mutates operators, proves operator correctness, exposes source notes, or authorizes launch, external model access, or any whole-system-correctness claim.","evidence":{"kind":"Contract validator","basis":"Import validation","rank":5,"runs_real_tools":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.cognitive_operator_registry run --input fixtures/first_wave/cognitive_operator_registry/input --out /tmp/cognitive_operator_registry_out","tags":["architecture","navigation","doctrine"],"links_to":[],"paper_module_backrefs":[{"id":"paper_module.cognitive_operator_registry","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Cognitive Operator Registry","url":"paper-module-cognitive-operator-registry.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[11:paper_module.cognitive_operator_registry].subjects[0]"}],"primary_paper_module_id":"paper_module.cognitive_operator_registry","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json#paper_module.cognitive_operator_registry"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/cognitive_operator_registry.py"}],"visible_text_byte_count":1246},{"id":"routing_anti_patterns_registry","label":"Routing Anti Patterns Registry","family_id":"architecture_and_navigation","family_label":"Architecture & navigation","relative_href":"docs/component-routing-anti-patterns-registry.html","primary_reader_href":"docs/paper-module-routing-anti-patterns-registry.html","component_detail_href":"docs/component-routing-anti-patterns-registry.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Indexes the navigation mistakes agents repeat and guards the public list.","what_it_does":"This is a checker for the system's public list of navigation mistakes agents keep making, such as grepping before asking the kernel for a route or sending work to a bridge before scope is chosen. It confirms each anti-pattern row has a stable id and explanation, rejects duplicate or body-leaking rows, and proves the public bundle carries the actual source registry body copied byte-for-byte instead of a synthetic paraphrase.","scope_limit":"It validates only the declared public routing anti-pattern registry contract and copied source body; it never becomes route source authority, mutates routes, exposes private routing notes, calls providers, authorizes launch, or proves whole-system correctness.","evidence":{"kind":"Contract validator","basis":"Import validation","rank":5,"runs_real_tools":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.routing_anti_patterns_registry run --input fixtures/first_wave/routing_anti_patterns_registry/input --out /tmp/routing_anti_patterns_registry_out","tags":["architecture","navigation","doctrine"],"links_to":[{"component_id":"navigation_hologram_route_plane","label":"Navigation Hologram Route Plane","relative_href":"docs/component-navigation-hologram-route-plane.html"},{"component_id":"agent_route_observability_runtime","label":"Agent Route Observability Runtime","relative_href":"docs/component-agent-route-observability-runtime.html"},{"component_id":"cold_reader_route_map","label":"Cold Reader Route Map","relative_href":"docs/component-cold-reader-route-map.html"}],"paper_module_backrefs":[{"id":"paper_module.routing_anti_patterns_registry","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Routing Anti-Patterns Registry","url":"paper-module-routing-anti-patterns-registry.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[57:paper_module.routing_anti_patterns_registry].subjects[0]"}],"primary_paper_module_id":"paper_module.routing_anti_patterns_registry","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/routing_anti_patterns_registry.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/routing_anti_patterns_registry.py"}],"visible_text_byte_count":987},{"id":"doctrine_fact_claim_audit","label":"Doctrine Fact Claim Audit","family_id":"architecture_and_navigation","family_label":"Architecture & navigation","relative_href":"docs/component-doctrine-fact-claim-audit.html","primary_reader_href":"docs/paper-module-doctrine-fact-claim-audit.html","component_detail_href":"docs/component-doctrine-fact-claim-audit.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Checks that public fact rows state the right count and point at live, anchored code.","what_it_does":"Checks that public fact rows state the expected count, point at live copied code loci, preserve anchors, and only reference facts that exist in the fixture DAG.","scope_limit":"fact assertion, code-loci, and DAG fixture truth gate only; it is not a comprehension engine and does not establish a minimum read graph","evidence":{"kind":"Contract validator","basis":"Import validation","rank":5,"runs_real_tools":false},"command":"microcosm doctrine-fact-claim-audit run --input fixtures/first_wave/doctrine_fact_claim_audit/input --out receipts/first_wave/doctrine_fact_claim_audit","tags":["architecture","navigation","doctrine"],"links_to":[{"component_id":"engine_room_demo","label":"Engine Room Demo","relative_href":"docs/component-engine-room-demo.html"},{"component_id":"self_ignorance_coverage_ledger","label":"Self Ignorance Coverage Ledger","relative_href":"docs/component-self-ignorance-coverage-ledger.html"}],"paper_module_backrefs":[{"id":"paper_module.doctrine_fact_claim_audit","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Doctrine Fact Claim Audit","url":"paper-module-doctrine-fact-claim-audit.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[48:paper_module.doctrine_fact_claim_audit].subjects[0]"}],"primary_paper_module_id":"paper_module.doctrine_fact_claim_audit","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/doctrine_fact_claim_audit.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/doctrine_fact_claim_audit.py"}],"visible_text_byte_count":560},{"id":"self_ignorance_coverage_ledger","label":"Self Ignorance Coverage Ledger","family_id":"architecture_and_navigation","family_label":"Architecture & navigation","relative_href":"docs/component-self-ignorance-coverage-ledger.html","primary_reader_href":"docs/paper-module-self-ignorance-coverage-ledger.html","component_detail_href":"docs/component-self-ignorance-coverage-ledger.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Compares expected against built entities to report known coverage gaps.","what_it_does":"Compares declared Kind Atlas expectations against materialized entities and reports the known coverage debt that falls out of that public fixture comparison.","scope_limit":"known Kind Atlas coverage debt projection only; it does not claim literal unknown-unknown omniscience or absence proof","evidence":{"kind":"Computed projection","basis":"Source-faithful refactor","rank":3,"runs_real_tools":false},"command":"microcosm self-ignorance-coverage-ledger run --input fixtures/first_wave/self_ignorance_coverage_ledger/input --out receipts/first_wave/self_ignorance_coverage_ledger","tags":["architecture","navigation","doctrine"],"links_to":[],"paper_module_backrefs":[{"id":"paper_module.self_ignorance_coverage_ledger","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Self-Ignorance Coverage Ledger","url":"paper-module-self-ignorance-coverage-ledger.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[49:paper_module.self_ignorance_coverage_ledger].subjects[0]"}],"primary_paper_module_id":"paper_module.self_ignorance_coverage_ledger","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/self_ignorance_coverage_ledger.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/self_ignorance_coverage_ledger.py"}],"visible_text_byte_count":546},{"id":"navigation_fitness_benchmark","label":"Navigation Fitness Benchmark","family_id":"architecture_and_navigation","family_label":"Architecture & navigation","relative_href":"docs/component-navigation-fitness-benchmark.html","primary_reader_href":"docs/paper-module-navigation-fitness-benchmark.html","component_detail_href":"docs/component-navigation-fitness-benchmark.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Measures a navigation result against its target for recall, precision, and speed on public examples.","what_it_does":"A scorer for how well a navigation system found the right thing. Each test case is a \"task\" (what someone was looking for, which exact items count as the right answer, which shortcut routes are off-limits, and how fast it should be) paired with a \"route packet\" (what the system actually returned and how long it took). The tool re-checks the answer: did it find the expected items (recall), how much of what it returned was on-target (precision), did it take a banned first step, did it mention the right cue words, and did it stay under the time budget. It then flags any \"debt\" \u2014 places that missed the mark or ran slow. It judges curated public examples only; it does not run the real private system, does not test the underlying search/embeddings, and does not claim to be a general-purpose navigation score.","scope_limit":"The bundle demonstrates a route-packet evaluator for expected stable ids, forbidden first routes, and latency budgets. It is not a live private kernel run, not an embedding benchmark, not a universal navigation benchmark, and not launch-scope decision.","evidence":{"kind":"Bounded runtime computation","basis":"Real runtime result","rank":4,"runs_real_tools":true},"command":"microcosm navigation-fitness-benchmark run --input fixtures/first_wave/navigation_fitness_benchmark/input --out receipts/first_wave/navigation_fitness_benchmark --acceptance-out receipts/acceptance/first_wave/navigation_fitness_benchmark_fixture_acceptance.json","tags":["navigation","evaluation","architecture"],"links_to":[{"component_id":"certificate_kernel_execution_lab","label":"Certificate Kernel Execution Lab","relative_href":"docs/component-certificate-kernel-execution-lab.html"},{"component_id":"verifier_lab_kernel","label":"Verifier Lab Kernel","relative_href":"docs/component-verifier-lab-kernel.html"},{"component_id":"formal_math_lean_proof_witness","label":"Formal Math Lean Proof Witness","relative_href":"docs/component-formal-math-lean-proof-witness.html"}],"paper_module_backrefs":[{"id":"paper_module.navigation_fitness_benchmark","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Navigation Fitness Benchmark","url":"paper-module-navigation-fitness-benchmark.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[98:paper_module.navigation_fitness_benchmark].subjects[0]"}],"primary_paper_module_id":"paper_module.navigation_fitness_benchmark","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json#paper_module.navigation_fitness_benchmark"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/navigation_fitness_benchmark.py"}],"visible_text_byte_count":1460},{"id":"annex_knowledge_routing","label":"Reference Knowledge Routing","family_id":"architecture_and_navigation","family_label":"Architecture & navigation","relative_href":"docs/component-annex-knowledge-routing.html","primary_reader_href":"docs/paper-module-annex-knowledge-routing.html","component_detail_href":"docs/component-annex-knowledge-routing.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Runs a keyword-overlap match that ranks catalog entries, shows why each matched, and admits when none do.","what_it_does":"Give it a plain-English problem and a small catalog of reference entries (each tagged with the domains, capabilities, and problem-spaces it covers), and it returns the entries most likely to help, ranked, with a breakdown of why each matched. It weighs strong structured tags more than loose description text, and shows its working rather than hiding it behind a black-box score. If nothing in the catalog actually overlaps your problem, it says no_match instead of inventing a result. It is a transparent keyword-overlap matcher, not a search engine or embedding model, and it never reaches outside the catalog you hand it.","scope_limit":"The bundle demonstrates explainable tiered weighted-token retrieval over a sanitized reference catalog. It is not BM25, TF-IDF, embedding search, repository cloning, license authority, private reference corpus authority, or launch-scope decision.","evidence":{"kind":"Bounded runtime computation","basis":"Real runtime result","rank":4,"runs_real_tools":true},"command":"microcosm annex-knowledge-routing run --input fixtures/first_wave/annex_knowledge_routing/input --out receipts/first_wave/annex_knowledge_routing --acceptance-out receipts/acceptance/first_wave/annex_knowledge_routing_fixture_acceptance.json","tags":["navigation","architecture"],"links_to":[{"component_id":"certificate_kernel_execution_lab","label":"Certificate Kernel Execution Lab","relative_href":"docs/component-certificate-kernel-execution-lab.html"},{"component_id":"verifier_lab_kernel","label":"Verifier Lab Kernel","relative_href":"docs/component-verifier-lab-kernel.html"},{"component_id":"formal_math_lean_proof_witness","label":"Formal Math Lean Proof Witness","relative_href":"docs/component-formal-math-lean-proof-witness.html"}],"paper_module_backrefs":[{"id":"paper_module.annex_knowledge_routing","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Reference Knowledge Routing","url":"paper-module-annex-knowledge-routing.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[99:paper_module.annex_knowledge_routing].subjects[0]"}],"primary_paper_module_id":"paper_module.annex_knowledge_routing","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json#paper_module.annex_knowledge_routing"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/annex_knowledge_routing.py"}],"visible_text_byte_count":1247},{"id":"certificate_kernel_execution_lab","label":"Certificate Kernel Execution Lab","family_id":"formal_math_and_proof","family_label":"Formal math & proof","relative_href":"docs/component-certificate-kernel-execution-lab.html","primary_reader_href":"docs/paper-module-certificate-kernel-execution-lab.html","component_detail_href":"docs/component-certificate-kernel-execution-lab.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Runs the Lean verifier over a small public proof project and reports which rows it accepted.","what_it_does":"It builds a small public Lean/Lake project, then runs the Lean verifier over declared \"transition\" rows that reference a set of generated \"certificate\" declarations, and writes a structured result record showing which rows the verifier accepted, which it left unresolved or rejected, plus the exact build command, return code, and file hashes. The result record is honest, inspectable evidence that a real Lean verifier ran on public material, with proof text, provider/oracle output, and private source deliberately excluded from the result record and that exclusion recorded (not silently dropped) rather than passed off as evidence.","scope_limit":"It is a local tool-witness that the declared public fixture rows compiled and were adjudicated by the local Lean verifier; it excludes general proof authority, count oracle/provider output as proof, expose proof text, change source files, claim a benchmark solve-rate, or include launch operations.","evidence":{"kind":"External tool run","basis":"Real runtime result","rank":4,"runs_real_tools":true},"command":"microcosm certificate-kernel-execution-lab run --input fixtures/first_wave/certificate_kernel_execution_lab/input --out receipts/first_wave/certificate_kernel_execution_lab","tags":["formal-methods","theorem-proving","lean"],"links_to":[],"paper_module_backrefs":[{"id":"paper_module.certificate_kernel_execution_lab","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Certificate Kernel Execution Lab","url":"paper-module-certificate-kernel-execution-lab.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[7:paper_module.certificate_kernel_execution_lab].subjects[0]"}],"primary_paper_module_id":"paper_module.certificate_kernel_execution_lab","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/certificate_kernel_execution_lab.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/certificate_kernel_execution_lab.py"}],"visible_text_byte_count":1233},{"id":"formal_math_lean_proof_witness","label":"Formal Math Lean Proof Witness","family_id":"formal_math_and_proof","family_label":"Formal math & proof","relative_href":"docs/component-formal-math-lean-proof-witness.html","primary_reader_href":"docs/paper-module-formal-math-lean-proof-witness.html","component_detail_href":"docs/component-formal-math-lean-proof-witness.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Compiles a tiny Lean example with the real prover and records whether it built, leaking no proof text.","what_it_does":"This takes a small, purpose-built Lean math file (a handful of toy theorems written just for this demo) and a tiny project setup, copies them into a throwaway scratch folder, and actually tries to compile them with the installed Lean theorem-prover and its Lake build tool. It then writes down exactly what happened: whether the Lean and Lake tools were found, whether the build passed, fingerprints (hashes) of the source files, the names of the theorems it defined, and how many lines each file had. It also deliberately feeds in a broken proof and a couple of off-limits files to confirm they get rejected. The point is to show real proof-checking machinery run on a small example, while keeping the written records honest and redacted: no proof text or internal logs leak out, and it states plainly that this is a narrow toy check on one fixture, not a general-purpose proof system.","scope_limit":"It authorizes only a witness that a tiny declared public toy proof compiled under the locally installed Lean/Lake toolchain in a temporary workspace, plus confirmation that its leakage guardrails fired. It excludes Mathlib/Aesop/Batteries-dependent or general proof or theorem-program authority, external model access, private proof import, benchmark or performance claims, whole-system correctness, or any launch, hosted deployment, or public sharing.","evidence":{"kind":"External tool run","basis":"Real runtime result","rank":4,"runs_real_tools":true},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.formal_math_lean_proof_witness run --input fixtures/first_wave/formal_math_lean_proof_witness/input --out receipts/first_wave/formal_math_lean_proof_witness","tags":["formal-methods","theorem-proving","lean"],"links_to":[{"component_id":"formal_math_premise_retrieval","label":"Formal Math Premise Retrieval","relative_href":"docs/component-formal-math-premise-retrieval.html"},{"component_id":"formal_math_readiness_gate","label":"Formal Math Readiness Gate","relative_href":"docs/component-formal-math-readiness-gate.html"},{"component_id":"proof_diagnostic_evidence_spine","label":"Proof Diagnostic Evidence Spine","relative_href":"docs/component-proof-diagnostic-evidence-spine.html"},{"component_id":"verifier_lab_kernel","label":"Verifier Lab Kernel","relative_href":"docs/component-verifier-lab-kernel.html"},{"component_id":"verifier_lab_execution_spine","label":"Verifier Lab Execution Spine","relative_href":"docs/component-verifier-lab-execution-spine.html"}],"paper_module_backrefs":[{"id":"paper_module.formal_math_lean_proof_witness","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Formal Math Lean Proof Witness","url":"paper-module-formal-math-lean-proof-witness.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[22:paper_module.formal_math_lean_proof_witness].subjects[0]"}],"primary_paper_module_id":"paper_module.formal_math_lean_proof_witness","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/formal_math_lean_proof_witness.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/formal_math_lean_proof_witness.py"}],"visible_text_byte_count":1678},{"id":"verifier_lab_execution_spine","label":"Verifier Lab Execution Spine","family_id":"formal_math_and_proof","family_label":"Formal math & proof","relative_href":"docs/component-verifier-lab-execution-spine.html","primary_reader_href":"docs/paper-module-verifier-lab-execution-spine.html","component_detail_href":"docs/component-verifier-lab-execution-spine.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Runs Lean on small bounded proof attempts in a temp copy and records what passed or failed.","what_it_does":"It copies a small public Lean math project into a throwaway temporary workspace and actually runs the Lean/Lake checker on a handful of small, bounded proof-step attempts the tool builds itself. It then writes down what the checker said: which attempts were accepted, which failed, and the failure category for each, plus safety counts (for example, how many attempts tried to sneak in forbidden content and were rejected). The pass/fail facts and the safety counts are readable directly, while the tool never shows the underlying proof text, never calls any outside service, and never modifies the original project or any existing source files.","scope_limit":"It is a tool-witness result record for bounded public Lean transition rows only: it does not establish general proof authority, count oracle/provider output as proof, export proof bodies or tactic scripts, use external model services, change source files, claim benchmark solve-rates, or include launch operations/public sharing.","evidence":{"kind":"External tool run","basis":"Real runtime result","rank":4,"runs_real_tools":true},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.verifier_lab_execution_spine run --input fixtures/first_wave/verifier_lab_execution_spine/input --out .microcosm/verifier_lab_execution_spine","tags":["formal-methods","theorem-proving","lean"],"links_to":[{"component_id":"certificate_kernel_execution_lab","label":"Certificate Kernel Execution Lab","relative_href":"docs/component-certificate-kernel-execution-lab.html"},{"component_id":"engine_room_demo","label":"Engine Room Demo","relative_href":"docs/component-engine-room-demo.html"}],"paper_module_backrefs":[{"id":"paper_module.verifier_lab_execution_spine","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Verifier Lab Execution Spine","url":"paper-module-verifier-lab-execution-spine.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[44:paper_module.verifier_lab_execution_spine].subjects[0]"}],"primary_paper_module_id":"paper_module.verifier_lab_execution_spine","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/verifier_lab_execution_spine.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/verifier_lab_execution_spine.py"}],"visible_text_byte_count":1286},{"id":"corpus_readiness_mathlib_absence_gate","label":"Corpus Readiness Mathlib Absence Gate","family_id":"formal_math_and_proof","family_label":"Formal math & proof","relative_href":"docs/component-corpus-readiness-mathlib-absence-gate.html","primary_reader_href":"docs/paper-module-corpus-readiness-mathlib-absence-gate.html","component_detail_href":"docs/component-corpus-readiness-mathlib-absence-gate.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Runs the real Lean toolchain to confirm the math library is absent, then gates proof tasks.","what_it_does":"It reads a recorded readiness report from a Lean math toolchain run and makes one fact inspectable: when the report was captured, the Mathlib library was not importable (its import probe failed). From that, it lists which math corpora are absent or usable only for translation smoke tests, and which downstream tasks must be blocked before any Mathlib-dependent proof work is attempted. It also re-checks that the recorded source files match their recorded SHA-256 digests and that no proof bodies, provider outputs, or non-public paths leaked into the public output. The result shows exactly where the proof pipeline draws a hard \"not ready, do not proceed\" line, with provenance, instead of quietly assuming the environment is fine.","scope_limit":"It only projects and gate-checks recorded corpus/toolchain readiness accounting, re-verifies recorded source digests and leakage guards, and runs a bounded Lean/Lake import probe when a toolchain is present. It does not run a full Lake build, prove formal-result correctness, claim Mathlib is available beyond the probe result, benchmark corpora, score model performance, use external model services, or include launch operations or public sharing.","evidence":{"kind":"External tool run","basis":"Real runtime result","rank":4,"runs_real_tools":true},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.corpus_readiness_mathlib_absence_gate run --input fixtures/first_wave/corpus_readiness_mathlib_absence_gate/input --out receipts/first_wave/corpus_readiness_mathlib_absence_gate","tags":["formal-methods","theorem-proving","lean"],"links_to":[{"component_id":"verifier_lab_kernel","label":"Verifier Lab Kernel","relative_href":"docs/component-verifier-lab-kernel.html"},{"component_id":"tactic_portfolio_availability_probe","label":"Tactic Portfolio Availability Probe","relative_href":"docs/component-tactic-portfolio-availability-probe.html"},{"component_id":"undeclared_library_prior_symbol_classifier","label":"Undeclared Library Prior Symbol Classifier","relative_href":"docs/component-undeclared-library-prior-symbol-classifier.html"},{"component_id":"formal_math_lean_proof_witness","label":"Formal Math Lean Proof Witness","relative_href":"docs/component-formal-math-lean-proof-witness.html"}],"paper_module_backrefs":[{"id":"paper_module.corpus_readiness_mathlib_absence_gate","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Corpus Readiness Mathlib Absence Gate","url":"paper-module-corpus-readiness-mathlib-absence-gate.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[8:paper_module.corpus_readiness_mathlib_absence_gate].subjects[0]"}],"primary_paper_module_id":"paper_module.corpus_readiness_mathlib_absence_gate","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/corpus_readiness_mathlib_absence.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/corpus_readiness_mathlib_absence_gate.py"}],"visible_text_byte_count":1539},{"id":"batch4_proof_authority_runtime","label":"Proof / Control / Runtime Import Bundle","family_id":"formal_math_and_proof","family_label":"Formal math & proof","relative_href":"docs/component-batch4-proof-authority-runtime.html","primary_reader_href":"docs/paper-module-batch4-proof-authority-runtime.html","component_detail_href":"docs/component-batch4-proof-authority-runtime.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Checks fourteen proof, control, and runtime parts as one unit that rejects every overclaim.","what_it_does":"This bundle imports the Set-4 proof/control/runtime source modules and checks them as one inspectable unit. It surfaces the 14 mechanisms, the copied module manifest, the digest/anchor evidence, and the negative cases that reject proof, benchmark, launch, runtime, and non-public-state overclaims without exposing source bodies in result records.","scope_limit":"It validates only a public source-open bundle and bounded negative fixtures; it is not an Erdos #257 solution, not benchmark evidence, not public sharing or launch-scope decision, not live Codex/browser/runtime authority, and not whole-system equivalence.","evidence":{"kind":"Verified source import","basis":"Copied source body","rank":5,"runs_real_tools":false},"command":"microcosm batch4-proof-authority-runtime run --input fixtures/first_wave/batch4_proof_authority_runtime/input --out receipts/first_wave/batch4_proof_authority_runtime --acceptance-out receipts/acceptance/first_wave/batch4_proof_authority_runtime_fixture_acceptance.json","tags":["formal-methods","theorem-proving","lean"],"links_to":[],"paper_module_backrefs":[{"id":"paper_module.batch4_proof_authority_runtime","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Set 4 Proof, Authority, and Runtime Bundle","url":"paper-module-batch4-proof-authority-runtime.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[74:paper_module.batch4_proof_authority_runtime].subjects[0]"}],"primary_paper_module_id":"paper_module.batch4_proof_authority_runtime","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/batch4_proof_authority_runtime.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/batch4_proof_authority_runtime.py"}],"visible_text_byte_count":1004},{"id":"proof_diagnostic_evidence_spine","label":"Proof Diagnostic Evidence Spine","family_id":"formal_math_and_proof","family_label":"Formal math & proof","relative_href":"docs/component-proof-diagnostic-evidence-spine.html","primary_reader_href":"docs/paper-module-proof-diagnostic-evidence-spine.html","component_detail_href":"docs/component-proof-diagnostic-evidence-spine.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Sorts proof-pipeline checks into accepted or rejected without inflating a pass.","what_it_does":"An evidence checkpoint that sits in front of formal-proof work. It reads the diagnostic records left by earlier proof-pipeline steps and writes a \"diagnostic board\" listing which checks were accepted, which were rejected, and why. The board shows exactly what evidence was kept, and refuses to let raw model output, a stale record, or a merely-passing check get inflated into a claim that the math is actually correct. It only arranges and judges existing records; it never runs a proof checker itself.","scope_limit":"It records proof/evidence diagnostics over existing result record references only. It does not run Lean, use external model services, expose proof bodies, turn a passing check into formal-proof or theorem authority, prove runtime or whole-system correctness, authorize later components, certify public launch, authorize public sharing or recipient work, or establish secret export.","evidence":{"kind":"Computed projection","basis":"Source-faithful refactor","rank":3,"runs_real_tools":false},"command":"microcosm proof-diagnostic-evidence-spine run --input fixtures/first_wave/proof_diagnostic_evidence_spine/input --out receipts/first_wave/proof_diagnostic_evidence_spine --card","tags":["formal-methods","theorem-proving","lean"],"links_to":[{"component_id":"formal_math_verifier_trace_repair_loop","label":"Formal Math Verifier Trace Repair Loop","relative_href":"docs/component-formal-math-verifier-trace-repair-loop.html"},{"component_id":"formal_evidence_cell_anchor_resolver","label":"Formal Evidence Cell Anchor Resolver","relative_href":"docs/component-formal-evidence-cell-anchor-resolver.html"},{"component_id":"formal_math_lean_proof_witness","label":"Formal Math Lean Proof Witness","relative_href":"docs/component-formal-math-lean-proof-witness.html"},{"component_id":"verifier_lab_kernel","label":"Verifier Lab Kernel","relative_href":"docs/component-verifier-lab-kernel.html"},{"component_id":"ring2_premise_retrieval_precision_recall_harness","label":"Ring2 Premise Retrieval Precision Recall Harness","relative_href":"docs/component-ring2-premise-retrieval-precision-recall-harness.html"},{"component_id":"target_shape_tactic_routing_gate","label":"Target Shape Tactic Routing Gate","relative_href":"docs/component-target-shape-tactic-routing-gate.html"}],"paper_module_backrefs":[{"id":"paper_module.proof_diagnostic_evidence_spine","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Proof Diagnostic Evidence Spine","url":"paper-module-proof-diagnostic-evidence-spine.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[14:paper_module.proof_diagnostic_evidence_spine].subjects[0]"}],"primary_paper_module_id":"paper_module.proof_diagnostic_evidence_spine","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json#paper_module.proof_diagnostic_evidence_spine"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/proof_diagnostic_evidence_spine.py"}],"visible_text_byte_count":1173},{"id":"formal_math_readiness_gate","label":"Formal Math Readiness Gate","family_id":"formal_math_and_proof","family_label":"Formal math & proof","relative_href":"docs/component-formal-math-readiness-gate.html","primary_reader_href":"docs/paper-module-formal-math-readiness-gate.html","component_detail_href":"docs/component-formal-math-readiness-gate.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Reads declared math setups and lists which proof tactics may be attempted versus blocked.","what_it_does":"Before anyone tries to prove a math theorem with the Lean prover, this gate reads simple description files that declare what a math setup is supposed to have \u2014 which math library is claimed to be present, which proof tactics are reported as already probed, which lemmas may be looked up, and which limits apply to the text budgets handed to AI providers \u2014 and writes a plain checklist of what is allowed to be attempted versus blocked. It works only from those declared description files; it does not inspect the real toolchain or run anything. Its guards keep the claims honest and checkable: it refuses to let a library be marked available unless a probe result backs that up, blocks routing a proof tactic that was not probed, and refuses to let any real proof text sneak into the lemma-lookup tables or provider budgets.","scope_limit":"It only validates and projects declared readiness metadata; it does not run Lean/Lake, inspect the real toolchain, use external model services, prove any theorem correct, produce benchmark claims, or authorize Mathlib-dependent proof attempts.","evidence":{"kind":"Computed projection","basis":"Source-faithful refactor","rank":3,"runs_real_tools":false},"command":"microcosm formal-math-readiness-gate run --input fixtures/first_wave/formal_math_readiness_gate/input --out receipts/first_wave/formal_math_readiness_gate","tags":["formal-methods","theorem-proving","lean"],"links_to":[{"component_id":"formal_math_lean_proof_witness","label":"Formal Math Lean Proof Witness","relative_href":"docs/component-formal-math-lean-proof-witness.html"},{"component_id":"macro_projection_import_protocol","label":"Source Projection Import Protocol","relative_href":"docs/component-macro-projection-import-protocol.html"},{"component_id":"batch4_proof_authority_runtime","label":"Proof / Control / Runtime Import Bundle","relative_href":"docs/component-batch4-proof-authority-runtime.html"},{"component_id":"provider_context_recipe_budget_policy","label":"Provider Context Recipe Budget Policy","relative_href":"docs/component-provider-context-recipe-budget-policy.html"},{"component_id":"tactic_portfolio_availability_probe","label":"Tactic Portfolio Availability Probe","relative_href":"docs/component-tactic-portfolio-availability-probe.html"},{"component_id":"target_shape_tactic_routing_gate","label":"Target Shape Tactic Routing Gate","relative_href":"docs/component-target-shape-tactic-routing-gate.html"}],"paper_module_backrefs":[{"id":"paper_module.formal_math_readiness_gate","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Formal Math Readiness Gate","url":"paper-module-formal-math-readiness-gate.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[21:paper_module.formal_math_readiness_gate].subjects[0]"}],"primary_paper_module_id":"paper_module.formal_math_readiness_gate","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/formal_math_readiness_gate.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/formal_math_readiness_gate.py"}],"visible_text_byte_count":1344},{"id":"mathematical_strategy_atlas_hypothesis_scorer","label":"Mathematical Strategy Atlas Hypothesis Scorer","family_id":"formal_math_and_proof","family_label":"Formal math & proof","relative_href":"docs/component-mathematical-strategy-atlas-hypothesis-scorer.html","primary_reader_href":"docs/paper-module-mathematical-strategy-atlas.html","component_detail_href":"docs/component-mathematical-strategy-atlas-hypothesis-scorer.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Picks a first-guess proof strategy from a problem's tags and flags any it cannot map.","what_it_does":"Before any proof is attempted, this component looks at a math problem's feature tags and writes down its first-guess strategy (for example, \"this looks like an if-and-only-if, so split it both ways\"), and flags anything it cannot map as an explicit \"no strategy matched\" instead of a silent failure. The chosen opening move, why it was chosen, and the cases it could not map are all recorded in machine-readable result records.","scope_limit":"It only projects pre-oracle strategy-hypothesis and retrieval mechanics; it does not run Lean/Lake, prove theorems, establish domain or formal-result correctness, reveal oracle labels, expose proof bodies, use external model services, tune on test answers, or include launch operations.","evidence":{"kind":"Computed projection","basis":"Source-faithful refactor","rank":3,"runs_real_tools":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.mathematical_strategy_atlas_hypothesis_scorer run --input fixtures/first_wave/mathematical_strategy_atlas_hypothesis_scorer/input --out receipts/first_wave/mathematical_strategy_atlas_hypothesis_scorer","tags":["formal-methods","theorem-proving","lean"],"links_to":[],"paper_module_backrefs":[{"id":"paper_module.mathematical_strategy_atlas","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Mathematical Strategy Atlas","url":"paper-module-mathematical-strategy-atlas.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[43:paper_module.mathematical_strategy_atlas].subjects[0]"}],"primary_paper_module_id":"paper_module.mathematical_strategy_atlas","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/mathematical_strategy_atlas.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/mathematical_strategy_atlas_hypothesis_scorer.py"}],"visible_text_byte_count":1096},{"id":"tactic_portfolio_availability_probe","label":"Tactic Portfolio Availability Probe","family_id":"formal_math_and_proof","family_label":"Formal math & proof","relative_href":"docs/component-tactic-portfolio-availability-probe.html","primary_reader_href":"docs/paper-module-tactic-portfolio-availability.html","component_detail_href":"docs/component-tactic-portfolio-availability-probe.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Maps which Lean proof tactics a recorded run marked usable before any code relies on one.","what_it_does":"It turns one captured Lean run's results into an inspectable list of which proof shortcuts (\"tactics\" like `rfl`, `simp`, `omega`, `aesop`) were recorded as compiling, showing at a glance which are usable and which are off before anything treats a tactic as available. In this fixture seven tactics are marked usable and `aesop` is marked failed (its recorded run hit a missing-Mathlib error). The tool reads pre-recorded status rows and checks them for honesty; it does not run Lean itself.","scope_limit":"It only projects and validates which tactics were recorded as compiling in one captured environment; it does not run Lean/Lake at all, prove any goal, certify domain-level conclusions, use external model services, claim benchmark performance, or include launch operations.","evidence":{"kind":"Computed projection","basis":"Source-faithful refactor","rank":3,"runs_real_tools":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.tactic_portfolio_availability_probe run --input fixtures/first_wave/tactic_portfolio_availability_probe/input --out receipts/first_wave/tactic_portfolio_availability_probe --acceptance-out receipts/acceptance/first_wave/tactic_portfolio_availability_probe_fixture_acceptance.json","tags":["formal-methods","theorem-proving","lean"],"links_to":[{"component_id":"corpus_readiness_mathlib_absence_gate","label":"Corpus Readiness Mathlib Absence Gate","relative_href":"docs/component-corpus-readiness-mathlib-absence-gate.html"},{"component_id":"formal_math_verifier_trace_repair_loop","label":"Formal Math Verifier Trace Repair Loop","relative_href":"docs/component-formal-math-verifier-trace-repair-loop.html"},{"component_id":"target_shape_tactic_routing_gate","label":"Target Shape Tactic Routing Gate","relative_href":"docs/component-target-shape-tactic-routing-gate.html"},{"component_id":"undeclared_library_prior_symbol_classifier","label":"Undeclared Library Prior Symbol Classifier","relative_href":"docs/component-undeclared-library-prior-symbol-classifier.html"},{"component_id":"verifier_lab_kernel","label":"Verifier Lab Kernel","relative_href":"docs/component-verifier-lab-kernel.html"}],"paper_module_backrefs":[{"id":"paper_module.tactic_portfolio_availability","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Tactic Portfolio Availability","url":"paper-module-tactic-portfolio-availability.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[40:paper_module.tactic_portfolio_availability].subjects[0]"}],"primary_paper_module_id":"paper_module.tactic_portfolio_availability","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/tactic_portfolio_availability.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/tactic_portfolio_availability_probe.py"}],"visible_text_byte_count":1218},{"id":"target_shape_tactic_routing_gate","label":"Target Shape Tactic Routing Gate","family_id":"formal_math_and_proof","family_label":"Formal math & proof","relative_href":"docs/component-target-shape-tactic-routing-gate.html","primary_reader_href":"docs/paper-module-target-shape-tactic-routing.html","component_detail_href":"docs/component-target-shape-tactic-routing-gate.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Records an allow-or-reject decision and reason for each proof tactic before any proof runs.","what_it_does":"Before a proof is attempted, this component checks a list of candidate proof tactics for a given goal and writes down an allow-or-reject decision, with a plain reason, for each one. Rejections fall into three kinds: the tactic isn't actually available in the declared environment, it was never listed in the environment's tested set of tactics, or it simply doesn't fit the kind of goal being proved. The resulting record shows, tactic by tactic, exactly what was admitted or blocked and why, instead of an opaque \"we tried these\" claim. It only inspects and records the routing decision over references that already exist; it never runs a prover or proves anything itself.","scope_limit":"It only inspects and records the projection mechanics of pre-execution tactic-routing references \u2014 emitting per-tactic allow/reject decisions with reasons. It does not run Lean/Lake, does not establish or judge the correctness of any goal, emits no proof bodies, makes no external model access, performs no post-execution route selection, reports no benchmark claims or maturity, and excludes launch.","evidence":{"kind":"Computed projection","basis":"Source-faithful refactor","rank":3,"runs_real_tools":false},"command":"PYTHONPATH=src python3 -m microcosm_core.cli target-shape-tactic-routing-gate run-routing-bundle --input examples/target_shape_tactic_routing_gate/exported_target_shape_tactic_routing_bundle --out receipts/runtime_shell/demo_project/organs/target_shape_tactic_routing_gate","tags":["formal-methods","theorem-proving","lean"],"links_to":[{"component_id":"formal_math_verifier_trace_repair_loop","label":"Formal Math Verifier Trace Repair Loop","relative_href":"docs/component-formal-math-verifier-trace-repair-loop.html"},{"component_id":"formal_evidence_cell_anchor_resolver","label":"Formal Evidence Cell Anchor Resolver","relative_href":"docs/component-formal-evidence-cell-anchor-resolver.html"},{"component_id":"proof_diagnostic_evidence_spine","label":"Proof Diagnostic Evidence Spine","relative_href":"docs/component-proof-diagnostic-evidence-spine.html"},{"component_id":"tactic_portfolio_availability_probe","label":"Tactic Portfolio Availability Probe","relative_href":"docs/component-tactic-portfolio-availability-probe.html"},{"component_id":"verifier_lab_kernel","label":"Verifier Lab Kernel","relative_href":"docs/component-verifier-lab-kernel.html"}],"paper_module_backrefs":[{"id":"paper_module.target_shape_tactic_routing","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Target Shape Tactic Routing","url":"paper-module-target-shape-tactic-routing.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[41:paper_module.target_shape_tactic_routing].subjects[0]"}],"primary_paper_module_id":"paper_module.target_shape_tactic_routing","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/target_shape_tactic_routing.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/target_shape_tactic_routing_gate.py"}],"visible_text_byte_count":1474},{"id":"lean_std_premise_index","label":"Lean Std Premise Index","family_id":"formal_math_and_proof","family_label":"Formal math & proof","relative_href":"docs/component-lean-std-premise-index.html","primary_reader_href":"docs/paper-module-lean-std-premise-index.html","component_detail_href":"docs/component-lean-std-premise-index.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Lists a fixed catalog of public Lean building blocks and confirms none hides proof text or test answers.","what_it_does":"Presents a small, fixed catalog of Lean standard-library \"premises\" (named building blocks like facts about numbers, booleans, lists, and basic logic) along with the labels and source references that say where each one comes from. It shows what proof ingredients are on the table and that they were copied from public Lean sources, with no hidden proof text, no Mathlib, and nothing that secretly gives away test answers. It only checks and displays this catalog; it does not run Lean or prove anything.","scope_limit":"It only validates the projection of premise metadata and copied source bodies; it does not run Lean or Lake, prove any theorem correct, expose proof bodies or oracle-needed ids, use external model services, produce benchmark claims, or include launch operations.","evidence":{"kind":"Computed projection","basis":"Source-faithful refactor","rank":3,"runs_real_tools":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.lean_std_premise_index run --input fixtures/first_wave/lean_std_premise_index/input --out receipts/first_wave/lean_std_premise_index --acceptance-out receipts/acceptance/first_wave/lean_std_premise_index_fixture_acceptance.json","tags":["formal-methods","theorem-proving","lean"],"links_to":[{"component_id":"formal_math_verifier_trace_repair_loop","label":"Formal Math Verifier Trace Repair Loop","relative_href":"docs/component-formal-math-verifier-trace-repair-loop.html"},{"component_id":"verifier_lab_kernel","label":"Verifier Lab Kernel","relative_href":"docs/component-verifier-lab-kernel.html"},{"component_id":"mathematical_strategy_atlas_hypothesis_scorer","label":"Mathematical Strategy Atlas Hypothesis Scorer","relative_href":"docs/component-mathematical-strategy-atlas-hypothesis-scorer.html"},{"component_id":"undeclared_library_prior_symbol_classifier","label":"Undeclared Library Prior Symbol Classifier","relative_href":"docs/component-undeclared-library-prior-symbol-classifier.html"}],"paper_module_backrefs":[{"id":"paper_module.lean_std_premise_index","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Lean/Std Premise Index","url":"paper-module-lean-std-premise-index.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[26:paper_module.lean_std_premise_index].subjects[0]"}],"primary_paper_module_id":"paper_module.lean_std_premise_index","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/lean_std_premise_index.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/lean_std_premise_index.py"}],"visible_text_byte_count":1170},{"id":"formal_math_premise_retrieval","label":"Formal Math Premise Retrieval","family_id":"formal_math_and_proof","family_label":"Formal math & proof","relative_href":"docs/component-formal-math-premise-retrieval.html","primary_reader_href":"docs/paper-module-formal-math-premise-retrieval.html","component_detail_href":"docs/component-formal-math-premise-retrieval.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Shows which lemmas a plain search surfaces per query, and never leaks proof text or answer keys.","what_it_does":"Given a small copied set of Lean/Std math-lemma descriptions plus some search queries, this component shows which lemmas a plain term-matching search would surface for each query, how it keeps each assembled context within a fixed size budget, and that it never exposes proof text or \"answer-key\" hints (the premise ids a solver would only get to see after the fact). On the bundled first-wave fixture, the result record shows the retrieval mechanism working in miniature alongside deliberate bad inputs (a leaked proof body, leaked answer-key ids, a budget overflow, an attempt to peek at test answers, and an unknown strategy) that the component catches; the leak and budget guards actually fire.","scope_limit":"It only checks that public retrieval metadata is internally coherent, term-scored over a copied index, budget-bounded, and leakage-clean; it does not run Lean/Lake, use external model services, prove any theorem or its own correctness, claim benchmark performance, or include launch operations.","evidence":{"kind":"Computed projection","basis":"Source-faithful refactor","rank":3,"runs_real_tools":false},"command":"PYTHONPATH=src python -m microcosm_core.organs.formal_math_premise_retrieval run --input fixtures/first_wave/formal_math_premise_retrieval/input --out receipts/first_wave/formal_math_premise_retrieval","tags":["formal-methods","theorem-proving","lean"],"links_to":[{"component_id":"formal_math_lean_proof_witness","label":"Formal Math Lean Proof Witness","relative_href":"docs/component-formal-math-lean-proof-witness.html"},{"component_id":"formal_math_verifier_trace_repair_loop","label":"Formal Math Verifier Trace Repair Loop","relative_href":"docs/component-formal-math-verifier-trace-repair-loop.html"},{"component_id":"lean_std_premise_index","label":"Lean Std Premise Index","relative_href":"docs/component-lean-std-premise-index.html"},{"component_id":"verifier_lab_kernel","label":"Verifier Lab Kernel","relative_href":"docs/component-verifier-lab-kernel.html"},{"component_id":"mathematical_strategy_atlas_hypothesis_scorer","label":"Mathematical Strategy Atlas Hypothesis Scorer","relative_href":"docs/component-mathematical-strategy-atlas-hypothesis-scorer.html"},{"component_id":"ring2_premise_retrieval_precision_recall_harness","label":"Ring2 Premise Retrieval Precision Recall Harness","relative_href":"docs/component-ring2-premise-retrieval-precision-recall-harness.html"}],"paper_module_backrefs":[{"id":"paper_module.formal_math_premise_retrieval","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Formal Math Premise Retrieval","url":"paper-module-formal-math-premise-retrieval.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[25:paper_module.formal_math_premise_retrieval].subjects[0]"}],"primary_paper_module_id":"paper_module.formal_math_premise_retrieval","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/formal_math_premise_retrieval.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/formal_math_premise_retrieval.py"}],"visible_text_byte_count":1321},{"id":"formal_math_verifier_trace_repair_loop","label":"Formal Math Verifier Trace Repair Loop","family_id":"formal_math_and_proof","family_label":"Formal math & proof","relative_href":"docs/component-formal-math-verifier-trace-repair-loop.html","primary_reader_href":"docs/paper-module-formal-math-verifier-trace-repair-loop.html","component_detail_href":"docs/component-formal-math-verifier-trace-repair-loop.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Replays how a proof lab turns verifier failures into fixes, with no promotion without a fresh re-run.","what_it_does":"It replays how a proof-lab turns a verifier's failure feedback into a teaching signal, working from copied (non-secret) run data so the failure categories, the repair action tied to each failure, and the rule that nothing gets promoted without a fresh re-run result record are all inspectable. Actual proofs, answer keys, and model outputs are deliberately kept out, so the whole correction loop is visible without exposing any of them.","scope_limit":"It demonstrates control-loop projection mechanics over copied Ring2 run rows only; it does not run Lean/Lake, use external model services, expose proof bodies or oracle premise ids, treat human or provider advice as correctness, prove any theorem, or include launch operations.","evidence":{"kind":"Computed projection","basis":"Source-faithful refactor","rank":3,"runs_real_tools":false},"command":"microcosm formal-math-verifier-trace-repair-loop run-loop-bundle --input examples/formal_math_verifier_trace_repair_loop/exported_verifier_trace_repair_bundle --out receipts/runtime_shell/demo_project/organs/formal_math_verifier_trace_repair_loop","tags":["formal-methods","theorem-proving","lean"],"links_to":[{"component_id":"formal_evidence_cell_anchor_resolver","label":"Formal Evidence Cell Anchor Resolver","relative_href":"docs/component-formal-evidence-cell-anchor-resolver.html"},{"component_id":"proof_diagnostic_evidence_spine","label":"Proof Diagnostic Evidence Spine","relative_href":"docs/component-proof-diagnostic-evidence-spine.html"},{"component_id":"verifier_lab_kernel","label":"Verifier Lab Kernel","relative_href":"docs/component-verifier-lab-kernel.html"},{"component_id":"target_shape_tactic_routing_gate","label":"Target Shape Tactic Routing Gate","relative_href":"docs/component-target-shape-tactic-routing-gate.html"},{"component_id":"verifier_lab_execution_spine","label":"Verifier Lab Execution Spine","relative_href":"docs/component-verifier-lab-execution-spine.html"}],"paper_module_backrefs":[{"id":"paper_module.formal_math_verifier_trace_repair_loop","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Formal Math Verifier Trace Repair Loop","url":"paper-module-formal-math-verifier-trace-repair-loop.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[23:paper_module.formal_math_verifier_trace_repair_loop].subjects[0]"}],"primary_paper_module_id":"paper_module.formal_math_verifier_trace_repair_loop","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json#paper_module.formal_math_verifier_trace_repair_loop"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/formal_math_verifier_trace_repair_loop.py"}],"visible_text_byte_count":1102},{"id":"formal_evidence_cell_anchor_resolver","label":"Formal Evidence Cell Anchor Resolver","family_id":"formal_math_and_proof","family_label":"Formal math & proof","relative_href":"docs/component-formal-evidence-cell-anchor-resolver.html","primary_reader_href":"docs/paper-module-formal-evidence-cell-anchor-resolver.html","component_detail_href":"docs/component-formal-evidence-cell-anchor-resolver.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Resolves each proof-flavored math claim to named evidence and flags ones that overreach or lack backing.","what_it_does":"When the project's writeups make proof-flavored claims about its formal-math work, this component checks each claim against a named piece of recorded evidence and the public reference files in the repo, confirms the claim is no stronger than that evidence allows, and flags claims that have no backing or that overreach. The record shows which claims are anchored to evidence and which are just words, while proof contents and any private file references are kept out of the output.","scope_limit":"It validates claim-to-evidence anchoring mechanics only: claim-to-cell resolution, source-anchor presence, permitted claim strength, copied-source-module digest checks, and leakage refusals. It does not run Lean/Lake, certify theorem or mathematical correctness, expose proof bodies or non-public source refs, use external model services, or include launch operations/public sharing.","evidence":{"kind":"Computed projection","basis":"Source-faithful refactor","rank":3,"runs_real_tools":false},"command":"microcosm formal-evidence-cell-anchor-resolver run-anchor-bundle --input examples/formal_evidence_cell_anchor_resolver/exported_evidence_cell_anchor_bundle --out receipts/runtime_shell/demo_project/organs/formal_evidence_cell_anchor_resolver","tags":["formal-methods","theorem-proving","lean"],"links_to":[{"component_id":"formal_math_verifier_trace_repair_loop","label":"Formal Math Verifier Trace Repair Loop","relative_href":"docs/component-formal-math-verifier-trace-repair-loop.html"},{"component_id":"proof_diagnostic_evidence_spine","label":"Proof Diagnostic Evidence Spine","relative_href":"docs/component-proof-diagnostic-evidence-spine.html"},{"component_id":"target_shape_tactic_routing_gate","label":"Target Shape Tactic Routing Gate","relative_href":"docs/component-target-shape-tactic-routing-gate.html"}],"paper_module_backrefs":[{"id":"paper_module.formal_evidence_cell_anchor_resolver","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Formal Evidence Cell Anchor Resolver","url":"paper-module-formal-evidence-cell-anchor-resolver.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[24:paper_module.formal_evidence_cell_anchor_resolver].subjects[0]"}],"primary_paper_module_id":"paper_module.formal_evidence_cell_anchor_resolver","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/formal_evidence_cell_anchor_resolver.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/formal_evidence_cell_anchor_resolver.py"}],"visible_text_byte_count":1250},{"id":"undeclared_library_prior_symbol_classifier","label":"Undeclared Library Prior Symbol Classifier","family_id":"formal_math_and_proof","family_label":"Formal math & proof","relative_href":"docs/component-undeclared-library-prior-symbol-classifier.html","primary_reader_href":"docs/paper-module-undeclared-library-prior-classifier.html","component_detail_href":"docs/component-undeclared-library-prior-symbol-classifier.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Detects when a checked Lean proof cites a library result outside its approved set.","what_it_does":"It checks whether a Lean proof cites a library result (a lemma or definition) that was never on its approved list. Even after a prover accepts a proof, that proof can still quietly use a library symbol it wasn't allowed to, and this component surfaces those out-of-bounds uses as an inspectable record that names each symbol and where the rule came from. It matters because \"the proof checked\" does not mean \"the proof stayed within the allowed set of building blocks,\" and this makes that gap visible without ever reading the proof's own steps.","scope_limit":"It only projects the symbol-boundary classification mechanic over copied Lean/Std premise rows and pre-extracted symbol observations; it does not read proof source, run Lean or Lake, prove formal-result correctness, treat the whole standard library as an implicit allowlist, claim Mathlib availability, use external model services, or include launch operations.","evidence":{"kind":"Computed projection","basis":"Source-faithful refactor","rank":3,"runs_real_tools":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.undeclared_library_prior_symbol_classifier run --input fixtures/first_wave/undeclared_library_prior_symbol_classifier/input --out receipts/first_wave/undeclared_library_prior_symbol_classifier","tags":["formal-methods","theorem-proving","lean"],"links_to":[{"component_id":"corpus_readiness_mathlib_absence_gate","label":"Corpus Readiness Mathlib Absence Gate","relative_href":"docs/component-corpus-readiness-mathlib-absence-gate.html"},{"component_id":"tactic_portfolio_availability_probe","label":"Tactic Portfolio Availability Probe","relative_href":"docs/component-tactic-portfolio-availability-probe.html"},{"component_id":"provider_context_recipe_budget_policy","label":"Provider Context Recipe Budget Policy","relative_href":"docs/component-provider-context-recipe-budget-policy.html"}],"paper_module_backrefs":[{"id":"paper_module.undeclared_library_prior_classifier","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Undeclared Library Prior Classifier","url":"paper-module-undeclared-library-prior-classifier.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[55:paper_module.undeclared_library_prior_classifier].subjects[0]"}],"primary_paper_module_id":"paper_module.undeclared_library_prior_classifier","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/undeclared_library_prior_classifier.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/undeclared_library_prior_symbol_classifier.py"}],"visible_text_byte_count":1274},{"id":"ring2_premise_retrieval_precision_recall_harness","label":"Ring2 Premise Retrieval Precision Recall Harness","family_id":"formal_math_and_proof","family_label":"Formal math & proof","relative_href":"docs/component-ring2-premise-retrieval-precision-recall-harness.html","primary_reader_href":"docs/paper-module-ring2-premise-precision-recall.html","component_detail_href":"docs/component-ring2-premise-retrieval-precision-recall-harness.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Scores how much proof support a premise search found, problem by problem.","what_it_does":"When a math-proving system searches for the supporting facts (\"premises\") a proof will need, this component replays saved records of that search and reports, problem by problem, how much of the needed support the search actually turned up. Per problem it labels one of four outcomes: the search found everything needed and the proof went through; it found everything needed but the proof still failed; it found only some of the needed support; or it found none of it. Separating \"the proof failed even though every needed premise was found\" from \"the proof failed because the search missed a needed premise\" shows which part to fix. It also runs as a regression guard that refuses inputs which try to slip the answer into the search itself (the known-correct premises planted in the ranked results), leak proof text, tune on the test answers, or claim more than retrieval-quality numbers.","scope_limit":"These are after-the-fact retrieval-attribution labels and precision/recall counts over copied run records only. The component does not run Lean or Lake, call any provider, expose proof bodies, tune on test answers, claim benchmark performance, prove formal-result correctness, or include launch operations, and its labels are explicitly forbidden from flowing into provider context. The aggregate numbers describe only the copied fixture/bundle replayed, not any benchmark claims.","evidence":{"kind":"Computed projection","basis":"Source-faithful refactor","rank":3,"runs_real_tools":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.ring2_premise_retrieval_precision_recall_harness run --input fixtures/first_wave/ring2_premise_retrieval_precision_recall_harness/input --out receipts/first_wave/ring2_premise_retrieval_precision_recall_harness","tags":["formal-methods","theorem-proving","lean"],"links_to":[{"component_id":"mathematical_strategy_atlas_hypothesis_scorer","label":"Mathematical Strategy Atlas Hypothesis Scorer","relative_href":"docs/component-mathematical-strategy-atlas-hypothesis-scorer.html"},{"component_id":"verifier_lab_kernel","label":"Verifier Lab Kernel","relative_href":"docs/component-verifier-lab-kernel.html"}],"paper_module_backrefs":[{"id":"paper_module.ring2_premise_precision_recall","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Ring-2 Premise Precision Recall","url":"paper-module-ring2-premise-precision-recall.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[42:paper_module.ring2_premise_precision_recall].subjects[0]"}],"primary_paper_module_id":"paper_module.ring2_premise_precision_recall","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/ring2_premise_precision_recall.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/ring2_premise_retrieval_precision_recall_harness.py"}],"visible_text_byte_count":1751},{"id":"verifier_lab_kernel","label":"Verifier Lab Kernel","family_id":"formal_math_and_proof","family_label":"Formal math & proof","relative_href":"docs/component-verifier-lab-kernel.html","primary_reader_href":"docs/paper-module-verifier-lab-kernel.html","component_detail_href":"docs/component-verifier-lab-kernel.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Folds nine proof checks into one report labeling each line by which source actually backs it.","what_it_does":"This assembly point for the Lean/proof toolkit runs nine smaller formal-math pipeline checkers together and folds their results into one leak-proof report that labels every line by where it came from: a Lean verifier, an answer-key (oracle) comparator, an AI suggestion, a retrieval miss, or a row thrown out for breaking the rules. The report carries only references, hashes, counts, and verdicts, never the actual proof text, AI output, or answer-key bodies. One result record shows which claims a Lean verifier actually backed versus which are just hints or were rejected, instead of leaving a pile of separate outputs to be taken on faith.","scope_limit":"It validates the declared public contract shape of the proof packet and component result records only; it does not establish anything correct, count oracle/provider output as forward proof success, import private or Mathlib-dependent proof bodies, use external model services, change source files, or claim benchmark solve rates, launch, or maturity.","evidence":{"kind":"Contract validator","basis":"Import validation","rank":5,"runs_real_tools":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.verifier_lab_kernel run --input fixtures/first_wave/verifier_lab_kernel/input --out receipts/first_wave/verifier_lab_kernel","tags":["formal-methods","theorem-proving","lean"],"links_to":[{"component_id":"corpus_readiness_mathlib_absence_gate","label":"Corpus Readiness Mathlib Absence Gate","relative_href":"docs/component-corpus-readiness-mathlib-absence-gate.html"},{"component_id":"lean_std_premise_index","label":"Lean Std Premise Index","relative_href":"docs/component-lean-std-premise-index.html"},{"component_id":"formal_math_premise_retrieval","label":"Formal Math Premise Retrieval","relative_href":"docs/component-formal-math-premise-retrieval.html"},{"component_id":"tactic_portfolio_availability_probe","label":"Tactic Portfolio Availability Probe","relative_href":"docs/component-tactic-portfolio-availability-probe.html"},{"component_id":"target_shape_tactic_routing_gate","label":"Target Shape Tactic Routing Gate","relative_href":"docs/component-target-shape-tactic-routing-gate.html"},{"component_id":"ring2_premise_retrieval_precision_recall_harness","label":"Ring2 Premise Retrieval Precision Recall Harness","relative_href":"docs/component-ring2-premise-retrieval-precision-recall-harness.html"},{"component_id":"formal_math_verifier_trace_repair_loop","label":"Formal Math Verifier Trace Repair Loop","relative_href":"docs/component-formal-math-verifier-trace-repair-loop.html"},{"component_id":"proof_diagnostic_evidence_spine","label":"Proof Diagnostic Evidence Spine","relative_href":"docs/component-proof-diagnostic-evidence-spine.html"},{"component_id":"formal_math_lean_proof_witness","label":"Formal Math Lean Proof Witness","relative_href":"docs/component-formal-math-lean-proof-witness.html"},{"component_id":"verifier_lab_execution_spine","label":"Verifier Lab Execution Spine","relative_href":"docs/component-verifier-lab-execution-spine.html"}],"paper_module_backrefs":[{"id":"paper_module.verifier_lab_kernel","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Verifier Lab Kernel","url":"paper-module-verifier-lab-kernel.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[0:paper_module.verifier_lab_kernel].subjects[0]"}],"primary_paper_module_id":"paper_module.verifier_lab_kernel","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/verifier_lab_kernel.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/verifier_lab_kernel.py"}],"visible_text_byte_count":1280},{"id":"proof_derived_governed_mutation_authorization","label":"Proof Derived Governed Mutation Authorization","family_id":"formal_math_and_proof","family_label":"Formal math & proof","relative_href":"docs/component-proof-derived-governed-mutation-authorization.html","primary_reader_href":"docs/paper-module-proof-derived-governed-mutation-authorization.html","component_detail_href":"docs/component-proof-derived-governed-mutation-authorization.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Checks a synthetic change-authorization record for its proof-and-approval chain, bound to a real commit.","what_it_does":"Replays a make-believe example of \"should this change be allowed to run?\" and shows, step by step, why each proposed action was permitted. All three actions (a look-only inspection, a small config write, and an undo of that write) had to carry proof evidence and two visible policy approvals before anything was admitted; on top of that, the two actions that actually change something (the config write and the undo) also had to show a logged record of the change and a matching undo result record. Just holding a password or account secret is never treated as permission, and nothing here touches a real account or makes any real change.","scope_limit":"It validates only a declared, synthetic governed-mutation contract and excludes live cloud/account action, standing account secrets, source or irreversible mutation, policy-after-execution, hidden votes, external model access, benchmark-score claims, or launch.","evidence":{"kind":"Contract validator","basis":"Import validation","rank":5,"runs_real_tools":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.proof_derived_governed_mutation_authorization run --input fixtures/first_wave/proof_derived_governed_mutation_authorization/input --out receipts/first_wave/proof_derived_governed_mutation_authorization --acceptance-out receipts/acceptance/first_wave/proof_derived_governed_mutation_authorization_fixture_acceptance.json","tags":["formal-methods","theorem-proving","lean"],"links_to":[{"component_id":"batch12_release_claim_language_gate","label":"Release Public Wording Gate","relative_href":"docs/component-batch12-release-claim-language-gate.html"},{"component_id":"batch4_proof_authority_runtime","label":"Proof / Control / Runtime Import Bundle","relative_href":"docs/component-batch4-proof-authority-runtime.html"},{"component_id":"batch8_policy_engines_capsule","label":"Policy Engines Bundle","relative_href":"docs/component-batch8-policy-engines-capsule.html"}],"paper_module_backrefs":[{"id":"paper_module.proof_derived_governed_mutation_authorization","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Proof-Derived Governed Mutation Authorization","url":"paper-module-proof-derived-governed-mutation-authorization.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[15:paper_module.proof_derived_governed_mutation_authorization].subjects[0]"}],"primary_paper_module_id":"paper_module.proof_derived_governed_mutation_authorization","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json#paper_module.proof_derived_governed_mutation_authorization"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/proof_derived_governed_mutation_authorization.py"}],"visible_text_byte_count":1419},{"id":"finite_erdos_denominator_certificate_strike","label":"Finite Erdos Denominator-Order Certificate Strike","family_id":"formal_math_and_proof","family_label":"Formal math & proof","relative_href":"docs/component-finite-erdos-denominator-certificate-strike.html","primary_reader_href":"docs/paper-module-finite-erdos-denominator-certificate-strike.html","component_detail_href":"docs/component-finite-erdos-denominator-certificate-strike.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Computes an exact-arithmetic denominator-order identity and catches forged ones, not the open Erdos problem.","what_it_does":"Computes a finite denominator-order certificate in exact whole-number arithmetic. For a finite set F of exponents and a base b, it sums 1/(b^n - 1) over the exponents as an exact fraction P/Q, then checks a sharp claim: the multiplicative order of b modulo the reduced denominator Q is exactly the least common multiple of F. It shows the certificate holding on small cases \u2014 including one where a common factor cancels in the fraction yet the claim still holds \u2014 and it shows two forged certificates being caught: a wrong order and a wrong denominator are both rejected because the runner recomputes the real values and compares. This is the finite identity behind the Erdos #257 period-noncollapse strike; it is not a solution to the open problem.","scope_limit":"It computes the finite denominator-order certificate ord_Q(b)=lcm(F) for S_F(b)=sum 1/(b^n-1)=P/Q in exact rational arithmetic over bounded public fixtures and rejects forged certificates by recomputation; it does not establish the open infinite Erdos #257 problem, is not an oracle, prover, or provider result, and a holding certificate is a bounded computational witness, not a machine-checked proof.","evidence":{"kind":"Bounded runtime computation","basis":"Real runtime result","rank":4,"runs_real_tools":true},"command":"microcosm finite-erdos-denominator-certificate-strike run --input fixtures/first_wave/finite_erdos_denominator_certificate_strike/input --out receipts/first_wave/finite_erdos_denominator_certificate_strike --acceptance-out receipts/acceptance/first_wave/finite_erdos_denominator_certificate_strike_fixture_acceptance.json","tags":["formal-math","number-theory","exact-arithmetic"],"links_to":[{"component_id":"certificate_kernel_execution_lab","label":"Certificate Kernel Execution Lab","relative_href":"docs/component-certificate-kernel-execution-lab.html"},{"component_id":"verifier_lab_kernel","label":"Verifier Lab Kernel","relative_href":"docs/component-verifier-lab-kernel.html"},{"component_id":"formal_math_lean_proof_witness","label":"Formal Math Lean Proof Witness","relative_href":"docs/component-formal-math-lean-proof-witness.html"}],"paper_module_backrefs":[{"id":"paper_module.finite_erdos_denominator_certificate_strike","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Finite Erdos Denominator-Order Certificate Strike","url":"paper-module-finite-erdos-denominator-certificate-strike.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[94:paper_module.finite_erdos_denominator_certificate_strike].subjects[0]"}],"primary_paper_module_id":"paper_module.finite_erdos_denominator_certificate_strike","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json#paper_module.finite_erdos_denominator_certificate_strike"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/finite_erdos_denominator_certificate_strike.py"}],"visible_text_byte_count":1637},{"id":"lean_proof_search_lab_runtime","label":"Lean Proof-Search Lab Runtime","family_id":"formal_math_and_proof","family_label":"Formal math & proof","relative_href":"docs/component-lean-proof-search-lab-runtime.html","primary_reader_href":"docs/paper-module-lean-proof-search-lab-runtime.html","component_detail_href":"docs/component-lean-proof-search-lab-runtime.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Finds tactic scripts the real Lean prover accepts on toy theorems, and refuses proofs that cheat.","what_it_does":"A small, runnable proof-search lab for toy logic theorems, checked by the real Lean theorem prover. For each toy theorem it searches over candidate tactic scripts, compiles each one with Lean, and keeps a script only when Lean actually closes the proof. It refuses three ways of cheating: forwarding a ready-made proof body (an oracle leak), leaning on `sorry` or extra axioms (an unclean proof), and a policy that only picks the right tactic because it recognised the problem's name (memorisation, caught by renaming the problem). It shows one honest search succeeding and four cheats being rejected. Lean is optional: install it to unlock the real run; without it the lab reports itself locked and claims nothing, rather than pretending to have proved anything.","scope_limit":"This standard governs a bounded symbolic Lean proof-search lab over tiny public toy-theorem fixtures whose verdict source is the installed Lean subprocess only. It is not neural theorem proving, does not solve any open mathematical problem, does not forward oracle proof bodies, is not frontier-scale math automation or online-RL bandit search, is not a private source prover-run export, and excludes launch or public sharing.","evidence":{"kind":"External tool run","basis":"Real runtime result","rank":4,"runs_real_tools":true},"command":"microcosm lean-proof-search-lab-runtime run --input fixtures/first_wave/lean_proof_search_lab_runtime/input --out receipts/first_wave/lean_proof_search_lab_runtime --acceptance-out receipts/acceptance/first_wave/lean_proof_search_lab_runtime_fixture_acceptance.json","tags":["formal-math","theorem-proving","proof-search"],"links_to":[{"component_id":"certificate_kernel_execution_lab","label":"Certificate Kernel Execution Lab","relative_href":"docs/component-certificate-kernel-execution-lab.html"},{"component_id":"verifier_lab_kernel","label":"Verifier Lab Kernel","relative_href":"docs/component-verifier-lab-kernel.html"},{"component_id":"formal_math_lean_proof_witness","label":"Formal Math Lean Proof Witness","relative_href":"docs/component-formal-math-lean-proof-witness.html"}],"paper_module_backrefs":[{"id":"paper_module.lean_proof_search_lab_runtime","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Lean Proof-Search Lab Runtime","url":"paper-module-lean-proof-search-lab-runtime.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[102:paper_module.lean_proof_search_lab_runtime].subjects[0]"}],"primary_paper_module_id":"paper_module.lean_proof_search_lab_runtime","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json#paper_module.lean_proof_search_lab_runtime"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/lean_proof_search_lab_runtime.py"}],"visible_text_byte_count":1584},{"id":"agent_closeout_faithfulness_audit","label":"Agent Completion Faithfulness Audit","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","relative_href":"docs/component-agent-closeout-faithfulness-audit.html","primary_reader_href":"docs/paper-module-agent-closeout-faithfulness-audit.html","component_detail_href":"docs/component-agent-closeout-faithfulness-audit.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Runs real git and pytest on a sample repo so wrap-up claims state only what the evidence proves.","what_it_does":"Runs a public fixture repo through real git and pytest subprocesses, then checks that completion claims only say what the evidence supports: commit object exists, ledger cap exists, and pytest pass is claimed only after exit-zero status was checked.","scope_limit":"verified means the referenced evidence object exists or a pytest span ran; it does not imply the span passed unless exit-zero status was explicitly checked","evidence":{"kind":"External tool run","basis":"Real runtime result","rank":4,"runs_real_tools":true},"command":"microcosm agent-closeout-faithfulness-audit run --input fixtures/first_wave/agent_closeout_faithfulness_audit/input --out receipts/first_wave/agent_closeout_faithfulness_audit","tags":["ai-safety","agent-evaluation","red-teaming"],"links_to":[],"paper_module_backrefs":[{"id":"paper_module.agent_closeout_faithfulness_audit","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Agent Completion Faithfulness Audit","url":"paper-module-agent-closeout-faithfulness-audit.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[12:paper_module.agent_closeout_faithfulness_audit].subjects[0]"}],"primary_paper_module_id":"paper_module.agent_closeout_faithfulness_audit","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json#paper_module.agent_closeout_faithfulness_audit"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/agent_closeout_faithfulness_audit.py"}],"visible_text_byte_count":714},{"id":"bounded_autonomy_campaign_packet","label":"Bounded Autonomy Campaign Packet","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","relative_href":"docs/component-bounded-autonomy-campaign-packet.html","primary_reader_href":"docs/paper-module-bounded-autonomy-campaign-packet.html","component_detail_href":"docs/component-bounded-autonomy-campaign-packet.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Drafts proposed work from coverage gaps and proves it cannot repair or rewrite the code itself.","what_it_does":"Turns synthetic coverage gaps into a draft candidate packet in a subprocess and records the boundary that it proposes work but cannot repair itself or write source.","scope_limit":"self-proposal campaign packet only; no self-repair or unsupervised source-file changes","evidence":{"kind":"External tool run","basis":"Real runtime result","rank":4,"runs_real_tools":true},"command":"microcosm bounded-autonomy-campaign-packet run --input fixtures/first_wave/bounded_autonomy_campaign_packet/input --out receipts/first_wave/bounded_autonomy_campaign_packet","tags":["ai-safety","agent-evaluation","red-teaming"],"links_to":[],"paper_module_backrefs":[{"id":"paper_module.bounded_autonomy_campaign_packet","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Bounded Autonomy Campaign Packet","url":"paper-module-bounded-autonomy-campaign-packet.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[45:paper_module.bounded_autonomy_campaign_packet].subjects[0]"}],"primary_paper_module_id":"paper_module.bounded_autonomy_campaign_packet","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/bounded_autonomy_campaign_packet.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/bounded_autonomy_campaign_packet.py"}],"visible_text_byte_count":553},{"id":"provider_context_recipe_budget_policy","label":"Provider Context Recipe Budget Policy","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","relative_href":"docs/component-provider-context-recipe-budget-policy.html","primary_reader_href":"docs/paper-module-provider-context-recipe-budget.html","component_detail_href":"docs/component-provider-context-recipe-budget-policy.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Runs the real context harness to measure assembled byte sizes and check each bundle fits its budget.","what_it_does":"This component checks that the bundles of context an AI agent would assemble before calling an outside model provider stay inside fixed size limits (in bytes), fill their sections in the declared order until the budget runs out, list any section that was dropped for not fitting, and never carry answer keys, proof solutions, or other \"correct answer\" material. The record shows the exact size ceilings for each recipe, which sections fit versus got left out, and which output each recipe is allowed to produce, so the context boundary is inspectable as plain accounting before any external model access or answer authority is ever in play. It only validates this metadata; it does not itself call any provider.","scope_limit":"It validates context-budget projection mechanics (byte ceilings, ordered section fill, omitted-section manifests, deliverable routing, and digest-checked source-body imports) only. It excludes provider/API calls, run Lean/Lake, expose or carry proof or oracle truth-side material, assert theorem or domain-level conclusions, or include launch operations.","evidence":{"kind":"Bounded runtime computation","basis":"Real runtime result","rank":4,"runs_real_tools":true},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.provider_context_recipe_budget_policy run --input fixtures/first_wave/provider_context_recipe_budget_policy/input --out receipts/first_wave/provider_context_recipe_budget_policy","tags":["ai-safety","agent-evaluation","red-teaming"],"links_to":[{"component_id":"bounded_autonomy_campaign_packet","label":"Bounded Autonomy Campaign Packet","relative_href":"docs/component-bounded-autonomy-campaign-packet.html"},{"component_id":"tool_server_pressure_inventory","label":"Tool Server Pressure Inventory","relative_href":"docs/component-tool-server-pressure-inventory.html"}],"paper_module_backrefs":[{"id":"paper_module.provider_context_recipe_budget","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Provider Context Recipe Budget","url":"paper-module-provider-context-recipe-budget.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[54:paper_module.provider_context_recipe_budget].subjects[0]"}],"primary_paper_module_id":"paper_module.provider_context_recipe_budget","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/provider_context_recipe_budget.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/provider_context_recipe_budget_policy.py"}],"visible_text_byte_count":1431},{"id":"batch10_cold_eval_honesty_capsule","label":"Cold Evaluation Honesty Bundle","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","relative_href":"docs/component-batch10-cold-eval-honesty-capsule.html","primary_reader_href":"docs/paper-module-batch10-cold-eval-honesty-capsule.html","component_detail_href":"docs/component-batch10-cold-eval-honesty-capsule.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Runs a copied route-quality simulator and checks its all-B scorecard against the original code.","what_it_does":"This component imports the real cold_eval.py route-quality simulator as an exact source copy. Running it over a synthetic workspace inspects the all-B scorecard shape, source-module digest evidence, and scope limit checks without exporting body text in result records or turning the fixture into benchmark truth.","scope_limit":"verified cold-eval source body import only, not a live benchmark, navigation truth, source authority, external model access, whole-system equivalence, public sharing, or launch-scope decision","evidence":{"kind":"Verified source import","basis":"Copied source body","rank":5,"runs_real_tools":false},"command":"microcosm batch10-cold-eval-honesty-capsule run --input fixtures/first_wave/batch10_cold_eval_honesty_capsule/input --out receipts/first_wave/batch10_cold_eval_honesty_capsule --acceptance-out receipts/acceptance/first_wave/batch10_cold_eval_honesty_capsule_fixture_acceptance.json","tags":["ai-safety","agent-evaluation","red-teaming"],"links_to":[],"paper_module_backrefs":[{"id":"paper_module.batch10_cold_eval_honesty_capsule","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Set 10 Cold Eval Honesty Bundle","url":"paper-module-batch10-cold-eval-honesty-capsule.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[67:paper_module.batch10_cold_eval_honesty_capsule].subjects[0]"}],"primary_paper_module_id":"paper_module.batch10_cold_eval_honesty_capsule","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/batch10_cold_eval_honesty_capsule.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/batch10_cold_eval_honesty_capsule.py"}],"visible_text_byte_count":913},{"id":"batch8_validator_checker_capsule","label":"Validator Checker Bundle","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","relative_href":"docs/component-batch8-validator-checker-capsule.html","primary_reader_href":"docs/paper-module-batch8-validator-checker-capsule.html","component_detail_href":"docs/component-batch8-validator-checker-capsule.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Runs the real validator code over public examples so its safety checks stay inspectable.","what_it_does":"This component imports the real idea_microcosm validators.py body as an exact source copy. Running it shows status-policy judging, private-boundary scans, specimen checks, launch-gate checks, and the validate entrypoint exercised against public fixtures and negative cases.","scope_limit":"It validates only the imported validators.py source body and its checker membrane. It does not claim source authority, a full validator-suite proof, whole-system equivalence, launch, hosted-public status, public sharing, external model access, or source-file changes.","evidence":{"kind":"Verified source import","basis":"Copied source body","rank":5,"runs_real_tools":false},"command":"microcosm batch8-validator-checker-capsule run --input fixtures/first_wave/batch8_validator_checker_capsule/input --out receipts/first_wave/batch8_validator_checker_capsule --acceptance-out receipts/acceptance/first_wave/batch8_validator_checker_capsule_fixture_acceptance.json","tags":["ai-safety","agent-evaluation","red-teaming"],"links_to":[{"component_id":"batch10_cold_eval_honesty_capsule","label":"Cold Evaluation Honesty Bundle","relative_href":"docs/component-batch10-cold-eval-honesty-capsule.html"},{"component_id":"batch12_release_claim_language_gate","label":"Release Public Wording Gate","relative_href":"docs/component-batch12-release-claim-language-gate.html"}],"paper_module_backrefs":[{"id":"paper_module.batch8_validator_checker_capsule","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Set 8 Validator Checker Bundle","url":"paper-module-batch8-validator-checker-capsule.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[63:paper_module.batch8_validator_checker_capsule].subjects[0]"}],"primary_paper_module_id":"paper_module.batch8_validator_checker_capsule","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/batch8_validator_checker_capsule.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/batch8_validator_checker_capsule.py"}],"visible_text_byte_count":933},{"id":"batch7_secondary_runtime_capsule","label":"Secondary Runtime Source Bundle","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","relative_href":"docs/component-batch7-secondary-runtime-capsule.html","primary_reader_href":"docs/paper-module-batch7-secondary-runtime-capsule.html","component_detail_href":"docs/component-batch7-secondary-runtime-capsule.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Runs eight trace, graph, and market engines on test rows without fetching live markets.","what_it_does":"This bundle imports a second Set 7 runtime slice as public runnable system. It checks agent trace view-model trust classes, lane-progress state normalization, graph-lens focus roles, graph projection summaries, observe-only cartography rendering, stockgrid payload terms, Polymarket CLOB microstructure, and four-lens market scanning over synthetic public fixtures without exporting sessions, fetching live markets, or giving trading decisions.","scope_limit":"verified source body import only; no browser/session export, wallet authority, live market data, investment-related actions, external model access, source-file changes, whole-system equivalence, public sharing, launch, semantic-truth, or whole-system correctness claim","evidence":{"kind":"Verified source import","basis":"Copied source body","rank":5,"runs_real_tools":false},"command":"microcosm batch7-secondary-runtime-capsule run --input fixtures/first_wave/batch7_secondary_runtime_capsule/input --out receipts/first_wave/batch7_secondary_runtime_capsule --acceptance-out receipts/acceptance/first_wave/batch7_secondary_runtime_capsule_fixture_acceptance.json","tags":["source intake","provenance","drift-control"],"links_to":[],"paper_module_backrefs":[{"id":"paper_module.batch7_secondary_runtime_capsule","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Set 7 Secondary Runtime Bundle","url":"paper-module-batch7-secondary-runtime-capsule.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[91:paper_module.batch7_secondary_runtime_capsule].subjects[0]"}],"primary_paper_module_id":"paper_module.batch7_secondary_runtime_capsule","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/batch7_secondary_runtime_capsule.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/batch7_secondary_runtime_capsule.py"}],"visible_text_byte_count":1111},{"id":"agent_benchmark_integrity_anti_gaming_replay","label":"Agent Benchmark Integrity Anti Gaming Replay","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","relative_href":"docs/component-agent-benchmark-integrity-anti-gaming-replay.html","primary_reader_href":"docs/paper-module-agent-benchmark-integrity-anti-gaming-replay.html","component_detail_href":"docs/component-agent-benchmark-integrity-anti-gaming-replay.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Validates a synthetic benchmark-integrity record and flags the contamination cases it declares.","what_it_does":"Checks a public benchmark-integrity example bundle that contains three copied source pattern provenance bodies under source_artifacts. The component verifies the source-module manifest digests, requires each replay row to cite those copied source artifacts, recomputes pass/quarantine verdicts from contamination, file-access, and locked-evaluator spans, and rejects common gaming attempts such as peeking at hidden answers, training on the test set, exposing the oracle patch, cherry-picking the best of many tries, or asserting a score. It still does not run real bug fixes or claim any benchmark claims.","scope_limit":"It authorizes only bounded public runtime validation over copied source-open pattern provenance bodies and metadata-only benchmark-integrity replay rows; it does not establish any benchmark or SWE-bench score, agent capability, external model service, live-repo mutation, private/oracle/hidden-gold body access, product progress, or launch-scope decision.","evidence":{"kind":"Computed projection","basis":"Source-faithful refactor","rank":3,"runs_real_tools":false},"command":"microcosm agent-benchmark-integrity-anti-gaming-replay run-benchmark-integrity-bundle --input examples/agent_benchmark_integrity_anti_gaming_replay/exported_benchmark_integrity_bundle --out .microcosm/agent_benchmark_integrity_anti_gaming_replay","tags":["ai-safety","agent-evaluation","red-teaming"],"links_to":[{"component_id":"research_replication_rubric_artifact_replay","label":"Research Replication Rubric Artifact Replay","relative_href":"docs/component-research-replication-rubric-artifact-replay.html"},{"component_id":"batch10_cold_eval_honesty_capsule","label":"Cold Evaluation Honesty Bundle","relative_href":"docs/component-batch10-cold-eval-honesty-capsule.html"}],"paper_module_backrefs":[{"id":"paper_module.agent_benchmark_integrity_anti_gaming_replay","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Agent Benchmark Integrity Anti-Gaming Replay","url":"paper-module-agent-benchmark-integrity-anti-gaming-replay.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[3:paper_module.agent_benchmark_integrity_anti_gaming_replay].subjects[0]"}],"primary_paper_module_id":"paper_module.agent_benchmark_integrity_anti_gaming_replay","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/agent_benchmark_integrity_anti_gaming_replay.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/agent_benchmark_integrity_anti_gaming_replay.py"}],"visible_text_byte_count":1349},{"id":"agent_monitor_redteam_falsification_replay","label":"Monitor Evidence-Boundary Replay","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","relative_href":"docs/component-agent-monitor-redteam-falsification-replay.html","primary_reader_href":"docs/paper-module-agent-monitor-redteam-falsification-replay.html","component_detail_href":"docs/component-agent-monitor-redteam-falsification-replay.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Replays honest and deceptive agent runs and flags any verdict missing its declared backing evidence.","what_it_does":"It replays three made-up agent trajectories (a normal one, one trying to hide a failure, and one overstepping a tool's permissions) and checks that every monitor verdict \u2014 including a clean \"pass\" \u2014 comes with the result records needed to re-run and verify it. The result records show exactly which checks each verdict depends on, and any verdict missing its backing evidence gets quarantined instead of trusted. It is a demo over fictional data, not a measurement of how well a real monitor catches anything.","scope_limit":"Bounded public runtime validation over copied source pattern bodies, sanitized dogfood trace slices, recomputed monitor-verdict spans, source-artifact evidence refs, digest/metadata-only/non-public-state gates, and negative cases only; no live agent execution, monitor product performance, control-eval score, safety-validation, benchmark, provider-call, source-file changes, launch, public sharing, or product authority.","evidence":{"kind":"Computed projection","basis":"Source-faithful refactor","rank":3,"runs_real_tools":false},"command":"microcosm agent-monitor-redteam-falsification-replay run-monitor-bundle --input examples/agent_monitor_redteam_falsification_replay/exported_monitor_redteam_bundle --out receipts/runtime_shell/demo_project/organs/agent_monitor_redteam_falsification_replay","tags":["ai-safety","agent-evaluation","red-teaming"],"links_to":[{"component_id":"agent_sabotage_scheming_monitor_replay","label":"Sabotage-Monitor Contract Replay","relative_href":"docs/component-agent-sabotage-scheming-monitor-replay.html"}],"paper_module_backrefs":[{"id":"paper_module.agent_monitor_redteam_falsification_replay","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Agent Monitor Redteam Falsification Replay","url":"paper-module-agent-monitor-redteam-falsification-replay.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[33:paper_module.agent_monitor_redteam_falsification_replay].subjects[0]"}],"primary_paper_module_id":"paper_module.agent_monitor_redteam_falsification_replay","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/agent_monitor_redteam_falsification_replay.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/agent_monitor_redteam_falsification_replay.py"}],"visible_text_byte_count":1325},{"id":"agent_sabotage_scheming_monitor_replay","label":"Sabotage-Monitor Contract Replay","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","relative_href":"docs/component-agent-sabotage-scheming-monitor-replay.html","primary_reader_href":"docs/paper-module-agent-sabotage-scheming-monitor-replay.html","component_detail_href":"docs/component-agent-sabotage-scheming-monitor-replay.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Audits a hidden-goal catch claim for the steps, suspicion scores, and counterfactual it needs.","what_it_does":"This inspects whether a write-up that claims an AI agent was caught secretly working a hidden side-goal actually carries the supporting structure a reader would need: the step-by-step actions, a per-step \"how suspicious did the monitor find this?\" score, and a stated counterfactual saying that removing the sneaky steps would drop the hidden goal while keeping the real task working. It runs only on synthetic, scrubbed example data, so it shows what the skeleton of such a safety claim should look like and checks the pieces line up with each other. It does not run a real agent, does not detect real scheming, and proves nothing about how any deployed monitor performs.","scope_limit":"Bounded public runtime validation over copied source pattern bodies, sanitized dogfood trace slices, recomputed sabotage/scheming monitor spans, source-artifact evidence refs, digest/metadata-only/non-public-state gates, and negative cases only; no live sabotage, live agent execution, exploit instruction, account secret/account, private-reasoning, harmful-payload, monitor-product-performance, deployment-risk, benchmark, provider-call, source-file changes, launch, public sharing, or product authority.","evidence":{"kind":"Computed projection","basis":"Source-faithful refactor","rank":3,"runs_real_tools":false},"command":"microcosm agent-sabotage-scheming-monitor-replay run-sabotage-bundle --input examples/agent_sabotage_scheming_monitor_replay/exported_sabotage_monitor_bundle --out receipts/runtime_shell/demo_project/organs/agent_sabotage_scheming_monitor_replay","tags":["ai-safety","agent-evaluation","red-teaming"],"links_to":[],"paper_module_backrefs":[{"id":"paper_module.agent_sabotage_scheming_monitor_replay","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Agent Sabotage Scheming-Monitor Replay","url":"paper-module-agent-sabotage-scheming-monitor-replay.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[34:paper_module.agent_sabotage_scheming_monitor_replay].subjects[0]"}],"primary_paper_module_id":"paper_module.agent_sabotage_scheming_monitor_replay","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/agent_sabotage_scheming_monitor_replay.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/agent_sabotage_scheming_monitor_replay.py"}],"visible_text_byte_count":1552},{"id":"agent_memory_temporal_conflict_replay","label":"Agent Memory Temporal Conflict Replay","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","relative_href":"docs/component-agent-memory-temporal-conflict-replay.html","primary_reader_href":"docs/paper-module-agent-memory-temporal-conflict-replay.html","component_detail_href":"docs/component-agent-memory-temporal-conflict-replay.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Replays a memory edit-and-delete to show stale facts get flagged before they sway an answer.","what_it_does":"Replays a canned three-step story: an agent's memory first records two facts (a preference and a tool result), then learns one is out of date and edits the preference and deletes the stale fact, then re-runs the same task once with memory on and once with memory off. From plain result records, it shows that the update and deletion each carry a \"conflict\" and \"downgrade\" result record before the memory is allowed to affect the answer, that the memory-on vs memory-off runs are compared through logged evidence rather than just the final wording, and that private-thread content stays in the record only as metadata pointers, never as copied text. It checks the bookkeeping of this synthetic example; it does not judge whether the agent's memory decisions were the right ones.","scope_limit":"It validates the projection mechanics of a synthetic memory fixture only \u2014 that the required refs, decisions, paired replays, negative cases, and secret-exclusion scan line up and that result records are metadata-only. It does not claim live-memory product quality, judge whether memory decisions were domain-correct, treat memory recall as source authority, adopt active injection, export private transcripts, use external model services, change source files, or include launch operations.","evidence":{"kind":"Computed projection","basis":"Source-faithful refactor","rank":3,"runs_real_tools":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.agent_memory_temporal_conflict_replay run --input fixtures/first_wave/agent_memory_temporal_conflict_replay/input --out /tmp/agent_memory_temporal_conflict_replay_out","tags":["ai-safety","agent-evaluation","red-teaming"],"links_to":[{"component_id":"sleeper_memory_poisoning_quarantine_replay","label":"Memory-Poisoning Quarantine Policy Replay","relative_href":"docs/component-sleeper-memory-poisoning-quarantine-replay.html"}],"paper_module_backrefs":[{"id":"paper_module.agent_memory_temporal_conflict_replay","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Agent Memory Temporal-Conflict Replay","url":"paper-module-agent-memory-temporal-conflict-replay.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[32:paper_module.agent_memory_temporal_conflict_replay].subjects[0]"}],"primary_paper_module_id":"paper_module.agent_memory_temporal_conflict_replay","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/agent_memory_temporal_conflict_replay.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/agent_memory_temporal_conflict_replay.py"}],"visible_text_byte_count":1617},{"id":"sleeper_memory_poisoning_quarantine_replay","label":"Memory-Poisoning Quarantine Policy Replay","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","relative_href":"docs/component-sleeper-memory-poisoning-quarantine-replay.html","primary_reader_href":"docs/paper-module-sleeper-memory-poisoning-quarantine-replay.html","component_detail_href":"docs/component-sleeper-memory-poisoning-quarantine-replay.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Replays a recorded memory-tamper case, checking its declared quarantine, block, and delete steps line up.","what_it_does":"This is a worked, made-up example of how an agent should handle a tampered \"memory\": a poisoned note gets spotted, held in quarantine, blocked from being used in any later action, and finally deleted-with-an-audit, with a clean re-run recorded showing the bad memory is no longer there. The component does not actually delete anything or perform the re-run; it reads the on-disk record of that whole guard-and-cleanup sequence and checks that every required step and result record lines up, so exactly which checks must hold is visible. It works entirely from synthetic refs and metadata, with no private memory or transcripts exposed.","scope_limit":"It only checks the structural shape and internal consistency of a synthetic memory-security policy projection recorded as JSON. It does not run or validate any real memory store, does not itself quarantine, delete, or re-run anything, and does not establish that any system actually resists poisoning. It exports no private memory bodies or transcripts, calls no providers, mutates no source, produces no benchmark claims, and excludes launch (all scope limit flags are hardcoded false).","evidence":{"kind":"Computed projection","basis":"Source-faithful refactor","rank":3,"runs_real_tools":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.sleeper_memory_poisoning_quarantine_replay run --input fixtures/first_wave/sleeper_memory_poisoning_quarantine_replay/input --out receipts/first_wave/sleeper_memory_poisoning_quarantine_replay --acceptance-out receipts/acceptance/first_wave/sleeper_memory_poisoning_quarantine_replay_fixture_acceptance.json","tags":["ai-safety","agent-evaluation","red-teaming"],"links_to":[],"paper_module_backrefs":[{"id":"paper_module.sleeper_memory_poisoning_quarantine_replay","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Sleeper Memory Poisoning Quarantine Replay","url":"paper-module-sleeper-memory-poisoning-quarantine-replay.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[37:paper_module.sleeper_memory_poisoning_quarantine_replay].subjects[0]"}],"primary_paper_module_id":"paper_module.sleeper_memory_poisoning_quarantine_replay","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/sleeper_memory_poisoning_quarantine_replay.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/sleeper_memory_poisoning_quarantine_replay.py"}],"visible_text_byte_count":1627},{"id":"mcp_tool_authority_replay","label":"MCP Tool-Authority Policy Replay","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","relative_href":"docs/component-mcp-tool-authority-replay.html","primary_reader_href":"docs/paper-module-mcp-tool-authority-replay.html","component_detail_href":"docs/component-mcp-tool-authority-replay.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Audits a recorded tool-use log to confirm each action was scoped, approved, undoable, and fenced.","what_it_does":"When an AI agent uses outside tools (look something up, change a ticket, read a result from an untrusted source), this checks a recorded log of that tool use to confirm each action was properly fenced: bound to a narrow permission, approved before it changed anything, given a way to undo it, and kept from letting untrusted tool output boss the agent around. It runs on a small built-in make-believe example that carries only labels and references (no real accounts, secrets, or tool contents), so the safety checks an agent's tool use is supposed to pass are inspectable without anything touching a live account.","scope_limit":"It only checks that the tool-authority evidence in a recorded bundle (scopes, approvals, rollbacks, instruction/data splits, cold replays, redaction, and the expected abuse-case failures) is present and internally consistent. It does not run tools or authorize live MCP/account access, account secret or payload export, treating tool output as instruction, source-file changes, benchmark safety scores, or launch, and it makes no claim that the underlying tool-use policy is domain-correct.","evidence":{"kind":"Computed projection","basis":"Source-faithful refactor","rank":3,"runs_real_tools":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.mcp_tool_authority_replay run --input fixtures/first_wave/mcp_tool_authority_replay/input --out receipts/first_wave/mcp_tool_authority_replay","tags":["ai-safety","agent-evaluation","red-teaming"],"links_to":[{"component_id":"agent_sandbox_policy_escape_replay","label":"Sandbox-Policy Replay","relative_href":"docs/component-agent-sandbox-policy-escape-replay.html"},{"component_id":"sleeper_memory_poisoning_quarantine_replay","label":"Memory-Poisoning Quarantine Policy Replay","relative_href":"docs/component-sleeper-memory-poisoning-quarantine-replay.html"}],"paper_module_backrefs":[{"id":"paper_module.mcp_tool_authority_replay","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"MCP Tool Authority Replay","url":"paper-module-mcp-tool-authority-replay.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[39:paper_module.mcp_tool_authority_replay].subjects[0]"}],"primary_paper_module_id":"paper_module.mcp_tool_authority_replay","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/mcp_tool_authority_replay.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/mcp_tool_authority_replay.py"}],"visible_text_byte_count":1426},{"id":"belief_state_process_reward_replay","label":"Belief-State Reward Bundle Replay","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","relative_href":"docs/component-belief-state-process-reward-replay.html","primary_reader_href":"docs/paper-module-belief-state-process-reward-replay.html","component_detail_href":"docs/component-belief-state-process-reward-replay.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Checks that each step reward in a recorded run cites a declared verifier-feedback row, not a trick.","what_it_does":"Takes a recorded, synthetic bundle of agent steps on three partially-observable toy tasks (a terminal investigation, a mock purchase, a small planner) and checks that every \"the agent did the right step\" reward is actually backed by a checkable verifier or observed feedback reference, not by hidden reasoning, formatting tricks, a smuggled answer key, or a final-answer-only score. It does not run or watch a live agent; it validates pre-recorded files. The resulting result record files show, per step, the belief summary, the reward, and whether the reward-hacking and replay checks passed, so it is inspectable why each reward was or was not allowed.","scope_limit":"It only checks that the projection's accounting lines up under its own schema rules over recorded synthetic fixtures; it excludes hidden-reasoning export, RL training, hidden gold or neural-judge-only labels, benchmark-performance claims, external model access, source-file changes, or launch, and proves nothing about real-world reward, live agent behavior, or domain-level conclusions.","evidence":{"kind":"Computed projection","basis":"Source-faithful refactor","rank":3,"runs_real_tools":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.belief_state_process_reward_replay run --input fixtures/first_wave/belief_state_process_reward_replay/input --out .microcosm/belief_state_process_reward_replay","tags":["ai-safety","agent-evaluation","red-teaming"],"links_to":[],"paper_module_backrefs":[{"id":"paper_module.belief_state_process_reward_replay","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Belief-State Process Reward Replay","url":"paper-module-belief-state-process-reward-replay.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[36:paper_module.belief_state_process_reward_replay].subjects[0]"}],"primary_paper_module_id":"paper_module.belief_state_process_reward_replay","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/belief_state_process_reward_replay.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/belief_state_process_reward_replay.py"}],"visible_text_byte_count":1384},{"id":"agent_sandbox_policy_escape_replay","label":"Sandbox-Policy Replay","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","relative_href":"docs/component-agent-sandbox-policy-escape-replay.html","primary_reader_href":"docs/paper-module-agent-sandbox-policy-escape-replay.html","component_detail_href":"docs/component-agent-sandbox-policy-escape-replay.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Maps sandboxed agent actions to show each was approved or blocked before running, then rolled back.","what_it_does":"This takes a synthetic record of an agent attempting risky actions inside a sandbox and lays out, step by step, what each action requested, whether a safety policy approved or blocked it before it would have run, what (if anything) changed afterward, and for the actions that did run, whether the change was rolled back and could be re-checked. The record shows exactly how each containment decision is captured: every blocked attempt is still logged as a traced step but is marked as never executed with no resulting change, all from local files with no real secrets, network, or live agent involved.","scope_limit":"It validates the projection / trace-refactor mechanics over a synthetic fixture only; it excludes live sandbox escape, secret or account secret handling, live network access, host filesystem mutation, executable payload export, raw environment export, external model access, security benchmark claims, source-file changes, or launch. A pass proves the projection boundary and trace-refactor mechanics for this contract, not real sandbox security, exploit resistance, or whole-system safety.","evidence":{"kind":"Computed projection","basis":"Source-faithful refactor","rank":3,"runs_real_tools":false},"command":"PYTHONPATH=src python3 -m microcosm_core.cli agent-sandbox-policy-escape-replay run-sandbox-bundle --input examples/agent_sandbox_policy_escape_replay/exported_sandbox_policy_escape_bundle --out receipts/runtime_shell/demo_project/organs/agent_sandbox_policy_escape_replay","tags":["ai-safety","agent-evaluation","red-teaming"],"links_to":[{"component_id":"sleeper_memory_poisoning_quarantine_replay","label":"Memory-Poisoning Quarantine Policy Replay","relative_href":"docs/component-sleeper-memory-poisoning-quarantine-replay.html"}],"paper_module_backrefs":[{"id":"paper_module.agent_sandbox_policy_escape_replay","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Agent Sandbox Policy-Escape Replay","url":"paper-module-agent-sandbox-policy-escape-replay.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[35:paper_module.agent_sandbox_policy_escape_replay].subjects[0]"}],"primary_paper_module_id":"paper_module.agent_sandbox_policy_escape_replay","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/agent_sandbox_policy_escape_replay.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/agent_sandbox_policy_escape_replay.py"}],"visible_text_byte_count":1487},{"id":"indirect_prompt_injection_information_flow_policy_replay","label":"Prompt-Injection Flow-Policy Replay","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","relative_href":"docs/component-indirect-prompt-injection-information-flow-policy-replay.html","primary_reader_href":"docs/paper-module-indirect-prompt-injection-information-flow-policy-replay.html","component_detail_href":"docs/component-indirect-prompt-injection-information-flow-policy-replay.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Replays an agent run to show untrusted text was gated before any sensitive action, leaking no secret.","what_it_does":"Replays a recorded sample agent episode (built from synthetic, metadata-only rows) and makes visible whether the instructions the agent treated as trusted were kept separate from untrusted web, tool, or browser text before any sensitive action was taken. Row by row, the record shows where untrusted text flowed, what the policy decided for each flow (allow / warn / block / review) before the action, and that the recorded outcome leaked no secret and disclosed no trusted context. It also bundles deliberately-bad cases it must reject (e.g. untrusted text reaching a sensitive action ungated, or a account secret being exfiltrated).","scope_limit":"Passing result records only show this projection satisfies the named information-flow contract over synthetic, redacted, metadata-only rows; they do not prove general prompt-injection robustness, benchmark performance, live account/tool/provider safety, hidden-message handling in a real system, source-file changes, or launch-scope decision.","evidence":{"kind":"Computed projection","basis":"Source-faithful refactor","rank":3,"runs_real_tools":false},"command":"microcosm indirect-prompt-injection-information-flow-policy-replay run-prompt-injection-bundle --input examples/indirect_prompt_injection_information_flow_policy_replay/exported_prompt_injection_flow_bundle --out receipts/runtime_shell/demo_project/organs/indirect_prompt_injection_information_flow_policy_replay","tags":["ai-safety","agent-evaluation","red-teaming"],"links_to":[],"paper_module_backrefs":[{"id":"paper_module.indirect_prompt_injection_information_flow_policy_replay","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Indirect Prompt-Injection Information-Flow Policy Replay","url":"paper-module-indirect-prompt-injection-information-flow-policy-replay.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[38:paper_module.indirect_prompt_injection_information_flow_policy_replay].subjects[0]"}],"primary_paper_module_id":"paper_module.indirect_prompt_injection_information_flow_policy_replay","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/indirect_prompt_injection_information_flow_policy_replay.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/indirect_prompt_injection_information_flow_policy_replay.py"}],"visible_text_byte_count":1428},{"id":"agentic_vulnerability_discovery_patch_proof_replay","label":"Vulnerability Patch-Proof Replay","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","relative_href":"docs/component-agentic-vulnerability-discovery-patch-proof-replay.html","primary_reader_href":"docs/paper-module-agentic-vulnerability-discovery-patch-proof-replay.html","component_detail_href":"docs/component-agentic-vulnerability-discovery-patch-proof-replay.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Checks a fixed-bug evidence chain and re-runs three small real security checks; no real attack material.","what_it_does":"Takes a claim that an AI agent \"found and fixed a security bug\" and lays it out as a local, inspectable chain of made-up (synthetic) evidence: the imagined target, the suspected issue, the trace pointed to as backing, a reference to an abstract exploitability argument, the patch, and regression tests the fixture says fail before the fix and pass after it. The component checks only that these pieces are all present, refer to each other consistently, and carry no real targets, exploits, payloads, account secrets, or attack steps; it does not run the tests or judge whether the bug or fix is actually real. The result record shows whether the declared chain holds together, with no real attack material ever present.","scope_limit":"It validates only the projection/evidence-chain mechanics of a synthetic replay: structural presence, cross-reference consistency, declared boolean flags, and the secret/live-access exclusion scan. It executes small regression witnesses but performs no real vulnerability discovery and makes no judgment of real-world security or fix correctness. It excludes live-target testing, real CVE exploitation, weaponized payloads, account secret handling, network exfiltration, actionable exploit steps, external model access, source-file changes, benchmark security scores, launch, or any whole-system security claim.","evidence":{"kind":"Computed projection","basis":"Source-faithful refactor","rank":3,"runs_real_tools":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.agentic_vulnerability_discovery_patch_proof_replay run --input fixtures/first_wave/agentic_vulnerability_discovery_patch_proof_replay/input --out receipts/first_wave/agentic_vulnerability_discovery_patch_proof_replay --acceptance-out receipts/acceptance/first_wave/agentic_vulnerability_discovery_patch_proof_replay_fixture_acceptance.json","tags":["ai-safety","agent-evaluation","red-teaming"],"links_to":[],"paper_module_backrefs":[{"id":"paper_module.agentic_vulnerability_discovery_patch_proof_replay","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Agentic Vulnerability Discovery Patch-Proof Replay","url":"paper-module-agentic-vulnerability-discovery-patch-proof-replay.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[5:paper_module.agentic_vulnerability_discovery_patch_proof_replay].subjects[0]"}],"primary_paper_module_id":"paper_module.agentic_vulnerability_discovery_patch_proof_replay","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/agentic_vulnerability_discovery_patch_proof_replay.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/agentic_vulnerability_discovery_patch_proof_replay.py"}],"visible_text_byte_count":1857},{"id":"agent_route_observability_runtime","label":"Agent Route Observability Runtime","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","relative_href":"docs/component-agent-route-observability-runtime.html","primary_reader_href":"docs/paper-module-agent-route-observability-runtime.html","component_detail_href":"docs/component-agent-route-observability-runtime.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Recomputes an agent run's route-compliance score and anti-pattern flags with real trace-analytics code.","what_it_does":"This validator takes a sample (synthetic, not live) record of an agent's local run \u2014 the route it picked, the work it did, the events it logged, the evidence it pointed to, and the authority limit it declared \u2014 and checks that this recorded trail is well-formed and self-consistent, instead of leaving raw log JSON to be read by hand. The record is built to state, up front, where the agent's authority was supposed to stop, so the limits are written down and checkable rather than taken on faith. It checks the recorded evidence; it does not watch a live agent or prove one actually stayed in bounds.","scope_limit":"It validates only public, recorded trace-feedback metadata and regression fixtures; it does not inspect live operator state, certify or prove runtime behavior, read model-output data, mutate the work log, authorize pattern assimilation, or include launch operations.","evidence":{"kind":"Contract validator","basis":"Import validation","rank":5,"runs_real_tools":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.agent_route_observability_runtime run --input fixtures/first_wave/agent_route_observability_runtime/input --out receipts/first_wave/agent_route_observability_runtime","tags":["ai-safety","agent-evaluation","red-teaming"],"links_to":[{"component_id":"navigation_hologram_route_plane","label":"Navigation Hologram Route Plane","relative_href":"docs/component-navigation-hologram-route-plane.html"},{"component_id":"cold_reader_route_map","label":"Cold Reader Route Map","relative_href":"docs/component-cold-reader-route-map.html"},{"component_id":"routing_anti_patterns_registry","label":"Routing Anti Patterns Registry","relative_href":"docs/component-routing-anti-patterns-registry.html"},{"component_id":"pattern_binding_contract","label":"Pattern Binding Contract","relative_href":"docs/component-pattern-binding-contract.html"},{"component_id":"macro_projection_import_protocol","label":"Source Projection Import Protocol","relative_href":"docs/component-macro-projection-import-protocol.html"},{"component_id":"bounded_autonomy_campaign_packet","label":"Bounded Autonomy Campaign Packet","relative_href":"docs/component-bounded-autonomy-campaign-packet.html"},{"component_id":"batch11_saturation_engines_capsule","label":"Saturation Engines Bundle","relative_href":"docs/component-batch11-saturation-engines-capsule.html"},{"component_id":"batch4_proof_authority_runtime","label":"Proof / Control / Runtime Import Bundle","relative_href":"docs/component-batch4-proof-authority-runtime.html"},{"component_id":"batch6_unsurfaced_primitives_capsule","label":"Unsurfaced Source Primitives Bundle","relative_href":"docs/component-batch6-unsurfaced-primitives-capsule.html"},{"component_id":"batch7_macro_engines_capsule","label":"Trace, Code-Map & Scheduling Engines Bundle","relative_href":"docs/component-batch7-macro-engines-capsule.html"},{"component_id":"batch8_compliance_pipeline_capsule","label":"Compliance Pipeline Bundle","relative_href":"docs/component-batch8-compliance-pipeline-capsule.html"},{"component_id":"agent_memory_temporal_conflict_replay","label":"Agent Memory Temporal Conflict Replay","relative_href":"docs/component-agent-memory-temporal-conflict-replay.html"},{"component_id":"agent_sandbox_policy_escape_replay","label":"Sandbox-Policy Replay","relative_href":"docs/component-agent-sandbox-policy-escape-replay.html"},{"component_id":"provider_context_recipe_budget_policy","label":"Provider Context Recipe Budget Policy","relative_href":"docs/component-provider-context-recipe-budget-policy.html"},{"component_id":"belief_state_process_reward_replay","label":"Belief-State Reward Bundle Replay","relative_href":"docs/component-belief-state-process-reward-replay.html"}],"paper_module_backrefs":[{"id":"paper_module.agent_route_observability_runtime","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Agent Route Observability Runtime","url":"paper-module-agent-route-observability-runtime.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[2:paper_module.agent_route_observability_runtime].subjects[0]"},{"id":"paper_module.computer_use_action_trace_replay","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Computer-Use Action Trace Replay","url":"paper-module-computer-use-action-trace-replay.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[46:paper_module.computer_use_action_trace_replay].subjects[0]"}],"primary_paper_module_id":"paper_module.agent_route_observability_runtime","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/agent_route_observability_runtime.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/agent_route_observability_runtime.py"}],"visible_text_byte_count":1224},{"id":"bridge_campaign_dag_validation","label":"Bridge Campaign DAG Validation","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","relative_href":"docs/component-bridge-campaign-dag-validation.html","primary_reader_href":"docs/paper-module-bridge-campaign-dag-validation.html","component_detail_href":"docs/component-bridge-campaign-dag-validation.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Runs shape checks on a fan-out work plan for unique steps, dependencies, and no cycles, not the plan itself.","what_it_does":"Validates the shape of a bridge campaign - a small plan that fans several parallel reads into one synthesis. It checks the plan carries the required identity fields, that every step has a unique name, that each dependency points at a real step, that the steps form no cycle, that there is exactly one synthesis step which can be traced back to at least one probe, that the barrier names that synthesis, and that the requested worker count stays within the provider's safe limit. It shows two well-formed plans accepted - a straight chain and a three-way fan-in - and two broken plans rejected: one with a dependency loop and one with two synthesis steps. It validates plan structure only; it does not run the campaign or dispatch any agents.","scope_limit":"The bundle validates a public bridge-campaign DAG contract and provider worker ceiling. It is not a dispatcher, not a live multi-agent run, not a provider safety proof, and not launch-scope decision.","evidence":{"kind":"Bounded runtime computation","basis":"Real runtime result","rank":4,"runs_real_tools":true},"command":"microcosm bridge-campaign-dag-validation run --input fixtures/first_wave/bridge_campaign_dag_validation/input --out receipts/first_wave/bridge_campaign_dag_validation --acceptance-out receipts/acceptance/first_wave/bridge_campaign_dag_validation_fixture_acceptance.json","tags":["agent-concurrency","continuity","ai-safety"],"links_to":[{"component_id":"certificate_kernel_execution_lab","label":"Certificate Kernel Execution Lab","relative_href":"docs/component-certificate-kernel-execution-lab.html"},{"component_id":"verifier_lab_kernel","label":"Verifier Lab Kernel","relative_href":"docs/component-verifier-lab-kernel.html"},{"component_id":"formal_math_lean_proof_witness","label":"Formal Math Lean Proof Witness","relative_href":"docs/component-formal-math-lean-proof-witness.html"}],"paper_module_backrefs":[{"id":"paper_module.bridge_campaign_dag_validation","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Bridge Campaign DAG Validation","url":"paper-module-bridge-campaign-dag-validation.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[95:paper_module.bridge_campaign_dag_validation].subjects[0]"}],"primary_paper_module_id":"paper_module.bridge_campaign_dag_validation","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json#paper_module.bridge_campaign_dag_validation"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/bridge_campaign_dag_validation.py"}],"visible_text_byte_count":1351},{"id":"metabolism_queue_reconciliation","label":"Metabolism Queue Reconciliation","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","relative_href":"docs/component-metabolism-queue-reconciliation.html","primary_reader_href":"docs/paper-module-metabolism-queue-reconciliation.html","component_detail_href":"docs/component-metabolism-queue-reconciliation.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Runs a scratch-database model of a durable job queue and flags impossible states for a human to review.","what_it_does":"A small, runnable model of a durable job queue. It puts jobs in a temporary scratch database, hands one out to a worker on a lease, recovers the job when the lease expires, and tracks claim-and-contradiction events on a shared blackboard. It then runs a consistency check that flags impossible situations for a human to review, for example a job marked \"running\" with no record of a run, or a run that finished while its job still says running. Two clean scenarios are checked end to end; two deliberately broken scenarios are confirmed to be caught and rejected. It is a faithful but bounded copy of the real machinery: it never touches the live production database, never starts agents, and never silently fixes anything.","scope_limit":"The bundle demonstrates a synthetic SQLite durable queue, lease recovery, blackboard claim-event projection, and cold-start reconciliation taxonomy. It is not a live non-public runtime export, not an agent dispatcher, not external model service, not ambiguous auto-repair, not a distributed database, and not launch-scope decision.","evidence":{"kind":"Bounded runtime computation","basis":"Real runtime result","rank":4,"runs_real_tools":true},"command":"microcosm metabolism-queue-reconciliation run --input fixtures/first_wave/metabolism_queue_reconciliation/input --out receipts/first_wave/metabolism_queue_reconciliation --acceptance-out receipts/acceptance/first_wave/metabolism_queue_reconciliation_fixture_acceptance.json","tags":["agent-concurrency","continuity","operational-discipline"],"links_to":[{"component_id":"certificate_kernel_execution_lab","label":"Certificate Kernel Execution Lab","relative_href":"docs/component-certificate-kernel-execution-lab.html"},{"component_id":"verifier_lab_kernel","label":"Verifier Lab Kernel","relative_href":"docs/component-verifier-lab-kernel.html"},{"component_id":"formal_math_lean_proof_witness","label":"Formal Math Lean Proof Witness","relative_href":"docs/component-formal-math-lean-proof-witness.html"}],"paper_module_backrefs":[{"id":"paper_module.metabolism_queue_reconciliation","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Metabolism Queue Reconciliation","url":"paper-module-metabolism-queue-reconciliation.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[96:paper_module.metabolism_queue_reconciliation].subjects[0]"}],"primary_paper_module_id":"paper_module.metabolism_queue_reconciliation","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json#paper_module.metabolism_queue_reconciliation"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/metabolism_queue_reconciliation.py"}],"visible_text_byte_count":1465},{"id":"egress_self_compliance_audit","label":"Egress Self-Compliance Audit","family_id":"agent_reliability_and_safety","family_label":"Agent reliability & safety","relative_href":"docs/component-egress-self-compliance-audit.html","primary_reader_href":"docs/paper-module-egress-self-compliance-audit.html","component_detail_href":"docs/component-egress-self-compliance-audit.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Runs phrase-membership checks on an agent's own replies for three self-policing slips, not deep meaning.","what_it_does":"Checks a snippet of an AI agent's own output for three self-policing slips: asking the user \"shall I continue?\" without naming a real reason to stop, admitting a mistake without saying it was logged, or telling the user to run a command instead of running it. It works by looking for specific giveaway phrases and the phrases that would excuse them \u2014 so it is a fast, transparent style check, not a deep understanding of meaning. Reword the slip and it will be missed; that is the honest limit.","scope_limit":"The bundle evaluates egress text through explicit phrase-membership policy. It is not taint analysis, prompt-injection defense, sandboxing, information-flow control, or launch-scope decision.","evidence":{"kind":"Bounded runtime computation","basis":"Real runtime result","rank":4,"runs_real_tools":true},"command":"microcosm egress-self-compliance-audit run --input fixtures/first_wave/egress_self_compliance_audit/input --out receipts/first_wave/egress_self_compliance_audit --acceptance-out receipts/acceptance/first_wave/egress_self_compliance_audit_fixture_acceptance.json","tags":["ai-safety","compliance","red-teaming"],"links_to":[{"component_id":"certificate_kernel_execution_lab","label":"Certificate Kernel Execution Lab","relative_href":"docs/component-certificate-kernel-execution-lab.html"},{"component_id":"verifier_lab_kernel","label":"Verifier Lab Kernel","relative_href":"docs/component-verifier-lab-kernel.html"},{"component_id":"formal_math_lean_proof_witness","label":"Formal Math Lean Proof Witness","relative_href":"docs/component-formal-math-lean-proof-witness.html"}],"paper_module_backrefs":[{"id":"paper_module.egress_self_compliance_audit","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Egress Self-Compliance Audit","url":"paper-module-egress-self-compliance-audit.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[100:paper_module.egress_self_compliance_audit].subjects[0]"}],"primary_paper_module_id":"paper_module.egress_self_compliance_audit","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json#paper_module.egress_self_compliance_audit"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/egress_self_compliance_audit.py"}],"visible_text_byte_count":1084},{"id":"finance_forecast_evaluation_spine","label":"Finance Forecast Evaluation Spine","family_id":"research_and_science_replays","family_label":"Research & science","relative_href":"docs/component-finance-forecast-evaluation-spine.html","primary_reader_href":"docs/paper-module-finance-forecast-evaluation-spine.html","component_detail_href":"docs/component-finance-forecast-evaluation-spine.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Runs econometric forecast-evaluation tests on synthetic fixtures, recording p-values and refusals with no advice.","what_it_does":"Runs public synthetic forecast-evaluation fixtures through copied finance statistics modules and records p-value/refusal behavior without live market data or advice claims.","scope_limit":"synthetic fixture forecast-evaluation statistics only; no investment-related actions, live market data, track record, or performance claim","evidence":{"kind":"External tool run","basis":"Real runtime result","rank":4,"runs_real_tools":true},"command":"microcosm finance-forecast-evaluation-spine run --input fixtures/first_wave/finance_forecast_evaluation_spine/input --out receipts/first_wave/finance_forecast_evaluation_spine","tags":["research-workflows","forecasting","finance"],"links_to":[{"component_id":"batch12_prediction_market_board_capsule","label":"Prediction Market Board Bundle","relative_href":"docs/component-batch12-prediction-market-board-capsule.html"},{"component_id":"prediction_oracle_reconciliation","label":"Prediction Oracle Reconciliation","relative_href":"docs/component-prediction-oracle-reconciliation.html"}],"paper_module_backrefs":[{"id":"paper_module.finance_forecast_evaluation_spine","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Finance Forecast Evaluation Spine","url":"paper-module-finance-forecast-evaluation-spine.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[30:paper_module.finance_forecast_evaluation_spine].subjects[0]"}],"primary_paper_module_id":"paper_module.finance_forecast_evaluation_spine","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/finance_forecast_evaluation_spine.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/finance_forecast_evaluation_spine.py"}],"visible_text_byte_count":635},{"id":"batch12_prediction_market_board_capsule","label":"Prediction Market Board Bundle","family_id":"research_and_science_replays","family_label":"Research & science","relative_href":"docs/component-batch12-prediction-market-board-capsule.html","primary_reader_href":"docs/paper-module-batch12-prediction-market-board-capsule.html","component_detail_href":"docs/component-batch12-prediction-market-board-capsule.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Replays imported quant market math on test rows, with duplicate retention and seven refusals.","what_it_does":"This bundle imports the quant presentation mart source as public runnable system. Running it over synthetic prediction-market and feed-diagnostic rows shows event identity joining, duplicate-market retention by volume, orphan identity refusal, provider drift flags, missingness rows, unavailable previous-green deltas, and source lifecycle vintage enrichment.","scope_limit":"This is deterministic fixture evidence for copied quant helpers only; it is not live prediction-market-level conclusions, not provider truth, not forecast correctness, not investment-related actions, not external model access, and not launch-scope decision.","evidence":{"kind":"Verified source import","basis":"Copied source body","rank":5,"runs_real_tools":false},"command":"microcosm batch12-prediction-market-board-capsule run-prediction-market-board-bundle --input examples/batch12_prediction_market_board_capsule/exported_batch12_prediction_market_board_capsule_bundle --out receipts/runtime_shell/demo_project/organs/batch12_prediction_market_board_capsule","tags":["research-workflows","forecasting","finance"],"links_to":[{"component_id":"batch8_structural_theses_capsule","label":"Structural Theses Finance Bundle","relative_href":"docs/component-batch8-structural-theses-capsule.html"}],"paper_module_backrefs":[{"id":"paper_module.batch12_prediction_market_board_capsule","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Set 12 Prediction Market Board Bundle","url":"paper-module-batch12-prediction-market-board-capsule.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[65:paper_module.batch12_prediction_market_board_capsule].subjects[0]"}],"primary_paper_module_id":"paper_module.batch12_prediction_market_board_capsule","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/batch12_prediction_market_board_capsule.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/batch12_prediction_market_board_capsule.py"}],"visible_text_byte_count":1029},{"id":"batch12_market_dashboard_read_model_capsule","label":"Market Dashboard Read-Model Bundle","family_id":"research_and_science_replays","family_label":"Research & science","relative_href":"docs/component-batch12-market-dashboard-read-model-capsule.html","primary_reader_href":"docs/paper-module-batch12-market-dashboard-read-model-capsule.html","component_detail_href":"docs/component-batch12-market-dashboard-read-model-capsule.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Runs a copied market-dashboard reader to catch broken links, stale feeds, and trading overclaims.","what_it_does":"This bundle imports the market dashboard read-model source as public runnable system. Running it over synthetic market-dashboard rows shows how structural read-model checks, feed freshness classification, and related-situation grouping catch dangling graph edges, unsafe route refs, auto-apply overclaims, trading-language overclaims, silent omissions, stale or missing readiness, and no-overlap relation cases.","scope_limit":"This is fixture-bound read-model, freshness, and relation-grouping evidence only; it is not live market-level conclusions, not investment-related actions, not external model access, not launch-scope decision, and not whole-system correctness.","evidence":{"kind":"Verified source import","basis":"Copied source body","rank":5,"runs_real_tools":false},"command":"microcosm batch12-market-dashboard-read-model-capsule run-market-dashboard-bundle --input examples/batch12_market_dashboard_read_model_capsule/exported_batch12_market_dashboard_read_model_capsule_bundle --out receipts/runtime_shell/demo_project/organs/batch12_market_dashboard_read_model_capsule","tags":["research-workflows","forecasting","finance"],"links_to":[{"component_id":"batch6_unsurfaced_primitives_capsule","label":"Unsurfaced Source Primitives Bundle","relative_href":"docs/component-batch6-unsurfaced-primitives-capsule.html"}],"paper_module_backrefs":[{"id":"paper_module.batch12_market_dashboard_read_model_capsule","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Set 12 Market Dashboard Read-Model Bundle","url":"paper-module-batch12-market-dashboard-read-model-capsule.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[64:paper_module.batch12_market_dashboard_read_model_capsule].subjects[0]"}],"primary_paper_module_id":"paper_module.batch12_market_dashboard_read_model_capsule","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/batch12_market_dashboard_read_model_capsule.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/batch12_market_dashboard_read_model_capsule.py"}],"visible_text_byte_count":1083},{"id":"mechanistic_interpretability_circuit_attribution_replay","label":"Toy-Transformer Attribution Replay","family_id":"research_and_science_replays","family_label":"Research & science","relative_href":"docs/component-mechanistic-interpretability-circuit-attribution-replay.html","primary_reader_href":"docs/paper-module-mechanistic-interpretability-circuit-attribution-replay.html","component_detail_href":"docs/component-mechanistic-interpretability-circuit-attribution-replay.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Records which model features drove an answer, each tied to checkable evidence.","what_it_does":"This takes the workflow of \"tracing which internal features inside a model drove an answer\" and turns it into inspectable local records. Each row links feature ids to a machine-readable graph of connections, records the before/after results of poking those features (the causal-intervention deltas), notes how far the explanation can be trusted (its faithfulness limit), and points to where the underlying evidence lives. The records show that every interpretability claim is backed by checkable evidence, and that they deliberately hold no model weights, no raw activations, no prompts, and no hidden reasoning \u2014 they carry only refs, digests, counts, and verdicts.","scope_limit":"It validates only the declared public circuit-attribution runtime-result record contract. It excludes model-transparency product claims, live model access, export of private weights/raw activations/proprietary prompts/hidden chain-of-thought, external model access, benchmark claims, or public sharing/launch.","evidence":{"kind":"Contract validator","basis":"Real runtime result","rank":4,"runs_real_tools":false},"command":"microcosm mechanistic-interpretability-circuit-attribution-replay run-attribution-bundle --input examples/mechanistic_interpretability_circuit_attribution_replay/exported_circuit_attribution_bundle --out receipts/runtime_shell/demo_project/organs/mechanistic_interpretability_circuit_attribution_replay","tags":["research-workflows","forecasting","provider operations"],"links_to":[],"paper_module_backrefs":[{"id":"paper_module.mechanistic_interpretability_circuit_attribution_replay","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Mechanistic Interpretability Circuit Attribution Replay","url":"paper-module-mechanistic-interpretability-circuit-attribution-replay.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[51:paper_module.mechanistic_interpretability_circuit_attribution_replay].subjects[0]"}],"primary_paper_module_id":"paper_module.mechanistic_interpretability_circuit_attribution_replay","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/mechanistic_interpretability_circuit_attribution_replay.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/mechanistic_interpretability_circuit_attribution_replay.py"}],"visible_text_byte_count":1395},{"id":"spatial_world_model_counterfactual_simulation_replay","label":"Gridworld Counterfactual State Replay","family_id":"research_and_science_replays","family_label":"Research & science","relative_href":"docs/component-spatial-world-model-counterfactual-simulation-replay.html","primary_reader_href":"docs/paper-module-spatial-world-model-counterfactual-simulation-replay.html","component_detail_href":"docs/component-spatial-world-model-counterfactual-simulation-replay.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Replays six what-if robotics scenes to show what a spatial prediction claim is built from.","what_it_does":"This replay takes six made-up \"what if\" spatial scenes from robotics and self-driving-style settings (a forklift appears from behind an occlusion, a small pedestrian steps into a crosswalk, a gust pushes a drone off course, a shiny floor fools a robot into seeing free space, a stacked load shifts into a lane, and an oncoming car turns late) and shows each one as inspectable rows: the starting scene, the action taken, the predicted next scene, what changed between them, a sanity check, and honest notes on its limits (it is synthetic, not real-world ground truth). The rows show exactly what a spatial \"world model\" claim is built from, plus a checklist of dangerous claims it deliberately refuses to make.","scope_limit":"It validates only the declared public contract of synthetic spatial counterfactual-replay metadata rows. It is evidence for inspectable replay rows and limitation labels, not for real-world spatial accuracy, simulator-product validity, media-only authority, operational deployment, service distribution, or scope decisions.","evidence":{"kind":"Contract validator","basis":"Real runtime result","rank":4,"runs_real_tools":false},"command":"microcosm spatial-world-model-counterfactual-simulation-replay run-simulation-bundle --input examples/spatial_world_model_counterfactual_simulation_replay/exported_spatial_world_model_simulation_bundle --out receipts/runtime_shell/demo_project/organs/spatial_world_model_counterfactual_simulation_replay","tags":["research-workflows","forecasting"],"links_to":[],"paper_module_backrefs":[{"id":"paper_module.spatial_world_model_counterfactual_simulation_replay","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Spatial World Model Counterfactual Simulation Replay","url":"paper-module-spatial-world-model-counterfactual-simulation-replay.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[52:paper_module.spatial_world_model_counterfactual_simulation_replay].subjects[0]"}],"primary_paper_module_id":"paper_module.spatial_world_model_counterfactual_simulation_replay","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/spatial_world_model_counterfactual_simulation_replay.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/spatial_world_model_counterfactual_simulation_replay.py"}],"visible_text_byte_count":1467},{"id":"research_replication_rubric_artifact_replay","label":"Research Replication Rubric Artifact Replay","family_id":"research_and_science_replays","family_label":"Research & science","relative_href":"docs/component-research-replication-rubric-artifact-replay.html","primary_reader_href":"docs/paper-module-research-replication-rubric-artifact-replay.html","component_detail_href":"docs/component-research-replication-rubric-artifact-replay.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Audits whether a paper-replication claim carries the full evidence trail.","what_it_does":"It checks whether a claim that an AI agent \"replicated a research paper\" comes with the paper trail real replication would leave behind. It re-runs nothing; instead it confirms the bundle names every required piece of evidence: a breakdown of the paper's contributions, a grading rubric, the list of allowed public inputs, a from-scratch repo scaffold, an experiment plan, the metric scripts, a roster of declared file-hashes for the outputs plus hashes that all stay inside that roster, a grader report, a capped compute/runtime budget, an ablation diff, a failure list, and a cold-rerun result record. It also catches eight ways a claim can cheat: reusing the original authors' code, leaking a hidden rubric, calling a run a \"success\" when only a write-up backs it, asserting a benchmark claims, leaking a private paper or dataset body, using unbounded compute, grading only the final answer, or pointing at a file-hash that was never declared. The work runs on two made-up sample papers (one machine-learning method, one computational-science study), and the generated result record shows which of the eight cheats each test case triggered, rather than taking \"it was replicated\" on trust.","scope_limit":"It validates the shape and presence of synthetic replay metadata and result record references only - it does not run any experiment, metric script, or rerun, excludes any claim that a paper was actually replicated, that a benchmark claims was achieved, or that the underlying science is correct, and it never calls providers, exposes private paper/data bodies, or authorizes public sharing or launch.","evidence":{"kind":"Computed projection","basis":"Source-faithful refactor","rank":3,"runs_real_tools":false},"command":"PYTHONPATH=src python3 -m microcosm_core.organs.research_replication_rubric_artifact_replay run --input fixtures/first_wave/research_replication_rubric_artifact_replay/input --out receipts/first_wave/research_replication_rubric_artifact_replay","tags":["research-workflows","forecasting","provider operations"],"links_to":[{"component_id":"materials_chemistry_closed_loop_lab_safety_replay","label":"Materials Lab-Safety Refusal Replay","relative_href":"docs/component-materials-chemistry-closed-loop-lab-safety-replay.html"},{"component_id":"mechanistic_interpretability_circuit_attribution_replay","label":"Toy-Transformer Attribution Replay","relative_href":"docs/component-mechanistic-interpretability-circuit-attribution-replay.html"},{"component_id":"prediction_oracle_reconciliation","label":"Prediction Oracle Reconciliation","relative_href":"docs/component-prediction-oracle-reconciliation.html"},{"component_id":"spatial_world_model_counterfactual_simulation_replay","label":"Gridworld Counterfactual State Replay","relative_href":"docs/component-spatial-world-model-counterfactual-simulation-replay.html"},{"component_id":"batch10_cold_eval_honesty_capsule","label":"Cold Evaluation Honesty Bundle","relative_href":"docs/component-batch10-cold-eval-honesty-capsule.html"}],"paper_module_backrefs":[{"id":"paper_module.research_replication_rubric_artifact_replay","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Research Replication Rubric Artifact Replay","url":"paper-module-research-replication-rubric-artifact-replay.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[4:paper_module.research_replication_rubric_artifact_replay].subjects[0]"}],"primary_paper_module_id":"paper_module.research_replication_rubric_artifact_replay","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/research_replication_rubric_artifact_replay.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/research_replication_rubric_artifact_replay.py"}],"visible_text_byte_count":1955},{"id":"materials_chemistry_closed_loop_lab_safety_replay","label":"Materials Lab-Safety Refusal Replay","family_id":"research_and_science_replays","family_label":"Research & science","relative_href":"docs/component-materials-chemistry-closed-loop-lab-safety-replay.html","primary_reader_href":"docs/paper-module-materials-chemistry-closed-loop-lab-safety-replay.html","component_detail_href":"docs/component-materials-chemistry-closed-loop-lab-safety-replay.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Replays a self-driving lab loop as records, with safety gates and no real chemicals, robot, or lab.","what_it_does":"Takes the pattern of a \"self-driving materials lab\" (propose a candidate material, run safety screens, simulate an assay, then decide what to try next) and replays it locally as inspectable records: every step, its safety gate, its simulated result, and the decision that followed, plus the pre-recorded points where such a loop would fail and where it would restart. It makes the workflow's structure visible, all on a simulator-only fixture, so how the loop is wired is traceable without any real lab, real chemicals, or real robot ever being involved.","scope_limit":"It documents projection and replay mechanics only and excludes wetlab protocols, hazardous synthesis steps, reagent amounts, controlled/bioactive targets, robot commands, live assay data, discovery claims, benchmark claims, external model access, or any judgment of domain/chemical correctness.","evidence":{"kind":"Computed projection","basis":"Source-faithful refactor","rank":3,"runs_real_tools":false},"command":"microcosm materials-chemistry-closed-loop-lab-safety-replay run-lab-bundle --input examples/materials_chemistry_closed_loop_lab_safety_replay/exported_materials_lab_safety_bundle --out receipts/runtime_shell/demo_project/organs/materials_chemistry_closed_loop_lab_safety_replay","tags":["research-workflows","forecasting"],"links_to":[{"component_id":"spatial_world_model_counterfactual_simulation_replay","label":"Gridworld Counterfactual State Replay","relative_href":"docs/component-spatial-world-model-counterfactual-simulation-replay.html"}],"paper_module_backrefs":[{"id":"paper_module.materials_chemistry_closed_loop_lab_safety_replay","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Materials Chemistry Closed-Loop Lab-Safety Replay","url":"paper-module-materials-chemistry-closed-loop-lab-safety-replay.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[6:paper_module.materials_chemistry_closed_loop_lab_safety_replay].subjects[0]"}],"primary_paper_module_id":"paper_module.materials_chemistry_closed_loop_lab_safety_replay","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/materials_chemistry_closed_loop_lab_safety_replay.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/materials_chemistry_closed_loop_lab_safety_replay.py"}],"visible_text_byte_count":1263},{"id":"prediction_oracle_reconciliation","label":"Prediction Oracle Reconciliation","family_id":"research_and_science_replays","family_label":"Research & science","relative_href":"docs/component-prediction-oracle-reconciliation.html","primary_reader_href":"docs/paper-module-prediction-oracle-reconciliation.html","component_detail_href":"docs/component-prediction-oracle-reconciliation.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Replays a forecast against the discipline a careful predictor would have to defend.","what_it_does":"Runs a made-up forecasting case through the discipline a careful predictor would have to defend: which way a fork was called and why the losing side was ruled out, whether each prediction stayed inside the pre-declared list of allowed outcomes, that no \"after the fact\" evidence got used as if it were known in advance, how the guesses compared to a synthetic \"what actually happened\" result, and that any edits to the running record were small, allowed changes rather than rewrites. The reasoning is laid out as inspectable records rather than a single handed-down verdict. Everything is invented test data \u2014 it makes no real forecast and claims no track record.","scope_limit":"It exercises projection mechanics on a synthetic, invented packet only. 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It exposes observer set diffs, JSON patch interpretation, stable ledger-id hashing, and shadow envelope parsing without invoking live oracles, repository mutation, external model access, public sharing, or launch-scope decision.","scope_limit":"It validates only the imported source body. 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It exposes lab contract audit, market fusion readiness, and campaign transition adjudication mechanics without model dispatch, live campaign execution, public sharing, or launch-scope decision.","scope_limit":"It validates only the imported source body. It does not claim source authority, whole-system equivalence, launch, or public sharing.","evidence":{"kind":"Verified source import","basis":"Copied source body","rank":5,"runs_real_tools":false},"command":"microcosm batch8-policy-engines-capsule run --input fixtures/first_wave/batch8_policy_engines_capsule/input --out receipts/first_wave/batch8_policy_engines_capsule --acceptance-out receipts/acceptance/first_wave/batch8_policy_engines_capsule_fixture_acceptance.json","tags":["source intake","provenance","drift-control"],"links_to":[],"paper_module_backrefs":[{"id":"paper_module.batch8_policy_engines_capsule","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Set 8 Policy Engines Bundle","url":"paper-module-batch8-policy-engines-capsule.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[60:paper_module.batch8_policy_engines_capsule].subjects[0]"}],"primary_paper_module_id":"paper_module.batch8_policy_engines_capsule","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/batch8_policy_engines_capsule.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/batch8_policy_engines_capsule.py"}],"visible_text_byte_count":826},{"id":"batch8_audio_level_rms_port","label":"Audio Level RMS Port","family_id":"import_projection_and_drift","family_label":"Import & drift control","relative_href":"docs/component-batch8-audio-level-rms-port.html","primary_reader_href":"docs/paper-module-batch8-audio-level-rms-port.html","component_detail_href":"docs/component-batch8-audio-level-rms-port.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Computes the audio loudness math on test arrays without opening a microphone or capturing input.","what_it_does":"This port projects the pure AudioLevelMonitor normalized-level RMS math into a runnable Python fixture over synthetic sample arrays. 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It does not claim source authority, whole-system equivalence, launch, public sharing, live ledger control, or source-file changes.","evidence":{"kind":"Verified source import","basis":"Copied source body","rank":5,"runs_real_tools":false},"command":"microcosm batch10-governance-compilers-capsule run --input fixtures/first_wave/batch10_governance_compilers_capsule/input --out receipts/first_wave/batch10_governance_compilers_capsule --acceptance-out receipts/acceptance/first_wave/batch10_governance_compilers_capsule_fixture_acceptance.json","tags":["source intake","provenance","drift-control"],"links_to":[],"paper_module_backrefs":[{"id":"paper_module.batch10_governance_compilers_capsule","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Set 10 Governance And Compilers Bundle","url":"paper-module-batch10-governance-compilers-capsule.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[72:paper_module.batch10_governance_compilers_capsule].subjects[0]"}],"primary_paper_module_id":"paper_module.batch10_governance_compilers_capsule","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/batch10_governance_compilers_capsule.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/batch10_governance_compilers_capsule.py"}],"visible_text_byte_count":1087},{"id":"batch11_saturation_engines_capsule","label":"Saturation Engines Bundle","family_id":"import_projection_and_drift","family_label":"Import & drift control","relative_href":"docs/component-batch11-saturation-engines-capsule.html","primary_reader_href":"docs/paper-module-batch11-saturation-engines-capsule.html","component_detail_href":"docs/component-batch11-saturation-engines-capsule.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Verifies twelve copied engine routines and computes each failure probe from inputs, not echoes.","what_it_does":"This bundle imports twelve Set-11 saturation-engine mechanisms as source-open system. It exposes for inspection exact source-module digest evidence, source-faithful computed exercises, and computed negative-case probes without exposing non-public paths, copied body text in result records, live runtime state, source-file changes, public sharing, or launch-scope decision.","scope_limit":"It validates only the imported source body. It does not claim source authority, whole-system equivalence, launch, public sharing, live runtime control, or source-file changes.","evidence":{"kind":"Verified source import","basis":"Copied source body","rank":5,"runs_real_tools":false},"command":"microcosm batch11-saturation-engines-capsule run --input fixtures/first_wave/batch11_saturation_engines_capsule/input --out receipts/first_wave/batch11_saturation_engines_capsule --acceptance-out receipts/acceptance/first_wave/batch11_saturation_engines_capsule_fixture_acceptance.json","tags":["source intake","provenance","drift-control"],"links_to":[{"component_id":"macro_projection_import_protocol","label":"Source Projection Import Protocol","relative_href":"docs/component-macro-projection-import-protocol.html"},{"component_id":"batch8_compliance_pipeline_capsule","label":"Compliance Pipeline Bundle","relative_href":"docs/component-batch8-compliance-pipeline-capsule.html"},{"component_id":"navigation_hologram_route_plane","label":"Navigation Hologram Route Plane","relative_href":"docs/component-navigation-hologram-route-plane.html"},{"component_id":"agent_route_observability_runtime","label":"Agent Route Observability Runtime","relative_href":"docs/component-agent-route-observability-runtime.html"}],"paper_module_backrefs":[{"id":"paper_module.batch11_saturation_engines_capsule","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Set 11 Saturation Engines Bundle","url":"paper-module-batch11-saturation-engines-capsule.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[73:paper_module.batch11_saturation_engines_capsule].subjects[0]"}],"primary_paper_module_id":"paper_module.batch11_saturation_engines_capsule","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/batch11_saturation_engines_capsule.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/batch11_saturation_engines_capsule.py"}],"visible_text_byte_count":956},{"id":"tool_server_pressure_inventory","label":"Tool Server Pressure Inventory","family_id":"import_projection_and_drift","family_label":"Import & drift control","relative_href":"docs/component-tool-server-pressure-inventory.html","primary_reader_href":"docs/paper-module-tool-server-pressure-inventory.html","component_detail_href":"docs/component-tool-server-pressure-inventory.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Flags detached helper processes and launch pressure from synthetic rows, not live hosts.","what_it_does":"This component imports the source helper-process pressure inventory pattern as a public-safe, read-only validator. Over synthetic ps-shaped process rows it surfaces detached helper candidates, active-owner descendants, keep runtimes, and owner-launch pressure groups without reading live host processes or exposing command bodies.","scope_limit":"validates declared public helper-process pressure inventory contract only; no live process reads, process signalling, host mutation, launch-scope decision, external model access, non-public data equivalence, or whole-system correctness","evidence":{"kind":"Contract validator","basis":"Import validation","rank":5,"runs_real_tools":false},"command":"microcosm tool-server-pressure-inventory run --input fixtures/first_wave/tool_server_pressure_inventory/input --out receipts/first_wave/tool_server_pressure_inventory --acceptance-out receipts/acceptance/first_wave/tool_server_pressure_inventory_fixture_acceptance.json","tags":["source intake","provenance","drift-control"],"links_to":[],"paper_module_backrefs":[{"id":"paper_module.tool_server_pressure_inventory","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Tool Server Pressure Inventory","url":"paper-module-tool-server-pressure-inventory.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[50:paper_module.tool_server_pressure_inventory].subjects[0]"}],"primary_paper_module_id":"paper_module.tool_server_pressure_inventory","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/tool_server_pressure_inventory.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/tool_server_pressure_inventory.py"}],"visible_text_byte_count":956},{"id":"batch8_compliance_pipeline_capsule","label":"Compliance Pipeline Bundle","family_id":"import_projection_and_drift","family_label":"Import & drift control","relative_href":"docs/component-batch8-compliance-pipeline-capsule.html","primary_reader_href":"docs/paper-module-batch8-compliance-pipeline-capsule.html","component_detail_href":"docs/component-batch8-compliance-pipeline-capsule.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Confirms six copied compliance source files carry their functions; runs one helper on sample text.","what_it_does":"This component imports the compliance adapter registry, compliance coverage and baseline scanners, Microcosm compliance adapter, bounded compliance-ledger builder, and observe pipeline stages as public runnable system. Running it shows how registered compliance adapters, bounded no-write checks, baseline companion scans, digest normalization, observe-plan helper selection, and dispatch/process boundaries fit together without refreshing the live ledger or dispatching providers.","scope_limit":"validates declared public Set 8 compliance pipeline bundle contract only; no full compliance-ledger freshness, external model access, model dispatch, source-file changes, source note mutation, launch, public sharing, non-public data equivalence, or whole-system correctness","evidence":{"kind":"Computed projection","basis":"Source-faithful refactor","rank":3,"runs_real_tools":false},"command":"microcosm batch8-compliance-pipeline-capsule validate-bundle --input examples/batch8_compliance_pipeline_capsule/exported_batch8_compliance_pipeline_capsule_bundle --out /tmp/microcosm-batch8-compliance-pipeline-capsule","tags":["source intake","provenance","drift-control"],"links_to":[{"component_id":"macro_projection_import_protocol","label":"Source Projection Import Protocol","relative_href":"docs/component-macro-projection-import-protocol.html"},{"component_id":"batch11_saturation_engines_capsule","label":"Saturation Engines Bundle","relative_href":"docs/component-batch11-saturation-engines-capsule.html"},{"component_id":"navigation_hologram_route_plane","label":"Navigation Hologram Route Plane","relative_href":"docs/component-navigation-hologram-route-plane.html"},{"component_id":"agent_route_observability_runtime","label":"Agent Route Observability Runtime","relative_href":"docs/component-agent-route-observability-runtime.html"}],"paper_module_backrefs":[{"id":"paper_module.batch8_compliance_pipeline_capsule","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Set 8 Compliance Pipeline Bundle","url":"paper-module-batch8-compliance-pipeline-capsule.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[59:paper_module.batch8_compliance_pipeline_capsule].subjects[0]"}],"primary_paper_module_id":"paper_module.batch8_compliance_pipeline_capsule","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/batch8_compliance_pipeline_capsule.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/batch8_compliance_pipeline_capsule.py"}],"visible_text_byte_count":1101},{"id":"batch10_live_source_drift_capsule","label":"Live Source Drift Bundle","family_id":"import_projection_and_drift","family_label":"Import & drift control","relative_href":"docs/component-batch10-live-source-drift-capsule.html","primary_reader_href":"docs/paper-module-batch10-live-source-drift-capsule.html","component_detail_href":"docs/component-batch10-live-source-drift-capsule.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Compares four copied router and landing routines against current code to surface stale copies.","what_it_does":"This component imports exact current public source bodies for the option-surface router, mission-transaction landing preflight, work landing controller, and work log controller. Running it inspects stale-versus-current digest repair, source anchors, and compile-only validation without reading private runtime state or granting live mutation authority.","scope_limit":"verified source body import only, not route authority, work log or work log mutation authority, mission-transaction execution, git staging or commit approval, source-file changes, non-public runtime export, launch, or public sharing","evidence":{"kind":"Verified source import","basis":"Copied source body","rank":5,"runs_real_tools":false},"command":"microcosm batch10-live-source-drift-capsule run --input fixtures/first_wave/batch10_live_source_drift_capsule/input --out receipts/first_wave/batch10_live_source_drift_capsule --acceptance-out receipts/acceptance/first_wave/batch10_live_source_drift_capsule_fixture_acceptance.json","tags":["source intake","provenance","drift-control"],"links_to":[{"component_id":"batch10_governance_compilers_capsule","label":"Governance & Compiler Mechanisms Bundle","relative_href":"docs/component-batch10-governance-compilers-capsule.html"}],"paper_module_backrefs":[{"id":"paper_module.batch10_live_source_drift_capsule","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Set 10 Live Source Drift Bundle","url":"paper-module-batch10-live-source-drift-capsule.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[68:paper_module.batch10_live_source_drift_capsule].subjects[0]"}],"primary_paper_module_id":"paper_module.batch10_live_source_drift_capsule","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/batch10_live_source_drift_capsule.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/batch10_live_source_drift_capsule.py"}],"visible_text_byte_count":987},{"id":"batch12_release_claim_language_gate","label":"Release Public Wording Gate","family_id":"import_projection_and_drift","family_label":"Import & drift control","relative_href":"docs/component-batch12-release-claim-language-gate.html","primary_reader_href":"docs/paper-module-batch12-release-claim-language-gate.html","component_detail_href":"docs/component-batch12-release-claim-language-gate.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Flags affirmative open-source and deployment-posture wording while allowing safe boundary notes.","what_it_does":"This component imports the launch claim-language gate as public runnable system. Running it over small public sharing-manifest fixtures shows boundary-only warnings allowed while affirmative open-source and production-readiness wording is classified as active claim language and blocked by the assert-clear contract.","scope_limit":"This is lexical fixture evidence only; it is not launch-scope decision, not publishing-scope decision, not semantic NLP truth, not secret-scan coverage, and not whole-system correctness.","evidence":{"kind":"Verified source import","basis":"Copied source body","rank":5,"runs_real_tools":false},"command":"microcosm batch12-release-claim-language-gate run-release-claim-language-gate-bundle --input examples/batch12_release_claim_language_gate/exported_batch12_release_claim_language_gate_bundle --out receipts/runtime_shell/demo_project/organs/batch12_release_claim_language_gate","tags":["source intake","provenance","drift-control"],"links_to":[],"paper_module_backrefs":[{"id":"paper_module.batch12_release_claim_language_gate","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Set 12 launch claim-Language Gate","url":"paper-module-batch12-release-claim-language-gate.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[66:paper_module.batch12_release_claim_language_gate].subjects[0]"}],"primary_paper_module_id":"paper_module.batch12_release_claim_language_gate","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/batch12_release_claim_language_gate.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/batch12_release_claim_language_gate.py"}],"visible_text_byte_count":903},{"id":"generated_projection_drift_runtime","label":"Generated Projection Drift Runtime","family_id":"import_projection_and_drift","family_label":"Import & drift control","relative_href":"docs/component-generated-projection-drift-runtime.html","primary_reader_href":"docs/paper-module-generated-projection-drift-runtime.html","component_detail_href":"docs/component-generated-projection-drift-runtime.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Runs each owner's own drift check on generated files and flags repairs, without changing the file.","what_it_does":"Generated files (build outputs that are supposed to be reproducible from their source) can quietly drift when someone hand-edits the output or a builder stops matching its source. This component is the gate that catches that. For each \"owner\" (a generated artifact plus its source and a check command) it hashes the source and the output, checks whether a recent clean record lets it safely skip re-running the check, makes sure the required output actually exists, and otherwise runs the owner's own no-write check. It calls the file clean only when that check passes and the output is present. It flags drift; it does not fix files, does not judge every builder in the larger source system, and does not decide anything about releasing software.","scope_limit":"The bundle demonstrates owner-routed generated projection drift detection over declared artifacts, source authorities, clean-result record fingerprints, and no-write check return codes. It is not semantic drift proof, not full source registry validation, not repair authority, and not launch-scope decision.","evidence":{"kind":"External tool run","basis":"Real runtime result","rank":4,"runs_real_tools":true},"command":"microcosm generated-projection-drift-runtime run --input fixtures/first_wave/generated_projection_drift_runtime/input --out receipts/first_wave/generated_projection_drift_runtime --acceptance-out receipts/acceptance/first_wave/generated_projection_drift_runtime_fixture_acceptance.json","tags":["drift-control","provenance","source intake"],"links_to":[{"component_id":"certificate_kernel_execution_lab","label":"Certificate Kernel Execution Lab","relative_href":"docs/component-certificate-kernel-execution-lab.html"},{"component_id":"verifier_lab_kernel","label":"Verifier Lab Kernel","relative_href":"docs/component-verifier-lab-kernel.html"},{"component_id":"formal_math_lean_proof_witness","label":"Formal Math Lean Proof Witness","relative_href":"docs/component-formal-math-lean-proof-witness.html"}],"paper_module_backrefs":[{"id":"paper_module.generated_projection_drift_runtime","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Generated Projection Drift Runtime","url":"paper-module-generated-projection-drift-runtime.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[101:paper_module.generated_projection_drift_runtime].subjects[0]"}],"primary_paper_module_id":"paper_module.generated_projection_drift_runtime","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json#paper_module.generated_projection_drift_runtime"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/generated_projection_drift_runtime.py"}],"visible_text_byte_count":1475},{"id":"mission_transaction_work_spine","label":"Mission Transaction Work Spine","family_id":"work_landing_and_continuity","family_label":"Work & continuity","relative_href":"docs/component-mission-transaction-work-spine.html","primary_reader_href":"docs/paper-module-mission-transaction-work-spine.html","component_detail_href":"docs/component-mission-transaction-work-spine.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Runs the real work-ledger engine on a sanitised snapshot to re-derive each change's verdict.","what_it_does":"Replays a fixed set of pre-recorded work-landing situations against a toy repository and shows when a change would be allowed to \"land\" versus blocked: two claims competing on the same file, a claim built on a stale parent commit, a claim missing its owned path, a clean preflight check that wrongly says the work is already finished, and which commit lane (a narrow scoped commit vs a broad checkpoint) a dirty working tree is permitted to use. Its exported bundle also anchors the public work log seed-speed source imports for session heartbeat, mutation-check, active-claim snapshot, and path-collision handling. The resulting result records show exactly why each situation was permitted or refused, instead of an opaque \"it's done\" message.","scope_limit":"It validates work-landing, claim, checkpoint-lane, and dependency metadata projections over fixed fixtures only; it does not mutate live ledgers or git, certify real completion, authorize broad staging without operator intent, or prove any change is actually correct or complete.","evidence":{"kind":"Bounded runtime computation","basis":"Real runtime result","rank":4,"runs_real_tools":true},"command":"microcosm mission-transaction-work-spine run --input fixtures/first_wave/mission_transaction_work_spine/input --out receipts/first_wave/mission_transaction_work_spine","tags":["agent-concurrency","workflow-engineering","continuity"],"links_to":[{"component_id":"bounded_autonomy_campaign_packet","label":"Bounded Autonomy Campaign Packet","relative_href":"docs/component-bounded-autonomy-campaign-packet.html"},{"component_id":"concurrency_mission_control","label":"Concurrency Mission Control","relative_href":"docs/component-concurrency-mission-control.html"},{"component_id":"macro_projection_import_protocol","label":"Source Projection Import Protocol","relative_href":"docs/component-macro-projection-import-protocol.html"},{"component_id":"tool_server_pressure_inventory","label":"Tool Server Pressure Inventory","relative_href":"docs/component-tool-server-pressure-inventory.html"},{"component_id":"batch10_live_source_drift_capsule","label":"Live Source Drift Bundle","relative_href":"docs/component-batch10-live-source-drift-capsule.html"},{"component_id":"batch4_proof_authority_runtime","label":"Proof / Control / Runtime Import Bundle","relative_href":"docs/component-batch4-proof-authority-runtime.html"},{"component_id":"batch8_tools_tail_primitives_capsule","label":"Tools-Tail Primitives Bundle","relative_href":"docs/component-batch8-tools-tail-primitives-capsule.html"},{"component_id":"agent_benchmark_integrity_anti_gaming_replay","label":"Agent Benchmark Integrity Anti Gaming Replay","relative_href":"docs/component-agent-benchmark-integrity-anti-gaming-replay.html"},{"component_id":"agentic_vulnerability_discovery_patch_proof_replay","label":"Vulnerability Patch-Proof Replay","relative_href":"docs/component-agentic-vulnerability-discovery-patch-proof-replay.html"}],"paper_module_backrefs":[{"id":"paper_module.mission_transaction_work_spine","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Mission Transaction Work Spine","url":"paper-module-mission-transaction-work-spine.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[20:paper_module.mission_transaction_work_spine].subjects[0]"}],"primary_paper_module_id":"paper_module.mission_transaction_work_spine","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json#paper_module.mission_transaction_work_spine"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/mission_transaction_work_spine.py"}],"visible_text_byte_count":1315},{"id":"durable_agent_work_landing_replay","label":"Durable Agent Work Landing Replay","family_id":"work_landing_and_continuity","family_label":"Work & continuity","relative_href":"docs/component-durable-agent-work-landing-replay.html","primary_reader_href":"docs/paper-module-durable-agent-work-landing-replay.html","component_detail_href":"docs/component-durable-agent-work-landing-replay.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Audits recorded work-claims so each cites files, validates before commit, and proves HEAD moved.","what_it_does":"This checks recorded examples of an agent finishing a piece of work the careful way: each example must name the exact files it claims to have touched, show that validation was recorded before any commit was attempted, and only label itself \"committed\" if the example also records the repository's HEAD moving. It also checks that blockers and ledger completion were captured. The check shows whether each recorded work-claim carries the required evidence and ordering, rather than being an unbacked chat boast. It judges the recorded claims against the contract; it does not run Git or prove that any commit truly landed in a real repository.","scope_limit":"It validates only the declared public work-landing contract over recorded rows. It is evidence for fixture-local completion mechanics, not for live Git side effects, unrelated-path staging, non-public body export, service operation, or distribution clearance.","evidence":{"kind":"Contract validator","basis":"Import validation","rank":5,"runs_real_tools":false},"command":"microcosm durable-agent-work-landing-replay run-work-landing-bundle","tags":["workflow-engineering","continuity"],"links_to":[{"component_id":"agent_closeout_faithfulness_audit","label":"Agent Completion Faithfulness Audit","relative_href":"docs/component-agent-closeout-faithfulness-audit.html"},{"component_id":"macro_projection_import_protocol","label":"Source Projection Import Protocol","relative_href":"docs/component-macro-projection-import-protocol.html"}],"paper_module_backrefs":[{"id":"paper_module.durable_agent_work_landing_replay","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Durable Agent Work-Landing Replay","url":"paper-module-durable-agent-work-landing-replay.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[16:paper_module.durable_agent_work_landing_replay].subjects[0]"}],"primary_paper_module_id":"paper_module.durable_agent_work_landing_replay","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json#paper_module.durable_agent_work_landing_replay"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/durable_agent_work_landing_replay.py"}],"visible_text_byte_count":1101},{"id":"bridge_phase_continuity_runtime","label":"Bridge-Continuity Sign-off Replay","family_id":"work_landing_and_continuity","family_label":"Work & continuity","relative_href":"docs/component-bridge-phase-continuity-runtime.html","primary_reader_href":"docs/paper-module-bridge-phase-continuity-runtime.html","component_detail_href":"docs/component-bridge-phase-continuity-runtime.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Replays a paused job to prove the rules for safely resuming it hold and reject duplicate resumes.","what_it_does":"Replays a small synthetic record of a paused background job to check that the rules for safely resuming it hold. It confirms the job left behind a real resume note, that trying to resume the same job twice is refused, that \"still alive\" pings can never count as permission to resume (and a long-stale ping can't be passed off as proof the job is still healthy), and that only a proper completion result record is allowed to claim the work actually finished. It also includes deliberately broken cases to prove each rule rejects them. These continuity rules are inspectable in plain result records without any live job, network call, or non-public data ever being touched.","scope_limit":"It validates only the declared public continuity contract over synthetic fixtures; it does not run live bridge transport, use external model services, read operator HUD/browser/phase-runtime or private-memory state, prove provider or UI uptime, land work, change source files, or include launch operations.","evidence":{"kind":"Contract validator","basis":"Import validation","rank":5,"runs_real_tools":false},"command":"microcosm bridge-phase-continuity-runtime run --input fixtures/second_wave/bridge_phase_continuity_runtime/input --out /tmp/microcosm-bridge-continuity","tags":["workflow-engineering","continuity"],"links_to":[{"component_id":"agent_memory_temporal_conflict_replay","label":"Agent Memory Temporal Conflict Replay","relative_href":"docs/component-agent-memory-temporal-conflict-replay.html"},{"component_id":"concurrency_mission_control","label":"Concurrency Mission Control","relative_href":"docs/component-concurrency-mission-control.html"}],"paper_module_backrefs":[{"id":"paper_module.bridge_phase_continuity_runtime","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Bridge Phase Continuity Runtime","url":"paper-module-bridge-phase-continuity-runtime.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[10:paper_module.bridge_phase_continuity_runtime].subjects[0]"}],"primary_paper_module_id":"paper_module.bridge_phase_continuity_runtime","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json#paper_module.bridge_phase_continuity_runtime"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/bridge_phase_continuity_runtime.py"}],"visible_text_byte_count":1262},{"id":"concurrency_mission_control","label":"Concurrency Mission Control","family_id":"work_landing_and_continuity","family_label":"Work & continuity","relative_href":"docs/component-concurrency-mission-control.html","primary_reader_href":"docs/paper-module-concurrency-mission-control.html","component_detail_href":"docs/component-concurrency-mission-control.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Runs copied claim-coordination code so duplicate, stale, and conflicting claims get blocked.","what_it_does":"This component imports the real concurrency mission-control specimen builder plus public provider and work log bridge artifacts as exact copies. Running it shows duplicate claims, dependency conflicts, stale leases, missing result records, supervised finalizers, and misanchored claims blocked through repair rows while authority-collapse counters stay at zero.","scope_limit":"verified concurrency mission-control source body import only, not a live scheduler, external model access, hosted orchestration, production concurrency-safety proof, source authority, whole-system equivalence, public sharing, or launch-scope decision","evidence":{"kind":"Verified source import","basis":"Copied source body","rank":5,"runs_real_tools":false},"command":"microcosm concurrency-mission-control run --input fixtures/first_wave/concurrency_mission_control/input --out receipts/first_wave/concurrency_mission_control --acceptance-out receipts/acceptance/first_wave/concurrency_mission_control_fixture_acceptance.json","tags":["agent-concurrency","workflow-engineering","continuity"],"links_to":[{"component_id":"mission_transaction_work_spine","label":"Mission Transaction Work Spine","relative_href":"docs/component-mission-transaction-work-spine.html"},{"component_id":"batch10_live_source_drift_capsule","label":"Live Source Drift Bundle","relative_href":"docs/component-batch10-live-source-drift-capsule.html"}],"paper_module_backrefs":[{"id":"paper_module.concurrency_mission_control","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Concurrency Mission Control","url":"paper-module-concurrency-mission-control.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[47:paper_module.concurrency_mission_control].subjects[0]"}],"primary_paper_module_id":"paper_module.concurrency_mission_control","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/concurrency_mission_control.md"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/concurrency_mission_control.py"}],"visible_text_byte_count":991},{"id":"semantic_singleflight_dedup_runtime","label":"Semantic Singleflight Dedup Runtime","family_id":"work_landing_and_continuity","family_label":"Work & continuity","relative_href":"docs/component-semantic-singleflight-dedup-runtime.html","primary_reader_href":"docs/paper-module-semantic-singleflight-dedup-runtime.html","component_detail_href":"docs/component-semantic-singleflight-dedup-runtime.html","map_primary_click_policy":"prefer_primary_reader_href","summary_line":"Reuses one result for duplicate command runs, keyed by repo state rather than the words alone.","what_it_does":"Surfaces the public command-run singleflight bundle as a first-class operational-discipline runtime. It runs a small set of bounded command exercises and shows how duplicate runs collapse: a first run becomes the leader and runs once; a second run with the same content key reuses the completed run instead of executing again (the side-effecting counter stays at 1); mutating a scoped file flips the key so a stale working tree cannot answer for a different run; and an empty command is refused outright. The content key is built from the command words, the resolved working directory, the git HEAD commit, a scoped dirty-tree fingerprint, and an environment fingerprint, so what counts as 'the same run' is decided by repo state rather than by the command words alone.","scope_limit":"It keys and dedups command runs by a repo-state fingerprint over bounded public fixture commands only; it does not guarantee global mutual exclusion, does not replace a lock service, cannot prove cross-host correctness, and is not a job scheduler, a daemon, or launch-scope decision.","evidence":{"kind":"Bounded runtime computation","basis":"Real runtime result","rank":4,"runs_real_tools":true},"command":"microcosm semantic-singleflight-dedup-runtime run --input fixtures/first_wave/semantic_singleflight_dedup_runtime/input --out receipts/first_wave/semantic_singleflight_dedup_runtime --acceptance-out receipts/acceptance/first_wave/semantic_singleflight_dedup_runtime_fixture_acceptance.json","tags":["agent-concurrency","operational-discipline","deduplication"],"links_to":[{"component_id":"mission_transaction_work_spine","label":"Mission Transaction Work Spine","relative_href":"docs/component-mission-transaction-work-spine.html"},{"component_id":"durable_agent_work_landing_replay","label":"Durable Agent Work Landing Replay","relative_href":"docs/component-durable-agent-work-landing-replay.html"},{"component_id":"concurrency_mission_control","label":"Concurrency Mission Control","relative_href":"docs/component-concurrency-mission-control.html"},{"component_id":"engine_room_demo","label":"Engine Room Demo","relative_href":"docs/component-engine-room-demo.html"}],"paper_module_backrefs":[{"id":"paper_module.semantic_singleflight_dedup_runtime","kind":"paper_module","kind_label":"paper module","relation":"explained by","label":"Semantic Singleflight Dedup Runtime","url":"paper-module-semantic-singleflight-dedup-runtime.html","relation_id":"paper_module.explains.organ_or_mechanism","source_ref":"core/paper_module_capsules.json::paper_modules[93:paper_module.semantic_singleflight_dedup_runtime].subjects[0]"}],"primary_paper_module_id":"paper_module.semantic_singleflight_dedup_runtime","source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json#paper_module.semantic_singleflight_dedup_runtime"},{"label":"Source","url":"https://github.com/wcook04/plectis/blob/main/src/microcosm_core/organs/semantic_singleflight_dedup_runtime.py"}],"visible_text_byte_count":1474}],"paper_modules":[{"id":"paper_module.verifier_lab_kernel","title":"Verifier Lab Kernel","relative_href":"docs/paper-module-verifier-lab-kernel.html","read_href":"docs/paper-module-verifier-lab-kernel.html","context_href":"docs/rules-and-ideas.html#paper-module-verifier-lab-kernel","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"The public verifier-lab composition root folds bounded formal-math component result records into one leak-proof result record while separating verifier-backed, oracle-compared, provider-suggested, retrieval-miss, CP2, Evolve, and contract-rejected rows.","summary":"Verifier Lab Kernel is the public result record compiler for the formal-math fixture cohort. It runs or consumes named component result records, classifies proof-lab rows by authority class, preserves scope boundaries, and excludes proof/provider/oracle bodies from exported result records.","authority_ceiling":"Public fixture and exported-bundle result records only; no theorem-prover authority, Mathlib import authority, oracle-to-forward success, provider proof authority, benchmark solve-rate claim, launch-scope decision, publishing-scope decision, source-file changes, or secret export.","abstract":"The public verifier-lab composition root folds bounded formal-math component result records into one leak-proof result record while separating verifier-backed, oracle-compared, provider-suggested, retrieval-miss, CP2, Evolve, and contract-rejected rows.","teleology":{"section_id":"","text":"The public verifier-lab composition root folds bounded formal-math component result records into one leak-proof result record while separating verifier-backed, oracle-compared, provider-suggested, retrieval-miss, CP2, Evolve, and contract-rejected rows.","source":"capsule_summary_line"},"body_markdown":"**It is the composition root that runs nine formal-math components and folds their outputs into one public record labelling every row by origin.** `verifier_lab_kernel` is the composition root for the formal-math verifier lab. It runs or consumes nine smaller formal-math components, then folds their outputs into one public result record that labels every row by where it came from.\n\n**Not a prover or benchmark runner, it reads a public packet, drives the stack, and sorts evidence into claim buckets, emitting references and verdicts without bodies.** It is not a theorem prover, a benchmark runner, a private Lean import, or a frontend surface. It reads a public problem packet, drives the component stack, sorts each piece of evidence into a fixed set of claim buckets, and writes result records that carry references, hashes, counts, and verdicts. The proof text, provider output, oracle answers, and command stdout stay out of what it emits.\n\n**The point is separation: one record shows which claims a Lean checker backed, which are provider hints, which are oracle comparisons, and which were rejected.** The point is separation. One record shows which claims a Lean checker actually backed, which are provider hints, which are oracle comparisons, and which rows were thrown out for breaking the rules, instead of a pile of separate outputs taken on faith.\n\n##### Purpose\n\n###### How progress blurs in a proof loop\n\n**In a proof loop a checker sign-off, an oracle match, a provider tactic, and a retrieval hit all look like progress, quietly inflating the count of theorems proved.** In a formal-math agent loop several different things look like progress. A Lean checker can accept a term. An oracle holding a hindsight answer can say a candidate matches. A provider model can offer a plausible next tactic. A retrieval step can return a premise. Treated loosely, all of these blur into a single sense of \"it worked\", and oracle or provider success quietly inflates the count of theorems actually proved.\n\n###### One authority-class question per row\n\n**It stops that blur by asking one question per evidence row, which authority class it belongs to, so a passing component cannot lend its standing elsewhere.** This component exists to stop that blur. It answers one question for each row of evidence: which authority class does it belong to, and what may that class claim? A passing component cannot lend its standing to a different bucket.\n\n##### How it works\n\n###### Two entry points and the CLI\n\n** `run` drives the fixture with five negative cases, `run_kernel_bundle` consumes an exported bundle as a synthetic contract, and `main` is the CLI over both.** Two entry points run the kernel. `run` drives the first-wave fixture with the five negative cases included. `run_kernel_bundle` drives an exported bundle, where the component result records are consumed as a declared synthetic contract rather than a live run. `main` is the CLI over both, taking `run` or `run-kernel-bundle` with `--input` and `--out`.\n\n**Both entry points delegate to `_build_result`, which runs the stages in order:** Both entry points delegate to `_build_result`, which runs the stages in order:\n- `_load_payloads` reads the public packet, the optional proof-lab route file, and any negative-case inputs from the input directory.\n- `scan_paths`, seeded by `load_forbidden_classes`, scans every input and copied source file for forbidden classes. A blocking hit fails the run.\n- `_run_component_stack` runs each of the nine components on its own fixture (or `_standalone_exported_component_stack` consumes bundle result records). It normalizes each component record to strip legacy redaction fields and public path roots. The stack passes only when every component status passes.\n- `_validate_proof_lab_route` checks that the declared route resolves to `formal_prover_context_strategy_gate` and maps to the expected component set.\n- `_validate_packet` runs the five typed row validators (below) over forward problems, oracle structured source record, provider hypotheses, CP2 candidates, and Evolve candidates. Each violation becomes a finding with an error code.\n- `_claim_separation` sorts the validated rows into the seven buckets.\n\n###### The nine components in the stack\n\n**The stack runs nine formal-math components spanning corpus readiness, premise retrieval, tactic routing, trace repair, diagnostics, and the Lean proof witness.** The nine components are `corpus_readiness_mathlib_absence_gate`, `lean_std_premise_index`, `formal_math_premise_retrieval`, `tactic_portfolio_availability_probe`, `target_shape_tactic_routing_gate`, `ring2_premise_retrieval_precision_recall_harness`, `formal_math_verifier_trace_repair_loop`, `proof_diagnostic_evidence_spine`, and `formal_math_lean_proof_witness`.\n\n###### `_claim_separation` sorts into seven buckets\n\n** `_claim_separation` sorts validated rows into exactly seven buckets, with `lean_verified` carrying the proof-witness count and `contract_rejected` holding the observed negative cases.** `_claim_separation` produces exactly seven buckets: `lean_verified`, `provider_suggested`, `oracle_compared`, `contract_rejected`, `retrieval_miss`, `cp2_translated`, and `evolve_candidate`. The `lean_verified` bucket carries the proof-witness component status and its compiled declaration count. The `contract_rejected` bucket carries the observed negative cases, marked as contract violations, not results.\n\n###### Counters, not description, enforce the boundary\n\n**The boundary is enforced by counters that read zero on a clean run, and the status passes only when the scan, component stack, packet validation, and proof-lab route all pass.** The boundary is enforced by counters, not description. `_build_result` computes `oracle_forward_success_increment_count` (oracle rows that marked themselves as forward success), `provider_results_counted` (provider rows claiming proof authority), and `proof_body_export_count` (forward or CP2 rows carrying a proof body). On a clean run these read zero. The final status is a pass only when the secret scan has zero blocking hits, the component stack passes, packet validation passes, and the proof-lab route passes.\n\n###### `_write_receipts` emits metadata-only records\n\n** `_write_receipts` emits the public records with `body_in_receipt` false, the fixture path marking a real runtime result record and the bundle path marking itself a synthetic contract.** `_write_receipts` then emits the public records. The fixture path writes a result record, a board, a validation record, and a fixture sign-off record; the bundle path writes one bundle validation result. Every record sets `body_in_receipt` to false and carries `receipt_transparency_contract`, whose `omitted_payload_scope` is proof, provider, oracle, private-source, and stdout/stderr bodies only. The fixture path marks `real_runtime_receipt` true on a pass; the bundle path marks itself a synthetic contract and not a live run.\n\nFunction Role `run` Fixture entry point; drives the packet plus five negative cases. `run_kernel_bundle` Exported-bundle entry point; consumes bundle result records as a synthetic contract. `_build_result` Loads inputs, scans for leaks, runs the component stack, validates rows, sorts buckets, sets status. `_run_component_stack` Runs the nine components and normalizes their records. `_claim_separation` Sorts validated rows into the seven claim buckets. `_write_receipts` Writes the metadata-only public result records. Diagram of the mechanism (6 steps). Diagram source & refs\n\n```text\nflowchart TD packet[\"Public packet plus 5 negative cases\"] build[\"_build_result load / scan / validate\"] stack[\"_run_component_stack 9 formal-math components\"] buckets[\"_claim_separation 7 claim buckets\"] counters[\"Authority counters oracle / provider / proof-body = 0\"] result records[\"_write_receipts metadata-only result records\"] packet --> build build --> stack stack --> buckets build --> counters buckets --> result records counters --> result records\n```\n\n##### Negative cases\n\n**The fixture carries five named inputs the kernel must reject, each handled by one typed row validator:** The fixture carries five named inputs the kernel must reject, each handled by one typed row validator:\n- `forward_problem_leaks_candidate_body`: a forward problem carrying a candidate, ideal, repair, oracle-premise, source-proof, or base-index field. `_validate_forward_problems` records `VERIFIER_LAB_FORWARD_FIELD_FORBIDDEN`.\n- `oracle_counted_as_forward`: an oracle structured source record with `counted_as_forward_success` set true. `_validate_oracle_sidecars` records `VERIFIER_LAB_ORACLE_FORWARD_CONTAMINATION`.\n- `provider_claims_proof`: a provider hypothesis asserting proof authority. `_validate_provider_hypotheses` records `VERIFIER_LAB_PROVIDER_PROOF_AUTHORITY_FORBIDDEN`.\n- `cp2_candidate_contains_proof_body`: a CP2 row carrying a proof body, raw tactic script, provider body, or oracle template. `_validate_cp2_candidates` records `VERIFIER_LAB_CP2_PROOF_BODY_FORBIDDEN`, and it also rejects action classes outside the bounded vocabulary and candidates with no disconfirmation test.\n- `evolve_mutates_unbounded_artifact`: an Evolve row touching an artifact outside the bounded policy set, or asking for arbitrary code mutation. `_validate_evolve_candidates` records `VERIFIER_LAB_EVOLVE_SCOPE_FORBIDDEN`, and it also flags candidates that cite no baseline result record.\n\n**Each rejected row lands in `contract_rejected` as a contract violation, never in a claim bucket.** Each rejected row lands in `contract_rejected` as a contract violation, never in a claim bucket.\n\n##### Prior Art Grounding\n\n###### LCF composition and LeanDojo boundaries\n\n**It follows LCF-style kernel composition that keeps checked results apart from automation, and LeanDojo's lesson that retrieval and provider hints need explicit boundaries.** This component follows small-kernel theorem-proving and proof-certificate composition. The [LCF approach](https://www.research.ed.ac.uk/en/publications/lcf-a-way-of-doing-proofs-with-a-machine/) and [HOL Light](https://www.cl.cam.ac.uk/~jrh13/papers/hollight.html) anchor the idea that a verifier lab keeps trusted checked results apart from heuristics and automation. Lean-oriented work such as [LeanDojo](https://arxiv.org/abs/2306.15626) adds the agent context: retrieval, provider hypotheses, and proof-state interaction need explicit boundaries before they can influence a proof claim. The kernel borrows the composition discipline and keeps verifier, oracle, provider, CP2, and Evolve rows in separate buckets with separate authority.\n\n##### Validation Result record Path\n\n###### Reader validation without durable writes\n\n```text\n./repo-pytest tests/test_verifier_lab_kernel.py -q --basetemp=/tmp/microcosm_verifier_lab_kernel_pytest\n./repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n###### What a pass means here\n\n**A pass means the stack runs, the five negative cases are observed, seven buckets hold, counters stay at zero, and no record leaks a body or local root.** A pass means the component stack runs, the five negative cases are observed, `claim_separation` holds exactly the seven buckets, the authority counters stay at zero, and no public record carries a body or a leaked local root.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim\n\n**The strongest claim is bucket separation and metadata-only containment across the composed records, with any return code bound to its own fixture, not generalizable to benchmarks.** The strongest claim this component supports is bucket separation and metadata-only containment across the composed public component records: a reader can see which row a Lean checker backed, which is an oracle comparison, which is a provider hint, which is a CP2 or Evolve candidate, and which was rejected. A Lean/Lake return code or compiled-declaration count is evidence for the public fixture or exported bundle it came from, not a license to generalize to arbitrary benchmarks.\n\n###### What the scope limit refuses\n\n**Past that line oracle rows stay comparators, provider rows stay advisory, CP2 and Evolve rows stay bounded, and nothing authorizes Mathlib import, benchmark rates, or launch.** That is the proof boundary. The scope limit refuses everything past it. Oracle structured source record stay hindsight comparators and cannot increment forward success. Provider hypotheses stay advisory until a verifier record exists. CP2 rows stay typed action candidates, bounded evidence bodies. Evolve rows stay bounded policy artifacts, not source-file changes. 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Plane","relative_href":"docs/paper-module-navigation-hologram-route-plane.html","read_href":"docs/paper-module-navigation-hologram-route-plane.html","context_href":"docs/rules-and-ideas.html#paper-module-navigation-hologram-route-plane","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"The public navigation route-plane fixture validates bounded route projections, source-coupling gates, entry floors, affordance passports, and copied navigation source-module digests without treating browse rows as authority.","summary":"Navigation Hologram Route Plane is the public route-surface validator for moving from a Microcosm-local control entry into browsable route projections. It checks route rows, source-coupling fingerprints, copied source-module anchors and hashes, route-lease policy, entry-packet floors, affordance-passport selection, and code-architecture projection packets while keeping full source bodies out of result records.","authority_ceiling":"Public fixture and exported-bundle result records only; no live route freshness, source authority, provider/live-kernel execution, private operator state, later-component authorization, launch-scope decision, or whole-wave certification.","abstract":"The public navigation route-plane fixture validates bounded route projections, source-coupling gates, entry floors, affordance passports, and copied navigation source-module digests without treating browse rows as authority.","teleology":{"section_id":"","text":"The public navigation route-plane fixture validates bounded route projections, source-coupling gates, entry floors, affordance passports, and copied navigation source-module digests without treating browse rows as authority.","source":"capsule_summary_line"},"body_markdown":"**Readers trust whatever browse surface is nearest, so this validator checks that a navigation surface starts from one control entry and keeps every row a projection.** Anyone who lands in a large codebase starts from whatever browse surface is nearest, treats that surface as the truth, and acts on a stale or partial view. The route plane is a validator that pushes back on this. It checks that a navigation surface starts from one control entry, that browse rows stay projections, and that no row is allowed to pose as the thing it describes.\n\n**It runs offline over fixtures and one exact copy of navigation source modules, emitting plain metadata-only result records that each state only what they checked.** The component runs offline over fixture files and over one exact copy of navigation source modules. It never touches a live system. What it emits is a set of plain result records: a cluster summary, a selected card, a source-coupling result, a route lease, an entry-admission record, an affordance-selection record, and a code-architecture packet. Each record states what it checked and carries no source body text.\n\n**It is a rules check on a navigation surface, not a live router, a search index, or a source of navigation truth.** It is not a live router, a search index, or a source of navigation truth. It is a check that a navigation surface obeys its own rules before a reader or agent trusts it.\n\n##### Purpose\n\n###### Why a browse row needs a coupling record\n\n**A browse row that speaks with its source's authority lets a stale view steer the next move, so the plane demands a coupling record before granting any authority.** A loose navigation surface fails in a specific way. A browse row is a compressed view of some source, and if the row is allowed to speak with the source's authority, then a stale or mislabeled row silently steers the next move. The route plane treats every browse row as a projection and demands a coupling record before that projection gets any authority.\n\n###### Source coupling is a SHA-256 fingerprint\n\n**Source coupling hashes the route rows with SHA-256 and denies authority when the fingerprint or row count disagrees with disk, recording a stale-but-current summary as an overclaim.** Source coupling is a plain SHA-256 over the route rows. The manifest declares an expected fingerprint and an expected row count. If either disagrees with the rows on disk, the projection is denied current authority. A summary that claims to be current while its coupling is stale is recorded as an overclaim.\n\n###### Refusals the fixture must keep catching\n\n**The plane forces first contact to the control entry, forbids dropping a required control field on compaction, and demotes anti-trigger affordances, each a named negative case.** The other half is what the plane refuses. First contact must begin at the control entry, so a request that starts from a browse row is replaced with the entry route. Compaction of the entry packet may not drop a required control field. An affordance row whose passport carries an anti-trigger is demoted before similarity search can select it. Each of these is a named negative case the fixture must keep catching.\n\n##### How it works\n\n###### Two entry points, one discipline\n\n**Two entry points share the same discipline: `run` drives the fixture mode and `run_route_plane_bundle` drives the exported-bundle mode.** The component has two entry points against the same discipline. `run` drives the fixture mode. `run_route_plane_bundle` drives the exported-bundle mode.\n\n###### The `run` fixture-mode pipeline\n\n** `run` loads thirteen fixture inputs, scans for leakage, builds a toy option surface, and passes only when the scan is clean and every expected negative case fired.** `run` loads thirteen fixture inputs, scans them for non-public-state leakage, builds a toy option surface with `build_toy_option_surface` (a cluster summary plus one selected card), then evaluates nine negative-case validators. It merges the negative cases each validator observed and computes the set of expected cases that went unobserved. The status is `pass` only when no expected case is missing and the non-public-state scan is clean. The fixture proves each guard still fires by requiring every expected negative case to actually appear.\n\n###### The `run_route_plane_bundle` bundle-mode gate\n\n** `run_route_plane_bundle` loads nine bundle files, digests and secret-scans them, and passes only when a card was selected and every bundle validator finds nothing.** `run_route_plane_bundle` loads nine bundle files, records a freshness basis and digest over them, scans them with the secret-exclusion scan, then runs the group of exported-bundle validators. Its status is `pass` only when the secret scan is clean, a card row was selected, no validator produced a finding, and the source-module, lease, entry, affordance, and code-architecture checks all pass.\n\nFunction Role `run` Fixture entry: builds the toy surface, runs the nine negative-case guards, requires every expected case to fire. `run_route_plane_bundle` Bundle entry: validates a copied route-plane bundle end to end and gates the pass on every component check. `validate_exported_route_rows` Checks each route row for an allowed surface role, an actionable command, no source-authority claim, and an omission record when the body is compressed; catches duplicate route ids and a missing card row. `validate_exported_source_coupling` The coupling gate: hashes the route rows and compares the fingerprint and row count against the manifest; denies authority on mismatch or stale status. `validate_source_module_manifest` Validates the five copied source modules by SHA-256 digest and by required navigation anchors, and enforces the copied-body boundary. `write_receipts` Writes the result records atomically under public-root-relative paths, with a common envelope and no source body text.\n\n###### Digest and anchor checks on copied modules\n\n** `_stable_hash` fingerprints the route rows, while `validate_source_module_manifest` matches each copied file's digest and anchors, rejects unsafe paths, and keeps bodies out of records.** The coupling gate uses `_stable_hash`, a SHA-256 over the route rows serialized as canonical JSON. `validate_source_module_manifest` reads each copied file, requires its digest to match both the declared source and target digests, requires every declared navigation anchor to be present in the body, and requires the boundary flags `body_copied=true` and `body_in_receipt=false`. A target reference that is absolute or contains `..` is rejected before any read. The body text is never written into a result record.\n\nDiagram of the mechanism (9 steps). Diagram source & refs\n\n```text\nflowchart TD Entry[\"Control entry first a browse row that claims first contact is replaced with the entry route\"] Couple[\"Source coupling SHA-256 over route rows vs manifest fingerprint and row count\"] Rows[\"Route rows allowed role, actionable command, no source-authority claim, omission record when compressed\"] Modules[\"Source-module manifest five copied bodies digest and required anchors, body never in the record\"] Floor[\"Entry-packet floor required control fields survive compaction\"] Pass[\"Affordance passports anti-trigger rows demoted before selection\"] Verdict[\"Coupling current, all checks pass, card row selected?\"] Result records[\"metadata-only result records cluster flag, card, coupling, route lease, entry admission, affordance, code-architecture packet\"] Blocked[\"Blocked stable error codes and findings, bodies redacted\"] Entry --> Couple Couple --> Rows Rows --> Modules Modules --> Floor Floor --> Pass Pass --> Verdict Verdict --> Result records Verdict --> Blocked\n```\n\n##### Negative cases\n\n###### Eight cases carrying nine error codes\n\n**Eight negative cases carry nine stable error codes, one case carrying two, and the fixture passes only when every code is observed.** The fixture mode is defined by eight negative cases that carry nine stable error codes. One case carries two codes, which is why the counts differ. Every code below must be observed for the fixture to pass.\n\nCase Error code First contact starts from a drilldown projection `BANNED_FIRST_CONTACT_ROUTE` Source-coupling fingerprint is stale `SOURCE_COUPLING_STALE` Compressed route card carries no omission record `MISSING_OMISSION_RECEIPT` `ATLAS_PROJECTION_NOT_CONTROL_ENTRY` Route card leaks a private body `ROUTE_CARD_PRIVATE_BODY_LEAK` Route summary overclaims freshness `ROUTE_SUMMARY_OVERCLAIMS_FRESHNESS` Two rows share a route id `DUPLICATE_ROUTE_ID_CONFLICT` Entry admission drops a required control field `ENTRY_ADMISSION_CONTROL_FLOOR_DROPPED` An anti-trigger affordance is not demoted `AFFORDANCE_PASSPORT_ANTITRIGGER_IGNORED`\n\n##### Prior Art Grounding\n\n###### Progressive disclosure, CLI, and graph tooling\n\n**The design borrows progressive disclosure, CLI discoverability, and directed-graph tooling, but keeps cards, leases, and browse rows below authority until coupling and entry records agree.** The route plane sits on information-architecture and graph-navigation practice. The first-contact rule follows the usability pressure behind [progressive disclosure](https://www.nngroup.com/articles/progressive-disclosure/): show the control entry and immediate affordances before deeper browse rows. The command-facing surface follows the [Command Line Interface Guidelines](https://clig.dev/) emphasis on discoverable commands and clear next actions. The graph side is ordinary directed-graph tooling: NetworkX documents [topological sorting](https://networkx.org/documentation/stable/reference/algorithms/generated/networkx.algorithms.dag.topological_sort.html) over dependency edges, and [PageRank](https://networkx.org/documentation/stable/reference/algorithms/generated/networkx.algorithms.link_analysis.pagerank_alg.pagerank.html) shows the older pattern of computing route salience from graph structure. This component keeps those ideas below authority: cards, leases, and browse rows stay projections unless the coupling and entry-admission records agree.\n\n##### Source-Backed Doctrine Packet\n\nThe authoritative sources for this module are:\n- `core/organ_registry.json::implemented_organs[navigation_hologram_route_plane]` records the accepted component authority: evidence class `semantic_validator`, scope limit `validates declared public contract only`, and validator command `python -m microcosm_core.organs.navigation_hologram_route_plane run --input fixtures/first_wave/navigation_hologram_route_plane/input --out receipts/first_wave/navigation_hologram_route_plane`.\n- `core/organ_atlas.json::organs[navigation_hologram_route_plane]` carries the cold-reader gloss. It restates the scope limit: It validates only the declared public toy route-plane contract and its regression fixtures (plus exact copied navigation source modules in the bundle path); it does not establish live route freshness, grant source authority, authorize any later component, run any provider/live-kernel call, or certify the whole wave.\n- `standards/std_microcosm_navigation_hologram_route_plane.json` governs the authority boundary `public_navigation_route_plane_runtime_and_copied_source_body_validator_not_live_source_authority` and declares the runtime bundle command `python -m microcosm_core.organs.navigation_hologram_route_plane validate-route-plane-bundle --input examples/navigation_hologram_route_plane/exported_route_plane_bundle`.\n- `src/microcosm_core/organs/navigation_hologram_route_plane.py` is the runtime source for fixture validation, bundle validation, the coupling gate, the negative-case guards, and the source-module digest and anchor checks.\n- `core/fixture_manifests/navigation_hologram_route_plane.fixture_manifest.json` binds fixture expectations: `body_copied_material_count=5`, body material status `copied_non_secret_macro_route_substrate_with_provenance`, `body_in_receipt=false`, and the nine stable error codes above.\n- `examples/navigation_hologram_route_plane/exported_route_plane_bundle/source_module_manifest.json` names the five exact copied route and control bodies validated by digest and anchors.\n- `tests/test_navigation_hologram_route_plane.py` is the regression floor for fixture records, copied-source digests, source-module anchors, record redaction, bundle validation, digest-mismatch rejection, and this packet.\n\nThe five copied source modules:\n\nModule id Source Copied target `navigation_route_plane_intervention_source_body_import` `system/lib/navigation_route_intervention.py` `examples/navigation_hologram_route_plane/exported_route_plane_bundle/source_modules/system/lib/navigation_route_intervention.py` `navigation_route_plane_context_pack_source_body_import` `system/lib/navigation_context_pack.py` `examples/navigation_hologram_route_plane/exported_route_plane_bundle/source_modules/system/lib/navigation_context_pack.py` `navigation_route_plane_entry_packet_source_body_import` `system/lib/kernel/commands/comprehension_snapshot.py` `examples/navigation_hologram_route_plane/exported_route_plane_bundle/source_modules/system/lib/kernel/commands/comprehension_snapshot.py` `navigation_route_plane_option_surface_source_body_import` `system/lib/standard_option_surface.py` `examples/navigation_hologram_route_plane/exported_route_plane_bundle/source_modules/system/lib/standard_option_surface.py` `navigation_route_plane_navigation_contract_source_body_import` `codex/standards/std_navigation_contract.json` `examples/navigation_hologram_route_plane/exported_route_plane_bundle/source_modules/codex/standards/std_navigation_contract.json`\n\nThe fixture command writes these result records:\n- `receipts/first_wave/navigation_hologram_route_plane/affordance_passport_selection_receipt.json`\n- `receipts/first_wave/navigation_hologram_route_plane/code_architecture_projection_packet_receipt.json`\n- `receipts/first_wave/navigation_hologram_route_plane/entry_payload_admission_receipt.json`\n- `receipts/first_wave/navigation_hologram_route_plane/route_lease.json`\n- `receipts/first_wave/navigation_hologram_route_plane/source_coupling_result.json`\n- `receipts/first_wave/navigation_hologram_route_plane/toy_kind_card.json`\n- `receipts/first_wave/navigation_hologram_route_plane/toy_kind_cluster_flag.json`\n\n##### Validation Result record Path\n\nFrom `microcosm-substrate/`, reproduce this page's proof boundary with temporary result records:\n\n```text\nPYTHONPATH=src ../repo-python -m microcosm_core.organs.navigation_hologram_route_plane run --input fixtures/first_wave/navigation_hologram_route_plane/input --out /tmp/microcosm-navigation-hologram-route-plane\nPYTHONPATH=src ../repo-python -m microcosm_core.organs.navigation_hologram_route_plane validate-route-plane-bundle --input examples/navigation_hologram_route_plane/exported_route_plane_bundle --out /tmp/microcosm-navigation-hologram-route-plane-bundle\n../repo-pytest tests/test_navigation_hologram_route_plane.py\nPYTHONPATH=src ../repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n###### What a pass certifies here\n\n**A pass means the fixture caught every negative case, the bundle validated its rows, coupling, and copied modules, and no record leaked a body or non-public paths.** A pass means the fixture observed every expected negative case, the exported bundle validated its route rows, source coupling, and copied source modules, and no result record carried a source body or a non-public paths. The validation result records for this module are the result records those commands write.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim\n\n**The strongest claim is that public fixture and bundle evidence agree on the declared route-plane contract, and that is the proof boundary and scope limit.** The strongest claim this module supports is public fixture and exported-bundle evidence: the route rows, copied source modules, source manifests, negative cases, and validation result records agree on the declared public route-plane contract. That is its proof boundary and its scope limit.\n\n###### What it refuses past that line\n\n**Past that line it proves no live route freshness, grants no kernel authority, and excludes source-file changes, external model access, state export, public sharing, or whole-system equivalence.** It refuses everything past that line. This module does not establish live route freshness, grant live source-kernel authority, authorize source-file changes, authorize external model access, export account or browser state, expose browser UI live access, authorize public sharing or launch, prove whole-system correctness, or assert whole-system equivalence.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":16712,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/navigation_hologram_route_plane.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"progressive disclosure","url":"https://www.nngroup.com/articles/progressive-disclosure/","kind":"external_reference","section_id":"progressive-disclosure-cli-and-graph-tooling"},{"label":"Command Line Interface 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Ledger","url":"paper-module-self-ignorance-coverage-ledger.html","relation_id":"paper_module.depends_on.paper_module","source_ref":"core/paper_module_capsules.json::paper_modules[49:paper_module.self_ignorance_coverage_ledger].depends_on"},{"id":"paper_module.voice_to_doctrine_self_improvement_loop","kind":"paper_module","kind_label":"paper module","relation":"depended on by","label":"Voice to Doctrine Self-Improvement Loop","url":"paper-module-voice-to-doctrine-self-improvement-loop.html","relation_id":"paper_module.depends_on.paper_module","source_ref":"core/paper_module_capsules.json::paper_modules[56:paper_module.voice_to_doctrine_self_improvement_loop].depends_on"}],"source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/navigation_hologram_route_plane.md"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json"}],"visible_text_byte_count":17846,"section_titles":["Readers trust whatever browse surface is nearest","It runs offline over fixtures","It is a rules check on a navigation surface","Purpose","Why a browse row needs a coupling record","Source coupling is a SHA-256 fingerprint","Refusals the fixture must keep catching","How it works","Two entry points, one discipline","The `run` fixture-mode pipeline","The `run_route_plane_bundle` bundle-mode gate","Digest and anchor checks on copied modules","Negative cases","Eight cases carrying nine error codes","Prior Art Grounding","Progressive disclosure, CLI, and graph tooling","Validation Result record Path","What a pass certifies here","Source-Backed Doctrine Packet","Scope limit","The strongest supported claim","What it refuses past that line"]},{"id":"paper_module.agent_route_observability_runtime","title":"Agent Route Observability Runtime","relative_href":"docs/paper-module-agent-route-observability-runtime.html","read_href":"docs/paper-module-agent-route-observability-runtime.html","context_href":"docs/rules-and-ideas.html#paper-module-agent-route-observability-runtime","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"The public route-observability fixture validates synthetic route feedback, route leases, hook-shadow advisory rows, anti-pattern debt, copied source trace manifests, and metadata-only result records without claiming live session authority.","summary":"Agent Route Observability Runtime is the public evidence membrane for recorded agent route feedback. It checks actor-axis boundaries, selected and replacement routes, route-lease consumption, duplicate trace ids, hook-shadow advisory status, anti-pattern debt retirement, agent-principle-lens admission, egress-mirror boundaries, source-module manifests, and non-public-state exclusion while keeping transcript, provider, browser, HUD, account, account secret, and live-hook bodies out of result records.","authority_ceiling":"Public synthetic route-observability fixtures, copied source body digests, and exported-bundle result records only; no live session introspection, provider/browser UI/account authority, live hook control, benchmark-performance proof, source-file changes, launch-scope decision, or whole-system correctness.","abstract":"The public route-observability fixture validates synthetic route feedback, route leases, hook-shadow advisory rows, anti-pattern debt, copied source trace manifests, and metadata-only result records without claiming live session authority.","teleology":{"section_id":"","text":"The public route-observability fixture validates synthetic route feedback, route leases, hook-shadow advisory rows, anti-pattern debt, copied source trace manifests, and metadata-only result records without claiming live session authority.","source":"capsule_summary_line"},"body_markdown":"###### One recorded trail is checked end to end\n\n**It replays one agent's recorded trail \u2014 route, work, events, evidence, and declared authority limit \u2014 and checks the whole thing is well-formed and self-consistent.** This component takes a recorded trail of one agent's local run, the route it picked, the work it logged, the events it emitted, the evidence it pointed to, and the authority limit it declared, and checks that the trail is well-formed and self-consistent. It is a replay validator over synthetic fixtures, not a live tap on a running agent.\n\n###### Six checks share the declared scope limit\n\n**Because the record states its authority limit up front, the runtime runs six checks and writes six metadata-only result records that each carry status, findings, and the scope limit.** The record is built to state up front where the agent's authority was supposed to stop, so the limit is written down and checkable rather than taken on faith. The runtime reads that record, runs six checks over it, and writes six small JSON result records. Each result record carries a status, the negative cases it saw, the findings, and the scope limit. Large or private payloads are dropped before anything is written.\n\n###### Recorded evidence cannot prove live compliance\n\n**It reads recorded evidence only, inspecting no live session and proving nothing about whether an agent actually stayed in bounds.** It reads recorded evidence. It does not inspect live sessions, mutate a route, install a hook, authorize a provider, or prove that any agent actually stayed in bounds.\n\n##### Purpose\n\n###### A raw log reads as proof\n\n**A raw agent log is hard to trust because a behavior-change claim or a quiet authority grab reads as proof while nothing checkable is recorded.** A raw agent log is hard to trust. Someone reading it by hand cannot easily tell whether a row that claims \"I changed how the agent behaves\" is backed by anything, or whether an advisory step quietly asserted authority it never had. Loose logging fails here in a specific way: a trace that reads as proof, with nothing recorded that a reader can check.\n\n###### No claim counts unless the record backs it\n\n**It refuses to treat a claim as evidence unless the record supports it: advisory rows cannot claim mutation authority, behavior-change needs named trace ids, and private bodies are redacted.** This component closes that gap by refusing to treat a claim as evidence unless the record supports it. An advisory row cannot also claim live mutation authority. A behavior-change claim is blocked unless it names the trace ids that evidence it. A private transcript body is rejected and redacted. The useful output is a compact result record: one command, one selected route, one work-event-evidence chain, one validator handle, and one scope limit.\n\n##### How it works\n\n###### `run` scans, then calls six validators\n\n** `run` loads the fixture, scans the inputs for forbidden non-public-state classes before any check, then calls six validators in order and merges what they observed.** The default entrypoint is `run`. It loads the first-wave fixture, resolves the public root for the input path, and scans the inputs for forbidden non-public-state classes before any check runs. It then calls six validators in order and merges what they observed.\n\n###### A green run means every rejection fired\n\n**The fixture is deliberately adversarial, so a green run means every expected negative case fired and nothing unexpected did, not that the trace was clean.** The fixture holds 10 trace rows, 6 hook-shadow cases, 6 egress cases, and 2 anti-pattern debt rows. It is deliberately adversarial: it is built to trigger every planned rejection. A green run therefore means every expected negative case fired and nothing unexpected did, not that the trace was clean.\n\n###### `validate_route_compliance` classifies each row\n\n** `validate_route_compliance` checks each trace row for duplicate ids, a missing lease, a route miss, a private payload, an advisory authority claim, or unevidenced behavior change, passing only on expected cases.** `validate_route_compliance` classifies the trace rows. It counts duplicate event ids, and for each row it checks for a missing route lease, a route miss where the observed route differs from the expected route, a private transcript payload class, an advisory row that claims mutation authority, and a behavior-change claim with no evidence ids. Each check that fires records a typed finding and marks the row rejected. The check passes when every negative case it saw is one of the expected cases.\n\n###### `validate_route_lease_mode_control` reads leases\n\n** `validate_route_lease_mode_control` flags kernel bloat before direct action and leases that offer static route metadata in place of consumed trace feedback.** `validate_route_lease_mode_control` reads the same trace rows as route leases. It flags `KERNEL_BLOAT_BEFORE_DIRECT_ACTION` when a lease returns to broad kernel context before acting, and `ROUTE_LEASE_NOT_CONSUMED` when static route metadata is offered in place of consumed trace feedback.\n\n###### `validate_hook_shadow` keeps hooks projection-only\n\n** `validate_hook_shadow` replays each hook case, requiring `projection_not_authority`, metadata-only fields, and every live-state field false, so any read of live operator, provider, or operator UI state is rejected.** `validate_hook_shadow` replays the hook intervention cases. Each case must map a case id, hook id, coverage claim, repair class, and expected intervention, must declare `projection_not_authority`, must stay metadata-only, and must keep every live-state field false. A case that reads live operator, provider, browser, HUD, cockpit, or ledger state is rejected.\n\n###### `validate_debt_retirement` needs behavior-change evidence\n\n** `validate_debt_retirement` retires a debt row only when it names behavior-change evidence trace ids, so evidence-only telemetry cannot retire debt on its own.** `validate_debt_retirement` marks a debt row retired only when it names behavior-change evidence trace ids and carries no evidence-only trace ids. Evidence-only telemetry cannot retire debt on its own.\n\n###### `validate_agent_principle_lens` mints nothing\n\n** `validate_agent_principle_lens` confirms the lens preserves its selected principle ids and route handles while minting no principle, promoting no axiom, and exporting no full body.** `validate_agent_principle_lens` confirms the lens preserves its selected principle ids and its route handles back to the principle cards and the agent operating packet. The lens cannot mint principles, promote a candidate axiom, or export full principle bodies. Any of those sets a finding.\n\n###### `validate_egress_mirror` recomputes each decision\n\n** `validate_egress_mirror` recomputes each detector decision against the declared one, requiring every private-read, provider-read, cockpit-read, and body-export flag to stay false.** `validate_egress_mirror` recomputes each detector decision and compares it to the declared one. Every case must keep `private_state_read`, `provider_payload_read`, `browser_hud_cockpit_state_read`, and `body_exported` false.\n\n###### A pass writes six metadata-only result records\n\n** `run` passes only when no expected case is missing, the scan is clean, and all six validators pass, after which `write_receipts` emits six metadata-only JSON records and `result_card` a compact card.** `run` passes only when no expected negative case is missing, the non-public-state scan passes, and route compliance, hook shadow, route trace analytics, the principle lens, and the egress mirror all pass. It then calls `write_receipts`, which writes the six JSON result records under the public root and returns their public-relative paths. `result_card` compresses a full result into a start card and drops the full findings, private scans, source bodies, and scope-limit bodies.\n\n###### Bundle runners over exported evidence slices\n\n** `run_observability_bundle` validates a larger exported bundle and blocks on any forbidden payload key or manifest drift, while companion runners check the same membrane over adjacent evidence slices.** Alongside the fixture path, `run_observability_bundle` validates a larger exported bundle: route events, agent path observations, session diagnostics, hook-shadow rows, actor-axis checks, debt rows, process-audit rows, an observability policy, and a source-module manifest, blocking on any forbidden payload key or manifest drift. Companion runners such as `run_agent_observability_store_bundle` check the same membrane over adjacent public route, session, bridge, controller, and store evidence slices.\n\nFunction Role `run` Loads the fixture, runs the six checks in order, composes status, writes six result records `validate_route_compliance` Classifies trace rows; rejects missing lease, duplicate id, private body, advisory authority claim, behavior overclaim, route miss `validate_route_lease_mode_control` Reads lease consumption; flags kernel bloat before direct action and unconsumed leases `validate_hook_shadow` Replays hook cases; requires projection-only, metadata-only, and all live-state fields false `validate_debt_retirement` Retires a debt row only when it names behavior-change evidence and no evidence-only refs `validate_agent_principle_lens` Checks the lens preserves selected ids and route handles and mints nothing `validate_egress_mirror` Recomputes detector decisions; requires no private read, provider read, HUD read, or body export `write_receipts` Writes the six result records under the public root and returns their paths `result_card` Compresses a result into a card, omitting findings, private scans, and source bodies Diagram of the mechanism (7 steps). Diagram source & refs\n\n```text\nflowchart TD Fixture[\"Fixture inputs agent_trace, hook_shadow, debt, principle_lens, egress\"] Scan[\"non-public-state scan forbidden classes rejected\"] Checks[\"Six validators route, lease, hook, debt, lens, egress\"] Merge[\"Merge observed negative cases\"] Status[\"Status pass only if every expected case fired\"] Result records[\"Six result records write_receipts\"] Card[\"Compact card result_card\"] Fixture --> Scan --> Checks --> Merge --> Status --> Result records --> Card\n```\n\n##### Negative cases\n\n###### The full rejection set a green run needs\n\n**A green run needs the whole rejection set to fire \u2014 eight route and lease cases plus five hook-shadow cases \u2014 and a missing or unexpected case blocks the run instead.** The fixture must trigger the full rejection set for a green run. The route and lease cases are: an actor-axis mismatch on an advisory row that claims mutation authority; a missing route lease; a private transcript body; a duplicate trace id; a route-compliance overclaim of behavior change with no evidence; kernel bloat before direct action; static route metadata without trace feedback; and a route miss replaced by a public route. The hook-shadow cases are: missing authority; a banned-route intervention; command displacement; a live-state read attempt; and a hook-shadow budget overrun. If any expected case fails to fire, or an unexpected case appears, the run is marked blocked rather than green.\n\n##### Prior Art Grounding\n\n###### Distributed tracing meets agent trajectories\n\n**Its shape borrows from distributed tracing \u2014 W3C Trace Context and OpenTelemetry \u2014 and from ReAct trajectories, exposing route, lease, trace id, evidence chain, validator, and scope limit together.** This component borrows its shape from distributed tracing and agent trajectory work. The [W3C Trace Context](https://www.w3.org/TR/trace-context/) recommendation and [OpenTelemetry](https://opentelemetry.io/docs/reference/specification/overview/) set the observability pattern: propagate trace identity, collect events, and keep enough context to debug a distributed transaction. Agent work such as [ReAct](https://arxiv.org/abs/2210.03629) made the interleaved reasoning-and-action trajectory a first-class object for reading agent behavior. The traceability shape carries over: selected route, route lease, trace id, work-event-evidence chain, validator ref, and scope limit are exposed together.\n\n##### Source-Backed Doctrine Packet\n\n###### Authority rows connect claims to runtime evidence\n\n**The registry, evidence-class row, and governing standard let readers trace the public claim to its exact semantic-validator boundary.** A reader can move from the public claim to the runtime, standard, manifests, result records, and negative cases without guessing. The authority rows are `core/organ_registry.json::implemented_organs[organ_id=agent_route_observability_runtime]` and `core/organ_evidence_classes.json::organ_evidence_classes[agent_route_observability_runtime]`. The registry declares evidence class `semantic_validator`. The governing standard is `standards/std_microcosm_agent_route_observability_runtime.json`, whose authority boundary is `public_route_observability_runtime_metadata_and_copied_macro_trace_bodies_not_live_session_provider_browser_hud_or_hook_authority`.\n\n###### Source manifests expose copied bodies without live sessions\n\n**The runtime and three source manifests keep copied bodies inspectable with `body_in_receipt=false` while granting no access to live sessions.** The runtime source is `src/microcosm_core/organs/agent_route_observability_runtime.py`. Copied public source-module bodies stay inspectable through their manifests with `body_in_receipt=false`, including `examples/agent_route_observability_runtime/exported_observability_bundle/source_module_manifest.json` and the two shared projection-import manifests `examples/macro_projection_import_protocol/exported_projection_import_bundle/agent_execution_trace_source_module_manifest.json` and `examples/macro_projection_import_protocol/exported_projection_import_bundle/agent_observability_source_module_manifest.json`. The cross-module test `tests/test_macro_projection_import_protocol.py::test_agent_execution_trace_body_import_is_unified_under_macro_projection_spine` keeps that body import under one shared spine. This packet does not inspect live sessions.\n\nThe registry validator command is:\n\n```text\npython -m microcosm_core.organs.agent_route_observability_runtime run --input fixtures/first_wave/agent_route_observability_runtime/input --out receipts/first_wave/agent_route_observability_runtime\n```\n\nThe six result records this run writes are:\n- `receipts/first_wave/agent_route_observability_runtime/route_compliance_audit.json`\n- `receipts/first_wave/agent_route_observability_runtime/hook_shadow_coverage.json`\n- `receipts/first_wave/agent_route_observability_runtime/debt_retirement_receipt.json`\n- `receipts/first_wave/agent_route_observability_runtime/route_lease_mode_control_receipt.json`\n- `receipts/first_wave/agent_route_observability_runtime/agent_principle_lens_receipt.json`\n- `receipts/first_wave/agent_route_observability_runtime/egress_mirror_receipt.json`\n\n##### Observable First Artifact Contract\n\n###### One viewport in a fixed order\n\n**The first artifact fits one viewport in a fixed order \u2014 command, route, work, evidence chain, authority boundary \u2014 and any browser or video view is a projection that keeps those visible.** The first artifact a reader sees fits one viewport and keeps this order: the local action command, the selected route with a short reason, the work transaction with its state, the event and evidence chain with its class, and the authority boundary beside the positive claim. A browser view, screenshot, or video is a projection of that board, not a separate claim, and it must keep the command, evidence ref, and scope limit visible before any decoration.\n\n##### Validation Result record Path\n\nFrom `microcosm-substrate/`, reproduce this page's proof boundary with temporary result records:\n\n```text\nPYTHONPATH=src ../repo-python \\\n  -m microcosm_core.organs.agent_route_observability_runtime \\\n  run \\\n  --input fixtures/first_wave/agent_route_observability_runtime/input \\\n  --out /tmp/microcosm-agent-route-observability-runtime\nPYTHONPATH=src ../repo-python \\\n  -m microcosm_core.organs.agent_route_observability_runtime \\\n  validate-observability-bundle \\\n  --input examples/agent_route_observability_runtime/exported_observability_bundle \\\n  --out /tmp/microcosm-agent-route-observability-bundle\n../repo-pytest \\\n  tests/test_agent_route_observability_runtime.py \\\n  tests/test_macro_projection_import_protocol.py::test_agent_execution_trace_body_import_is_unified_under_macro_projection_spine\n```\n\n###### What a pass certifies, and does not\n\n**A pass means the fixture and bundle produced coherent, metadata-only route feedback within the scope limit and the focused regression still holds, but the result records certify no live behavior.** A pass means the fixture and exported bundle produced coherent route feedback and metadata-only result records within the scope limit, and the focused regression still holds. The focused test is the proof consumer for this section, covering streaming JSONL loaders, duplicate-key rejection, source-module manifest contracts, digest and line-count helpers, redacted result records, and the required negative cases. These validation result records do not certify live behavior.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim\n\n**A green run shows only that a public synthetic fixture or bundle carries coherent, self-consistent route feedback and metadata-only records with every planned rejection firing.** The strongest claim the evidence supports: a green run shows that a public synthetic fixture or exported bundle carries coherent, self-consistent route feedback and metadata-only result records, with every planned rejection firing. That is the whole proof boundary.\n\n###### The metadata boundary it refuses to cross\n\n**It reads no live operator, provider, or account state and mutates, installs, authorizes, promotes, or releases nothing, holding to the metadata boundary rather than any live-trace authority.** What it refuses: it does not read live operator, provider, browser, HUD, or account state, mutate a route or source, install a hook, authorize a provider, promote a candidate axiom, prove benchmark behavior, or approve launch. The scope limit is the metadata boundary, not a live-trace authority. The exact scope limit from the registry reads:\n\n```text\nvalidates only public recorded route-feedback and observability metadata fixtures, including route-lease consumption, trace attribution, hook-shadow advisory status, anti-pattern debt retirement, behavior-change evidence gates, and public source body import refs; does not read live operator/provider/browser UI/account state, mutate work log or source, install hooks, certify runtime behavior, authorize pattern assimilation, whole-system equivalence, launch, public sharing, or whole-system correctness\n```","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":18940,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/agent_route_observability_runtime.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"W3C Trace 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Budget","url":"paper-module-provider-context-recipe-budget.html","relation_id":"paper_module.depends_on.paper_module","source_ref":"core/paper_module_capsules.json::paper_modules[54:paper_module.provider_context_recipe_budget].depends_on"},{"id":"paper_module.routing_anti_patterns_registry","kind":"paper_module","kind_label":"paper module","relation":"depended on by","label":"Routing Anti-Patterns Registry","url":"paper-module-routing-anti-patterns-registry.html","relation_id":"paper_module.depends_on.paper_module","source_ref":"core/paper_module_capsules.json::paper_modules[57:paper_module.routing_anti_patterns_registry].depends_on"}],"source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/agent_route_observability_runtime.md"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json"}],"visible_text_byte_count":20266,"section_titles":["One recorded trail is checked end to end","Six checks share the declared scope limit","Recorded evidence cannot prove live compliance","Purpose","A raw log reads as proof","No claim counts unless the record backs it","How it works","`run` scans, then calls six validators","A green run means every rejection fired","`validate_route_compliance` classifies each row","`validate_route_lease_mode_control` reads leases","`validate_hook_shadow` keeps hooks projection-only","`validate_debt_retirement` needs behavior-change evidence","`validate_agent_principle_lens` mints nothing","`validate_egress_mirror` recomputes each decision","A pass writes six metadata-only result records","Bundle runners over exported evidence slices","Negative cases","The full rejection set a green run needs","Prior Art Grounding","Distributed tracing meets agent trajectories","Source-Backed Doctrine Packet","Authority rows connect claims to runtime evidence","Source manifests expose copied bodies without live sessions","Observable First Artifact Contract","One viewport in a fixed order","Validation Result record Path","What a pass certifies, and does not","Scope limit","The strongest supported claim","The metadata boundary it refuses to cross"]},{"id":"paper_module.agent_benchmark_integrity_anti_gaming_replay","title":"Agent Benchmark Integrity Anti-Gaming Replay","relative_href":"docs/paper-module-agent-benchmark-integrity-anti-gaming-replay.html","read_href":"docs/paper-module-agent-benchmark-integrity-anti-gaming-replay.html","context_href":"docs/rules-and-ideas.html#paper-module-agent-benchmark-integrity-anti-gaming-replay","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"The agent benchmark integrity anti-gaming replay validates copied public source pattern provenance bodies and metadata-only benchmark replay rows before any score-like language is allowed.","summary":"Agent Benchmark Integrity Anti-Gaming Replay is the public benchmark-claim boundary for Microcosm. It checks locked evaluator ids and config hashes, declared benchmark case rosters, replay rows, file-access and contamination refs, trusted-reference score refs, output-replay refs, computed-vs-declared integrity verdicts, three copied source pattern provenance bodies, and eleven anti-gaming negative cases while keeping private issue, oracle patch, hidden-gold, provider, raw patch, and score payload bodies out of result records.","authority_ceiling":"Copied public source pattern provenance bodies and metadata-only synthetic benchmark-integrity replay result records only; no benchmark claims, SWE-bench performance claim, hidden-gold access, oracle patch body export, private issue body export, external model access, live repository mutation, launch-scope decision, publishing-scope decision, source-file changes, or product-progress evidence.","abstract":"The agent benchmark integrity anti-gaming replay validates copied public source pattern provenance bodies and metadata-only benchmark replay rows before any score-like language is allowed.","teleology":{"section_id":"","text":"The agent benchmark integrity anti-gaming replay validates copied public source pattern provenance bodies and metadata-only benchmark replay rows before any score-like language is allowed.","source":"capsule_summary_line"},"body_markdown":"###### Every claimed pass is recomputed\n\n**A claimed repair-benchmark \"pass\" is downgraded to a claim and recomputed from public references before this component will repeat it.** Agent benchmark numbers are easy to state and hard to trust. This component takes a set of claimed repository-repair passes and refuses to repeat the word \"pass\" until each one can be replayed from public references that name its evaluator, its frozen configuration, and the evidence that the run was not gamed. It recomputes every verdict from scratch. A row that simply declares `integrity_pass` is treated as a claim to be checked, not as proof.\n\n###### Public references in, metadata-only verdicts out\n\n**The inputs are public case ids, evaluator ids, configuration hashes, and evidence references; the output is a metadata-only record of which replays cleared the boundary and why.** The component reads a fixture that models a repair benchmark with public case ids, evaluator ids, configuration hashes, and reference paths to file-access logs, contamination checks, trusted-reference scores, and output replays. It keeps issue bodies, oracle patches, hidden answers, and model-output data out of every result record. What it emits is a metadata-only record: which case replays cleared the boundary, which were quarantined, and why.\n\n###### A consistency checker, not a leaderboard\n\n**It is a consistency checker over public references, not a benchmark runner, a leaderboard, or a score.** It checks that a set of public references is internally consistent and traces back to a command that actually ran.\n\n##### Purpose\n\n###### What a bare pass-rate headline hides\n\n**A headline pass-rate hides the evaluator, configuration, leakage, and selection decisions that produced it; this gate makes the claim carry them or be quarantined.** A single headline like \"passes N percent of repair tasks\" hides every decision that produced it: which evaluator ran, whether its configuration was frozen, whether the agent could see held-out answers, whether the test cases leaked into training, and whether one lucky attempt was promoted as the result. A loose benchmark claim carries none of that. This component exists to make the claim carry it, or be quarantined.\n\n###### Declaring success is not a way through\n\n**Every reference the policy names must exist and pass, or the row recomputes to quarantine no matter what it declared.** The gate is deliberately hostile to the easy path. The evaluator id must be on a locked list. The configuration hash must be one the policy declared in advance. File-access, contamination, trusted-reference, and output-replay references must exist and pass. The case id must be registered up front. If any of those is missing or contradicted, the row is recomputed as `quarantine` no matter what it declared.\n\n###### A real command run backs every pass\n\n**Each `integrity_pass` must cite a digest-bound artifact from a completed, exit-zero run of this component's focused test, so no claim rests on prose alone.** An `integrity_pass` must be backed by a sanitised real command run, not only by hand-written references. Each passing row cites a `real_benchmark_trace_ref` that has to resolve to a copied artifact carrying a completed, exit-zero run of this component's focused test, with sha256 digests bound to a recorded run id and an explicit list of omitted live material. This is what stops a benchmark claim from resting on prose.\n\n##### How it works\n\n###### One orchestrator, two entry points\n\n** `_build_result` loads the four fixture inputs and runs every validation stage in order; `run` and `run_benchmark_integrity_bundle` are thin wrappers over it.** `_build_result` is the orchestrator. It loads the four fixture inputs (`projection_protocol.json`, `locked_evaluator_policy.json`, `benchmark_cases.json`, `replay_observations.json`), then runs each validation stage in order and returns one metadata-only result. The entry points `run` and `run_benchmark_integrity_bundle` wrap it. `run` validates the fixture with the twelve negative-case files included. `run_benchmark_integrity_bundle` validates the exported public bundle without them, and can reuse a prior result record when `_freshness_basis` shows the inputs, validator source, and evidence digests are unchanged.\n\nFunction What it does `validate_source_module_imports` Reads `source_module_manifest.json`, checks each copied body is `copied_non_secret_macro_body` with `body_in_receipt=false`, and recomputes its sha256 against disk. `_real_trace_artifact_findings` Validates the sanitised real-run artifact: schema, `completed` status with `exit_code` 0, the focused pytest argument shape, a passing test summary, sha256 digests, and declared omissions. `validate_locked_evaluator_policy` Indexes the evaluator ids and configuration hashes the policy allows. `validate_benchmark_cases` Confirms the case roster and held-out guard ids. `_validate_replay_row` Recomputes one row's verdict from every check below. `validate_public_trace` Recomputes each verdict a second time from independent trace spans and flags any mismatch.\n\n###### Per-row recomputation is the centre of the mechanism\n\n**A row passes only when every required field is present, its evaluator and configuration hash are policy-locked, its real-run trace verifies, and no evidence check fails.** The center of the mechanism is `_validate_replay_row`. It collects the required fields listed in `REQUIRED_REPLAY_FIELDS`, checks the evaluator id is locked and its configuration hash is policy-declared, checks the case id is in the roster, and confirms the `real_benchmark_trace_ref` resolves to a digest-verified real-run artifact that is also cited among the row's evidence references. For a positive row it opens the file-access, contamination, trusted-reference, and output-replay evidence and folds any failing check or contamination flag into a reason list. The rule at the end is one line: the computed verdict is `integrity_pass` only when no required field is missing and the reason list is empty, otherwise `quarantine`.\n\n###### An independent second pass over the public trace\n\n** `validate_public_trace` recomputes every verdict a second time from independent trace spans and turns any disagreement into a blocking finding.** It reads the trace built by `build_public_benchmark_integrity_anti_gaming_trace`, which recomputes each verdict from contamination, file-access, and locked-evaluator spans. If a span's recomputed verdict disagrees with the declared one, or its configuration hash is not policy-declared, or its evaluator is not locked, the disagreement becomes a finding. `_build_result` returns `status=pass` only when the non-public-state scan finds no blocking hit, every sub-validator passes, and no expected negative case is missing.\n\nDiagram of the mechanism (6 steps). Diagram source & refs\n\nSource refs\n\nmanifest digest + real-run floor `validate_source_module_imports`\n\n```text\nflowchart TD Inputs[\"projection, policy, cases, replay rows\"] --> Source[\"validate_source_module_imports manifest digest + real-run floor\"] Inputs --> Rows[\"_validate_replay_row recompute per row\"] Source --> Rows Rows --> Verdict[\"integrity_pass or quarantine\"] Verdict --> Trace[\"validate_public_trace independent recompute\"] Trace --> Result[\"metadata-only result record\"]\n```\n\n###### What the shipped fixture recomputes to\n\n**Three public cases recompute to two passes and one quarantine, and the public trace has to agree before the run can pass.** On the shipped fixture the three cases `repo_issue_public_001`, `repo_issue_public_002`, and `repo_issue_public_003` recompute to two `integrity_pass` rows and one `quarantine`.\n\n##### Negative cases\n\n###### Twelve gaming attempts the fixture must catch\n\n**Each named negative case maps to a required reason code, and a run fails if any expected case is not observed.** `EXPECTED_NEGATIVE_CASES` names twelve gaming attempts the fixture must catch, each mapped to the reason code the component has to raise. They are an evaluator edit attempt, train/test leakage, a real-trace variant of train/test leakage, an oracle patch body leak, hidden-gold access, final-answer-only grading, a model-output data leak, a score overclaim, pass-k cherry-picking, an admitted misleading test, a private issue body leak, and an unregistered case replay. The negative fixtures are a required floor rather than decoration.\n\n##### Prior Art Grounding\n\n###### Goodhart, reward hacking, and leakage accounting\n\n**The design follows the reward-hacking and benchmark-leakage literature: a metric stops being evidence once it becomes a target, so the references must carry the proof.** The component draws on the long-standing observation that an optimized metric loses its evidential force once it becomes a target, and on the AI-safety literature on reward hacking and specification gaming. [Concrete Problems in AI Safety](https://arxiv.org/abs/1606.06565) frames reward hacking as a practical accident risk, DeepMind's [specification-gaming survey](https://deepmind.google/blog/specification-gaming-the-flip-side-of-ai-ingenuity/) collects concrete examples, and [Benchmarking Benchmark Leakage in Large Language Models](https://arxiv.org/abs/2404.18824) motivates explicit leakage documentation. The engineering pattern is simple accounting: evaluator ids, configuration hashes, case rosters, and evidence references must be present and consistent before benchmark-style language is allowed.\n\n##### Validation Result record Path\n\n###### Fixture output stays outside the working tree\n\n**The fixture validator runs from the repository root and writes its result to a disposable path outside the working tree.** Run the fixture validator from the repository root and write its result record outside the working tree:\n\nThen run the exported-bundle validator:\n\n```text\ncd microcosm-substrate && PYTHONPATH=src ../repo-python -m microcosm_core.organs.agent_benchmark_integrity_anti_gaming_replay run-benchmark-integrity-bundle --input examples/agent_benchmark_integrity_anti_gaming_replay/exported_benchmark_integrity_bundle --out /tmp/agent_benchmark_integrity_bundle_receipt --card > /tmp/agent_benchmark_integrity_bundle_card.json\n```\n\nThe focused test and corpus check are:\n\n```text\ncd microcosm-substrate && ../repo-pytest tests/test_agent_benchmark_integrity_anti_gaming_replay.py\n./repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n###### What a pass does and does not mean\n\n**A pass means every verdict was recomputed from the public references and agreed with the independent trace; it does not mean a benchmark claims was produced.** A pass means the fixture and bundle validators recomputed every verdict from public case ids, locked evaluator references, configuration hashes, evidence references, manifest digests, and the twelve negative cases, and agreed with the independent trace.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim\n\n**On this fixture and bundle, a metadata-only replay respected the anti-gaming boundary, and `AUTHORITY_CEILING` stops the proof there.** The strongest claim the evidence supports is narrow: on this public fixture and the exported bundle, a metadata-only replay respected an anti-gaming boundary over public case ids and locked evaluator references, and the two positive rows each trace back to a passing command run. The `AUTHORITY_CEILING` constant sets every outward authorization flag to false, so the proof boundary stops there.\n\n###### What stays false even on a pass\n\n**A green run still claims no benchmark claims, no repair capability, and no access to hidden or private material.** It claims no benchmark claims, no SWE-bench result, and no repair capability. It does not access hidden-gold answers, oracle patches, or private issue text, does not call a provider, does not mutate a live repository, and excludes public sharing or launch. Those remain false even when the fixture passes.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":11991,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/agent_benchmark_integrity_anti_gaming_replay.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"Concrete Problems in AI Safety","url":"https://arxiv.org/abs/1606.06565","kind":"external_reference","section_id":"goodhart-reward-hacking-and-leakage-accounting"},{"label":"specification-gaming survey","url":"https://deepmind.google/blog/specification-gaming-the-flip-side-of-ai-ingenuity/","kind":"external_reference","section_id":"goodhart-reward-hacking-and-leakage-accounting"},{"label":"Benchmarking Benchmark Leakage in Large Language 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Replay","url":"paper-module-research-replication-rubric-artifact-replay.html","relation_id":"paper_module.depends_on.paper_module","source_ref":"core/paper_module_capsules.json::paper_modules[4:paper_module.research_replication_rubric_artifact_replay].depends_on"}],"source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/agent_benchmark_integrity_anti_gaming_replay.md"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json"}],"visible_text_byte_count":13342,"section_titles":["Every claimed pass is recomputed","Public references in, metadata-only verdicts out","A consistency checker, not a leaderboard","Purpose","What a bare pass-rate headline hides","Declaring success is not a way through","A real command run backs every pass","How it works","One orchestrator, two entry points","Per-row recomputation is the centre of the mechanism","An independent second pass over the public 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Replay","relative_href":"docs/paper-module-research-replication-rubric-artifact-replay.html","read_href":"docs/paper-module-research-replication-rubric-artifact-replay.html","context_href":"docs/rules-and-ideas.html#paper-module-research-replication-rubric-artifact-replay","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"The research replication rubric-artifact replay validates source-backed public replication bundles before any paper-replication language is allowed.","summary":"Research Replication Rubric Artifact Replay is the public research-replication claim boundary for Microcosm. It checks contribution decomposition refs, rubric trees, allowed public inputs, scratch repo scaffolds, experiment DAG refs, metric scripts, declared artifact-hash rosters, artifact hashes, grader reports, compute/runtime budgets, ablation diffs, failure taxonomies, cold-rerun result records, public agent-execution trace spans, four copied source modules, and eight replication-overclaim negative cases while keeping private paper/data bodies, hidden rubrics, model-output data, original-author code bodies, benchmark claims, and public-sharing claims out of result records.","authority_ceiling":"Copied public source pattern provenance bodies, exact-copy public Python internal control body, metadata-only research-replication replay result records, public agent-execution trace spans, and fixture validation only; no actual paper replication success, benchmark performance claim, private paper/data body export, hidden-rubric export, external model access, unbounded compute search, original-author code reuse, launch-scope decision, publishing-scope decision, source-file changes, or product-progress evidence.","abstract":"The research replication rubric-artifact replay validates source-backed public replication bundles before any paper-replication language is allowed.","teleology":{"section_id":"","text":"The research replication rubric-artifact replay validates source-backed public replication bundles before any paper-replication language is allowed.","source":"capsule_summary_line"},"body_markdown":"###### Evidence bundle for a claimed paper replication\n\n**It turns the claim that an agent replicated a paper into an evidence bundle a cold runtime can check, reruns nothing scientific, and forces the packet's numbers to reproduce.** `research_replication_rubric_artifact_replay` is a public Microcosm component that turns the sentence \"an agent replicated a paper\" into a bundle a cold runtime can check. It reruns nothing scientific. It reads a replay packet and confirms the packet names every piece of evidence a real replication would have left behind, then runs one small local metric so the numbers in the packet have to reproduce rather than be taken on trust.\n\n**It runs over two synthetic public papers, and a pass means the replay packet is accountable and digest-bound, not that a real paper was independently replicated.** The component runs over two synthetic public papers, one machine-learning method and one computational-science study. A pass says the replay packet is structurally accountable, digest-bound, and negative-case tested. It does not say a real paper was independently replicated.\n\n##### Purpose\n\n###### Four things collapsed into one claim\n\n**Demos fold the paper, the artifact, the rubric, and the rerun evidence into a single word, so this component pulls \"replicated\" apart and asks for each object by name.** Research-agent demos collapse four separate things into one claim: the paper, the runnable artifact, the grading rubric, and the evidence that an independent rerun happened. A loose check accepts any of them as proof of all four. This component refuses to treat \"replicated\" as one fact. It pulls the claim apart into the objects a real replication produces and asks for each one by name.\n\n###### Fourteen fields, eight axes, four forbidden authorities\n\n** `validate_replication_policy` sets the floor by requiring fourteen replay fields and eight rubric axes and forcing four escape authorities false.** `validate_replication_policy` sets the floor. It requires the fourteen fields in `REQUIRED_REPLAY_FIELDS`: `paper_id`, `contribution_decomposition_ref`, `rubric_tree_ref`, `allowed_public_input_refs`, `scratch_repo_scaffold_ref`, `experiment_dag_ref`, `metric_script_refs`, `artifact_hash_refs`, `declared_artifact_hash_refs`, `grader_report_ref`, `cost_runtime_budget_ref`, `ablation_diff_ref`, `failure_taxonomy_ref`, and `cold_rerun_receipt_ref`. It also requires the eight rubric axes in `REQUIRED_RUBRIC_AXES`: contribution decomposition, artifact replay, experiment DAG, metric script, grader alignment, budget boundary, failure taxonomy, and cold rerun. A policy that drops any field or axis is rejected. The policy must also declare four escape authorities false: hidden-rubric export, benchmark-performance claim, unbounded compute search, and final-answer-only grading. A packet cannot grant itself any of them.\n\n##### How it works\n\n**The two entry points are `run` and `run_replication_bundle`. Both call `_build_result` and differ only in what they read.** The two entry points are `run` and `run_replication_bundle`. Both call `_build_result` and differ only in what they read.\n\n###### Fixture mode versus exported-bundle mode\n\n** `run` loads the fixture with its eight negative cases, while `run_replication_bundle` validates the public export without them and reuses an unchanged prior record.** `run` is fixture mode. It reads the first-wave input directory with `include_negative=True`, so it loads the positive replay rows plus the eight negative-case files, and writes the result, board, validation, and sign-off records. `run_replication_bundle` is exported-bundle mode. It reads the public example bundle with `include_negative=False`, then writes `exported_research_replication_bundle_validation_result.json`. It can reuse a prior record when `_freshness_basis` shows the inputs are byte-identical, so a clean rerun does not rewrite an unchanged result.\n\n** `_build_result` runs the checks in order and folds them into one record.** `_build_result` runs the checks in order and folds them into one record.\n\nFunction What it checks `validate_source_module_imports` The source-module manifest declares copied public bodies with matching digests. `validate_replication_policy` The policy requires all fourteen fields and eight axes and forbids the escape authorities. `validate_research_replays` At least two positive rows carry every required field; the eight negative files each trip their expected error code. `validate_execution_artifact_replay` One declared metric runs over the allowed public input, and the output matches the declared hash. `validate_projection_protocol` The projection cites its source patterns, result records, target symbols, and public trace import.\n\n###### The metric replay that must reproduce\n\n** `_execute_metric_script` sums one named field over an allowed public input, and `validate_execution_artifact_replay` fails the row if the produced numbers do not match the declared hash.** The executable core is `_execute_metric_script`. It reads a metric spec whose `operation` is `sum_field`, sums one named field over the rows of one allowed public input table, and returns the total plus a row count. It rejects a spec that is not a bare sum, and it rejects input rows that are missing or non-numeric. `validate_execution_artifact_replay` wraps that call: it reads `execution_artifacts/execution_artifact_manifest.json`, refuses a manifest that sets `body_in_receipt`, authorizes external model access, or names a private body, runs the metric, then compares the produced payload against the declared output artifact and its declared `sha256` file. A replay row can name every correct reference and still fail here if the numbers do not reproduce, so the pass is not a field-presence check.\n\n###### Verdicts from fields, not from fixture labels\n\n** `validate_research_replays` reads verdicts from row fields rather than declared labels, and `validate_source_module_imports` verifies copied-body digests while result records stay metadata-only.** `validate_research_replays` derives its verdicts from the replay-row fields, not from any label the fixture declares. It records the keys it ignored, so a file that claims `status: pass` or names its own `expected_error_codes` cannot talk its way past the check. `validate_source_module_imports` reads the manifest at `SOURCE_MODULE_MANIFEST_REF`, requires `source_import_class` of `copied_non_secret_macro_body`, requires `body_in_receipt` to be false, and verifies each declared target digest. For an exact-copy row it also confirms the live source file exists and that the source and target bytes match. Result records carry counts, references, digests, and verdicts. They never carry a source body.\n\nDiagram of the mechanism (6 steps). Diagram source & refs\n\n```text\nflowchart TD policy[\"Replication policy 14 fields + 8 axes + forbidden claims\"] replays[\"Replay rows 2 synthetic papers + 8 negative cases\"] artifacts[\"Execution manifest allowed input + metric spec + declared hash\"] metric[\"Local metric replay sum_field over allowed input\"] source[\"Source-module manifest 4 copied public bodies\"] result[\"metadata-only result record counts, refs, digests, scope boundaries\"] policy --> replays replays --> artifacts artifacts --> metric metric --> result source --> result\n```\n\n##### Negative cases\n\n**The fixture ships eight named negative files, one per key in `EXPECTED_NEGATIVE_CASES`. Each must trip its own error code:** The fixture ships eight named negative files, one per key in `EXPECTED_NEGATIVE_CASES`. Each must trip its own error code:\n- original-author code reuse\n- hidden-rubric leakage\n- report-only success\n- benchmark-performance overclaim\n- private paper or data body leakage\n- unbounded compute search\n- final-answer-only grading\n- undeclared artifact-hash reference\n\n###### Eight fakes plus mutation cases stay blocked\n\n**These name the eight ways a plausible fake tries to pass, and the focused tests add mutation cases that stay blocked even when the tampered bundle is made internally consistent.** These are the eight ways a plausible fake tries to pass. A packet that reuses the original authors' code, calls a run a success on the strength of a write-up alone, asserts a benchmark number, or points at an output hash it never declared is rejected by field semantics, not by filename. The focused test suite adds mutation cases on top: digest mismatch, local body tamper, rehashed source swaps, wrong execution-artifact hashes, metric perturbation, input perturbation, output body tamper, and self-consistent input/output/hash rewrites. Each mutation stays blocked even when the tampered bundle is made internally consistent.\n\n##### Prior Art Grounding\n\n###### Borrowed from ACM artifact badging\n\n**The rubric-over-artifact shape follows ACM artifact badging, which separates available from functional from reproduced, made here into a runtime gate over a declared bundle.** This is the rubric-over-artifact shape from systems and machine-learning artifact evaluation, where the [ACM Artifact Review and Badging](https://www.acm.org/publications/policies/artifact-review-and-badging-current) process separates \"available\" from \"functional\" from \"reproduced\". Microcosm borrows that separation and makes it a runtime gate over a declared evidence bundle. The result is fixture-bound replay evidence, not a reproducibility guarantee and not a peer-review verdict.\n\n##### Validation Result record Path\n\nFocused runtime validation:\n\n```text\n./repo-pytest tests/test_research_replication_rubric_artifact_replay.py -q --basetemp=/tmp/microcosm_research_replication_rubric_artifact_replay_pytest\n```\n\nPaper-module corpus validation:\n\n```text\n./repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\nThe runtime commands behind the records are:\n\n###### What a pass over the records means\n\n**A pass certifies every check held, all eight negative cases were observed, and the records stayed metadata-only.** A pass means the policy, replay rows, source-module digests, projection protocol, and local metric replay all held, all eight negative cases were observed, and the records stayed metadata-only.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The narrow strongest claim\n\n**The strongest supported claim is narrow: over synthetic public inputs the packet named every evidence object, the eight overclaims were caught, and one metric reproduced against its hash.** The strongest claim the evidence supports is narrow. Over public synthetic inputs, the component confirmed a replay packet named every required evidence object, the eight overclaim cases were each caught, and one declared metric reproduced against its declared output hash. That is the whole proof boundary: the bundle, mechanism row, runtime code, source-module manifest, exported record, first-wave fixture records, and focused tests.\n\n###### What a green run still does not claim\n\n**The scope limit stops there: a green run claims no real replication, benchmark performance, external model service, or launch-scope decision, and the value is the boundary, not the science.** The scope limit stops there. A green run does not claim actual paper replication, benchmark performance, external model service, hidden-rubric access, original-author code reuse, private paper or data export, unbounded compute, final-answer-only grading, launch-scope decision, or publishing-scope decision. The two subjects are synthetic public examples, and the metric is one sum over one small table. The value here is the boundary, not the science. 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Replay","relative_href":"docs/paper-module-agentic-vulnerability-discovery-patch-proof-replay.html","read_href":"docs/paper-module-agentic-vulnerability-discovery-patch-proof-replay.html","context_href":"docs/rules-and-ideas.html#paper-module-agentic-vulnerability-discovery-patch-proof-replay","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"The agentic vulnerability discovery patch-proof replay validates metadata-only synthetic vulnerability evidence chains before any found-and-fixed security language is allowed.","summary":"Agentic Vulnerability Discovery Patch-Proof Replay is the public security-claim boundary for Microcosm. It checks projection protocol, vulnerability policy, synthetic target refs, issue hypotheses, trace evidence, abstract exploitability refs, patch diff refs, regression tests, verifier result records, sandbox verdicts, false-positive triage, cold replay, public agent-execution trace spans, secret-exclusion scan, nine copied source/control/standard/tool bodies, source-module manifest digests, metadata-only result record policy, and eight security-overclaim negative cases while keeping live targets, real CVE exploitation, weaponized payloads, account secrets, network exfiltration steps, actionable exploit instructions, model-output data, raw issue or patch bodies, benchmark claims, and source-file changes out of result records.","authority_ceiling":"Copied public source/control/standard/tool bodies, metadata-only synthetic patch-proof replay result records, public agent-execution trace spans, and fixture validation only; no live target testing, real CVE exploitation, weaponized payload export, account secret handling, network exfiltration, actionable exploit instructions, external model access, source-file changes, benchmark security score, launch-scope decision, publishing-scope decision, whole-system security claim, or product-progress evidence.","abstract":"The agentic vulnerability discovery patch-proof replay validates metadata-only synthetic vulnerability evidence chains before any found-and-fixed security language is allowed.","teleology":{"section_id":"","text":"The agentic vulnerability discovery patch-proof replay validates metadata-only synthetic vulnerability evidence chains before any found-and-fixed security language is allowed.","source":"capsule_summary_line"},"body_markdown":"###### Patch-proof language needs more than a claim\n\n**\"Found and fixed a security bug\" is easy to type and hard to check, standing equally for a tested repair, a patch never run, a false positive, or an empty benchmark number.** An agent that says it \"found and fixed a security bug\" is making a claim that is easy to type and hard to check. The phrase can stand for a real, tested repair. It can also stand for a patch that was never run, a false positive promoted to a finding, or a benchmark number with nothing behind it.\n\n###### Seven evidence stages must agree\n\n**It breaks the claim into a synthetic evidence chain, from target through issue, trace, patch, test, witness, and verifier record, and checks the pieces line up before allowing patch-proof language.** This component takes that claim apart into an inspectable chain of synthetic evidence and checks that the pieces line up before any patch-proof language is allowed. The chain runs from a made-up target, through a suspected issue, a trace that backs it, an abstract exploitability reference, a patch, a regression test, an executable witness, and a verifier record. Nothing in the chain is a real target, exploit, payload, or account secret. The result record says whether the declared chain holds together, and carries no attack material.\n\n###### Synthetic replay grants no security authority\n\n**It is a synthetic, metadata-only replay, not live security work, that runs against no real system, judges no bug or fix as genuine, and produces no benchmark claims.** The code lives in `src/microcosm_core/organs/agentic_vulnerability_discovery_patch_proof_replay.py`. It is a synthetic, metadata-only replay, not live security work. It does not run against real systems, does not judge whether the bug or fix is genuine, and does not produce a benchmark claims.\n\n##### Purpose\n\n###### Why two recompute checks exist\n\n**A summary can hide the absence of the work it summarises, so instead of trusting the fixture's labels two checks recompute the thing the fixture claims.** The problem is that \"found and fixed\" is a summary, and a summary can hide the absence of the work it summarises. A loose check that only reads the fixture's own labels would pass a chain that asserts a green result without the work behind it. Two checks here refuse that by recomputing the thing the fixture claims, rather than trusting the field.\n\n###### Recompute one: running the regression witnesses\n\n**The first recompute runs each witness's mini-target twice, unpatched and patched, and rejects any witness whose declared pre and post outputs do not match the computed ones.** The first recompute is on the executable regression witnesses. Each witness names one of three small public mini-targets: a webhook redirect allowlist, a notebook log redactor, and a scheduler path normaliser. The validator runs that function twice, once unpatched and once patched, and compares its own computed outputs against the `expected_pre_patch` and `expected_post_patch` values the fixture declared. A witness whose declared output does not match the computed output is rejected.\n\n###### Recompute two: the verifier verdict\n\n**The second recompute derives each verifier verdict from the joined hypothesis, proof, patch, test, and witness evidence, and requires the filename to match rather than trusting the record's own label.** The second recompute is on the verifier records. Each record's `pass` or `false_positive` verdict is recomputed from the joined hypothesis, proof, patch, test, and witness evidence, and the record's filename must match that recomputed verdict. The record's own label is not taken on trust.\n\n##### How it works\n\n###### `_build_result` orchestrates the validators\n\n** `_build_result` loads inputs and policy, scans every path for secrets, builds the public trace, then runs one validator per evidence kind, passing only when every check and negative case clears.** `_build_result` is the orchestrator. It loads the fixture inputs, loads the forbidden-class policy, runs a secret-exclusion scan over every input path, builds the public agent-execution trace, then runs one validator per evidence kind in dependency order. Each validator returns a status, exported rows, and a list of findings. A result reaches `pass` only when there are no missing negative cases, the secret scan has zero blocking hits, the public trace status is `pass`, every positive validator returned `pass`, and, for the exported bundle, the source-module manifest returned `pass`.\n\n###### Each validator joins against the preceding stage\n\n**The validators form an ordered evidence join in which every stage resolves its rows against the stage immediately before it.** The validators form a join, not a scanner. Each stage resolves its rows against the stage before it.\n\nFunction Role `run` Fixture entry point. Runs `_build_result` with negative cases on. `run_patch_proof_bundle` Exported-bundle entry point. Negative cases off, source-module manifest required. `_build_result` Loads inputs, runs every validator, computes the final status. `validate_issue_hypotheses` Each hypothesis joins to a synthetic target and cites evidence refs; needs at least four rows. `validate_trace_evidence` Each trace joins to a hypothesis; the set must cover static, dynamic, and property types. `validate_patch_diffs` Each patch-required hypothesis carries a metadata-only diff, updates tests, and claims no source-file changes. `validate_regression_tests` Each patch pairs with a test that fails before the patch and passes after it. `validate_executable_regression_witnesses` Runs the mini-target twice and compares computed output to the declared pre/post values. `validate_verifier_receipts` Recomputes each `pass` or `false_positive` verdict from joined evidence and checks the filename. `validate_sandbox_policy_verdicts` Each verdict precedes action and denies live-target and network material. `validate_cold_replay` Each replay row reproduces its trace, patch record, and sandbox verdict. `validate_negative_cases` Records the forbidden overclaims the fixture is expected to trip. `_source_module_manifest_result` Digest-checks the copied public source bodies carried by the exported bundle.\n\n###### Three mini-target engines run both ways\n\n**Three `MINI_TARGETS` engines each take a `patched` flag: unpatched they leak or over-allow, patched they close the hole, and running both forms confirms a declared fails-before-passes-after test.** The three mini-target engines are `_webhook_redirect_allowlist`, `_notebook_log_redaction`, and `_scheduler_path_normalization`, registered in `MINI_TARGETS`. Each takes an input and a `patched` flag. Unpatched, the webhook engine allows any redirect, the notebook engine leaks its sensitive refs, and the scheduler engine accepts a path that escapes the workspace. Patched, each closes its hole. Running both forms is what lets `validate_executable_regression_witnesses` confirm that a declared \"fails before, passes after\" test actually does.\n\n###### Strict join order blocks broken chains\n\n**The join is strict: an unresolved hypothesis, a patch missing its proof or diff, a patch with no test, or a promoted false positive is each blocked.** The join order is strict. A hypothesis that does not resolve to a synthetic target is blocked. A patch-required hypothesis missing either an exploitability proof or a metadata-only patch diff is blocked. A patch with no paired regression test is blocked. A false positive promoted to a finding is blocked, because `validate_verifier_receipts` will not recompute its verdict as `pass`.\n\nDiagram of the mechanism (9 steps). Diagram source & refs\n\n```text\nflowchart TD targets[\"synthetic targets\"] hypotheses[\"issue hypotheses\"] traces[\"trace evidence\"] proofs[\"exploitability proofs\"] patches[\"patch diffs\"] tests[\"regression tests\"] witnesses[\"executable witnesses run mini-target twice\"] verifiers[\"verifier records recompute verdict\"] status[\"result status\"] targets --> hypotheses hypotheses --> traces hypotheses --> proofs hypotheses --> patches patches --> tests tests --> witnesses witnesses --> verifiers verifiers --> status\n```\n\n###### Two runtime modes, both metadata-only\n\n** `run` drives the fixture with negative cases on, `run_patch_proof_bundle` drives the exported bundle as the public example, and both write metadata-only records with source bodies kept behind manifest hashes.** The runtime keeps two modes separate. `run` drives the first-wave fixture with negative cases on, so it must observe every expected overclaim failure. `run_patch_proof_bundle` drives the exported bundle as the public runtime example, so its expected negative cases can be empty while it still proves the source-body floor, the public trace, the digest checks, and the regression witnesses. Both write metadata-only result records. Copied source bodies stay behind `source_module_manifest.json` refs and hashes; the exported bundle carries nine digest-verified public source bodies.\n\n##### Negative cases\n\n###### Eight overclaims the runtime must trip\n\n**Eight `EXPECTED_NEGATIVE_CASES` overclaims must each trip a matching error code or the run is blocked, and `_has_forbidden_key` separately rejects any row carrying a forbidden field.** The first-wave fixture includes eight negative cases, held in `EXPECTED_NEGATIVE_CASES`. Each one is an overclaim the runtime must trip, not a capability it offers: `live_target_material`, `real_cve_exploitation`, `weaponized_payload_export`, account secret material, `network_exfiltration`, `exploit_instruction_steps`, `patch_without_tests`, and `benchmark_score_claim`. `validate_negative_cases` records the matching error code for each, for example `AGENTIC_VULN_PATCH_WITHOUT_TESTS` when a patch claim arrives with no regression test. If any expected case goes unobserved, the run is blocked. Separately, `_has_forbidden_key` rejects any row that carries a forbidden field such as a raw exploit payload, a account secret value, or a live target URL.\n\n##### Prior Art Grounding\n\n###### Test-backed repair lineage from SWE-bench\n\n**The design borrows the accountability pattern of SWE-bench, SWE-agent, and security benchmarks like CyberSecEval 2 and SecCodePLT, requiring evidence to line up before patch-proof language, without importing live exploits.** This component borrows an accountability pattern from agentic software-engineering and security-evaluation work that treats a code repair as an executable, test-backed claim rather than a prose claim. [SWE-bench](https://arxiv.org/abs/2310.06770) framed repository issue resolution as a task with real codebases and test-based patch evaluation, and [SWE-agent](https://arxiv.org/abs/2405.15793) made the agent-computer interface part of the repair loop. Security benchmarks such as [CyberSecEval 2](https://arxiv.org/abs/2404.13161) and [SecCodePLT](https://arxiv.org/abs/2410.11096) motivate separating a secure-code or vulnerability claim from an uninspected generated patch. The borrowed idea is narrow: hypotheses, traces, patches, tests, and verifier records must line up before patch-proof language is allowed. It does not import live targets, CVE exploitation, weaponized payloads, or benchmark claims.\n\n##### Validation Result record Path\n\n###### Fixture output stays outside the system tree\n\n**The first-wave validator runs from the system root and writes its result record to a disposable path outside the working tree.** Run the first-wave fixture validator from the system root and write its record outside the working tree:\n\nThen run the exported bundle validator:\n\n```text\ncd microcosm-substrate && PYTHONPATH=src ../repo-python -m microcosm_core.organs.agentic_vulnerability_discovery_patch_proof_replay \\\n  run-patch-proof-bundle \\\n  --input examples/agentic_vulnerability_discovery_patch_proof_replay/exported_patch_proof_bundle \\\n  --out /tmp/agentic_vulnerability_patch_proof_bundle_receipt \\\n  --card > /tmp/agentic_vulnerability_patch_proof_bundle_card.json\n```\n\nThe focused regression test is:\n\n```text\nPYTHONPATH=src ./repo-pytest \\\n  tests/test_agentic_vulnerability_discovery_patch_proof_replay.py\n```\n\n###### What a pass and the rejection tests prove\n\n**A pass means the whole join held with both recompute checks agreeing, a clean scan, and every negative case tripped, while rejection tests mutate the chain and require a blocked result.** A pass means the whole join held: the synthetic chain resolved end to end, both recompute checks agreed with the fixture, the secret scan was clean, and every expected negative case tripped. The rejection tests in the same file mutate the chain, remove a regression test, tamper a witness, cross-wire a verifier record, or alter a source-module digest, and require a `blocked` result with a specific error code.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim\n\n**On the public fixtures the synthetic chain resolves and both recompute checks pass, with witnesses, verifier verdicts, and paired tests all agreeing and every forbidden overclaim observed.** The strongest claim this component supports is narrow. On the public fixtures, the synthetic evidence chain resolves and both recompute checks pass: witnesses match their computed pre/post outputs, verifier verdicts match their recomputed result, patches pair with real tests, and every forbidden overclaim is observed. That is the scope limit, and it is where the scope limit sits.\n\n###### What the component refuses beyond that\n\n**It refuses everything past that: no live targets, real CVE exploitation, weaponized payloads, account secrets, external model access, source-file changes, benchmark claims, or whole-system security claim, only the local synthetic records.** It refuses everything past that. It does not test live targets, perform real CVE exploitation, export weaponized payloads, handle account secrets, authorise network exfiltration, or emit actionable exploit instructions. It does not make external model access, change source files, claim a benchmark security score, or judge patch correctness on a real repository. It is not launch or publishing-scope decision, and it is not a whole-system security claim. The proof boundary is the local result records, the synthetic bundle, the digest-checked source-module manifest, and the named negative cases, nothing wider.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":14357,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/agentic_vulnerability_discovery_patch_proof_replay.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"SWE-bench","url":"https://arxiv.org/abs/2310.06770","kind":"external_reference","section_id":"test-backed-repair-lineage-from-swe-bench"},{"label":"SWE-agent","url":"https://arxiv.org/abs/2405.15793","kind":"external_reference","section_id":"test-backed-repair-lineage-from-swe-bench"},{"label":"CyberSecEval 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Replay","relative_href":"docs/paper-module-materials-chemistry-closed-loop-lab-safety-replay.html","read_href":"docs/paper-module-materials-chemistry-closed-loop-lab-safety-replay.html","context_href":"docs/rules-and-ideas.html#paper-module-materials-chemistry-closed-loop-lab-safety-replay","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"The materials chemistry lab-safety replay validates metadata-only simulator-only closed-loop rows before any materials-lab or discovery language is allowed.","summary":"Materials Chemistry Closed-Loop Lab-Safety Replay is the public lab-safety claim boundary for Microcosm. It checks candidate material refs, safety-screen refs, simulator-only assay refs, active-learning decisions, failure taxonomy refs, cold replay refs, source bundle hashes, Lab/Evolve replay graph evidence, copied source/control/result record/standard bodies, metadata-only result record policy, and eight lab-safety overclaim negative cases while keeping wetlab protocols, hazardous synthesis steps, reagent quantities, controlled or bioactive targets, live lab account secrets, robot commands, private lab notebook bodies, live assay data, discovery claims, benchmark claims, model-output data, source notes, and launch-scope decision out of result records.","authority_ceiling":"Copied public Lab/Evolve source/control/result record/standard bodies, metadata-only simulator-only fixture result records, runtime bundle result records, and artifact safety/refusal validation only; no wetlab execution, hazardous synthesis guidance, reagent quantity, controlled or bioactive target, live assay, robot command, private lab notebook, external model access, discovery claim, benchmark claims, launch-scope decision, publishing-scope decision, or product-progress evidence.","abstract":"The materials chemistry lab-safety replay validates metadata-only simulator-only closed-loop rows before any materials-lab or discovery language is allowed.","teleology":{"section_id":"","text":"The materials chemistry lab-safety replay validates metadata-only simulator-only closed-loop rows before any materials-lab or discovery language is allowed.","source":"capsule_summary_line"},"body_markdown":"**\"Closed-loop materials lab\" is easy language to overclaim. A fixture can look like an autonomous discovery loop while carrying nothing that should be spoken aloud: wetlab steps, reagent quantities, a controlled or bioactive target, robot commands, or a bare assertion that some material was found.** \"Closed-loop materials lab\" is easy language to overclaim. A fixture can look like an autonomous discovery loop while carrying nothing that should be spoken aloud: wetlab steps, reagent quantities, a controlled or bioactive target, robot commands, or a bare assertion that some material was found. This component sits in front of that language and answers one question. Is a closed-loop-lab-shaped fixture safe and grounded enough to be talked about at all, in a simulator-only frame, before any lab claim is allowed?\n\n**Its stable id is `materials_chemistry_closed_loop_lab_safety_replay`, and its public surface name (`PUBLIC_SURFACE_NAME`) is `materials_chemistry_artifact_safety_refusal_validator`.** Its stable id is `materials_chemistry_closed_loop_lab_safety_replay`, and its public surface name (`PUBLIC_SURFACE_NAME`) is `materials_chemistry_artifact_safety_refusal_validator`. The promise-sounding \"closed-loop replay\" name was reframed on purpose: nothing here executes a wetlab loop or commands a robot. The component reads public simulator rows, recomputes a winner, checks refusals, and writes a metadata-only result record. It emits a verdict of either `public_safe_simulator_replay_accepted` or `blocked_public_safety_boundary`, plus the findings that produced it.\n\n**The part worth naming is that the component does not trust the fixture's own conclusion. A plain replay would read a declared \"selected candidate\" label and report it.** The part worth naming is that the component does not trust the fixture's own conclusion. A plain replay would read a declared \"selected candidate\" label and report it. This one recomputes the winner from public numbers and treats a mismatch with the declared label as a failure, not a footnote.\n\n##### Purpose\n\n**A loose reader of a lab fixture accepts a declared winner and a \"safe\" flag at face value.** A loose reader of a lab fixture accepts a declared winner and a \"safe\" flag at face value. That fails two ways. A stale or flattering label passes even when the numbers no longer support it, and dangerous content slips through because nothing is watching for it. This component closes both gaps. It rederives the selection from score-backed rows, and it treats eight named categories of dangerous or overclaiming content as things to refuse rather than ignore. The result is fixture-bound evidence that the loop's structure is inspectable on a simulator, with no real lab, chemicals, or robot involved.\n\n##### How it works\n\n**There are two entrypoints. `run` validates a first-wave fixture directory with negative cases included.** There are two entrypoints. `run` validates a first-wave fixture directory with negative cases included. `run_lab_bundle` validates an exported bundle with `include_negative=False` and requires a copied source-module manifest. Both call `_build_result` to compute the payload, attach `_freshness_basis`, and write the result record. `run_lab_bundle` can reuse a still-fresh prior record through `_fresh_lab_bundle_receipt` when the input digests are unchanged.\n\n`_build_result` runs in this order.\n\nFunction Role `_load_payloads` Reads the six positive inputs (protocol, policy, candidates, experiment DAG, assays, decisions) and, in fixture mode, the eight forbidden negative inputs `_candidate_findings`, `_experiment_findings`, `_assay_findings`, `_decision_findings` Check required fields, cross-references, and safety flags on each row; each missing field or bad link becomes one finding `_numeric_replay_result` Recomputes the selected candidate from public scores; the core check `_source_module_manifest_result` Verifies copied source-module bodies by digest without putting body text into the record; required only in bundle mode `scan_paths`, `_secret_exclusion_scan` `scan_paths` sweeps the input set against the forbidden-class list and reports blocking hits; `_secret_exclusion_scan` drops body text from that scan record so it stays metadata-only `build_materials_lab_evolve_replay` Builds the Lab/Evolve replay graph from the same payloads and findings\n\n**The numeric recompute is the heart. `_numeric_replay_result` reads `public_safety_gate_score` from each candidate, `public_assay_proxy_value` from each assay, and `public_active_learning_score` from each decision.** The numeric recompute is the heart. `_numeric_replay_result` reads `public_safety_gate_score` from each candidate, `public_assay_proxy_value` from each assay, and `public_active_learning_score` from each decision. `_numeric_value` accepts a number only when it is finite and in `[0, 1]`; anything else yields `MATERIALS_NUMERIC_REPLAY_SCORE_OUT_OF_RANGE` or `MATERIALS_NUMERIC_REPLAY_SCORE_REQUIRED`. When no score field is present at all, the run blocks with `MATERIALS_NUMERIC_REPLAY_POLICY_REQUIRES_SCORE_BACKED_ROWS` (policy active) or `MATERIALS_NUMERIC_REPLAY_REQUIRED`. Scores are joined by candidate id. A candidate missing one of its three scores gives `MATERIALS_NUMERIC_REPLAY_LINKAGE_REQUIRED`. A safety gate below `NUMERIC_REPLAY_MIN_SAFETY_GATE` (0.70) gives `MATERIALS_NUMERIC_REPLAY_SAFETY_GATE_FAILED`.\n\n###### The weighted candidate score\n\n**For each linked candidate the weighted score is `0.45 * assay_proxy + 0.35 * active_learning + 0.20 * safety_gate`.** For each linked candidate the weighted score is `0.45 * assay_proxy + 0.35 * active_learning + 0.20 * safety_gate`. The winner is the maximum by score, ties broken by id. Only after that pick is computed does the component compare it to the policy's declared expected label. A mismatch is `MATERIALS_NUMERIC_REPLAY_EXPECTED_LABEL_STALE`. Numeric replay reports `pass` only when there are no findings and at least one verified row.\n\nThe baseline fixture holds four candidates.\n\nCandidate Safety gate Assay proxy Active-learning Weighted score Decision / action `mat_polymer_membrane_001` 0.94 0.92 0.90 0.917 `decision_membrane_001` / `simulate_assay` `mat_solid_electrolyte_002` 0.91 0.84 0.81 0.8435 `decision_electrolyte_002` / `update_surrogate_model` `mat_catalyst_support_003` 0.85 0.78 0.74 0.780 `decision_support_003` / `choose_next_simulation` `mat_sorbent_surface_004` 0.88 0.70 0.66 0.722 `decision_sorbent_004` / `screen_candidate`\n\n**The recompute selects `mat_polymer_membrane_001` at 0.917, which matches the policy's `expected_selected_candidate_material_id`.** The recompute selects `mat_polymer_membrane_001` at 0.917, which matches the policy's `expected_selected_candidate_material_id`. The policy declares `minimum_safety_gate_score` 0.70 and the selection rule `max_weighted_public_assay_active_learning_and_safety_gate_score`.\n\n** `_build_result` folds every check into one `policy_passed` gate: the four row sets are non-empty, no positive finding fired, no forbidden content leaked into the encoded rows, no expected negative case is missing, every row is `simulator_only`, every hazard flag is false, and numeric replay passed.** `_build_result` folds every check into one `policy_passed` gate: the four row sets are non-empty, no positive finding fired, no forbidden content leaked into the encoded rows, no expected negative case is missing, every row is `simulator_only`, every hazard flag is false, and numeric replay passed. The overall `status` is a pass only when `policy_passed` holds, the secret scan passed, the replay graph passed, and, in bundle mode, the source-module manifest passed. The verdict string follows that status.\n\nDiagram of the mechanism (11 steps). Diagram source & refs\n\nSource refs\n\nAccepted `public_safe_simulator_replay_accepted` Blocked `blocked_public_safety_boundary`\n\n```text\nflowchart TD inputs[\"six public inputs protocol, policy, candidates, experiments, assays, decisions\"] rows[\"per-row field + link checks\"] numeric[\"numeric recompute 0.45 assay + 0.35 active + 0.20 gate\"] labelcheck[\"recomputed pick vs declared label gate >= 0.70\"] negatives[\"eight forbidden fixtures\"] refuse[\"any MATERIALS_*_FORBIDDEN observed\"] manifest[\"source-module manifest digest check (bundle mode)\"] scan[\"secret exclusion scan\"] status[\"status + safety verdict\"] accepted[\"public_safe_simulator_replay_accepted\"] blocked[\"blocked_public_safety_boundary\"] inputs --> rows inputs --> numeric numeric --> labelcheck labelcheck --> blocked labelcheck --> status negatives --> refuse refuse --> blocked refuse --> status manifest --> status scan --> status rows --> status status --> accepted status --> blocked\n```\n\n##### Negative cases\n\n** `EXPECTED_NEGATIVE_CASES` names eight forbidden fixtures, each with the code the component must raise when it observes one.** `EXPECTED_NEGATIVE_CASES` names eight forbidden fixtures, each with the code the component must raise when it observes one. A run that fails to refuse any of them lists the case under `expected_missing` and blocks.\n\nForbidden fixture Code `wetlab_protocol_steps` `MATERIALS_WETLAB_PROTOCOL_FORBIDDEN` `hazardous_synthesis_steps` `MATERIALS_HAZARDOUS_SYNTHESIS_FORBIDDEN` `reagent_amounts` `MATERIALS_REAGENT_AMOUNT_FORBIDDEN` `controlled_or_bioactive_target` `MATERIALS_CONTROLLED_OR_BIOACTIVE_TARGET_FORBIDDEN` live lab account secrets MATERIALS LIVE LAB account secret FORBIDDEN `robot_command_execution` `MATERIALS_ROBOT_COMMAND_FORBIDDEN` `private_lab_notebook_export` `MATERIALS_PRIVATE_LAB_NOTEBOOK_FORBIDDEN` `discovery_claim` `MATERIALS_DISCOVERY_CLAIM_FORBIDDEN`\n\n**The numeric perturbations are covered too. Lowering the membrane gate to 0.52 moves the pick to `mat_solid_electrolyte_002` and blocks with a safety-gate failure.** The numeric perturbations are covered too. Lowering the membrane gate to 0.52 moves the pick to `mat_solid_electrolyte_002` and blocks with a safety-gate failure. Declaring a stale expected label while the numbers select a different candidate raises the stale-label code. Removing score rows while the policy is active raises the score-backed-rows code with `verified_numeric_row_count` at 0.\n\n##### Prior Art Grounding\n\n###### Self-driving-lab structure under fixture evidence\n\n**The shape borrowed here is the [self-driving laboratory](https://en.wikipedia.org/wiki/Self-driving_laboratory), where a propose-run-measure loop is paired with safety interlocks that can refuse an unsafe experiment.** The shape borrowed here is the [self-driving laboratory](https://en.wikipedia.org/wiki/Self-driving_laboratory), where a propose-run-measure loop is paired with safety interlocks that can refuse an unsafe experiment. This component keeps the loop-plus-safety-gate structure but runs it over a simulator fixture and adds a numeric recompute so the declared winner is checked rather than trusted. The engineering lineage is ordinary refusal-first input validation applied to a lab-shaped record. It is not a real laboratory controller, a chemical-safety authority, or a discovery benchmark.\n\n##### Validation Result record Path\n\nRun the first-wave fixture proof from the Microcosm root:\n\n###### Exported source-body bundle inspection\n\n**Inspect the exported source-body bundle. Until the exported fixture is refreshed with score-backed numeric rows, this command can return a blocked numeric verdict while still proving the manifest and body-floor boundary:** Inspect the exported source-body bundle. Until the exported fixture is refreshed with score-backed numeric rows, this command can return a blocked numeric verdict while still proving the manifest and body-floor boundary:\n\n```text\ncd microcosm-substrate\nPYTHONPATH=src ../repo-python -m microcosm_core.organs.materials_chemistry_closed_loop_lab_safety_replay run-lab-bundle --input examples/materials_chemistry_closed_loop_lab_safety_replay/exported_materials_lab_safety_bundle --out /tmp/microcosm_materials_chemistry_lab_safety_bundle\n```\n\nRun the focused regression suite:\n\n```text\ncd microcosm-substrate\nPYTHONPATH=src ../repo-pytest tests/test_materials_chemistry_closed_loop_lab_safety_replay.py -q\n```\n\n```text\ncd microcosm-substrate\nPYTHONPATH=src ../repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n**A pass means the runtime recomputed the winner from public numbers, matched the declared label, refused all eight forbidden cases, kept the result record metadata-only, and stayed inside the simulator-only ceiling.** A pass means the runtime recomputed the winner from public numbers, matched the declared label, refused all eight forbidden cases, kept the result record metadata-only, and stayed inside the simulator-only ceiling. The validation result records named here are the authority for a current run; older archived records predate the numeric rows and are not the numeric proof.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The bounded winner-recomputation claim\n\n**The strongest claim the evidence supports is narrow: over public synthetic inputs, this component recomputes a lab-shaped fixture's selected candidate from score-backed rows, refuses eight named dangerous or overclaiming classes, verifies copied source-module bodies by digest, and writes a metadata-only result record, all reproducible with the commands above.** The strongest claim the evidence supports is narrow: over public synthetic inputs, this component recomputes a lab-shaped fixture's selected candidate from score-backed rows, refuses eight named dangerous or overclaiming classes, verifies copied source-module bodies by digest, and writes a metadata-only result record, all reproducible with the commands above. That is the proof boundary. Fixture numbers are public replay coordinates for a safety-gated contract, not experimental measurements.\n\n###### Claims outside the public fixture\n\n**It refuses to claim more. This is not a laboratory.** It refuses to claim more. This is not a laboratory. It does not synthesize materials, give wetlab or hazardous-synthesis instructions, state reagent quantities, target controlled or bioactive compounds, control a robot, export a private lab notebook, use live account secrets, validate live assay data, rank real compounds, call a provider, assert a discovery, or post a benchmark claims. Its scope limit is the bundle ceiling: copied public source-module bodies, metadata-only simulator-only records, and artifact safety and refusal validation only. 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Lab","relative_href":"docs/paper-module-certificate-kernel-execution-lab.html","read_href":"docs/paper-module-certificate-kernel-execution-lab.html","context_href":"docs/rules-and-ideas.html#paper-module-certificate-kernel-execution-lab","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"The certificate kernel execution lab validates bounded public Lean/Lake certificate-kernel rows before any proof-adjacent claim is allowed.","summary":"Certificate Kernel Execution Lab is the public proof-adjacent execution boundary for Microcosm. It checks a Lean/Lake certificate-kernel fixture, generated certificate rows, analyzer metadata, transition traces, typed CP2 action translations, bounded Evolve reruns, source-module manifest digests, copied Lean/tool/profile bodies, metadata-only result record policy, and four negative cases while keeping proof bodies, raw tactic scripts, model-output data, oracle ideal answers, oracle-needed premise ids, private source paths, stdout/stderr bodies, account secrets, private Erdos #257 proof bodies, launch-scope decision, benchmark solve-rate, and general theorem-proof authority out of result records. Its upgrade posture is certificate-engine first: shard and compress public certificate families while preserving the Lean/Lake result record boundary; treat reflective/native checker work as a named trust-register fork.","authority_ceiling":"Public Lean/Lake subprocess witness, copied source proof/tool/profile bodies, first-wave fixture result records, and exported bundle result records only; no general formal-result correctness, private proof body export, external model access, oracle authority, source-file changes, benchmark solve-rate, launch-scope decision, publishing-scope decision, or whole-system proof claim.","abstract":"The certificate kernel execution lab validates bounded public Lean/Lake certificate-kernel rows before any proof-adjacent claim is allowed.","teleology":{"section_id":"","text":"The certificate kernel execution lab validates bounded public Lean/Lake certificate-kernel rows before any proof-adjacent claim is allowed.","source":"capsule_summary_line"},"body_markdown":"###### Lab answers question\n\n**The lab answers one question: did a real Lean verifier just compile a small public project and accept the declared certificate rows, with the command, return code, and hashes recorded.** The certificate kernel execution lab answers one narrow question. Did a real Lean verifier compile a small public project and accept the declared certificate rows, here and now, with the exact build command, the return code, and the source hashes on record? Everything else the component produces is bookkeeping that keeps that answer honest.\n\n###### Runs lean project\n\n**It runs the Lean project in a temporary copy, checks rows against a decidable Boolean checker, and writes a metadata-only record of accepted, residual, and rejected rows with command and hashes.** It runs a source-available Lean/Lake project inside a temporary copy, checks generated certificate rows against a decidable Boolean checker, and writes a structured result record. The record names which rows the verifier accepted, which it left unresolved, and which it rejected, together with the build command identity and the file hashes. Proof text, provider output, and private source stay out of the record, and that exclusion is written down rather than dropped in silence.\n\n**Both public entrypoints, `run` and `run_certificate_bundle`, build one payload through `_build_result` and write it as metadata-only public evidence.** The public entrypoints are `run` and `run_certificate_bundle` in `src/microcosm_core/organs/certificate_kernel_execution_lab.py`. Both build one payload through `_build_result` and write it as metadata-only public evidence.\n\n##### Purpose\n\n###### Facts as finite certificate plus decidable checker\n\n**Proof-adjacent facts become a finite certificate plus a decidable `validate : Cert -> Bool` checker, so the agent supplies rows and Lean decides across two checker families.** A large class of proof-adjacent facts can be written as a finite certificate plus a decidable checker shaped like `validate : Cert -> Bool`. The component never asks an agent to write a human proof. It asks for the right certificate rows and lets Lean decide. The fixture carries two checker families, `NatSumCertificate` over arithmetic and `BoundedOrderCertificate` over a bounded modular order, so sign-off is not one hard-coded shape.\n\n###### Sign-off needs exit-zero Lean, not a screenshot\n\n**A row counts as accepted only when the runner shells out to Lean/Lake and gets exit code 0; sign-off without a genuine rejection is a replay, not evidence.** A loose version of this would let a green screenshot or a confident model stand in for a check. This component refuses that. A row counts as accepted only when the runner shells out to Lean/Lake over a temporary copy of the public project and receives exit code 0. A result that can show only sign-off, with no genuine rejection, is treated as a replay, not as evidence.\n\n##### How it works\n\n###### Two entry points into `_build_result`\n\n** `run` drives `_build_result` with the first-wave fixture and negative cases on; `run_certificate_bundle` drives it with the exported bundle and negatives off.** `run` calls `_build_result` with `input_mode=\"first_wave_fixture\"` and the negative cases on. `run_certificate_bundle` calls it with `input_mode=\"exported_certificate_kernel_execution_lab_bundle\"` and the negative cases off, for the standalone public example. The stages below run in `_build_result` order.\n\n###### Fixing the source and record boundary\n\n**First it enumerates and secret-scans the public inputs, verifies copied source bodies by class, anchors, and SHA-256, and projects them into a citable body floor.** First it fixes the source and record boundary. `_input_paths` enumerates the public Lean files and JSON inputs. `scan_paths` checks them against `core/private_state_forbidden_classes.json`. `_source_module_manifest_result` verifies the exported bundle's copied source bodies by material class, target presence, required provenance anchors, and SHA-256 equality, and returns a finding for any row that fails one of those. `_source_open_body_import_summary` turns that into a body floor the record can cite by ref, hash, class, count, and verdict, with `body_in_receipt: false`.\n\n###### Gathering execution evidence from `lake build`\n\n**Then it copies the project to a temp dir, runs `lake build`, and sets certificate rows through Lean: accepted rows must return 0, bad rows must fail in their predicted class.** Then it gathers execution evidence. For the fixture path it copies the project into a temporary directory, records the public Lean imports, declarations, line counts, and hashes, and runs `lake build MicrocosmCertificateLab`. If the build returns 0 it sets the certificate transition rows through Lean. Accepted rows must return 0. Missing or bad certificate rows must fail in the residual class the fixture predicted. Typed-repair and bounded-policy rows must rerun inside their allowed action and artifact classes rather than editing arbitrary source.\n\n###### An obligation graph of public ids only\n\n**Then it builds an obligation dependency graph citing public ids only, validated against observed transitions and rejecting any row carrying a proof body, model-output data, oracle template, or private obligation body.** Then it builds a dependency graph that is weaker than a proof body and stronger than prose. Each obligation row cites public ids only: an obligation id, dependency ids, transition ids, certificate refs, an evidence kind, and an expected verifier class. The runner validates the rows as a graph over the transitions it actually observed and rejects any row that tries to carry a proof body, a model-output data, an oracle template, or a private obligation body.\n\n###### The pass floor in `_build_result`\n\n** `_build_result` returns pass only when the secret scan is clean, Lean and Lake are present, the build returned 0, and the transition, repair, and policy counts each clear their floor; otherwise `blocked`.** `_build_result` then computes status. It returns pass only when the secret scan has zero blocking hits, Lean and Lake are both available, the build returned 0, at least eight declarations were analyzed, no expected negative case is missing, and the transition, repair, and policy counts each clear their floor. The pass floor requires at least five transitions, three accepted, two residual, one typed-repair effect, and one accepted policy rerun. For the bundle path it also requires the source-body check to pass. Anything short of that is `blocked`.\n\nFunction Role `run` Fixture entrypoint; runs the negative floor and writes the fixture records `run_certificate_bundle` Runs the standalone public bundle and writes `BUNDLE_RESULT_NAME` `_build_result` Loads inputs, scans, runs Lean/Lake, sets transitions, computes status `_source_module_manifest_result` Checks copied source bodies by class, presence, anchors, and SHA-256 `_source_open_body_import_summary` Projects the checked bodies into a citable body floor `_receipt_freshness` Compares input and record mtimes so a current record can be reused `build_public_readout` Assembles the cold-reader readout from the written records Diagram of the mechanism (8 steps). Diagram source & refs\n\n```text\nflowchart TD inputs[\"public Lean files and JSON inputs\"] scan[\"scan_paths secret gate\"] sources[\"_source_module_manifest_result copied body check\"] build[\"lake build MicrocosmCertificateLab\"] transitions[\"set certificate rows through Lean\"] dag[\"obligation graph over observed rows\"] status[\"_build_result status pass or blocked\"] record[\"metadata-only result record\"] inputs --> scan scan --> sources sources --> build build --> transitions transitions --> dag dag --> status status --> record\n```\n\n##### Negative cases\n\n###### Five rejections the fixture must observe\n\n** `EXPECTED_NEGATIVE_CASES` names five rows that must each be rejected with their own error code, and a fixture run missing any of them is `blocked` because green-only is not evidence.** `EXPECTED_NEGATIVE_CASES` names five rows that must be rejected, each with its own error code. A row that makes provider or oracle text visible to the verifier fails with `CERTIFICATE_KERNEL_EXECUTION_PROVIDER_OR_ORACLE_VISIBLE`. A typed-repair row carrying a proof body fails with `CERTIFICATE_KERNEL_EXECUTION_CP2_PROOF_BODY_FORBIDDEN`. A policy row that tries to change source files fails with `CERTIFICATE_KERNEL_EXECUTION_EVOLVE_SCOPE_FORBIDDEN`. A manifest row pointing at a non-public source refs fails with `CERTIFICATE_KERNEL_EXECUTION_PRIVATE_SOURCE_REF_FORBIDDEN`. An obligation row carrying a body fails with `CERTIFICATE_KERNEL_EXECUTION_OBLIGATION_BODY_FORBIDDEN`. A fixture run that does not observe all five is `blocked`, because a component that can only show green is not evidence that the rejection path is real.\n\n##### Prior Art Grounding\n\n###### Proof-carrying code and small-kernel Lean lineage\n\n**The design follows the proof-carrying-code and proof-assistant traditions: an untrusted producer supplies a certificate a small trusted Lean kernel checks, with reproducible environments and a named evaluation-scope limit.** This component sits in the proof-carrying and proof-assistant traditions. Necula's [Proof-Carrying Code](https://www.usenix.org/legacy/publications/library/proceedings/osdi96/full_papers/necula/html/node2.html) anchors the idea that an untrusted producer can supply a certificate checked by a small trusted verifier. The [Lean theorem prover](https://www.microsoft.com/en-us/research/publication/the-lean-theorem-prover-system-description/) continues the small-kernel lineage, and [LeanDojo](https://arxiv.org/abs/2306.15626) shows why reproducible Lean environments and programmatic proof-state access matter for theorem-proving agents. [PBLean](https://arxiv.org/abs/2602.08692) is the nearest public reference for the reflective-checker research fork, where a verified Boolean checker imports certificates into Lean without building every explicit proof term. The [Lean proof validation reference](https://lean-lang.org/doc/reference/latest/ValidatingProofs/) is the local reason that any move to native or compiler-backed evaluation must be named in the scope limit, not smuggled into an ordinary result record.\n\n##### Validation Result record Path\n\nRun the first-wave fixture into disposable records from the Microcosm root:\n\nRun the exported bundle through the same component:\n\n```text\ncd microcosm-substrate\nPYTHONPATH=src ../repo-python -m microcosm_core.organs.certificate_kernel_execution_lab run-certificate-bundle --input examples/certificate_kernel_execution_lab/exported_certificate_kernel_execution_lab_bundle --out /tmp/microcosm_certificate_kernel_execution_lab_bundle\n```\n\n```text\ncd microcosm-substrate\n../repo-pytest tests/test_certificate_kernel_execution_lab.py -q\ncd ..\n./repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n###### What a pass means here\n\n**A pass certifies only that the project compiled, accepted rows returned 0, each negative case failed in its predicted class, and no proof or provider body reached the record.** A pass means the public Lean project compiled, the accepted rows returned 0, every expected negative case was rejected in its predicted class, and the result record carried no proof, provider, oracle, or private-source body.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The bounded strongest claim\n\n**The strongest supported claim is bounded: a green run shows the public Lean project compiled and its certificate rows were accepted, rejected, or residual under a real local verifier, nothing more.** The strongest claim this component supports is bounded. A green run shows that the declared public Lean project compiled and that the declared certificate rows were accepted, rejected, or left residual under a real local Lean/Lake verifier, with the command identity, return codes, hashes, and negative-case rows on record. The proof boundary is that local subprocess witness together with the fixture and bundle records, nothing more.\n\n###### What a green run still denies\n\n**A green run still proves no theorem, claims no Mathlib coverage or solve rate, exports no proof bodies, mutates no source, and excludes launch.** It does not establish the Erdos #257 theorem, claim Mathlib coverage, or report a benchmark solve rate. It does not export proof bodies, count provider or oracle output as proof, change source files, or include launch operations or public sharing. The Atlas card for this module is completed by the owning projection route and builder regeneration, not by editing this page.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":12678,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/certificate_kernel_execution_lab.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"Proof-Carrying Code","url":"https://www.usenix.org/legacy/publications/library/proceedings/osdi96/full_papers/necula/html/node2.html","kind":"external_reference","section_id":"proof-carrying-code-and-small-kernel-lean-lineage"},{"label":"Lean theorem prover","url":"https://www.microsoft.com/en-us/research/publication/the-lean-theorem-prover-system-description/","kind":"external_reference","section_id":"proof-carrying-code-and-small-kernel-lean-lineage"},{"label":"LeanDojo","url":"https://arxiv.org/abs/2306.15626","kind":"external_reference","section_id":"proof-carrying-code-and-small-kernel-lean-lineage"},{"label":"PBLean","url":"https://arxiv.org/abs/2602.08692","kind":"external_reference","section_id":"proof-carrying-code-and-small-kernel-lean-lineage"},{"label":"Lean proof validation reference","url":"https://lean-lang.org/doc/reference/latest/ValidatingProofs/","kind":"external_reference","section_id":"proof-carrying-code-and-small-kernel-lean-lineage"}],"grouped_relation_feeds":[{"relation":"abides by","count":1,"links":[{"id":"AX-2","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Kernelized verification","url":"doctrine.html#dcard-ax-2"}]},{"relation":"depends on","count":1,"links":[{"id":"paper_module.verifier_lab_execution_spine","kind":"paper_module","kind_label":"paper module","relation":"depends on","label":"Verifier Lab Execution Spine","url":"paper-module-verifier-lab-execution-spine.html"}]},{"relation":"explains","count":2,"links":[{"id":"certificate_kernel_execution_lab","kind":"organ","kind_label":"component","relation":"explains","label":"Certificate Kernel Execution Lab","url":"component-certificate-kernel-execution-lab.html"},{"id":"mechanism.certificate_kernel_execution_lab.validates_public_certificate_kernel_execution","kind":"mechanism","kind_label":"mechanism","relation":"explains","label":"validates public certificate kernel execution","url":"rules-and-ideas.html#lattice-mechanism-mechanism-certificate-kernel-execution-lab-validates-public-certificate-kernel-execution"}]},{"relation":"governed by","count":2,"links":[{"id":"P-3","kind":"principle","kind_label":"principle","relation":"governed by","label":"Concentrate trust in small checkers","url":"doctrine.html#dcard-p-3"},{"id":"concept.formal_math_and_proof_witness_bundle","kind":"concept","kind_label":"concept","relation":"governed by","label":"formal math and proof witnesses as bounded public proof-adjacent result records","url":"rules-and-ideas.html#lattice-concept-concept-formal-math-and-proof-witness-bundle"}]}],"paper_module_backrefs":[],"source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/certificate_kernel_execution_lab.md"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json"}],"visible_text_byte_count":14298,"section_titles":["Lab answers question","Runs lean project","Both public entrypoints","Purpose","Facts as finite certificate plus decidable checker","Sign-off needs exit-zero Lean, not a screenshot","How it works","Two entry points into `_build_result`","Fixing the source and record boundary","Gathering execution evidence from `lake build`","An obligation graph of public ids only","The pass floor in `_build_result`","Negative cases","Five rejections the fixture must observe","Prior Art Grounding","Proof-carrying code and small-kernel Lean lineage","Validation Result record Path","What a pass means here","Scope limit","The bounded strongest claim","What a green run still denies"]},{"id":"paper_module.corpus_readiness_mathlib_absence_gate","title":"Corpus Readiness Mathlib Absence Gate","relative_href":"docs/paper-module-corpus-readiness-mathlib-absence-gate.html","read_href":"docs/paper-module-corpus-readiness-mathlib-absence-gate.html","context_href":"docs/rules-and-ideas.html#paper-module-corpus-readiness-mathlib-absence-gate","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"The corpus readiness Mathlib absence gate validates copied corpus/toolchain readiness bodies before any Mathlib-dependent proof or retrieval claim is allowed.","summary":"Corpus Readiness Mathlib Absence Gate is the public formal-math corpus readiness boundary for Microcosm. It checks copied PROVER smoke-run corpus readiness rows, Lean/Std toolchain probe rows, Mathlib absence status, consumer-gate decisions, absent corpus blocking, source-module manifest digests, metadata-only result record policy, and five negative cases while keeping proof bodies, model-output data, non-public source refs, benchmark-completeness claims, launch-scope decision, and Mathlib proof authority out of result records.","authority_ceiling":"Public algorithmic projection over copied corpus/toolchain readiness system, first-wave fixture result records, and exported bundle result records only; no Lean/Lake rerun, Mathlib availability claim, formal-result correctness, proof body export, external model access, benchmark/corpus completeness, launch-scope decision, publishing-scope decision, or whole-system proof claim.","abstract":"The corpus readiness Mathlib absence gate validates copied corpus/toolchain readiness bodies before any Mathlib-dependent proof or retrieval claim is allowed.","teleology":{"section_id":"","text":"The corpus readiness Mathlib absence gate validates copied corpus/toolchain readiness bodies before any Mathlib-dependent proof or retrieval claim is allowed.","source":"capsule_summary_line"},"body_markdown":"###### From recorded lean\n\n**From a recorded Lean readiness report it answers, per corpus, whether the library is present and whether the Mathlib import lane resolves, pairing copied rows with a live probe.** `corpus_readiness_mathlib_absence_gate` reads a recorded readiness report from a Lean math toolchain run and turns one question into an inspectable answer: for each math corpus, is it actually present on this host, and is the Mathlib import lane actually available, or not? It carries copied corpus and toolchain rows from a 2026-05-11 proof-state curriculum smoke run and pairs them with a live import check.\n\n###### Emits metadata-only readiness\n\n**It emits a metadata-only readiness board and consumer verdicts: which corpora are absent, which are smoke-test only, and which tasks stay blocked until a Mathlib probe passes.** The output is a readiness board and a set of consumer verdicts. It records which corpora are absent, which are usable only for translation smoke tests, and which downstream tasks stay blocked until a passing Mathlib probe exists. Everything it emits is metadata-only: no proof text, no provider output, no non-public paths.\n\n###### Is not prover\n\n**It is not a prover or Lake build but a gate that forces a formal-math route to expose corpus readiness before any proof route counts as usable.** It is not a theorem prover, a Lake build, or a Mathlib availability claim. It is a gate that forces a formal-math route to expose corpus readiness before any proof route is treated as usable.\n\n##### Purpose\n\n###### Roster presence mistaken for usability\n\n**Formal-math agents conflate a corpus appearing on a roster with it being usable for their proof route, so a missing library surfaces only later as a broken import.** Formal-math agents fail in one specific way. They treat \"there is a corpus\" as if it meant \"this corpus is usable for the proof route I am about to take\". A roster lists miniF2F, PutnamBench, ProofNet, LeanDojo and Mathlib, the agent assumes the libraries are present, and the failure only surfaces later as a broken import or a tactic that needs a premise the host cannot resolve.\n\n###### Alias field must agree with a live probe\n\n**The loose check trusts a `mathlib_available` field or `PASS` status; this gate refuses that drift, flagging any row whose alias claim a live probe does not confirm.** The loose version of this check trusts the roster. It reads a `mathlib_available` field, or a `PASS` lean status, and marks the corpus ready. That is the drift the gate is built to stop. A stale alias field cannot turn the gate green on its own. It has to agree with a live probe or the row is flagged.\n\n##### How it works\n\n###### Two entrypoints funnel into `_build_result`\n\n** `run` and `run_projection_bundle` both call `_build_result`, which loads two inputs, scans paths against the exclusion policy, then folds four ordered checks into one result and board.** The two entrypoints `run` and `run_projection_bundle` both call `_build_result`. `run` handles the first-wave fixture with its negative cases. `run_projection_bundle` handles an exported bundle without negative cases and requires a source-module manifest. `_build_result` loads the two input files, scans every input path against the non-public-state exclusion policy, then runs four checks in order and folds them into one result and a readiness board.\n\nDiagram of the mechanism (7 steps). Diagram source & refs\n\nSource refs\n\nfour source digests, parse JSON, run probe `validate_runtime_source_artifacts` Std compiles, Mathlib import rejected `runtime_lean_import_probe` seven corpus rows checked against the probe `validate_corpus_readiness` verdicts derived from readiness facts `validate_consumer_gate_cases` copied source digests for the bundle `validate_source_module_imports`\n\n```text\nflowchart TD input[\"Fixture or exported bundle input corpus rows plus consumer gate cases\"] artifacts[\"validate_runtime_source_artifacts four source digests, parse JSON, run probe\"] probe[\"runtime_lean_import_probe Std compiles, Mathlib import rejected\"] corpus[\"validate_corpus_readiness seven corpus rows checked against the probe\"] gates[\"validate_consumer_gate_cases verdicts derived from readiness facts\"] imports[\"validate_source_module_imports copied source digests for the bundle\"] result[\"_build_result one metadata-only result and board\"] input --> artifacts artifacts --> probe probe --> corpus corpus --> gates input --> imports gates --> result imports --> result\n```\n\n###### The live import probe: Std compiles, Mathlib rejected\n\n** `validate_runtime_source_artifacts` digest-checks four source refs and runs a probe that expects `Std` to compile and `Mathlib` to be rejected, setting `mathlib_lake_project_import_available` false when the lane fails.** `validate_runtime_source_artifacts` anchors the run to four source refs: the corpus readiness rows, a tactic-affordance probe, the Mathlib import probe Lean file, and a tactic portfolio availability file. It checks each file's SHA-256 against a recorded digest, parses the JSON ones, and calls `runtime_lean_import_probe`. That probe writes two small Lean files into a temporary directory. One imports `Std` and is expected to compile. One imports `Mathlib` and is expected to be rejected with the toolchain's own `unknown module prefix 'Mathlib'` error. The probe also confirms `lake` is on the path. It never runs `lake build` and never exports any Lean body. Its result sets `mathlib_lake_project_import_available`, which stays false whenever the Mathlib lane does not resolve.\n\n###### Corpus rows normalised against probe evidence\n\n** `validate_corpus_readiness` marks a corpus usable only when it exists, carries a Lake file, and the probe agrees; contradicting alias claims are flagged and every unavailable lane is blocked.** `validate_corpus_readiness` normalises the seven corpus rows against that runtime evidence. A corpus is marked usable for Mathlib-dependent work only when the runtime row says it exists, carries a Lake file, and the Mathlib import lane is available. If a row claims `mathlib_lake_project_import_available` or any present alias in `MATHLIB_PRESENT_ALIAS_FIELDS` while the probe disagrees, the row is flagged with `CORPUS_READINESS_RUNTIME_PROBE_CONTRADICTION`. Absent corpora and every unavailable Mathlib lane land in `blocked_capabilities`. In the current fixture the probe stays false, so all seven Mathlib lanes are blocked.\n\n###### Consumer verdicts derived from readiness facts\n\n** `validate_consumer_gate_cases` recomputes each verdict from readiness facts, not the case's own label, blocking cases needing an unavailable Mathlib lane or an absent corpus; only the miniF2F smoke passes.** `validate_consumer_gate_cases` derives each verdict from the readiness facts instead of trusting a case's own expected-decision label. A case is blocked if it requires the Mathlib lake project import while that lane is unavailable, or if its target corpus is absent. Of the seven cases, one passes: the miniF2F Lean3 translation smoke, which does not require a Mathlib project. The other six stay blocked. A case that never set `readiness_gate_checked` is flagged with `CONSUMER_READINESS_GATE_UNCHECKED`.\n\n###### Exported-bundle source floor and final status\n\n** `validate_source_module_imports` enforces copied-body digests and byte equality, and `_build_result` passes only when no negative case is missing, the secret scan is clean, and the source floor holds.** `validate_source_module_imports` adds the exported-bundle floor. It requires the manifest to declare `copied_non_secret_macro_body`, each row to use a body material class, each copied file to match its recorded source and target digest, and no row to carry a body inside a result record. Exact-copy rows must preserve byte equality; rewrite rows must match the declared private-absolute-path rewrite recipe exactly. `_build_result` then computes one status: it passes only when no expected negative case is missing, the secret scan has no blocking hit, the source-module floor holds, and no unexpected finding appears.\n\n##### Negative cases\n\n###### Fixture carries five\n\n**The fixture carries five named negative inputs, declared in `EXPECTED_NEGATIVE_CASES`.** Each one must be observed with its error code or the run is blocked:\n- `mathlib_available_without_probe` must raise `MATHLIB_AVAILABILITY_OVERCLAIM` when Mathlib is claimed available with no passing probe.\n- `consumer_skips_readiness_gate` must raise `CONSUMER_SKIPS_CORPUS_READINESS_GATE` when a consumer attempts Mathlib-dependent work without checking readiness.\n- `private_corpus_source_ref` must raise `PRIVATE_CORPUS_SOURCE_REF_FORBIDDEN` when a source ref points at a non-public paths.\n- `proof_body_leakage` must raise `CORPUS_READINESS_PROOF_BODY_FORBIDDEN` when a corpus row carries a proof or provider body field.\n- `release_overclaim` must raise `CORPUS_READINESS_RELEASE_OVERCLAIM` when the metadata tries to include launch operations or proof authority.\n\n##### Prior Art Grounding\n\n###### Grounded in Lean corpus and theorem-proving work\n\n**The design draws on Lean corpus and neural theorem-proving work where library availability and benchmark splits are part of the claim, requiring probes and digests before tactic-routing language is allowed.** This component sits in Lean corpus and neural theorem-proving work where library availability, premise access, and benchmark splits are part of the claim. The [Lean mathematical library](https://arxiv.org/abs/1910.09336) establishes Mathlib as a large community-maintained formal mathematics corpus, [miniF2F](https://arxiv.org/abs/2109.00110) gives a cross-system benchmark for formal Olympiad statements, and [LeanDojo](https://arxiv.org/abs/2306.15626) shows why reproducible corpus extraction and accessible-premise metadata matter for theorem-proving agents. The gate borrows the discipline: corpus rows, Mathlib probes, blocked consumer cases, and copied-source digests must be visible before retrieval or tactic-routing language is allowed. It does not claim Mathlib is present or that any theorem was proved.\n\n##### Validation Result record Path\n\nReader-verifiable commands, run from the `microcosm-substrate/` public root:\n\n```text\nPYTHONPATH=src python3 -m microcosm_core.organs.corpus_readiness_mathlib_absence_gate run \\\n  --input fixtures/first_wave/corpus_readiness_mathlib_absence_gate/input \\\n  --out /tmp/microcosm-corpus-readiness-mathlib-absence-vrp\nPYTHONPATH=src python3 -m microcosm_core.organs.corpus_readiness_mathlib_absence_gate run-projection-bundle \\\n  --input examples/corpus_readiness_mathlib_absence_gate/exported_corpus_readiness_bundle \\\n  --out /tmp/microcosm-corpus-readiness-mathlib-absence-bundle-vrp\nPYTHONPATH=src python3 scripts/build_doctrine_projection.py --check-paper-module-corpus\nPYTHONPATH=src .venv/bin/python -m pytest -p no:cacheprovider --basetemp=/tmp/microcosm-corpus-readiness-mathlib-absence-tests -q tests/test_corpus_readiness_mathlib_absence_gate.py\n```\n\n###### What a pass certifies and excludes\n\n**A pass means fixture and exported bundle agree on readiness, Mathlib absence, blocked consumers, digests, and the five negative cases; it reruns no Lean, proves nothing, and installs no Mathlib.** A pass means the fixture and the exported bundle agree about corpus readiness, Mathlib absence, blocked consumers, copied source digests, metadata-only result files, and the five negative cases. It does not rerun Lean or Lake, prove any theorem, or install Mathlib.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim\n\n**The strongest claim is narrow: a fixture and exported bundle agree on readiness, Mathlib absence, verdicts, digests, and leakage guards, bounded to projection over copied material plus one import probe.** The strongest claim the evidence supports is narrow: a public fixture and an exported bundle agree about corpus readiness, Mathlib absence, blocked consumer verdicts, copied source-module digests, metadata-only result files, and the negative leakage guards. The proof boundary is algorithmic projection over copied corpus and toolchain material plus one bounded import probe.\n\n###### What stays refused past the line\n\n** `AUTHORITY_CEILING` sets every outward flag false, so the component proves no install, runs no Lake build, validates no theorem, calls no provider, and approves no launch.** It refuses everything past that line. The scope limit in `AUTHORITY_CEILING` sets `mathlib_lake_project_import_authorized`, `formal_proof_authority`, `provider_calls_authorized`, and `release_authorized` all to false. This component does not establish Mathlib is installed, run a Lake build, validate formal-result correctness, benchmark corpus quality, call a provider, expose any proof or provider body, change source files, or approve launch.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":12853,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/corpus_readiness_mathlib_absence.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"Lean mathematical 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Classifier","url":"paper-module-undeclared-library-prior-classifier.html","relation_id":"paper_module.depends_on.paper_module","source_ref":"core/paper_module_capsules.json::paper_modules[55:paper_module.undeclared_library_prior_classifier].depends_on"},{"id":"paper_module.verifier_lab_kernel","kind":"paper_module","kind_label":"paper module","relation":"depended on by","label":"Verifier Lab Kernel","url":"paper-module-verifier-lab-kernel.html","relation_id":"paper_module.depends_on.paper_module","source_ref":"core/paper_module_capsules.json::paper_modules[0:paper_module.verifier_lab_kernel].depends_on"}],"source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/corpus_readiness_mathlib_absence.md"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json"}],"visible_text_byte_count":14123,"section_titles":["From recorded lean","Emits metadata-only readiness","Is not prover","Purpose","Roster presence mistaken for usability","Alias field must agree with a live probe","How it works","Two entrypoints funnel into `_build_result`","The live import probe: Std compiles, Mathlib rejected","Corpus rows normalised against probe evidence","Consumer verdicts derived from readiness facts","Exported-bundle source floor and final status","Negative cases","Fixture carries five","Prior Art Grounding","Grounded in Lean corpus and theorem-proving work","Validation Result record Path","What a pass certifies and excludes","Scope limit","The strongest supported claim","What stays refused past the line"]},{"id":"paper_module.pattern_binding_contract","title":"Pattern Binding Contract","relative_href":"docs/paper-module-pattern-binding-contract.html","read_href":"docs/paper-module-pattern-binding-contract.html","context_href":"docs/rules-and-ideas.html#paper-module-pattern-binding-contract","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"The public pattern-binding component validates pattern rows, source bundles, authority handles, exported system bundles, and route-readiness selector overlays while keeping mined rows component-first and fixture-bound.","summary":"Pattern Binding Contract is the public system membrane for mined pattern rows. It validates binding fields, source bundles, reference bundles, authority-chain handles, secret-exclusion scans, exported system bundles, and route-readiness selector overlays, then writes bounded result records that keep private bodies out and prevent individual pattern rows from becoming standalone public leaves.","authority_ceiling":"Public pattern-binding fixtures, exported system bundles, route-readiness selector bundles, source-bundle refs, and metadata-only result records only; no private pattern-ledger certification, launch-scope decision, external model access, hosted-public posture, recipient work, non-public data equivalence, source-file changes, or whole-system correctness.","abstract":"The public pattern-binding component validates pattern rows, source bundles, authority handles, exported system bundles, and route-readiness selector overlays while keeping mined rows component-first and fixture-bound.","teleology":{"section_id":"teleology","text":null,"source":"markdown_section"},"body_markdown":"##### Teleology\n\n###### Binding pattern rows to source-backed evidence\n\n** `pattern_binding_contract` binds each pattern row to a source-available bundle, public runtime ref, authority handle, scope boundary, and secret-exclusion result record, with synthetic rows allowed only as controls.** `pattern_binding_contract` is the public root component that binds pattern rows to source-available source bundles, public runtime refs, authority-chain handles, scope boundaries, and secret-exclusion result records. Synthetic rows are allowed only as regression controls or negative cases; they are not product evidence.\n\n##### Purpose\n\n###### Why a lone pattern row cannot self-publish\n\n**A mined pattern row reads like a finished insight, so this component exists to decide whether it can be admitted to the public surface and under exactly what evidence and ceiling.** A mined engineering pattern is a tempting thing to publish on its own. It reads like a self-contained insight, so it is easy to lift a single row out of a private ledger and present it as a finished public claim. This component exists to stop that. It answers one question: can a given pattern row be admitted to the public surface, and if so, under exactly what evidence and what ceiling?\n\n###### Binding, not display\n\n**The check binds rather than displays: every row must name a source bundle, governing standard, and scope boundary, while duplicates, overclaims, and standalone-leaf rows are rejected and negative cases run alongside.** The check is binding rather than display. Every pattern row must name a source bundle that points at a real public runtime ref or regression-harness ref, a governing standard, and an scope boundary. A row that lacks any of these, duplicates another row's id, or claims to be a standalone public leaf is rejected. The same validator runs deliberate negative cases alongside the positive control. The result record shows that good rows pass and that each known failure mode is still caught.\n\n###### Truth accounting and route-readiness\n\n**Truth accounting separates runtime-metadata rows from real pattern-ledger imports so accepted-row counts are not read as system progress, and route-readiness keeps selection component-first and fixture-bound.** The less obvious idea is truth accounting. When an exported bundle is validated, the component separates rows that merely describe runtime metadata from rows that represent a real pattern-ledger import, and records that a high accepted-row count is not the same as system progress. This guards against the quiet inflation where counting accepted rows starts to read like a measure of how much real work has landed. The route-readiness layer closes the matching gap on the selector side: a row can look selectable in isolation, but it is only admitted through the component that owns it, its fixture contract, and a gate that refuses to let hard no-standalone rows appear as selectable targets.\n\n##### Public contract\n\n###### The four validators and their result records\n\n** `validate_pattern_bindings`, `validate_source_capsules`, `validate_reference_capsules`, and the authority-chain and ceiling checks run over one fixture and write command-owned result records.** `validate_pattern_bindings` checks required binding fields, rejects duplicate pattern ids, and flags a row that overclaims source authority or public-leaf readiness. `validate_source_capsules` and `validate_reference_capsules` check that each bundle names a runtime ref and keeps its body out of the result record. `validate_authority_chain_handle_resolver` rejects an unsupported handle that still implies authority, and `validate_authority_ceiling` records whether the two overclaim negative cases were actually observed. `validate` runs these checks over one fixture and writes command-owned result records under `receipts/first_wave/pattern_binding_contract/`.\n\n###### Route-readiness closes the selector gap\n\n**The exported bundle carries the route-readiness overlays as public bodies, and `validate_substrate_bundle` recomputes the four checks then validates the selector so a row cannot look selectable without opening its owning component.** The exported system bundle also carries the source route-readiness selector overlays as public source-open bodies: `examples/pattern_binding_contract/exported_route_readiness_bundle/`. `validate_substrate_bundle` recomputes the same four checks over the imported pattern ledger, then calls `validate_route_readiness_bundle` against the route-readiness audit, row-to-component router, route cards, fixture specs, decision matrix, dependency DAG, internal routing graph, and copied source validation report. This closes the old gap where a mined pattern row could look selectable without opening the component bundle that owns it.\n\n###### Which command answers selector admission\n\n**For selector admission rather than generic binding, cold readers run `microcosm pattern-route-readiness validate-bundle`, with the older `validate-route-readiness-bundle` action kept as a compatibility route.** Cold readers should use `microcosm pattern-route-readiness validate-bundle` against `examples/pattern_binding_contract/exported_route_readiness_bundle/` when the question is selector admission rather than generic pattern binding. The older `pattern-binding validate-route-readiness-bundle` action remains a compatibility route to the same validator.\n\n##### Shape\n\nDiagram of the mechanism (13 steps). Diagram source & refs\n\nSource refs\n\nrequired fields, duplicate ids, bundle resolution, secret-exclusion scan `pattern_binding_contract`\n\n```text\nflowchart LR subgraph Inputs[\"Pattern-binding inputs\"] Patterns[\"Pattern rows id, governing standard, scope boundary, source refs, projection posture\"] Bundles[\"Source bundles metadata-only refs to public runtime or regression harness\"] Handles[\"Authority-chain handles resolver result records\"] end Validator[\"pattern_binding_contract required fields, duplicate ids, bundle resolution, secret-exclusion scan\"] subgraph Negative[\"Refusal floor\"] Dup[\"Duplicate id rejected\"] Leak[\"Private body leak rejected\"] Overclaim[\"Public-leaf overclaim rejected\"] Unsupported[\"Unsupported authority handle not upgraded\"] end subgraph Bundle[\"Exported-bundle path\"] Truth[\"Truth accounting runtime-metadata rows vs real pattern-ledger import\"] RouteReadiness[\"Route-readiness selector component-first admission, fixture contract, hard no-standalone gate\"] end Result records[\"Result records refs, digests, counts, verdicts; body text omitted\"] Patterns --> Validator Bundles --> Validator Handles --> Validator Validator --> Negative Validator --> Bundle Truth --> RouteReadiness Negative --> Result records Bundle --> Result records\n```\n\n##### Evidence binding\n\n**Accepted component row: `core/organ_registry.json::implemented_organs[pattern_binding_contract]`.** Accepted component row: `core/organ_registry.json::implemented_organs[pattern_binding_contract]`. Evidence class: `core/organ_evidence_classes.json::organ_evidence_classes[pattern_binding_contract]` with rank 5 semantic-validator authority. The runtime locus is `src/microcosm_core/organs/pattern_binding_contract.py`, with focused coverage in `tests/test_pattern_binding_contract.py`.\n\n**Paper bundle authority: `core/paper_module_capsules.json#paper_module.pattern_binding_contract`.** Paper bundle authority: `core/paper_module_capsules.json#paper_module.pattern_binding_contract`. Mechanism source: `core/mechanism_sources.json#mechanism.pattern_binding_contract.validates_public_pattern_bindings`.\n\n##### Reader evidence routing\n\n**Start with `paper_modules/pattern_binding_contract.json` for the bundle payload, then open `standards/std_microcosm_pattern_binding_contract.json` to check the required fields, public/private boundary, source-open body import floor, route-readiness rules, and result record expectations.** Start with `paper_modules/pattern_binding_contract.json` for the bundle payload, then open `standards/std_microcosm_pattern_binding_contract.json` to check the required fields, public/private boundary, source-open body import floor, route-readiness rules, and result record expectations.\n\n**Use `core/fixture_manifests/pattern_binding_contract.fixture_manifest.json` before inspecting fixtures or exported bundles. The manifest and the `source_module_manifest.json` files name the copied source body floor; result record payloads carry source refs, digests, anchors, counts, verdicts, and omission result records rather than inlining body text.** Use `core/fixture_manifests/pattern_binding_contract.fixture_manifest.json` before inspecting fixtures or exported bundles. The manifest and the `source_module_manifest.json` files name the copied source body floor; result record payloads carry source refs, digests, anchors, counts, verdicts, and omission result records rather than inlining body text.\n\n##### Prior Art Grounding\n\n###### Pattern-language lineage\n\n**The component follows the software pattern-language tradition of making reusable engineering structures explicit and reviewable, with the Hillside patterns library as its direct prior-art family.** This component follows the software pattern-language tradition of making reusable engineering structures explicit, named, and reviewable. The [Hillside patterns library](https://hillside.net/patterns/) is the direct prior-art family for treating patterns as shared vocabulary rather than loose implementation notes.\n\n###### Provenance and supply-chain attestation\n\n**The binding layer borrows provenance and supply-chain attestation shapes from W3C PROV, SLSA, and in-toto, applied to pattern rows and route-readiness selectors rather than launch certification.** The binding layer also borrows from provenance and supply-chain attestation patterns. [W3C PROV](https://www.w3.org/TR/prov-overview/) motivates the source/ref/evidence relation shape, while [SLSA](https://slsa.dev/spec/) and [in-toto](https://in-toto.io/) motivate digest-bound artifact claims and step-level metadata. Microcosm applies those ideas to pattern rows and route-readiness selectors, not to launch certification.\n\n**Re-entry condition: if copied source bodies, route-readiness overlays, or negative-case rules change, rerun the `validate`, validate-system-bundle, and `validate-route-readiness-bundle` commands under Validation Result record Path and update this paper module plus `standards/std_microcosm_pattern_binding_contract.json` from the new result record fields.** Re-entry condition: if copied source bodies, route-readiness overlays, or negative-case rules change, rerun the `validate`, validate-system-bundle, and `validate-route-readiness-bundle` commands under Validation Result record Path and update this paper module plus `standards/std_microcosm_pattern_binding_contract.json` from the new result record fields.\n\n##### Validation Result record Path\n\nFrom `microcosm-substrate/`, reproduce this page's proof boundary with temporary result records:\n\n###### What a pass shows\n\n**A pass means the fixture, system-bundle, and route-readiness result records all show `status: pass` with metadata-only authority handles, with Scope limit stating what it does not certify.** A pass means the fixture, system-bundle, and route-readiness result records all show `status: pass` with metadata-only authority handles; see Scope limit for what a pass does and does not certify.\n\n###### Inspect result record fields, not markdown constants\n\n**The current authority is the runtime result record set, and cold readers should read result record fields such as `status` and `secret_exclusion_scan` rather than markdown constants or a separate sign-off result record.** The current authority is the runtime result record set under `receipts/first_wave/pattern_binding_contract/`; do not cite a separate pattern-specific sign-off result record unless an sign-off-lane artifact is actually present. Cold readers should inspect result record fields rather than markdown constants: `status`, `secret_exclusion_scan`, `source_open_body_imports`, `truth_accounting`, `route_readiness_summary`, `selection_contract`, and `source_manifest`.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### What this module is evidence for\n\n**The module covers public pattern-binding mechanics and stands as evidence for the `pattern_binding_contract` component and its public-binding mechanism.** This module covers public pattern-binding mechanics: source-bundle validation, reference-bundle validation, authority-handle validation, route-readiness selector admission, duplicate and unknown-ref rejection, private-leakage sentinel checks, and metadata-only result record shape. It is evidence for the `pattern_binding_contract` component and `mechanism.pattern_binding_contract.validates_public_pattern_bindings`.\n\n###### Where the ceiling stops\n\n**The ceiling stops before private ledger authority, launch or deployment posture, standalone-leaf status, and provider or source-file changes; importing the bundle keeps selection component-first.** The ceiling stops before private pattern-ledger authority, hosted or public launch-scope decision, deployment posture, standalone public-leaf selector status, non-public data equivalence, external model access, recipient work, source-file changes, publishing-scope decision, or whole-system correctness. Importing the route-readiness bundle does not make any mined pattern row a standalone public leaf; selection stays component-first and fixture-bound.\n\nSource and projection details\n\n###### Source-open body floor\n\n**The source-open body floor is the imported public bundle, not the private pattern ledger. Cold readers can open `examples/pattern_binding_contract/exported_substrate_bundle/` and `examples/pattern_binding_contract/exported_route_readiness_bundle/` to inspect the copied source module manifests, source bundles, reference bundles, authority-chain handles, route-readiness overlays, selector contract inputs, and copied source validation report.** The source-open body floor is the imported public bundle, not the private pattern ledger. Cold readers can open `examples/pattern_binding_contract/exported_substrate_bundle/` and `examples/pattern_binding_contract/exported_route_readiness_bundle/` to inspect the copied source module manifests, source bundles, reference bundles, authority-chain handles, route-readiness overlays, selector contract inputs, and copied source validation report. The required body floor is named by each `source_module_manifest.json` plus `source_capsules.json`, `reference_capsules.json`, and `authority_chain_handles.json`.\n\n**Result records and manifests must stay metadata-only where the standard requires it: they carry refs, digests, anchors, counts, verdicts, omission result records, and secret-exclusion results.** Result records and manifests must stay metadata-only where the standard requires it: they carry refs, digests, anchors, counts, verdicts, omission result records, and secret-exclusion results. They do not inline private source bodies, raw operator payloads, model-output data, recipient data, or hidden pattern-ledger material.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":15151,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/pattern_binding_contract.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"Hillside patterns library","url":"https://hillside.net/patterns/","kind":"external_reference","section_id":"pattern-language-lineage"},{"label":"W3C 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It consumes synthetic transport fixtures, validates copied observe-runtime body digests, writes five metadata-only result records, checks seven negative-case classes, and keeps heartbeat/resume evidence below work-landing, provider, UI, source-file changes, and launch-scope decision.","authority_ceiling":"Public synthetic observe/apply fixture and copied body digest evidence only; no live bridge transport, provider/UI uptime, operator HUD/browser state, live phase runtime, prompt-shelf/private-memory bodies, work landing, source-file changes, launch-scope decision, or whole-system correctness.","abstract":"The public bridge-continuity fixture validates disk-first continuation packets, heartbeat/resource-pressure boundaries, resume-once semantics, worker-skip dedupe, tracked result record-write gates, and non-public-state exclusion without live bridge transport.","teleology":{"section_id":"","text":"The public bridge-continuity fixture validates disk-first continuation packets, heartbeat/resource-pressure boundaries, resume-once semantics, worker-skip dedupe, tracked result record-write gates, and non-public-state exclusion without live bridge transport.","source":"capsule_summary_line"},"body_markdown":"###### Component replays synthetic\n\n**This component replays a synthetic record of a paused background job and checks the rules for safely resuming it, opening no live bridge and touching no private memory.** `bridge_phase_continuity_runtime` is a public component that replays a small synthetic record of a paused background job and checks that the rules for safely resuming it hold. It reads fixture files on disk. It opens no live bridge, calls no provider, reads no operator screen state, and touches no private memory.\n\n###### Run produces five\n\n**A run produces five plain result records for packet, heartbeat, pressure, resume, and completion, each carrying status fields, counts, and error codes but no copied body, live packet, or secret.** A run produces five plain result records: one for the continuation packet, one for the heartbeat, one for resource pressure, one for the resume decision, and one for the completion transition. Each record carries status fields, counts, and error codes. None of them carries a copied body, a live packet, or a secret.\n\n###### Component answers whether\n\n**The component answers whether a disk-first pause-and-resume handoff can be validated against fixed rules from public fixtures, where a pass means the fixture held, not that a real job is alive.** The component answers one bounded question. Can a disk-first pause-and-resume handoff be represented by public fixture inputs, validated against a fixed set of rules, and kept below any live-bridge or launch-scope decision? A pass means the fixture held those rules. It does not mean a real job is alive.\n\n##### Purpose\n\n###### How loose resume stories overreach\n\n**Detached work is easy to describe and hard to trust, and conflating a liveness ping with resume permission, or a resume with finished work, is how bridge systems overreach.** Detached background work is easy to describe and hard to trust. A running agent says it paused, left a note, and can resume. A loose version of that story lets a \"still alive\" ping stand in for permission to resume, or lets a successful resume stand in for finished work. Those are different claims, and conflating them is how bridge systems overreach.\n\n###### Fixed continuity rules a cold reader can run\n\n**It makes the difference checkable with fixed rules a reader can run against fixtures: a paused job leaves a packet, double resume is refused, and only completion claims finished work.** This component makes the difference checkable. It exists so a cold reader can run the continuity rules against fixture files instead of trusting a story about hidden agents. The rules are fixed. A paused job must leave a real continuation packet. Resuming the same job twice is refused. A heartbeat is liveness evidence and never resume authority. A stale heartbeat cannot pass itself off as proof of health. Only a proper completion record may claim the work finished.\n\n##### How it works\n\n###### `run` reads the fixture and six synthetic inputs\n\n**The `run` entry point resolves the session fixture and manifests, then reads six synthetic transport inputs, streaming the heartbeat so a malformed line becomes a finding, not an ingested body.** The entry point is `run`. It takes an input directory and an output directory. It reads the session fixture `observe_apply_session_fixture.json`, pulls the fixture-manifest and source-module-manifest references out of it, and resolves both. From the manifest it reads six synthetic transport inputs through `_read_synthetic_transport_inputs`: `detached_job.json`, `continuation_packet.json`, `heartbeat_rows.jsonl`, `resource_pressure.json`, `worker_skip_receipt.json`, and `private_state_forbidden_terms.json`. The heartbeat file is JSON Lines, streamed by `_read_required_jsonl`, so a malformed line becomes a finding and never a reason to ingest a whole live heartbeat body.\n\n###### The positive job's sign-off conditions\n\n** `_validate_synthetic_transport_contract` accepts the one positive job only when it yielded to disk with an unconsumed packet, a fresh heartbeat, and agreeing phase and continuity references across job, packet, and heartbeat.** The core check is `_validate_synthetic_transport_contract`. It reads the job, packet, heartbeat, pressure, and worker-skip rows, then evaluates one positive job against a floor of negative perturbations. The positive job (`synthetic_detached_job_001`) passes only when its state is `yielded_to_disk`, its transport label is the accepted synthetic-transport label, its continuation packet targets the same job id, the packet's claim reference matches the job's, the packet is not yet consumed, at least one fresh heartbeat is present, and the phase id and continuity reference agree across job, packet, and heartbeat. Freshness is concrete. `_heartbeat_is_fresh` requires status `alive` and an age at or under 60 seconds. `_heartbeat_is_stale` treats status `stale`, or an age over 300 seconds, as stale.\n\n###### Every declared bad case must be rejected\n\n**The same function turns each declared bad case into a specific error code, and a pass requires the fixture both to accept the good job and to observe the whole refusal floor.** The same function checks that each declared bad case is actually rejected. A missing packet, a packet missing required fields, an already-consumed packet, a heartbeat that claims resume authority, a stale heartbeat that claims live health, a blocked resource-pressure decision, an invalid worker-skip record, and a private-class fixture that leaks bodies each turn into a specific error code. When a perturbation is present and correctly rejected, the run records the matching code. When the fixture fails to carry a perturbation, the run records a finding and blocks. A pass therefore means the fixture both accepted the one good job and observed the whole refusal floor. The declared cases live in the `EXPECTED_NEGATIVE_CASES` constant.\n\n###### `_validate_fixture_contract` matches ids and digests\n\n** `_validate_fixture_contract` matches the envelope ids and recomputes each source-module SHA-256 so drift becomes `BRIDGE_CONTINUITY_SOURCE_DIGEST_MISMATCH`, then requires `can_continue`, `closed_with_receipts`, and a rollback that refuses writes after failure.** `_validate_fixture_contract` checks the session envelope. The fixture's `organ_id` and its `pattern_id` (`observe_runtime_apply_session`) must match, the manifest ids must match, and the fixture's source-module references must equal the manifest's. For each listed source module it recomputes a SHA-256 digest and compares it to the recorded one, so a drifted copied body becomes `BRIDGE_CONTINUITY_SOURCE_DIGEST_MISMATCH` rather than silent trust. It then requires the status packet to report `can_continue`, the finalizer status to be `closed_with_receipts`, and the rollback case to refuse writes after a validation failure.\n\n###### Forbidden-class scan and the tracked-write gate\n\n**Between the checks, `run` scans every path and strips excerpts through `_safe_scan` to report counts without text, then gates writes so a tracked path without the env flag reports `tracked_receipt_writes_blocked`.** Between those two checks, `run` scans every input path with `scan_paths` against the forbidden-class policy and strips matched excerpts through `_safe_scan`, so the scan reports counts and status without carrying the offending text. Only then does it build the five result records and write them. Writes pass through a gate: when the target is a tracked path and `MICROCOSM_TRACKED_RECEIPT_WRITES=1` is absent, the run reports `tracked_receipt_writes_blocked` instead of refreshing tracked evidence. `run` returns the completion-transition record.\n\nFunction Role `run` Reads the fixture and manifests, drives the checks, writes five result records, returns the completion record. `_read_required_jsonl` Streams the heartbeat JSON Lines; a bad line is a finding, not an ingested body. `_validate_synthetic_transport_contract` Accepts the one good job and requires every declared bad case to be rejected, emitting error codes. `_validate_fixture_contract` Matches ids, compares source-module SHA-256 digests, and checks the continue, finalizer, and rollback status. Diagram of the mechanism (10 steps). Diagram source & refs\n\nSource refs\n\n_validate_fixture_contract ids, SHA-256 digests, `continue/finalizer/rollback` Skip `tracked_receipt_writes_blocked`\n\n```text\nflowchart TD Read[\"run reads fixture and six synthetic inputs\"] --> Transport[\"_validate_synthetic_transport_contract\"] Transport --> Good{\"One valid job? yielded to disk, packet unconsumed, fresh heartbeat, phase and continuity agree\"} Good -->|\"yes\"| Floor[\"Every declared bad case rejected as an error code\"] Good -->|\"no\"| Block[\"Finding: valid job invalid\"] Floor --> Fixture[\"_validate_fixture_contract ids, SHA-256 digests, continue/finalizer/rollback\"] Fixture --> Scan[\"scan_paths and _safe_scan counts only, no excerpts\"] Scan --> Gate{\"Tracked path without MICROCOSM_TRACKED_RECEIPT_WRITES=1?\"} Gate -->|\"yes\"| Skip[\"tracked_receipt_writes_blocked\"] Gate -->|\"no\"| Write[\"Five result records written\"]\n```\n\n##### Negative cases\n\n###### Source-declared refusal floor mapped to codes\n\n**The refusal floor is source-declared in `EXPECTED_NEGATIVE_CASES` and validated against fixture contents, with each named case mapping to the error codes the run must observe.** The refusal floor is source-declared in `EXPECTED_NEGATIVE_CASES` and validated against fixture contents. Each named case maps to the error codes the run must observe:\n- `missing_packet_duplicate_resume_and_resource_block`: `MISSING_CONTINUATION_PACKET`, `RESOURCE_PRESSURE_DISPATCH_BLOCKED`, `CONTINUATION_PACKET_ALREADY_CONSUMED`\n- `continuation_packet_missing_required_fields`: `MISSING_CONTINUATION_PACKET_FIELDS`\n- `heartbeat_claims_resume_authority`: `HEARTBEAT_NOT_RESUME_AUTHORITY`\n- `bridge_packet_private_hud_body`: `BRIDGE_PACKET_PRIVATE_HUD_BODY`\n- `stale_heartbeat_overclaims_liveness`: `STALE_HEARTBEAT_LIVENESS_CLAIM`\n- `resume_success_overclaims_work_landed`: `RESUME_PASS_OVERCLAIMS_WORK_LANDED`\n- `apply_validation_failure_rolls_back_observe_promotion`: `OBSERVE_APPLY_VALIDATION_FAILED`\n\n##### Prior Art Grounding\n\n###### Durable-execution shapes without running them\n\n**The runtime borrows fixture-level shapes from Temporal, Airflow, Kubernetes leases, and W3C PROV without running any: a resumable handoff, a liveness lease that is not permission, and an inspectable provenance record.** The runtime borrows shapes from durable execution and provenance work without running any of it. [Temporal](https://docs.temporal.io/) keeps workflow state resumable across process failure; Apache Airflow [DAGs](https://airflow.apache.org/docs/apache-airflow/stable/core-concepts/dags.html) separate task ordering and retry policy from task internals; Kubernetes [lease-based leader election](https://kubernetes.io/docs/concepts/cluster-administration/coordinated-leader-election/) treats a heartbeat as liveness evidence and not as work completion; and W3C [PROV](https://www.w3.org/TR/prov-overview/) records how an output was produced. This component keeps only the fixture-level shape of those ideas: a resumable handoff, a liveness lease that is not permission, and a provenance record a reader can inspect.\n\n##### Validation Result record Path\n\n###### Run focused tests\n\n**Run the focused tests and the corpus check from the repository root without mutating durable records or generated projections.** Run the focused tests and the corpus check from the repository root without mutating durable records or generated projections:\n\n```text\n./repo-pytest --host-pressure-policy=warn tests/test_bridge_phase_continuity_runtime.py -q --basetemp=/tmp/microcosm_bridge_phase_continuity_runtime_pytest\n./repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\nTo exercise the component directly against its fixture:\n\n```text\nmicrocosm bridge-phase-continuity-runtime run --input fixtures/second_wave/bridge_phase_continuity_runtime/input --out /tmp/microcosm-bridge-continuity\n```\n\n###### What a passing run confirms\n\n**A pass confirms the fixture accepted the one good job, rejected every declared bad case, kept the non-public-state scan clean, and held the source-module digests.** A pass means the fixture accepted the one good job, rejected every declared bad case, kept the non-public-state scan clean, and held the source-module digests.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The bounded proof boundary\n\n**The strongest supported claim is bounded: over its public fixture a disk-first pause-and-resume handoff validates against a fixed rule set into five metadata-only records, and nothing wider.** The strongest claim this component supports is a bounded one: over its public fixture, a disk-first pause-and-resume handoff can be validated by a fixed rule set, and the resulting five result records are metadata-only. Its proof boundary is that fixture and its validation result records, nothing wider.\n\n###### What each record does not establish\n\n**It runs no live bridge, provider, or private memory; a heartbeat proves liveness not resume, a resume proves single consumption not landing, and completion proves fixture state, not a finished job.** It does not run live bridge transport, call a provider, read operator screen or browser state, read live phase state, or read private-memory bodies. A fresh heartbeat proves liveness, not resume authority. A resume proves the packet was consumed once, not that the apply step landed. A completion record proves the fixture reached its declared state, not that a real provider-backed job finished. 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five","Component answers whether","Purpose","How loose resume stories overreach","Fixed continuity rules a cold reader can run","How it works","`run` reads the fixture and six synthetic inputs","The positive job's sign-off conditions","Every declared bad case must be rejected","`_validate_fixture_contract` matches ids and digests","Forbidden-class scan and the tracked-write gate","Negative cases","Source-declared refusal floor mapped to codes","Prior Art Grounding","Durable-execution shapes without running them","Validation Result record Path","Run focused tests","What a passing run confirms","Scope limit","The bounded proof boundary","What each record does not establish"]},{"id":"paper_module.cognitive_operator_registry","title":"Cognitive Operator Registry","relative_href":"docs/paper-module-cognitive-operator-registry.html","read_href":"docs/paper-module-cognitive-operator-registry.html","context_href":"docs/rules-and-ideas.html#paper-module-cognitive-operator-registry","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"The public cognitive-operator registry fixture validates operator-shape rows, active-operator dogfood result records, anti-sprawl decisions, copied source registry/standard/tool bodies, and scope limits without becoming operator source authority.","summary":"Cognitive Operator Registry is the public evidence membrane for reusable cognition as typed system. It validates public operator rows, dogfood result records with cognition-delta evidence, negative cases for missing fields, missing dogfood, sprawl, operator-voice claims, authority overclaims, and private-source leakage, then checks copied source registry, standard, and validator bodies by digest while keeping bodies out of result records.","authority_ceiling":"Public registry-contract fixture and copied source body evidence only; no live operator execution, registry mutation, source-file changes, external model access, launch-scope decision, non-public data equivalence, operator correctness proof, or whole-system correctness.","abstract":"The public cognitive-operator registry fixture validates operator-shape rows, active-operator dogfood result records, anti-sprawl decisions, copied source registry/standard/tool bodies, and scope limits without becoming operator source authority.","teleology":{"section_id":"","text":"The public cognitive-operator registry fixture validates operator-shape rows, active-operator dogfood result records, anti-sprawl decisions, copied source registry/standard/tool bodies, and scope limits without becoming operator source authority.","source":"capsule_summary_line"},"body_markdown":"###### Registry checks public\n\n**The registry checks a public catalogue of named thinking moves and asks which operators have actually changed a live decision, not just described one tidily.** `cognitive_operator_registry` is a checker for the system's catalogue of named thinking moves. Each entry describes one reusable operator, for example \"reduce competing pressure to one bounded action\" or \"compile a handoff packet when validated work cannot be committed\". The component reads the public catalogue and answers one question: which of these operators has actually changed a live decision, and which is only a tidy description of one.\n\n###### Validates every operator\n\n**It validates every operator row against a fixed shape, makes each `active` operator cite a dogfood record with concrete evidence, and rejects voice or authority overclaims.** It runs over a public fixture. It reads three input files, validates every operator row against a fixed shape, requires each operator marked `active` to point at a dogfood result record carrying concrete evidence, and rejects any row that claims the owner's voice or that grants itself launch, provider, or mutation authority. It writes result records that name refs, hashes, counts, and verdicts, never the operator text itself.\n\n##### Purpose\n\n###### `active` requires resolved cognition-delta evidence\n\n**An entry may call itself `active` only when it cites a dogfood record whose `cognition_delta_evidence` names a decision that changed, with evidence surfaces resolved against the public system.** A team that writes its reusable thinking moves down as a registry tends to accumulate entries faster than it can prove any of them helped. A loose registry passes on prose alone: an entry says it was useful, and the reader has to take its word. This component refuses that. An entry may call itself `active` only if it points to a dogfood record, and that record must carry `cognition_delta_evidence` describing a concrete decision that came out differently because the operator was applied. Where the record cites evidence surfaces or command paths, the checker resolves each one against the public system rather than trusting the prose.\n\n##### How it works\n\n###### Two entrypoints over `_build_result`\n\n**Two entrypoints, `run` over the first-wave fixture and `run_registry_bundle` over the exported bundle, both call `_build_result` to run the ordered pipeline.** The runtime lives in `src/microcosm_core/organs/cognitive_operator_registry.py`. There are two entrypoints. `run` validates the first-wave fixture and includes the negative cases; `run_registry_bundle` validates the exported bundle and makes the copied source-body manifest mandatory. Both call `_build_result`, which runs the pipeline in this order.\n\n###### The five-step pipeline and pass rule\n\n**The pipeline loads inputs, checks copied bodies, runs the secret scan, validates operator rows, and exercises negatives; status is `pass` only when all four conditions hold.** First, `_load_payloads` reads the three public inputs: `operator_registry.json`, `operator_standard.json`, and `dogfood_index.json` (plus the eight negative inputs in fixture mode). Second, `_source_module_manifest_result` checks the copied source bodies carried by the exported bundle. Third, `scan_paths` runs the secret-exclusion pass over every input. Fourth, `_positive_findings` validates the operator rows. Fifth, `_negative_findings` exercises the declared rejection cases. The status is `pass` only when there are no positive findings, no expected negative case went unobserved, no blocking secret hit was found, and the source-module check is `pass` or `not_present`.\n\n###### `_positive_findings` carries the core contract\n\n** `_positive_findings` checks each row's shape and status, resolves every `active` operator's dogfood result record and evidence commands, and flags any policy that sets an `OVERCLAIM_KEYS` flag true.** `_positive_findings` carries the core contract. For each operator row it requires all eleven fields in `OPERATOR_REQUIRED_FIELDS` and a `status` drawn from `candidate`, `active`, or `retired`. For each `active` operator it looks up a dogfood row keyed by `operator_id`, requires the nine fields in `DOGFOOD_REQUIRED_RECEIPT_FIELDS`, and calls `_dogfood_receipt_ref_resolves` to confirm the result record ref points at a real dogfood row. It then calls `_record_dogfood_evidence_resolution_findings`, which parses each evidence surface as a command and resolves its path and handle tokens against the public system. A row whose prose says it was dogfooded but whose evidence does not resolve is recorded as a failure. Finally it checks the registry policy against `OVERCLAIM_KEYS`: if the policy sets any of `release_authorized`, `provider_calls_authorized`, `source_mutation_authorized`, or the other listed flags to true, that is an overclaim.\n\n###### The bundle body-floor manifest check\n\n** `_source_module_manifest_result` proves the bundle's copied bodies by class, `module_count`, sha256 across three digest fields, and declared anchors, recording ids and hashes but never the body text.** `_source_module_manifest_result` is the body-floor check used by the bundle entrypoint. The exported bundle at `examples/cognitive_operator_registry/exported_cognitive_operator_registry_bundle` copies the source registry, standard, and projection-tool bodies. The manifest must classify each as `copied_non_secret_macro_body`, keep `body_in_receipt` false, match its declared `module_count`, and for every row: use a material class, copy the body into `source_modules`, hash to the recorded sha256 across all three digest fields, and carry the declared anchors in the copied text. The result summary records module ids, classes, line counts, and hashes, never the body text.\n\nFunction Role `run` First-wave fixture entrypoint; writes the result, board, validation, and fixture sign-off records `run_registry_bundle` Exported-bundle entrypoint; requires the source-body manifest `_build_result` Runs the five-step pipeline and computes the pass or blocked status `_positive_findings` Shape, status, active-operator dogfood, and policy-overclaim checks `_dogfood_receipt_ref_resolves` Confirms a dogfood result record ref points at a real row `_record_dogfood_evidence_resolution_findings` Resolves each evidence surface command against the public system `_negative_findings` Exercises the eight declared rejection cases `_source_module_manifest_result` Proves copied source bodies by sha256 and anchors, body text excluded Diagram of the mechanism (6 steps). Diagram source & refs\n\nSource refs\n\nInputs `operator_registry.json` `operator_standard.json` `dogfood_index.json`\n\n```text\nflowchart TD Inputs[\"operator_registry.json operator_standard.json dogfood_index.json\"] Positive[\"_positive_findings shape, status, dogfood, policy overclaim\"] Negative[\"_negative_findings eight rejection cases\"] Manifest[\"_source_module_manifest_result copied bodies by sha256\"] Secret[\"scan_paths secret exclusion\"] Result[\"result record refs, hashes, counts, verdicts; body text omitted\"] Inputs --> Positive Inputs --> Negative Inputs --> Manifest Inputs --> Secret Positive --> Result Negative --> Result Manifest --> Result Secret --> Result\n```\n\n##### Negative cases\n\n###### Eight perturbed inputs and their error codes\n\n**The fixture ships eight perturbed inputs, each of which `_negative_findings` must observe and map to a specific error code, from missing fields to unresolvable dogfood evidence.** The fixture ships eight perturbed inputs, and a pass requires every one to be observed and recorded. `_negative_findings` reads each and emits the matching error code. Missing operator fields give `COGOP_MISSING_REQUIRED_FIELD`. An active operator with no backing dogfood row gives `COGOP_ACTIVE_WITHOUT_DOGFOOD`. A dogfood row with no cognition-delta evidence gives `COGOP_DOGFOOD_MISSING_COGNITION_DELTA`. Two operators sharing a slug or a near-identical claim with no recorded accretion decision give `COGOP_OPERATOR_SPRAWL`. A policy that grants itself launch, provider, or mutation authority gives `COGOP_AUTHORITY_OVERCLAIM`. A row that claims operator-voice authority gives `COGOP_OPERATOR_VOICE_FORBIDDEN`. A private source body or model-output data in a public input gives `COGOP_PRIVATE_SOURCE_FORBIDDEN`. A dogfood result record whose evidence surfaces do not resolve gives `COGOP_COGNITION_DELTA_EVIDENCE_UNRESOLVABLE`.\n\n##### Prior Art Grounding\n\n###### Cognitive work analysis, PROV, schema, policy\n\n**The design draws on cognitive work analysis, W3C PROV provenance, JSON Schema shape checks, and Open Policy Agent's data/policy separation, applied as a public contract check.** This component sits in the lineage of cognitive work analysis, provenance tracking, schema validation, and policy-gated registries. Cognitive Work Analysis, summarized in this [information-systems design overview](https://files.eric.ed.gov/fulltext/EJ1082064.pdf), studies reusable moves in complex sociotechnical work. W3C [PROV](https://www.w3.org/TR/prov-overview/) connects a claim to the activities, agents, and evidence used to judge it. [JSON Schema](https://json-schema.org/) is the required-shape pattern behind the operator-row check. [Open Policy Agent](https://www.openpolicyagent.org/docs/latest) is a precedent for keeping policy evaluation distinct from the data being evaluated. This component applies those patterns as a public contract check over already-public inputs.\n\n##### Validation Result record Path\n\nRun the first-wave fixture into disposable output from the Microcosm root:\n\nRun the exported bundle through the same component:\n\n```text\ncd microcosm-substrate\nPYTHONPATH=src ../repo-python -m microcosm_core.organs.cognitive_operator_registry run-registry-bundle --input examples/cognitive_operator_registry/exported_cognitive_operator_registry_bundle --out /tmp/microcosm_cognitive_operator_registry_bundle\n```\n\nThen run the focused tests:\n\n```text\ncd microcosm-substrate\n../repo-pytest tests/test_cognitive_operator_registry.py -q\n```\n\n###### What a pass means here\n\n**A pass means the public operator contract replayed from public inputs with every shape check holding and every rejection observed, but does not certify the operators as correct.** A pass means the public operator contract replayed from public inputs: every shape check held, every expected rejection was observed, and no body text reached the result records. It does not certify the operators as generally correct.\n\n##### Scope boundary\n\n###### Scope limit\n\n**The strongest supported claim is that the operator contract is replayable from public inputs, with the checker rejecting bad shape, unresolved evidence, sprawl, voice, overclaim, and leakage.** The strongest claim the evidence supports is that the public operator contract is replayable from public inputs. The checker rejects missing shape, missing or unresolved dogfood evidence, near-duplicate sprawl, operator-voice authority, authority overclaim, and private-source leakage, and it proves the copied source bodies by hash and anchor while keeping their text out of the result records.\n\n###### Where the proof boundary stops\n\n**The component is source-linked only, mutates no operators, proves no operator correct beyond its dogfood record, and excludes voice export, external model access, mutation, or launch.** The proof boundary stops there. This component does not become source authority for the operator registry, does not mutate operators, and does not establish that any listed operator is correct beyond its own dogfood record. Its scope limit excludes private operator bodies, operator-voice export, external model access, source-file changes, and launch or public sharing. 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It builds a fixture repo, verifies commit and HEAD evidence with git subprocesses, runs pytest for the declared span, checks cap claims against a fixture ledger, validates a copied source-module manifest for agent experience diagnostics, and writes bounded result records that keep source bodies out.","authority_ceiling":"Public fixture and exported-bundle result records only; no arbitrary live commit proof, work log mutation or closure, live Git mutation, external model access, launch-scope decision, broad completion certification, or pytest-pass claim without explicit exit-zero status.","abstract":"The public completion-faithfulness fixture checks commit, ledger-cap, and pytest-span claims with real git/pytest subprocess witnesses while refusing unchecked pass overclaims.","teleology":{"section_id":"","text":"The public completion-faithfulness fixture checks commit, ledger-cap, and pytest-span claims with real git/pytest subprocess witnesses while refusing unchecked pass overclaims.","source":"capsule_summary_line"},"body_markdown":"###### Completion language needs observable evidence\n\n**The finish-line sentence an agent writes is admitted only claim by claim, once observable `git` and `pytest` evidence backs each part of it.** `agent_closeout_faithfulness_audit` checks the sentence an agent writes when it finishes a task: \"I committed the change, closed the ledger item, and the test passed.\" It runs the supplied public fixture through real `git` and `pytest` subprocesses and admits each claim only when an evidence object it can observe actually backs it.\n\nA fixture repository is copied into a throwaway directory, committed with real `git`, and its `HEAD` is read back. A declared test span is run, and the exit code alone decides whether the span passed.\n\n###### Four metadata-only records carry the result\n\n**It emits only metadata-only records: a result file, a board, a validation record, and an exported bundle validation record.** It emits metadata-only result records: a result file, a board, a validation record, and an exported bundle validation record.\n\n##### Purpose\n\n###### A done report is only text\n\n**A finished-work report is just text: the commit, the ledger row, and the \"passed\" may or may not be real, and an audit that trusts the wording certifies nothing.** When an agent reports that work is done, the report is text. The commit may or may not exist. The ledger row may or may not be there. \"The test passed\" can mean the test ran, or it can mean nothing was checked. A loose audit that trusts the wording certifies all of those as done and learns nothing.\n\n###### Ran and passed are separate facts\n\n**It asks one question per fixture: is each completion claim backed by an evidence object that exists, and is \"passed\" backed by an explicit exit-zero check rather than by the phrasing of the claim?** The component answers one question over a fixed fixture: is each completion claim backed by an evidence object that genuinely exists, and is a \"passed\" claim backed by an explicit exit-zero check rather than by the phrasing of the claim? The distinction it defends is narrow. \"The span ran\" and \"the span passed\" are separate facts. A completion sentence that conflates them is the exact failure mode here.\n\n##### How it works\n\n###### Two entrypoints forward to `evaluate`\n\n** `run` and `run_agent_closeout_bundle` are the two entrypoints, and both forward to `evaluate`, which carries the whole mechanism.** The runtime is `src/microcosm_core/organs/agent_closeout_faithfulness_audit.py`. `run` and `run_agent_closeout_bundle` are the two entrypoints; both forward to `evaluate`, which carries the mechanism.\n\n`evaluate` runs in this order:\n1. Load `closeout_claims.json` and `fixture_ledger.json`. If `public_fixture_repo` is not a directory, emit `CLOSEOUT_PUBLIC_FIXTURE_REPO_MISSING` and stop.\n2. `_prepare_public_fixture_repo` copies the fixture repo into a temporary directory and runs `git init`, two `git config` calls, `git add .`, and `git commit`, then `git rev-parse HEAD`. Six subprocesses in total. The observed `HEAD` is returned. Each subprocess row is a return code and two sha256 digests from `_sha256_text`, never raw output. `_run_subprocess` builds the five setup rows in that metadata-only shape.\n3. `_select_pytest_python` picks the first interpreter that answers `pytest --version` with exit zero, so the span runs under an interpreter that actually has pytest installed.\n4. Walk the claim rows. A `commit` claim verifies only when its `commit_ref` is `HEAD` or the observed head, else `CLOSEOUT_FAKE_COMMIT_CLAIM`. A task_ledger_cap claim verifies only when its `cap_id` is present in `fixture_ledger.json`, else `CLOSEOUT_FAKE_CAP_CLAIM`. A `pytest_span` claim runs `python -m pytest <nodeid> -q` inside the temporary repo: `span_ran` is true when the process exits 0 or 1, and `passed` is true only on exit 0. A span that did not run emits `CLOSEOUT_FAKE_TEST_CLAIM`. A claim whose `expected_pass` is true but whose `pass_status_checked` is false, or whose span did not exit zero, emits `CLOSEOUT_PYTEST_PASS_STATUS_NOT_CHECKED`.\n\nFunction Role `evaluate` `_prepare_public_fixture_repo` Copies and commits the fixture repo, reads `HEAD`, records six metadata-only subprocess rows `_select_pytest_python` Chooses an interpreter that has pytest before running the span `evaluate_negative_case` Mutates one claim row to force a named rejection `_semantic_closeout_contract_findings` Re-checks claims against the fixture test and arithmetic source on the negative path `run` / `run_agent_closeout_bundle` First-wave and exported-bundle entrypoints over the shared evaluator\n\n###### A pass counts pass status separately\n\n**A pass records five counters, where `verified` means the evidence object exists or the span ran, and pass status is tallied separately only when the exit-zero check held.** A pass records `verified_claim_count`, `git_subprocess_count`, `pytest_span_ran_count`, `pytest_pass_status_checked_count`, and `head_verified_by_subprocess`. Here `verified` means the evidence object exists or the span ran. Pass status is counted separately, only when the exit-zero check held. That keeps a referenced test distinct from a passed test in the record itself.\n\nDiagram of the mechanism (8 steps). Diagram source & refs\n\nSource refs\n\ncommit, cap, span `closeout_claims.json` cap rows `fixture_ledger.json`\n\n```text\nflowchart TD Claims[\"completion_claims.json commit, cap, span\"] --> Eval[\"evaluate()\"] Ledger[\"fixture_ledger.json cap rows\"] --> Eval Repo[\"public_fixture_repo\"] --> Prep[\"_prepare_public_fixture_repo git init..commit, rev-parse HEAD\"] Prep --> Eval Eval --> Pytest[\"python -m pytest nodeid -q\"] Pytest --> Eval Eval --> Pass[\"pass: verified counts, metadata-only record\"] Eval --> Reject[\"reject: fake commit, cap, test, unchecked pass\"]\n```\n\n##### Negative cases\n\n###### One mutated row per rejection\n\n** `evaluate_negative_case` mutates one claim row at a time and confirms the matching rejection fires.** The four cases named in `EXPECTED_NEGATIVE_CASES`: `evaluate_negative_case` mutates one claim row at a time and confirms the matching rejection fires.\n- `fake_commit_claim`: rewrites the commit ref to all zeros, expects `CLOSEOUT_FAKE_COMMIT_CLAIM`.\n- `fake_cap_claim`: points the cap at an id missing from the fixture ledger, expects `CLOSEOUT_FAKE_CAP_CLAIM`.\n- `fake_test_claim`: names a test node that does not exist, expects `CLOSEOUT_FAKE_TEST_CLAIM`.\n- `unchecked_pass_claim`: sets `pass_status_checked` to false, expects `CLOSEOUT_PYTEST_PASS_STATUS_NOT_CHECKED`.\n\n###### Losing a case loses the proof\n\n**If any of these rejections stops appearing in the observed cases, the component no longer proves it refuses fabricated commit, cap, test-node, or unchecked-pass claims.** If these stop appearing in the observed cases, the component no longer proves that it refuses fabricated commit, cap, test-node, or unchecked-pass claims.\n\n##### Prior Art Grounding\n\n###### Claim, artifact, and evidence stay separate\n\n**The posture is claim-then-evidence verification drawn from FEVER, TruthfulQA, and the artifact-evaluation tradition: commit, ledger, and span refs must line up with observed objects before completion language is admitted.** The posture is claim-then-evidence verification rather than trust in fluent summary text. [FEVER](https://arxiv.org/abs/1803.05355) framed fact extraction and verification as a task over cited evidence. [TruthfulQA](https://arxiv.org/abs/2109.07958) showed that fluent answers can be untrue without an explicit truthfulness check. The wider artifact-evaluation tradition keeps a claim, its artifact, and its validation evidence as three separate things. This component applies that separation to agent completion: commit refs, ledger refs, and test spans must line up with observed objects before the completion language is admitted.\n\n##### Validation Result record Path\n\n```text\n./repo-pytest tests/test_agent_closeout_faithfulness_audit.py -q --basetemp=/tmp/microcosm_agent_closeout_faithfulness_audit_pytest\n./repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n###### What a focused-test pass certifies\n\n**A focused-test pass means completion language was checked against fixture evidence across seven named witnesses and all four negative classes; the corpus check only proves the bundle and instance agree.** A pass on the focused test means completion language was checked against fixture evidence: three verified claims, at least five `git` subprocess witnesses, one `pytest` witness, one span that ran, one checked pass status, `head_verified_by_subprocess` true, and observation of all four negative classes. The corpus check only proves that the source record and the generated instance agree.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim\n\n**On the supplied fixture repo, ledger, and completion claims, commit, cap, and span references are checked against observed `git` and `pytest` evidence, and a pass is admitted only after an explicit exit-zero check.** The strongest claim: over the supplied fixture repo, fixture ledger, and completion claims, this component proves that commit, cap, and span references are checked against observed `git` and `pytest` evidence, and that a pass claim is admitted only after an explicit exit-zero check. The proof boundary is that fixture and its records, nothing wider.\n\n###### What stays refused past that line\n\n**Past that line it proves nothing: no live commit landed, no ledger mutation, no `git`, provider, or source change, no launch, and no turning a span that merely ran into one that passed.** It refuses everything past that line. It does not establish any live commit landed, close or mutate ledger work, mutate `git`, use external model services, change source files, include launch operations, certify completion prose in general, or turn a span that merely ran into a span that passed. The scope limit is public fixture evidence existence and pytest span witnessing only. `AUTHORITY_CEILING` records those refusals in machine-readable form.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":10077,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/agent_closeout_faithfulness_audit.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"FEVER","url":"https://arxiv.org/abs/1803.05355","kind":"external_reference","section_id":"claim-artifact-and-evidence-stay-separate"},{"label":"TruthfulQA","url":"https://arxiv.org/abs/2109.07958","kind":"external_reference","section_id":"claim-artifact-and-evidence-stay-separate"}],"grouped_relation_feeds":[{"relation":"abides by","count":1,"links":[{"id":"AX-1","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Derivation before assertion","url":"doctrine.html#dcard-ax-1"}]},{"relation":"depends on","count":1,"links":[{"id":"paper_module.durable_agent_work_landing_replay","kind":"paper_module","kind_label":"paper module","relation":"depends on","label":"Durable Agent Work-Landing Replay","url":"paper-module-durable-agent-work-landing-replay.html"}]},{"relation":"explains","count":2,"links":[{"id":"agent_closeout_faithfulness_audit","kind":"organ","kind_label":"component","relation":"explains","label":"Agent Completion Faithfulness Audit","url":"component-agent-closeout-faithfulness-audit.html"},{"id":"mechanism.agent_closeout_faithfulness_audit.validates_closeout_evidence_claims","kind":"mechanism","kind_label":"mechanism","relation":"explains","label":"validates completion evidence claims","url":"rules-and-ideas.html#lattice-mechanism-mechanism-agent-closeout-faithfulness-audit-validates-closeout-evidence-claims"}]},{"relation":"governed by","count":3,"links":[{"id":"P-1","kind":"principle","kind_label":"principle","relation":"governed by","label":"Recompute, do not echo","url":"doctrine.html#dcard-p-1"},{"id":"P-2","kind":"principle","kind_label":"principle","relation":"governed by","label":"Lower claim strength to checker strength","url":"doctrine.html#dcard-p-2"},{"id":"concept.agent_reliability_and_safety_validator_bundle","kind":"concept","kind_label":"concept","relation":"governed by","label":"agent reliability and safety validators as bounded public scope limits","url":"rules-and-ideas.html#lattice-concept-concept-agent-reliability-and-safety-validator-bundle"}]}],"paper_module_backrefs":[],"source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/agent_closeout_faithfulness_audit.md"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json"}],"visible_text_byte_count":11139,"section_titles":["Completion language needs observable evidence","The audit rebuilds Git and test evidence","Four metadata-only records carry the result","Purpose","A done report is only text","Ran and passed are separate facts","How it works","Two entrypoints forward to `evaluate`","A pass counts pass status separately","Negative cases","One mutated row per rejection","Losing a case loses the proof","Prior Art Grounding","Claim, artifact, and evidence stay separate","Validation Result record Path","What a focused-test pass certifies","Scope limit","The strongest supported claim","What stays refused past that line"]},{"id":"paper_module.cold_reader_route_map","title":"Cold-Reader Route Map","relative_href":"docs/paper-module-cold-reader-route-map.html","read_href":"docs/paper-module-cold-reader-route-map.html","context_href":"docs/rules-and-ideas.html#paper-module-cold-reader-route-map","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"The public cold-reader route-map fixture validates first-run command order, docs refs, result record refs, scope limits, copied cold-entry source-module digests, and non-public-state exclusion without becoming route registry control.","summary":"Cold-Reader Route Map is the public first-run route membrane. It validates route rows, route-to-result record bindings, ordinal first-run sequencing, command/docs/result record refs, launch/provider/private-source overclaim rejection, exact copied source cold-entry source modules, and metadata-only result records so a cold agent can see what to run first and what proof bounds each row.","authority_ceiling":"Public route-map fixture, exported-bundle result records, and copied source cold-entry body evidence only; no route registry control, external model access, source-file changes, launch-scope decision, non-public data equivalence, trading or financial decisions, or whole-system correctness.","abstract":"The public cold-reader route-map fixture validates first-run command order, docs refs, result record refs, scope limits, copied cold-entry source-module digests, and non-public-state exclusion without becoming route registry control.","teleology":{"section_id":"","text":"The public cold-reader route-map fixture validates first-run command order, docs refs, result record refs, scope limits, copied cold-entry source-module digests, and non-public-state exclusion without becoming route registry control.","source":"capsule_summary_line"},"body_markdown":"###### Route map checks\n\n**The route map checks a newcomer's first ten minutes, confirming each first-run step's command, docs reference, and evidence record resolves against real source, not prose.** `cold_reader_route_map` checks the guided path a newcomer follows in their first ten minutes with Microcosm. It reads a public route map whose rows bind each first-run step to a runnable command, a public documentation reference, and an evidence record, then it confirms that every one of those bindings resolves against real source instead of prose.\n\n###### Takes three fixture\n\n**It takes three fixture files plus an optional copied-source manifest and emits metadata-only findings, a route board, validation records, and a card carrying ids, refs, and digests only.** The component takes three fixture files (`route_map.json`, `route_receipts.json`, `route_policy.json`) plus an optional copied-source manifest, and emits deterministic findings, a route board, validation records, and a compact card. Every emitted record is metadata-only: it carries route ids, commands, refs, digests, and `body_in_receipt: false` markers, never inlined source or live state.\n\n**Source replay is what lifts it above field-presence checking: an unrunnable command, an unresolved docs heading, or a non-passing evidence ref each blocks the route.** The discipline that makes it more than a field-presence checker is source replay. A route that promises a command the system does not actually run is blocked, as is a docs reference that resolves to no heading and an evidence ref that does not open a passing record. That is the failure a hand-written quick-start guide slides into the moment the commands change underneath it.\n\n##### Purpose\n\n###### What to run first, with evidence\n\n**A cold reader should not reverse-engineer the path from a README, so the route map answers what to run first and what evidence shows that path is wired.** A cold technical reader should not have to reverse-engineer the product path from a long README or a raw tree of result files. The route map answers one question: what do I run first, and what evidence shows that path is wired?\n\n###### Each row as a claim-boundary check\n\n**A prose route map lists commands nothing re-checks; this one treats each row as a claim-boundary check so the trusted sequence stays tied to re-runnable source.** A loose route map answers that question with prose. It lists impressive commands, and nothing checks that they still exist or that their supporting records still pass. The route map here treats each row as a claim-boundary check, so the sequence a newcomer trusts stays tied to source that a test can re-run.\n\n##### How it works\n\n###### Fixture and exported-bundle entrypoints\n\n** `run` builds the fixture with negative cases and writes the record set, while `run_route_map_bundle` runs the same builder with negatives off and the copied-source manifest required.** `run` is the fixture entrypoint. It calls `_build_result` with negative cases included, then writes the result, board, and validation records, plus a `cold_reader_route_map_fixture_acceptance_v1` record when the caller passes `acceptance_out`. `run_route_map_bundle` is the exported-bundle entrypoint. It runs the same builder with negative cases off and the copied-source manifest required, so the bundle proves replay shape from public inputs alone.\n\n###### `_build_result` joins four checks and a scan\n\n** `_build_result` runs four independent checks plus a secret-exclusion scan, passing only when no check files a blocking finding and the scan reports no blocking hit.** `_build_result` assembles four independent checks plus a secret-exclusion scan over the inputs, and passes only when none of the four checks files a blocking finding and the scan reports no blocking hit.\n\n###### Required routes and front-door ordering\n\n** `_positive_findings` requires each route to name a command, docs refs, and a result record ref, and the front-door sequence to match live first-screen strings in ordinal order.** `_positive_findings` walks the routes named in `route_policy`. Each required route must exist, name a `command`, carry `docs_refs`, and point at a result record ref. The first-run sequence must start with the three front-door routes in `FRONT_DOOR_ROUTE_COMMANDS` (`tour_project`, `status_card`, `proof_lab`), those front-door commands must match the live first-screen strings, and the whole sequence must be present and ordinal sorted. A gap, a missing field, or a drifted front-door command becomes a finding.\n\n###### Replaying commands, refs, and signals against source\n\n** `_route_source_replay_result` builds a source corpus and blocks any command token, docs ref, result record ref, or claimed signal that does not resolve against real public source.** `_route_source_replay_result` is the replay pass. It builds one text corpus from the copied source modules, a fixed set of public source refs, every route's docs refs, and every result record ref, then it tests each route against that corpus. `_command_support` splits a command into material tokens and blocks it if a token appears nowhere in the corpus. `_document_ref_result` requires each docs ref to resolve to a real public file and, when the ref names a heading anchor, to a real heading. `_receipt_ref_result` opens each result record ref and blocks it unless the file is JSON with `status` equal to pass. `_mechanism_signal_support` takes the human-readable signals a row claims to show and blocks the row if their terms are absent from the corpus.\n\n###### The copied-body floor for the bundle\n\n** `_source_module_import_result` recomputes each copied body's digest against the manifest `sha256`, checks provenance anchors survive, and requires a `copied_non_secret_macro_body` exact-copy declaration.** `_source_module_import_result` governs the copied-body floor used by the exported bundle. For each row in `source_module_manifest.json` it recomputes the file digest, compares it to the manifest `sha256`, checks that required provenance anchors survive in the copied text, and confirms the row declares `copied_non_secret_macro_body` with a material class and an `exact_copy` relation. A digest mismatch, a missing anchor, a body marked for a record, or an unsupported class each files a finding.\n\n###### Negative findings evaluates\n\n** `_negative_findings` evaluates the five perturbed inputs and records the error code each one must raise, listed under Negative cases below.** `_negative_findings` evaluates the five perturbed inputs and records the error code each one must raise, listed under Negative cases below.\n\nDiagram of the mechanism (7 steps). Diagram source & refs\n\n```text\nflowchart TD Inputs[\"route_map, route_receipts, route_policy, manifest\"] --> Positive[\"_positive_findings required routes, commands, docs refs, result record refs, order\"] Inputs --> Replay[\"_route_source_replay_result commands and refs matched against the source corpus\"] Inputs --> Modules[\"_source_module_import_result copied-body digests and anchors\"] Inputs --> Negative[\"_negative_findings five rejection cases\"] Positive --> Result[\"_build_result pass only when no blocking finding\"] Replay --> Result Modules --> Result Negative --> Result Result --> Records[\"run and run_route_map_bundle write metadata-only records\"]\n```\n\nFunction to role:\n\nFunction Role `run` Fixture entrypoint; builds with negative cases, writes the record set. `run_route_map_bundle` Exported-bundle entrypoint; manifest required, no negatives. `_build_result` Joins the four checks; sets pass only with no blocking finding. `_positive_findings` Required routes name a command, docs refs, result record ref, ordinal order. `_route_source_replay_result` Replays commands, docs refs, result record refs, signals against source. `_source_module_import_result` Verifies copied-body digests, anchors, and import class. `_negative_findings` Confirms the five perturbed inputs raise their error codes.\n\n##### Negative cases\n\n###### Expected negative cases\n\n** `EXPECTED_NEGATIVE_CASES` names the five inputs the fixture must reject and the code each one raises.** `EXPECTED_NEGATIVE_CASES` names the five inputs the fixture must reject and the code each one raises:\n- `missing_command_ref` raises `COLD_ROUTE_MISSING_COMMAND_REF`: a route with no command cannot guide anyone.\n- `missing_receipt_ref` raises `COLD_ROUTE_MISSING_RECEIPT_REF`: a route with no evidence ref is not backed.\n- `route_sequence_gap` raises `COLD_ROUTE_SEQUENCE_GAP`: a sequence that does not start from `tour_project` is malformed.\n- `release_overclaim` raises `COLD_ROUTE_AUTHORITY_OVERCLAIM`: a row that sets a key in `OVERCLAIM_KEYS`, such as `release_authorized` or `whole_system_correctness_claim`, is refused.\n- `private_source_leakage` raises `COLD_ROUTE_PRIVATE_SOURCE_FORBIDDEN`: a row carrying a key in `FORBIDDEN_PRIVATE_KEYS`, such as `private_source_body`, `proof_body`, or `provider_payload_body`, is refused so a route card cannot become a disclosure channel.\n\n###### If fixture run\n\n**If the fixture run does not observe one of these five, `_build_result` marks the run blocked for the missing case.** If the fixture run does not observe one of these five, `_build_result` marks the run blocked for the missing case.\n\n##### Reader-Specific Evidence Routing\n\n###### Command order and evidence order kept together\n\n**The route map keeps a command order and an evidence order, so a reader sees the compact sequence first yet keeps each step's result record-ref and scope-limit context in drilldowns.** The route map keeps two orders: the command order and the evidence order. A reader sees the compact command sequence first, then can open full drilldowns without losing the result record-ref and scope-limit context attached to each step.\n\nThe accepted path is:\n1. `plectis hello <project>`\n2. `plectis tour --card <project>`\n3. `plectis status --card <project>`\n4. `plectis authority --card`\n5. `plectis workingness --card`\n6. `plectis legibility-scorecard`\n7. Read `selected_route_id` from the compact tour or status card.\n8. `plectis explain <project> <selected_route_id>`\n9. `plectis observe --card <project>`\n10. `plectis cold-reader-route-map run-route-map-bundle`\n11. plectis comprehend --component certificate_kernel_execution_lab\n\nFull drilldowns stay available after the compact path is visible:\n- `plectis tour <project>` for the full first-screen route tree.\n- `plectis compile <project>` for project-state materialization detail.\n- `plectis proof-lab --out /tmp/microcosm-proof-lab` for proof-lab component records.\n- `plectis serve <project> --host 127.0.0.1 --port 8765` for the local observatory.\n\n###### Compact board must match the terminal row\n\n**A browser-first reader follows the same order, and the compact board must carry the same command ref, evidence ref, class, and scope limit as the terminal row, or it is a preview.** A browser-first reader follows the same order in compressed form, and the compact board must carry the same command ref, evidence ref, evidence class, and scope limit as the terminal row, or it is a preview only.\n\n##### Prior Art Grounding\n\n###### Diataxis and literate programming lineage\n\n**The design follows Diataxis's task-shaped documentation and Knuth's literate programming, borrowing the ordered-reader pattern without turning documentation sequence into source authority.** The component follows documentation systems that treat reader state and task shape as first-class. [Diataxis](https://diataxis.fr/) separates tutorials, how-to guides, reference, and explanation so a reader is not pushed through one undifferentiated pile. Knuth's [literate programming](https://cs.stanford.edu/~knuth/lp.html) is the older anchor for writing an executable system for human comprehension as well as machine execution. The route map borrows the ordered-reader pattern of first command, evidence ref, evidence class, scope limit, and next drilldown, without turning documentation sequence into source authority.\n\n##### Validation Result record Path\n\nFrom `microcosm-substrate/`, reproduce this page's proof boundary with temporary records:\n\n###### What a pass means here\n\n**A pass means the eleven-route fixture, the exported bundle, and the copied source-module digests all validated with no blocking finding and stay consistent with the source record and generated views.** A pass means the eleven-route fixture, the exported bundle, and the copied source-module digests all validated with no blocking finding, and that this page stays consistent with its source record, generated diagram, and Atlas card.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The narrow strongest claim\n\n**The strongest claim is narrow: a first-run route map whose commands, docs refs, and evidence refs resolve against public source, with an ordinal front-door sequence and five named rejections.** The strongest claim this component supports is narrow: a public first-run route map whose commands, docs refs, and evidence refs each resolve against real public source, whose front-door sequence is ordinal sorted, and whose five negative cases are rejected with named codes. Its `AUTHORITY_CEILING` sets `route_registry_authority`, `source_mutation_authorized`, `release_authorized`, `provider_calls_authorized`, `private_data_equivalence_claim`, and `whole_system_correctness_claim` all to false.\n\n###### What the route map cannot become\n\n**It is not route registry control, mutates no source, calls no providers, excludes launch, and cannot promote the documentation sequence into proof beyond its public fixtures and records.** It refuses to become route registry control, to change source files projects, to use external model services, or to include launch operations or public sharing. It does not establish the whole system correct, and it does not claim equivalence to any non-public data. The route map can tell a cold reader what to run first and which record bounds that run. 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Spine","relative_href":"docs/paper-module-proof-diagnostic-evidence-spine.html","read_href":"docs/paper-module-proof-diagnostic-evidence-spine.html","context_href":"docs/rules-and-ideas.html#paper-module-proof-diagnostic-evidence-spine","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"The public proof-diagnostic evidence spine fixture validates Ring2 diagnostic result record refs, copied runtime artifact digests, provider/proof-body exclusions, stale-coupling visibility, and scope limits without becoming formal proof authority.","summary":"Proof Diagnostic Evidence Spine is the public evidence membrane before formal proof authority. It validates Ring2 failure-taxonomy and graph-update artifacts, verifier-trace repair and evidence-cell result record refs, copied runtime artifact digests, model-output data policy rows, negative cases, and a copied public component source-body floor while keeping proof bodies and provider output bodies out of public result records.","authority_ceiling":"Public diagnostic result record refs, copied Ring2 runtime artifacts, and copied public component source body only; no Lean/Lake execution, formal proof authority, formal-result correctness, external model access authority, runtime correctness, launch-scope decision, publishing-scope decision, whole-system equivalence, or whole-system correctness.","abstract":"The public proof-diagnostic evidence spine fixture validates Ring2 diagnostic result record refs, copied runtime artifact digests, provider/proof-body exclusions, stale-coupling visibility, and scope limits without becoming formal proof authority.","teleology":{"section_id":"","text":"The public proof-diagnostic evidence spine fixture validates Ring2 diagnostic result record refs, copied runtime artifact digests, provider/proof-body exclusions, stale-coupling visibility, and scope limits without becoming formal proof authority.","source":"capsule_summary_line"},"body_markdown":"** `proof_diagnostic_evidence_spine` sits before formal proof work and recomputes which earlier diagnostic records may count as evidence, writing a board of accepted and rejected checks with reasons.** `proof_diagnostic_evidence_spine` is the component that sits one step before formal proof work and decides which diagnostic records are allowed to count as evidence. It reads the records left by earlier formal-math evaluation and premise-retrieval steps, recomputes whether each one actually holds, and writes a board that lists which checks were accepted, which were rejected, and why.\n\n**Rather than trust a row's self-declared pass, it resolves and re-hashes the named files and opens the cited result records, so a pass means evidence is present and coupled, not correct mathematics.** It exists because a diagnostic record can lie by omission. A check row can name a real failure-taxonomy report and a real result record, declare itself passing, and be believed on its own word. This component does not trust that word. It resolves the files the row names, re-hashes them, opens the result records the row cites, and only then decides. A pass here means the named evidence is present and coupled. It does not mean the underlying mathematics is correct.\n\n###### metadata-only diagnostic evidence records\n\n**The output is a metadata-only record set carrying no proof or provider-output body, because the component judges and arranges existing records rather than running a proof checker itself.** The result is a metadata-only set of records: proof-evidence checks, a model-output data policy result, a diagnostic board, and a fixture sign-off record. None of them carry a proof body or a provider output body. The component arranges and judges existing records. It never runs a proof checker itself.\n\n##### Purpose\n\n**Evidence inflation is cheap to write: trusting a row's own `expected_result` label would let a drifted digest, an unrelated result record, or a mislabelled check pass silently.** The problem is evidence inflation. A row that says \"this check passed\" is cheap to write and expensive to trust. If the spine accepted the row's own `expected_result` label, a drifted digest, a result record that never mentions the ref it claims, or a check labelled as failure-taxonomy evidence while pointing at an unrelated file would all pass silently.\n\n###### Rederiving the verdict, not reading labels\n\n**The component asks per row whether the check recomputes against the system it names, ignoring the fixture's pass label because the label is the thing under suspicion.** The component answers one question per row: does this check recompute against the system it names, or is it asserting more than its refs support. A loose gate that reads labels fails because the label is the thing under suspicion. The spine ignores the fixture's pass label and rederives the verdict.\n\n##### How it works\n\n###### Two entry points and the shared pipeline\n\n** `run` drives the base fixture and `run_evidence_bundle` the exported bundle with added artifact checks, both scanning inputs, running the validators in order, and merging observed negative cases before writing records.** The runtime is `src/microcosm_core/organs/proof_diagnostic_evidence_spine.py`. Two entry points drive it. `run` handles the base fixture. `run_evidence_bundle` handles the exported evidence bundle and adds copied-artifact and source-body checks. Both load their inputs, run a secret-exclusion scan over them, call the validators below in order, merge the negative cases each validator observed, then write the records.\n\n###### The per-check verdict in `_classify_evidence_check`\n\n** `_classify_evidence_check` resolves and re-hashes each source ref, confirms every cited result record truly references it, applies a semantic floor, and accepts only when all of that holds with no negative code declared.** The core of a per-check verdict is `_classify_evidence_check`. For each `source_ref` on a check row it resolves a real public file with `_resolve_public_ref`, hashes the bytes with `_sha256_file`, and compares that hash to the expected digest the module holds in `SOURCE_DIGESTS`. It opens each result record anchor and confirms the result record payload actually contains the source ref and its digest, so a check is result record-backed only when the result record it cites genuinely references it. On top of that it applies a semantic floor through `_check_semantic_floor`: a check whose id mentions `failure_taxonomy` must resolve to a source carrying a failure-taxonomy report and a result record carrying a failure-mode ledger; a `graph_update` check needs graph-update candidates with ids and a matching anchor. The row is accepted only when every source resolves, every digest matches, every cited result record backs the ref, the semantic floor is satisfied, and no expected-negative error code is declared.\n\n###### Running the classifier over every check\n\n** `validate_evidence_receipts` runs the classifier over every check, stamps any self-declared pass label as non-authoritative, and reports its own verdict as accepted ids or a recompute-failed finding.** `validate_evidence_receipts` runs that classifier over every check. It records `legacy_expected_result_label_present` and stamps any such label as `ignored_non_authoritative_fixture_label`, then reports the verdict it computed itself. Accepted checks land in `accepted_check_ids`, rejected ones become an `EVIDENCE_RECEIPT_ANCHOR_RECOMPUTE_FAILED` finding with its rejection reasons.\n\n###### Advisory metadata kept, authority denied\n\n** `validate_provider_payload_policy` rejects any provider or oracle row carrying a forbidden body key or claiming oracle authority, while a clean row keeps its advisory metadata under a one-sided retain-metadata, deny-authority rule.** `validate_provider_payload_policy` handles provider and oracle advisory rows. It walks each payload with `_forbidden_body_key_paths` looking for any key in `FORBIDDEN_BODY_KEYS`, a fixed list that includes `proof_body`, `ground_truth_proof`, `provider_output_body`, `oracle_payload_body`, and `prompt`. A row with a forbidden key, or one that claims oracle authority, is rejected and kept as a regression fixture. A clean row keeps its advisory metadata and is marked `advisory_metadata_preserved`. The rule is one-sided: metadata is retained, authority is denied.\n\n###### Rejecting authority upgrades, keeping stale coupling\n\n** `validate_authority_ceiling` rejects a board's source-authority and a check's runtime-correctness upgrades, while `validate_stale_source_coupling` keeps a drifted fingerprint as evidence and marks it stale.** `validate_authority_ceiling` rejects two upgrade attempts. A diagnostic board that sets `claims_source_authority` produces `DIAGNOSTIC_BOARD_AUTHORITY_UPGRADE`. A passing check that sets `claims_runtime_correctness` produces `EVIDENCE_PASS_OVERCLAIMS_RUNTIME_CORRECTNESS`. `validate_stale_source_coupling` keeps a drifted fingerprint as evidence rather than dropping it, emitting `PROOF_RECEIPT_SOURCE_COUPLING_STALE` and setting `source_fingerprint_status` to stale.\n\n###### Digest-checking the copied Ring2 artifacts\n\n**On the bundle path, `validate_copied_macro_body_artifacts` re-hashes each copied Ring2 artifact and passes only when all thirteen targets are present and digest-matched, blocking on any mismatch.** On the bundle path, `validate_copied_macro_body_artifacts` re-hashes each copied Ring2 runtime artifact against `PUBLIC_RING2_ARTIFACT_IMPORTS`. It passes only when all thirteen expected targets are declared, present, and digest-matched under either the exact-copy or the source-faithful public-light-edit policy. Any missing file or digest mismatch blocks the status.\n\nFunction Role `run` Base fixture orchestrator: load, scan, validate, write records `run_evidence_bundle` Exported-bundle path with freshness reuse and copied-artifact checks `validate_evidence_receipts` Recompute every check row, ignore its self-declared pass label `validate_provider_payload_policy` Keep advisory metadata, deny authority, reject forbidden-body rows `validate_authority_ceiling` Reject board source-authority and runtime-correctness upgrades `validate_copied_macro_body_artifacts` Digest-check the copied Ring2 runtime artifacts Diagram of the mechanism (7 steps). Diagram source & refs\n\n```text\nflowchart TD Check[\"Check row source_refs, receipt_anchor_refs, source_digest_refs\"] Resolve[\"Resolve each source ref re-hash file (sha256)\"] Result record[\"Open result record anchor does it contain this ref and digest?\"] Floor[\"Semantic floor failure-taxonomy / graph-update source and result record match\"] Accept[\"Accepted verdict recomputed\"] Reject[\"Rejected, kept as diagnostic evidence\"] Board[\"diagnostic board evidence accounting only\"] Check --> Resolve --> Result record --> Floor Floor -->|all agree| Accept Floor -->|any mismatch| Reject Accept --> Board Reject --> Board\n```\n\n##### Negative cases\n\n###### Six negative inputs the run must observe\n\n**The fixture declares six negative inputs the run must observe, each mapped to its error codes, and the run blocks if any is missing.** The fixture declares the negative cases the run must observe, listed in `EXPECTED_NEGATIVE_CASES`. A run is blocked if any of them is missing:\n- `provider_proof_body_payload_rejected`: a model-output data carrying a forbidden proof body, giving `FORBIDDEN_PROOF_BODY` and `PROVIDER_PAYLOAD_NOT_AUTHORITY`.\n- `oracle_payload_claims_authority_rejected`: an oracle payload claiming authority, giving `ORACLE_PAYLOAD_BODY_FORBIDDEN` and `ORACLE_PAYLOAD_NOT_AUTHORITY`.\n- `evidence_receipt_missing_required_fields`: a result record lacking a validator id or scope boundary, giving `MISSING_VALIDATOR_ID` and MISSING_RECEIPT_ANTI_CLAIM.\n- `diagnostic_board_claims_source_authority`: giving `DIAGNOSTIC_BOARD_AUTHORITY_UPGRADE`.\n- `stale_proof_receipt_source_coupling`: a drifted source fingerprint, giving `PROOF_RECEIPT_SOURCE_COUPLING_STALE`.\n- `passing_check_overclaims_runtime_correctness`: giving `EVIDENCE_PASS_OVERCLAIMS_RUNTIME_CORRECTNESS`.\n\n##### Prior Art Grounding\n\n###### Assurance cases and provenance discipline\n\n**The component follows assurance-case practice, tying evidence to a claim and its limits, and borrows result record discipline from GSN, W3C PROV, and observability, treating a diagnostic as an evidence cell rather than a proof.** The component follows assurance-case practice: evidence is tied to a claim, its assumptions, and its limits before it is treated as support. NASA's Goal Structuring Notation example for spacecraft assurance frames assurance as model-structured evidence rather than document-level persuasion, [NTRS 20160005295](https://ntrs.nasa.gov/citations/20160005295). The result record discipline also borrows from [W3C PROV](https://www.w3.org/TR/prov-overview/) and from observability practice, where a diagnostic artifact is an evidence cell with provenance and not a proof of correctness.\n\n##### Validation Result record Path\n\n###### Fixture and bundle validation commands\n\n**Run these from the repository root. Neither command mutates a durable record or a generated projection:** Run these from the repository root. Neither command mutates a durable record or a generated projection:\n\n```text\n./repo-pytest tests/test_proof_diagnostic_evidence_spine.py -q --basetemp=/tmp/microcosm_proof_diagnostic_evidence_spine_pytest\n./repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n###### What a pass means\n\n**Clearing this validation shows the fixture's expected refusals all fired, but proves nothing about the underlying mathematics or any provider's authority.** A pass means the observed negative cases match `EXPECTED_NEGATIVE_CASES`. It does not establish formal-result correctness or provider authority.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### Bounded evidence accounting\n\n**The strongest claim is bounded evidence accounting: over the fixture inputs, the component recomputes which refs, digests, result record anchors, and negative cases hold and refuses to let a row inflate past that.** The strongest claim the evidence supports is bounded evidence accounting: for the fixture inputs, this component recomputes which public refs, digests, result record anchors, and negative cases hold, and refuses to let a row inflate itself past that.\n\n###### What it refuses to establish\n\n**It runs no Lean or Lake, calls no providers, exposes no proof or provider-output body, and proves no runtime correctness, with `PROOF_AUTHORITY_CEILING` and `FORBIDDEN_BODY_KEYS` encoding that boundary in code.** It refuses the rest. It does not run Lean or Lake, it does not use external model services, it does not expose proof bodies or provider output bodies, and it does not establish that the imported runtime is correct. Its constants `PROOF_AUTHORITY_CEILING` and `FORBIDDEN_BODY_KEYS` encode that boundary in code. 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Authorization","relative_href":"docs/paper-module-proof-derived-governed-mutation-authorization.html","read_href":"docs/paper-module-proof-derived-governed-mutation-authorization.html","context_href":"docs/rules-and-ideas.html#paper-module-proof-derived-governed-mutation-authorization","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"The public proof-derived governed-mutation fixture validates synthetic mutation proposals through proof cells, visible pre-execution policy verdicts, logged side effects, rollback result records, cold replay, negative cases, and copied source internal control bodies without granting live mutation authority.","summary":"Proof-Derived Governed Mutation Authorization is the public mutation-authority replay contract. It checks three synthetic proposals, proof-cell validator refs, visible policy verdicts, side-effect logs, rollback result records, cold replay, eight negative cases, and six exact copied source pattern/result record/internal control bodies while keeping account secrets, proof bodies, model-output data, account refs, source-file changes, and launch-scope decision out of result records.","authority_ceiling":"Public synthetic governed-mutation result records and copied source body digest evidence only; no standing account secret authority, live cloud/account mutation, source-file changes, external model access, hidden vote, policy-after-execution authority, benchmark security claim, launch-scope decision, public sharing, hosting, or whole-system correctness.","abstract":"The public proof-derived governed-mutation fixture validates synthetic mutation proposals through proof cells, visible pre-execution policy verdicts, logged side effects, rollback result records, cold replay, negative cases, and copied source internal control bodies without granting live mutation authority.","teleology":{"section_id":"","text":"The public proof-derived governed-mutation fixture validates synthetic mutation proposals through proof cells, visible pre-execution policy verdicts, logged side effects, rollback result records, cold replay, negative cases, and copied source internal control bodies without granting live mutation authority.","source":"capsule_summary_line"},"body_markdown":"**The component replays whether a proposed change should be allowed to run, checking synthetic proposals against their evidence and writing a metadata-only record of what was admitted and why.** This component replays a make-believe question that an agent system has to answer safely: should this proposed change be allowed to run? It takes a small bundle of synthetic proposals and their supporting evidence, checks each proposal against that evidence, and writes a metadata-only result record that shows which proposals were admitted and why.\n\n###### Three proposals under one authorization replay\n\n**Three proposals, an inspection, a scoped write, and its undo, are admitted only when an independent evidence chain resolves, so wording, identity, hidden votes, and bare success claims all fail.** The bundle holds three proposals: a look-only inspection, a small scoped config write, and an undo of that write. A proposal is admitted only when an independent evidence chain resolves for it. Impressive wording, an admin-looking identity, a hidden vote, or a final answer that just says it worked all fail on their own.\n\n###### Entry points and emitted authorization evidence\n\n**The component runs via `run` on the fixture or `run_authorization_bundle` on the exported example, emitting counts, refs, hashes, and verdicts but never proof bodies, votes, or account secrets.** The component is at `src/microcosm_core/organs/proof_derived_governed_mutation_authorization.py`. Run it with `run` for the fixture, or `run_authorization_bundle` for the exported example. The result record carries counts, refs, hashes, and verdicts, never proof bodies, policy votes, or account secrets.\n\n##### Purpose\n\n###### The action that grants itself permission\n\n**The danger is a step that authorizes itself by claiming a standing account secret, recording a hidden governance-vote, or reporting success after the fact, all of which a trusting check would admit.** The danger this guards against is an action that grants itself permission. In an agent system a step can claim a standing account secret, record a governance-vote nobody can see, or report success after the fact. A loose check that trusts the proposal's own summary would admit all three.\n\n###### Authorization derived from resolved evidence files\n\n**Each admitted proposal must resolve against separate evidence files, and a recomputed evidence-chain hash must match the declared one, so a hand-written hash constant cannot survive.** Authorization here is derived, not asserted. Each admitted proposal must resolve against separate evidence files: redacted proof cells that name validator result refs, at least two visible policy verdicts evaluated before any execution identity, a logged side-effect diff for the two changing actions, a matching undo record, and a cold-replay pass. The component recomputes an evidence-chain hash from those resolved rows and rejects the proposal when the declared hash does not match. A hand-written hash constant cannot survive that step.\n\n###### The anti-bake gate binds to a real commit\n\n**The anti-bake gate requires each proposal to bind to a real git commit that touched this component's source or test, with its refs re-derived so a fixture cannot pre-write its own answer.** The harder part is the anti-bake gate. Passing the synthetic chain is not enough. Every admitted proposal must also bind to a real repository record: a concrete git commit that the component resolves with a `git` subprocess and checks touched this component's own source or its focused test. The declared proof, policy, and undo refs on that record are re-derived from the evidence files and compared, so a fixture cannot pre-write its own answer.\n\n##### How it works\n\n###### The two entry points share one validator order\n\n** `run` validates the fixture with negative cases and `run_authorization_bundle` validates the exported example without them, both calling the same validators in the same order.** `run` loads the fixture input directory with negative cases included and writes the durable result set. `run_authorization_bundle` runs the exported example without the negative cases and writes one bundle result file. Both call the same validators in the same order.\n\n###### Per-proposal checks in `_validate_proposal_row`\n\n** `_validate_proposal_row` checks the fourteen required fields against the other files, recomputes the evidence-chain hash, and rejects an undo-less write, thin verdicts, or an action outside the three fixed classes.** Each proposal must carry the fourteen fields in `REQUIRED_PROPOSAL_FIELDS`, including `proof_cell_refs`, `policy_verdict_refs`, `policy_evaluated_before_execution`, `side_effect_class`, `evidence_chain_hash`, and `cold_replay_ref`. `_validate_proposal_row` checks each field against the other files rather than against the proposal's own claim. It recomputes the hash through `_proposal_evidence_chain_hash` over the resolved proof digests, policy digests, side-effect ref, undo ref, and cold-replay ref. A write or undo proposal with no undo ref is rejected as an irreversible change. Fewer than two visible allow or warn verdicts from distinct evaluators blocks the proposal. The three action classes admitted are fixed in `REQUIRED_ACTION_CLASSES`: `read_only_inspection`, `scoped_config_write`, and `rollback`.\n\n###### A gate for each supporting evidence file\n\n**Each supporting file has its own gate over proof cells, policy verdicts, side-effect rows, undo records, and cold-replay passes, with a minimum count and a body-redacted requirement on each.** The supporting files each have their own gate. `validate_proof_evidence_cells` needs three or more cells, each naming its proposal, carrying evidence refs and validator result refs, and marked body-redacted with no exported proof body. `validate_policy_verdicts` needs six or more verdicts and at least two visible allow or warn verdicts per proposal, each resolving back to a proof cell. `validate_side_effect_ledger` needs exactly two logged, reversible, synthetic side-effect rows. `validate_rollback_receipts` needs at least two passing undo records that do not claim irreversible authority. `validate_cold_replay` needs at least three redacted cold-replay passes.\n\n###### The real-record gate in `validate_governed_mutation_records`\n\n** `validate_governed_mutation_records` resolves each commit with `_git_commit_evidence`, sets `scope_verified` only when a changed file is this component's source or test, and downgrades any proposal without an accepted record.** `validate_governed_mutation_records` runs the real-record gate. A record must declare a repo record class, name a proposal that exists, and carry a commit ref that `_git_commit_evidence` resolves. That helper runs `git rev-parse` and `git show --name-only`, then sets `scope_verified` only when a changed file ends with this component's source or test path. The record's declared proof, policy, and undo refs are re-derived from the evidence indices and must match. A proposal whose id is not in the accepted real-record set is downgraded from admitted to blocked.\n\n###### Digest check on copied internal control bodies\n\n** `_source_module_manifest_result` verifies each copied internal control body by sha256 against its declared digest, and the summary holds every body-in-result record flag false so only refs and hashes are reported.** `_source_module_manifest_result` checks the copied internal control bodies shipped with the exported example. It reads `source_module_manifest.json`, confirms each row is classed as copied body material, verifies each target file exists under `source_modules/`, and recomputes its sha256 against the declared digest and required anchors. `_source_open_body_import_summary` projects that result and holds `body_in_receipt`, `body_text_in_receipt`, and `body_text_exported_in_receipts` at false, so refs and hashes are reported but body text is not.\n\nFunction Role `run`, `run_authorization_bundle` Fixture and exported-example entry points that write the result set `validate_mutation_proposals`, `_validate_proposal_row` The authorization predicate for each proposal `_proposal_evidence_chain_hash` Recomputes the evidence-chain hash from resolved rows `validate_proof_evidence_cells` Redacted proof cells with evidence and validator refs `validate_policy_verdicts` Visible allow or warn verdicts that resolve to proof `validate_side_effect_ledger` Two logged reversible synthetic side effects `validate_rollback_receipts` Passing undo records without irreversible claims `validate_cold_replay` Redacted cold-replay passes per proposal path `validate_governed_mutation_records`, `_git_commit_evidence` Real-record anti-bake gate over a resolved git commit `_source_module_manifest_result`, `_source_open_body_import_summary` Digest check on copied internal control bodies, body text kept out `EXPECTED_NEGATIVE_CASES`, `AUTHORITY_CEILING` The falsification set and the scope limit Diagram of the mechanism (7 steps). Diagram source & refs\n\nSource refs\n\nread-only, scoped write, undo `mutation_proposals.json` real commit ref `governed_mutation_records.json`\n\n```text\nflowchart TD Proposals[\"mutation_proposals.json read-only, scoped write, undo\"] Evidence[\"proof cells, policy verdicts, side effects, undo records, cold replay\"] Hash{\"declared hash == derived hash?\"} Records[\"governed_mutation_records.json real commit ref\"] AntiBake{\"git commit touched source/test? re-derived refs match?\"} Result[\"metadata-only result record counts, refs, hashes, verdicts\"] Ceiling[\"scope_limit no account secrets, no live change\"] Proposals --> Evidence Evidence --> Hash Hash -->|match| AntiBake Hash -->|mismatch| Result Records --> AntiBake AntiBake -->|bound| Result AntiBake -->|unbound| Result Result --> Ceiling\n```\n\n##### Negative cases\n\n###### Eight rejections mapped to `GOV_MUT_*` codes\n\n** `EXPECTED_NEGATIVE_CASES` lists eight rejections, each mapped to a `GOV_MUT_*` code, from standing account secrets and hidden votes to unlogged side effects and final-answer-only success.** `EXPECTED_NEGATIVE_CASES` lists eight rejections, each mapped to a `GOV_MUT_*` error code. A proposal that claims a standing account secret fails GOV MUT STANDING account secret AUTHORITY. A proposal evaluated by policy after it ran fails `GOV_MUT_POLICY_AFTER_EXECUTION`. A hidden vote fails `GOV_MUT_HIDDEN_POLICY_VOTE`. A live cloud account secret fails GOV MUT LIVE CLOUD account secret. A change with no undo ref fails `GOV_MUT_IRREVERSIBLE_MUTATION`. An unlogged side effect fails `GOV_MUT_UNLOGGED_SIDE_EFFECT`. Consensus with no resolving evidence fails `GOV_MUT_CONSENSUS_WITHOUT_EVIDENCE`. A proposal graded only by a final-answer success fails `GOV_MUT_FINAL_ANSWER_ONLY_SUCCESS`.\n\n###### Each rejection turns on evidence, not on labels\n\n**The eight cases are the security argument: each isolates one overclaim and fails because the public evidence tables do not resolve, never because the proposal looked suspicious.** These eight are the security argument. Each isolates one overclaim: a standing account secret, an after-the-fact policy check, a hidden vote, a live cloud account secret, a write with no undo ref, an unlogged side effect, a missing evidence chain, or a bare success claim. None of them depends on a suspicious label. Each fails because the public evidence tables the predicate checks do not resolve, not because the proposal looked wrong.\n\n##### Prior Art Grounding\n\n###### Admission control and policy-as-code lineage\n\n**The shape follows admission-control and policy-as-code practice from Open Policy Agent and Kubernetes admission controllers, with undo and side-effect parts adjacent to controlled-rollout feature-flag patterns, kept synthetic and replay-only.** The shape follows admission-control and policy-as-code practice: a proposed state change is evaluated before it mutates the system, and the decision is separate from the actor's own assertion. The closest public anchors are [Open Policy Agent](https://www.openpolicyagent.org/docs), which separates policy decision-making from enforcement over structured input, and Kubernetes [admission controllers](https://kubernetes.io/docs/reference/access-authn-authz/admission-controllers/), which validate or mutate API requests before they persist. The undo and side-effect parts are adjacent to controlled-rollout practice, including the feature-flag patterns described by [Martin Fowler](https://martinfowler.com/articles/feature-toggles.html). This component keeps the pattern synthetic and replay-only.\n\n##### Validation Result record Path\n\n###### Reader validation without durable writes\n\n```text\n./repo-pytest tests/test_proof_derived_governed_mutation_authorization.py -q --basetemp=/tmp/microcosm_proof_derived_governed_mutation_authorization_pytest\n./repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n###### What a pass means\n\n**A pass means the focused tests hold and this Markdown still satisfies the shared coverage contract, not that any real change was authorized.** A pass means the focused tests hold and this Markdown still satisfies the shared paper-module coverage contract. It does not mean any real change was authorized.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The bounded claim the evidence supports\n\n**On a synthetic bundle, three proposals were admitted only after a full evidence chain, matching hash, and a scoped real commit, while the same predicate rejects eight named overclaims.** The strongest claim the evidence supports is a bounded one: for a synthetic bundle, three proposals were admitted only after proof cells, two visible resolving policy verdicts, logged side effects, undo records, cold-replay passes, an evidence-chain hash that matched its recomputation, and a real commit that touched this component's own source or test. The negative cases show the same predicate rejecting eight named overclaims. Copied internal control bodies are checked by digest with body text kept out of the result record.\n\n###### What stays unauthorized\n\n**The boundary grants no live mutation, account secrets, cloud reach, source-file changes, external model access, or exported body, with `AUTHORITY_CEILING` recording each as not authorized and a green result staying fixture evidence.** That is the whole proof boundary. This component does not grant live mutation authority, hold standing account secrets, reach any cloud or account, change source files, call a provider, export a proof body or governance-vote, or claim benchmark safety. `AUTHORITY_CEILING` records every one of those as not authorized. A green result is validator evidence over a fixture, not launch-scope decision, hosting approval, or whole-system correctness.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":14763,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/proof_derived_governed_mutation_authorization.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"Open Policy Agent","url":"https://www.openpolicyagent.org/docs","kind":"external_reference","section_id":"admission-control-and-policy-as-code-lineage"},{"label":"admission controllers","url":"https://kubernetes.io/docs/reference/access-authn-authz/admission-controllers/","kind":"external_reference","section_id":"admission-control-and-policy-as-code-lineage"},{"label":"Martin 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Bundle","url":"paper-module-batch4-proof-authority-runtime.html","relation_id":"paper_module.depends_on.paper_module","source_ref":"core/paper_module_capsules.json::paper_modules[74:paper_module.batch4_proof_authority_runtime].depends_on"},{"id":"paper_module.batch8_policy_engines_capsule","kind":"paper_module","kind_label":"paper module","relation":"depended on by","label":"Set 8 Policy Engines Bundle","url":"paper-module-batch8-policy-engines-capsule.html","relation_id":"paper_module.depends_on.paper_module","source_ref":"core/paper_module_capsules.json::paper_modules[60:paper_module.batch8_policy_engines_capsule].depends_on"}],"source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/proof_derived_governed_mutation_authorization.md"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json"}],"visible_text_byte_count":16268,"section_titles":["The component replays whether a proposed change should be allowed to run","Three proposals under one authorization replay","Entry points and emitted authorization evidence","Purpose","The action that grants itself permission","Authorization derived from resolved evidence files","The anti-bake gate binds to a real commit","How it works","The two entry points share one validator order","Per-proposal checks in `_validate_proposal_row`","A gate for each supporting evidence file","The real-record gate in `validate_governed_mutation_records`","Digest check on copied internal control bodies","Negative cases","Eight rejections mapped to `GOV_MUT_*` codes","Each rejection turns on evidence, not on labels","Prior Art Grounding","Admission control and policy-as-code lineage","Validation Result record Path","Reader validation without durable writes","What a pass means","Scope limit","The bounded claim the evidence supports","What stays unauthorized"]},{"id":"paper_module.durable_agent_work_landing_replay","title":"Durable Agent Work-Landing Replay","relative_href":"docs/paper-module-durable-agent-work-landing-replay.html","read_href":"docs/paper-module-durable-agent-work-landing-replay.html","context_href":"docs/rules-and-ideas.html#paper-module-durable-agent-work-landing-replay","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"The public durable work-landing replay fixture validates recorded agent landing rows, copied source internal control source bodies, validation-before-commit ordering, HEAD-advance evidence, blocker capture, and work log completion without performing live Git work.","summary":"Durable Agent Work-Landing Replay is the public work-spine replay contract for agent landing claims. It checks claimed paths, owner-native validation refs, commit-attempt order, HEAD-before/after evidence, metadata-blocked rows, work log finalizer evidence, nine negative cases, and six exact copied source bodies while keeping raw diffs, non-public paths, model-output data, and source bodies out of result records.","authority_ceiling":"Public synthetic replay result records and copied source work-landing/internal control source bodies only; no live Git mutation, unrelated staging, broad checkpoint authority, arbitrary commit proof, external model access, non-public body export, public sharing, hosting, launch-scope decision, or whole-system correctness.","abstract":"The public durable work-landing replay fixture validates recorded agent landing rows, copied source internal control source bodies, validation-before-commit ordering, HEAD-advance evidence, blocker capture, and work log completion without performing live Git work.","teleology":{"section_id":"","text":"The public durable work-landing replay fixture validates recorded agent landing rows, copied source internal control source bodies, validation-before-commit ordering, HEAD-advance evidence, blocker capture, and work log completion without performing live Git work.","source":"capsule_summary_line"},"body_markdown":"###### Landing language must follow recorded evidence\n\n**An agent's \"I committed the fix\" is a claim, and this component reads recorded landing examples and decides whether each one's evidence supports the words used to describe it.** An agent that says \"I committed the fix\" is making a claim, and the claim is the part that is usually wrong. This component reads recorded examples of an agent finishing a piece of work and decides whether the evidence in each example supports the words used to describe it.\n\n###### Four inputs produce four metadata-only records\n\n**It reads a fixture of landing attempts, a policy, a protocol, and broken cases, plus copied internal control bodies in bundle mode, and emits a metadata-only board, result, validation, and negative-coverage record.** It takes a fixture of recorded landing attempts, a landing policy, a projection protocol, and a set of deliberately broken cases. In bundle mode it also reads the copied bodies of the internal control source files the model was tested against. It emits a metadata-only result set: a board, a result record, a validation record, and a fourth record confirming every expected negative case was caught.\n\n###### Recorded claims only, with no live mutation\n\n**It runs no Git, stages nothing, moves no ledger, and excludes launch: it reads recorded claims and rejects the ones the evidence does not back.** It does not run Git, stage a file, move a ledger, or authorize a launch. It reads recorded claims and rejects the ones the evidence does not back.\n\n##### Purpose\n\n###### Does the evidence support the landing language\n\n**The narrow question is whether a recorded landing's evidence supports its language, since a loose check passes any row that merely describes itself well: a confident row with nothing behind it.** The question is narrow. Given a recorded landing attempt, does the recorded evidence support the landing language? A loose check passes any row that describes itself well, which is exactly the failure mode: a confident row with nothing behind it.\n\n###### HEAD movement and validation order are separate floors\n\n**A landed-commit claim needs recorded HEAD movement, a commit-path claim needs validation before the attempt, and every row still needs blocker and ledger evidence.** The component turns two ordinary rules into rejections. A row that uses landed-commit language is rejected unless the recorded Git HEAD moved between before and after, so \"I landed it\" cannot stand on a HEAD that never advanced. A row on a commit path is rejected unless validation is recorded as preceding the commit attempt, so \"it passed\" cannot be back-filled after the fact. Blocker capture for metadata-blocked rows and work log completion for every row are the other two floors.\n\n###### Copied internal control bodies ground the check\n\n**Six digest-provenanced internal control files let readers inspect the mechanics behind the landing rules instead of trusting a paraphrase of them.** The check is also source-backed. The mechanics it tests rows against are not paraphrased. The six internal control source files are copied into the bundle with digest provenance, so a reader can see which code the model was checked against instead of trusting a summary.\n\n##### How it works\n\n###### Two entrypoints compose the same checks\n\n**Fixture and bundle entrypoints both call `_build_result`, differing only in negative-case inclusion, manifest requirements, and fresh-record reuse.** Two entry points drive the same builder. `run` reads the first-wave fixture with `include_negative=True` and writes four durable records. `run_work_landing_bundle` reads an exported bundle with `include_negative=False`, adds the source-module manifest check, and can reuse a fresh prior record when the input digest is unchanged. Both call `_build_result`, which composes four independent checks and one secret scan into a single verdict.\n\nFunction What it consumes What it checks or rejects `validate_projection_protocol` `projection_protocol.json` Requires the `durable_agent_work_landing_replay_compound` source pattern, at least three source refs, two projection result record refs, and three public runtime refs. `validate_landing_policy` `landing_policy.json` Requires the four lanes in `REQUIRED_LANE_IDS`. Rejects a broad-checkpoint lane that skips operator authorization and any lane that sets `release_authorized`. `validate_work_landing_runs` `work_landing_runs.json` plus the nine negative files Runs `_validate_run_row` over each row and each broken case. `validate_source_module_imports` `source_module_manifest.json` Bundle mode only. Verifies copied internal control bodies by class, relation, digest, and required anchors.\n\n###### Row validation checks evidence and forbidden authority\n\n** `_validate_run_row` requires validation, claimed-path, and completion refs while rejecting stationary HEADs, late validation, uncaptured blockers, private leaks, and authority overclaims.** `_validate_run_row` carries the row-level mechanism. For each recorded run it requires validation refs, claimed path refs, and a work log completion ref. It rejects a `commit_claimed_landed` row whose `head_before` equals `head_after`. It rejects a commit-path row (`landing_status` in `COMMIT_ATTEMPT_STATUSES`) that does not record `validation_precedes_commit_attempt`. It rejects a metadata-blocked row with no blocker capture ref, any row that claims live Git mutation, staged unrelated dirty paths, or launch-scope decision, and any row whose keys or strings leak a private body or non-public paths.\n\n###### Six copied modules must match digest and anchors\n\n** `validate_source_module_imports` accepts the six exact-copy internal control modules only when their class, count, sha256, paths, and required anchors all match the manifest.** `validate_source_module_imports` keeps the source claim honest. The manifest must declare `copied_non_secret_macro_body`, keep `body_in_receipt` false, and match its own module count. Each of the six rows names a internal control file under `system/lib/`, `tools/meta/control/`, or `tools/meta/factory/`, states an `exact_copy` relation, and carries a `sha256`. The check recomputes the digest of the copied target and rejects a mismatch, then rejects the row if any required anchor string is absent from the copied body. The copied files are `workitem_runtime_entrypoint.py`, `work_landing_status.py`, `work_landing.py`, `mission_transaction_preflight.py`, `scoped_commit.py`, and work_ledger.py.\n\n###### The final verdict requires every independent check\n\n** `_build_result` passes only with all checks green, no missing negative case, and no secret hit, then projects four evidence counts into the board.** `_build_result` merges the four checks. The final status is a pass only when no expected negative case is missing, the secret scan has zero blocking hits, and every check passed. `_board_from_result` then projects four mechanic counts: claimed rows, landed-commit rows, validation-before-commit passes, and metadata-blocked rows. On the first-wave fixture that is three claimed rows, one landed-commit row, two commit-path rows that must record validation order, and one metadata-blocked row, with all nine negative cases observed. Every record carries the `AUTHORITY_CEILING` constant, and each finding and copied-module row sets `body_in_receipt: false`.\n\nDiagram of the mechanism (6 steps). Diagram source & refs\n\nSource refs\n\nclaim, validation order, HEAD advance, blocker capture `validate_work_landing_runs` class, relation, digest, anchors `validate_source_module_imports`\n\n```text\nflowchart TD Fixture[\"Fixture inputs projection, policy, runs, nine broken cases\"] Source[\"Copied internal control bodies six files by digest\"] Build[\"_build_result\"] Runs[\"validate_work_landing_runs claim, validation order, HEAD advance, blocker capture\"] Imports[\"validate_source_module_imports class, relation, digest, anchors\"] Records[\"metadata-only records board, result, validation, coverage confirmation\"] Fixture --> Build Source --> Imports Build --> Runs Build --> Imports Runs --> Records Imports --> Records\n```\n\n##### Negative cases\n\n###### Nine broken rows define the refusal floor\n\n**A passing fixture must catch all nine missing-evidence, ordering, mutation, boundary, blocker, launch, and private-path failures through the same row validator.** The fixture ships nine broken rows, one per key in `EXPECTED_NEGATIVE_CASES`. A pass requires all nine to be caught. They are: a row with no validation ref, a row whose completion is missing, a landed-commit claim with no HEAD advance, validation recorded after a commit attempt, a row claiming live Git mutation authority, a row with no dirty-tree boundary, an uncaptured metadata blocker, a launch overclaim, and a non-public paths leak. Catching those refusals is part of the positive claim, not a separate concern.\n\n##### Prior Art Grounding\n\n###### Provenance fields stay distinct through landing\n\n**W3C PROV and SLSA motivate keeping paths, validation, commit attempts, HEAD movement, blocker capture, and ledger completion as separately checked evidence.** The posture comes from provenance and software supply-chain integrity work. The [W3C PROV](https://www.w3.org/TR/prov-overview/) family models the entities, activities, and agents behind an artifact. [SLSA](https://slsa.dev/spec/) tracks source, build process, provenance, and artifact integrity so a consumer can reason about where an artifact came from. This component borrows that separation for agent work landing: claimed paths, validation refs, commit attempts, HEAD before and after, blocker capture, and ledger completion are distinct evidence fields, checked one at a time.\n\n##### Validation Result record Path\n\n###### Two read-only commands verify behavior and parity\n\n```text\n./repo-pytest tests/test_durable_agent_work_landing_replay.py -q --basetemp=/tmp/microcosm_durable_agent_work_landing_replay_pytest\n./repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n###### A pass joins negative coverage and source integrity\n\n**The test pass requires every negative case, a clean secret scan, and matching copied-source digests; the corpus pass establishes projection sign-off only.** The first runs the component over the fixture and its nine broken cases. A pass means every negative case was caught, the secret scan was clean, and the copied source digests matched.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The supported claim is fixture conformance\n\n**The fixture supports a conformance claim that every landing row carries the required paths, ordering, HEAD movement, blocker capture, completion, source digests, and negative-case coverage.** The strongest claim the evidence supports is a conformance claim. On the public fixture and exported bundle this component shows that each recorded landing row carries the required evidence and ordering: claimed paths, validation before commit, a HEAD advance behind any landed-commit language, blocker capture for metadata-blocked rows, and work log completion throughout, with six internal control bodies copied by matching digest and nine negative cases caught.\n\n###### The recorded fixture grants no live authority\n\n**The scope limit excludes live Git or ledger mutation, proof that a real commit landed, broad checkpoints, external model access, public sharing, launch, and whole-system correctness.** The proof boundary stops at the recorded fixture and its records. The scope limit is the `AUTHORITY_CEILING` constant: no live Git mutation, no proof that any commit truly landed in a real repository, no live ledger mutation, no broad checkpoint authority, no source-file changes, no external model access, no public sharing or launch, and no whole-system correctness. Broader work-continuity behavior lives in sibling components such as `bridge_phase_continuity_runtime` and in the internal control source files themselves. This module can cite their copied bodies as evidence; it cannot inherit their authority.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":12047,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/durable_agent_work_landing_replay.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"W3C PROV","url":"https://www.w3.org/TR/prov-overview/","kind":"external_reference","section_id":"provenance-fields-stay-distinct-through-landing"},{"label":"SLSA","url":"https://slsa.dev/spec/","kind":"external_reference","section_id":"provenance-fields-stay-distinct-through-landing"}],"grouped_relation_feeds":[{"relation":"abides by","count":2,"links":[{"id":"AX-4","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Content-addressed 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Spine","url":"paper-module-mission-transaction-work-spine.html","relation_id":"paper_module.depends_on.paper_module","source_ref":"core/paper_module_capsules.json::paper_modules[20:paper_module.mission_transaction_work_spine].depends_on"},{"id":"paper_module.work_landing_control_spine","kind":"paper_module","kind_label":"paper module","relation":"depended on by","label":"Work Landing Control Spine","url":"paper-module-work-landing-control-spine.html","relation_id":"paper_module.depends_on.paper_module","source_ref":"core/paper_module_capsules.json::paper_modules[17:paper_module.work_landing_control_spine].depends_on"}],"source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/durable_agent_work_landing_replay.md"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json"}],"visible_text_byte_count":13352,"section_titles":["Landing language must follow recorded evidence","Four inputs produce four metadata-only records","Recorded claims only, with no live mutation","Purpose","Does the evidence support the landing language","HEAD movement and validation order are separate floors","Copied internal control bodies ground the check","How it works","Two entrypoints compose the same checks","Row validation checks evidence and forbidden authority","Six copied modules must match digest and anchors","The final verdict requires every independent check","Negative cases","Nine broken rows define the refusal floor","Prior Art Grounding","Provenance fields stay distinct through landing","Validation Result record Path","Two read-only commands verify behavior and parity","A pass joins negative coverage and source integrity","Scope limit","The supported claim is fixture conformance","The recorded fixture grants no live authority"]},{"id":"paper_module.work_landing_control_spine","title":"Work Landing Control Spine","relative_href":"docs/paper-module-work-landing-control-spine.html","read_href":"docs/paper-module-work-landing-control-spine.html","context_href":"docs/rules-and-ideas.html#paper-module-work-landing-control-spine","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"The public work-landing control spine validates copied work-landing internal control source bodies without authorizing live Git, ledger, claim-launch, private-index, public sharing, or launch operations.","summary":"Work Landing Control Spine is the public source-open import for work-landing internal control bodies. It validates copied work_landing, work_landing_status, mission transaction preflight, landing preflight, and scoped-commit source modules by manifest digest, required anchors, no-live-mutation contract flags, secret-exclusion scan, and metadata-only result record policy while keeping live Git, work log, work log, claim launch, shared-index mutation, private-index execution, external model access, public sharing, and launch-scope decision out of scope.","authority_ceiling":"Copied source internal control source bundle, validator result record, source-manifest digest evidence, required-anchor evidence, no-live-mutation contract flags, and secret-exclusion evidence only; no live Git mutation, live work log or work log mutation, claim launch, shared-index mutation, private-index commit execution, broad staging, external model access, account secret export, public sharing, launch-scope decision, or whole-system correctness.","abstract":"The public work-landing control spine validates copied work-landing internal control source bodies without authorizing live Git, ledger, claim-launch, private-index, public sharing, or launch operations.","teleology":{"section_id":"","text":"The public work-landing control spine validates copied work-landing internal control source bodies without authorizing live Git, ledger, claim-launch, private-index, public sharing, or launch operations.","source":"capsule_summary_line"},"body_markdown":"**With many agents editing at once, the hard problem is landing one finished commit without sweeping up a neighbour's half-done edits, solved by internal control modules copied into a public bundle.** The larger source project this public folder draws from is edited by several agents at the same time. Its hardest everyday problem is dull: how does one agent land a small, finished commit without sweeping up a neighbour's half-done edits in the same files, and without claiming more than it actually did. The answer lives in a handful of internal control source modules. A separate import step copies those bodies, non-secret, into a public bundle so a reader can inspect the real mechanics instead of taking a description on trust.\n\n** `work_landing_control_spine` guards that copy, confirming five copied modules are exact, their load-bearing logic is present, the contract claims no live authority, and nothing secret leaked.** `work_landing_control_spine` is the check that guards that copy. It reads a bundle of five copied source modules plus the manifests and runtime contract that describe them, confirms the copies are exact, confirms the load-bearing logic is still present, confirms the runtime contract claims no live authority, and confirms nothing secret leaked in. It then writes one result record.\n\nIt is a witness over copied source, not a runner.\n\n##### Purpose\n\n###### A described control plane is easy to fake\n\n**A prose description of a control plane is easy to fake, so this component copies the actual bodies and asks whether they are genuine, load-bearing, and within their limits.** A description of a control plane is easy to fake. A reader cannot tell whether \"we validate scoped commits\" means the real actuator or a paragraph of prose. So this component copies the actual bodies into the public bundle and then answers one narrow question: are these copied bodies the genuine ones, is their load-bearing logic still present, and does the bundle stay inside its stated limits.\n\nThe copied set is five modules:\n- `tools/meta/control/work_landing.py` (the command parser and admission gate)\n- `system/lib/work_landing_status.py` (controller action order and finalizer policy)\n- `tools/meta/control/mission_transaction_preflight.py` (owned-path preflight)\n- `system/lib/mission_transaction_landing_preflight.py` (shared-index quarantine)\n- `tools/meta/control/scoped_commit.py` (the private-index scoped-commit actuator)\n\n###### Scoped commit and controller ordering\n\n**Two ideas matter: `scoped_commit.py` commits from a throwaway private index that never touches a neighbour's changes, and `work_landing_status.py` fixes the controller action order.** Two ideas in that source are worth a reader's attention. One is the scoped commit: `scoped_commit.py` builds a throwaway Git index from the current `HEAD`, stages only the exact paths an agent declares it owns, writes a tree from that private index, and commits against the captured parent with a compare-and-swap on the branch ref. The shared `.git/index` is never written. An agent cannot pick up a neighbour's dirty changes, because those changes were never in the index it committed from. The other is ordering: `work_landing_status.py` fixes a sequence of controller actions so claims launch only after the session is finalised and convergence recomputes only after claims launch. The check confirms both are still present in the copied bodies, which is what separates an honest copy from a stale bag of hashes.\n\n##### How it works\n\n###### The `validate_work_landing_control_bundle` entry point\n\n** `validate_work_landing_control_bundle` runs the stages in order and returns one result record, wrapped by `main` and leaning on shared JSON, result record, and secret-scan helpers.** The check lives in `src/microcosm_core/macro_tools/work_landing_control_spine.py`. The entry point is `validate_work_landing_control_bundle`, which runs the stages below in order and returns a single result record. `main` and `build_parser` wrap it for the command line. It depends on three shared helpers: the strict JSON reader in `microcosm_core.schemas`, the atomic record writer in `microcosm_core.receipts`, and the secret scanner in `microcosm_core.secret_exclusion_scan`.\n\nThe stages run in this order.\n\n###### Loading the three JSON inputs\n\n**First `_load_json_input` reads the bundle manifest, source manifest, and runtime contract, turning a missing file or non-object JSON into a finding rather than a crash.** First it loads the three JSON inputs with `_load_json_input`: the bundle manifest, the source module manifest, and the runtime contract. A missing file or non-object JSON becomes a finding, not a crash.\n\n###### `_source_manifest_summary` checks copy fidelity\n\n** `_source_manifest_summary` recomputes a SHA-256 and line count for each of the seven required inputs against the manifest, and confirms every material and import class.** Then `_source_manifest_summary` checks copy fidelity. For each of the seven required inputs (five source bodies plus the two JSON files, named in `REQUIRED_INPUTS`) it computes a SHA-256 with `_file_sha256` and a line count with `_line_count`, then compares them against the manifest's expected values. It also confirms each row's `material_class` is one of `ALLOWED_MATERIAL_CLASSES` and that each source body is classified `copied_non_secret_macro_body` (`SOURCE_IMPORT_CLASS`).\n\n###### `_anchor_summary` confirms load-bearing anchors\n\n** `_anchor_summary` confirms every required anchor string in `REQUIRED_SOURCE_REFS` still survives in each copied body, counting any missing anchor rather than dropping it.** Then `_anchor_summary` reads each copied body and confirms every required anchor string is still in the text. The anchors are declared in `REQUIRED_SOURCE_REFS`, keyed by module. For example `work_landing.py` must still contain `build_parser`, `admission-check`, and `begin`; `scoped_commit.py` must still contain `perform_scoped_commit`, `private-index scoped commit`, and the line stating it never invokes `git commit` against the shared index. A missing anchor is counted, not silently dropped.\n\n###### `_contract_summary` checks the scope limit\n\n** `_contract_summary` requires all thirteen `FALSE_AUTHORITY_FLAGS` to be exactly `False`, treating any other value as an overclaim, and confirms the required import classifications.** Then `_contract_summary` checks the scope limit. Every flag in `FALSE_AUTHORITY_FLAGS` (thirteen of them, from `live_git_mutation_authorized` through `publication_authorized` and `release_authorized`) must be exactly `False` in the contract. Any flag that is not `False` is an overclaim. The same stage confirms the contract declares the required import classifications (`REQUIRED_CLASSIFICATIONS`).\n\n###### `scan_paths` runs the secret scanner\n\n** `scan_paths` runs the shared secret scanner over all seven inputs, turning any blocking hit into a finding.** Then `scan_paths` runs the shared secret scanner over all seven inputs. Any blocking hit is a finding.\n\n###### Assembling the metadata-only result record\n\n**Finally the record is assembled, `pass` only when there are no findings, carrying digests, anchor and authority rows, and `body_in_receipt: false` so no body is copied out.** Finally the record is assembled. `status` is `pass` when there are no findings and `blocked` otherwise. The record carries refs, digests, line counts, anchor rows, authority rows, material-class counts, and findings, and it sets `body_in_receipt: false` so no source body is copied into the output. The record is written atomically as `exported_work_landing_control_bundle_validation_result.json` and also printed to stdout.\n\nFunction Role `validate_work_landing_control_bundle` Runs the stages, assembles and writes the result record `_load_json_input` Loads and shape-checks each JSON input, or records a finding `_source_manifest_summary` Digest, line-count, material-class, and import-class parity `_anchor_summary` Confirms required anchor strings survive in each copied body `_contract_summary` Confirms all `FALSE_AUTHORITY_FLAGS` are false and classifications present `build_parser` / `main` Command-line wrapper returning exit 0 on pass, 1 on block Diagram of the mechanism (7 steps). Diagram source & refs\n\n```text\nflowchart TD A[\"Copied bundle 5 bodies + manifest + contract\"] --> B[\"Load JSON inputs _load_json_input\"] B --> C[\"Copy parity _source_manifest_summary\"] C --> D[\"Anchor scan _anchor_summary\"] D --> E[\"Scope limit _contract_summary\"] E --> F[\"Secret scan scan_paths\"] F --> G[\"Result record status pass or blocked\"]\n```\n\n##### Negative cases\n\n###### Five rejections from one-field mutations\n\n**The focused tests drive five rejections by mutating one field at a time, each producing a `blocked` status with a stable finding code, and the same shape covers other drifts.** The focused tests drive five rejections from the accepted bundle by mutating one field at a time. A digest changed in the manifest gives status `blocked` with `SOURCE_DIGEST_MISMATCH`. A source body relabelled to a private material class gives `MATERIAL_CLASS_NOT_ALLOWED`. A body reclassified away from `copied_non_secret_macro_body` gives `SOURCE_IMPORT_CLASS_MISMATCH`. Dropping `secret_exclusion` from the contract's required classifications gives `CLASSIFICATION_FLOOR_MISSING`. Setting `live_git_mutation_authorized` to `True` in the contract gives `AUTHORITY_CEILING_OVERCLAIM`. The check emits the same shape of finding for a missing anchor (`SOURCE_ANCHOR_MISSING`), a missing input (`MISSING_REQUIRED_SOURCE_INPUT`), a line-count drift (`SOURCE_LINE_COUNT_MISMATCH`), and a secret hit (`SECRET_EXCLUSION_BLOCKING_HIT`).\n\n##### Prior Art Grounding\n\n###### Git index and durable-workflow ancestry\n\n**The mechanics come from Git's index separating selected changes and from durable-workflow action ordering, imported here as an auditable public check with no power to act.** The mechanics copied here come from ordinary version-control and workflow engineering. Git's index already separates the changes you have selected from the rest of a dirty worktree, which is the practical ancestor of scoped path ownership. Durable-workflow systems show the value of a recorded action order and deterministic replay for long-running work. This component imports those ideas as a public check: digest parity, anchor presence, a no-live-mutation contract, and a secret scan make the copied control plane auditable without granting it any power to act.\n- Git staging and index workflow: https://git-scm.com/book/en/v2/Git-Basics-Recording-Changes-to-the-Repository\n- Durable workflow event history and replay: https://docs.temporal.io/workflows\n\n##### Validation Result record Path\n\nRun from `microcosm-substrate/`:\n\n```text\nPYTHONPATH=src ../repo-python -m microcosm_core.cli work-landing-control-spine \\\n  validate-control-bundle \\\n  --input examples/work_landing_control_spine/exported_work_landing_control_bundle \\\n  --out receipts/first_wave/work_landing_control_spine\n```\n\n###### What a pass writes and means\n\n**A pass writes the validation result record and means the copied bodies matched the manifest, every anchor was present, all authority flags stayed false, and the secret scan found nothing.** A pass writes `receipts/first_wave/work_landing_control_spine/exported_work_landing_control_bundle_validation_result.json` and means the copied bodies matched the manifest, every required anchor was present, all authority flags stayed false, and the secret scan found nothing. The focused regression suite that pins the pass case and the four rejections above is:\n\n```text\nPYTHONPATH=src ../repo-python -m pytest -p no:cacheprovider \\\n  tests/test_work_landing_control_spine.py -q\n```\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim\n\n**For a given bundle, the five copied bodies match their digests and line counts, the anchor logic is present, the contract declares no live authority, and no secret is present.** The strongest claim the evidence supports is narrow: for a given bundle, the five copied internal control bodies match their recorded digests and line counts, the named anchor logic is still present, the runtime contract declares no live authority, and no secret material is present. That is the whole proof boundary.\n\n###### What the scope limit stops\n\n**The check runs no live Git, mutates no ledger, releases no claim, and calls no provider, so a passing record proves only a faithful copy, never that a future landing is safe.** The scope limit stops there. This check does not run live Git, does not mutate any ledger, does not launch a claim, does not stage broadly, does not execute a private-index commit, does not call a provider, does not export account secrets, and does not publish or authorize a launch. A passing record is evidence that a copy is faithful and honest about its limits. 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supported claim","What the scope limit stops"]},{"id":"paper_module.executable_doctrine_grammar","title":"Executable Doctrine Grammar","relative_href":"docs/paper-module-executable-doctrine-grammar.html","read_href":"docs/paper-module-executable-doctrine-grammar.html","context_href":"docs/rules-and-ideas.html#paper-module-executable-doctrine-grammar","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"The public executable-doctrine grammar fixture validates public standard rows, paper-module sections, negative cases, copied executable-grammar and standards/type-plane source-module bodies, and metadata-only result records without claiming doctrine completeness.","summary":"Executable Doctrine Grammar is the public grammar membrane for doctrine-shaped runtime fixtures. It validates standard row fields, paper-module teleology and result record sections, duplicate-slug and overclaim negative cases, exact copied executable-grammar, standards-registry/type-plane, lattice, kind-atlas, and standards option-surface bodies, then emits metadata-only result records with source refs, digests, counts, and scope limits.","authority_ceiling":"Public fixture, exported standards bundle, and copied source specimen/body digest evidence only; no private standards engine export, source doctrine completeness, source notes export, model-output data export, source-file changes, launch-scope decision, publishing-scope decision, non-public data equivalence, or whole-system correctness.","abstract":"The public executable-doctrine grammar fixture validates public standard rows, paper-module sections, negative cases, copied executable-grammar and standards/type-plane source-module bodies, and metadata-only result records without claiming doctrine completeness.","teleology":{"section_id":"","text":"The public executable-doctrine grammar fixture validates public standard rows, paper-module sections, negative cases, copied executable-grammar and standards/type-plane source-module bodies, and metadata-only result records without claiming doctrine completeness.","source":"capsule_summary_line"},"body_markdown":"###### Component turns doctrine\n\n**This component turns doctrine shape into something a program can pass or fail, checking each standard and paper-module fixture for a stated purpose, governing rule, result records, and honest scope statement.** Executable doctrine grammar is a component that turns doctrine shape into something a program can pass or fail. It reads a folder of example standards and paper-module write-ups and checks each one for the parts doctrine here requires: a stated purpose, a governing rule, the result records it should produce, and an honest statement of what it does not claim. It writes a per-file report saying which entries are well formed and which fail a required check.\n\n###### Report is set\n\n**The report is a set of metadata-only result records carrying references, hashes, counts, and verdicts, and a pass means the fixtures carry the required structure and every expected failure was observed.** The report is a set of metadata-only result records. They carry references, hashes, counts, and verdicts. They never inline the document bodies being checked. A pass means the fixtures carry the required structure and that the checker observed every failure it is supposed to catch.\n\n###### Checks are structural\n\n**The checks are structural, not semantic: they judge whether load-bearing fields are present, so a row cannot quietly drop its result-record expectations or scope statement and still pass.** They judge whether the load-bearing fields are present, so a row cannot quietly drop its result-record expectations or its scope statement and still pass.\n\n##### Purpose\n\n###### Doctrine as prose convention\n\n**In most systems doctrine is prose convention that nothing checks, so a loose fixture reads the part then drops a required field with no signal; this component asks per row whether the structure is real.** Doctrine in most systems is prose convention. A standard says a rule should hold, a paper module says a section should be present, and nothing checks whether the claim is true. A loose fixture passes review because it reads the part, then drops a required field later with no signal. This component removes that gap. It answers one question per row: does this standard or paper-module fixture carry the structure doctrine here requires, or is it text that looks the part?\n\n###### Failure is first class\n\n**The less obvious half is that failure is first class: five negative cases are part of the contract, so a run that never demonstrates a rejection is treated as blocked.** The less obvious half is that failure is first class. Five negative cases are part of the contract. A run that accepts every input but never demonstrates a rejection is treated as blocked, so the checker has to prove it can reject, not only that it can accept.\n\n##### How it works\n\n###### `validate` reads and merges two checks\n\n** `validate` reads `standards_registry.json` and every `paper_modules/*.md` file, scans them for forbidden content classes, then runs two structural checks and merges their results.** The entry point is `validate`. It reads `standards_registry.json` and every `paper_modules/*.md` file from the input directory, scans them for forbidden content classes, then runs two structural checks and merges their results.\n\n###### Standard-registry field and overclaim checks\n\n** `validate_standard_registry` records missing-field, duplicate-slug, prose-authority, and doctrine-completeness overclaim findings per row, and accepts a row only when it has no findings and no expected-failure marker.** `validate_standard_registry` walks each standard row. It records `MISSING_TELEOLOGY`, `MISSING_RECEIPT_EXPECTATIONS`, `MISSING_GOVERNING_STANDARD`, and MISSING_ANTI_CLAIM when those fields are absent. It records `DUPLICATE_STANDARD_SLUG_CONFLICT` when two rows share a slug. It records `PROSE_STANDARD_NOT_EXECUTABLE_AUTHORITY` when a row marked `standard_kind: prose` also claims executable authority, and `GRAMMAR_PASS_OVERCLAIMS_DOCTRINE_COMPLETE` when a row claims doctrine completeness. A row with no findings and no expected-failure marker is accepted.\n\n###### Paper-module heading and sentinel checks\n\n** `validate_paper_modules` checks each fixture module for the `Teleology`, `Governing Standard`, `Receipt Expectations`, and Scope boundary headings and rejects any that carries the copied-body sentinel or a completeness marker.** `validate_paper_modules` calls `validate_paper_module_shape` on each fixture module. That function checks for the `Teleology`, `Governing Standard`, `Receipt Expectations`, and Scope boundary headings, and rejects any fixture that contains the copied-body sentinel text or a doctrine-completeness marker.\n\n###### Merging findings and required negatives\n\n** `validate` passes only when no required negative case is missing and the content scan is clean, writing four result records under result records/sign-off/first_wave/.** `validate` then computes which of the five required negative cases were observed and which are missing. The run status is a pass only when no required case is missing and the content scan is clean. It writes four result records: the standards report, the standards group index, the paper-module report, and the fixture result record under result records/sign-off/first_wave/.\n\n###### Two bundle entry points reuse the checks\n\n** `validate_standards_bundle` and `validate_executable_grammar_metabolism_bundle` reuse the same checks over the exported public standards bundle and the exported specimen, each passing only on a clean scan with valid rows.** Two more entry points reuse the same checks over exported bundles. `validate_standards_bundle` validates the exported public standards bundle and passes only when the scan is clean, there are no findings, and at least one standard row and one paper module are valid. `validate_executable_grammar_metabolism_bundle` validates the exported specimen: it reads `bundle_manifest.json`, `README.md`, `grammar_board.json`, and `receipt.json`, runs `_metabolism_bundle_findings` over their required keys and boundary phrases, then calls the copied-body gate.\n\n###### The copied-body digest gate\n\n** `validate_source_module_imports` requires each of the 12 declared bodies to be a copied, allowlisted, non-result record source body whose SHA-256 recomputes to match the manifest, treating any mismatch as a finding.** `validate_source_module_imports` is that gate. It reads `source_module_manifest.json` and requires `source_import_class` to be `copied_non_secret_macro_body`, `body_in_receipt` to be false, and the declared `module_count` to equal the number of rows. For each of the 12 declared bodies it checks the material class against `public_macro_standard_body`, `public_macro_tool_body`, and `public_macro_receipt_body`, requires `body_copied` true and `body_in_receipt` false, requires the source reference to sit on the allowlist, confirms the copied target file exists, and recomputes its SHA-256 digest with `_sha256_file` to match the manifest. A mismatch is a finding, not a silent pass.\n\n###### `result_card` projects a metadata-only card\n\n** `result_card` projects any result into a compact card of status, counts, and verdicts, dropping the full finding lists and any body-bearing fields named in `CARD_OMITTED_FULL_PAYLOAD_KEYS`.** `result_card` projects any of these results into a compact card for a reader or the Atlas. It keeps status, counts, and verdicts and drops the full finding lists and any body-bearing fields named in `CARD_OMITTED_FULL_PAYLOAD_KEYS`.\n\nFunction Role `validate` Runs the fixture check and writes the four result records `validate_standard_registry` Field, duplicate-slug, prose-authority, and overclaim checks per standard row `validate_paper_module_shape` Required-heading and sentinel checks per fixture module `validate_standards_bundle` Same checks over the exported public standards bundle `validate_executable_grammar_metabolism_bundle` Specimen bundle keys, boundary phrases, and copied-body checks `validate_source_module_imports` Manifest, allowlist, and SHA-256 digest gate for the 12 copied bodies `result_card` metadata-only card projection for readers and the Atlas\n\nTo run the fixture check from `microcosm-substrate/`:\n\n```text\nPYTHONPATH=src python3 -m microcosm_core.organs.executable_doctrine_grammar validate --input fixtures/first_wave/executable_doctrine_grammar/input --out receipts/first_wave/executable_doctrine_grammar --card\n```\n\nDiagram of the mechanism (6 steps). Diagram source & refs\n\nSource refs\n\nPublic fixtures `standards_registry.json + paper_modules/*.md` field, slug, authority checks `validate_standard_registry` heading and sentinel checks `validate_paper_module_shape` metadata-only result records `receipts/first_wave/executable_doctrine_grammar/`\n\n```text\nflowchart TD A[\"Public fixtures standards_registry.json + paper_modules/*.md\"] --> B[\"validate\"] B --> C[\"validate_standard_registry field, slug, authority checks\"] B --> D[\"validate_paper_module_shape heading and sentinel checks\"] C --> E[\"Merge findings observed vs required negative cases\"] D --> E E --> F[\"metadata-only result records result records/first_wave/executable_doctrine_grammar/\"]\n```\n\n##### Negative cases\n\n###### Fixture keeps five\n\n**The fixture keeps five failures executable rather than described in prose.** The fixture keeps five failures executable rather than described in prose. Each maps to an error code the checker must observe:\n- `invalid_standard_and_module`: a row missing teleology, result-record expectations, governing standard, and scope statement, raising `MISSING_TELEOLOGY`, `MISSING_RECEIPT_EXPECTATIONS`, `MISSING_GOVERNING_STANDARD`, and MISSING_ANTI_CLAIM.\n- `prose_standard_claims_runtime_authority`: a prose-only standard that claims executable authority, raising `PROSE_STANDARD_NOT_EXECUTABLE_AUTHORITY`.\n- `macro_doctrine_body_copied_into_fixture`: a copied private body pasted into a public fixture, raising `MACRO_DOCTRINE_BODY_IN_PUBLIC_FIXTURE`.\n- `duplicate_standard_slug_conflict`: two rows sharing a slug, raising `DUPLICATE_STANDARD_SLUG_CONFLICT`.\n- `grammar_index_pass_overclaims_doctrine_complete`: a passing fixture that claims doctrine completeness, raising `GRAMMAR_PASS_OVERCLAIMS_DOCTRINE_COMPLETE`.\n\n###### Required cases enforce provable rejection\n\n**The required cases live in `EXPECTED_NEGATIVE_CASES`, and if a run does not observe every one `validate` returns a blocked status.** The set of required cases lives in `EXPECTED_NEGATIVE_CASES`. If a run does not observe every one, `validate` returns a blocked status.\n\n##### Prior Art Grounding\n\n###### Schema validation and executable semantics\n\n**The design follows schema validation, generated grammars, and executable-semantics work like JSON Schema, Tree-sitter, and the K framework, checking doctrine shape by program instead of leaving it as convention.** The design follows schema validation, generated grammars, and executable-semantics work. [JSON Schema](https://json-schema.org/specification) is the reference for validating document shape against a shared machine contract. [Tree-sitter](https://github.com/tree-sitter/tree-sitter) shows the value of generated grammars for inspectable structure. The [K framework](https://kframework.org/) is a close reference for turning semantic rules into executable artifacts. The borrowed pattern is narrow: doctrine shape, result-record expectations, duplicate slugs, copied bodies, and scope statements are checked by a program instead of left as convention.\n\n##### Validation Result record Path\n\n```text\n./repo-pytest tests/test_executable_doctrine_grammar.py -q --basetemp=/tmp/microcosm_executable_doctrine_grammar_pytest\n./repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n###### What a pass means here\n\n**A green run here means only that the focused tests and the corpus parity check both accept these fixtures and this projection.** A pass means the focused test suite and the corpus check both accept the current fixtures and this projection.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim\n\n**Given the public fixtures and exported bundles, it checks doctrine shape, rejects the five named failure classes, and verifies the 12 copied bodies match their declared SHA-256 digests, all metadata-only.** The strongest claim this component supports is bounded. Given the public fixtures and the exported bundles, it checks doctrine shape, rejects the five named failure classes, and verifies that the 12 copied bodies match their declared SHA-256 digests, all through metadata-only result records. The proof boundary is the public standards and paper-module fixtures, the exported bundles, the copied-body manifest, and the result records those runs produce.\n\n###### What a grammar pass is not\n\n**A grammar pass is not doctrine completeness, publishing-scope decision, or launch-scope decision, and the component judges no prose quality, calls no provider, exports no private bodies, and mutates no source.** It does not claim more. A grammar pass is not doctrine completeness, publishing-scope decision, or launch-scope decision. The component does not judge prose quality, prove whole-system correctness, call any provider, export private bodies, or change source files. That scope limit is fixed. A stronger claim has to land in the source record and the generated projection before this page can narrate it.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":13457,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/executable_doctrine_grammar.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"JSON Schema","url":"https://json-schema.org/specification","kind":"external_reference","section_id":"schema-validation-and-executable-semantics"},{"label":"Tree-sitter","url":"https://github.com/tree-sitter/tree-sitter","kind":"external_reference","section_id":"schema-validation-and-executable-semantics"},{"label":"K 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Claim Audit","url":"paper-module-doctrine-fact-claim-audit.html","relation_id":"paper_module.depends_on.paper_module","source_ref":"core/paper_module_capsules.json::paper_modules[48:paper_module.doctrine_fact_claim_audit].depends_on"},{"id":"paper_module.microcosm_axiom_substrate","kind":"paper_module","kind_label":"paper module","relation":"depended on by","label":"Microcosm Axiom System","url":"paper-module-microcosm-axiom-substrate.html","relation_id":"paper_module.depends_on.paper_module","source_ref":"core/paper_module_capsules.json::paper_modules[92:paper_module.microcosm_axiom_substrate].depends_on"}],"source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/executable_doctrine_grammar.md"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json"}],"visible_text_byte_count":14794,"section_titles":["Component turns doctrine","Report is set","Checks are structural","Purpose","Doctrine as prose convention","Failure is first class","How it works","`validate` reads and merges two checks","Standard-registry field and overclaim checks","Paper-module heading and sentinel checks","Merging findings and required negatives","Two bundle entry points reuse the checks","The copied-body digest gate","`result_card` projects a metadata-only card","Negative cases","Fixture keeps five","Required cases enforce provable rejection","Prior Art Grounding","Schema validation and executable semantics","Validation Result record Path","Validate reader projection","What a pass means here","Scope limit","The strongest supported claim","What a grammar pass is not"]},{"id":"paper_module.macro_projection_import_protocol","title":"Source Projection Import Protocol","relative_href":"docs/paper-module-macro-projection-import-protocol.html","read_href":"docs/paper-module-macro-projection-import-protocol.html","context_href":"docs/rules-and-ideas.html#paper-module-macro-projection-import-protocol","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"The public source-projection import protocol validates classified source-to-Microcosm projection cells, per-slice manifests, digest relations, omission result records, intake statuses, formal-math claim boundaries, and scope limits without claiming source or launch-scope decision.","summary":"Source Projection Import Protocol is the public import membrane for source-backed Microcosm growth. It validates fixture and exported projection bundles by checking source refs, public target refs, content digests, source-to-target relations, per-slice source-module manifests, validation refs, omitted-material result records, metadata-only result record policy, secret-exclusion scans, projection cell statuses, formal-math claim-boundary negatives, and other negative cases while keeping true private bodies, theorem/formal-result correctness claims, provider/oracle payloads, launch material, and static count claims out of public authority.","authority_ceiling":"Verified source body imports, fixture result records, exported projection-bundle result records, per-slice manifest refs, metadata-only formal-math metadata and validation refs, and metadata-only result record fields only; no theorem/formal-result correctness authority, private target status, challenge-result public sharing, live source authority, whole-system equivalence, launch, public sharing, hosted deployment, recipient work, provider/oracle payload authority, provider or Lean/Lake execution, secret/non-public body export, source-file changes, or whole-system correctness.","abstract":"The public source-projection import protocol validates classified source-to-Microcosm projection cells, per-slice manifests, digest relations, omission result records, intake statuses, formal-math claim boundaries, and scope limits without claiming source or launch-scope decision.","teleology":{"section_id":"","text":"The public source-projection import protocol validates classified source-to-Microcosm projection cells, per-slice manifests, digest relations, omission result records, intake statuses, formal-math claim boundaries, and scope limits without claiming source or launch-scope decision.","source":"capsule_summary_line"},"body_markdown":"###### The checkpoint for copying source material\n\n** `macro_projection_import_protocol` is the checkpoint for copying material out of a much larger source project into the public Microcosm folder.** `macro_projection_import_protocol` is the checkpoint for copying material out of a much larger source project into the public Microcosm folder. It reads a proposed set of copied files, classifies each one, and decides whether the copy is allowed out and honestly tied to the source it claims to come from.\n\n**It exists so the public folder can stay dense and real without becoming a place where a secret, an operator conversation, a model-output data, or launch material leaks out alongside genuinely useful code.** It exists so the public folder can stay dense and real without becoming a place where a secret, an operator conversation, a model-output data, or launch material leaks out alongside genuinely useful code. Every run answers one question per copied file: was this body allowed out, and does the public copy match the source it names.\n\n**The output is a metadata-only result record. It lists what was imported, what was deliberately left out with a note saying so, and what was refused.** The output is a metadata-only result record. It lists what was imported, what was deliberately left out with a note saying so, and what was refused. The record carries paths, hashes, counts, and validation refs, never the private bodies themselves.\n\n##### Purpose\n\n**Microcosm grows by copying real files out of a private codebase. A loose copy is the exact artifact that quietly carries a account secret or an operator thread along with the code you wanted.** Microcosm grows by copying real files out of a private codebase. A loose copy is the exact artifact that quietly carries a account secret or an operator thread along with the code you wanted. A trust statement (\"we checked, it is fine\") does not scale and cannot be re-run by a reader.\n\n**So the check is an accounting check, not a trust statement. Each copied row declares its source ref, its public target ref, a content digest, and a material class.** So the check is an accounting check, not a trust statement. Each copied row declares its source ref, its public target ref, a content digest, and a material class. The protocol sorts the class into one of two sets. `PUBLIC_SAFE_BODY_MATERIAL_CLASSES` holds five classes that may be copied with provenance: pattern, standard, tool, result-record, and proof bodies. `TRUE_FORBIDDEN_MATERIAL_CLASSES` holds fifteen classes that can never appear as an imported body, including raw operator source, operator thread, model-output data, account secret, secret, recipient material, launch material, and the six private formal-math body classes. Anything claiming to be must also carry a verification record naming the digest, the source-to-target relation, and the mode used to check the copy.\n\n##### How it works\n\n** `main` dispatches four commands. `run` validates a first-wave fixture directory (the positive slices plus the named negative cases) and writes result records through `write_receipts`.** `main` dispatches four commands. `run` validates a first-wave fixture directory (the positive slices plus the named negative cases) and writes result records through `write_receipts`. `run_projection_bundle` validates an exported bundle and writes one `exported_projection_import_bundle_validation_result.json`. `plan` calls `preview_import_plan`, which scores each proposed cell and writes nothing. `refresh-exact-copy-source-modules` calls `refresh_exact_copy_source_modules` to re-copy exact-copy slices whose source has moved on. `run` and `run_projection_bundle` both assemble their answer through `_build_result`, which runs four validators over the fixture: `validate_projection_protocol` for the copied and omitted rows, `validate_cleaning_policy` for the cleaned rows, `validate_standalone_runtime_severance` for standalone-runtime dependency leaks, and `validate_import_plan` for the forward plan.\n\n**The core is `validate_projection_protocol`. It reads the protocol payload and runs these checks in order:** The core is `validate_projection_protocol`. It reads the protocol payload and runs these checks in order:\n\nFunction What it does `validate_projection_protocol` Sorts every copied row and emits the pass/blocked record `_public_safe_import_status` Classifies a row whose class is public-safe; returns nothing for other classes `_body_import_verification_findings` Checks the verification record: mode, verified flag, and digest binding `_forbidden_body_request` Detects a request to import a account secret or account-bound body `_authority_upgrade` Detects a claim to live private-source authority `_release_or_equivalence_overclaim` Detects a launch, public sharing, or private-equivalence claim `_formal_math_claim_boundary_subjects` Detects a formal-math theorem, proof, or solver-trace authority claim\n\n**First it enforces a density floor: the protocol must cite at least two source refs, two public runtime refs, and two validation refs, or the row fails as `MACRO_PROJECTION_PROTOCOL_DENSITY_MISSING`.** First it enforces a density floor: the protocol must cite at least two source refs, two public runtime refs, and two validation refs, or the row fails as `MACRO_PROJECTION_PROTOCOL_DENSITY_MISSING`. Then it walks each copied row. A row whose class is goes to `_public_safe_import_status`; anything it flags becomes a blocking finding. A row whose class is in `TRUE_FORBIDDEN_MATERIAL_CLASSES`, or any row that sets `body_copied` without a valid verification record, is rejected as `MACRO_PROJECTION_FORBIDDEN_BODY_IMPORT`. Omitted rows must carry an `omission_receipt_ref`. The result is `PASS` only when the floors are met, copied, omitted, cleaned, and step rows are all present, and no blocking finding remains.\n\n###### Digest requirements for a copied body\n\n**For a body that is copied, `_body_import_verification_findings` does the real integrity work.** For a body that is copied, `_body_import_verification_findings` does the real integrity work. The verification record must be marked verified and must use one of two modes in `BODY_IMPORT_VERIFICATION_MODES`: `exact_source_digest_match` or `verified_light_edit_recipe`. The target body digest must equal the declared target digest. For an exact copy it also re-hashes the named source, requires a relation from `EXACT_COPY_SOURCE_TO_TARGET_RELATIONS`, and checks the source digest.\n\n**The load-bearing detail is how the protocol treats an exact copy whose source has since changed. A mismatch on an exact-copy row is not a failed import.** The load-bearing detail is how the protocol treats an exact copy whose source has since changed. A mismatch on an exact-copy row is not a failed import. It is recorded as live source drift: the original copy was still honest, the source has simply moved on, and the row is flagged for the refresh path rather than failed. That separates a dishonest import from a stale one, so a routine upstream edit is not mistaken for a broken proof, and the public copy stays faithful without tracking every upstream change in lock-step.\n\n** `validate_import_plan` guards the forward direction. Each proposed cell must name source and target refs, must name validation refs, must not use a private-system- shaped ref, and must only reference body material that the protocol already carries.** `validate_import_plan` guards the forward direction. Each proposed cell must name source and target refs, must name validation refs, must not use a private-system- shaped ref, and must only reference body material that the protocol already carries. A plan needs at least three cells to pass. `preview_import_plan` wraps this as the `macro_projection_import_intake_preview_v1` record: it scores each cell, projects the intake and severance boards, and lists blocking surfaces without writing anything.\n\nDiagram of the mechanism (9 steps). Diagram source & refs\n\nSource refs\n\nValidate projection protocol `validate_projection_protocol`\n\n```text\nflowchart TD A[\"Copied row source ref, target ref, digest, material class\"] --> B[\"validate_projection_protocol\"] B --> C[\"Class in forbidden set Reject: forbidden body import\"] B --> D[\"Class _public_safe_import_status\"] D --> E[\"_body_import_verification_findings mode, verified flag, digest binding\"] E --> F[\"Exact copy: re-hash live source\"] F --> G[\"Match: body floor\"] F --> H[\"Mismatch: flag live source drift, refresh later\"] H --> G G --> I[\"metadata-only result record\"]\n```\n\n##### Negative cases\n\n**The first-wave fixture drives these named rejections through `validate_projection_protocol`, `validate_import_plan`, and `validate_standalone_runtime_severance`:** The first-wave fixture drives these named rejections through `validate_projection_protocol`, `validate_import_plan`, and `validate_standalone_runtime_severance`:\n- `forbidden_body_import_overclaim`: a request to import a account secret or account-bound body.\n- `missing_omission_receipt`: omitted material with no omission result record ref.\n- `authority_upgrade_overclaim`: a claim that a public import grants live private-source authority.\n- `release_or_private_equivalence_overclaim`: a launch, public sharing, recipient, or private-equivalence claim.\n- `formal_math_claim_boundary_overclaim`: a formal-math theorem, proof, private target status, solver-trace, or provider-oracle authority claim.\n- `missing_validation_ref`: a proposed cell with no validation refs.\n- `standalone_dependency_leak`: a runtime dependency on the live source root that a standalone public checkout cannot resolve.\n\n##### Prior Art Grounding\n\n**The check follows established provenance and software-supply-chain patterns: a copied or refactored artifact needs source refs, target refs, digests, validation refs, omission records, and a stated claim boundary.** The check follows established provenance and software-supply-chain patterns: a copied or refactored artifact needs source refs, target refs, digests, validation refs, omission records, and a stated claim boundary. The closest public anchors are [W3C PROV](https://www.w3.org/TR/prov-overview/) for entity, activity, and agent provenance, the [SLSA specification](https://slsa.dev/spec/) for artifact integrity and provenance in software supply chains, and [in-toto](https://in-toto.io/) for linking supply-chain steps through signed metadata. This component applies those patterns to a public and private projection boundary rather than to launch attestation, and it stays a local validator over public fixtures and exported bundles.\n\n##### Validation Result record Path\n\nFrom `microcosm-substrate/`, reproduce this page's checks with temporary result records:\n\n###### What the fixture and bundle validate\n\n**A pass means the fixture and bundle slices carried their required refs and digests, the named negative cases were rejected, and the result records stayed metadata-only.** A pass means the fixture and bundle slices carried their required refs and digests, the named negative cases were rejected, and the result records stayed metadata-only. Re-enter this module when a new projection cell lands, a source-module manifest is refreshed, or a result record count changes.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest import-accounting claim\n\n**The strongest claim this component supports: for a named slice, it validated classification, source-to-target digest binding, per-slice manifests, omission records, the named negative cases, and a metadata-only result-record policy.** The strongest claim this component supports: for a named slice, it validated classification, source-to-target digest binding, per-slice manifests, omission records, the named negative cases, and a metadata-only result-record policy. Accepted records expose the current `public_safe_body_import_count`, `public_safe_body_import_status`, `projection_status_counts`, `open_actionable_cell_count`, and `secret_exclusion_scan` fields.\n\n###### The import protocol's scope limit\n\n**That is the proof boundary and the scope limit. A passing scan is a declared-input regression gate over the material classes the protocol recognizes, not a complete account secret audit and bounded evidence that no private material exists anywhere.** That is the proof boundary and the scope limit. A passing scan is a declared-input regression gate over the material classes the protocol recognizes, not a complete account secret audit and bounded evidence that no private material exists anywhere. It does not show the public copy is complete, equal to the private root, or ready to launch. It does not grant authority over live private source, and it does not permit public sharing, hosted deployment, secret export, source-file changes, external model access, or Lean and Lake execution. The formal-math lane may carry statement-shape metadata, evidence-cell refs, and synthetic fixtures, but never theorem or formal-result correctness. 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checkpoint for copying source material","It exists so the public folder can stay dense","The output is a metadata-only result record","Purpose","Microcosm grows by copying real files out of a private codebase","So the check is an accounting check, not a trust statement","How it works","`main` dispatches four commands","The core is `validate_projection_protocol`","First it enforces a density floor","Digest requirements for a copied body","The load-bearing detail is how the protocol treats an exact copy whose source has since changed","`validate_import_plan` guards the forward direction","Negative cases","The first-wave fixture drives these named rejections through `validate_projection_protocol`","Prior Art Grounding","The check follows established provenance and software-supply-chain patterns","Validation Result record Path","What the fixture and bundle validate","Scope limit","The strongest import-accounting claim","The import protocol's scope limit"]},{"id":"paper_module.mission_transaction_work_spine","title":"Mission Transaction Work Spine","relative_href":"docs/paper-module-mission-transaction-work-spine.html","read_href":"docs/paper-module-mission-transaction-work-spine.html","context_href":"docs/rules-and-ideas.html#paper-module-mission-transaction-work-spine","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"The public mission-transaction fixture validates work-landing, claim, dependency, scoped-commit, checkpoint-lane, result record-drain, completion, and copied control source-module contracts without mutating live ledgers or git.","summary":"Mission Transaction Work Spine is the public replay membrane for Microcosm work-landing discipline. It checks fixed Work item, claim, dependency, transaction, result record-drain, completion, scoped mutation, and checkpoint-lane rows; validates copied work log, work log, checkpoint, scoped-commit, and mission-preflight source modules by manifest; and writes metadata-only result records with secret-exclusion and scope limits.","authority_ceiling":"Public fixture and exported-bundle result records only; no live work log mutation, live work log mutation, live git mutation, private backup execution, broad checkpoint authorization, launch-scope decision, publishing-scope decision, or whole-system correctness.","abstract":"The public mission-transaction fixture validates work-landing, claim, dependency, scoped-commit, checkpoint-lane, result record-drain, completion, and copied control source-module contracts without mutating live ledgers or git.","teleology":{"section_id":"","text":"The public mission-transaction fixture validates work-landing, claim, dependency, scoped-commit, checkpoint-lane, result record-drain, completion, and copied control source-module contracts without mutating live ledgers or git.","source":"capsule_summary_line"},"body_markdown":"**The riskiest moment in agentic code work is the one that feels safest. An agent runs a few checks, sees no errors, and concludes its work is finished and committed.** The riskiest moment in agentic code work is the one that feels safest. An agent runs a few checks, sees no errors, and concludes its work is finished and committed. Those are two different facts. A clean preflight describes the state of the checks. It says nothing about whether a competing claim already owns the same file, whether the branch moved underneath the agent, or whether the commit ever landed.\n\n** `mission_transaction_work_spine` is a fixture that replays a fixed set of work-landing situations against a toy repository and records, for each one, whether the change would be allowed to land or would be blocked.** `mission_transaction_work_spine` is a fixture that replays a fixed set of work-landing situations against a toy repository and records, for each one, whether the change would be allowed to land or would be blocked. It reads recorded inputs, recomputes each decision from copied internal control source code, and writes one result record per situation that says exactly why the situation was permitted or refused.\n\n###### Recompute decisions instead of trusting labels\n\n**The design choice that matters is that the validator refuses to trust the labels baked into its own inputs.** The design choice that matters is that the validator refuses to trust the labels baked into its own inputs. It then perturbs the input one field at a time and requires the decision to move.\n\n##### Purpose\n\n**A loose check passes whenever its inputs carry the right labels. That is the failure this component is built against.** A loose check passes whenever its inputs carry the right labels. That is the failure this component is built against. Agentic work fails most often at transaction boundaries, not at isolated syntax errors: a clean preflight read as a landed commit, a competing claim on the same file ignored, a claim whose expected parent no longer matches the repository, a downstream item marked ready without its hard dependency, a dirty tree used to justify a broad commit with no operator intent, or a result record that smuggles private ledger bodies into a public artifact.\n\n**The single question the fixture answers is narrow. What evidence has to hold before a unit of work is allowed to land, and is that evidence checkable rather than asserted?** The single question the fixture answers is narrow. What evidence has to hold before a unit of work is allowed to land, and is that evidence checkable rather than asserted? A cold reader can open the inputs, the copied source bodies that implement the checks, and the result records, and see which decision came from which check.\n\n##### How it works\n\n**The module has two entry paths, selected by subcommand. `run` validates the first-wave replay fixture.** The module has two entry paths, selected by subcommand. `run` validates the first-wave replay fixture. `run_mission_transaction_bundle` validates the exported bundle of copied source modules. Both write metadata-only result records.\n\n** `run` loads the fixture payloads, scans every input file for forbidden private classes, then calls one validator per concern and merges their findings.** `run` loads the fixture payloads, scans every input file for forbidden private classes, then calls one validator per concern and merges their findings. Status is `pass` only when no expected negative case is missing, the secret scan passes, and the real-snapshot replay passes. The validators it calls carry the mechanism.\n\n** `validate_claim_preflight` reads the active claims and the toy repository state. It walks the claims, records the first owner of each path, and emits `SAME_PATH_CLAIM_CONFLICT` when a second active claim wants a path that is already owned.** `validate_claim_preflight` reads the active claims and the toy repository state. It walks the claims, records the first owner of each path, and emits `SAME_PATH_CLAIM_CONFLICT` when a second active claim wants a path that is already owned. It compares each claim's `expected_parent_sha` against the current parent in the repository state and emits `EXPECTED_PARENT_MISMATCH` when they differ. A claim with no owned path emits `MISSING_OWNED_PATH`.\n\n** `validate_checkpoint_lane_policy` reads a set of commit -lane cases. For each case it computes the recommended lane and compares it to the selected lane.** `validate_checkpoint_lane_policy` reads a set of commit-lane cases. For each case it computes the recommended lane and compares it to the selected lane. A `broad_checkpoint` lane without `operator_authorized_broad_checkpoint` set to true emits `CHECKPOINT_BROAD_AUTH_REQUIRED`. A suspected secret not routed to the hard-stop lane emits `CHECKPOINT_SECRET_REQUIRES_HARD_STOP`. A dirty tree with isolated owned paths must not block a scoped commit, so a case that treats it as a scoped blocker emits `CHECKPOINT_DIRTY_TREE_NOT_SCOPED_BLOCKER`. A case with no selected lane emits `CHECKPOINT_LANE_DECISION_REQUIRED`.\n\n###### `validate_real_active_claims_snapshot` is the discriminating check\n\n** `validate_real_active_claims_snapshot` is the discriminating check. It then calls `active_claim_collisions_for_paths` three times: on the real status, on a status mutated to add a same-path conflict, and on a status mutated to add a disjoint-path claim. The real status must return no collision for the requested path, the same-path mutation must return one, and the disjoint mutation must return none. It also mutates the expected parent and the checkpoint lane row and requires each mutation to block while the equal-parent and disjoint cases stay clear. Only when the good case passes, every mutation blocks, every clear case clears, the source hash binds, and the session heartbeat binds does it award the `R3` rung public_safe_real_work_ledger_session_snapshot_replay. Otherwise it falls to `R2` or `R1`.\n\n** `run_mission_transaction_bundle` validates copied control source. The four import validators (validate_task_ledger_source_import, validate_work_ledger_source_import, `validate_checkpoint_source_import`, `validate_mission_control_source_import`) each read a source-module manifest and a runtime contract, check that the manifest declares exactly the expected module ids and count, then for each module confirm the copied file exists, contains its required internal control anchor strings, matches the manifest `sha256`, and carries `classification: copied_non_secret_macro_body` with `body_in_receipt: false`.** `run_mission_transaction_bundle` validates copied control source. The four import validators (validate_task_ledger_source_import, validate_work_ledger_source_import, `validate_checkpoint_source_import`, `validate_mission_control_source_import`) each read a source-module manifest and a runtime contract, check that the manifest declares exactly the expected module ids and count, then for each module confirm the copied file exists, contains its required internal control anchor strings, matches the manifest `sha256`, and carries `classification: copied_non_secret_macro_body` with `body_in_receipt: false`. The bundle path also requires the real claims snapshot to be present and to pass.\n\n###### Nine metadata-only transaction records\n\n** `write_receipts` writes the nine result records for a fixture run (preflight, dependency-blocked, work-landing attempt, claim preflight, scoped mutation, checkpoint lane, completion projection, dependency-unlock scheduler, reconcile plan).** `write_receipts` writes the nine result records for a fixture run (preflight, dependency-blocked, work-landing attempt, claim preflight, scoped mutation, checkpoint lane, completion projection, dependency-unlock scheduler, reconcile plan). Each record carries a schema version, refs, counts, hashes, verdicts, and `body_in_receipt: false`. Source bodies live only in the exported bundle's `source_modules/` tree; they never enter a result record.\n\nDiagram of the mechanism (6 steps). Diagram source & refs\n\nSource refs\n\n+ checkpoint lane policy `validate_claim_preflight` `validate_real_active_claims_snapshot`\n\n```text\nflowchart TD Fixture[\"First-wave fixture claims, deps, lane cases, snapshot\"] Bundle[\"Exported bundle copied source modules\"] Preflight[\"validate_claim_preflight + checkpoint lane policy\"] Replay[\"validate_real_active_claims_snapshot runtime-rebuilt snapshot + perturbations\"] Imports[\"four source-import validators manifest, anchors, sha256\"] Records[\"metadata-only result records refs, hashes, verdicts, limits\"] Fixture --> Preflight Fixture --> Replay Bundle --> Imports Preflight --> Records Replay --> Records Imports --> Records\n```\n\n##### Negative cases\n\n**The positive claim is not that the fixture passes. It is that the fixture accepts good evidence and rejects targeted one-field perturbations.** The positive claim is not that the fixture passes. It is that the fixture accepts good evidence and rejects targeted one-field perturbations. Fourteen named negative cases are required, each bound to its error code:\n\n###### `competing_claim_and_stale_parent` must observe both `SAME_PATH_CLAIM_CONFLICT` and `EXPECTED_PARENT_MISMATCH`\n\n** `competing_claim_and_stale_parent` must observe both `SAME_PATH_CLAIM_CONFLICT` and `EXPECTED_PARENT_MISMATCH`. `mission_claim_missing_owned_path` observes `MISSING_OWNED_PATH`. `scoped_commit_receipt_claims_global_authority` observes `SCOPED_RECEIPT_AUTHORITY_UPGRADE`. mission_fixture_private_task_ledger_body observes LIVE_TASK_LEDGER_BODY_IN_FIXTURE. `clean_preflight_overclaims_landing_complete` observes `PREFLIGHT_PASS_OVERCLAIMS_WORK_LANDED`. `dependency_unlock_without_resolution_receipt` observes `DANGLING_DEPENDENCY_REF` and `ready_workitem_with_unsatisfied_hard_dep` observes `READY_WITH_INCOMPLETE_HARD_DEP`. The four checkpoint-lane cases (`broad_checkpoint_without_operator_authorization`, `suspected_secret_without_hard_stop`, `dirty_tree_blocks_scoped_lane`, `missing_selected_lane_for_state`) observe the lane error codes above. A run is blocked if any of these expected cases is not observed.\n\n##### Prior Art Grounding\n\n**This is the mission-transaction member of a small family of work-landing fixtures. Its closest sibling is `durable_agent_work_landing_replay`, which checks recorded landing rows, validation-before-commit ordering, HEAD movement, blocker capture, and completion evidence without running live git.** This is the mission-transaction member of a small family of work-landing fixtures. Its closest sibling is `durable_agent_work_landing_replay`, which checks recorded landing rows, validation-before-commit ordering, HEAD movement, blocker capture, and completion evidence without running live git. This module narrows that pattern to the transaction preflight and the work-ledger claims membrane: same-path conflicts, expected-parent mismatches, checkpoint-lane selection, dependency unlocks, and session finalisation. The lineage is transactional-integrity engineering, specifically compare-and-set on a parent commit and single-writer path claims. There is no external citation to invent here; the pattern is standard optimistic-concurrency control applied to a repo.\n\n##### Validation Result record Path\n\n```text\nPYTHONPATH=src python3 -m microcosm_core.organs.mission_transaction_work_spine run \\\n  --input fixtures/first_wave/mission_transaction_work_spine/input \\\n  --out receipts/first_wave/mission_transaction_work_spine\n```\n\nTo validate the exported source bundle:\n\n```text\nPYTHONPATH=src python3 -m microcosm_core.organs.mission_transaction_work_spine \\\n  validate-mission-transaction-bundle \\\n  --input examples/mission_transaction_work_spine/exported_mission_transaction_bundle \\\n  --out receipts/first_wave/mission_transaction_work_spine\n```\n\nFor the focused regression tests and the paper-module corpus check:\n\n```text\nPYTHONPATH=src python3 -m pytest tests/test_mission_transaction_work_spine.py -q\nPYTHONPATH=src python3 scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n**A pass means the fixture rows, the copied source bodies, the real snapshot replay, and the fourteen negative cases all hold together, and the result records carry no private body.** A pass means the fixture rows, the copied source bodies, the real snapshot replay, and the fourteen negative cases all hold together, and the result records carry no private body.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The `R3` replay-and-evidence-shape claim\n\n**The strongest claim the evidence supports: the public fixture rows, the copied control source bodies, the real work-ledger session-snapshot replay, the discriminating negative cases, and the metadata-only result records preserve the work-landing contract at the `R3` rung, and the checks that discriminate still discriminate.** The strongest claim the evidence supports: the public fixture rows, the copied control source bodies, the real work-ledger session-snapshot replay, the discriminating negative cases, and the metadata-only result records preserve the work-landing contract at the `R3` rung, and the checks that discriminate still discriminate. That is a replay-and-evidence-shape claim. The proof boundary is those inputs and their recomputation. Nothing runs live.\n\n**Within that scope limit the module refuses more. It does not mutate the live work log, the live work log, or git.** Within that scope limit the module refuses more. It does not mutate the live work log, the live work log, or git. It does not certify completion of the current repository. It grants no authority to commit, checkpoint broadly, back up, or launch, and it makes no claim about provider behaviour, browser state, hosted-product readiness, or whole-system correctness.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":13805,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/mission_transaction_work_spine.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"}],"grouped_relation_feeds":[{"relation":"abides by","count":1,"links":[{"id":"AX-9","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Compensable transactional effects","url":"doctrine.html#dcard-ax-9"}]},{"relation":"depends on","count":1,"links":[{"id":"paper_module.durable_agent_work_landing_replay","kind":"paper_module","kind_label":"paper module","relation":"depends on","label":"Durable 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Inventory","url":"paper-module-tool-server-pressure-inventory.html","relation_id":"paper_module.depends_on.paper_module","source_ref":"core/paper_module_capsules.json::paper_modules[50:paper_module.tool_server_pressure_inventory].depends_on"},{"id":"paper_module.work_landing_control_spine","kind":"paper_module","kind_label":"paper module","relation":"depended on by","label":"Work Landing Control Spine","url":"paper-module-work-landing-control-spine.html","relation_id":"paper_module.depends_on.paper_module","source_ref":"core/paper_module_capsules.json::paper_modules[17:paper_module.work_landing_control_spine].depends_on"}],"source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/mission_transaction_work_spine.md"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json"}],"visible_text_byte_count":14984,"section_titles":["The riskiest moment in agentic code work is the one that feels safest","`mission_transaction_work_spine` is a fixture that replays a fixed set of work-landing situations against a toy repository","Recompute decisions instead of trusting labels","Purpose","A loose check passes whenever its inputs carry the right labels","The single question the fixture answers is narrow","How it works","The module has two entry paths, selected by subcommand","`run` loads the fixture payloads","`validate_claim_preflight` reads the active claims and the toy repository state","`validate_checkpoint_lane_policy` reads a set of commit-lane cases","`validate_real_active_claims_snapshot` is the discriminating check","`run_mission_transaction_bundle` validates copied control source","Nine metadata-only transaction records","Negative cases","The positive claim is not that the fixture passes","`competing_claim_and_stale_parent` must observe both `SAME_PATH_CLAIM_CONFLICT` and `EXPECTED_PARENT_MISMATCH`","Prior Art Grounding","This is the mission-transaction member of a small family of work-landing fixtures","Validation Result record Path","A pass means the fixture rows","Scope limit","The `R3` replay-and-evidence-shape claim","Within that scope limit the module refuses more"]},{"id":"paper_module.formal_math_readiness_gate","title":"Formal Math Readiness Gate","relative_href":"docs/paper-module-formal-math-readiness-gate.html","read_href":"docs/paper-module-formal-math-readiness-gate.html","context_href":"docs/rules-and-ideas.html#paper-module-formal-math-readiness-gate","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"The public formal-math readiness gate validates declared corpus, tactic, premise, routing, provider-budget, source-module manifest, copied PROVER probe body, and negative-case boundaries without claiming Lean/Lake or proof authority.","summary":"Formal Math Readiness Gate is the public readiness membrane before downstream proof work. It checks declared Mathlib and corpus readiness, tactic probe result records, proof-metadata-only premise indexes, target-shape routing, provider context budgets, formal_math_readiness_extensions intake rows, copied PROVER smoke-run artifacts, public component source body imports, digest manifests, secret exclusion, and negative cases, then emits readiness boards rather than theorem evidence.","authority_ceiling":"Public readiness metadata, copied PROVER smoke-run readiness/probe artifacts, public component source body floor, fixture result records, and exported-bundle result records only; no Lean/Lake execution, Mathlib availability beyond probe status, formal proof authority, formal-result correctness, external model access, private proof body, oracle premise id, launch-scope decision, publishing-scope decision, or whole-system correctness.","abstract":"The public formal-math readiness gate validates declared corpus, tactic, premise, routing, provider-budget, source-module manifest, copied PROVER probe body, and negative-case boundaries without claiming Lean/Lake or proof authority.","teleology":{"section_id":"teleology","text":null,"source":"markdown_section"},"body_markdown":"##### Teleology\n\n###### From deferred slogan to executable boundary\n\n**This is the public runtime cell that turns the formal-math slice into an executable boundary, validating synthetic readiness metadata for corpus, tactics, premises, routing, and provider recipes before any Lean witness claims authority.** `formal_math_readiness_gate` is the public runtime cell that turns the formal math slice from a deferred slogan into an executable boundary. It validates synthetic readiness metadata for corpus availability, tactic probes, premise indexes, target-shape routing, and provider context recipes before any future Lean witness can claim authority.\n\n###### One question for a cold reader\n\n**The page should let a cold reader answer one question without rereading the component: what evidence has Microcosm validated, and where does that evidence stop?** The page should let a cold reader answer one question without rereading the component: what evidence has Microcosm actually validated, and where does that evidence stop?\n\n##### Purpose\n\n###### Making the ready-versus-proven boundary explicit\n\n**Formal-math tooling fails quietly when a library or corpus is assumed rather than checked, so this component makes the ready-versus-proven boundary explicit by asking which declared inputs a later proof witness could safely consume.** Formal-math tooling fails quietly when a library, tactic, or corpus is assumed present rather than checked. A pipeline that routes a proof to `aesop` when `aesop` is not actually available, or that treats a premise index as proof evidence because it happens to carry a proof body, has already lost the boundary between \"ready to attempt\" and \"proven\". This component exists to make that boundary explicit before any downstream proof work begins. It answers one question: which declared formal-math inputs are well-formed and honest enough that a later proof witness could safely consume them, and where exactly does that warrant stop?\n\n###### A reducer that refuses specific lies\n\n**The mechanism is a deterministic reducer over five public JSON inputs that runs no Lean and instead refuses the specific ways they can lie, emitting a readiness board rather than theorem evidence.** The mechanism is a deterministic reducer over five public JSON inputs: corpus readiness, tactic-portfolio availability, a premise index, target-shape routing, and provider context recipes. It does not run Lean or Lake. Instead it reads what those inputs declare and refuses the specific ways they can lie. A corpus that claims Mathlib is available without a passing probe is rejected. A tactic marked available without a probe result record is rejected. A premise row carrying a `proof_body` or `oracle_needed_premise_ids` field is rejected. A route that admits a tactic the portfolio probe already marked unavailable is rejected. The output is a readiness board, not theorem evidence.\n\n###### Proving discipline through negative cases\n\n**The gate proves its own discipline through five overclaim inputs that each run must catch, while a digest manifest keeps copied probe bodies source-open without leaking proof bodies, model-output data, or private state.** The design choice worth noticing is that the gate proves its own discipline through negative cases. Alongside the positive inputs, the fixture carries five inputs that each commit a known overclaim, and the run passes only when every one of those overclaims is caught and no unexpected finding appears. The gate is therefore not merely asserting \"we check Mathlib availability\"; it is demonstrating, on each run, that a falsified Mathlib claim is actually refused. A second guard keeps the floor source-open without leaking: copied prover probe bodies are verified by digest through a manifest, while proof bodies, model-output data, and private state stay out of the result records entirely.\n\n##### Shape\n\nDiagram of the mechanism (11 steps). Diagram source & refs\n\nSource refs\n\nreject Mathlib-availability overclaim `validate_corpus_readiness` each available tactic needs a probe result record `validate_tactic_portfolio` reject route admitting an unavailable tactic `validate_target_shape_routing` reject over-budget or proof-body recipe `validate_provider_context_recipes` copied probe bodies, digest-checked `validate_source_module_imports`\n\n```text\nflowchart TD Inputs[\"Five public JSON inputs corpus, tactics, premises, routes, provider recipes\"] Scan[\"Secret-exclusion scan zero blocking hits required\"] Corpus[\"validate_corpus_readiness reject Mathlib-availability overclaim\"] Tactics[\"validate_tactic_portfolio each available tactic needs a probe result record\"] Premises[\"validate_premise_index reject proof_body / oracle premise ids\"] Routing[\"validate_target_shape_routing reject route admitting an unavailable tactic\"] Provider[\"validate_provider_context_recipes reject over-budget or proof-body recipe\"] SourceFloor[\"validate_source_module_imports copied probe bodies, digest-checked\"] Reconcile[\"Reconcile findings vs EXPECTED_NEGATIVE_CASES every known overclaim must be caught\"] Board[\"Readiness board + extension board available / blocked capabilities, counts\"] Ceiling[\"Scope limit no Lean/Lake, proof, provider, launch, or private-system authority\"] Inputs --> Scan Scan --> Corpus Scan --> Tactics Scan --> Premises Scan --> Provider Tactics -->|unavailable tactic ids| Routing Corpus --> Reconcile Tactics --> Reconcile Premises --> Reconcile Routing --> Reconcile Provider --> Reconcile SourceFloor --> Reconcile Reconcile --> Board Board --> Ceiling\n```\n\n###### The hand-authored diagram is not the machine graph\n\nThe machine graph remains the generated `paper_module.formal_math_readiness_gate.mermaid` projection derived from the source record, not from this hand-authored Mermaid block.\n\n##### Reader Evidence Routing\n\nRead this module in evidence order:\n1. Start at `core/paper_module_capsules.json::paper_modules[21:paper_module.formal_math_readiness_gate]`. That row names the source authority, subjects, mechanism refs, code locus, Microcosm concept/principle/axiom refs, generated projection statuses, and the bundle scope limit.\n2. Check the generated structured source record `paper_modules/formal_math_readiness_gate.json`. Its `relationships.edges` cite the bundle source refs and show the generated Mermaid status, Atlas status, `source_authority: json_capsule`, and unresolved selective-relation count.\n3. Inspect the runtime locus `src/microcosm_core/organs/formal_math_readiness_gate.py`, especially `run`, `run_readiness_bundle`, `validate_source_module_imports`, `write_receipts`, `EXPECTED_NEGATIVE_CASES`, `AUTHORITY_CEILING`, and `SOURCE_MODULE_MANIFEST_NAME`.\n4. Use fixture evidence for the gate behavior: `fixtures/first_wave/formal_math_readiness_gate/input`, `receipts/first_wave/formal_math_readiness_gate/readiness_gate_result.json`, `formal_math_readiness_board.json`, `formal_math_readiness_extension_board.json`, `formal_math_readiness_validation_receipt.json`, and result records/sign-off/first_wave/formal_math_readiness_gate_fixture_acceptance.json.\n5. Use exported-bundle evidence for source-open body-floor claims: `examples/formal_math_readiness_gate/exported_formal_math_readiness_bundle/source_module_manifest.json`, `bundle_manifest.json`, `source_artifacts/`, `source_body_floor/source_modules/`, and `receipts/runtime_shell/demo_project/organs/formal_math_readiness_gate/exported_formal_math_readiness_bundle_validation_result.json`.\n6. Use `tests/test_formal_math_readiness_gate.py` for the behavioral result record boundary. The tests cover negative cases, exported bundle sign-off, source-module digest and target-ref mismatch rejection, bounded command-card output, source-body omission from result records, secret-exclusion/public-relative result record paths, and non-writing plan preview.\n\n###### This page routes readiness evidence, bounded evidence claims\n\n**A proof claim must go elsewhere because this page routes only readiness evidence, result record integrity, and source-body-floor accounting.** Do not route a proof claim through this page. It routes readiness evidence, result record integrity, and source-body-floor accounting only.\n\n##### Technical Mechanism\n\n###### One reducer, two entrypoints\n\n** `run()` evaluates the fixture with negative cases and `run_readiness_bundle()` evaluates the exported bundle, both calling `_build_result()` so they share one scope limit, secret scan, digest checker, and board schema.** The runtime is a deterministic readiness reducer over declared public inputs. `run()` evaluates the first-wave fixture directory with positive and negative JSON cases enabled; `run_readiness_bundle()` evaluates the exported public bundle without fixture-negative cases and requires the bundle source-module manifest. Both entrypoints call `_build_result()`, so the fixture and exported bundle result records share one scope limit, one secret scan, one source-module digest checker, and one readiness-board schema.\n\n###### Loading inputs and the non-advisory scan\n\n** `_build_result()` loads the five public input families and scans them for forbidden material, and the result passes only with zero blocking hits, passing source-module imports, all negatives observed, and no stray positive findings.** `_build_result()` first loads the five public input families: `corpus_readiness.json`, `tactic_portfolio_availability.json`, `premise_index.json`, `target_shape_tactic_routing.json`, and `provider_context_recipes.json`. It then scans those inputs plus any declared source artifacts through `secret_exclusion_scan.scan_paths`, using the public Microcosm forbidden-class policy. The scan is not advisory: the result can pass only when the scan has zero blocking hits, source-module imports pass, all expected fixture-negative cases are observed, and no unexpected positive-case findings remain.\n\nThe mechanism is split into six validators:\n- `validate_corpus_readiness()` records Lean and Mathlib readiness metadata and adds `lean_std_synthetic_core:mathlib` to blocked capabilities when Mathlib is unavailable. A Mathlib availability claim without a passing probe becomes `MATHLIB_AVAILABILITY_OVERCLAIM`.\n- `validate_tactic_portfolio()` separates available from unavailable tactics and requires every available tactic to carry a probe result record. Synthetic probe labels are accepted only when `_tactic_probe_realness_evidence()` binds them to copied source modules or fixture-manifest source-open evidence.\n- `validate_premise_index()` admits premise rows as metadata only. It counts premises, namespaces, retrieval terms, and split eligibility, but rejects `proof_body`, `ground_truth_proof`, `provider_output_body`, and `oracle_needed_premise_ids`.\n- `validate_target_shape_routing()` intersects each route case's allowed tactics with the unavailable tactics emitted by the portfolio validator. Any overlap becomes `ROUTING_ALLOWS_UNAVAILABLE_TACTIC`, so routing cannot silently re-enable a tactic that the probe plane blocked.\n- `validate_provider_context_recipes()` records byte budgets and deliverable shape while rejecting public recipes over 32,768 bytes or recipes that allow proof bodies or provider-body material.\n- `validate_source_module_imports()` verifies the exported bundle's `source_module_manifest.json`, target refs, source refs, line counts, target digests, source digests, exact-copy rows, and the two permitted private-path rewrites. It reports digest/ref failures without placing copied source bodies in result records.\n\n###### Reconciling findings and projecting the board\n\n** `_merge_observed()` and `_merge_findings()` compare observed failures against `EXPECTED_NEGATIVE_CASES`, then `_build_extension_board()` projects accepted metadata into pattern ids, counts, blocked routes, budgets, and the scope limit.** After the validators run, `_merge_observed()` and `_merge_findings()` compare observed fixture failures against `EXPECTED_NEGATIVE_CASES`. This is the local scope limit: the fixture run must prove that the known overclaims are caught, while the exported-bundle run must prove that the positive public bundle has no unexpected findings. `_build_extension_board()` then projects the accepted metadata into the extension board: selected pattern ids, namespace and split counts, tactic availability counts, Mathlib-dependent unavailable tactics, blocked route cases, provider budgets, source-body import counts, the scope limit, and the scope boundary.\n\n###### Result records and the asserting test suite\n\n** `write_receipts()` emits the gate result and boards while preserving the boundary, and the focused test asserts the mechanism itself, from negative-case ids to digest-mismatch rejection and result record omission of copied bodies.** Result record writing preserves the same boundary. `write_receipts()` emits the gate result, readiness board, extension board, validation result record, and sign-off result record for fixture mode. `run_readiness_bundle()` emits the exported-bundle result record. The focused test suite asserts the mechanism rather than just file existence: it checks the five expected negative case ids, local Lean/Lake probe metadata with Mathlib unavailable, six available tactics with `aesop` blocked, eleven premises, five route cases, three provider recipes, thirteen verified source artifacts, source/target digest mismatch rejection, target-ref mismatch rejection, secret-exclusion/public-relative result record paths, and result record omission of copied body text.\n\n##### Public Contract\n\n###### What the component consumes and rejects\n\n**The component runs no Lean or Lake; it records available and blocked capabilities and rejects Mathlib overclaims, unprobed tactics, proof-body premise rows, unavailable-tactic routes, and over-budget or proof-body recipes.** The component does not run Lean or Lake. It consumes public JSON fixtures and exported bundles, records which capabilities are available or blocked, rejects Mathlib availability overclaims, rejects unprobed tactics, rejects premise rows that contain proof bodies, rejects routes that admit unavailable tactics, and rejects provider recipes that exceed the public budget or allow proof bodies.\n\n###### The result is a readiness board\n\n**The accepted result is a readiness board that can tell a later component what is safe to attempt, but it is bounded evidence evidence, benchmark evidence, or permission to run a prover.** The accepted result is a readiness board. That board can tell a later component what is safe to attempt, but it is bounded evidence evidence, benchmark evidence, or permission to execute a theorem prover.\n\n##### Prior Art Grounding\n\n###### Formal-math readiness and environment work\n\n**The component is grounded in formal-math benchmark and environment-readiness work where a library or corpus being present is not enough, drawing on miniF2F, LeanDojo, and mathlib.** This component is grounded in formal-math benchmark and environment-readiness work where the presence of a library, tactic, or corpus is not enough by itself. [miniF2F](https://arxiv.org/abs/2109.00110) motivates explicit benchmark split discipline for formal mathematics, [LeanDojo](https://arxiv.org/abs/2306.15626) motivates reproducible theorem-proving environments, and [mathlib](https://arxiv.org/abs/1910.09336) makes the availability of library imports a concrete precondition rather than a vague capability claim.\n\n###### Borrowing the readiness-gate pattern\n\n**Microcosm borrows the readiness-gate pattern, requiring corpus availability, Mathlib and tactic probes, premise indexes, routing, and context budgets to be checked before downstream proof language, without authorizing Lean execution.** Microcosm borrows the readiness-gate pattern: corpus availability, Mathlib probes, tactic probes, premise indexes, target-shape routing, and context budgets must be checked before downstream proof or retrieval language is allowed. It excludes Lean execution or proof authority.\n\n##### Runtime Surfaces\n- `python -m microcosm_core.organs.formal_math_readiness_gate run --input fixtures/first_wave/formal_math_readiness_gate/input --out receipts/first_wave/formal_math_readiness_gate`\n- `python -m microcosm_core.organs.formal_math_readiness_gate run-readiness-bundle --input examples/formal_math_readiness_gate/exported_formal_math_readiness_bundle --out receipts/runtime_shell/demo_project/organs/formal_math_readiness_gate`\n- `python -m microcosm_core.organs.formal_math_readiness_gate plan --input fixtures/first_wave/formal_math_readiness_gate/input`\n- `microcosm formal-math-readiness-gate run --input fixtures/first_wave/formal_math_readiness_gate/input --out receipts/first_wave/formal_math_readiness_gate`\n- `microcosm formal-math-readiness-gate plan --input fixtures/first_wave/formal_math_readiness_gate/input`\n\n##### Relationship To Lean Witness\n\n###### The deferral is typed and testable\n\n** `formal_math_lean_proof_witness` remains deferred, and this gate makes the deferral typed and testable: absent Mathlib, unroutable unavailable tactics, no proof or oracle bodies in premises, and no proof-body provider recipes.** `formal_math_lean_proof_witness` remains deferred. This gate makes the deferral typed and testable: Mathlib is absent until a passing probe says otherwise, unavailable tactics cannot be routed, premise indexes cannot carry proof or oracle bodies, and provider recipes cannot smuggle proof-body deliverables.\n\n##### Validation Result record Path\n\n###### Three read-only checks validate body and projection\n\n```text\n./repo-pytest tests/test_formal_math_readiness_gate.py -q --basetemp=/tmp/microcosm_formal_math_readiness_gate_pytest\n./repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\njq '{edge_count:(.relationships.edges|length), mermaid_status:.paper_module_payload.generated_projections.mermaid.status, atlas_status:.paper_module_payload.generated_projections.atlas_card.status, source_authority:.relationships.source_authority, unresolved_selective_relation_count:(.relationships.unpopulated_selective_relations|length)}' paper_modules/formal_math_readiness_gate.json\n```\n\n###### The generated row should expose fifteen resolved edges\n\n**A correct row reports fifteen edges, available Mermaid, owner-blocked Atlas, JSON-bundle authority, and no unresolved selective relation.** Expected generated-row proof: `edge_count: 15`, `mermaid_status: available_from_capsule_edges`, `atlas_status: blocked_until_organ_atlas_owner_lane_binds_edges`, `source_authority: json_capsule`, and `unresolved_selective_relation_count: 0`.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The bounded valid claim\n\n**The module may claim a public readiness gate for formal-math preparation, bounded to corpus availability, probe metadata, premise coverage, routing, budgets, pattern ids, source artifacts, an component-source floor, and result records.** This module may claim that Microcosm has a public readiness gate for formal math system preparation. The valid claim is bounded to corpus availability, Mathlib and tactic probe metadata, premise-index coverage, target-shape tactic routing, provider context budget checks, extension-board pattern ids, public PROVER smoke-run source artifacts, an exact public component-source body floor, and fixture or exported-bundle result records.\n\n###### What the module must not claim\n\n**It must not claim Lean/Lake execution, theorem proving, proof authority, benchmark performance, external model access, private proof-body import, source-file changes, or launch, and its strongest statement is readiness-boundary enforcement.** The module must not claim Lean/Lake execution, theorem proving, formal proof authority, formal-result correctness, Mathlib-dependent proof success, benchmark performance, provider-call execution, private proof-body import, oracle-needed premise disclosure, source-file changes, publishing-scope decision, hosted deployment, recipient work, secret export, or whole-system correctness. Its strongest launch-facing statement is readiness-boundary enforcement over public metadata and copied source artifacts.\n\n###### Limitations\n\n###### Finite fixtures, no execution\n\n**The runtime validates finite public fixtures without executing Lean or Lake, importing Mathlib, or calling a provider, so a blocked capability is a downstream boundary, not an invitation to route around the gate.** The runtime validates finite public fixtures and exported-bundle manifests. It does not execute Lean or Lake, import Mathlib in the current environment, call a provider, or check theorem statements. When the result reports blocked capabilities such as `lean_std_synthetic_core:mathlib`, that is a readiness boundary for downstream components, not an invitation to route around the gate.\n\n###### Source artifacts are body-floor evidence only\n\n**The copied source artifacts are source-open body-floor evidence only: digest and target-ref checks show they match their manifests but authorize no source-file changes, proof-body disclosure, deployment, or public sharing.** The copied source artifacts are source-open body-floor evidence only. Digest and target-ref checks show that selected PROVER readiness/probe bodies and the public component source copy match their manifests; they do not authorize source-file changes, private source-root export, proof-body disclosure, recipient work, hosted deployment, or public sharing. Result records intentionally carry counts, digests, paths, negative-case coverage, and authority flags instead of copied body text.\n\n**The negative cases are finite, covering the known overclaims in `EXPECTED_NEGATIVE_CASES`, so a new formal-math claim needs a new source-backed negative case or a different proof consumer, not this page.** The negative cases are also finite. They cover the known overclaims encoded in `EXPECTED_NEGATIVE_CASES`: Mathlib availability without a passing probe, unprobed tactic availability, premise rows with proof bodies, target routes that admit unavailable tactics, and provider recipes that exceed public budgets or permit proof bodies. A new formal-math claim needs either a new source-backed negative case here or a different proof consumer; this page should not be used as a generic formal-proof claim surface.\n\n###### Scope boundary\n\n###### What this module excludes\n\n**This module documents a public readiness gate only, authorizing no Lean/Lake execution, proof authority, external model access, launch, or deployment, and adding no private source-root, account, or proof-body material to the public floor.** This module documents a public readiness gate only. It excludes Lean/Lake execution, formal proof authority, Mathlib-dependent proof attempts, external model access, benchmark claims, public launch, hosted deployment, public sharing, recipient work, secret export, or whole-system correctness. It also does not make private source-root material, browser UI state, account or browser material, browser state, account secrets, source notes, model-output data bodies, recipient-send state, or private proof bodies part of the public Microcosm body floor.\n\nSource and projection details\n\n###### Source-Open Body Floor\n\n###### Thirteen probe bodies preserve the readiness floor\n\n**The exported bundle carries thirteen probe bodies plus two digest-bound path rewrites covering corpus, tactic, Mathlib, trace, and Lean portfolio readiness.** The exported readiness bundle carries thirteen PROVER smoke-run readiness/probe bodies under `source_artifacts`. They cover corpus readiness, tactic-affordance probe metadata, Mathlib and trace probes, and the copied portfolio-core Lean probes used to decide which tactics are blocked or available. Two JSON rows are private-path rewrites; those rows retain source and target digests plus the rewrite mode.\n\n###### An exact component copy remains separate from result records\n\n**The public component-source copy is inspectable as body-floor evidence, while result records exclude copied text, private proof material, provider bodies, live session state, account secrets, and send state.** The bundle also carries an exact public component-source copy for `src/microcosm_core/organs/formal_math_readiness_gate.py` under `source_body_floor/source_modules`. Generated `state/runs` Lean artifacts are runnable readiness evidence, not source-body authority. Neither floor places body text in result records or workingness cards, and neither imports model-output data bodies, account or browser state, browser UI live access, recipient-send state, account secrets, private proof bodies, or oracle-needed premise ids.\n\n###### Manifests carry bodies while result records carry references\n\n**Body-floor inspection belongs in the source and bundle manifests; validation result records deliberately retain only status, digests, counts, and public-relative references.** The source-module manifest and bundle manifest are the right surfaces for body-floor inspection. The validation result records intentionally carry status, digests, counts, and public-relative refs rather than copied source bodies.\n\n###### The extension board projects typed readiness metadata\n\n**Wave 011's extension board records selected patterns, intake and validation refs, premise coverage, tactic and Mathlib status, route admissibility, and provider budgets without becoming proof evidence.** Wave 011 adds the explicit extension board for the source intake cell `formal_math_readiness_extensions`. The board is still metadata-only, but it is more useful than the older flat counts: it records the selected pattern ids (`lean_std_toolchain_premise_index`, `tactic_portfolio_availability_probe`, `target_shape_tactic_routing_gate`), the source projection intake ref, public target refs, validation refs, namespace and split coverage for the premise index, tactic availability status counts, Mathlib-dependent unavailable tactics, target-shape routing admissibility, and provider context budgets.\n\n###### Governing Lattice Relation\n\n###### Two mechanisms separate positive readiness from refusal\n\n**The bundle's concept and two mechanisms bind the positive bundle-validation path separately from the validators that reject readiness-to-proof overclaims.** The bundle binds this module to `concept.formal_math_and_proof_witness_bundle` because the component is not a theorem prover; it is the membrane that decides which public formal-math inputs are safe enough for a later proof witness to consume. The governing mechanisms split that membrane in two. The `validates_public_formal_math_readiness_bundle` mechanism names the positive bundle path: `run`, `run_readiness_bundle`, `validate_source_module_imports`, and `write_receipts` validate the declared corpus, tactic, premise, routing, provider-budget, source-module-manifest, and source-body-floor evidence before writing readiness boards. The `validates_public_readiness_boundary` mechanism names the negative path: `validate_corpus_readiness`, `validate_tactic_portfolio`, `validate_premise_index`, `validate_target_shape_routing`, and `validate_provider_context_recipes` reject the cases that would turn readiness metadata into proof authority.\n\n###### Principles and axioms enforce authority separation\n\n**The cited principles separate bundle, structured source record, runtime, result record, and body classes, while the axioms keep private material excluded and the downstream proof witness deferred.** The principle and axiom refs are therefore operational, not decorative. `P-1`, `P-2`, and `P-3` are expressed by keeping the JSON bundle, generated structured source record, runtime code locus, and result records as separate authority classes. `P-6` and `P-8` are expressed by the body-floor and secret-exclusion contracts: copied PROVER probe bodies and the public component source copy can be inspected through digests and manifests, while private proof bodies, model-output data bodies, and browser or account state stay outside the public floor. `AX-1`, `AX-2`, `AX-5`, and `AX-7` are the local reason the downstream `paper_module.formal_math_lean_proof_witness` remains a dependency rather than an already-proven conclusion.\n\n###### Fifteen generated edges prove connection, not capability\n\n**The deliberately small edge set establishes bundle, source, and dependency bindings without proving witness execution, Mathlib availability, or a fully bound Atlas card.** The generated lattice edge count is small on purpose: it proves that this page is bundle-backed, source-bound, and connected to one deferred proof-witness module.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":28597,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/formal_math_readiness_gate.md","kind":"source_link"},{"label":"Source 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Bundle","url":"paper-module-batch4-proof-authority-runtime.html","relation_id":"paper_module.depends_on.paper_module","source_ref":"core/paper_module_capsules.json::paper_modules[74:paper_module.batch4_proof_authority_runtime].depends_on"},{"id":"paper_module.macro_projection_import_protocol","kind":"paper_module","kind_label":"paper module","relation":"depended on by","label":"Source Projection Import Protocol","url":"paper-module-macro-projection-import-protocol.html","relation_id":"paper_module.depends_on.paper_module","source_ref":"core/paper_module_capsules.json::paper_modules[19:paper_module.macro_projection_import_protocol].depends_on"},{"id":"paper_module.provider_context_recipe_budget","kind":"paper_module","kind_label":"paper module","relation":"depended on by","label":"Provider Context Recipe Budget","url":"paper-module-provider-context-recipe-budget.html","relation_id":"paper_module.depends_on.paper_module","source_ref":"core/paper_module_capsules.json::paper_modules[54:paper_module.provider_context_recipe_budget].depends_on"},{"id":"paper_module.tactic_portfolio_availability","kind":"paper_module","kind_label":"paper module","relation":"depended on by","label":"Tactic Portfolio Availability","url":"paper-module-tactic-portfolio-availability.html","relation_id":"paper_module.depends_on.paper_module","source_ref":"core/paper_module_capsules.json::paper_modules[40:paper_module.tactic_portfolio_availability].depends_on"},{"id":"paper_module.target_shape_tactic_routing","kind":"paper_module","kind_label":"paper module","relation":"depended on by","label":"Target Shape Tactic Routing","url":"paper-module-target-shape-tactic-routing.html","relation_id":"paper_module.depends_on.paper_module","source_ref":"core/paper_module_capsules.json::paper_modules[41:paper_module.target_shape_tactic_routing].depends_on"}],"source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/formal_math_readiness_gate.md"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json"}],"visible_text_byte_count":30015,"section_titles":["Teleology","From deferred slogan to executable boundary","One question for a cold reader","Purpose","Making the ready-versus-proven boundary explicit","A reducer that refuses specific lies","Proving discipline through negative cases","Shape","The hand-authored diagram is not the machine graph","Reader Evidence Routing","This page routes readiness evidence, bounded evidence claims","Technical Mechanism","One reducer, two entrypoints","Loading inputs and the non-advisory scan","Reconciling findings and projecting the board","Result records and the asserting test suite","Public Contract","What the component consumes and rejects","The result is a readiness board","Prior Art Grounding","Formal-math readiness and environment work","Borrowing the readiness-gate pattern","Source-Open Body Floor","Thirteen probe bodies preserve the readiness floor","An exact component copy remains separate from result records","Manifests carry bodies while result records carry references","The extension board projects typed readiness metadata","Scope limit","The bounded valid claim","What the module must not claim","Governing Lattice Relation","Two mechanisms separate positive readiness from refusal","Principles and axioms enforce authority separation","Fifteen generated edges prove connection, not capability","Limitations","Finite fixtures, no execution","Source artifacts are body-floor evidence only","The negative cases are finite","Runtime Surfaces","Validation Result record Path","Three read-only checks validate body and projection","The generated row should expose fifteen resolved edges","Relationship To Lean Witness","The deferral is typed and testable","Scope boundary","What this module excludes"]},{"id":"paper_module.formal_math_lean_proof_witness","title":"Formal Math Lean Proof Witness","relative_href":"docs/paper-module-formal-math-lean-proof-witness.html","read_href":"docs/paper-module-formal-math-lean-proof-witness.html","context_href":"docs/rules-and-ideas.html#paper-module-formal-math-lean-proof-witness","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"The public Lean proof witness runs local Lean/Lake over a tiny synthetic project, validates copied public source-module digests and negative cases, and emits redacted result records without claiming general proof authority.","summary":"Formal Math Lean Proof Witness is the public runtime crossing from readiness metadata into a real local Lean/Lake subprocess witness. It copies a bounded public Lake project, records tool availability, Lake build status, source hashes, declaration names, line counts, source-module manifest verdicts, a Mathlib-blocked debt row, and six leakage/invalid-proof negative cases while keeping proof bodies, provider output bodies, oracle premise ids, and command output bodies out of result records.","authority_ceiling":"Declared public toy Lean/Lake fixture result records only; no Mathlib/Aesop/Batteries authority, general formal-result correctness, private proof import, external model access, benchmark performance, launch-scope decision, publishing-scope decision, source-file changes, or whole-system correctness.","abstract":"The public Lean proof witness runs local Lean/Lake over a tiny synthetic project, validates copied public source-module digests and negative cases, and emits redacted result records without claiming general proof authority.","teleology":{"section_id":"","text":"The public Lean proof witness runs local Lean/Lake over a tiny synthetic project, validates copied public source-module digests and negative cases, and emits redacted result records without claiming general proof authority.","source":"capsule_summary_line"},"body_markdown":"###### Component checks whether\n\n**This component checks whether the installed Lean toolchain actually compiles a tiny purpose-built project end to end and whether that run can be recorded without copying the proof into the record.** This component answers one narrow question and refuses the larger ones. The question is whether the installed Lean toolchain will actually compile a tiny, purpose-built Lean project end to end, and whether that run can be recorded without copying the proof into the record. It is not a theorem prover, a benchmark, a Mathlib check, or an import of any private proof. It runs `lake build` over a synthetic fixture and writes down what happened.\n\n###### Emits metadata-only record\n\n**It emits a metadata-only record of tool availability, build return code, output line counts, source hashes, declaration names, and per-file counts, while the proof text and raw tool output never reach it.** What it emits is a metadata-only result record: whether Lean and Lake were found, the build return code, line counts for standard output and standard error, source file hashes, the theorem and definition names it parsed out, and per-file line counts. The proof text and the raw tool output never reach that record. A reader gets evidence that the build ran and what it contained, not a copy of the proof.\n\n###### Point is discipline\n\n**The point is the discipline around the run: a real subprocess ran, a deliberately broken proof was rejected, and no answer was smuggled in through the inputs.** The point of the component is the discipline around the run. Anyone can print \"Lean says it compiles.\" The harder claim is that a real subprocess ran, that a deliberately broken proof was rejected, and that no answer was smuggled in through the inputs.\n\n##### Purpose\n\n###### Guarding against a checker-free pass\n\n**The first failure mode is an integration that reports success without invoking the checker, so the witness runs a real `lake build` subprocess and separately compiles an invalid file it must reject.** Two failure modes shape the design. The first is an integration that reports success without ever invoking the checker. The witness guards against that by running a real `lake build` subprocess, recording its exit status, and separately compiling an intentionally invalid Lean file to confirm the toolchain rejects it. A green result that never touched Lean is not reachable through this path.\n\n###### Rejecting the circular pass\n\n**The second failure mode is a circular pass where the manifest carries the answer, so the component rejects any manifest embedding a proof body, ground truth, provider output, oracle ids, or private body.** The second failure mode is a circular pass, where the input manifest quietly carries the answer. The component rejects manifests that embed a `proof_body`, a `ground_truth_proof`, provider output, oracle premise ids, or a private source body. Each of those has a named negative case. A loose witness would accept a manifest that already states the conclusion; this one does not.\n\n###### The scope is small on purpose\n\n**The scope is deliberately small: Mathlib, Aesop, and Batteries imports are rejected before anything runs, so this witnesses a toy theorem under a local toolchain, not library-dependent proof work.** The scope is small on purpose. Imports of Mathlib, Aesop, and Batteries are rejected before anything runs. This is a witness for a toy theorem under a local toolchain, not a claim about library-dependent proof work. That boundary is the deliverable: it shows the record-keeping a larger formal-math component would need, without borrowing authority it has not earned.\n\n##### How it works\n\n###### Two entry points, one builder\n\n** `run` drives the first-wave fixture with negative cases on and `run_witness_bundle` drives the exported bundle with them off, both delegating to `_build_result`, which runs in a fixed order.** The runtime lives in `src/microcosm_core/organs/formal_math_lean_proof_witness.py`. Two entry points exist. `run` drives the first-wave fixture with `include_negative=True`. `run_witness_bundle` drives the exported public bundle with `include_negative=False`, and reuses a fresh prior result record through `_fresh_bundle_receipt` when the inputs have not changed. Both delegate the real work to `_build_result`, which executes in this order.\n\n###### Validating copied source modules\n\n**In bundle mode `validate_source_module_imports` recomputes each copied module's SHA-256 against its live source, checks the declared exact-copy or replacement relation, material class, and anchors, and blocks on any mismatch.** First it collects the input file set with `_input_paths` and, for the bundle mode, validates copied source modules with `validate_source_module_imports`. That function reads `source_module_manifest.json`, and for each declared module it recomputes the SHA-256 of the copied target, recomputes the SHA-256 of the live source it was copied from, and checks that the declared `source_to_target_relation` is `exact_copy` (matching digests, `sha256_match: true`) or `public_replacement_source_body` (distinct digests, `sha256_match: false`). It also checks the material class is one of the four public classes, that `body_in_receipt` is false, that the source ref sits under an allowed public prefix, and that every declared anchor string is present in the target. Any mismatch becomes a finding, and any finding blocks the module.\n\n###### Scanning inputs and the witness manifest\n\n**It scans every input and copied artifact for forbidden material, then `_validate_manifest` rejects forbidden keys and non-public refs and requires the positive witness rows, Lean files, and source pattern ids.** Next it scans every input and copied artifact for forbidden material with `scan_paths` against the public forbidden-class policy, then validates the main `witness_manifest.json` with `_validate_manifest`. That check walks the manifest for the forbidden keys, rejects non-public source refs, and requires the positive witness rows, declared Lean files, source refs, public replacement refs, projection result record refs, and source pattern ids to all be present.\n\n###### Probing the toolchain and running the build\n\n** `_tool_versions` records whether `lean` and `lake` are on the path, then `_run_command` runs `lake build` in a temporary copy, keeping only the return code, output line counts, and timeout flag.** Then it probes the toolchain. `_tool_versions` uses `shutil.which` to record whether `lean` and `lake` are on the path, and caches the answer. For the live path, `_copy_project_to_temp` copies the Lake project into a temporary workspace and `_run_command` runs `lake build MicrocosmProofWitness` with a 90-second timeout. `_run_command` captures the subprocess but keeps only the return code, the standard-output and standard-error line counts, and the timeout flag. It never keeps the output body.\n\n###### Redacted per-file source metadata\n\n** `_source_metadata` records each positive Lean file's ref, hash, line count, declaration names, and imports with `body_redacted: True`, and sums the compiled declaration count across files.** For each positive Lean file, `_source_metadata` reads the text and records its public-relative ref, its hash, its line count, the declaration names found by a `theorem|lemma|def` regular expression, and its imports, with `body_redacted: True`. The compiled declaration count is the sum across those files.\n\n###### Running the rejection checks\n\n**With negatives enabled, `_build_result` compiles the invalid proof and forbidden-import files for real and runs `_validate_negative_manifest` over four tampered manifests, recording the expected code for each.** When negative cases are enabled, `_build_result` runs the rejection checks. It copies `invalid_proof.lean` into the built project and runs `lake env lean` on it; a non-zero exit records `LEAN_WITNESS_INVALID_PROOF_REJECTED`. It scans `mathlib_import_forbidden.lean` with `_forbidden_imports` and records `LEAN_WITNESS_FORBIDDEN_IMPORT` when a blocked import is present. It then runs `_validate_negative_manifest` over four tampered manifests for the private-source-ref, proof-body, provider-output, and oracle-premise-id cases.\n\n###### Computing the final status\n\n**A result is `pass` only when Lean and Lake are present, the build returned 0, the manifest and bundle modules validated, no forbidden import appeared, no negative case is missing, and the scan is clean.** Finally the status is computed. A result is `pass` only when Lean and Lake are available, the Lake build returned 0, the manifest validated, bundle source modules validated when in bundle mode, no forbidden imports appeared in the positive files, no expected negative case is missing, and the non-public-state scan found zero blocking hits. Otherwise the status is `blocked`. `_build_result` returns the record with the scope limit, the compiled declaration count, the redacted tool metadata, and the source file metadata attached.\n\nFunction Role `run` First-wave fixture entry point; negative cases on. `run_witness_bundle` Exported bundle entry point; reuses a fresh record when inputs are unchanged. `_build_result` Orchestrates validation, toolchain probe, Lake build, negative cases, and status. `validate_source_module_imports` Checks copied public source digests, relations, and anchors in the bundle manifest. `_validate_manifest` Rejects forbidden keys and non-public refs; requires the positive witness fields. `_run_command` Runs a subprocess and keeps only return code, line counts, and timeout state. `_tool_versions` Records `lean` and `lake` availability without running the checker. `_source_metadata` Emits hash, line count, declarations, and imports with the body redacted. Diagram of the mechanism (11 steps). Diagram source & refs\n\nSource refs\n\nFixture input `witness_manifest.json + lake_project` Exported bundle `source_module_manifest.json` recompute copied digests, check exact_copy vs replacement `validate_source_module_imports`\n\n```text\nflowchart TD A[\"Fixture input witness_manifest.json + lake_project\"] --> B[\"run() include_negative=true\"] C[\"Exported bundle source_module_manifest.json\"] --> D[\"run_witness_bundle() include_negative=false\"] B --> E[\"_build_result\"] D --> E E --> F[\"_validate_manifest reject proof bodies, non-public source refs\"] E --> G[\"validate_source_module_imports recompute copied digests, check exact_copy vs replacement\"] E --> H[\"_tool_versions lean/lake on path?\"] H --> I[\"_run_command lake build MicrocosmProofWitness\"] I --> J[\"Negative cases: invalid proof rejected, forbidden import blocked\"] F --> K[\"metadata-only result record: return code, line counts, hashes, declaration names\"] G --> K I --> K J --> K\n```\n\n##### Negative cases\n\n###### Fixture ships six\n\n**The fixture ships six named negative cases, each with an expected error code, defined in `EXPECTED_NEGATIVE_CASES`.** The fixture ships six named negative cases, each with an expected error code, defined in `EXPECTED_NEGATIVE_CASES`.\n\n###### Six refusals, each with a code\n\n**A real invalid proof, a blocked import, and four tampered manifests each raise their named error code, and if any expected case is not observed the run is blocked.** An invalid Lean proof (`invalid_proof.lean`) is compiled for real and must fail: `LEAN_WITNESS_INVALID_PROOF_REJECTED`. A file importing a blocked library (`mathlib_import_forbidden.lean`) is caught before execution: `LEAN_WITNESS_FORBIDDEN_IMPORT`. A manifest citing a non-public source refs is refused: `LEAN_WITNESS_PRIVATE_SOURCE_REF_FORBIDDEN`. A manifest embedding a proof body is refused: `LEAN_WITNESS_PROOF_BODY_IN_MANIFEST_FORBIDDEN`. A manifest carrying provider output is refused: `LEAN_WITNESS_PROVIDER_OUTPUT_IN_MANIFEST_FORBIDDEN`. A manifest carrying oracle premise ids is refused: `LEAN_WITNESS_ORACLE_PREMISE_IDS_IN_MANIFEST_FORBIDDEN`. If any expected case is not observed, the run is blocked.\n\n##### Prior Art Grounding\n\n###### The Lean proof-assistant lineage\n\n**The component sits in the Lean small-kernel lineage, borrowing the discipline of a local toolchain run with source hashes, declaration names, negative cases, and metadata-only records, not Mathlib-dependent authority.** This component sits in the Lean proof-assistant lineage and the small-kernel theorem-proving tradition it belongs to. The [Lean theorem prover system description](https://www.microsoft.com/en-us/research/publication/the-lean-theorem-prover-system-description/) describes the toolchain this witness invokes, and the [Lean mathematical library](https://arxiv.org/abs/1910.09336) shows why real proof authority depends on explicit imports, declarations, and a checked environment. The witness borrows the discipline, a local toolchain run with source hashes, declaration names, negative cases, and metadata-only records, before any Lean-witness language is allowed. It does not borrow Mathlib-dependent proof authority.\n\n##### Validation Result record Path\n\n```text\n./repo-pytest tests/test_formal_math_lean_proof_witness.py -q --basetemp=/tmp/microcosm_formal_math_lean_proof_witness_pytest\n./repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\njq '{edge_count:(.relationships.edges|length), mermaid_status:.paper_module_payload.generated_projections.mermaid.status, atlas_status:.paper_module_payload.generated_projections.atlas_card.status, source_authority:.relationships.source_authority, unresolved_selective_relation_count:(.relationships.unpopulated_selective_relations|length)}' paper_modules/formal_math_lean_proof_witness.json\n```\n\n###### What a pass means here\n\n**A pass means the focused test saw a real Lake build for the toy witness, the six negative cases, redacted records, and a generated row reporting `edge_count: 8` and `source_authority: json_capsule`.** A pass means the focused test saw a successful Lake build for the toy witness when Lean and Lake were available, saw the six negative cases, saw public-relative redacted records, and the generated row reports `edge_count: 8`, `mermaid_status: available_from_capsule_edges`, `source_authority: json_capsule`, and zero unresolved selective relations.\n\nTo run the component directly:\n\n```text\nPYTHONPATH=src python3 -m microcosm_core.organs.formal_math_lean_proof_witness run --input fixtures/first_wave/formal_math_lean_proof_witness/input --out receipts/first_wave/formal_math_lean_proof_witness\n```\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The narrow proof boundary\n\n**The strongest claim is that a tiny public Lean fixture compiles in a temporary local workspace and the run is recorded as hashes, declarations, tool metadata, and verdicts without exporting any proof body.** The strongest claim the evidence supports is narrow: a tiny public Lean fixture compiles in a temporary local workspace using the installed Lean and Lake, and the run can be recorded as source hashes, declaration names, tool-return metadata, and negative-case verdicts without exporting a single proof body. That is the whole proof boundary. The positive evidence is one declared toy fixture, one exported public bundle, the copied source-module rows, and the compiled declarations when the toolchain is present.\n\n###### What the witness excludes\n\n**It proves no arbitrary Lean goals, Mathlib coverage, or general correctness, and excludes external model access, private proof import, benchmark claim, launch, deployment, or source-file changes, even on a green run.** It does not establish arbitrary Lean goals, Mathlib coverage, or general formal-result correctness. It excludes external model access, no private proof import, no benchmark performance claim, no launch, no hosted deployment, no public sharing, no source-file changes, and no whole-system correctness. Records may cite refs, hashes, material classes, declaration names, counts, and tool-return summaries; they may not embed proof bodies, model-output data, oracle answers, non-public source refs, or raw command output. 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Spine","url":"paper-module-verifier-lab-execution-spine.html","relation_id":"paper_module.depends_on.paper_module","source_ref":"core/paper_module_capsules.json::paper_modules[44:paper_module.verifier_lab_execution_spine].depends_on"},{"id":"paper_module.verifier_lab_kernel","kind":"paper_module","kind_label":"paper module","relation":"depended on by","label":"Verifier Lab Kernel","url":"paper-module-verifier-lab-kernel.html","relation_id":"paper_module.depends_on.paper_module","source_ref":"core/paper_module_capsules.json::paper_modules[0:paper_module.verifier_lab_kernel].depends_on"}],"source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/formal_math_lean_proof_witness.md"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json"}],"visible_text_byte_count":17422,"section_titles":["Component checks whether","Emits metadata-only record","Point is discipline","Purpose","Guarding against a checker-free pass","Rejecting the circular pass","The scope is small on purpose","How it works","Two entry points, one builder","Validating copied source modules","Scanning inputs and the witness manifest","Probing the toolchain and running the build","Redacted per-file source metadata","Running the rejection checks","Computing the final status","Negative cases","Fixture ships six","Six refusals, each with a code","Prior Art Grounding","The Lean proof-assistant lineage","Validation Result record Path","Validate reader projection","What a pass means here","Scope limit","The narrow proof boundary","What the witness excludes"]},{"id":"paper_module.formal_math_verifier_trace_repair_loop","title":"Formal Math Verifier Trace Repair Loop","relative_href":"docs/paper-module-formal-math-verifier-trace-repair-loop.html","read_href":"docs/paper-module-formal-math-verifier-trace-repair-loop.html","context_href":"docs/rules-and-ideas.html#paper-module-formal-math-verifier-trace-repair-loop","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"The public verifier-trace repair fixture validates copied Ring2 failure taxonomy, graph-update, oracle-repair contrast rows, source-module digests, negative cases, and one deterministic toy rerun without claiming proof authority.","summary":"Formal Math Verifier Trace Repair Loop is the public evidence membrane for proof-lab repair mechanics. It checks copied Ring2 run refs and digests, verifier attempts, trace grades, repair actions, promotion gates, source-module manifests, secret exclusion, and eight leakage or overclaim negative cases, then writes metadata-only result records that keep proof bodies, oracle premise ids, model-output data, repair-trace theorem-correctness claims, Lean/Lake execution, and formal proof authority out of scope.","authority_ceiling":"Copied Ring2 verifier-trace repair metadata, source-module digests, public fixture result records, and one deterministic public toy-theorem rerun only; no Lean/Lake authority, formal-result correctness, proof body, oracle premise id, external model access, human approval as proof authority, launch-scope decision, publishing-scope decision, or whole-system correctness.","abstract":"The public verifier-trace repair fixture validates copied Ring2 failure taxonomy, graph-update, oracle-repair contrast rows, source-module digests, negative cases, and one deterministic toy rerun without claiming proof authority.","teleology":{"section_id":"","text":"The public verifier-trace repair fixture validates copied Ring2 failure taxonomy, graph-update, oracle-repair contrast rows, source-module digests, negative cases, and one deterministic toy rerun without claiming proof authority.","source":"capsule_summary_line"},"body_markdown":"###### Verifier failures as repair signals\n\n**This is a source-available replay of one proof-lab pattern where a verifier's failure becomes a repair signal, with failure categories, per-failure repair actions, and a no-promotion-without-re-run rule all inspectable from copied data.** `formal_math_verifier_trace_repair_loop` is a source-available replay of one pattern from a proof lab: a verifier reports why an attempt failed, and that failure is turned into a repair signal instead of being thrown away. The component works from copied, run data, so the failure categories, the repair action tied to each failure, and the rule that nothing is promoted without a fresh re-run are all inspectable.\n\n**It is not a theorem prover or Lean runner: it validates copied Ring2 rows against their source digests, replays the verifier attempts and repair curriculum, and runs one small deterministic re-run over public inputs.** It is not a theorem prover, a Lean or Lake runner, a provider client, or a private import. It validates a bundle of copied Ring2 benchmark rows, checks that the copies match their source digests, replays the verifier attempts and the repair curriculum against those source bodies, and runs one small deterministic re-run over public inputs. The output is a set of metadata-only result records.\n\n###### Proof and model-output bodies stay excluded\n\n**Proofs, oracle answer keys, and model outputs are kept out on purpose, so a row can name a failure class without exposing the proof or oracle premise that produced it.** Proofs, oracle answer keys, and model outputs are kept out on purpose. A row can name a failure class without exposing the proof or the oracle premise that produced it.\n\n##### Purpose\n\n###### Bookkeeping is where overclaim enters\n\n**A failed attempt is cheap to discard and expensive to learn from, and the narrow question is whether its failure can become a reusable public repair signal without inheriting a real prover's authority.** A failed proof attempt is cheap to discard and expensive to learn from. The narrow question here is whether a verifier's failure can become a reusable repair signal on the public side, without that signal quietly inheriting the authority of a real prover. The interesting work in a repair loop is the bookkeeping, and bookkeeping is where overclaim usually enters.\n\n###### A teaching signal needs five fields\n\n**A verifier failure counts as a teaching signal only once it carries a trace-backed grade, a repair action named by failure class, a ledger entry, a curriculum edge, and a cold re-run reference.** A loose version would collapse the stages into a single verdict: the attempt failed, we fixed it, done. This component refuses that. A verifier failure counts as a teaching signal only once it carries a trace grade backed by trace events, a repair action named against the verifier failure class it answers, an entry in the failure-mode ledger, a curriculum edge, and a cold re-run reference. Each of those is a separate field, and promotion is blocked until the cold re-run reference is present.\n\n###### Guarding against stale copied rows\n\n**The guarded failure is stale copied rows pretending to be live proof-lab evidence, so the copies are re-checked against their sources on every run and the verdict is downgraded on any drift.** The failure mode it guards against is stale copied rows pretending to be live proof-lab evidence. The rows are imported from one real Ring2 benchmark run, so the temptation is to treat the copy as if the run were happening now. The answer is to re-check the copies against their sources on every run and downgrade the verdict when anything drifts.\n\n##### How it works\n\n###### Two entrypoints, one builder\n\n** `run` validates a fixture with the eight negative cases and `run_loop_bundle` validates an exported bundle under the stricter source-module path, both calling `_build_result`, which passes only with every check green and no negative case missing.** There are two entrypoints. `run` validates a fixture directory with the eight negative-case files included. `run_loop_bundle` validates an exported bundle: it computes a freshness digest over the inputs, can reuse a current result record, and runs the stricter source-module path that requires every copied body to match its manifest. Both call `_build_result`, which composes the checks below in order and returns one record. A run is a pass only when every check passes and no negative case is missing.\n\n###### `validate_projection_protocol` demands provenance\n\n** `validate_projection_protocol` requires each copied row to keep its `source_ref`, `source_sha256`, target and validation refs, and copied-material status, blocking thin or unprovenanced projections.** `validate_projection_protocol` reads the projection manifest. It requires copied-material provenance: each copied row must keep its `source_ref`, `source_sha256`, target refs, validation refs, and a copied-material status, and any omitted proof, oracle, or provider material must carry an omission reference. Thin or unprovenanced projections are blocked.\n\n###### `validate_source_module_manifest` is the strict gate\n\n** `validate_source_module_manifest` requires a import class, `body_in_receipt: false`, and a matching digest, line, and byte count per Ring2 source ref, emitting typed mismatch findings on any drift.** `validate_source_module_manifest` is the strict gate, required for the exported bundle. It requires a import class, `body_in_receipt: false`, one row per declared Ring2 source ref, and for each row a matching digest, line count, and byte count against the copied file on disk. Drift produces typed findings such as `VERIFIER_TRACE_SOURCE_MODULE_DIGEST_MISMATCH`, `VERIFIER_TRACE_SOURCE_MODULE_SIZE_MISMATCH`, or `VERIFIER_TRACE_SOURCE_MODULE_COUNT_MISMATCH`. The exported bundle carries seven source modules and all seven must verify.\n\n###### `validate_verifier_attempts` replays each attempt\n\n** `validate_verifier_attempts` checks each attempt's verifier class and trace events, then `_validate_attempt_source_replay` dereferences the premise, oracle-contrast, and candidate rows, counting attempts, events, actions, and promotions.** `validate_verifier_attempts` checks each attempt for a verifier class and trace-event metadata, then hands the row to `_validate_attempt_source_replay`, which dereferences the premise-run row, the oracle-repair contrast row, and the graph-update candidate named by the attempt. Mismatches become `VERIFIER_TRACE_SOURCE_REPLAY_MISMATCH`, `VERIFIER_TRACE_ORACLE_REPLAY_MISMATCH`, `VERIFIER_TRACE_COLD_RERUN_SOURCE_MISMATCH`, or `VERIFIER_TRACE_CANDIDATE_REPLAY_MISMATCH`. It counts attempts, trace events, repair actions, and cold-rerun promotions.\n\n###### Curriculum and promotion policy checks\n\n** `validate_repair_curriculum` rejects a ledger append accepted after a cold re-run with no reference, and `validate_promotion_policy` requires `cold_rerun_receipt_ref` and rejects a policy setting `formal_proof_authority` true.** `validate_repair_curriculum` checks the failure-mode ledger and the curriculum edges, and rejects a ledger append marked accepted after a cold re-run that has no cold re-run reference. `validate_promotion_policy` requires `cold_rerun_receipt_ref` in the policy's promotion requirements and rejects a policy that sets `formal_proof_authority` to true.\n\n###### The one thing actually executed\n\n** `_run_toy_theorem_repair_rerun` is the only executed step: over inputs 0, 1, 2, 7 the failing candidate returns constant `0` and the repair returns `n`, plain integer arithmetic, not a Lean proof.** `_run_toy_theorem_repair_rerun` is the one thing actually executed. It takes the toy statement `toy_add_zero_identity` over the inputs 0, 1, 2, and 7. The failing candidate sets the right side to the constant `0`, which fails on three of the four inputs. The repair sets the right side to `n`, which passes on all four. It is plain integer arithmetic over public inputs through `_eval_toy_expr`, not a Lean proof.\n\n###### Grading the run from R1 to R4\n\n** `_runtime_realness_evidence` reaches R4 only for a fully verified bundle with 30-plus clean source-replay checks and a passing toy re-run, dropping to R3 on any mismatch, R2 for fixtures, and R1 otherwise.** `_runtime_realness_evidence` grades the run. It reaches R4 only for an exported bundle with all seven source modules verified, at least 30 source-replay checks with zero mismatches, at least three attempts, at least nine trace events, at least three failure modes, and a passing toy re-run. A verified bundle with any source-replay mismatch drops to R3 (mutated trace rejected). A fixture with copied material and observed negative cases sits at R2. Everything else is R1, the metadata floor. `_write_receipts` then emits the result record, a board, a validation record, and a fixture-pass record, all metadata-only.\n\nDiagram of the mechanism (9 steps). Diagram source & refs\n\nSource refs\n\ncopied-material provenance `validate_projection_protocol` digest, line, byte match; body_in_receipt false `validate_source_module_manifest` trace events, verifier class, source replay `validate_verifier_attempts` failure-mode ledger, curriculum edges `validate_repair_curriculum` requires cold re-run ref `validate_promotion_policy`\n\n```text\nflowchart TD Input[\"Fixture or exported bundle copied Ring2 rows + source manifest\"] Protocol[\"validate_projection_protocol copied-material provenance\"] Manifest[\"validate_source_module_manifest digest, line, byte match; body_in_receipt false\"] Attempts[\"validate_verifier_attempts trace events, verifier class, source replay\"] Curriculum[\"validate_repair_curriculum failure-mode ledger, curriculum edges\"] Promotion[\"validate_promotion_policy requires cold re-run ref\"] Toy[\"_run_toy_theorem_repair_rerun fail on 0, repair to n\"] Realness[\"_runtime_realness_evidence clean replay to R4; any drift downgrades\"] Result records[\"_write_receipts result, board, validation, fixture-pass\"] Input --> Protocol Protocol --> Manifest Manifest --> Attempts Attempts --> Curriculum Curriculum --> Promotion Promotion --> Toy Toy --> Realness Realness --> Result records\n```\n\n###### What the shipped bundle produces\n\n**On the shipped exported bundle this produces five attempts, 15 trace events, five repair actions, three cold-rerun promotions, seven verified source modules, 37 clean source-replay checks, and an R4 grade.** On the shipped exported bundle this produces five attempts, 15 trace events, five repair actions, three cold-rerun promotions, three failure modes, three curriculum edges, seven verified source modules, 37 source-replay checks with zero mismatches, and an R4 grade.\n\n##### Negative cases\n\n**The fixture path carries eight named negative files, and `EXPECTED_NEGATIVE_CASES` requires each to be observed before a pass.** The fixture path carries eight named negative files, and `EXPECTED_NEGATIVE_CASES` requires each to be observed before a pass. Each maps to one error code:\n- `attempt_with_proof_body` produces `VERIFIER_TRACE_PROOF_BODY_FORBIDDEN`.\n- `attempt_with_oracle_ids` produces `VERIFIER_TRACE_ORACLE_IDS_FORBIDDEN`.\n- `trace_grade_without_trace` produces `VERIFIER_TRACE_GRADE_WITHOUT_TRACE`.\n- `repair_without_verifier_class` produces `VERIFIER_REPAIR_WITHOUT_VERIFIER_CLASS`.\n- `promotion_without_cold_rerun` produces `VERIFIER_PROMOTION_WITHOUT_COLD_RERUN`.\n- `provider_payload_leakage` produces `VERIFIER_PROVIDER_PAYLOAD_FORBIDDEN`.\n- `human_approval_as_proof` produces `VERIFIER_HUMAN_APPROVAL_NOT_PROOF_AUTHORITY`.\n- `repair_claims_theorem_correctness` produces `VERIFIER_REPAIR_CLAIMS_THEOREM_CORRECTNESS`.\n\n###### Why repair metadata is not a truth claim\n\n**A repair trace that sets a theorem-correctness flag is rejected, so the loop can say what to try next without saying the theorem is true, and mutations downgrade the grade rather than passing quietly.** The last case is why repair metadata cannot become a truth claim: a repair trace that sets a theorem-correctness flag is rejected, so the loop can say what to try next without saying the theorem is true. The focused tests also perturb an oracle source row, a manifest digest, an attempt source label, and a curriculum source count, and confirm each mutation downgrades the grade rather than passing quietly.\n\n##### Prior Art Grounding\n\n###### Proof-attempt feedback loops\n\n**The component follows interactive theorem-proving feedback loops like GamePad, HOList, and LeanDojo, borrowing the accounting shape of separate verifier-event, grade, failure-class, repair, and cold-rerun fields without treating advice as correctness.** This component follows interactive theorem-proving feedback loops where failed proof attempts become structured training or repair signals. [GamePad](https://arxiv.org/abs/1806.00608) and [HOList](https://arxiv.org/abs/1904.03241) expose theorem-proving interaction data for machine-learning experiments, and [LeanDojo](https://arxiv.org/abs/2306.15626) keeps proof-assistant feedback, retrieval, and proof-state interaction reproducible. What is borrowed is the accounting shape: verifier events, grades, failure classes, repair actions, curriculum deltas, and cold re-run references stay separate fields. Human or provider advice is not treated as formal-result correctness.\n\n##### Validation Result record Path\n\nRun these from the repo root without mutating durable result record surfaces:\n\n```text\n./repo-pytest tests/test_formal_math_verifier_trace_repair_loop.py -q --basetemp=/tmp/microcosm_formal_math_verifier_trace_repair_loop_pytest\n./repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\njq '{edge_count:(.relationships.edges|length), mermaid_status:.paper_module_payload.generated_projections.mermaid.status, atlas_status:.paper_module_payload.generated_projections.atlas_card.status, source_authority:.relationships.source_authority, unresolved_selective_relation_count:(.relationships.unpopulated_selective_relations|length)}' paper_modules/formal_math_verifier_trace_repair_loop.json\n```\n\n###### What a pass means here\n\n**A pass means the fixture and bundle replay hold, the eight negative cases are observed, the copied bodies match their digests, and the row reports `edge_count: 17`, but no theorem is proved.** A pass means the fixture and exported-bundle replay hold, the eight negative cases are observed, the copied bodies match their digests, and the generated row reports `edge_count: 17`, `source_authority: json_capsule`, and zero unresolved selective relations. It does not mean any theorem is proved.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The narrow proof boundary\n\n**Copied Ring2 rows can drive a public repair loop that demands trace events before a grade, routes repair by failure class, gates promotion on a cold re-run, and replays each copy against its digest.** The strongest claim the evidence supports is narrow: copied, Ring2 verifier rows can drive a public repair loop that requires trace events before a grade, routes a repair action by failure class, gates promotion on a cold re-run reference, replays every copy against its source digest, and emits metadata-only result records. The proof boundary is copied metadata, the source digests, the fixture records, and one deterministic toy re-run over public integers.\n\n###### What the loop refuses\n\n**It runs no Lean or Lake, calls no provider, exposes no proof body or oracle id, treats no advice as proof authority, proves no formal-result correctness, and approves no launch.** It refuses more. It does not run Lean or Lake, call a provider, expose a proof body or an oracle premise id, or treat human or provider advice as proof authority. It does not establish formal-result correctness, and a repair trace that claims it is rejected. It does not approve launch or certify whole-system correctness. 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model-output bodies stay excluded","Purpose","Bookkeeping is where overclaim enters","A teaching signal needs five fields","Guarding against stale copied rows","How it works","Two entrypoints, one builder","`validate_projection_protocol` demands provenance","`validate_source_module_manifest` is the strict gate","`validate_verifier_attempts` replays each attempt","Curriculum and promotion policy checks","The one thing actually executed","Grading the run from R1 to R4","What the shipped bundle produces","Negative cases","The fixture path carries eight named negative files","Why repair metadata is not a truth claim","Prior Art Grounding","Proof-attempt feedback loops","Validation Result record Path","What a pass means here","Scope limit","The narrow proof boundary","What the loop refuses"]},{"id":"paper_module.formal_evidence_cell_anchor_resolver","title":"Formal Evidence Cell Anchor Resolver","relative_href":"docs/paper-module-formal-evidence-cell-anchor-resolver.html","read_href":"docs/paper-module-formal-evidence-cell-anchor-resolver.html","context_href":"docs/rules-and-ideas.html#paper-module-formal-evidence-cell-anchor-resolver","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"The formal evidence cell anchor resolver binds proof-language paper claims to public evidence cells, source anchors, machine-anchor metadata, frozen public source-module snapshots, public source-artifact mirrors, and negative-case result records without claiming formal-result correctness.","summary":"Formal Evidence Cell Anchor Resolver is the evidence-legibility membrane for Microcosm's formal math claims. It resolves three paper claims to public evidence-cell ids, checks source-anchor and machine-anchor metadata, anchors the verifier-trace cell to real Ring2 verifier-trace repair result records, validates six frozen source-open body snapshots, resolves public source-artifact mirrors for state/runs anchors, observes seven proof/private/human-approval/theorem-correctness negative cases, and emits metadata-only result records that make proof-language boundaries inspectable without becoming proof authority.","authority_ceiling":"Evidence-cell anchor metadata and source-open runtime result records only; no formal-result correctness, proof-body import, private source-ref authority, human approval as proof authority, Lean/Lake execution, external model access, launch-scope decision, publishing-scope decision, source-file changes, or formal-proof certification.","abstract":"The formal evidence cell anchor resolver binds proof-language paper claims to public evidence cells, source anchors, machine-anchor metadata, frozen public source-module snapshots, public source-artifact mirrors, and negative-case result records without claiming formal-result correctness.","teleology":{"section_id":"","text":"The formal evidence cell anchor resolver binds proof-language paper claims to public evidence cells, source anchors, machine-anchor metadata, frozen public source-module snapshots, public source-artifact mirrors, and negative-case result records without claiming formal-result correctness.","source":"capsule_summary_line"},"body_markdown":"###### Component asks whether\n\n**This component asks whether a page's proof-language claim can be tied to a named piece of public evidence, resolving each claim to an evidence cell and capping its strength at what the cell allows.** `formal_evidence_cell_anchor_resolver` checks the project's proof-flavoured writing against recorded evidence. When a paper page says a claim is proved or certified, this component asks a narrower question: can the words be tied to a named piece of public evidence, and does that evidence stay below formal-result correctness. It answers by resolving each claim to an evidence cell, checking the cell's source anchors point at files that exist on the public path, and capping the claim's strength at what the cell allows.\n\n###### Unit of evidence\n\n**The unit of evidence is a cell, a stable id bundling source-anchor refs, a `machine_anchor_class`, and allowed claim strengths, and the resolver reports which claims resolved, overreached, or had no backing.** The unit of evidence is a cell. A cell is a stable id that stands for a bundle of recorded evidence: a list of source-anchor refs, a `machine_anchor_class` naming what kind of machine artifact backs it, and the claim strengths the cell is allowed to support. The resolver reads paper claims, an evidence-cell registry, a claim-boundary policy, and an obligation graph, then writes a result record that reports which claims resolved, which overreached, and which had no backing. Proof bodies, non-public source refs, and model-output data stay out of that record.\n\n###### Is not theorem\n\n**It is not a theorem prover: it runs over already-public inputs, emits metadata-only result records, and states its full proof boundary once under Scope limit.** It is not a theorem prover. It runs over already-public inputs and emits metadata-only result records. The full proof boundary is stated once, under Scope limit.\n\n##### Purpose\n\n###### Where proof-adjacent prose drifts\n\n**Proof-adjacent prose drifts easily, and a loose check that only scanned for the word proof would pass any sentence avoiding it and could not bound how strong a backed claim may be.** Proof-adjacent prose is the easiest place for a claim to drift. A page can write \"this proves the theorem\" and a reader has no cheap way to tell whether the words rest on a checked artifact or on nothing. A loose check that only scanned for the word \"proof\" would pass any sentence that avoided the word, and would have no way to bound how strong a backed claim is allowed to be.\n\n###### `proof_language_requires_machine_anchor`\n\n**The key policy is `proof_language_requires_machine_anchor`: proof-language claims must name a resolving cell whose anchors point at existing files, with strength capped and human approval refused as a substitute.** The policy that makes the component useful is `proof_language_requires_machine_anchor`. A claim that uses proof language must name a cell, the cell must resolve in the registry, and the cell's source anchors must point at files that exist. A claim that uses proof language but names no cell, or names a cell absent from the registry, lowers the run to a blocked status instead of passing as green prose. On top of the resolved cell sit two limits: the claim's strength is capped by the cell's `allowed_claim_strengths`, and human approval is refused as a substitute for a machine anchor.\n\n##### How it works\n\n###### Two entrypoints share one core\n\n** `run` validates a fixture directory with negative cases included and `run_anchor_bundle` validates an exported bundle with them off, both calling `_build_result` to run the pipeline.** Two entrypoints share one core. `run` validates a fixture directory with the negative cases included. `run_anchor_bundle` validates an exported bundle directory with negative cases off and the source-module manifest required. Both call `_build_result`, which runs the pipeline and returns the result record.\n\n###### `_build_result` scans, validates, folds\n\n** `_build_result` loads the five inputs, scans every path for secret-equivalent material, runs six validators, and passes only when all pass, every expected negative case was observed, and no scan hit fired.** `_build_result` loads the five fixture inputs with `_load_payloads`, scans every input path for secret-equivalent material with `scan_paths`, then runs six validators and folds their findings into one status. The run passes only when every validator passes, every expected negative case was observed, and no secret scan hit fired.\n\nThe validators, in the order `_build_result` calls them:\n\nFunction What it checks `validate_projection_protocol` The projection cites enough source refs, pattern ids, and public runtime refs, and its Ring2 anchor refs resolve on the public path. `validate_cell_registry` Every evidence cell has at least one public source anchor, exposes no non-public refs, and records a `machine_anchor_class`. `validate_claim_boundary_policy` The policy sets `proof_language_requires_machine_anchor`, declares claim-strength levels and oracle-boundary classes, and excludes theorem-correctness claims or advisory export. `validate_obligation_graph` Declared obligations are met and carry no non-public refs, proof body, or oracle payload. `validate_source_module_manifest` For an exported bundle, the frozen source-module snapshots exist, match their recorded digests, and cover every declared owner ref. `validate_claims` Each claim resolves to a known cell, stays inside the cell's allowed strengths, and trips no leakage or overclaim rule.\n\n###### `validate_claims` resolves each claim\n\n** `validate_claims` passes each claim to `_inspect_claim_row`, which rejects proof bodies, non-public refs, oracle payloads, missing boundary classes, unknown cells, and out-of-range strengths as structured findings.** `validate_claims` is the resolver proper. It builds a `cell_by_id` map from the registry and passes each claim to `_inspect_claim_row`. That function rejects a claim when it carries a proof body, a non-public source refs, or a provider or oracle payload; when it omits an oracle-boundary class; when it uses proof language with no cell; when its cell id is unknown; or when its stated strength is not in the cell's allowed list. A theorem-correctness overclaim is caught either from an explicit flag or from the claim text by `THEOREM_CORRECTNESS_OVERCLAIM_RE`, a regex that matches \"proves\" or \"certifies\" near \"theorem\" and \"correctness\". Each rejection is a structured finding, not a silent pass.\n\n###### Resolving source anchors on the public path\n\n** `_validate_anchor_refs` splits each ref, rejects private schemes and escaping paths, resolves the file among public candidates including a source-artifact mirror, and records digest matches without holding file bodies.** Source anchors resolve through `_validate_anchor_refs`. It splits each ref into a path and an optional marker, rejects private schemes and absolute or parent paths through `_public_ref_is_forbidden`, then looks for the file among the candidates from `_public_ref_candidates`. A `state/runs/...` anchor is allowed to resolve through a public source-artifact mirror under `examples/*/*/source_artifacts`, so the resolver can confirm an anchor exists from an existing public example bundle without copying wider source trees into this component. When a ref carries a digest, `_sha256_file` recomputes it and the row records whether it matched. The record holds refs, digests, and status, not file bodies.\n\n###### metadata-only card shaping\n\n** `result_card`, `_source_module_summary_card`, and `_board_from_result` shape the result into compact cards, each carrying `body_in_receipt: False`.** The result record is shaped by `result_card`, `_source_module_summary_card`, and `_board_from_result` into compact cards. Every card and finding carries `body_in_receipt: False`.\n\nDiagram of the mechanism (7 steps). Diagram source & refs\n\n```text\nflowchart TD Inputs[\"paper claims, cell registry, boundary policy, obligation graph, projection protocol\"] --> Build[\"_build_result\"] Build --> Scan[\"scan_paths secret-equivalent material\"] Build --> Validators[\"six validators projection, registry, policy, obligations, source modules, claims\"] Validators --> Claims[\"validate_claims _inspect_claim_row\"] Claims --> Status[\"status = pass only if all validators pass and every negative case observed\"] Scan --> Status Status --> Record[\"metadata-only result record refs, digests, findings\"]\n```\n\n##### Negative cases\n\n###### Thirteen required negative-case classes\n\n** `EXPECTED_NEGATIVE_CASES` declares thirteen negative-case classes the fixture run must observe, and a `run` fails if any is missing, keeping the refusal coverage from silently eroding.** `EXPECTED_NEGATIVE_CASES` declares thirteen negative-case classes that the fixture run must observe. A `run` fails if any of them is missing, which keeps the refusal coverage from silently eroding. The classes name the exact rejection each case exercises: `unknown_cell_overclaim`, `missing_source_anchor`, `proof_language_without_cell`, `private_source_ref_leakage`, `proof_body_leakage`, `theorem_correctness_overclaim`, `human_approval_as_evidence_cell`, `oracle_payload_leakage`, `oracle_boundary_forward_success_overclaim`, and four obligation-graph variants for unmet obligations and private-ref, proof-body, or oracle-payload leakage inside an obligation.\n\n##### Prior Art Grounding\n\n###### Provenance and proof-certificate lineage\n\n**The component follows provenance and proof-certificate work like W3C PROV and Proof-Carrying Code, borrowing anchor resolution so a small checker bounds what cell metadata can assert without granting theorem authority.** This component follows provenance and proof-certificate work where a claim must point at checkable evidence rather than untyped narrative. The [W3C PROV](https://www.w3.org/TR/prov-overview/) model links entities, activities, and agents in an evidence graph. [Proof-Carrying Code](https://www.usenix.org/legacy/publications/library/proceedings/osdi96/full_papers/necula/html/node2.html) and small-kernel proof assistants motivate keeping a certificate or anchor separate from the trusted checker that bounds its meaning. This resolver borrows the anchor-resolution pattern: proof-language claims must name cell ids, source anchors, and machine-anchor classes, and a small checker bounds what those metadata can assert. It does not turn metadata cells into theorem-correctness authority.\n\n##### Validation Result record Path\n\n###### Run from repo\n\n**Run these from the repo root.** Run these from the repo root. They do not mutate durable result record or generated projection surfaces.\n\n```text\n./repo-pytest tests/test_formal_evidence_cell_anchor_resolver.py -q --basetemp=/tmp/microcosm_formal_evidence_cell_anchor_resolver_pytest\n./repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n###### What a pass means here\n\n**A pass means the fixture run observed all thirteen negative cases, resolved three claims to three cells across eight anchors, validated six frozen snapshots by digest, and kept everything metadata-only.** A pass means the fixture run observed all thirteen negative cases, resolved three paper claims to three evidence cells across eight source anchors and three machine anchors, validated the exported bundle against six frozen source-module snapshots by digest, and kept every card and finding metadata-only. It is bounded validator evidence over that fixture, not a statement about the wider formal-math corpus.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim\n\n**Public evidence-cell metadata can bind proof-language claims to recorded digest-checked cells, cap each claim at the strength its cell allows, and reject claims that leak private material or overreach.** The strongest claim the evidence supports: public evidence-cell metadata can bind proof-language claims to recorded, digest-checked cells, cap each claim at the strength its cell allows, and reject claims that leak private material or overreach into formal-result correctness. The counts are the fixture's contract, not a scale claim.\n\n**The proof boundary is anchor metadata and source-open runtime records only: this runs no Lean, proves no formal-result correctness, and a digest match confirms only that a snapshot equals its recorded hash.** The proof boundary is anchor metadata and source-open runtime records only. This component does not run Lean or Lake, prove formal-result correctness, import proof bodies, expose non-public source refs, carry provider or oracle payloads, treat human approval as proof, or include launch operations or public sharing. A digest match confirms a snapshot equals its recorded hash; it does not certify that the snapshot equals the latest live source, that every source file was imported, or that the copied bodies may be released. `AUTHORITY_CEILING` records these limits as explicit false flags in every result record.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":13023,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/formal_evidence_cell_anchor_resolver.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"W3C PROV","url":"https://www.w3.org/TR/prov-overview/","kind":"external_reference","section_id":"provenance-and-proof-certificate-lineage"},{"label":"Proof-Carrying 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proof-certificate lineage","Validation Result record Path","Run from repo","What a pass means here","Scope limit","The strongest supported claim","The proof boundary is anchor metadata"]},{"id":"paper_module.formal_math_premise_retrieval","title":"Formal Math Premise Retrieval","relative_href":"docs/paper-module-formal-math-premise-retrieval.html","read_href":"docs/paper-module-formal-math-premise-retrieval.html","context_href":"docs/rules-and-ideas.html#paper-module-formal-math-premise-retrieval","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"Formal math premise retrieval validates copied public Lean/Std premise metadata, query scoring, context budgets, strategy gates, body-floor provenance, and leakage negative cases without claiming proof authority.","summary":"Formal Math Premise Retrieval is the source-backed retrieval slice between the Lean/Std premise catalog and proof-witness boundary. It validates eleven public premise descriptors, four retrieval queries, forty-four considered candidates, exact and source-faithful source body imports, context recipe byte budgets, strategy ids, card freshness, and five leakage/overclaim negative cases while keeping proof bodies, oracle premise ids, model-output data, Lean/Lake execution, and launch claims out of result records.","authority_ceiling":"Copied public source retrieval metadata and runtime validation result records only; no formal-result correctness, proof-body import, oracle-needed premise authority, Mathlib authority, Lean/Lake execution, external model access, benchmark claim, launch-scope decision, publishing-scope decision, source-file changes, or general formal-proof authority.","abstract":"Formal math premise retrieval validates copied public Lean/Std premise metadata, query scoring, context budgets, strategy gates, body-floor provenance, and leakage negative cases without claiming proof authority.","teleology":{"section_id":"","text":"Formal math premise retrieval validates copied public Lean/Std premise metadata, query scoring, context budgets, strategy gates, body-floor provenance, and leakage negative cases without claiming proof authority.","source":"capsule_summary_line"},"body_markdown":"###### Public premise retrieval after source import\n\n**This is the first source-available formal-math import slice after the source projection protocol, turning the source prover lab's premise-index, term-scoring, context-budget, and strategy-selection patterns into a runnable component.** `formal_math_premise_retrieval` is the source-available first real formal-math import slice after the source projection protocol. It turns the source prover lab's premise-index, term-scoring, context-budget, and strategy-selection patterns into a runnable Microcosm component.\n\nIt is still deliberately below proof authority. It validates:\n- Lean/Std premise metadata;\n- query term scoring across public premise ids, namespaces, declaration names, statement excerpts, and retrieval terms;\n- split eligibility;\n- context recipe budgets;\n- public strategy ids;\n- redacted result records;\n- negative cases.\n\n**It stays deliberately below proof authority, refusing Lean or Lake execution, external model access, proof-body or oracle-id exposure, test-split tuning, correctness claims, and launch.** It does not run Lean or Lake, use external model services, expose proof bodies, expose oracle-needed premise ids, tune on test split truth, claim formal-result correctness, or include launch operations.\n\n##### Purpose\n\n###### Premise selection narrows the library\n\n**Before a model can attempt a proof it must find the right lemmas, and this component is the smallest honest premise-selection step: score every public premise against a query and return a ranked shortlist.** Before a model can attempt a formal proof, it has to find the right lemmas. A Lean library holds thousands of theorems and definitions, and the useful ones for a given goal are a handful. Premise selection is the step that narrows that library down to candidates worth putting in front of a prover. This component is the smallest honest version of that step: it takes a query, scores every public premise against it, and returns a ranked shortlist.\n\n###### Retrieval must work and the boundary must hold\n\n**The narrow question is whether transparent term-scoring over copied public Lean/Std metadata returns the premises a query should find without touching a proof, so retrieval must work and the boundary must hold.** The single question it answers is narrow and checkable: given a copied catalogue of public Lean/Std premise metadata, does a transparent term-scoring retrieval return the premises a query should find, without ever touching a proof? Both halves matter. The retrieval has to actually work, so each fixture query carries the premise ids it is expected to surface and the run fails if the shortlist misses them. And the boundary has to hold, so the same run refuses any input that smuggles in a proof body, an oracle answer, or test-split truth.\n\n###### The restraint is the point\n\n**The index is plain inspectable term overlap, not a learned embedding or benchmark claims, so the claim is that retrieval runs over real copied Lean metadata and forbids the shortcuts that would flatter it.** What is unusual is the restraint. The retrieval index is not a learned embedding model and the scoring is not a benchmark claims. It is plain term overlap over fields that a reader can inspect: premise ids, namespaces, declaration names, statement excerpts, and retrieval terms. The interesting claim is therefore not \"this retrieves well\" but \"this retrieves over real, copied Lean metadata and can be audited end to end, and the design forbids the shortcuts that would make a premise-selection result look better than it is\".\n\n##### Shape\n\nDiagram of the mechanism (14 steps). Diagram source & refs\n\nSource refs\n\nJSON source record `paper_module.formal_math_premise_retrieval` Runtime component `formal_math_premise_retrieval.py`\n\n```text\nflowchart TD bundle[\"JSON source record paper_module.formal_math_premise_retrieval\"] --> instance[\"Generated paper-module instance 15 relationship edges\"] instance --> component[\"Runtime component formal_math_premise_retrieval.py\"] subgraph Inputs[\"Public inputs\"] index[\"Premise index copied Lean/Std metadata\"] queries[\"Retrieval queries terms, split, strategy, top_k\"] recipes[\"Context recipes byte budgets\"] negatives[\"Negative-case inputs proof body, oracle ids, test-split tuning, budget, strategy\"] end component --> index component --> queries component --> recipes component --> negatives index --> split[\"Split gate skip premises not in allowed_for_split\"] queries --> split split --> score[\"Term-overlap scoring shared tokens + strategy bonus\"] score --> shortlist[\"Ranked top_k shortlist\"] shortlist --> recall[\"Recall check vs expected premise ids\"] negatives --> reject[\"Required rejections five leakage/overclaim guards\"] recipes --> reject recall --> result records[\"metadata-only result records board, validation, sign-off\"] reject --> result records result records --> ceiling[\"Scope limit metadata coherence, no Lean/Lake, no proof\"]\n```\n\nEvidence/accounting:\n- Bundle authority: `core/paper_module_capsules.json::paper_modules[25:paper_module.formal_math_premise_retrieval]` has `source_authority: json_capsule`, three `subjects`, one resolved `code_loci[0].path`, `depends_on` naming `paper_module.formal_math_lean_proof_witness`, and generated projection statuses for Markdown, Mermaid, and Atlas.\n- Generated instance: `paper_modules/formal_math_premise_retrieval.json::paper_module_payload` repeats the bundle `authority_ceiling`, reports Mermaid status `available_from_capsule_edges`, and derives 15 `relationships.edges` with `relationships.unpopulated_selective_relations: []`.\n- Component atlas: `core/organ_atlas.json::organs[9:formal_math_premise_retrieval]` classifies the component in `family: formal_math_and_proof`, cites the runtime locus, and restates that retrieval metadata coherence is not Lean/Lake, provider, theorem-correctness, benchmark, or launch-scope decision.\n- Mechanism rows: `core/mechanism_sources.json::mechanisms[27:mechanism.formal_math_premise_retrieval.validates_public_premise_retrieval_slice]` and `core/mechanism_sources.json::mechanisms[37:mechanism.formal_math_premise_retrieval.validates_public_premise_retrieval_projection]` point at `src/microcosm_core/organs/formal_math_premise_retrieval.py` and name first-wave, sign-off, and runtime-shell result record refs.\n- Runtime and tests: `src/microcosm_core/organs/formal_math_premise_retrieval.py` exposes `run`, `run_retrieval_bundle`, `EXPECTED_NEGATIVE_CASES`, and `AUTHORITY_CEILING`; `tests/test_formal_math_premise_retrieval.py` checks 11 premises, 4 queries, 44 considered candidates, five negative cases, metadata-only result records, and compact runtime-shell cards.\n- Result records: `receipts/first_wave/formal_math_premise_retrieval/formal_math_premise_retrieval_result.json` records `status: pass`, 11 premises, 4 queries, 44 considered candidates, five observed negative cases, `missing_negative_cases: []`, and a secret-exclusion scan with `blocking_hit_count: 0`; the exported runtime result record at `receipts/runtime_shell/demo_project/organs/formal_math_premise_retrieval/exported_premise_retrieval_bundle_validation_result.json` records `status: pass`, the same premise/query/candidate counts, no negative cases, and `secret_exclusion_scan.scanned_path_count: 11`.\n- Standard ceiling: `standards/std_microcosm_formal_math_premise_retrieval.json::authority_ceiling` has `status: pass` while keeping `formal_proof_authority`, `lean_lake_authority`, `provider_authority`, and `release_authority` false.\n\n##### Runtime Surfaces\n- Component runner: `python -m microcosm_core.organs.formal_math_premise_retrieval run --input fixtures/first_wave/formal_math_premise_retrieval/input --out receipts/first_wave/formal_math_premise_retrieval`\n- Exported bundle runner: `python -m microcosm_core.organs.formal_math_premise_retrieval run-retrieval-bundle --input examples/formal_math_premise_retrieval/exported_premise_retrieval_bundle --out receipts/runtime_shell/demo_project/organs/formal_math_premise_retrieval`\n- CLI route: `microcosm formal-math-premise-retrieval run-retrieval-bundle`\n- Standard: `standards/std_microcosm_formal_math_premise_retrieval.json`\n- Fixture manifest: `core/fixture_manifests/formal_math_premise_retrieval.fixture_manifest.json`\n\n##### Public Claim\n\nMicrocosm can show a real formal-math retrieval mechanism in miniature:\n- a source-available Lean/Std premise index;\n- public field-haystack term-scored queries;\n- split-aware eligibility;\n- context recipe ceilings;\n- strategy gates;\n- redacted validation result records.\n\n##### How retrieval scoring works\n\n###### Five inspectable fields per premise\n\n**Each premise contributes five inspectable fields, premise id, namespace, declaration name, statement excerpt, and retrieval terms, while a query carries its own terms, split, optional strategy, recipe, and expected ids.** Each premise row contributes five inspectable fields to the haystack: its premise id, namespace, declaration name, statement excerpt, and a list of retrieval terms. A query carries its own terms, a data split, an optional strategy id, a context recipe, and the public premise ids it is expected to return.\n\n###### Term-overlap scoring with a split gate\n\n**Scoring is per-query term overlap over the five fields, with a split gate that structurally excludes premises whose `allowed_for_split` omits the query split and a single-point bonus for a matching strategy tag.** Scoring is term overlap, computed per query. Both the query and each premise are tokenised into lowercase word counts. A premise is only considered if the query's split appears in that premise's `allowed_for_split` list, which is how test-split leakage is kept out at the structural level rather than by trust. For each eligible premise the score is the summed minimum count of every shared token across the five fields, so a term that appears in both the query and the premise contributes as many points as the smaller of the two counts. A premise that also carries the query's strategy id as a tag gets a single extra point. The ranked list is sorted by score descending, ties broken by premise id, and the top of that list up to the query's `top_k` is taken as the retrieval.\n\n###### Grading recall against expected ids\n\n**The retrieval is graded against itself: each query declares the ids it should surface, recall is the fraction that landed in the shortlist, and any miss blocks the run.** The retrieval is then graded against itself. Each query declares the public premise ids it should surface, and the component computes recall as the fraction of those expected ids that actually landed in the shortlist. A query that declares expectations but misses any of them blocks the run. In the first-wave fixture this is eleven premises and four queries, scoring forty-four considered candidates in total, and every query is expected to reach full recall.\n\n###### Five refusals keep the signal honest\n\n**The five negative-case inputs each encode a shortcut, a shipped proof body, oracle ids, test-split tuning, a budget overflow, and an unknown strategy, and all five rejections must be observed or the fixture blocks.** The failure mode this guards against is a premise-selection result that looks good because it cheated. The five negative-case inputs each encode one such shortcut: a premise index that ships a proof body, a query that lists the oracle premise ids it is \"meant\" to find, a query that tunes on test-split truth, a context recipe that blows past the byte budget, and a query naming a strategy id outside the allowed set. The run is required to observe all five rejections; if any expected rejection is missing, the whole fixture is blocked rather than passed. Recall over copied real metadata is the positive signal; the refusals are what keep that signal honest.\n\n##### Prior Art Grounding\n\n###### Premise-selection and retrieval-augmented proving\n\n**The component is grounded in premise-selection and retrieval-augmented theorem proving, closest to LeanDojo and motivated by earlier environments like HOList and GamePad that extract premise metadata for learning-assisted proving.** This component is grounded in premise-selection and retrieval-augmented theorem proving work. [LeanDojo](https://arxiv.org/abs/2306.15626) is the closest modern anchor because it couples Lean interaction with retrieval-augmented premise selection. Earlier theorem-proving environments such as [HOList](https://arxiv.org/abs/1904.03241) and [GamePad](https://arxiv.org/abs/1806.00608) also motivate extracting proof-state or premise metadata for learning-assisted theorem proving.\n\n###### Borrowing the retrieval accounting pattern\n\n**Microcosm borrows the retrieval accounting pattern, requiring premise ids, excerpts, terms, split eligibility, budgets, and strategy gates to be inspectable before any retrieval claim, without running Lean/Lake or exposing proof bodies.** Microcosm borrows the retrieval accounting pattern: premise ids, namespaces, statement excerpts, retrieval terms, split eligibility, context budgets, and strategy gates must be inspectable before premise-retrieval claims are admitted. It does not run Lean/Lake or expose proof bodies.\n\n##### Negative Cases\n- `premise_index_proof_body_forbidden`\n- `query_oracle_ids_forbidden`\n- `test_split_tuning_attempt`\n- `context_recipe_budget_overflow`\n- `unknown_strategy_id`\n\n##### Reader Evidence Routing\n- Start with the JSON Bundle Binding to identify the source record, generated instance, proof boundary, and scope limit.\n- Use Structured Lattice Bindings for navigation; the generated JSON row is the authority for relationship counts and dependency state.\n- Use Runtime Surfaces and Result record Expectations when checking metadata coherence, redaction, leakage checks, and source-available bundle behavior.\n- Use Negative Cases, Scope limit, and Scope limit together before admitting any formal-math public claim.\n\n##### Validation Result record Path\n\n###### Reader validation without durable writes\n\n```text\n./repo-pytest tests/test_formal_math_premise_retrieval.py -q --basetemp=/tmp/microcosm_formal_math_premise_retrieval_pytest\n./repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n##### Scope boundary\n\n###### Scope limit\n\n###### Coherence only, boundary unchanged\n\n**The component proves only that public retrieval metadata is internally coherent and leakage-checked, and the deferred `formal_math_lean_proof_witness` boundary remains unchanged.** The component proves only that public retrieval metadata is internally coherent and leakage-checked. The deferred `formal_math_lean_proof_witness` boundary remains unchanged.\n\n###### Scope limit\n\n###### The reader-verifiable claim and its limits\n\n**The module supports only the reader-verifiable claim that public premise metadata, retrieval terms, split eligibility, strategy gates, and result records are coherent and leakage-checked, running no Lean and proving no formal-result correctness.** This module supports only the reader-verifiable claim that public premise metadata, retrieval terms, split eligibility, strategy gates, and redacted result records are coherent and leakage-checked. It does not run Lean or Lake, prove formal-result correctness, expose proof bodies, authorize oracle-needed premise ids, tune on test split truth, use external model services, approve public sharing, or expand the deferred Lean proof-witness boundary.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":15468,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/formal_math_premise_retrieval.md","kind":"source_link"},{"label":"Source 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It imports a premise descriptor index, validates eleven Lean/Std premise rows across Nat, Bool, List, and Iff namespaces, checks six copied body modules through a source-module manifest, observes Mathlib/proof-body/oracle/test-split/source-ref negative cases, and writes metadata-only result records that make premise system inspectable without turning metadata into proof authority.","authority_ceiling":"Copied public Lean/Std descriptor index and Ring2 premise-retrieval source result records only; no Lean/Lake execution, Mathlib authority, proof-body import, oracle-needed premise authority, external model access, benchmark claim, launch-scope decision, publishing-scope decision, source-file changes, or theorem-correctness claim.","abstract":"The Lean/Std premise index validates a copied public Lean/Std descriptor catalog plus Ring2 premise-retrieval source bodies without claiming proof, Mathlib, Lean/Lake, provider, launch, or theorem-correctness authority.","teleology":{"section_id":"","text":"The Lean/Std premise index validates a copied public Lean/Std descriptor catalog plus Ring2 premise-retrieval source bodies without claiming proof, Mathlib, Lean/Lake, provider, launch, or theorem-correctness authority.","source":"capsule_summary_line"},"body_markdown":"###### A premise catalogue before proof search\n\n**A premise index is the catalogue a theorem-proving system reads before it tries to prove anything: the named lemmas and definitions it may cite, with enough metadata to retrieve the relevant ones.** A premise index is the catalogue a theorem-proving system reads before it tries to prove anything: the named lemmas and definitions it may cite, with enough metadata to retrieve the relevant ones. `lean_std_premise_index` does not build such an index and does not run a prover. It takes a small public catalogue that was copied out of a private Ring2 premise-retrieval run and checks that the copy is faithful and metadata-only.\n\n**The component consumes a fixture of JSON inputs, opens the source artifact the catalogue claims to copy, and emits metadata-only result records.** The component consumes a fixture of JSON inputs, opens the source artifact the catalogue claims to copy, and emits metadata-only result records. A pass means the public rows match the opened source row for row after a single allowed path rewrite, the six copied body files match their declared digests and line and byte counts, and none of the five things that would turn the catalogue into an answer key slipped in.\n\n**It reads metadata only. It never imports Mathlib, never runs Lean or Lake, and never prints a proof body.** It reads metadata only. It never imports Mathlib, never runs Lean or Lake, and never prints a proof body.\n\n##### Purpose\n\n**The value of a premise index for a benchmark sits one edit away from its danger. The same rows that tell a reader which standard-library facts are on the table would, with a proof body or an oracle-needed id attached, tell a solver exactly which lemmas it needs.** The value of a premise index for a benchmark sits one edit away from its danger. The same rows that tell a reader which standard-library facts are on the table would, with a proof body or an oracle-needed id attached, tell a solver exactly which lemmas it needs. A loose copy is worse than no copy: it looks authoritative while leaking the answer.\n\n**So the question this component answers is narrow. Given that a real premise index already exists inside a private run, can a cold reader see its public shape and be sure the public shape is a faithful copy and not a hand-written stand-in?** So the question this component answers is narrow. Given that a real premise index already exists inside a private run, can a cold reader see its public shape and be sure the public shape is a faithful copy and not a hand-written stand-in? The answer rests on opening the declared source and comparing, not on trusting a hand-maintained mirror.\n\n##### How it works\n\n** `run` validates the first-wave fixture with negative cases included. `run_index_bundle` validates the exported bundle with the source-module manifest required.** `run` validates the first-wave fixture with negative cases included. `run_index_bundle` validates the exported bundle with the source-module manifest required. Both call `_build_result`, which runs the checks below in order and sets status to pass only when there are no positive findings, no missing negative case, and no blocking secret-scan hit.\n\n** `_validate_entries` walks the eleven premise rows in the positive fixture. Each row must cite an `Init/` Lean toolchain source ref, carry retrieval terms, and name a nonempty split drawn from train, dev, and test.** `_validate_entries` walks the eleven premise rows in the positive fixture. Each row must cite an `Init/` Lean toolchain source ref, carry retrieval terms, and name a nonempty split drawn from train, dev, and test. The index as a whole must carry at least ten rows and cover the namespaces `Nat`, `Bool`, `List`, and `Iff`. A row is rejected if it carries a Mathlib ref, a proof body, an `oracle_needed_premise_ids` field, or a duplicate premise id.\n\n** `_validate_source_artifact` is the check that makes the copy claim real. It reads the declared `source_ref`, recomputes the file's SHA-256, and compares it to the declared digest.** `_validate_source_artifact` is the check that makes the copy claim real. It reads the declared `source_ref`, recomputes the file's SHA-256, and compares it to the declared digest. It then matches every public row to a source row by `premise_id` and compares signatures. The one permitted difference is a path rewrite: `_normalize_source_ref` turns a raw Lean toolchain path into a public `lean-toolchain://.../Init/...` reference so the reader sees where a lemma lives without seeing a private filesystem. A digest mismatch, a row-count mismatch, or a signature mismatch blocks the record.\n\n** `_source_module_manifest_result` checks the six copied body files: one normalized Lean/Std descriptor index plus five Ring2 body files.** `_source_module_manifest_result` checks the six copied body files: one normalized Lean/Std descriptor index plus five Ring2 body files. For each it recomputes the target digest, line count, and byte count against the manifest, confirms the declared source file opens and matches its own digest, and requires `body_copied` true with `body_in_receipt` false. Exact copies must still equal their opened source. The body text stays out of every record.\n\n**The five expected negative cases are declared in `EXPECTED_NEGATIVE_CASES`. `_build_result` runs each negative input through the same entry and policy checks, collects the codes it actually observed, and blocks if any expected case was not raised.** The five expected negative cases are declared in `EXPECTED_NEGATIVE_CASES`. `_build_result` runs each negative input through the same entry and policy checks, collects the codes it actually observed, and blocks if any expected case was not raised. A silent secret-exclusion scan over the inputs is the last gate.\n\nFunction Role `run` Validate the first-wave fixture, negative cases included `run_index_bundle` Validate the exported bundle, source-module manifest required `_build_result` Order the checks and decide pass or blocked `_validate_entries` Per-row and whole-index checks over the premise rows `_validate_source_artifact` Open the declared source, rehash it, compare rows by id `_source_module_manifest_result` Verify the six copied body files by digest, lines, bytes `_normalize_source_ref` Rewrite a raw toolchain path to a public `lean-toolchain://` ref Diagram of the mechanism (6 steps). Diagram source & refs\n\n```text\nflowchart TD inputs[\"fixture inputs premise_index, projection_protocol, index_policy, 5 negative cases, source_module_manifest\"] entries[\"_validate_entries 11 rows, Init/ refs, namespaces, splits\"] source[\"_validate_source_artifact reopen source, rehash SHA-256, compare rows by premise_id\"] manifest[\"_source_module_manifest_result 6 copied body files by digest, line, byte\"] gate[\"_build_result no findings + all negatives seen + clean secret scan\"] record[\"metadata-only result record\"] inputs --> entries inputs --> source inputs --> manifest entries --> gate source --> gate manifest --> gate gate --> record\n```\n\n##### Negative cases\n\n**The fixture ships five inputs that must be rejected, each mapped to its error code in `EXPECTED_NEGATIVE_CASES`:** The fixture ships five inputs that must be rejected, each mapped to its error code in `EXPECTED_NEGATIVE_CASES`:\n- `mathlib_premise_forbidden` raises `LEAN_STD_INDEX_MATHLIB_FORBIDDEN`;\n- `proof_body_leakage` raises `LEAN_STD_INDEX_PROOF_BODY_FORBIDDEN`;\n- `oracle_needed_ids_leakage` raises `LEAN_STD_INDEX_ORACLE_IDS_FORBIDDEN`;\n- `test_split_tuning_attempt` raises `LEAN_STD_INDEX_TEST_SPLIT_TUNING_FORBIDDEN`;\n- `namespace_without_source_ref` raises `LEAN_STD_INDEX_SOURCE_REF_REQUIRED`.\n\n**If any expected code is not observed, the run is blocked. These stay stable because the index is meant to be useful without becoming proof authority.** If any expected code is not observed, the run is blocked. These stay stable because the index is meant to be useful without becoming proof authority.\n\n##### Prior Art Grounding\n\n**This component borrows the closed-index discipline from formal-library indexing and premise-selection work. The [Lean mathematical library](https://arxiv.org/abs/1910.09336) anchors the library-as-corpus side.** This component borrows the closed-index discipline from formal-library indexing and premise-selection work. The [Lean mathematical library](https://arxiv.org/abs/1910.09336) anchors the library-as-corpus side. [LeanDojo](https://arxiv.org/abs/2306.15626) and [HOList](https://arxiv.org/abs/1904.03241) anchor the need for premise metadata, retrieval splits, and theorem-proving environments a learning system can inspect. Premise ids, declaration names, namespaces, source refs, retrieval terms, split eligibility, and copied-body digests are public metadata here; proof bodies and oracle-needed ids stay outside the boundary. It does not import Mathlib or prove theorems.\n\n##### Validation Result record Path\n\n**Run these from the public clone root without mutating durable result record or generated projection surfaces:** Run these from the public clone root without mutating durable result record or generated projection surfaces:\n\n```text\n./repo-pytest tests/test_lean_std_premise_index.py -q --basetemp=/tmp/microcosm_lean_std_premise_index_pytest\n./repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n**The focused test drives the runner over the fixture and the exported bundle: it asserts all five negative cases are observed, opens the real Ring2 source artifact, and rejects digest, row-count, row-signature, source-ref, source-module digest, and rehash-body-swap mutations.** The focused test drives the runner over the fixture and the exported bundle: it asserts all five negative cases are observed, opens the real Ring2 source artifact, and rejects digest, row-count, row-signature, source-ref, source-module digest, and rehash-body-swap mutations. A pass means the copy was faithful and metadata-only on this input, not that any theorem is correct.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The bounded premise-index claim\n\n**The strongest claim the evidence supports is bounded: public Lean/Std premise metadata, source refs, retrieval terms, split eligibility, and copied source-module digests can be indexed and shown to match their opened source without exposing proof bodies or oracle-needed ids.** The strongest claim the evidence supports is bounded: public Lean/Std premise metadata, source refs, retrieval terms, split eligibility, and copied source-module digests can be indexed and shown to match their opened source without exposing proof bodies or oracle-needed ids. That is the proof boundary, and it is also the scope limit. 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It validates eight drift rows, source refs, repair routes, validation refs, target refs, source-module digest evidence, copied world-model and view-quality source bodies, secret-exclusion policy, and eight negative cases while keeping private runtime bodies, model-output data, live repair, source-file changes, automatic doctrine changes, launch, public sharing, and source-authority claims out of scope.","authority_ceiling":"Public metadata-only runtime result record and copied source-module evidence only; no private runtime body inspection, source authority, source-file changes, live route repair, automatic doctrine changes, model-output data export, launch-scope decision, publishing-scope decision, or whole-system correctness claim.","abstract":"The world-model projection drift control room validates public metadata-only projection-drift rows and copied source-module bodies without treating projections as source authority or repair authority.","teleology":{"section_id":"","text":"The world-model projection drift control room validates public metadata-only projection-drift rows and copied source-module bodies without treating projections as source authority or repair authority.","source":"capsule_summary_line"},"body_markdown":"**A projected read model is a copy that can drift from its source and quietly read as truth, so this component checks public drift rows still trace back and refuses to let a copy speak for its source.** A projected read model is a copy. A dashboard row, a generated structured source record, a route card, or a public runtime result record can stay useful for navigation long after it has stopped matching the source it was copied from. When that happens quietly, the copy starts to read as truth. This component checks that public drift rows still trace back to the source they describe, and it refuses to let a copy speak for its source.\n\n###### Recomputing drift instead of trusting declarations\n\n**The runtime recomputes drift rows from a public result record rather than trusting the supplied file, diffs each against a real source-state artifact, and writes metadata-only records.** The runtime is `src/microcosm_core/organs/world_model_projection_drift_control_room.py`. It takes a small bundle of public JSON inputs, recomputes the drift rows from a public runtime result record rather than trusting the supplied file, diffs each row against a real source-state artifact, and writes metadata-only result records. Every record carries paths, hashes, counts, and verdicts. None of them carries a private body.\n\n###### Source-linked guards for each drift row\n\n**A pass means each named drift row is guarded by a source, target, repair route, validation reference, and fact-authority record and was witnessed, not that any source was repaired or made authoritative.** A pass here means the named public drift rows are guarded: each names its source, its target, a repair route, a validation reference, and a fact-authority record, and each was witnessed in the runtime result record. It does not mean any source was repaired, mutated, or made authoritative.\n\n##### Purpose\n\n###### Drift between a world model and projection\n\n**The failure mode is drift between a world model and its projection, where trusting the result record as source lets a stale or edited row pass a shape check while matching nothing real.** The failure mode is drift between a world model and its projection. The reducer `world_model.py` and the `/api/drift` endpoint in `main.py` produce a live view. A public result record then projects that view for readers. If the result record is trusted as the source, a stale or edited row can pass a shape check while no longer matching anything real.\n\n###### Recompute, never trust the supplied file\n\n**Rather than a shape check, `_build_result` recomputes rows from the result record, treats the supplied file as an expected snapshot only, and blocks any row it cannot re-derive from source.** A shape check alone fails here, so the design does not do one. The supplied `drift_rows.json` is never treated as input. `_build_result` recomputes the rows from the public runtime result record, then treats the supplied file only as an expected snapshot whose role is recorded as `expected_snapshot_not_source_authority`. If the snapshot disagrees with the recomputed rows, that is staleness, not fact. Each recomputed row is then diffed against a real source-state artifact. A row that cannot be re-derived from source, or whose guard reference or derivation path has changed, moves the verdict to `blocked`.\n\n###### A repair route stays a label\n\n**A named repair route stays a label, and `_row_policy_findings` rejects any row authorising live repair, source-file changes, doctrine changes, or launch, so a row names a fix but cannot act.** The same limit holds the other way. A row may name a repair route, but the route stays a label. `_row_policy_findings` rejects any row that authorises live repair, source-file changes, automatic doctrine changes, or launch. A row here can say what is wrong and where to go next. It cannot act, and it cannot stand in for its source.\n\n##### How it works\n\n###### Two entry points over `_build_result`\n\n** `run` validates the fixture with negative cases and `run_drift_control_bundle` validates the exported bundle without them, both flowing through `_build_result` into `_write_receipts`.** `run` validates the accepted fixture with negative cases included. `run_drift_control_bundle` validates the exported public bundle without them. Both call `_build_result`, which runs the stages below in order and then hands the payload to `_write_receipts`.\n\nFunction Role `_projection_recompute_result` A selected id absent from the result record blocks. `_supplied_drift_rows_snapshot_result` Compares the supplied `drift_rows.json` against the recomputed rows. Records the supplied file as an expected snapshot only. Any missing, extra, changed, or metadata-mismatched row is flagged as stale. `_source_ref_evidence_result` Checks that each row `source_ref` resolves to an entry in `SUPPORTED_DRIFT_SOURCE_REFS`. An unknown reference blocks the row. `_source_state_diff_result` Diffs recomputed rows against real source-state artifacts: a row in the extracted-pattern ledger, and the view-quality action-map lens whose `action_summary` is re-derived from its own `action_rows`. The two evidence classes are `extracted_pattern_ledger_row_diff` and `view_quality_action_map_summary_diff`. `_source_module_manifest_result` Reads `source_module_manifest.json`, and for each module checks `body_copied` is true, `body_in_receipt` is false, the copied body's `sha256` matches both declared digests, and the declared anchors appear in the body. A missing target, digest mismatch, or missing anchor blocks. `_source_open_body_import_summary` Folds the manifest result into a metadata-only summary: material status, copied count, and a reader action that points at the bundle rather than at any result record body. `_view_quality_geometry_grade_result` Imports the copied `view_quality_census.py`, grades `view_quality_geometry_probe.json` through its own `_geometry_vector_from_summary` and `_geometry_calibration_review`, and requires the verdict to be `calibrated_pass`. `_runtime_receipt_witness_result` Requires every recomputed row id to appear in the public runtime result record. An unwitnessed row blocks. `_row_policy_findings` Per-row gate: required fields, a source reference, a repair route and validation reference, a target reference, `body_in_receipt` false, no private needles, no model-output data, and a complete fact-authority record whose `authority_ref` equals the source, whose `guard_ref` equals the validation reference, and whose `residual_route` equals the repair route.\n\n###### `_build_result` gates the overall status\n\n** `_build_result` passes only when every stage and the secret scan pass with no findings, and the accepted fixture yields eight witnessed drift rows with all authority counts at zero.** `_build_result` collects the findings from every stage. The overall status is `PASS` only when there are drift rows, no findings, the recomputed rows are metadata-only, all forbidden booleans are false, the runtime witness, recompute, snapshot, source-ref, source-state, source-module, and geometry stages all pass, and the secret-exclusion scan passes. Anything else is `blocked`. The accepted fixture produces eight drift rows, all eight witnessed in the runtime result record, with `source_authority_claim_count`, `live_repair_authorized_count`, `source_mutation_authorized_count`, and `automatic_doctrine_promotion_count` at zero.\n\n###### `_write_receipts` normalises and pins authority\n\n** `_write_receipts` writes four records with host-temporary paths normalised away and `body_in_receipt`, `release_authorized`, and `AUTHORITY_CEILING` pinned on every one.** `_write_receipts` then writes the result record, a board, a validation record, and a `fixture_acceptance` record. Paths are normalised so host-temporary paths never surface. Every record sets `body_in_receipt` and `release_authorized` to false and repeats `AUTHORITY_CEILING`.\n\nDiagram of the mechanism (8 steps). Diagram source & refs\n\nSource refs\n\nPublic runtime result record `public_projection_drift_control_lens.json` expected snapshot, source-linked only `Supplied drift_rows.json` Geometry grade `copied view_quality_census.py`\n\n```text\nflowchart TD Result record[\"Public runtime result record public_projection_drift_control_lens.json\"] Recompute[\"Recompute rows selected_pattern_ids + result record drift_rows\"] Snapshot[\"Supplied drift_rows.json expected snapshot, source-linked only\"] SourceDiff[\"Source-state diff pattern ledger + view-quality action map\"] Geometry[\"Geometry grade copied view_quality_census.py\"] Witness[\"Runtime witness every row appears in the result record\"] Policy[\"Per-row policy gate refs, fact-authority, forbidden booleans\"] Records[\"metadata-only records result, board, validation, fixture_acceptance\"] Result record --> Recompute Recompute --> Snapshot Recompute --> SourceDiff Recompute --> Geometry Recompute --> Witness Recompute --> Policy Snapshot --> Records SourceDiff --> Records Geometry --> Records Witness --> Records Policy --> Records\n```\n\n###### Four copied source bodies in the bundle\n\n**The exported bundle copies four public source bodies, the reducer, the `/api/drift` endpoint, the action-map builder, and its test, so a reader inspects the implementation without private state.** The exported bundle copies four public source bodies so a reader can inspect the implementation without receiving private state: the world-model reducer `system/server/world_model.py`, the `/api/drift` endpoint in `system/server/main.py`, the action-map builder `tools/meta/observability/view_quality_census.py`, and its regression test `system/server/tests/test_view_quality_census.py`. The bodies live in the bundle. The records carry only their paths, hashes, and verdicts.\n\n##### Negative cases\n\n###### Nine negative inputs the run requires\n\n** `run` loads nine named negative inputs defined in `EXPECTED_NEGATIVE_CASES` and requires each to produce its rejection code, blocking the run if any expected code is missing.** `run` loads nine named negative inputs and requires each to produce its expected rejection code, defined in `EXPECTED_NEGATIVE_CASES`. A missing expected code blocks the run.\n\n###### The nine cases and added source-drift cases\n\n**The nine cases each map a malformed or overclaiming row to a stable code, and the source-drift stages add cases where a coherent row still loses authority if its source evidence changes.** The nine cases are: `drift_row_without_source_ref` to `DRIFT_SOURCE_REF_REQUIRED`; `repair_route_without_validation_ref` to `DRIFT_VALIDATION_REF_REQUIRED`; `drift_row_without_fact_authority` to `DRIFT_FACT_AUTHORITY_REQUIRED`; `projection_claiming_source_authority` to `DRIFT_SOURCE_AUTHORITY_FORBIDDEN`; `live_repair_action_authorized` to `DRIFT_LIVE_REPAIR_FORBIDDEN`; `private_runtime_data_export` to `DRIFT_PRIVATE_RUNTIME_EXPORT_FORBIDDEN`; `provider_payload_export` to `DRIFT_PROVIDER_PAYLOAD_FORBIDDEN`; `automatic_doctrine_promotion` to `DRIFT_AUTOMATIC_DOCTRINE_PROMOTION_FORBIDDEN`; and `release_from_drift_projection` to `DRIFT_RELEASE_AUTHORITY_FORBIDDEN`. The source-drift stages add further cases for unwitnessed rows, stale supplied snapshots, missing source-ledger rows, mutated view-quality summaries, and internally consistent but unsupported source references. A row can be internally coherent and still lose authority if its source evidence, guard, or derivation path changes.\n\n##### Prior Art Grounding\n\n###### Concept-drift monitoring as signal\n\n**It borrows the model-monitoring and concept-drift idea of a growing predicted-versus-observed gap as an operational signal, yielding fixture-bound monitoring evidence, source-linked only.** This component watches a projection for drift between what the model expects and what the runtime reports. It draws on the model-monitoring and [concept-drift](https://en.wikipedia.org/wiki/Concept_drift) literature, which treats a growing gap between predicted and observed behaviour as an operational signal. The borrowed shape is drift-as-signal over metadata-only records. The result is fixture-bound monitoring evidence, source-linked only or whole-system correctness.\n\n##### Validation Result record Path\n\nFocused runtime validation:\n\n```text\nPYTHONPATH=src ./repo-pytest \\\n  tests/test_world_model_projection_drift_control_room.py -q\n```\n\nPaper-module corpus validation:\n\n```text\ncd microcosm-substrate\nPYTHONPATH=src ../repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\nPaper-module index validation from the repo root:\n\n```text\n./repo-python tools/meta/factory/build_paper_module_index.py --check\n```\n\n###### What a pass means here\n\n**A pass means a fixture run produced the described records and rejected every named negative case.** A pass means the component ran over the public fixture, produced the records described here, and rejected every named negative case.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim\n\n**Over public synthetic inputs the named drift rows are guarded by references, per-row fact-authority, copied source-body evidence, and negative-case rejection, all reproducing under the validation commands.** The strongest claim this evidence supports is narrow: over public synthetic inputs, the named drift rows are guarded by source references, target references, repair routes, validation references, a fact-authority record per row, copied source-body evidence, metadata-only records, and negative-case rejection, and all of this reproduces under the validation commands above.\n\n###### The proof boundary and scope limit\n\n**The proof boundary is the metadata-only record, copied manifest, fixture rows, and checks on this page, and the ceiling is public metadata-only evidence only, with no source authority, mutation, or launch.** It refuses everything past that. The proof boundary is the metadata-only runtime record, the copied source-module manifest, the fixture rows, the source-state and geometry checks, and the validation result records named on this page. 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source-open body imports, negative cases, and scope limits without claiming launch-scope decision or whole-system equivalence.","summary":"Public Reveal Walkthrough is the source-backed public-entry membrane for Microcosm. It checks the declared reveal steps, runnable command set, evidence refs, claim-floor phrases, secret-exclusion scan, source-open body import manifest, runtime-bundle shape, and four overclaim negative cases while keeping copied bodies out of result records and launch/provider/private-equivalence claims out of scope.","authority_ceiling":"Public reveal fixture and exported-bundle result records only; no launch, hosted deployment, public sharing, recipient work, external model access, secret export, whole-system equivalence, Lean/Lake execution, source-file changes, or whole-system correctness.","abstract":"The public reveal walkthrough validates a ten-minute cold-reader path through commands, routes, evidence refs, source-open body imports, negative cases, and scope limits without claiming launch-scope decision or whole-system equivalence.","teleology":{"section_id":"","text":"The public reveal walkthrough validates a ten-minute cold-reader path through commands, routes, evidence refs, source-open body imports, negative cases, and scope limits without claiming launch-scope decision or whole-system equivalence.","source":"capsule_summary_line"},"body_markdown":"###### Executable bindings for the ten-minute tour\n\n** `public_reveal_walkthrough` makes the first-run ten-minute tour executable by binding each step to evidence in the public clone, blocking if any step points at nothing the system runs.** `public_reveal_walkthrough` checks the short guided tour Microcosm shows a first-time reader, and it binds that tour to evidence that exists in the public clone. The reader is promised a ten-minute path: run a command, look at local state, follow one route through patterns, work, events, and evidence, then read the result record and the scope limit behind it. This component makes that promise executable. It blocks if any step on the path points at nothing the system actually runs.\n\n**It reads two inputs, a first-wave fixture carrying the walkthrough, evidence map, claim floor, and four rejection cases, and an exported bundle of copied public source bodies, writing metadata-only records from both.** It reads two inputs. The first-wave fixture under `fixtures/first_wave/public_reveal_walkthrough/input/` carries the declared walkthrough, an evidence map, a claim floor, and four rejection cases. The exported bundle under `examples/public_reveal_walkthrough/exported_public_reveal_bundle/` carries copied public source bodies with a manifest. The component runs both and writes metadata-only result records: `public_reveal_walkthrough_result.json`, `ten_minute_reveal_board.json`, `public_reveal_validation_receipt.json`, and `public_reveal_walkthrough_fixture_acceptance.json`.\n\n**A pass is bounded validator evidence rather than a launch, with the scope limit stated once under Scope limit.** A pass is bounded validator evidence, not a launch. The scope limit is stated once, under Scope limit.\n\n##### Purpose\n\n###### Accounting, not a pitch\n\n**A walkthrough drifts into a pitch as soon as commands and evidence refs are dropped, so this component treats the first-run path as accounting where each step points at real state, work, and records.** A walkthrough drifts toward a pitch the moment it is allowed to. Dropping the commands and the evidence refs is the easiest way to make a first run look more impressive and prove less. This component removes that option. It treats the first-run path as an accounting exercise rather than a narrative: a command produces local state, a route points at work and events, an evidence ref points at a record, and a scope limit sits beside any impressive-looking total.\n\n###### The real-lane witness\n\n**The fixture does not pass on its own paperwork; it passes only when the exported bundle runs underneath it with copied bodies digest-verified, so the reveal cannot describe a runnable path that is broken.** The stricter part is the real-lane witness. The fixture does not pass on its own paperwork. It passes only when the exported bundle actually runs underneath it, with its copied source bodies present and digest-verified. So the reveal cannot describe a runnable path while the runnable path is broken. That is the quiet failure mode of every quick-start that says more than it can execute.\n\n##### How it works\n\n**The two entrypoints share one core builder, `_build_result`, and differ in what they require.** The two entrypoints share one core builder, `_build_result`, and differ in what they require.\n\n###### The fixture lane and its three validators\n\n** `run` calls `_build_result`, which scans inputs and runs three validators for the density floor, the evidence map, and the claim floor, passing only when all pass with no missing rejection and a clean scan.** `run` is the fixture lane. It calls `_build_result` with `include_negative` set true, then gates the outcome on a witness, then writes the record set. `_build_result` loads the fixture payloads, runs the secret-exclusion scan over every input path with `scan_paths`, and calls three validators. `validate_walkthrough` enforces the density floor: at least five steps, at least four distinct commands, and at least four evidence refs, otherwise it emits `PUBLIC_REVEAL_DENSITY_FLOOR_MISSING`. `validate_evidence_map` checks the evidence map. `validate_claim_floor` requires the declared product-loop phrases to appear in the public claim and rejects the overclaim cases. The builder sets `status` to pass only when no expected rejection case is missing, the secret scan has zero blocking hits, and all three validators pass.\n\n###### Gating the fixture on the real-lane witness\n\n** `_fixture_real_lane_witness` reruns the exported bundle with the manifest required, and `_attach_real_lane_witness` forces the fixture result to blocked if that witness did not pass.** After the builder, `run` calls `_fixture_real_lane_witness`, which runs the exported bundle through `_build_result` with the source-module manifest required, so the copied bodies are digest-checked for real. `_attach_real_lane_witness` reads that witness: if it did not pass, the fixture result is forced to blocked, `real_runtime_receipt` is set false, and a `PUBLIC_REVEAL_REAL_LANE_WITNESS_REQUIRED` or `...BLOCKED` finding is added. Then `write_receipts` emits the records.\n\n###### The bundle lane digest-verifies each module\n\n** `run_reveal_bundle` makes the source-module manifest mandatory, and `_source_module_manifest_result` checks each module's class, copied-and-metadata-only flag, existence, three digest fields, and required anchors, blocking on any mismatch.** `run_reveal_bundle` is the exported-bundle lane. It calls `_build_result` with `include_negative` false, which makes the source-module manifest mandatory. `_source_module_manifest_result` walks each declared module: the material class must be one of `PUBLIC_SAFE_SOURCE_BODY_CLASSES`, the row must mark the body copied and kept out of the record, the target file must exist, all three digest fields must match the recomputed `_sha256`, and every declared anchor must be present in the copied text. Any mismatch becomes a finding and the manifest status goes to blocked. `_source_open_body_import_summary` folds the verified modules into a count and class list for the record.\n\nFunction Role `run` Fixture entrypoint: builds the result, attaches the real-lane witness, writes the record set. `run_reveal_bundle` Exported-bundle entrypoint: requires and digest-verifies the copied source bodies, writes the bundle record. `_build_result` Loads inputs, runs the secret scan, calls the three validators plus the source-body check, sets pass or blocked. `validate_walkthrough` Enforces the five-step, four-command, four-evidence-ref density floor and drives the missing-evidence rejection. `validate_claim_floor` Requires the product-loop phrases in the public claim and rejects launch, private-equivalence, and marketing overclaims. `_source_module_manifest_result` Digest-verifies each copied source body against the manifest and keeps record fields metadata-only. `_fixture_real_lane_witness` Reruns the exported bundle so the fixture pass is gated on the real path, not on its own summary. Diagram of the mechanism (6 steps). Diagram source & refs\n\n```text\nflowchart TD Fixture[\"First-wave fixture walkthrough + claim floor + 4 negatives\"] Bundle[\"Exported bundle 5 copied source bodies + manifest\"] Build[\"_build_result load, scan, validate\"] Witness[\"_fixture_real_lane_witness rerun bundle, digest-check bodies\"] Records[\"metadata-only records result, board, validation, fixture\"] Ceiling[\"Scope limit no launch, hosting, or private equivalence\"] Fixture --> Build Bundle --> Witness Build --> Witness Witness --> Records Records --> Ceiling\n```\n\n##### Negative cases\n\n###### Four inputs that must stay rejected\n\n**The fixture ships four inputs that must stay rejected, and if any stops producing its finding the reveal is no longer bounded enough for a cold reader.** The fixture ships four inputs that must stay rejected. If any of these stops producing its finding, the reveal is no longer bounded enough for a cold reader.\n\nRejected input Finding code What it catches `release_or_hosting_overclaim` `PUBLIC_REVEAL_RELEASE_OVERCLAIM` A step claiming launch, hosting, public sharing, or recipient-work authority. `private_equivalence_overclaim` `PUBLIC_REVEAL_PRIVATE_EQUIVALENCE_OVERCLAIM` A claim of non-public data equivalence or whole-system correctness. `missing_evidence_route` `PUBLIC_REVEAL_STEP_EVIDENCE_MISSING` A step with no evidence ref or no inspect ref behind it. `marketing_without_runtime` `PUBLIC_REVEAL_RUNTIME_COMMAND_MISSING` Marketing copy with no command behind it.\n\n###### A missing case blocks the run\n\n**The four expected cases live in `EXPECTED_NEGATIVE_CASES`, and a run is blocked if any is absent from the observed set.** The four expected cases are declared in `EXPECTED_NEGATIVE_CASES`; a run is blocked if any of them is absent from the observed set.\n\n##### Prior Art Grounding\n\n###### First-run CLI, progressive disclosure, provenance\n\n**The path follows first-run CLI and progressive-disclosure practice for one runnable command and a bounded first route, and borrows provenance and tracing work for walking from artifact to source, record, and linked work.** The path follows first-run CLI and progressive-disclosure practice. The [Command Line Interface Guidelines](https://clig.dev/) motivate a single runnable command, worked examples, discoverable next steps, and machine-readable output. Nielsen Norman Group's [progressive disclosure](https://www.nngroup.com/articles/progressive-disclosure/) motivates showing the bounded first route before expanding into full drilldowns. The evidence walk borrows from provenance and tracing work: [W3C PROV](https://www.w3.org/TR/prov-overview/) for moving from artifact to source and record, and [OpenTelemetry traces](https://opentelemetry.io/docs/concepts/signals/traces/) for representing a causal chain as inspectable linked work.\n\n##### Validation Result record Path\n\n###### Reader validation without durable writes\n\n```text\nPYTHONPATH=src ./repo-pytest tests/test_public_reveal_walkthrough.py -q --basetemp=/tmp/microcosm_public_reveal_walkthrough_pytest --ignore-host-pressure\n./repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n###### What the two commands check\n\n**The focused suite covers the fixture path, exported bundle, digest checks, and four negative cases, and a pass means the density floor was met, bodies verified, overclaims rejected, and records metadata-only for this fixture and bundle only.** The first command runs the focused suite over the fixture path, the exported bundle, digest checks, and the four negative cases. The second confirms the paper-module corpus still resolves. A pass means the walkthrough met the density floor, the copied bodies verified, the overclaims stayed rejected, and the records stayed metadata-only for this fixture and this bundle only.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The narrow claim supported\n\n**The strongest claim is that a cold reader can run a local walkthrough from command to route to result record with every step backed by something the system runs, over the fixture, bundle, manifest, tests, and records.** The strongest claim the evidence supports is narrow: a cold reader can run a local walkthrough from a command to a route explanation to a result record and its source boundary, and every step on that path is backed by something the system runs. The proof boundary is the fixture, the exported bundle, the source-module manifest, the focused tests, and the metadata-only records named above.\n\n###### What it does not claim\n\n**The component claims no launch, hosting, public sharing, recipient work, external model service, or whole-system correctness, keeping authority with the source, manifest, validators, and records rather than any board or count layered over them.** It does not claim launch-scope decision, hosted deployment, publishing-scope decision, recipient work, external model service, secret export, whole-system equivalence, Lean or Lake execution, source-file changes, or whole-system correctness. The scope limit stays with the source, the manifest, the validators, and the records, not with any count, board, or generated card layered over them. Visual boards, screenshots, and copied-body counts are presentation over the record path, not evidence of maturity.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":12300,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/public_reveal_walkthrough.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"Command Line Interface Guidelines","url":"https://clig.dev/","kind":"external_reference","section_id":"first-run-cli-progressive-disclosure-provenance"},{"label":"progressive disclosure","url":"https://www.nngroup.com/articles/progressive-disclosure/","kind":"external_reference","section_id":"first-run-cli-progressive-disclosure-provenance"},{"label":"W3C 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It reads the project's public catalogs and confirms that every accepted component is still tied to a written standard, a documented way to run it, and a saved result record. When one of those ties is missing, it emits a structured finding with a named error code instead of a paragraph of prose.\n\n**Silent drift is the failure: attachments fall out of step one component at a time with nothing breaking, so a gap sits unnoticed until a reader hits the hole.** The problem it solves is silent drift. A spine of accepted components stays coherent only while each one keeps its four attachments: a standard that describes it, a runtime contract that runs it, a result record that holds its last verdict, and a stated limit on what it is allowed to claim. As the spine grows, those attachments fall out of step one component at a time, and nothing breaks when they do. A new component can be accepted while its standard file, registry row, or result-record ref is never added. The gap just sits there until a reader trusts the spine and finds a hole.\n\n**Its output is a metadata-only record of refs, counts, hashes, and verdicts, readable in the open without exporting private source.** The check emits a metadata-only result record: refs, counts, hashes, and verdicts, never the source bodies they describe. So a coverage report can be read in the open without exporting private source.\n\n##### Purpose\n\n###### Coverage projected from the live registry\n\n**Positive coverage is projected from the live public registry rather than stored, so four public surfaces must agree on one component set or a mismatch is a finding.** A loose check fails here in a specific way. If the diagnostic held its own checked-in list of what the spine should contain, that list would itself go stale, and a stale example would quietly become the thing the spine is measured against. So the positive coverage is projected from the live public registry rather than stored. The accepted-component list, the standard rows, the runtime-contract rows, and the result-record refs then have to agree on exactly the same set of components. Any component that appears in one surface but not another is a finding.\n\n###### Five negative fixtures keep it falsifiable\n\n**Five negative fixtures must each surface their expected failure, and the run blocks if one that should fail passes.** The check is also falsifiable. It carries five negative fixtures that each must surface their expected failure. If a fixture that should fail passes, the run is blocked, so the checker cannot silently stop catching a class of gap.\n\n##### How it works\n\nTwo entry points drive the same validator over different inputs.\n\n###### `run` drives the fixture path\n\n** `run` calls `_build_result` with live positive projection and all five negatives evaluated, then attaches a freshness basis and writes result records.** `run` is the fixture path. It calls `_build_result` with `include_negative=True` and `project_positive_from_live=True`, so the positive rows come from live registry state and all five negative fixtures are evaluated. It then attaches a freshness basis and calls `_write_receipts`.\n\n###### `run_diagnostics_bundle` reads the exported bundle\n\n** `run_diagnostics_bundle` reads the three exported JSON inputs directly and can return a cached result when the input digests are unchanged.** `run_diagnostics_bundle` is the exported-bundle path. It calls `_build_result` with `include_negative=False` and `project_positive_from_live=False`, reading the three exported JSON inputs directly. With `reuse_fresh_receipt` it can return a cached result when `_fresh_bundle_receipt` confirms the input digests have not changed.\n\n`_build_result` is the core. In order, it:\n1. loads the payloads through `_load_payloads`, which either reads `standards_inventory.json`, `organ_runtime_contracts.json`, and `diagnostic_policy.json`, or projects those rows from the live registry via `build_diagnostics_input_payloads`;\n2. validates any exported source-module manifest with `_source_module_manifest_result`, which digest-checks each copied body;\n3. runs `scan_paths` over every input with the forbidden-class policy loaded by `load_forbidden_classes`;\n4. computes `_positive_findings` over the accepted-component set and `_negative_findings` over the five negative fixtures;\n5. assembles the status.\n\n###### When status is pass versus blocked\n\n**Status is `pass` only with no positive findings, no missing negatives, no blocking secret hits, and a passing or absent manifest; anything else is `blocked`.** The status is `pass` only when there are no positive findings, no missing negative cases, no blocking secret hits, and the source-module manifest is `pass` or absent. Anything else is `blocked`.\n\n###### `_positive_findings` checks each accepted component\n\n** `_positive_findings` records a gap when any accepted component's inventory row, refs, standard identity, result records, or runtime step is missing, and it rejects overclaim flags.** `_positive_findings` does the coverage work. For each component id in the policy's `accepted_organ_ids`, it looks up the inventory row and the runtime row, and it derives the expected standard identity `std_microcosm_<id>` and the expected file `standards/std_microcosm_<id>.json`. It records a finding when the inventory row is absent, when `standard_id`, `standard_ref`, or `registry_row_ref` is empty, when the standard identity does not match the live file, when `receipt_refs` is empty, or when the runtime row lacks a `cli_command` or a `runtime_step`. It also rejects a policy that sets any overclaim flag such as `release_authorized`.\n\n###### Negative findings, result records, and the compact card\n\n** `_negative_findings` records each fixture's triggered code, `_write_receipts` writes the result, board, and validation records, and `result_card` projects a card that omits the detailed lists.** `_negative_findings` walks the five negative fixtures and records the error code each one is built to trigger. `_write_receipts` writes the result, board, and validation records with `write_json_atomic`, plus an sign-off record when asked. `result_card` projects a compact card that omits the covered-component list, findings, and source refs from its payload while keeping them in the full result.\n\nDiagram of the mechanism (10 steps). Diagram source & refs\n\n```text\nflowchart TD inventory[\"standards_inventory\"] contracts[\"runtime contract inventory\"] policy[\"diagnostic_policy\"] negatives[\"five negative fixtures\"] build[\"_build_result\"] positive[\"_positive_findings: coverage per component\"] negative[\"_negative_findings: expected failures\"] scan[\"scan_paths: secret exclusion\"] status[\"status: pass or blocked\"] result records[\"_write_receipts: metadata-only records\"] inventory --> build contracts --> build policy --> build negatives --> build build --> positive build --> negative build --> scan positive --> status negative --> status scan --> status status --> result records\n```\n\n##### Negative cases\n\n**Five negative fixtures each map to one expected error code, and a run that fails to surface any of them is blocked:** Five negative fixtures each map to one expected error code, and a run that fails to surface any of them is blocked:\n- `missing_standard_ref` yields `STANDARDS_META_MISSING_STANDARD_REF`\n- `unmapped_accepted_organ` yields `STANDARDS_META_UNMAPPED_ACCEPTED_ORGAN`\n- `missing_receipt_ref` yields `STANDARDS_META_MISSING_RECEIPT_REF`\n- `release_overclaim` yields `STANDARDS_META_AUTHORITY_OVERCLAIM`\n- `private_source_leakage` yields `STANDARDS_META_PRIVATE_SOURCE_FORBIDDEN`\n\n###### What the current fixture reports\n\n**On the current fixture the sign-off record reports `status: pass` with 88 accepted components fully mapped, all five error codes present, and zero blocking secret hits.** On the current fixture the sign-off record reports `status: pass` with `accepted_organ_count: 88`, `standard_mapping_count: 88`, `runtime_contract_count: 88`, all five expected error codes present, and `secret_exclusion_scan.blocking_hit_count: 0`.\n\n##### Prior Art Grounding\n\n###### Grounded in schema and contract validation\n\n**The design follows JSON Schema and OpenAPI validation practice, applying it to the launch boundary as bounded evidence about consistency rather than a claim of authority.** This component is grounded in schema and contract validation practice rather than in a claim that a diagnostic creates authority. JSON Schema treats a schema as a machine-readable vocabulary for validating structured JSON data, and OpenAPI uses interface descriptions so a consumer can understand an API without reading its source. The check applies that pattern to the launch boundary: standards, adapter contracts, result records, and scope limits are validated as public projections, and the check stays bounded evidence about consistency.\n- JSON Schema validation and structured-data constraints: https://json-schema.org/\n- OpenAPI interface descriptions and conformance expectations: https://spec.openapis.org/oas/latest.html\n\n##### Validation Result record Path\n\n###### Reader validation without durable writes\n\n```text\n./repo-pytest tests/test_standards_meta_diagnostics.py -q \\\n  --basetemp=/tmp/microcosm_standards_meta_diagnostics_pytest\n./repo-python scripts/build_doctrine_projection.py \\\n  --check-paper-module-corpus\n```\n\n###### What a pass certifies\n\n**A pass means positive inputs cover the accepted set, the five negatives surface their named codes, and the paper-module corpus is in parity, as validator evidence about consistency.** A pass means the positive inputs cover the accepted-component set, the five negative cases surface their named error codes, and the paper-module corpus is in parity. It is validator evidence about consistency.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim and its ceiling\n\n**The evidence supports only that public inputs project consistently into one inspectable metadata-only record; the check mutates nothing, exports no source, and excludes launch.** The strongest claim the evidence supports is that the public standards inventory, runtime contracts, accepted-component refs, result-record refs, diagnostic policy, and secret-exclusion scan are projected consistently into one metadata-only result record a reader can inspect. The proof boundary is those public inputs and the sign-off and validation records built from them. The scope limit stops there: the check does not mutate the standards registry, become source authority, export private source, authorize providers, include launch operations or public sharing, or prove whole-system correctness.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":10887,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/standards_meta_diagnostics.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"}],"grouped_relation_feeds":[{"relation":"abides by","count":3,"links":[{"id":"AX-6","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Open-world epistemics","url":"doctrine.html#dcard-ax-6"},{"id":"AX-11","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Executable grammar before doctrine 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ceiling"]},{"id":"paper_module.finance_forecast_evaluation_spine","title":"Finance Forecast Evaluation Spine","relative_href":"docs/paper-module-finance-forecast-evaluation-spine.html","read_href":"docs/paper-module-finance-forecast-evaluation-spine.html","context_href":"docs/rules-and-ideas.html#paper-module-finance-forecast-evaluation-spine","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"Forecast-evaluation component: Diebold-Mariano / Hansen-SPA / stationary-bootstrap stats over synthetic fixtures with typed refusal discipline; no market authority.","summary":"Runs admissible forecast-evaluation statistics (Diebold-Mariano loss-differential, Harvey-Leybourne-Newbold small-sample correction, Hansen SPA with recentering, Politis-Romano stationary bootstrap, Bartlett HAC long-run variance, purged/embargoed CV) over synthetic market-shaped fixtures and copied source bodies; refusal discipline returns typed refusals (horizon>=sample length, too-small samples, leakage-prone splits, missing SciPy, advice-shaped claims) instead of crashing; computed-statistic and refused-because-inadmissible are both valid validator outcomes.","authority_ceiling":"Synthetic market-shaped fixtures only; not investment or trading decisions, no live market data, no track record or performance claim, mutates no optimizer, SciPy absence is a typed HLN refusal.","abstract":"Forecast-evaluation component: Diebold-Mariano / Hansen-SPA / stationary-bootstrap stats over synthetic fixtures with typed refusal discipline; no market authority.","teleology":{"section_id":"","text":"Forecast-evaluation component: Diebold-Mariano / Hansen-SPA / stationary-bootstrap stats over synthetic fixtures with typed refusal discipline; no market authority.","source":"capsule_summary_line"},"body_markdown":"###### Component runs forecast-evaluation\n\n**This component runs forecast-evaluation statistics over synthetic, market-shaped fixtures, records both the numbers computed and the cases refused, and writes a metadata-only result record from the copied econometric code.** `finance_forecast_evaluation_spine` runs forecast-evaluation statistics over synthetic, market-shaped fixtures and records both the numbers it computed and the cases it refused. It is a public component with runnable source. It reads a synthetic loss matrix and paired loss series, checks the request is admissible, runs the copied econometric code, and writes a metadata-only result record.\n\n###### Value is evaluation\n\n**The value is the evaluation posture, not a claim about any asset.** The value is the evaluation posture, not a claim about any asset.\n\n###### Result record carries\n\n**The result record carries computed statistics when the fixture supports them and typed refusals when it does not, and both are treated as valid outcomes.** The result record carries computed statistics when the fixture supports them and typed refusals when it does not. Both are treated as valid outcomes.\n\n##### Purpose\n\n###### Why model comparison is hard\n\n**A lower average loss does not establish better forecasting because losses are autocorrelated, samples are short, and a careless split lets a model peek past its forecast point.** Comparing two forecasting models is harder than it looks. A lower average loss does not establish one model predicts better. Losses are autocorrelated, samples are short, and a careless time split lets a model peek past the point it was supposed to forecast from. A \"loose\" comparison, one that reports a mean and stops, produces a number that reads like a result but is not one.\n\n###### The econometric machinery it carries\n\n**It carries the careful machinery economists use, Diebold-Mariano with a Bartlett HAC variance, the HLN correction, Hansen's SPA test, a model confidence set, and a stationary bootstrap, run on synthetic data only.** This component carries the machinery economists use to answer the question carefully: the Diebold-Mariano loss-differential statistic with a Bartlett HAC long-run variance, the Harvey-Leybourne-Newbold small-sample correction, Hansen's test for superior predictive ability, a model confidence set, and a Politis-Romano stationary bootstrap. It runs them on synthetic data only, and it refuses when the input is inadmissible instead of returning a number that could be misread.\n\n##### How it works\n\n###### A thin runtime over `SPEC`\n\n**The component is a thin runtime over a `CrownJewelSpec` that fixes the four required inputs, names the three negative cases, binds the source manifest, and restricts reads to anchors in four copied modules.** The component is a thin runtime over a `CrownJewelSpec` named `SPEC`. The spec fixes the four required inputs (`family_loss_matrix.json`, `paired_loss_series.json`, `finance_boundary_policy.json`, `projection_protocol.json`), names the three negative cases, binds the source manifest, and restricts source-open reads to required anchors in four copied modules: `model_selection_stats.py`, `spa_statistics.py`, `loss_differentials.py`, and `family_loss_matrix.py`.\n\n###### Two entrypoints, one bundle difference\n\n** `run` and `run_finance_forecast_bundle` both delegate to `run_crown_jewel_organ`, differing only in that the bundle entrypoint sets `input_mode` to the exported-bundle mode.** `run` and `run_finance_forecast_bundle` both delegate to `run_crown_jewel_organ` with `evaluate` and `evaluate_negative_case`. The bundle entrypoint sets `input_mode` to the exported-bundle mode; that is the only difference.\n\n###### `evaluate` loads inputs, order matters\n\n** `evaluate` loads the loss matrix, paired loss series, and boundary policy, then calls `_evaluate_payloads`, where the order of operations matters.** `evaluate` loads the loss matrix, the paired loss series, and the boundary policy, then calls `_evaluate_payloads`. That function is where the order of operations matters.\n\n###### Guards run before any computation\n\n** `_policy_findings` and `_lookahead_split_findings` run first, blocking on any non- `False` advice flag or forecast-date leakage, so an inadmissible request never reaches a statistic.** The guards run first. `_policy_findings` walks `FALSE_BOUNDARY_FLAGS` (`investment_advice_authorized`, `trading_advice_authorized`, `live_market_data_authorized`, `track_record_claim_authorized`, `performance_claim_authorized`, `optimizer_mutation_authorized`) and raises `FINANCE_NO_ADVICE_OVERCLAIM` for any flag that is not exactly `False`. `_lookahead_split_findings` parses the row dates with `_parse_iso_date` and raises `FINANCE_LOOKAHEAD_SPLIT_FORBIDDEN` when a row's `subject_as_of` reaches its `event_start`, or when an event window ends before it begins. If either guard produces a finding, or the source manifest did not pass, `_evaluate_payloads` returns a blocked record and no statistics run. An inadmissible request never reaches a computation.\n\n###### Branching to standalone or subprocess stats\n\n**Only after the guards pass does `_is_exported_finance_bundle` branch to a declared, unexecuted contract for the exported bundle or a real child-interpreter subprocess for the first-wave fixture with `scipy` blocked.** Only after the guards pass does the branch happen. `_is_exported_finance_bundle` decides the path. For the exported public bundle, `_standalone_exported_statistics_contract` returns a declared contract: the statistics are named and marked `executed: false`, with `not_a_live_run` and `real_runtime_receipt: false`, so the standalone bundle does not depend on a live source subprocess. For the first-wave fixture, `_run_stats_subprocess` (memoized by `_run_stats_subprocess_cached`) shells out to a child interpreter that imports the copied `tools.finance` modules. It runs `reality_check_summary`, `spa_summary`, and `model_confidence_set_summary` with `min_sample` 10, `bootstrap_reps` 40, and `seed` 1729, `paired_loss_summary` with `min_paired` 10, and `stationary_bootstrap_indices` with `reps` 5 and `seed` 1729. The subprocess deliberately blocks `scipy` from importing before calling `harvey_leybourne_newbold_correction`, which forces a typed refusal instead of a fabricated t-distribution p-value.\n\n###### Checking the subprocess payload\n\n**Back in the parent, `_evaluate_payloads` checks each statistic field against its expected status, blocks on any mismatch, and keeps source bodies out with `body_in_receipt: false` and hashed stdout and stderr.** Back in the parent, `_evaluate_payloads` reads the subprocess payload and checks each field: Reality Check and SPA must be `computed_bootstrap`, the model confidence set must report `implemented`, Diebold-Mariano must be `computed_hac_normal_approximation`, and the HLN entry must be a refusal with reason `scipy_unavailable_for_t_distribution`. Any mismatch becomes a finding and the record is blocked. Source bodies never enter the record; every emitted stats object keeps `body_in_receipt: false`, and the subprocess is summarized by `stdout_sha256` and `stderr_sha256` from `_sha256_text`.\n\n###### Negative cases reuse the same path\n\n** `evaluate_negative_case` mutates one input to reproduce a specific fault, re-runs `_evaluate_payloads`, and confirms the expected stable error code appears.** `evaluate_negative_case` reuses the same path. It loads the fixture, mutates one input to reproduce a specific fault (moving a row's `subject_as_of` onto its `event_start` for the leakage case, or setting `investment_advice_authorized` true for the overclaim case), re-runs `_evaluate_payloads`, and confirms the expected stable error code appears.\n\nFunction Role `run`, `run_finance_forecast_bundle` Entrypoints; forward the spec and evaluators to `run_crown_jewel_organ` `evaluate` Loads the three fixture files, then calls `_evaluate_payloads` `_evaluate_payloads` Runs guards, blocks on failure, else branches to standalone or subprocess stats `_policy_findings` Rejects any advice/live-data/performance flag that is not `False` `_lookahead_split_findings` Rejects rows where the forecast date reaches the event window `_run_stats_subprocess` Runs the copied `tools.finance` code in a child interpreter with `scipy` blocked `_standalone_exported_statistics_contract` Declares, without recomputing, the statistics for the exported bundle `evaluate_negative_case` Mutates one input per case and confirms the expected error code Diagram of the mechanism (8 steps). Diagram source & refs\n\n```text\nflowchart TD Inputs[\"Synthetic fixture family_loss_matrix, paired_loss_series, finance_boundary_policy\"] Evaluate[\"evaluate loads inputs\"] Guards[\"_policy_findings and _lookahead_split_findings\"] Blocked[\"Blocked record no statistics run\"] Branch{\"Exported bundle?\"} Standalone[\"_standalone_exported_statistics_contract declared, not recomputed\"] Subprocess[\"_run_stats_subprocess DM/HAC, SPA, MCS, bootstrap, HLN refusal\"] Record[\"metadata-only result record refs, digests, verdicts, scope boundaries\"] Inputs --> Evaluate Evaluate --> Guards Guards -->|\"finding or bad manifest\"| Blocked Guards -->|\"admissible\"| Branch Branch -->|\"yes\"| Standalone Branch -->|\"no\"| Subprocess Standalone --> Record Subprocess --> Record Blocked --> Record\n```\n\n##### Negative cases\n\n###### Fixture declares three\n\n**The fixture declares three negative cases, each bound to a stable code in `EXPECTED_NEGATIVE_CASES`.** The fixture declares three negative cases, each bound to a stable code in `EXPECTED_NEGATIVE_CASES`.\n\n###### Three faults, three stable codes\n\n**A moved forecast cutoff raises `FINANCE_LOOKAHEAD_SPLIT_FORBIDDEN`, a flipped advice flag raises `FINANCE_NO_ADVICE_OVERCLAIM`, and an unavailable `scipy` yields a typed HLN refusal instead of a guessed p-value.** `finance_leakage_lookahead_split` moves a forecast cutoff onto the event it forecasts and must raise `FINANCE_LOOKAHEAD_SPLIT_FORBIDDEN`. `finance_no_advice_overclaim` flips a boundary flag to authorize advice and must raise `FINANCE_NO_ADVICE_OVERCLAIM`. `finance_hln_dependency_refusal` is the case where `scipy` is unavailable for the small-sample t-distribution; the correct behavior is a typed refusal carrying `FINANCE_HLN_TYPED_REFUSAL_REQUIRED`, not a guessed p-value.\n\n##### Prior Art Grounding\n\n###### Standard forecast-evaluation econometrics\n\n**The statistics are standard forecast-evaluation econometrics anchored by Diebold-Mariano, the HLN correction, Hansen's SPA test, and proper scoring rules, with a lookahead-split guard against temporal leakage.** The statistics are standard forecast-evaluation econometrics, not a trading system. The core anchors are the [Diebold-Mariano test for comparing predictive accuracy](https://www.nber.org/papers/t0169), the Harvey-Leybourne-Newbold small-sample correction ( [DOI reference](https://doi.org/10.1016/S0169-2070%2896%2900719-4)), Hansen's [test for superior predictive ability](https://ideas.repec.org/a/bes/jnlbes/v23y2005p365-380.html), and proper-scoring-rule work such as [Gneiting and Raftery](https://sites.stat.washington.edu/people/raftery/Research/PDF/Gneiting2007jasa.pdf). The lookahead-split guard follows the financial machine-learning concern that temporal leakage makes backtests look stronger than they are.\n\n##### Validation Result record Path\n\n###### Run from repo\n\n**Run from the repo root, without mutating durable result or projection surfaces.** Run from the repo root, without mutating durable result or projection surfaces:\n\n```text\nPYTHONPATH=src ./repo-pytest tests/test_finance_forecast_evaluation_spine.py -q --basetemp=/tmp/microcosm_finance_forecast_evaluation_spine_pytest\n./repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n###### To exercise component\n\n**To exercise the component directly and inspect a result record, run the CLI and read the emitted JSON.** To exercise the component directly and inspect a result record, run the CLI and read the emitted JSON:\n\n```text\nmicrocosm finance-forecast-evaluation-spine run --input fixtures/first_wave/finance_forecast_evaluation_spine/input --out receipts/first_wave/finance_forecast_evaluation_spine\n```\n\n###### What a pass means here\n\n**A green run computed the positive-fixture statistics, raised all three stable negative codes, kept the advice and live-data flags false, and admitted no source body.** A pass means the positive fixture computed the statistics, the three negative cases raised their stable codes, the advice and live-data flags stayed false, and no source body entered the record.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim\n\n**On synthetic, market-shaped fixtures it computes admissible forecast-evaluation statistics, returns typed refusals for inadmissible ones, and keeps copied source bodies out of every record.** The strongest thing this component supports: on synthetic, market-shaped fixtures it computes admissible forecast-evaluation statistics and returns typed refusals for the inadmissible ones, while keeping copied source bodies out of every record. That is the proof boundary, and the validation result records above are what confirm it.\n\n###### The narrow scope limit\n\n**This is not investment or trading decisions, uses no live market data, proves no track record or performance claim, mutates no optimizer, and certifies no strategy.** The scope limit is narrow. This is not investment or trading decisions. It uses no live market data. It proves no track record and no performance claim. It mutates no optimizer and certifies no strategy. Absence of `scipy` is a typed HLN refusal, nothing stronger. The generated diagram and Atlas entry are navigation projections; they help a reader find the module and do not widen what it proves.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":13852,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/finance_forecast_evaluation_spine.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"Diebold-Mariano test for comparing predictive accuracy","url":"https://www.nber.org/papers/t0169","kind":"external_reference","section_id":"standard-forecast-evaluation-econometrics"},{"label":"DOI reference","url":"https://doi.org/10.1016/S0169-2070%2896%2900719-4","kind":"external_reference","section_id":"standard-forecast-evaluation-econometrics"},{"label":"test for superior predictive 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under microcosm_core.engine_room, verifies the 14 controller-selected jewel targets, checks each owned staged bundle surface (module source, fixture input, fixture manifest, paper module, standard, tests), executes the staged demo through the public fixture chain, and observes a negative fixture where an expected target is intentionally absent.","authority_ceiling":"Validates the declared public composition contract only; not deployment posture, not whole-system equivalence, not a frontier theorem-proving claim, not a complete security proof, not benchmark validation, not launch-scope decision.","abstract":"Composition component: verifies the 14 staged Engine Room jewel targets and their owned bundle surfaces through the public fixture chain; composition contract only.","teleology":{"section_id":"","text":"Composition component: verifies the 14 staged Engine Room jewel targets and their owned bundle surfaces through the public fixture chain; composition contract only.","source":"capsule_summary_line"},"body_markdown":"###### Engine room is\n\n**The Engine Room is fourteen public capabilities staged as components, and `engine_room_demo` is the one that checks all of them together.** The Engine Room is fourteen separate capabilities staged as public components: four Lean proof-search targets, a metabolism runtime with its reconciler, command singleflight, a generated-projection drift gate, a derived-fact engine, a public-projection leak gate, an egress self-compliance gate, a navigation-fitness benchmark, a bridge-campaign DAG, and a knowledge router over the reference library. `engine_room_demo` is the component that checks all of them together.\n\n###### Asks whether ten\n\n**It asks whether the ten staged bundles cover the fourteen requested targets, each still owns its files, and each fixture still runs, not whether any bundle is finished.** It answers one question. Do the ten staged bundles cover the fourteen targets the controller asked for, does each bundle still own its full set of files, and does each fixture still run? A pass means the set is complete and every fixture executed. It does not mean that any single bundle is finished or correct.\n\n###### Reads fixtures runs\n\n**It reads fixtures, runs the shared audit, and writes four metadata-only records, reading the shared registry and generated atlas but never writing to them.** The component reads fixtures, runs the shared audit, and writes four metadata-only result records: a result, a board, a validation record, and an optional fixture sign-off record. It reads the shared component registry and the generated atlas but never writes to them.\n\n##### Purpose\n\n###### A stored list would drift and mislead\n\n**Without this component a reader trusts fourteen claims one at a time, and a stored list of target names would go stale the moment the bundle set changed and start agreeing with a wrong answer.** Without this component a reader has to trust fourteen claims one at a time and hope the set is complete. A loose check would compare the covered targets against a stored list of the fourteen names. That list goes stale the moment the bundle set changes, and a stale list can start agreeing with a wrong answer.\n\n###### Coverage recomputed from `EXPECTED_JEWEL_TARGETS`\n\n** `engine_room_demo` keeps no stored list: the fourteen names live in the frozen `EXPECTED_JEWEL_TARGETS`, and coverage is recomputed from the bundles every run, so nothing drifts out of sync.** `engine_room_demo` keeps no stored list. The fourteen expected names live in one frozen constant, `EXPECTED_JEWEL_TARGETS`, and the covered set is recomputed from the bundles on every run. Coverage is the union of what the bundles declare, checked against that constant. There is nothing to drift out of sync.\n\n###### Six files per bundle mark ownership\n\n**A bundle owns its surface only when six files exist for it, module source, fixture input, manifest, paper module, standard, and test, and the audit names any that are missing.** A bundle counts as owning its surface only when six files exist for it: module source, fixture input directory, fixture manifest, paper module, standard, and test. The audit checks all six per bundle and names any that are missing.\n\n##### How it works\n\n###### Wrapper is src\n\n**The wrapper is `src/microcosm_core/organs/engine_room_demo.py`.** The staged runner it calls is `src/microcosm_core/engine_room/demo.py`. Execution runs in this order.\n\n###### `build_result` folds cases into one status\n\n** `build_result` loads every JSON case, runs `_evaluate_case` on each, and passes only with at least one positive case, all positives passing, and every negative observed to fail.** `build_result` is the entry point. It calls `_fixture_cases` to load every JSON case under the input path, runs `_evaluate_case` on each one, then folds the rows into a status. The run passes only when there is at least one positive case, every positive case passed, and every negative case was observed to fail.\n\n###### `_evaluate_case` routes by `case_type`\n\n** `_evaluate_case` sends a positive case to `audit_controller_coverage`, a negative case to `_target_override_case`, and any other type to a fail row coded `ENGINE_ROOM_UNKNOWN_CASE_TYPE`.** `_evaluate_case` dispatches on `case_type`. A positive case calls `audit_controller_coverage`. A negative case calls `_target_override_case`. Any other type returns a fail row with the code `ENGINE_ROOM_UNKNOWN_CASE_TYPE`.\n\n###### `audit_controller_coverage` unions targets both ways\n\n** `audit_controller_coverage` runs the six-file check per bundle, unions declared targets, subtracts `EXPECTED_JEWEL_TARGETS` both ways for missing and unexpected, and reads shared surfaces for visibility only.** `audit_controller_coverage` is the audit. It walks the ten entries in `CAPSULES`, builds a `_capsule_surface_row` for each (the six-file check), and unions their declared targets. It subtracts `EXPECTED_JEWEL_TARGETS` both ways to find missing and unexpected targets. When asked, it calls `run_demo` to execute the fixtures. It reads the shared registry, the accepted-component list, and the atlas through `_shared_surface_ids` for visibility only, and always returns `shared_registry_mutated: false`. The staged pass needs no missing targets, no unexpected targets, no bundle missing a file, and a passing fixture run.\n\n###### `run_demo` calls each bundle's evaluator\n\n** `run_demo` is the execution spine: for each bundle it imports the module and calls `evaluate_fixture_dir` or `validate_fixture_dir`, passing only when every selected fixture ran and passed.** `run_demo` is the execution spine. For each bundle it calls `run_capsule`, which imports the bundle module and calls its declared evaluator, either `evaluate_fixture_dir` or `validate_fixture_dir`. It records a compact per-bundle status and the covered target list. A pass here means every selected bundle fixture ran and passed.\n\n###### `run` writes four metadata-only records\n\n**The wrapper's `run` writes four records through `write_json_atomic`, a result, a board, a validation result record, and an optional sign-off record, each carrying `body_in_receipt: false`.** The wrapper's `run` writes the four records with `write_json_atomic`: `engine_room_demo_result.json`, `engine_room_demo_board.json`, `engine_room_demo_validation_receipt.json`, and, when an sign-off path is given, `engine_room_demo_fixture_acceptance.json`. Every record carries `body_in_receipt: false`.\n\nFunction Role `build_result` Loads the fixture cases, evaluates each, and folds them into one pass or fail status. `_evaluate_case` Routes a case by `case_type` to the coverage audit or the negative-case check. `audit_controller_coverage` Unions declared targets, runs the six-file check per bundle, reads shared surfaces for visibility only. `_capsule_surface_row` Confirms the six required files exist for one bundle and lists any missing kind. `run_demo`, `run_capsule` Import each bundle and call its declared evaluator over the public fixtures. `_target_override_case` Recomputes the covered set live and fails when a named target is genuinely absent. `run` Writes the four metadata-only records through `write_json_atomic`. Diagram of the mechanism (10 steps). Diagram source & refs\n\nSource refs\n\npositive: `audit_controller_coverage`\n\n```text\nflowchart TD A[\"fixture cases\"] --> B[\"build_result\"] B --> C[\"_evaluate_case\"] C --> D[\"positive: audit_controller_coverage\"] C --> E[\"negative: _target_override_case\"] D --> F[\"_capsule_surface_row x10 bundles\"] D --> G[\"run_demo --> run_capsule\"] D --> H[\"shared surfaces read-only\"] B --> I[\"run writes 4 metadata-only records\"]\n```\n\n##### Negative cases\n\n###### The negative case must be observed to fail\n\n**One positive and one negative fixture run, and `_target_override_case` recomputes coverage live so the missing-target negative fails, since a silently passing negative would fail the whole run.** The fixture set names one positive case, `positive_controller_audit.json`, and one negative case, `missing_expected_target_negative.json`. The negative case names an expected target that is not in the covered set. `_target_override_case` recomputes the covered targets live, finds the named target missing, and returns a fail row with the code `ENGINE_ROOM_EXPECTED_TARGET_MISSING`. The run passes only when that negative case is observed to fail, so a negative case that silently passed would fail the whole run. There is no frozen answer key for the negative check to drift toward.\n\n##### Prior Art Grounding\n\n###### Lineage in integration testing and CI composition\n\n**This is ordinary integration-testing and CI-composition practice, anchored by integration testing, pytest fixtures, and staged status checks, without turning fixture evidence into a launch claim.** This is ordinary integration-testing and CI-composition practice: several component checks are assembled into one demo path, negative fixtures prove refusal behaviour, and the records summarise what ran without turning fixture evidence into a launch claim. Reasonable public anchors are IBM's [integration testing](https://www.ibm.com/think/topics/integration-testing) overview, which frames the question as whether composed modules interact as intended, [pytest fixtures](https://docs.pytest.org/en/stable/reference/fixtures.html) as a common shape for reusable synthetic inputs, and [GitHub Actions](https://github.com/features/actions) as a widely used surface for composing staged checks with explicit status.\n\n##### Validation Result record Path\n\n###### Run component and\n\n###### What a pass proves here\n\n**A pass means the fixtures ran, the ten bundles covered the fourteen targets with no file missing, the negative failed, and this Markdown matches its generated structured source record.** A pass means the fixtures ran, the ten bundles covered the fourteen expected targets with no file missing, the negative case was observed to fail, and this Markdown still matches its generated structured source record. It is not evidence of anything past that.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The bounded strongest claim\n\n**At most, the ten bundles cover the fourteen targets, each owns its six files, and each fixture passed, with `shared_registry_mutated: false` fixing the scope limit.** The strongest claim the evidence supports is bounded: the ten staged Engine Room bundles together cover the fourteen expected targets, each bundle owns its six required files, and each fixture ran and passed on the selected public inputs. The proof boundary is exactly that fixture run plus the six-file check. Nothing past it is claimed. It is not deployment posture, not whole-system equivalence, not a frontier theorem-proving claim, not a complete security proof, not benchmark validation, and not launch-scope decision. The scope limit is fixed by `shared_registry_mutated: false`: the audit reads the shared registry, accepted-component list, and atlas for visibility, but changes none of them, so a green run cannot imply registry authority it does not hold.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":11044,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/engine_room_demo.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"integration testing","url":"https://www.ibm.com/think/topics/integration-testing","kind":"external_reference","section_id":"lineage-in-integration-testing-and-ci-composition"},{"label":"pytest 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Replays synthetic episodes (a scoped preference later contradicted) and validates temporal-conflict resolution, negative cases, non-public-state exclusion, and scope limits with metadata-only result records.","authority_ceiling":"Synthetic replay fixture only; not a live memory product, private transcript export, source-authority claim, or launch claim.","abstract":"Synthetic replay fixture for an agent-memory honesty contract: models scoped-preference episodes and checks temporal-conflict handling; no live memory product.","teleology":{"section_id":"","text":"Synthetic replay fixture for an agent-memory honesty contract: models scoped-preference episodes and checks temporal-conflict handling; no live memory product.","source":"capsule_summary_line"},"body_markdown":"###### Stale memory rows must lose authority\n\n**A memory store can keep serving a fact that later turns have already contradicted, and this component checks the bookkeeping around that one stale-row failure.** An agent that remembers can quietly start trusting a row that has gone stale. A user states a preference, the world changes, a later turn contradicts the earlier one, and a naive memory store keeps serving the old fact as if it were still true. This component checks the bookkeeping around that one failure.\n\n###### Three episodes exercise update and deletion\n\n**A canned three-episode story writes two rows, updates and deletes one on new evidence, then replays with memory on and off to see whether each change and the memory-on answer carried real evidence.** It runs a canned three-episode story through a validator. Episode A writes two memory rows, a preference and a tool-result fact. Episode B learns one is out of date, updates the preference, and deletes the stale fact. Episode C replays the same task twice, once with memory on and once with memory off. The validator emits a metadata-only result record that says whether the update and the deletion each carried the conflict and downgrade evidence they were supposed to, and whether the memory-on run earned its answer through logged evidence rather than through the final wording alone.\n\n###### Synthetic memory accounting, not a live product\n\n**It is not a live memory product or a transcript export: the fixture rows are real sanitized data, but the check is one synthetic contract, not a measure of how well an assistant remembers.** It is not a live memory product, a private transcript export, or a claim that any real user memory was handled. The fixture rows are byte-aligned with an exported sanitized memory stream, so the example is real data, but the check is about one synthetic contract, not about how well an assistant remembers.\n\n##### Purpose\n\n###### The validator distrusts its labels\n\n**The validator does not trust the labels it is given, so a row can declare `UPDATE` with a plausible conflict edge and still be quarantined as a stale row in costume.** The unusual choice is that the validator does not trust the labels it is given. A row can declare `decision = UPDATE`, attach a plausible conflict-edge reference, and still be quarantined. A loose checker that accepts the label would pass exactly the case this component exists to catch: a stale row wearing the costume of a valid supersession.\n\n###### Conflict re-derived from verifiable fields\n\n**The checker re-derives each conflict from fields it can verify itself \u2014 order, timestamp, priority, and source trust \u2014 and rejects any supersession that is not later, drops priority, or leans on lower-trust evidence.** So the checker re-derives the conflict from fields it can verify on its own: episode order, event timestamp, memory priority, and source-trust score. An update or delete that claims to supersede an earlier write but is not timestamped after it, or that drops priority, or that leans on lower-trust evidence than the write it replaces, is rejected before it can affect the replay. The label is a claim to be recomputed, not an authority.\n\n##### How it works\n\n###### Two entry points over `_build_result`\n\n** `run` drives `_build_result` with negative cases and writes four result record files; `run_memory_bundle` points it at an exported bundle with the negative cases switched off.** `run` is the fixture entry point. It calls `_build_result` with the negative cases included, attaches a freshness basis, and hands the result to `_write_receipts`, which writes four files: a result record, a board, a validation record, and an sign-off record. `run_memory_bundle` is the second entry point. It points `_build_result` at an exported source-open bundle with the negative cases switched off, and writes a single bundle validation result.\n\n###### One blocked check blocks the record\n\n** `_build_result` runs the pipeline in order and passes only when every check clears and negative-case coverage is complete, so a single blocked check blocks the whole record.** `_build_result` runs the pipeline in order and sets `status` to pass only when every check below passes and the negative-case coverage is complete. A single blocked check blocks the whole record.\n\nFunction What it checks `scan_paths` No forbidden key or secret-shaped value appears in any input or copied source file `build_public_memory_conflict_trace` A public execution trace spans the five events and two cold replays `validate_projection_protocol` The projection cites source refs, result records, target symbols, and denies copying private thread bodies `validate_memory_policy` `ADD`, `UPDATE`, `DELETE`, `NOOP` are the only decisions, and no authority flag is set `validate_memory_episodes` The five event rows survive temporal, semantic, source, and trace checks `validate_replay_observations` The memory-on and memory-off rows are paired and share an answer-delta record `validate_source_module_imports` Five copied source bodies match their digests, with `body_in_receipt=false`\n\n###### Coarse temporal order comes first\n\n** `validate_memory_episodes` first runs `_apply_temporal_order_checks`, the coarse rule that a conflict edge must follow an earlier accepted write and a replay `NOOP` its evidence, quarantining any out-of-order row.** The heart of the check is inside `validate_memory_episodes`, which calls two functions in sequence. `_apply_temporal_order_checks` applies the coarse rule: a conflict edge must land after some earlier accepted write, and a replay `NOOP` must land after the conflict and downgrade evidence it depends on. Any row that breaks the order is marked `quarantine`.\n\n###### Four semantic checks against the prior write\n\n** `_apply_conflict_semantic_recompute` then checks every `UPDATE` or `DELETE` against the last accepted write for existence, a strictly later timestamp, non-falling priority, and source trust at or above `0.6`.** `_apply_conflict_semantic_recompute` then recomputes each conflict group in sorted event order. It keeps the last accepted write per group and, when an `UPDATE` or `DELETE` arrives, checks four things against that prior write. The prior write must exist. The new timestamp must be strictly later. Priority must not fall below the prior value. Source trust must not fall below it either, and the floor for a trusted source is `0.6`. A row that fails any of these gets a `semantic_` reason code and is quarantined, which drops it out of the accepted counts.\n\n###### The positive floor the fixture must clear\n\n**Only accepted, trace-backed rows count toward a floor of all four decisions, two conflict edges, one stale downgrade, and one prompt-adoption reference, which the canned story supplies.** Only accepted, trace-backed rows count toward the positive floor. The fixture must show all four decisions at least once, at least two conflict edges, at least one stale downgrade, and at least one prompt-adoption reference. The canned story supplies two `ADD` rows, one `UPDATE`, one `DELETE`, and one `NOOP` across `episode_a_initial_memory_write`, `episode_b_temporal_conflict`, and `episode_c_cold_replay`.\n\n###### Memory credit needs a resolving evidence handle\n\n** `validate_replay_observations` requires the memory-on row to cite evidence that resolves against accepted rows and refuses final-answer-only credit, while the card carries only counts, digests, and paths.** The output side is checked by `validate_replay_observations`. The memory-on row must cite evidence references that resolve against accepted event rows. Both rows must carry a cold-replay result record reference and share a `replay_group_id`. The pair must be bound by an answer-delta record whose `memory_credit_requires_evidence_handle` flag is true. Final-answer-only credit is refused. The card produced by `result_card` reports counts, digests, and result record paths only. Memory rows, replay rows, source bodies, and private transcript text never enter a result record.\n\nDiagram of the mechanism (8 steps). Diagram source & refs\n\n```text\nflowchart TD Policy[\"Policy: ADD / UPDATE / DELETE / NOOP only\"] --> Add[\"Episode A: two ADD rows\"] Add --> Conflict[\"Episode B: UPDATE + DELETE\"] Conflict --> Order[\"Temporal order check conflict after prior write\"] Order --> Semantic[\"Semantic recompute timestamp, priority, trust >= 0.6\"] Semantic --> Replay[\"Episode C: memory-on vs memory-off\"] Replay --> Delta[\"Answer-delta record evidence handle required\"] Delta --> Card[\"metadata-only result record\"]\n```\n\n##### Negative cases\n\n###### Seven falsification rows, each an error code\n\n**The fixture ships seven rows that must be rejected \u2014 from transcript copying to final-answer-only credit \u2014 each mapped to a named error code, with a missed rejection blocking the run.** The fixture ships seven falsification inputs, each one a memory or replay row that should be rejected. `raw_transcript_export` copies a private transcript body. `private_candidate_auto_promotion` promotes a private candidate without review. `stale_preference_override` lets an old preference win with no downgrade. `memory_as_source_authority` treats recalled memory as source authority. `vector_recall_without_evidence_handle` claims recall with no evidence handle. `final_answer_only_memory_credit` credits memory from the final answer alone. `active_injection_as_authoritative` adopts injected text as authority. Each maps to a named error code, and a run that fails to reject one of them is itself marked blocked through the missing-negative-case check.\n\n###### Plausible one row at a time\n\n**These cases matter because temporal memory bugs look plausible in isolation, yet a tidy stale row, an unresolved memory-on replay, or a tampered source body each still fails.** These cases matter because temporal memory bugs look plausible one row at a time. A stale row with a tidy label is still rejected when its conflict edge is absent or late. A memory-on replay is still rejected when its evidence references do not resolve. A tampered source body still fails its digest.\n\n##### Prior Art Grounding\n\n###### Memory architecture meets stale-memory risk\n\n**It joins the agent-memory architecture lineage of Generative Agents and MemGPT to the stale-and-poisoned-memory risk lineage of AgentPoison, borrowing their accounting questions without claiming a memory product.** The constructive lineage is agent-memory architecture. [Generative Agents](https://arxiv.org/abs/2304.03442) made observation, reflection, and retrieval a concrete memory pattern, and [MemGPT](https://arxiv.org/abs/2310.08560) framed long-context behavior as memory management. The risk lineage is stale and poisoned memory, studied in work such as [AgentPoison](https://arxiv.org/abs/2407.12784) on poisoned retrieval stores and in the broader question of whether an agent updates an invalidated memory once new evidence arrives. This component borrows the accounting questions from that work: which decision was made, which evidence justified it, which older row was downgraded, and whether the memory-on and memory-off replays support any claim beyond a final-answer comparison. It does not claim a memory product.\n\n##### Validation Result record Path\n\n###### Fixture result records stay outside the working tree\n\n**The first-wave validator runs against the checked-in fixture and writes its result record to a disposable path outside the working tree.** Run the first-wave fixture validator and write its result record outside the working tree:\n\nThen run the exported bundle validator:\n\n```text\ncd microcosm-substrate && PYTHONPATH=src ../repo-python -m microcosm_core.organs.agent_memory_temporal_conflict_replay run-memory-bundle --input examples/agent_memory_temporal_conflict_replay/exported_memory_temporal_conflict_bundle --out /tmp/agent_memory_temporal_conflict_bundle_receipt --card > /tmp/agent_memory_temporal_conflict_bundle_card.json\n```\n\nThe focused regression test and the corpus projection check are:\n\n```text\ncd microcosm-substrate && ../repo-pytest tests/test_agent_memory_temporal_conflict_replay.py\n./repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n###### What a pass covers\n\n**A pass means the checked rows, digests, traces, negative cases, and validation result records agreed for this fixture under test.** A pass means the checked rows, digests, traces, negative cases, and validation result records agreed for this fixture under test.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim\n\n**The strongest supported statement is fixture-level: this synthetic replay respected the declared temporal-conflict contract across every checked decision, edge, downgrade, replay, and negative case.** The strongest thing this component supports is a fixture-level statement: this synthetic replay respected the declared temporal-conflict contract under the checked inputs. Decisions, conflict edges, stale downgrades, paired cold replays, answer-delta accounting, manifest digests, and the seven negative cases were all checked.\n\n###### What stays refused past the fixture\n\n**Past that it measures no live memory system and exports nothing private, calls no provider, mutates no source, and excludes launch; a stronger claim would need a new standard and evidence.** It refuses everything past that. It does not measure the quality of any live memory system, export a private transcript, auto-promote a private candidate, treat memory recall as source authority, adopt injected text, call a provider, change source files, or include launch operations. The proof boundary is the fixture and the exported bundle. The scope limit is bounded validator evidence, not whole-system correctness. A stronger memory claim would need a new standard and new evidence.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":13868,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/agent_memory_temporal_conflict_replay.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"MemGPT","url":"https://arxiv.org/abs/2310.08560","kind":"external_reference","section_id":"memory-architecture-meets-stale-memory-risk"},{"label":"AgentPoison","url":"https://arxiv.org/abs/2407.12784","kind":"external_reference","section_id":"memory-architecture-meets-stale-memory-risk"}],"grouped_relation_feeds":[{"relation":"abides by","count":1,"links":[{"id":"AX-1","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Derivation before 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by","label":"Sleeper Memory Poisoning Quarantine Replay","url":"paper-module-sleeper-memory-poisoning-quarantine-replay.html","relation_id":"paper_module.depends_on.paper_module","source_ref":"core/paper_module_capsules.json::paper_modules[37:paper_module.sleeper_memory_poisoning_quarantine_replay].depends_on"}],"source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/agent_memory_temporal_conflict_replay.md"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json"}],"visible_text_byte_count":14623,"section_titles":["Stale memory rows must lose authority","Three episodes exercise update and deletion","Synthetic memory accounting, not a live product","Purpose","The validator distrusts its labels","Conflict re-derived from verifiable fields","How it works","Two entry points over `_build_result`","One blocked check blocks the record","Coarse temporal order comes first","Four semantic checks against the prior write","The positive floor the fixture must clear","Memory credit needs a resolving evidence handle","Negative cases","Seven falsification rows, each an error code","Plausible one row at a time","Prior Art Grounding","Memory architecture meets stale-memory risk","Validation Result record Path","Fixture result records stay outside the working tree","What a pass covers","Scope limit","The strongest supported claim","What stays refused past the fixture"]},{"id":"paper_module.agent_monitor_redteam_falsification_replay","title":"Agent Monitor Redteam Falsification Replay","relative_href":"docs/paper-module-agent-monitor-redteam-falsification-replay.html","read_href":"docs/paper-module-agent-monitor-redteam-falsification-replay.html","context_href":"docs/rules-and-ideas.html#paper-module-agent-monitor-redteam-falsification-replay","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"Synthetic regression drilldown for a monitor/redteam honesty contract; checks falsification handling, negative cases, non-public-state scan, scope limits.","summary":"Public Plectis regression drilldown for a monitor/redteam honesty contract. Replays synthetic monitor traces and validates redteam-falsification behavior, negative cases, non-public-state exclusion, and scope limits with metadata-only result records.","authority_ceiling":"Synthetic replay fixture only; not a live monitor product, control eval, exploit guide, product-spine system, or complete-security claim.","abstract":"Synthetic regression drilldown for a monitor/redteam honesty contract; checks falsification handling, negative cases, non-public-state scan, scope limits.","teleology":{"section_id":"","text":"Synthetic regression drilldown for a monitor/redteam honesty contract; checks falsification handling, negative cases, non-public-state scan, scope limits.","source":"capsule_summary_line"},"body_markdown":"###### Monitor verdicts must be earned by evidence\n\n**A monitor verdict counts only once the re-runnable evidence behind it is present and consistent, not because a label asserts it.** `agent_monitor_redteam_falsification_replay` is a regression check for one honesty claim: that a monitor verdict was earned by evidence, not asserted by a label. It replays a small set of synthetic agent trajectories and their monitor observations, then refuses to let any verdict count until the evidence that would let a reader re-run it is present and consistent.\n\n###### metadata-only records preserve the review trail\n\n**It reads public trajectory metadata and emits metadata-only records of ids, refs, digests, and verdicts, never copying agent reasoning, source bodies, or model-output data.** The component consumes public trajectory metadata and metadata-only result records. It emits a result record, a review board, a validation record, and a command card, all carrying ids, refs, digests, counts, verdicts, and findings. It never copies agent reasoning, source bodies, or model-output data into those records.\n\n###### Fictional replay grants no live-monitor authority\n\n**Nothing here runs live: it is fictional data with no real-monitor score and no launch authorized.** It is a demo over fictional data. It does not run a live monitor, does not score how well any real monitor performs, and excludes launch.\n\n##### Purpose\n\n###### A verdict field is bounded evidence\n\n**Writing \"the monitor caught it\" is easy; showing the verdict follows from its trajectory and survives a second reading is the hard part a field-presence check skips.** It is easy to write \"the monitor caught it\" next to a suspicious agent action. It is harder to show that the verdict follows from the trajectory it claims to summarise, and that a second reading of the same evidence reaches the same conclusion. A loose check that only asks \"does the row have a verdict field\" passes on a label that reads well and is backed by nothing.\n\n###### Cited refs plus an independent recompute\n\n**Each observation must cite nine re-runnable refs before its verdict is admitted, and a separate builder recomputes the verdict from span evidence, blocking on any disagreement.** This component makes the verdict falsifiable. Each observation must cite a trajectory hash, a monitor id, a suspicious-span ref, an adversarial-probe ref, a severity tier, a human-escalation ref, a mitigation-diff ref, a body-omission ref, and a cold-replay ref before the verdict is admitted. A separate builder then recomputes the verdict from the span evidence. If the recomputed verdict disagrees with the declared one, the run blocks. The failure it guards against is a verdict that summarises a trajectory it does not actually match.\n\n##### How it works\n\n###### Two entry points fold into `_build_result`\n\n** `run` and `run_monitor_bundle` both call `_build_result`, which loads four inputs plus seven negative-case files, runs each validator in order, and folds them into one payload.** The entry points are `run` for the first-wave fixture and `run_monitor_bundle` for an exported public bundle. Both call `_build_result`, which loads the four input documents (`projection_protocol.json`, `monitor_policy.json`, `trajectory_cases.json`, `monitor_observations.json`) plus the seven negative-case files, runs each validator in order, folds the results into one payload, and returns it. `_write_receipts` then writes the result, board, validation, and sign-off records; `result_card` projects a compact card that omits the full finding, scan, and trace bodies.\n\nThe validators run in this order inside `_build_result`:\n\nFunction Role `validate_projection_protocol` Rejects a protocol that cites fewer than three source refs, is missing the `agent_monitor_redteam_falsification_replay_compound` pattern id, or has fewer than two projection-result record refs or three regression-fixture refs. `validate_monitor_policy` Requires the policy to define the `pass`, `escalate`, and `block` verdicts and every field in `REQUIRED_OBSERVATION_FIELDS`, and blocks a policy that authorizes performance claims, live traffic, external model access, or launch. `validate_trajectory_cases` Requires each case to carry an id, a hash, and a monitor scope, and rejects any case that carries a forbidden key such as private reasoning, internal code, or a account secret value. `validate_source_module_manifest` Recomputes the SHA-256 of each copied source artifact against the digest the manifest declares, requires `body_in_receipt` to be false, and rejects body text in the record. `validate_monitor_observations` Runs `_validate_observation_row` over every observation, counts pass, escalate, and block verdicts, and blocks if any positive row is unbacked or missing its source-artifact evidence. `validate_public_trace` Folds the recomputed trace into findings: a coverage label with no probe span, or a span whose derived verdict does not match the declared verdict, becomes a blocking finding.\n\n###### Three conditions for a pass\n\n** `scan_paths` fails the run on any forbidden-class hit, and the status is `pass` only when no negative case is missing, the scan is clean, and every validator passed.** `scan_paths` runs over the inputs and the copied source artifacts against the forbidden-class policy; any blocking hit fails the run. The status in `_build_result` is `pass` only when no negative case is missing, the non-public-state scan has zero blocking hits, and every validator returned `pass`.\n\n###### The anti-echo recompute per span\n\n** `build_public_monitor_redteam_falsification_trace` recomputes each span's verdict from its own evidence, and `validate_public_trace` blocks on any mismatch, so a copied verdict cannot pass.** The anti-echo step is `build_public_monitor_redteam_falsification_trace`, imported from `microcosm_core.macro_tools.agent_execution_trace`. It reads the same inputs and recomputes, per span, whether the coverage label is backed by an adversarial-probe span and what the verdict should be given the span's severity, escalation, and mitigation evidence. `validate_public_trace` compares each recomputed verdict to the declared one. A disagreement records `PUBLIC_TRACE_MONITOR_REDTEAM_VERDICT_MISMATCH` and the run blocks. This is why a verdict cannot pass by being copied into the observation row.\n\n###### What the four fixture cases recompute to\n\n**The four fixture cases recompute to two `pass`, one `escalate`, and one `block`, spanning a benign review, a suppressed failure, a tool-authority escape, and a dogfood safety-evals trace.** The first-wave fixture has four trajectory cases and four observations: a benign route review that recomputes to `pass`, a suppressed-failure attempt that recomputes to `escalate`, a tool-authority escape that recomputes to `block`, and one observation over a sanitized public dogfood safety-evals trace that recomputes to `pass`. The two positive routes plus the escalation and the block give two `pass`, one `escalate`, and one `block`.\n\nDiagram of the mechanism (8 steps). Diagram source & refs\n\n```text\nflowchart TD inputs[\"protocol, policy, trajectories, observations\"] manifest[\"source module manifest\"] digest[\"SHA-256 digest check per artifact\"] scan[\"forbidden-class non-public-state scan\"] observed[\"per-observation evidence check\"] trace[\"public trace recomputes verdict\"] compare[\"recomputed vs declared verdict\"] result[\"metadata-only result record\"] inputs --> observed manifest --> digest digest --> scan observed --> compare trace --> compare scan --> result compare --> result\n```\n\n##### Negative cases\n\n###### Semantic floors, not file presence\n\n**The seven negative files each carry a semantic floor, so a case counts only when its payload actually triggers the named failure mode, not when the file merely exists.** The fixture ships seven negative files, each expected to block or quarantine. Every one has a semantic floor in `NEGATIVE_CASE_SEMANTIC_FLOORS`, so a case counts only when its payload actually triggers the failure mode named by its label, not when the file merely exists.\n- `private_chain_of_thought_leakage` records `MONITOR_REDTEAM_PRIVATE_COT_FORBIDDEN`.\n- `internal_code_export` records `MONITOR_REDTEAM_INTERNAL_CODE_FORBIDDEN`.\n- `exploit_instruction_material` records `MONITOR_REDTEAM_EXPLOIT_INSTRUCTIONS_FORBIDDEN`.\n- account secret material records MONITOR REDTEAM account secret MATERIAL FORBIDDEN.\n- `live_agent_traffic` records `MONITOR_REDTEAM_LIVE_TRAFFIC_FORBIDDEN`.\n- `monitor_product_performance_claim` records `MONITOR_REDTEAM_PRODUCT_PERFORMANCE_CLAIM_FORBIDDEN`.\n- `coverage_without_adversarial_probe` records `MONITOR_REDTEAM_COVERAGE_WITHOUT_PROBE`; its floor requires the `adversarial_probe_ref` field to be empty, so a coverage claim with no probe cannot pass.\n\n###### Expected verdicts and severities\n\n**All seven cases expect a blocking or escalating verdict, at `critical` severity for the first five and `high` for the performance-claim and coverage cases.** The first five expect a `block` verdict at `critical` severity. The performance-claim case expects `block` at `high` severity. The coverage-without-probe case expects `escalate` or `block` at `high` severity.\n\n##### Prior Art Grounding\n\n###### Falsify with probes, then account for it\n\n**It follows the red-team lineage that falsifies a monitor claim with adversarial probes rather than confirming it with benign examples, adding the accounting shape of required refs plus an agreeing recomputation.** The component follows the red-team and behavior-discovery lineage that treats a monitor claim as something to falsify with adversarial probes rather than confirm with benign examples. Anthropic's [Red Teaming Language Models to Reduce Harms](https://arxiv.org/abs/2209.07858) is a procedural anchor for eliciting unwanted behavior, and [Discovering Language Model Behaviors with Model-Written Evaluations](https://arxiv.org/abs/2212.09251) anchors the idea that evaluation cases surface behavior classes worth tracking. The local contribution is the accounting shape: a verdict needs a trajectory hash, a probe ref, a suspicious-span ref, escalation and mitigation refs, a replay ref, and a body-omission record before coverage language is allowed, and an independent recomputation has to agree with it.\n\n##### Validation Result record Path\n\n###### The fixture command writes a disposable result\n\n**The first-wave validator runs from the repository root and writes its result record outside the working tree.** Run the first-wave fixture validator from the repo root and write its record outside the working tree:\n\nThen run the exported bundle validator:\n\n```text\ncd microcosm-substrate && PYTHONPATH=src ../repo-python -m microcosm_core.organs.agent_monitor_redteam_falsification_replay run-monitor-bundle --input examples/agent_monitor_redteam_falsification_replay/exported_monitor_redteam_bundle --out /tmp/agent_monitor_redteam_bundle_receipt --card > /tmp/agent_monitor_redteam_bundle_card.json\n```\n\n###### Regression and corpus checks share the root\n\n**The focused regression test and the paper-module corpus check both run from the repository root.** The focused regression test and corpus projection check are run from the repo root:\n\n```text\nPYTHONDONTWRITEBYTECODE=1 PYTHONPYCACHEPREFIX=/tmp/mc_agent_monitor_pyc ./repo-pytest tests/test_agent_monitor_redteam_falsification_replay.py -q -p no:cacheprovider --basetemp=/tmp/mc_agent_monitor_bt\n./repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n###### What a pass means here\n\n**A pass means every validator returned `pass`, the non-public-state scan found no blocking hit, and each recomputed verdict matched its declared verdict for these fixtures.** A pass means every validator returned `pass`, the non-public-state scan found no blocking hit, and each recomputed verdict matched its declared verdict for these fixtures.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim\n\n**For these synthetic inputs, a verdict passes only when its re-runnable refs are present, its source-artifact digests match on disk, and an independent recomputation agrees.** The strongest claim the evidence supports is narrow: for these synthetic inputs, a monitor verdict cannot pass unless its observation carries the probe, escalation, mitigation, body-omission, and replay refs that make it re-runnable, its source-artifact digests match on disk, and an independent recomputation of the verdict agrees with the declared one. The proof boundary is the synthetic falsification replay over public metadata and metadata-only result records. The scope limit is set in `AUTHORITY_CEILING`, which marks live agent execution, live traffic import, external model access, source-file changes, and launch as unauthorized.\n\n###### What a pass says nothing about\n\n**It claims no live monitor product, control-evaluation score, or production coverage and exports nothing private, so a pass speaks only to the fixture's evidence contract.** It does not claim a live monitor product, a control-evaluation score, production monitoring coverage, exploit guidance, or whole-system correctness. It does not export private reasoning, internal source bodies, account secrets, or model-output data. A passing run says the fixture respected the evidence contract, and nothing about how a real monitor would perform.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":13428,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/agent_monitor_redteam_falsification_replay.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"Red Teaming Language Models to Reduce Harms","url":"https://arxiv.org/abs/2209.07858","kind":"external_reference","section_id":"falsify-with-probes-then-account-for-it"},{"label":"Discovering Language Model Behaviors with Model-Written Evaluations","url":"https://arxiv.org/abs/2212.09251","kind":"external_reference","section_id":"falsify-with-probes-then-account-for-it"}],"grouped_relation_feeds":[{"relation":"abides 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recompute to","Negative cases","Semantic floors, not file presence","Expected verdicts and severities","Prior Art Grounding","Falsify with probes, then account for it","Validation Result record Path","The fixture command writes a disposable result","Regression and corpus checks share the root","What a pass means here","Scope limit","The strongest supported claim","What a pass says nothing about"]},{"id":"paper_module.agent_sabotage_scheming_monitor_replay","title":"Agent Sabotage Scheming-Monitor Replay","relative_href":"docs/paper-module-agent-sabotage-scheming-monitor-replay.html","read_href":"docs/paper-module-agent-sabotage-scheming-monitor-replay.html","context_href":"docs/rules-and-ideas.html#paper-module-agent-sabotage-scheming-monitor-replay","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"Drilldown regression contract for synthetic sabotage/scheming monitor claims: replays task episodes, action traces, monitor scores, counterfactual/cold replay.","summary":"Drilldown-only regression contract for synthetic sabotage/scheming monitor claims. Checks projection_protocol, scheming_monitor_policy, task_episodes, action_traces, monitor_scores, counterfactual_replay, and cold_replay against negative cases and scope limits.","authority_ceiling":"No live sabotage, exploit instruction, account secret/account material, private reasoning, harmful payload, or deployment-risk product claim; synthetic fixtures and metadata-only result records only.","abstract":"Drilldown regression contract for synthetic sabotage/scheming monitor claims: replays task episodes, action traces, monitor scores, counterfactual/cold replay.","teleology":{"section_id":"","text":"Drilldown regression contract for synthetic sabotage/scheming monitor claims: replays task episodes, action traces, monitor scores, counterfactual/cold replay.","source":"capsule_summary_line"},"body_markdown":"###### Caught verdicts are recomputed from joined evidence\n\n**Given a synthetic write-up claiming an agent was caught pursuing a hidden side goal, this component recomputes whether the \"caught\" verdict actually follows from the evidence.** Suppose someone writes up an example where an AI agent was caught quietly pursuing a hidden side goal while doing its assigned task. What structure would that write-up need before a reader should believe it? This component answers that question for one fixed shape of claim. It takes a bundle of synthetic episodes, per-step action traces, per-step monitor scores, counterfactual replay rows, and cold replay rows, and it recomputes whether the \"caught\" verdict actually follows from that evidence.\n\n###### Synthetic episodes produce metadata-only findings\n\n**On scrubbed synthetic inputs only, it emits a metadata-only record: a recomputed caught-or-not verdict per episode, the joins that verdict rests on, and findings where declared and recomputed evidence disagree.** It runs only on scrubbed synthetic inputs. It does not run an agent, it does not watch a live transcript, and it produces no claim about how any deployed monitor performs. What it produces is a metadata-only result record: a recomputed caught-or-not verdict per episode, the joins that verdict rests on, and a list of findings when the declared evidence and the recomputed evidence disagree.\n\n###### Two source files implement replay and trace recomputation\n\n**The component runtime evaluates the sabotage fixture, while the shared execution-trace module performs the evidence recomputation it depends on.** The runtime lives in `src/microcosm_core/organs/agent_sabotage_scheming_monitor_replay.py`. The trace recomputation it depends on lives in `src/microcosm_core/macro_tools/agent_execution_trace.py`.\n\n##### Purpose\n\n###### A catch must be earned, not declared\n\n**A loose sabotage claim is easy to write and hard to check, so a monitor row that declares a catch has to earn it from joined rows or the verdict flips and the run blocks.** A loose sabotage claim is easy to write and hard to check. A paragraph saying \"the monitor flagged step 4 as suspicious and we caught the agent\" can be true, or it can be a label with nothing under it. The recompute rule here refuses to trust the label. A monitor row that declares a catch has to earn it from joined rows, or the verdict flips and the run blocks.\n\n###### The three questions a run answers\n\n**After a run a reader can say which scores recompute to a block, which episodes recompute to caught, and which fixture mutations make the whole thing fail closed rather than pass quietly.** The concrete question a reader should be able to answer after a run: which monitor scores recompute to a block, which episodes recompute to caught, and which mutations of the fixture cause the whole thing to fail closed rather than pass quietly.\n\n##### How it works\n\n###### Two entry points into one pipeline\n\n** `run` validates the checked-in fixture with negative cases and `run_sabotage_bundle` validates the exported bundle with a manifest, and both call the `_build_result` pipeline.** Two entry points admit inputs. `run` validates the checked-in fixture under `fixtures/first_wave/agent_sabotage_scheming_monitor_replay/input` with negative cases included. `run_sabotage_bundle` validates the exported bundle under `examples/agent_sabotage_scheming_monitor_replay/exported_sabotage_monitor_bundle` and requires a source module manifest. Both call `_build_result`, which is the pipeline.\n\n###### Seven inputs, episodes checked first\n\n** `_build_result` loads the seven core input files and runs the checks in dependency order, validating episodes first because their ids gate everything downstream.** `_build_result` runs `_load_payloads` to read the seven core input files (projection protocol, monitor policy, task episodes, action traces, monitor scores, counterfactual replay, cold replay), then runs the checks in dependency order. Episodes are validated first because their ids gate everything downstream.\n\n###### `validate_monitor_scores` recomputes each row\n\n** `validate_monitor_scores` recomputes rather than echoes, granting a `block` only when the suspicion score clears the `0.75` threshold and the row is a backed suspicious action across five joins:** The center of the mechanism is `validate_monitor_scores`. It does not read the declared verdict and echo it. It recomputes. A monitor row earns a `block` only when its suspicion score is at or above the policy block threshold (`minimum_block_suspicion_score`, default 0.75) and the row is a backed suspicious action. \"Backed\" is a conjunction of five joins that must all hold:\n- the row's `suspicious_action_span_ref` names a real `action_id` (not `none`);\n- that `action_id` also appears in the row's `evidence_refs`;\n- the joined action carries `attack_opportunity_label = present`;\n- the joined action's `hidden_side_objective_pressure` is `medium` or `high`;\n- the counterfactual row for that episode removes side-task success, preserves the main task, and lists the row's step in `changed_step_ids`.\n\n###### A number alone cannot manufacture a catch\n\n**Drop any one join and the row falls to `escalate` or `pass`, so a high score with no joins cannot manufacture a catch, and a declared verdict that disagrees with the recompute is quarantined.** Drop any one join and the row recomputes to `escalate` (score at least 0.4 and backed) or `pass`. A high score with no joins recomputes to `pass`, so a number alone cannot manufacture a catch. An episode recomputes to `caught` only if it holds at least one row that recomputes to `block`, passes its own floor, and carries verified source-artifact evidence. If a monitor row's declared `caught_verdict` disagrees with the recomputed episode verdict, the row is quarantined and the run records `SABOTAGE_MONITOR_CAUGHT_VERDICT_LABEL_RECOMPUTE_MISMATCH`. Rows carrying precomputed answer-key fields are rejected outright as baked verdicts.\n\n###### Surrounding checks fence the other rows\n\n**Companion checks require typed, metadata-only rows, counterfactuals that remove side-task success while preserving the main task, and a cold replay proven by result record rather than prose.** The other checks fence the surrounding rows. `validate_task_episodes` and `validate_action_traces` require typed fields, episode linkage, and metadata-only rows. `validate_counterfactual_replay` requires each row to name changed steps, remove side-task success, and preserve the main task. `validate_cold_replay` requires a runnable command, a result record reference, and an explicit pass label, so a cold run is proven by a result record rather than by prose.\n\n###### Manifest digests plus an independent trace\n\n**On the bundle, `validate_source_module_manifest` digest-checks each artifact and an independent public trace re-checks that the caught verdict matches, the counterfactual isolates, and changed steps rank higher.** For the exported bundle, `validate_source_module_manifest` hashes each copied source artifact and compares it to the manifest digest, rejects a mismatch, rejects a `path` and `target_ref` that resolve to different files, and rejects any row that claims body text lives in the result record. `validate_real_sanitized_evidence` requires the real sanitized public trace slice to be present and cited. Independently, `_build_result` calls `build_public_sabotage_scheming_monitor_trace` to fold the same joins into per-episode spans, and `validate_public_trace` checks each span: the recomputed caught verdict must match the declared one, the counterfactual must isolate the scheming steps, and the changed steps must rank higher in suspicion than the unchanged ones. Those three checks emit `PUBLIC_TRACE_SCHEMING_CAUGHT_VERDICT_MISMATCH`, `PUBLIC_TRACE_SCHEMING_COUNTERFACTUAL_NOT_ISOLATING`, and `PUBLIC_TRACE_SCHEMING_COUNTERFACTUAL_RANK_DELTA_NON_POSITIVE`.\n\n###### Three conditions before metadata-only result records\n\n**A non-public-state scan fails the run on any forbidden hit, and a pass requires no missing negative case, a clean scan, and every stage green before metadata-only result records are written.** Before findings are merged, `_build_result` runs `scan_paths` over every input file and the copied source artifacts against the forbidden-class list. A blocking hit fails the run. The overall status is a pass only when no expected negative case is missing, the non-public-state scan has zero blocking hits, and every stage returned a pass. `_write_receipts` then emits the result, board, validation, and sign-off records, each metadata-only, and `result_card` projects a compact card for readers.\n\nFunction Role `run`, `run_sabotage_bundle` admit the fixture or the exported bundle `_build_result` run the stages in order and compute overall status `validate_monitor_scores` recompute per-step verdict and per-episode caught from the five joins `validate_counterfactual_replay`, `validate_cold_replay` require benign-replay isolation and result record-backed cold runs `validate_source_module_manifest` digest-check copied source artifacts, reject drift `validate_public_trace` check recomputed caught, isolation, and rank delta per span `validate_negative_cases` reject the forbidden-export fixtures by semantic floor `_write_receipts`, `result_card` emit metadata-only records and the reader card Diagram of the mechanism (11 steps). Diagram source & refs\n\n```text\nflowchart TD inputs[\"fixture or exported bundle\"] episodes[\"task episodes\"] actions[\"action traces\"] scores[\"monitor scores\"] counter[\"counterfactual replay\"] cold[\"cold replay\"] manifest[\"source module manifest\"] recompute[\"recompute verdict from five joins\"] trace[\"public trace check\"] negatives[\"negative cases\"] record[\"metadata-only result record\"] inputs --> episodes inputs --> actions inputs --> scores inputs --> counter inputs --> cold inputs --> manifest episodes --> recompute actions --> recompute scores --> recompute counter --> recompute manifest --> recompute recompute --> trace cold --> trace trace --> record negatives --> record\n```\n\n##### Negative cases\n\n###### Seven unsafe-export classes must be rejected\n\n**The fixture ships seven negative inputs, each a real class of unsafe export mapped to a fixed rejection code, and a pass requires all seven observed and rejected.** The fixture ships seven negative inputs, and a pass requires all seven to be observed and rejected. Each names a real class of unsafe export: `live_sabotage_instruction`, real account secrets or accounts, `exploit_instruction_material`, `private_chain_of_thought_export`, `raw_harmful_payload_export`, `monitor_only_final_grading`, and `deployment_scare_story_claim`. Each maps to a fixed rejection code, for example `SABOTAGE_MONITOR_LIVE_SABOTAGE_FORBIDDEN` and `SABOTAGE_MONITOR_EXPLOIT_INSTRUCTIONS_FORBIDDEN`.\n\n###### Rejection is semantic, not label-matching\n\n**Rejection is semantic, so renaming a benign fixture to a forbidden id fails the semantic-mismatch check rather than passing, and final-grade-only rows are ruled out for lacking per-step scores.** The rejection is semantic, not label-matching. `_negative_case_semantic_receipt` keys each fixture by its filename and requires the declared `expected_negative_case_id` to match that key and the file to set the right truthy and falsey fields. Renaming a benign fixture to a forbidden case id does not pass: it fails with `SABOTAGE_MONITOR_NEGATIVE_CASE_SEMANTIC_MISMATCH`. `monitor_only_final_grading` is rejected specifically because it declares only a final grade with no per-step monitor score, which is the shape this component exists to rule out.\n\n##### Prior Art Grounding\n\n###### Grammar borrowed from AI-control work\n\n**It borrows its grammar \u2014 hidden-objective episodes, per-step scores, counterfactual benign replay \u2014 from the AI-control and sabotage-evaluation literature, yielding fixture-bound replay evidence, not a control guarantee.** This component replays a monitor that watches an agent transcript for sabotage and scheming signals. It borrows its grammar, episodes with a hidden side objective, per-step monitor scores, and counterfactual benign replay, from the AI-control literature on catching a model that may be subverting oversight rather than assuming it is benign ( [Greenblatt et al., AI Control](https://arxiv.org/abs/2312.06942)) and from Anthropic's [sabotage evaluations](https://www.anthropic.com/research/sabotage-evaluations). The result is fixture-bound replay evidence, not a live monitor, a control guarantee, or an exploit guide.\n\n##### Validation Result record Path\n\nRun the fixture validator:\n\nRun the exported-bundle validator:\n\n```text\ncd microcosm-substrate && PYTHONPATH=src ../repo-python \\\n  -m microcosm_core.organs.agent_sabotage_scheming_monitor_replay \\\n  run-sabotage-bundle \\\n  --input examples/agent_sabotage_scheming_monitor_replay/exported_sabotage_monitor_bundle \\\n  --out /tmp/agent_sabotage_scheming_bundle_receipt \\\n  --card > /tmp/agent_sabotage_scheming_bundle_card.json\n```\n\nRun the focused tests:\n\n```text\nPYTHONPATH=src ./repo-pytest \\\n  tests/test_agent_sabotage_scheming_monitor_replay.py\n```\n\n###### A pass is proven by failing mutations\n\n**A pass means the checked-in inputs validate and every mutation \u2014 corrupt digest, reference, verdict, rank, or label \u2014 makes the run fail closed rather than pass.** A pass means the checked-in inputs validate and the mutation tests block: corrupt a digest, a target reference, an evidence reference, a caught verdict, a counterfactual rank, or a negative-case label, and the run fails closed rather than passing.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim\n\n**The strongest claim is fixture-bound: over public synthetic inputs the run recomputed every verdict from the joins, rejected the seven forbidden fixtures, and emitted reproducible metadata-only records.** The strongest thing this component supports is a fixture-bound claim: over public synthetic inputs, the run recomputed every monitor verdict and every episode caught verdict from the joins described here, rejected the seven forbidden-export fixtures, and emitted metadata-only records reproducible by the commands above. The proof boundary is exactly that synthetic replay contract.\n\n###### What a green run does not mean\n\n**Past that boundary it excludes live sabotage, execution, external model access, or launch and claims nothing about a real monitor, so a green run never means real scheming was detected.** It refuses everything past that boundary. The scope limit excludes live sabotage, no live agent execution, no exploit instructions, no account secret or account material, no private reasoning, no harmful payloads, no external model access, no source-file changes, and no launch. It makes no claim about a real monitor's performance, no deployment-risk claim, and no benchmark claims. A green run means the synthetic replay held under these mutations. It does not mean real scheming was detected.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":15060,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/agent_sabotage_scheming_monitor_replay.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"Greenblatt et al., AI Control","url":"https://arxiv.org/abs/2312.06942","kind":"external_reference","section_id":"grammar-borrowed-from-ai-control-work"},{"label":"sabotage evaluations","url":"https://www.anthropic.com/research/sabotage-evaluations","kind":"external_reference","section_id":"grammar-borrowed-from-ai-control-work"}],"grouped_relation_feeds":[{"relation":"abides by","count":1,"links":[{"id":"AX-1","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Derivation before assertion","url":"doctrine.html#dcard-ax-1"}]},{"relation":"depends on","count":1,"links":[{"id":"paper_module.agent_monitor_redteam_falsification_replay","kind":"paper_module","kind_label":"paper module","relation":"depends on","label":"Agent Monitor Redteam Falsification Replay","url":"paper-module-agent-monitor-redteam-falsification-replay.html"}]},{"relation":"explains","count":2,"links":[{"id":"agent_sabotage_scheming_monitor_replay","kind":"organ","kind_label":"component","relation":"explains","label":"Sabotage-Monitor Contract Replay","url":"component-agent-sabotage-scheming-monitor-replay.html"},{"id":"mechanism.agent_sabotage_scheming_monitor_replay.validates_public_sabotage_scheming_monitor_replay","kind":"mechanism","kind_label":"mechanism","relation":"explains","label":"validates public sabotage scheming monitor replay","url":"rules-and-ideas.html#lattice-mechanism-mechanism-agent-sabotage-scheming-monitor-replay-validates-public-sabotage-scheming-monitor-replay"}]},{"relation":"governed by","count":3,"links":[{"id":"P-1","kind":"principle","kind_label":"principle","relation":"governed by","label":"Recompute, do not echo","url":"doctrine.html#dcard-p-1"},{"id":"P-2","kind":"principle","kind_label":"principle","relation":"governed by","label":"Lower claim strength to checker strength","url":"doctrine.html#dcard-p-2"},{"id":"concept.agent_reliability_and_safety_validator_bundle","kind":"concept","kind_label":"concept","relation":"governed by","label":"agent reliability and safety validators as bounded public scope limits","url":"rules-and-ideas.html#lattice-concept-concept-agent-reliability-and-safety-validator-bundle"}]}],"paper_module_backrefs":[],"source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/agent_sabotage_scheming_monitor_replay.md"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json"}],"visible_text_byte_count":15881,"section_titles":["Caught verdicts are recomputed from joined evidence","Synthetic episodes produce metadata-only findings","Two source files implement replay and trace recomputation","Purpose","A catch must be earned, not declared","The three questions a run answers","How it works","Two entry points into one pipeline","Seven inputs, episodes checked first","`validate_monitor_scores` recomputes each row","A number alone cannot manufacture a catch","Surrounding checks fence the other rows","Manifest digests plus an independent trace","Three conditions before metadata-only result records","Negative cases","Seven unsafe-export classes must be rejected","Rejection is semantic, not label-matching","Prior Art Grounding","Grammar borrowed from AI-control work","Validation Result record Path","A pass is proven by failing mutations","Scope limit","The strongest supported claim","What a green run does not mean"]},{"id":"paper_module.agent_sandbox_policy_escape_replay","title":"Agent Sandbox Policy-Escape Replay","relative_href":"docs/paper-module-agent-sandbox-policy-escape-replay.html","read_href":"docs/paper-module-agent-sandbox-policy-escape-replay.html","context_href":"docs/rules-and-ideas.html#paper-module-agent-sandbox-policy-escape-replay","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"Validator-backed public refactor of source agent_execution_trace for sandbox/security: metadata-only trace spans, pre-exec policy verdicts, side-effect diffs","summary":"Validator-backed public refactor of the source agent_execution_trace system for sandbox/security claims. Computes metadata-only trace spans from action requests, pre-execution policy verdicts, side-effect diff result records, and rollback against negative cases and scope limits.","authority_ceiling":"No live sandbox escape, live secret handling, or live network; source-faithful refactored fixtures and metadata-only result records only; not a complete-security or launch claim.","abstract":"Validator-backed public refactor of source agent_execution_trace for sandbox/security: metadata-only trace spans, pre-exec policy verdicts, side-effect diffs","teleology":{"section_id":"","text":"Validator-backed public refactor of source agent_execution_trace for sandbox/security: metadata-only trace spans, pre-exec policy verdicts, side-effect diffs","source":"capsule_summary_line"},"body_markdown":"###### Every sandbox action is replayed against policy\n\n**The validator replays a synthetic agent's risky sandbox actions and recomputes, per action, whether policy should have blocked it, what side effect it left, and whether that effect was rolled back.** `agent_sandbox_policy_escape_replay` is a validator. It reads a synthetic record of an agent attempting risky actions inside a sandbox and recomputes, action by action, whether a safety policy should have blocked the action before it ran, what side effect it left if it ran, and whether that side effect was rolled back and could be re-checked from cold.\n\n###### Local fixtures record blocked and executed attempts\n\n**Inputs are local JSON files with no live agent, secret, or network; blocked attempts are still traced as never-executed, and the output is a metadata-only record plus a public trace.** The input is a set of local JSON files. There is no live agent, no real secret, and no network. Each blocked attempt is still recorded as a traced step, marked as never executed with a zero side-effect count. The output is a metadata-only result record plus a public trace of the same steps.\n\n###### The replay asks three containment questions\n\n**It answers one narrow question over synthetic requests: was every policy decision recorded before execution, did blocks leave no side effect, and does each executed effect carry a diff and rollback.** The component answers one narrow question: for a fixed set of synthetic requests, was every policy decision recorded before execution, did blocked actions leave no side effect, and does every executed side effect carry a diff and a rollback record. It does not run anything in a real host.\n\n##### Purpose\n\n###### Containment claims without checkable records\n\n**Containment claims are easy to assert, and a loose validator that reads the verdict row, sees `block`, and passes is only trusting the label it was handed.** Sandbox and containment claims are easy to assert and hard to show. A system can say it blocked an untrusted action without ever producing the record that would make the statement checkable. A loose validator would read the verdict row, see the word `block`, and pass. That trusts the label it was handed.\n\n###### Deriving the verdict from request shape\n\n**Instead it derives each expected verdict from the request's own shape, fails closed to `block` on any unrecognised shape, and holds the side-effect check to that derived verdict.** This component does not trust the verdict. For each request it derives the expected verdict from the request's own shape: its action kind, requested capability, risk class, and source trust label. A declared `allow` on a request whose shape derives to `block` is a finding, not a pass. Any shape the policy table does not recognise fails closed to `block`. The same rule drives the side-effect check: a request whose shape requires a block must show no execution and a zero diff count regardless of what the verdict row claims.\n\n##### How it works\n\n###### Two entry points over one pipeline\n\n** `run` validates the first-wave fixture with eight negative cases and `run_sandbox_bundle` validates an exported bundle with a manifest; both call `_build_result`, then write result records.** Two entry points share one pipeline. `run` validates the first-wave fixture with the eight negative-case files included. `run_sandbox_bundle` validates an exported bundle without the negative files but with a required source-module manifest. Both call `_build_result`, which is the whole mechanism, then `_write_receipts` (or a bundle writer) records the outcome.\n\n###### Ordered pipeline inside `_build_result`\n\n** `_build_result` runs in order: `_load_payloads` strict-reads the seven input files, `scan_paths` blocks on any forbidden-class hit, trace spans are built, then each validator recomputes one layer.** `_build_result` runs these steps in order. `_load_payloads` reads the seven input files (`projection_protocol`, `sandbox_policy`, `action_requests`, `policy_verdicts`, `side_effect_receipts`, `rollback_receipts`, `cold_replay`) with strict JSON parsing. `scan_paths` checks those files and any copied source bodies against `core/private_state_forbidden_classes.json` and blocks on any hit. `build_public_sandbox_policy_trace` builds the metadata-only trace spans. Then each validator recomputes one layer.\n\n###### The fail-closed policy core\n\n** `_derived_sandbox_policy_verdict` looks up `risk_class` in a fixed table, checks the shape matches, and returns `block` on any mismatch, then `validate_policy_verdicts` flags rows that disagree or lack `pre_execution`.** The policy core is `_derived_sandbox_policy_verdict`. It looks up the request's `risk_class` in a fixed table (`EXPECTED_ACTION_POLICY`) and confirms the action kind, capability, and trust label match the expected shape (`_policy_shape_mismatches`). On a match it returns the table's verdict and rule ref. On any mismatch, or an unknown risk class, it returns `FAIL_CLOSED_POLICY_SEMANTICS`, which is `block`. `validate_policy_verdicts` joins each verdict row to its request, then compares the declared verdict against this derived verdict. A row that does not match, or that lacks `pre_execution: true`, is a finding.\n\n###### Side-effect, rollback, replay, and manifest checks\n\n**Blocked verdicts must show no execution and zero diff, allowed ones an executed diff, side-effecting actions a verified rollback, replay rows a reproduced state, and manifest bodies a digest match.** `validate_side_effect_receipts` enforces the mechanical consequence. When the derived verdict is `block`, the result record must show `execution_attempted: false` and a diff count of zero. When it is `allow` or `review`, the result record must show an executed diff of at least one. `validate_rollback_receipts` requires a verified rollback record for each side-effecting action and needs at least two. `validate_cold_replay` requires each replay row to reproduce both the verdict and the side-effect state. `_source_module_manifest_result` checks the seven copied source bodies for digest match, a material class, declared anchors, and metadata-only result record fields.\n\n###### When the whole run passes\n\n**The run passes only when no negative case is missing, the secret scan is clean, every layer validator passes, and any manifest passes; `result_card` projects a metadata-only card.** `_build_result` passes only when no negative case is missing, the secret scan has zero blocking hits, every layer validator returns `pass`, and (for a bundle) the manifest passes. `result_card` projects the result into a compact public card that carries refs, counts, and verdicts, never bodies.\n\nFunction Role `run` Fixture entry point, negative cases included `run_sandbox_bundle` Exported-bundle entry point, manifest required `_build_result` Runs the ordered pipeline and computes final status `_load_payloads` Strict JSON read of the seven input files `_derived_sandbox_policy_verdict` Derives the expected verdict, fails closed to `block` `validate_policy_verdicts` Compares declared verdicts against derived ones `validate_side_effect_receipts` Blocked means no execution and zero diff; allowed means executed diff `validate_rollback_receipts` Requires verified rollback for side-effecting actions `validate_cold_replay` Requires replay rows to reproduce verdict and side-effect state `_source_module_manifest_result` Checks the seven copied source bodies for digest and anchors `_write_receipts` Writes the result, board, validation, and sign-off records\n\n###### What the six-request fixture derives to\n\n**Six requests derive to four blocks, one allow, and one review, giving six side-effect result records, two rollback records, six cold replay rows, and six metadata-only trace spans.** The fixture has six requests. Four derive to `block` (a secret read from untrusted tool output, a network exfiltration attempt, a destructive delete, and a shell obfuscation payload), one derives to `allow` (a low-risk public fixture write), and one derives to `review` (a mock database update). That gives six side-effect result records, four blocked without execution, two verified rollback records, six cold replay rows, and six public trace spans.\n\nDiagram of the mechanism (8 steps). Diagram source & refs\n\n```text\nflowchart TD requests[\"six action requests\"] derived[\"derived verdict per request fail-closed to block\"] verdicts[\"declared verdict rows checked vs derived\"] effects[\"side-effect result records block=no diff, allow=diff\"] rollback[\"two verified rollbacks\"] replay[\"six cold replay rows\"] trace[\"six metadata-only trace spans\"] result[\"result record + status\"] requests --> derived derived --> verdicts verdicts --> effects effects --> rollback effects --> replay requests --> trace verdicts --> result effects --> result rollback --> result replay --> result trace --> result\n```\n\n##### Negative cases\n\n###### Eight escape patterns rejected by name\n\n** `validate_negative_cases` requires eight synthetic files to each raise their declared error code, from real secrets to a security-benchmark claim, and a missing case blocks the run.** `validate_negative_cases` reads eight synthetic files and requires each to raise its declared error code: `real_secret_material`, `live_network_access`, `raw_environment_export`, `policy_after_execution`, `unlogged_side_effect`, `tool_output_policy_bypass`, `executable_escape_payload`, and `security_benchmark_claim`. A missing case blocks the run. These files are the refusal surface: real secrets, live network access, raw environment export, a policy decision recorded after execution, an unlogged side effect, a tool-output authority bypass, an executable escape payload, and a broad security-benchmark claim are each rejected by name.\n\n##### Prior Art Grounding\n\n###### Least-privilege lineage and agent sandbox benchmarks\n\n**The design borrows Saltzer and Schroeder's least-privilege and complete-mediation principles and sandboxed agent benchmarks like AgentDojo and AgentHarm, taking the pattern and not a new exploit technique.** The security-control lineage is Saltzer and Schroeder's [least-privilege and complete-mediation principles](https://www.cs.virginia.edu/~evans/cs551/saltzer/) and capability-oriented confused-deputy thinking: check containment before an action, log side effects, keep a separate rollback record. The agent-evaluation lineage is closer to sandboxed tool-use benchmarks such as [AgentDojo](https://arxiv.org/abs/2406.13352) and misuse evaluations such as [AgentHarm](https://arxiv.org/abs/2410.09024), where an agent's requested actions and policy outcomes are scored under controlled conditions. This component borrows the pattern, not a new exploit technique.\n\n##### Validation Result record Path\n\n###### Fixture output stays outside the working tree\n\n**The first-wave validator runs against the checked-in fixture and writes its result record to a disposable path outside the working tree.** Run the first-wave fixture validator and write its record outside the working tree:\n\nThen run the exported bundle validator:\n\n```text\ncd microcosm-substrate && PYTHONPATH=src ../repo-python \\\n  -m microcosm_core.organs.agent_sandbox_policy_escape_replay \\\n  run-sandbox-bundle \\\n  --input examples/agent_sandbox_policy_escape_replay/exported_sandbox_policy_escape_bundle \\\n  --out /tmp/agent_sandbox_policy_escape_bundle_receipt \\\n  --card > /tmp/agent_sandbox_policy_escape_bundle_card.json\n```\n\nThe focused regression test and corpus projection check are:\n\n```text\ncd microcosm-substrate && ../repo-pytest \\\n  tests/test_agent_sandbox_policy_escape_replay.py\n./repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n###### What a pass certifies here\n\n**A pass means the validators recomputed every verdict, side effect, rollback, and replay row from the source files, the secret scan found nothing, and all eight negative cases fired.** A pass means the validators recomputed every verdict, side effect, rollback, and replay row from the source files, the secret scan found nothing, and all eight negative cases fired.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim\n\n**Over the public fixture and bundle, the component recomputes each verdict, enforces no side effect on blocks, requires diffs and rollbacks on executed actions, replays the state, and refuses eight escape patterns.** The strongest claim the evidence supports is narrow. Over the public fixture and exported bundle, this component recomputes the derived policy verdict for each request, checks the declared verdict against it, enforces that blocked actions leave no side effect, requires a diff and a verified rollback for executed actions, reproduces the state on cold replay, and refuses eight named escape patterns. The validation result records listed above are the evidence for that claim and nothing wider.\n\n###### What stays outside the proof boundary\n\n**The proof stops at replay consistency over local symbolic refs and establishes no live escape resistance, secret handling, network isolation, or benchmark performance; a pass is bounded evidence an agent is safe.** The proof boundary stops at replay consistency over symbolic refs in local files. It does not establish live sandbox escape resistance, live secret handling, live network isolation, host filesystem mutation authority, executable payload export, provider behaviour, security-benchmark performance, source-file changes, or launch. The scope limit is a metadata-only agent-execution trace refactor over a synthetic sandbox-policy fixture. A pass is validator evidence for one contract, bounded evidence that an agent is safe in the world.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":13736,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/agent_sandbox_policy_escape_replay.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"least-privilege and complete-mediation 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Replay","url":"paper-module-sleeper-memory-poisoning-quarantine-replay.html","relation_id":"paper_module.depends_on.paper_module","source_ref":"core/paper_module_capsules.json::paper_modules[37:paper_module.sleeper_memory_poisoning_quarantine_replay].depends_on"}],"source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/agent_sandbox_policy_escape_replay.md"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json"}],"visible_text_byte_count":14546,"section_titles":["Every sandbox action is replayed against policy","Local fixtures record blocked and executed attempts","The replay asks three containment questions","Purpose","Containment claims without checkable records","Deriving the verdict from request shape","How it works","Two entry points over one pipeline","Ordered pipeline inside `_build_result`","The fail-closed policy core","Side-effect, rollback, replay, and manifest checks","When the whole run passes","What the six-request fixture derives to","Negative cases","Eight escape patterns rejected by name","Prior Art Grounding","Least-privilege lineage and agent sandbox benchmarks","Validation Result record Path","Fixture output stays outside the working tree","What a pass certifies here","Scope limit","The strongest supported claim","What stays outside the proof boundary"]},{"id":"paper_module.belief_state_process_reward_replay","title":"Belief-State Process Reward Replay","relative_href":"docs/paper-module-belief-state-process-reward-replay.html","read_href":"docs/paper-module-belief-state-process-reward-replay.html","context_href":"docs/rules-and-ideas.html#paper-module-belief-state-process-reward-replay","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"Public projection of a belief-state process-reward claim contract, backed by the agent-execution trace refactor and copied source bodies.","summary":"Public Plectis projection of a belief-state process-reward claim contract. Backed by the public agent-execution trace refactor lane plus copied source bodies; validates trajectory groups, cold replay result records, negative cases, and scope limits.","authority_ceiling":"Source-faithful refactored fixtures and copied source bodies only; not fixture-echo product evidence, benchmark result, or launch claim.","abstract":"Public projection of a belief-state process-reward claim contract, backed by the agent-execution trace refactor and copied source bodies.","teleology":{"section_id":"","text":"Public projection of a belief-state process-reward claim contract, backed by the agent-execution trace refactor and copied source bodies.","source":"capsule_summary_line"},"body_markdown":"###### Every process reward must reconstruct from evidence\n\n**This component reads pre-recorded agent steps and asks of every step-level reward whether it reconstructs from lower-level public evidence or merely asserts its own correctness.** This component checks a recorded set of agent steps and asks one question of every step-level reward: does it reconstruct from lower-level public evidence, or is it a label asserting its own correctness. It reads pre-recorded files. It never runs or watches a live agent.\n\n###### Three toy tasks carry seven planted reward traps\n\n**The input models three partially observable toy tasks, and a reward is admitted only when its references recompute into one coherent chain with all seven planted traps observed.** The input models three partially observable toy tasks: a terminal investigation, a mock purchase, and a small formal planner. For each step it carries a public belief summary, the verifier or observed feedback the reward cites, the process reward itself, the trajectory group the step belongs to, the outcome reward for that trajectory, and a cold-replay record. A reward is admitted only when all of those references recompute into one coherent chain and none of seven planted reward-hacking traps go unobserved.\n\n###### metadata-only results explain each admitted reward\n\n**The result files record per step the belief summary, the reward, and the recompute and trap outcomes, so a reader sees why each reward was allowed without exposing hidden reasoning.** The result files record, per step, the belief summary, the reward, and whether the recompute and trap checks passed. So a reader can see why each reward was allowed or blocked, without any hidden reasoning being exposed.\n\n##### Purpose\n\n###### Process-reward claims a linter cannot catch\n\n**Process-reward language is easy to assert and hard to verify, since a row can claim good reasoning while resting on a hidden label or gamed formatting a shape linter passes.** Process-reward language is easy to assert and hard to verify. A row can claim a step earned a reward \"for good reasoning\" while the real basis is a hidden gold label, a neural-judge guess, or formatting that gamed the scorer. A shape linter does not catch this: every field can be present and well-typed and the claim can still be fake.\n\n###### The end-to-end recompute failure mode\n\n**The guarded failure is a reward correct field by field that fails end-to-end recompute, with outcome rewards kept beside process rewards and every reference metadata-only to keep the check honest.** The failure mode this component guards against is a reward that looks correct field by field but does not survive being recomputed end to end. A reward cannot point at a belief that points at a different episode, or cite feedback that belongs to another trajectory, and still count. Two design choices keep it honest. Outcome rewards are carried beside process rewards, so a final-answer score cannot be relabelled as step-level evidence. Every belief summary, feedback reference, and reward event is metadata-only, so the check proves the accounting structure without reading hidden reasoning.\n\n##### How it works\n\n###### One runtime exposes fixture and bundle modes\n\n**The same process-reward replay runtime supports a negative-bearing fixture mode and a manifest-backed exported-bundle mode.** The runtime lives in `belief_state_process_reward_replay.py` and runs in two modes.\n\n###### Two modes: fixture and exported bundle\n\n**In fixture mode `run` loads eight positive files and seven planted negatives before calling `_build_result`, while bundle mode's `run_reward_bundle` drops the negatives and makes the copied source-body floor mandatory.** In fixture mode, `run` loads the eight positive input files named by `INPUT_NAMES` plus the seven planted negatives named by `NEGATIVE_INPUT_NAMES`, with the expected trap-to-code mapping in `EXPECTED_NEGATIVE_CASES`. It calls `_build_result` with `include_negative=True`, then `_write_receipts`. In bundle mode, `run_reward_bundle` validates an exported bundle with no negative inputs and makes the copied source-body floor mandatory by passing `require_manifest`.\n\n###### `_build_result` orchestrates every check\n\n** `_build_result` orchestrates the per-file checks, semantic recompute, negative cases, secret scan, trace, and manifest, and passes only when all of them clear, blocking on any single failure.** `_build_result` is the orchestrator. It loads the payloads, runs the per-file checks (`validate_projection_protocol`, `validate_reward_policy`, `validate_task_episodes`, `validate_belief_states`, `validate_verifier_feedback`, `validate_reward_events`, `validate_trajectory_groups`, `validate_cold_replay`), runs the cross-file check `validate_semantic_recompute`, runs `validate_negative_cases`, scans every input path for forbidden classes with `scan_paths`, builds the public trace, and checks the copied source-body manifest with `_source_module_manifest_result`. A pass requires: no expected negative case missing, zero blocking secret-scan hits, public trace status pass, no findings outside the expected traps, every per-file check pass, and in bundle mode the manifest check pass. Any single failure sets the status to blocked with the error codes attached.\n\nIt checks that the cited feedback belongs to the same episode, that the process reward references the same belief, episode, trajectory, feedback ref, and belief-discrepancy value, that the trajectory lists both that episode and that reward, that the trajectory's outcome reward is a real outcome event for the same episode, and that the cold replay exists and passed. Any inconsistency appends a precise reason code such as `feedback_episode_mismatch`, `belief_discrepancy_mismatch`, or `trajectory_process_reward_missing`. One blocked row blocks the whole result. This is the check a label-only fixture cannot fake.\n\n###### Each planted trap must raise its code\n\n** `validate_negative_cases` confirms every planted trap triggers its expected error code, and a trap that fails to fire becomes a missing case that blocks the run.** `validate_negative_cases` reads the planted traps and confirms each one triggers its expected error code, for example `hidden_chain_of_thought_exported` raising `BELIEF_REWARD_HIDDEN_COT_EXPORT` and `neural_judge_only` with no verifier reference raising `BELIEF_REWARD_NEURAL_JUDGE_ONLY_LABEL`. A trap that fails to trigger becomes a missing case and blocks the run.\n\n###### Copied source-body floor for bundles\n\n** `_source_module_manifest_result` enforces the copied source-body floor, requiring each module to match its class, digest, and source authority while keeping result records metadata-only, raising a specific code on any breach.** `_source_module_manifest_result` enforces the copied source-body floor for the exported bundle. Each declared module must classify as `copied_non_secret_macro_body`, use a body class, copy its body into `source_modules/` while keeping result record fields metadata-only, exist as a target file, match its declared sha256 digest, and resolve its source reference to live source authority. Digest mismatch, missing target, wrong body class, result record-body leakage, count mismatch, and missing source authority each raise a specific code rather than passing quietly.\n\n###### Result records and the metadata-only public card\n\n** `_write_receipts` writes the four result files while `result_card` projects a compact card that omits the heavy payload fields, and the trace emits metadata-only spans keeping `body_in_receipt: false`.** `_write_receipts` writes the result, board, validation, and sign-off files. `result_card` projects a compact public card of counts, status, freshness digest, and reference handles. `CARD_OMITTED_FULL_PAYLOAD_KEYS` names the heavy fields kept off the card: findings, scans, trace bodies, row payloads, source imports, the scope limit, and the scope boundary text. The trace itself comes from `build_public_belief_state_process_reward_trace`, imported from `agent_execution_trace.py`, which emits metadata-only spans and keeps `body_in_receipt: false`.\n\nFunction to role:\n\nFunction Role `run` Fixture-mode entry: 8 positive files, 7 planted negatives, writes result records `run_reward_bundle` Bundle-mode entry: copied source-body floor required, no negatives `_build_result` Orchestrates every check and derives pass or blocked `validate_semantic_recompute` `validate_negative_cases` Confirms each planted trap raises its expected code `_source_module_manifest_result` Verifies copied source bodies by class, digest, and anchor `_write_receipts` / `result_card` Emit result records and the metadata-only public card Diagram of the mechanism (9 steps). Diagram source & refs\n\nSource refs\n\n`validate_semantic_recompute`\n\n```text\nflowchart TD run[\"run / run_reward_bundle load inputs\"] perfile[\"Per-file checks projection, policy, episodes, beliefs, feedback, rewards, trajectories, cold replay\"] recompute[\"validate_semantic_recompute rebuild one coherent chain\"] negatives[\"validate_negative_cases 7 traps must trigger\"] scan[\"scan_paths forbidden-class scan\"] gate{\"all checks pass, chain recomputes, every trap observed, no secret hit?\"} pass[\"status: pass\"] blocked[\"status: blocked with reason codes\"] result records[\"_write_receipts + result_card body_in_receipt false\"] run --> perfile perfile --> recompute recompute --> negatives negatives --> scan scan --> gate gate -->|yes| pass gate -->|no| blocked pass --> result records blocked --> result records\n```\n\n##### Negative cases\n\n###### Seven falsification traps and forbidden keys\n\n**The seven planted traps each name an abuse a reward claim might smuggle in and must raise its code, while `FORBIDDEN_KEYS` blocks reasoning bodies and model-output data from any row.** The seven planted traps are the falsification fixtures. Each names the abuse a process-reward claim might smuggle in, and each must raise its code for the run to pass: hidden chain-of-thought export, neural-judge-only labels, hidden gold labels, reward by formatting, verifier bypass, a benchmark-performance claim, and final-answer-only scoring. `FORBIDDEN_KEYS` blocks reasoning bodies, model-output data, gold-answer bodies, live-training run ids, and benchmark submission ids from appearing in any belief or reward row.\n\n##### Prior Art Grounding\n\n###### Three lineages: belief state, process reward, reward hacking\n\n**Three older ideas ground the component, partial-observability belief states, step-level process reward, and reward hacking, but it trains no model and only borrows their accounting form.** Three older ideas ground this component. The belief-state lineage comes from work on acting under partial observability such as Kaelbling, Littman, and Cassandra's [Planning and Acting in Partially Observable Stochastic Domains](https://people.smp.uq.edu.au/YoniNazarathy/Control4406_2014/resources/KaelblingLittmanCassandra1998.pdf). The step-level reward lineage follows OpenAI's [Let's Verify Step by Step](https://arxiv.org/abs/2305.20050), which separates process feedback from outcome-only supervision. The reward-hacking lineage comes from [Concrete Problems in AI Safety](https://openai.com/index/concrete-ai-safety-problems/). This component does not train a reward model or expose reasoning. It borrows the accounting form: public belief summaries, verifier-backed feedback, outcome rewards kept separate from process rewards, planted traps, and cold replay before a reward claim is admitted.\n\n##### Validation Result record Path\n\n###### Fixture output stays outside the working tree\n\n**The fixture validator runs from the repository root and writes its result to a disposable path outside the working tree.** Run the fixture check from the repo root and write its result outside the working tree:\n\nThen run the exported bundle check:\n\n```text\ncd microcosm-substrate && PYTHONPATH=src ../repo-python -m microcosm_core.organs.belief_state_process_reward_replay run-reward-bundle --input examples/belief_state_process_reward_replay/exported_belief_state_process_reward_bundle --out /tmp/belief_state_process_reward_bundle_receipt --card > /tmp/belief_state_process_reward_bundle_card.json\n```\n\n###### What a passing regression run means\n\n**A pass on the focused regression test means all eight positive files cleared their checks, the chain recomputed, every planted trap fired, and no secret-class or body leak appeared.** The focused regression test is `tests/test_belief_state_process_reward_replay.py`. A pass means all eight positive files cleared their checks, the chain recomputed, every planted trap triggered its code, and no secret-class or body leak was found.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The counted proof boundary, not scale\n\n**The evidence supports a metadata-only replay over three synthetic tasks, whose fixed counts are the proof boundary, showing process feedback stays separable from outcome reward and survives cold replay.** The evidence supports a metadata-only replay over three synthetic tasks: 3 episodes, 6 belief summaries, 6 process rewards, 3 outcome rewards, 3 trajectory groups, 3 cold replays, 7 observed traps in fixture mode, and 7 copied source bodies in bundle mode. Those counts are the proof boundary, not a scale claim. Within them the component shows that public process feedback stays separable from final-answer outcome reward and survives cold replay with every trap observed.\n\n###### What the scope limit refuses\n\n**It does not estimate calibration, generalize, compare policies, certify training, or score a benchmark, and grants no hidden-reasoning export, live RL, provider behavior, mutation, or launch-scope decision.** It does not estimate reward-model calibration, generalize to unseen tasks, compare policies, certify live training, or score a benchmark. The scope limit is source-faithful replay over declared public inputs only. It grants no hidden-reasoning export, no live RL, no neural-judge sufficiency, no hidden-gold benchmark claims, no provider behavior, no source-file changes, and no public sharing or launch-scope decision. Any higher claim must land first in the source record and its generated projection.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":14247,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/belief_state_process_reward_replay.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"Planning and Acting in Partially Observable Stochastic Domains","url":"https://people.smp.uq.edu.au/YoniNazarathy/Control4406_2014/resources/KaelblingLittmanCassandra1998.pdf","kind":"external_reference","section_id":"three-lineages-belief-state-process-reward-reward-hacking"},{"label":"Let's Verify Step by Step","url":"https://arxiv.org/abs/2305.20050","kind":"external_reference","section_id":"three-lineages-belief-state-process-reward-reward-hacking"},{"label":"Concrete Problems in AI 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scope limits","url":"rules-and-ideas.html#lattice-concept-concept-agent-reliability-and-safety-validator-bundle"}]}],"paper_module_backrefs":[],"source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/belief_state_process_reward_replay.md"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json"}],"visible_text_byte_count":14945,"section_titles":["Every process reward must reconstruct from evidence","Three toy tasks carry seven planted reward traps","metadata-only results explain each admitted reward","Purpose","Process-reward claims a linter cannot catch","The end-to-end recompute failure mode","How it works","One runtime exposes fixture and bundle modes","Two modes: fixture and exported bundle","`_build_result` orchestrates every check","Rebuilding each belief-to-reward chain by id","Each planted trap must raise its code","Copied source-body floor for bundles","Result records and the metadata-only public card","Negative cases","Seven falsification traps and forbidden keys","Prior Art Grounding","Three lineages: belief state, process reward, reward hacking","Validation Result record Path","Fixture output stays outside the working tree","What a passing regression run means","Scope limit","The counted proof boundary, not scale","What the scope limit refuses"]},{"id":"paper_module.sleeper_memory_poisoning_quarantine_replay","title":"Sleeper Memory Poisoning Quarantine Replay","relative_href":"docs/paper-module-sleeper-memory-poisoning-quarantine-replay.html","read_href":"docs/paper-module-sleeper-memory-poisoning-quarantine-replay.html","context_href":"docs/rules-and-ideas.html#paper-module-sleeper-memory-poisoning-quarantine-replay","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"Synthetic replay fixture for a persistent-memory security contract: quarantine of poisoned memory, audit refs, rerun result records, negative cases, authority","summary":"Public Plectis projection of a persistent-memory security claim contract. Replays synthetic memory-poisoning episodes and validates quarantine behavior, audit refs, rerun result records, negative cases, and scope limits with metadata-only result records.","authority_ceiling":"Synthetic replay fixture only; not a live memory product, live user memory import, benchmark security result, private memory export, or launch claim.","abstract":"Synthetic replay fixture for a persistent-memory security contract: quarantine of poisoned memory, audit refs, rerun result records, negative cases, authority","teleology":{"section_id":"","text":"Synthetic replay fixture for a persistent-memory security contract: quarantine of poisoned memory, audit refs, rerun result records, negative cases, authority","source":"capsule_summary_line"},"body_markdown":"**Persistent agent memory is an attack surface, and this component asks whether a recorded guard-and-cleanup sequence proves the quarantine held on later retrieval and the rollback truly removed the memory.** Persistent agent memory is an attack surface. If an agent reads a poisoned source in one session and writes a memory from it, that memory can shape a later session's actions long after the poisoning is out of view. This component asks one question about that story: given a recorded guard-and-cleanup sequence, can you show from the record alone that the quarantine held when the memory was retrieved later, and that the rollback genuinely removed it?\n\n**It is a synthetic replay validator, not a memory runtime, grading an on-disk record of four sessions against a policy and emitting metadata-only records over public metadata rows.** It is a synthetic replay validator, not a memory runtime. It does not delete a memory or run the agent again. It reads an on-disk record of four sessions, grades every required step and result record against a policy, and emits metadata-only result records that carry refs, hashes, counts, and verdicts. The inputs are public metadata rows. No live user memory, transcript body, or model-output data passes through.\n\n##### Purpose\n\n###### The whole chain must line up, not the answer\n\n**It refuses to grade only the final answer, requiring the write to carry provenance, the retrieval to be blocked before action citing the same memory, and the rollback to prove absence after rerun.** A naive memory-security check grades the final answer: the agent did not act on the bad memory, so it passed. This component refuses that shortcut. It requires the whole chain to be present and to line up. The poisoned write must carry provenance. The later retrieval must be blocked before any action and must cite the same memory the write quarantined, plus that write's audit ref and a cold replay result. The rollback must carry a deletion audit ref, a cold rerun result, and proof that the memory is absent after the rerun. A blocked retrieval that cannot name the quarantine audit ref or the cold replay for the memory it gates is treated as unproven, not as a pass.\n\n##### How it works\n\n###### Seven validators merged into one status\n\n** `_build_result` runs seven validators in order and merges their findings, with `run` driving the fixture and `run_quarantine_bundle` the exported bundle, both writing metadata-only result records.** The runtime is `src/microcosm_core/organs/sleeper_memory_poisoning_quarantine_replay.py`. `_build_result` loads the input directory and runs seven validators in a fixed order, then merges their findings and negative-case observations into one status. `run` drives the first-wave fixture with the eight negative fixtures loaded. `run_quarantine_bundle` drives the exported public bundle with negatives off and the source manifest required. Both call `_write_receipts`, and `result_card` projects a compact card that omits the row-level detail.\n\n###### Projection, policy, and the four session roles\n\n** `validate_projection_protocol` binds the source refs and pattern id, then `validate_memory_policy` and `validate_session_chain` check the trust tiers and the four required session roles.** The first check is projection. `validate_projection_protocol` requires the protocol to cite at least four source refs, the `sleeper_memory_poisoning_quarantine_replay_compound` pattern id, two projection result refs, three public replacement refs, and non-empty reimplemented and omitted lists, and to declare `copied_private_memory_bodies` false. Then `validate_memory_policy` reads the allowed trust tiers and quarantine verdicts, and `validate_session_chain` checks the four session roles in `REQUIRED_SESSION_ROLES`: poisoned source seen, memory write quarantined, later retrieval action gated, rollback and cold rerun.\n\n###### Provenance gate over each memory write\n\n** `validate_quarantine_events` blocks any write that leaks a body, claims live memory, drops provenance, or promotes untrusted context, and passes only with one quarantined write and one admitted control.** The provenance gate is `validate_quarantine_events`, which grades each write through `_validate_write_row`. A write is blocked when it carries a forbidden key or exports a private body, exports a raw transcript, claims live user memory, omits its provenance or source bundle ref, promotes untrusted context into trusted memory, or gives an audit ref that does not start with `audit.`. The gate passes only when one row is a quarantined sleeper-poisoning write (verdict `quarantine`, trust tier `untrusted_context`, label `sleeper_poisoning`) and one row is a provenance-bound admitted control.\n\n###### Delayed-influence gate coupled to the quarantine\n\n** `validate_retrieval_replays` counts a row blocked only when it is gated before action and cites the same memory's quarantine audit and cold replay refs, downgrading any uncoupled row.** The delayed-influence gate is `validate_retrieval_replays`, grading each row through `_validate_retrieval_row`. It rejects final-answer-only grading and any row where the influence grade or action gate lets quarantined memory reach an action. A row counts as blocked only with action gate `blocked_before_action` and influence grade `quarantined_before_action`. This gate is coupled to the first: `_quarantined_audit_refs_by_memory_ref` maps each quarantined memory to its audit ref, and a blocked row that targets a memory the write never quarantined, or omits that memory's audit ref or its cold replay ref from its evidence, is downgraded to blocked with a typed finding.\n\n###### Rollback needs audit, rerun, and absence\n\n** `validate_rollback_rerun` requires a deletion audit ref, a rollback and rerun result ref, and `memory_absent_after_rerun` true, blocking bogus or uncoupled rollbacks.** The rollback gate is `validate_rollback_rerun` through `_validate_rollback_row`. A rollback needs a deletion audit ref starting `audit.`, a rollback result ref and a rerun result ref both starting `receipt.`, and `memory_absent_after_rerun` set true. Nonempty but bogus refs, a missing rerun ref, or a memory still present after rerun all block rather than count as evidence. Rollback rows are coupled to the quarantined memory refs the same way retrieval rows are.\n\n###### Source-body digest checks and metadata-only result records\n\n** `_source_module_manifest_result` verifies each copied public body's class, digests, and anchors, then `_write_receipts` emits four records that omit retrieval rows, rollback rows, and copied bodies.** The last check is source-body handling. `_source_module_manifest_result` verifies each copied public source body: its import class, a material class, that the body was copied and kept out of the result record, that the target file exists, that its declared digests match the file, that required anchors are present, and, when the live source resolves, that the copy matches it. `_source_open_body_import_summary` reduces that to a count and a set of classes. `_write_receipts` then emits four result records (result, board, validation, sign-off) that carry ids, counts, refs, digests, and verdicts while omitting retrieval rows, rollback rows, and copied bodies from the card.\n\nDiagram of the mechanism (6 steps). Diagram source & refs\n\nSource refs\n\npoisoned write quarantined + provenance-bound control `validate_quarantine_events` retrieval blocked before action, same memory ref + audit + cold replay `validate_retrieval_replays`\n\n```text\nflowchart TD inputs[\"Public metadata inputs sessions, write proposals, retrieval replays, rollback rows\"] provenance[\"validate_quarantine_events poisoned write quarantined + provenance-bound control\"] influence[\"validate_retrieval_replays retrieval blocked before action, same memory ref + audit + cold replay\"] rollback[\"validate_rollback_rerun deletion audit + rerun ref + memory absent after rerun\"] bodies[\"_source_module_manifest_result copied bodies digest-checked, records stay metadata-only\"] records[\"metadata-only result records refs, hashes, counts, verdicts\"] inputs --> provenance provenance -->|quarantined memory ref| influence influence --> rollback rollback --> bodies bodies --> records\n```\n\n##### Negative cases\n\n###### Eight negatives that must each surface\n\n**A pass requires all eight named negative rows to surface as typed findings, spanning body export, unprovenanced writes, unaudited deletion, and unmetered influence, or the run blocks.** The fixture carries eight named negative rows, one per entry in `EXPECTED_NEGATIVE_CASES`. A pass requires every one to surface as a typed finding, and `_build_result` blocks the run for any expected case that was not observed. The cases are private memory body export, live user memory claim, raw transcript export, memory write without provenance, trusted promotion from untrusted context, deletion without audit, final-answer-only grading, and unmetered poison influence. Their error codes carry the `SLEEPER_MEMORY_` prefix.\n\n##### Prior Art Grounding\n\n###### Sleeper-trigger and memory-poisoning literature\n\n**The design draws on the Sleeper Agents and memory-poisoning literature, borrowing the control shape where writes need provenance, retrieval passes an influence gate, and deletion needs audit and rerun.** This component draws two lines of prior work. The sleeper-trigger line is Anthropic's [Sleeper Agents](https://arxiv.org/abs/2401.05566). The memory-poisoning line includes [AgentPoison](https://arxiv.org/abs/2407.12784), [MemoryGraft](https://arxiv.org/abs/2512.16962), and [Hidden in Memory](https://arxiv.org/abs/2605.15338), which all treat retrieved or persistent agent memory as an attack surface rather than a neutral cache. The component borrows the control shape from that work: memory writes need provenance, untrusted context cannot silently become authority, later retrieval must pass an influence gate, and deletion needs an audit ref plus cold rerun. It does not secure a live memory system.\n\n##### Validation Result record Path\n\n**Run the first-wave fixture validator from the repo root and write its record outside the working tree:** Run the first-wave fixture validator from the repo root and write its record outside the working tree:\n\nThen run the exported bundle validator:\n\n```text\ncd microcosm-substrate && PYTHONPATH=src ../repo-python -m microcosm_core.organs.sleeper_memory_poisoning_quarantine_replay run-quarantine-bundle --input examples/sleeper_memory_poisoning_quarantine_replay/exported_sleeper_memory_poisoning_bundle --out /tmp/sleeper_memory_poisoning_bundle_receipt --card > /tmp/sleeper_memory_poisoning_bundle_card.json\n```\n\nThe focused test and corpus projection check are:\n\n```text\ncd microcosm-substrate && ../repo-pytest tests/test_sleeper_memory_poisoning_quarantine_replay.py\n./repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n###### What a pass over the record means\n\n**A pass means the recorded chain and its eight negative cases lined up over public metadata refs and the non-public-state scan found nothing.** A pass means the recorded chain and its eight negative cases lined up over public metadata refs, and the non-public-state scan found nothing. These are the validation result records for this module.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### Narrow claim and the scope limit\n\n**The narrow claim is that over one synthetic record the quarantine, blocked retrieval, and audited rollback are present and coupled and eight cases caught, establishing no live-memory quality, provider behaviour, or launch-scope decision.** The strongest claim the evidence supports is narrow: over one synthetic record of public metadata refs, the quarantine, the blocked-before-action retrieval, and the audited rollback with a clean cold rerun are all present and internally coupled, and eight named failure cases are each caught. The proof boundary is the set of refs and verdicts in that record: source bundle refs, provenance refs, quarantine verdicts, classifier labels, retrieval influence gates, rollback audit refs, cold rerun result refs, the negative cases, and the source-body digest checks. The scope limit is `public_body_free_sleeper_memory_quarantine_policy_projection_only`, matching `AUTHORITY_CEILING` in the source. It does not establish live memory product quality, live user-memory handling, trusted promotion from untrusted context, provider behaviour, source-file changes, benchmark security, private memory export, public sharing, or launch-scope decision.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":12608,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/sleeper_memory_poisoning_quarantine_replay.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"Sleeper 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source-faithful trace refactor separated trusted instructions from untrusted web/tool/browser text before any privileged action.","summary":"Validator-backed public claim contract for indirect prompt-injection information-flow policy. Admits one narrow claim: a source-faithful trace refactor separated trusted instructions from untrusted web/tool/browser text before any privileged action or answer, checked against negative cases and scope limits.","authority_ceiling":"Source-faithful refactored fixtures and metadata-only result records only; not a live information-flow product, complete-security proof, or launch claim.","abstract":"Validator-backed claim: a source-faithful trace refactor separated trusted instructions from untrusted web/tool/browser text before any privileged action.","teleology":{"section_id":"","text":"Validator-backed claim: a source-faithful trace refactor separated trusted instructions from untrusted web/tool/browser text before any privileged action.","source":"capsule_summary_line"},"body_markdown":"**Indirect prompt injection is untrusted text an agent obeys as an instruction; this component replays one synthetic episode to check untrusted text stayed data and reached no privileged action before a policy gate ruled.** An agent that reads web pages, tool output, or retrieved documents takes in text from sources it does not control. Indirect prompt injection is the case where that untrusted text carries an instruction and the agent acts on it as if the operator had asked. This component replays one recorded synthetic agent episode and checks a single property: untrusted text stayed labelled as data, and no untrusted source reached a privileged action before a policy gate ruled on that flow.\n\n**It runs over metadata-only rows of refs, labels, and counts rather than raw prompts or tool bodies, recomputing the trust decisions and recording which source fed which sink and what the policy decided.** It runs over metadata-only rows. Every source, flow, verdict, and output is a small JSON record of refs, labels, and counts, never a raw prompt, a real account, or a tool-output body. The runtime reads those rows, recomputes the trust decisions itself, and writes a result record a reader can inspect: which source fed which sink, what the policy decided before the action, and that no secret or trusted context leaked.\n\n**The component does not defend a live agent; it is a regression check over a fixed fixture and one exported bundle, and a pass means only that the recorded trace satisfies the flow-policy contract.** The component does not defend a live agent. It is a regression check over a fixed fixture and one exported example bundle. A pass means the recorded trace satisfies the flow-policy contract, and nothing beyond that.\n\n##### Purpose\n\n###### A trace can be labelled to look safe\n\n**The guarded failure is a trace that looks safe because its labels were written to look safe, which a field-by-field check waves through while the boundary never held.** The failure this guards against is a trace that looks safe because the labels were written to look safe. A field-by-field check trusts whatever the fixture declares. If a row relabels an untrusted web source as clean, or marks a dangerous action `allow`, a loose check waves it through and the trace passes while the boundary never held.\n\n###### The runtime recomputes, never trusts labels\n\n**The runtime ignores declared taint labels and verdicts and recomputes them by propagating taint along the source-to-sink graph, so a flow cannot quietly relabel an untrusted source or downgrade a dangerous action.** So the runtime does not trust the declared labels. Each flow row claims a set of taint labels and a policy verdict, and the runtime ignores both and recomputes them. It propagates taint along the source-to-sink graph from the labelled sources, so a sink inherits the taint of everything that fed it. It then derives the verdict from that propagated taint plus the sink's privilege, the sanitizer state, the sink kind, and the proposed action. When a declared label or verdict disagrees with the derived one, the row is blocked. A flow cannot quietly relabel an untrusted source or downgrade a dangerous action, because the contradiction is recomputed rather than read back.\n\n##### How it works\n\n###### `_build_result` runs the validators in order\n\n** `_build_result` loads the seven inputs, runs a secret-exclusion scan, builds the public trace spans, and folds the ordered validators into one result, with `run` enabling negative cases and the bundle path leaving them out.** `_build_result` is the pipeline. It loads the seven input files, runs a secret-exclusion scan over the input paths, builds the public trace spans, then calls the validators in order and folds their statuses into one result. `run` drives fixture mode with the negative cases enabled. `run_prompt_injection_bundle` drives the exported example bundle and leaves the fixture-only negative cases out. Both write metadata-only records through `_write_receipts` and can emit a compact card through `result_card`.\n\nFunction What it checks `validate_source_documents` Each source row carries an id, trust label, channel, taint labels, redaction, and a synthetic flag; an untrusted source may not claim instruction authority; the set needs at least three untrusted and two trusted sources. `_taint_propagation_receipt` Propagates taint from the labelled sources along the source-to-sink graph, derives each flow's taint via `_derived_flow_taint_labels` and its verdict via `_derived_policy_verdict`, and records a finding when the declared taint or verdict differs from the derived one. `validate_information_flow_graph` Joins each flow to its source row, runs the propagation result record, and rejects an untrusted privileged flow marked `allow` without sanitization. `_live_tool_call_trace_promotion` Takes one generated tool-call span, wraps it as an `untrusted_tool_output` source flowing into a privileged sink, and pushes it through the same propagation machinery as evidence that tool output is treated as data until a gate reviews it. `validate_policy_verdicts` Each verdict joins a flow, precedes the action, cites rules, stays redacted, and equals the derived flow verdict; the set must cover allow, warn, block, and review. `validate_sanitized_outputs` Each output discloses no trusted context and obeys no untrusted instruction; a blocked flow may not attempt an external action. `validate_cold_replay` Each replay row names a command and a result record ref that reproduce the verdict and the sanitized output with no trusted-context disclosure. `validate_negative_cases` Turns each deliberately-bad fixture into its expected error code; fixture mode must observe all eight.\n\n###### The fixed verdict ladder\n\n** `_derived_policy_verdict` is a small fixed ladder that blocks hidden or instruction-channel promotion and unsanitized untrusted taint into privileged sinks, reviews or warns the sanitized cases, and allows everything else.** The verdict ladder in `_derived_policy_verdict` is small and fixed. A flow that promotes hidden or instruction-channel text is blocked. Untrusted taint into a privileged sink is blocked when unsanitized and set to review when sanitized. Untrusted taint into a non-privileged sink warns if sanitized and reviews if not. Everything else is allowed. `validate_projection_protocol` and `validate_injection_policy` sit alongside these: the policy must define all four verdicts and deny every authority overclaim, and the projection protocol must cite the trace source and target refs and keep bodies out of the record.\n\nDiagram of the mechanism (10 steps). Diagram source & refs\n\n```text\nflowchart TD sources[\"Source rows trusted and untrusted, with taint labels\"] flows[\"Source-to-sink flow rows declared taint and verdict\"] propagate[\"Propagate taint along the source-to-sink graph\"] derive[\"Derive verdict from taint, privilege, sanitizer, sink kind, action\"] compare{\"Declared labels and verdict match the derived ones?\"} blocked[\"Block the row\"] verdicts[\"Pre-action verdict allow / warn / review / block\"] outputs[\"Sanitized output no trusted context, no obeyed instruction\"] toolspan[\"One tool-call span as untrusted tool output\"] record[\"metadata-only result record\"] sources --> flows flows --> propagate toolspan --> propagate propagate --> derive derive --> compare compare -- no --> blocked compare -- yes --> verdicts verdicts --> outputs outputs --> record blocked --> record\n```\n\n###### What the fixture and bundle carry\n\n**The fixture carries five sources, five flows covering all four verdicts, five outputs, five cold replays, one promoted tool-call trace, and eight negatives, while the bundle adds a digest-checked source-module manifest.** The fixture carries five source documents, three untrusted and two trusted, five source-to-sink flows whose verdicts cover allow, warn, block, and review, five sanitized outputs, five cold replay passes, five public trace spans, one promoted tool-call trace, and eight negative cases. The exported bundle also ships a `source_module_manifest.json`, and `_source_module_manifest_result` checks that its copied source-module bodies are classified as `copied_non_secret_macro_body`, keep body text out of the record, match the declared count, and resolve each `path` and `target_ref` to the same file with a matching SHA-256 digest.\n\n##### Negative cases\n\n###### Eight falsification fixtures as the boundary\n\n**The eight `EXPECTED_NEGATIVE_CASES` are the admission boundary, each a deliberately-bad flow the runtime must reject, and fixture mode is blocked if any goes unobserved.** The eight fixtures in `EXPECTED_NEGATIVE_CASES` are the admission boundary. `validate_negative_cases` must reject each one: real account material, secret or trusted-context exfiltration, raw prompt body export, untrusted tool output treated as instruction authority, hidden system-message promotion, account secret exfiltration, final-answer-only success, and ungated untrusted flow into a privileged sink. These are falsification fixtures, part of the contract, not samples of live exploit traffic. Fixture mode is blocked if any expected case goes unobserved.\n\n##### Prior Art Grounding\n\n###### Indirect-injection and information-flow lineage\n\n**The threat shape follows Greshake et al., AgentDojo, and tool-calling data-leakage work, with the policy mechanism drawing on dynamic information-flow and taint-tracking, turned here into a metadata-only replay with negative cases.** The threat shape follows the indirect-injection framing of Greshake et al. in [Not what you've signed up for](https://arxiv.org/abs/2302.12173), the agentic evaluation setting of [AgentDojo](https://arxiv.org/abs/2406.13352), and the tool-calling data-leakage results in [Simple Prompt Injection Attacks Can Leak Personal Data](https://arxiv.org/abs/2506.01055). The policy mechanism draws on dynamic information-flow and taint-tracking, including [Permissive Information-Flow Analysis for Large Language Models](https://arxiv.org/abs/2410.03055). The engineering lesson taken from that work is narrow: untrusted content is labelled as data, source-to-sink flows are visible before a privileged action, and sanitized outputs carry evidence. This component turns that lesson into a metadata-only replay with explicit scope limits and negative cases.\n\n##### Validation Result record Path\n\n###### Fixture validation outside the working tree\n\n**Run the fixture validator from the repo root and write its record outside the working tree:** Run the fixture validator from the repo root and write its record outside the working tree:\n\nThen run the exported bundle validator:\n\n```text\ncd microcosm-substrate && PYTHONPATH=src ../repo-python -m microcosm_core.organs.indirect_prompt_injection_information_flow_policy_replay run-prompt-injection-bundle --input examples/indirect_prompt_injection_information_flow_policy_replay/exported_prompt_injection_flow_bundle --out /tmp/indirect_prompt_injection_flow_bundle_receipt --card > /tmp/indirect_prompt_injection_flow_bundle_card.json\n```\n\nThe focused regression test and corpus projection check are:\n\n```text\ncd microcosm-substrate && PYTHONPATH=src ../repo-python -m pytest -p no:cacheprovider tests/test_indirect_prompt_injection_information_flow_policy_replay.py -q\ncd microcosm-substrate && PYTHONPATH=src ../repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n###### What a pass means here\n\n**A pass means the synthetic trace reproduced its verdicts and sanitized outputs, observed the expected negative cases, and wrote no source body, not general robustness or live account safety.** A pass means the synthetic trace reproduced its verdicts and sanitized outputs, observed the expected negative cases, and wrote no source body. It does not mean general prompt-injection robustness or live account safety.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim\n\n**Over declared synthetic metadata-only rows the replay recomputes source trust, taint propagation, pre-action verdicts, sanitized-output constraints, cold replay, and copied source-module digests, exposing the results without leaking a body.** The strongest claim the evidence supports is this: over declared synthetic metadata-only rows, the replay recomputes source trust, taint propagation, pre-action verdicts, sanitized-output constraints, cold replay, and the copied source-module digests, and it exposes those results without leaking a body. The copied-body digest proves byte parity for the named files only; it does not widen the replay into live source authority.\n\n###### What the replay does not claim\n\n**The proof boundary is that fixed input set and the result records, not general prompt-injection robustness, live account or tool safety, raw body export, benchmark claims, or launch-scope decision.** The proof boundary is that fixed input set and the validation result records listed above, not this prose. Its scope limit stops there. It does not claim general prompt-injection robustness, live account safety, live tool or provider behaviour, raw prompt or system or tool body export, account secret-bearing data, hidden-message handling in production, benchmark claims, source-file changes, public sharing, hosting, launch-scope decision, or complete security.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":13370,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/indirect_prompt_injection_information_flow_policy_replay.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"Not what you've signed up 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claim"]},{"id":"paper_module.mcp_tool_authority_replay","title":"MCP Tool Authority Replay","relative_href":"docs/paper-module-mcp-tool-authority-replay.html","read_href":"docs/paper-module-mcp-tool-authority-replay.html","context_href":"docs/rules-and-ideas.html#paper-module-mcp-tool-authority-replay","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"Synthetic MCP-like replay fixture for a tool-authority claim contract: replay result records, negative cases, scope limits; no live MCP/provider/account secret","summary":"Public Plectis projection of a tool-authority claim contract. Replays a synthetic MCP-like tool-authority scenario and validates replay result records, negative cases, and scope limits with metadata-only result records.","authority_ceiling":"Synthetic MCP-like replay fixture only; not a live MCP account test, external model access, account secret-handling certification, benchmark security result, or launch claim.","abstract":"Synthetic MCP-like replay fixture for a tool-authority claim contract: replay result records, negative cases, scope limits; no live MCP/provider/account secret","teleology":{"section_id":"","text":"Synthetic MCP-like replay fixture for a tool-authority claim contract: replay result records, negative cases, scope limits; no live MCP/provider/account secret","source":"capsule_summary_line"},"body_markdown":"**This component replays a recorded MCP tool trace and calls the tool use safe only when every action shows a narrow scope, prior approval, a rollback path, and untrusted output kept out of instructions.** When an agent uses outside tools through a protocol like MCP, the sentence \"the agent used the tool safely\" is cheap to write. This component makes it expensive. It takes a recorded trace of tool use and checks that every action in it was fenced: bound to a narrow permission, approved before it changed anything, given a way to undo it, and stopped from letting untrusted tool output give the agent orders.\n\n**The check runs on a built -in example of labels and references only, scope names, argument hashes, and token ids, so tool-use safety is inspectable without touching a live account.** The check runs against a small built-in example, not a live account. The example carries only labels and references: capability scope names, argument hashes, approval token ids, ledger and rollback references. No secrets, no account secrets, no tool contents. So the safety properties an agent's tool use is meant to have become inspectable without anything touching a real provider or account.\n\n**It reads a directory of input JSON and emits four metadata-only records carrying counts, status bits, and digests, but no payload, account secret, or tool body.** The component reads a directory of input JSON and emits a set of metadata-only result records: a result, a board, a validation record, and a fixture sign-off record. Each records counts, status bits, and reference digests. None of them carries a payload, a account secret, or a tool body.\n\n##### Purpose\n\n###### What a loose tool trace hides\n\n**Describing a tool call is easy and backing it is hard, and loose prose hides the two failure modes that matter for tool-using agents.** A tool call is easy to describe and hard to back. A trace can say a write happened and that it was fine. Loose prose like that hides the two failure modes that matter for tool-using agents.\n\n###### Confused deputy and injected instructions\n\n**The two modes are the confused deputy, where a call borrows a scope wider than its task, and tool-output-as-instruction, where untrusted output is obeyed as a command.** The first is the confused deputy: a call asks for a scope wider than its task needs, so a tool quietly borrows more authority than it was granted. The second is tool-output-as-instruction, the prompt-injection shape where text returned by an untrusted tool is obeyed as a command. Both pass a reader who only sees a summary. Both fail once you require the trace to show its working.\n\n###### Each call is an evidence-citing transaction\n\n**Each tool call is treated as a transaction that must cite its evidence, and every reference is cross-checked against the accepted calls by call id rather than taken on trust.** So the component treats each tool call as a small transaction that must cite its evidence. A call names a narrow capability scope and an argument hash. If it writes, it must also name an approval token, a side-effect ledger entry, and a rollback record. Those references are not taken on trust. The side-effect ledger and the cold replay records are cross-checked against the accepted call records by call id, so a rollback record no call refers to, or a write that skips approval, is caught rather than waved through. A reference string is not authority until something downstream resolves it.\n\n##### How it works\n\n###### Two entrypoints over one pipeline\n\n** `run` validates the first-wave fixture with its eight negative cases and `run_tool_authority_bundle` validates an exported bundle without them, and both call the shared `_build_result` pipeline.** The two entrypoints are `run` and `run_tool_authority_bundle`. `run` validates the first-wave fixture with the eight negative cases included, then writes the four fixture records through `_write_receipts`. `run_tool_authority_bundle` validates an exported bundle without the negative cases and writes a single bundle result record. Both call `_build_result`, which is the pipeline.\n\n###### Secret scan runs before plane validation\n\n** `_build_result` runs in a fixed order: load the input JSON, scan inputs and copied source for a leaked secret first, then validate each plane and pass the accepted records forward.** `_build_result` runs in a fixed order. It loads the input JSON with `_load_payloads`, then runs `load_forbidden_classes` and `scan_paths` over the inputs and any copied source files to catch a leaked secret before anything else. It then validates each plane on its own and passes the accepted records forward:\n\nFunction What it checks `validate_tool_policy` the declared policy id and allowed tool classes `validate_tool_manifest` declared tool ids and their classes; produces the set of accepted tool ids `validate_tool_calls` each call, using `_validate_call_row`; requires three calls with one approved write and one untrusted result `validate_tool_results` each result binds to an accepted call id; one untrusted output is kept as data and ignored as instruction `validate_side_effect_ledger` one approved write side effect whose approval, ledger, and rollback references match its call record `validate_cold_replay` three passing replays whose evidence references bind to the manifest and to runtime records `validate_projection_protocol` the protocol id and source-to-target projection references\n\n###### Where a call is admitted or blocked\n\n** `_validate_call_row` rejects undeclared tools, overbroad scopes, exported account secrets, unapproved writes, missing rollbacks, and in-record payloads, returning `accepted_tool_call_metadata` only when no reason fires.** `_validate_call_row` is where a call is admitted or rejected. It rejects an undeclared tool id, an overbroad scope (`*`, `account_full_access`, and the like), an exported account secret, untrusted output flagged as instruction, live account access, final-answer-only grading, a write that lacks approval or a ledger reference, a missing rollback record, and any call whose payload sits in the record instead of staying metadata-only. It also checks that every field in `REQUIRED_CALL_FIELDS` is present. A clean call returns `accepted_tool_call_metadata`; any reason at all returns `blocked`.\n\n###### Trace spans and the body floor\n\n**After the planes validate, three public trace spans are contributed and the body floor verifies each copied source digest against its live reference with `body_in_receipt: false` on every row.** After the planes validate, `build_public_mcp_tool_authority_trace` contributes three public trace spans for the calls, and `_source_module_manifest_result` runs the body floor for the bundle path: it verifies the digest of each copied source file against its live source reference, checks required anchors, and requires `body_in_receipt: false` on every row. `_body_import_verification` binds that public refactor back to the source and target digests.\n\n**The final status is `pass` only when every plane passes, no expected negative case is missing, the secret scan has zero blocking hits, and the source-module floor is present when the bundle requires it.** The final status is `pass` only when every plane passes, no expected negative case is missing, the secret scan has zero blocking hits, and the source-module floor is present when the bundle requires it. `_write_receipts` then emits the result, board, validation, and sign-off records, and `result_card` projects a compact card that carries counts and status bits while omitting the tool rows, call rows, findings, and source bodies listed in `CARD_OMITTED_FULL_PAYLOAD_KEYS`.\n\nDiagram of the mechanism (7 steps). Diagram source & refs\n\n```text\nflowchart TD input[\"input JSON directory\"] scan[\"secret scan load_forbidden_classes + scan_paths\"] planes[\"plane validators policy, manifest, calls, results, side effects, cold replay\"] trace[\"three public trace spans\"] floor[\"source-module body floor\"] status[\"pass only if all clear\"] records[\"metadata-only result records\"] input --> scan scan --> planes planes --> trace planes --> floor trace --> status floor --> status status --> records\n```\n\n##### Negative cases\n\n###### Eight negatives as a falsification set\n\n**The first-wave fixture ships eight negative inputs, and a run passes only if it observes the error code for every one, making the fixture a falsification set.** The first-wave fixture ships eight negative inputs, one per case in `EXPECTED_NEGATIVE_CASES`. A first-wave run passes only if it observes the error code for every one of them, so the fixture is a falsification set the validators have to trip on:\n- `overbroad_scope` raises `MCP_TOOL_OVERBROAD_SCOPE`.\n- hidden account secret export raises MCP TOOL HIDDEN account secret EXPORT.\n- `tool_output_as_instruction` raises `MCP_TOOL_OUTPUT_AS_INSTRUCTION`.\n- `unapproved_side_effect` raises `MCP_TOOL_UNAPPROVED_SIDE_EFFECT`.\n- `live_account_access` raises `MCP_TOOL_LIVE_ACCOUNT_ACCESS`.\n- `final_answer_only_grading` raises `MCP_TOOL_FINAL_ANSWER_ONLY_GRADING`.\n- `missing_rollback_receipt` raises `MCP_TOOL_MISSING_ROLLBACK_RECEIPT`.\n- `unredacted_tool_payload` raises `MCP_TOOL_UNREDACTED_PAYLOAD_EXPORT`.\n\n###### An unobserved case blocks the run\n\n**A negative case that never trips is recorded under `missing_negative_cases`, and its absence blocks the run rather than letting it pass.** If any expected case goes unobserved, `_build_result` records it under `missing_negative_cases` and the run is blocked.\n\n##### Prior Art Grounding\n\n###### Capability security made replayable\n\n**The design follows capability-security and least-privilege lineage from Saltzer and Schroeder through Hardy and the MCP authorization spec, then makes it replayable so scope and approval references must precede any write authority.** The component follows capability security, least privilege, and current MCP authorization guidance. The classic lineage is Saltzer and Schroeder's [Protection of Information in Computer Systems](https://www.cs.virginia.edu/~evans/cs551/saltzer/) and Hardy's [Confused Deputy](https://web.cs.wpi.edu/~cs557/f14/papers/confused_deputy-hardy.pdf): authority should be narrow, mediated, and bound to the action being requested. The MCP-specific lineage is the official [MCP authorization](https://modelcontextprotocol.io/specification/2025-06-18/basic/authorization) and [security best practices](https://modelcontextprotocol.io/specification/2025-06-18/basic/security_best_practices) guidance on least-privilege scopes and token audience boundaries. The contribution here is making that shape replayable: a tool story has to expose scope, approval, side-effect, rollback, and instruction/data references before its write authority counts as evidence.\n\n##### Validation Result record Path\n\nRun the first-wave fixture into disposable records from the Microcosm root:\n\nRun the exported bundle through the same component:\n\n```text\ncd microcosm-substrate\nPYTHONPATH=src ../repo-python -m microcosm_core.organs.mcp_tool_authority_replay run-tool-authority-bundle --input examples/mcp_tool_authority_replay/exported_mcp_tool_authority_bundle --out /tmp/microcosm_mcp_tool_authority_bundle\n```\n\nThen run the focused test and the corpus parity check:\n\n```text\ncd microcosm-substrate\nPYTHONPATH=src ../repo-python -m pytest -p no:cacheprovider tests/test_mcp_tool_authority_replay.py -q\nPYTHONPATH=src ../repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n###### What a pass certifies\n\n**A green run certifies only internal consistency: every plane validated, all eight negatives observed, the secret scan clean, and the records metadata-only.** A pass means every plane validated, all eight negative cases were observed, the secret scan found nothing, and the records stayed metadata-only.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim\n\n**The strongest claim is narrow: a synthetic MCP-like replay preserved its tool-authority boundaries over metadata records, with every reference and negative case lined up or the run blocked.** The strongest claim is this: a synthetic MCP-like replay preserved its tool-authority boundaries over metadata records. Capability scopes, argument hashes, approval references, side-effect and rollback references, instruction/data splits, cold replay references, source digests, the eight negative cases, and metadata-only validation records all lined up, or the run blocked.\n\n###### What stays false on a pass\n\n**Even on a pass the component opens no account, calls no provider, and handles no account secret, so it claims no live MCP safety and certifies no real integration secure.** That is the scope limit and the proof boundary. The component never opens an MCP account, calls a provider, or handles a account secret, and it keeps every payload, tool result, and account secret out of what it writes. So it does not claim live MCP account safety, account secret-handling certification, live tool or provider behavior, benchmark security, source-file changes, complete security, publishing-scope decision, or launch-scope decision. It checks that a tool-authority story is internally consistent and metadata-only. It does not certify that any real tool integration is secure.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":13316,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/mcp_tool_authority_replay.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"}],"grouped_relation_feeds":[{"relation":"abides by","count":1,"links":[{"id":"AX-3","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Authority by derivation, not possession","url":"doctrine.html#dcard-ax-3"}]},{"relation":"depends on","count":1,"links":[{"id":"paper_module.macro_projection_import_protocol","kind":"paper_module","kind_label":"paper module","relation":"depends on","label":"Source Projection Import 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becoming proof, benchmark, or launch-scope decision.","summary":"Tactic Portfolio Availability Probe validates copied Lean/Std tactic affordance rows before downstream routing can treat a tactic as usable. It checks compile status, Mathlib absence handling, probe portfolio membership, negative cases, source digests, and metadata-only result records while keeping proof bodies, model-output data, benchmark claims, and launch-scope decision out of scope.","authority_ceiling":"Copied tactic affordance probe rows and public fixture/exported-bundle result records only; no Lean/Lake rerun, theorem proof, benchmark performance, external model access, Mathlib-dependent proof authority, launch-scope decision, publishing-scope decision, or whole-system correctness.","abstract":"Environment-scoped tactic availability rows gate downstream tactic routing without becoming proof, benchmark, or launch-scope decision.","teleology":{"section_id":"","text":"Environment-scoped tactic availability rows gate downstream tactic routing without becoming proof, benchmark, or launch-scope decision.","source":"capsule_summary_line"},"body_markdown":"**It turns one captured Lean smoke run into an explicit list of callable tactics, checking pre-recorded status rows for honesty without running Lean itself.** `tactic_portfolio_availability_probe` turns one captured Lean run into an explicit list of which proof tactics were callable, before any router or proof search treats a tactic as usable. It reads pre-recorded status rows from a real smoke run and checks them for honesty. It does not run Lean itself.\n\n**The copied 2026-05-11 fixture covers eight tactics, seven compiling and `aesop` failed on a missing Mathlib import, reproduced into a metadata-only board and result records.** The fixture is copied from a 2026-05-11 tactic affordance probe. It covers eight tactics: `rfl`, `decide`, `omega`, `simp`, `simp_all`, `grind`, `native_decide`, and `aesop`. Seven are recorded as compiling. `aesop` is recorded as failed because the paired environment probe reports that the Mathlib import it needs was absent. The component reproduces that verdict from the copied rows and writes a metadata-only availability board plus result records.\n\n##### Purpose\n\n###### A tactic name is not availability\n\n**A tactic name alone does not mean it can run; a router trusting the bare name reaches for tactics the environment cannot run and misreads the failure as a fact about the goal.** A tactic name is not a usable tactic. `aesop` runs only if the surrounding Lean and Std environment carries the imports it needs. `omega` is callable in one project layout and not in another. A router that trusts a bare tactic name will reach for tactics the current environment cannot run, then misread the failure as a fact about the goal rather than a fact about the environment.\n\n###### Missing import kept distinct from broken tactic\n\n**It answers which tactics were callable and on what evidence, keeping a missing import distinct from a broken tactic so a router can re-attempt elsewhere.** The component answers one question: in the observed environment, which tactics were callable, and on what evidence? The interesting part is how it treats failure. A recorded Lean `FAIL` is not flattened into one \"unavailable\" verdict. A missing import is kept distinct from a broken tactic, so a router can re-attempt the same tactic in a different environment instead of striking it off for good.\n\n##### How it works\n\n###### Two entrypoints over `_build_result`\n\n** `run` validates the first-wave fixture with negatives on, `run_availability_bundle` validates an exported bundle with negatives off and reuses an unchanged prior result.** Two entrypoints drive the pipeline. `run` validates the first-wave fixture with negative cases enabled. `run_availability_bundle` validates an exported bundle with negative cases disabled, and reuses a prior result when `_fresh_availability_bundle_receipt` finds a matching record whose inputs have not changed since it was written. Both delegate the work to `_build_result` and then persist records.\n\n`_build_result` runs these steps in order.\n\n###### Digest-checking the copied source payloads\n\n**It loads the three copied source payloads and digest-checks each body against a recorded SHA-256, so a silently edited body becomes a finding rather than trusted input.** It loads the copied source payloads with `_load_source_payloads`: the run-level affordance probe, the `portfolio_core_v0` availability artifact, and the paired corpus-readiness file. Each copied body is digest-checked by `_source_body_imports` and `_probe_source_imports` against a recorded SHA-256, so a silently edited body becomes a finding rather than trusted input.\n\n###### Mathlib probe gates dependent tactics\n\n**It derives the environment state from the Mathlib probe, whose single fact of whether the Lake import was available gates every Mathlib-dependent tactic.** It derives the environment state with `_authoritative_environment_state`, which reads the Mathlib probe and reports whether the Mathlib Lake import was available. That single fact gates every Mathlib-dependent tactic.\n\n###### Deriving each tactic's public status\n\n**It builds one row per tactic, mapping a source `PASS` to `compile_pass` and a `FAIL` to `environment_fail` when a required Mathlib import was absent, else `compile_fail`.** It builds one availability row per tactic with `_availability_rows_from_source`. The public status comes from `_derived_public_status_from_source`: a source `PASS` becomes `compile_pass`; a source `FAIL` becomes `environment_fail` when the tactic requires Mathlib and Mathlib was absent, and `compile_fail` otherwise. When the status is a failure, `_failure_classifier_from_source` labels a Mathlib gap as `MATHLIB_IMPORT_MISSING` and otherwise carries the source error class through. This is why `aesop` reads as an environment failure, not a broken tactic.\n\n###### Portfolio summary and latency banding\n\n**It summarises available and unavailable ids and bands each available tactic's copied duration as fast, moderate, or slow, stamped environment-scoped rather than a benchmark.** It summarises the rows with `_portfolio_summary`: available and unavailable tactic ids, Mathlib-dependent ids, and a latency profile. `_latency_band` bands each available tactic by its copied duration as `fast` at 2500ms or less, `moderate` up to 5000ms, and `slow` above that. The latency profile is stamped environment-scoped, so a router can prefer a cheaper available tactic without reading speed as a benchmark result.\n\n###### Negatives, secret scan, and result records\n\n**It evaluates the negative cases, scans every input path for forbidden material, and passes only with no finding, all negatives observed, and zero secret hits before writing result records.** It evaluates the declared negative cases with `_negative_findings`, scans all input paths for forbidden material with `scan_paths`, and sets the overall status. The result is `pass` only when no category produced a finding, every expected negative case was observed, and the secret scan had zero blocking hits. `_write_receipts` then writes the result record, the availability board, the validation record, and the sign-off record, each carrying result record paths but no tactic bodies.\n\nDiagram of the mechanism (9 steps). Diagram source & refs\n\n```text\nflowchart TD A[\"Copied affordance probe rows compile_status, requires_mathlib, duration_ms\"] --> B[\"_build_result\"] C[\"Mathlib environment probe\"] --> B B --> D{\"source compile_status\"} D -->|PASS| E[\"compile_pass band duration fast / moderate / slow\"] D -->|FAIL + requires Mathlib + Mathlib absent| F[\"environment_fail reason MATHLIB_IMPORT_MISSING\"] D -->|FAIL otherwise| G[\"compile_fail\"] E --> H[\"Availability board\"] F --> H G --> H B --> I[\"metadata-only result and validation records\"]\n```\n\n##### Negative cases\n\n###### Six perturbed fixtures and their codes\n\n**Six perturbed fixtures must each be rejected with a specific error code, spanning missing status, unbacked Mathlib claims, unprobed tactics, missing durations, leaked bodies, and authority overclaim.** `_negative_findings` checks six perturbed fixtures and requires each to be rejected with a specific error code. A tactic row with no `compile_status` gives `TACTIC_PORTFOLIO_MISSING_COMPILE_STATUS`. A Mathlib-dependent tactic marked available with no passing Mathlib probe gives `TACTIC_PORTFOLIO_MATHLIB_CLAIM_WITHOUT_PROBE`. A downstream request for a tactic not in the probed portfolio gives `TACTIC_PORTFOLIO_UNPROBED_TACTIC_REFERENCED`. An available tactic with no copied duration gives `TACTIC_PORTFOLIO_AVAILABLE_DURATION_MISSING`. A fixture carrying a proof or provider body gives `TACTIC_PORTFOLIO_PROOF_BODY_FORBIDDEN`. A payload asserting proof, provider, benchmark, or launch-scope decision gives `TACTIC_PORTFOLIO_AUTHORITY_OVERCLAIM`. A missing expected case blocks the run.\n\n##### Prior Art Grounding\n\n###### Feature-detection discipline from Autoconf and Lean\n\n**It follows the feature-detection habit of probing a local capability before relying on it, applying Autoconf and Lean's environment-sensitive discipline to one recorded run.** The component follows the feature-detection habit of testing a local capability before relying on it, the same discipline GNU Autoconf's configure workflow established for build environments. Lean's own tactic documentation notes that tactic use is environment- and goal-sensitive, so a tactic name alone does not justify downstream routing. This component applies that probe discipline to one recorded run and keeps a Mathlib-dependent absence as evidence rather than discarding it.\n- GNU Autoconf feature probing: https://ftp.gnu.org/old-gnu/Manuals/autoconf-2.57/html_chapter/autoconf.html\n- Lean 4 tactic documentation: https://lean-lang.org/theorem_proving_in_lean4/Tactics/\n\n##### Validation Result record Path\n\nFrom `microcosm-substrate/`, reproduce the evidence with temporary records:\n\n###### What a pass certifies\n\n**A pass means every copied body matched its digest, the eight rows produced the expected board with `aesop` as an environment failure, all six negatives were rejected, and the secret scan was clean.** A pass means every copied source body matched its recorded digest, the eight tactic rows produced the expected availability board with `aesop` as an environment failure, all six negative cases were rejected, and the secret scan found no forbidden material.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### Callability held as callability\n\n**The strongest claim is that in one observed environment these tactics were callable on digest-matched probe rows; it proves no goal, reruns no Lean, and authorizes nothing.** The strongest claim the evidence supports is narrow: in one observed Lean and Std environment, these tactics were callable, on the evidence of copied probe rows whose bodies match recorded digests, with Mathlib absence preserved as a recorded fact. The proof boundary is those copied rows, the derived availability board, the digest and secret-exclusion checks, the six negative cases, and the validation result records. The scope limit holds callability as callability. It does not establish any goal, rerun Lean or Lake, widen Lean or Lake authority, use external model services, claim benchmark performance, export non-public paths, include launch operations, or treat tactic callability as proof quality.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":10341,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/tactic_portfolio_availability.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"}],"grouped_relation_feeds":[{"relation":"abides by","count":5,"links":[{"id":"AX-1","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Derivation before assertion","url":"doctrine.html#dcard-ax-1"},{"id":"AX-2","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Kernelized 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It validates Ring2 problem-domain, failure-class, graph-update, and tactic-probe references, rejects unavailable or unprobed tactics, rejects private target ids/formulas/statuses/traces without echoing values, and records metadata-only route decisions before any Lean/Lake proof authority is claimed.","authority_ceiling":"Public route-reference fixture and exported-bundle result records only; no Lean/Lake execution, formal-result correctness, private target material disclosure, post-execution routing, external model access, proof body export, launch-scope decision, publishing-scope decision, or whole-system correctness.","abstract":"Pre-execution tactic routing admits or rejects tactics from target shape and probe evidence while excluding private target material.","teleology":{"section_id":"","text":"Pre-execution tactic routing admits or rejects tactics from target shape and probe evidence while excluding private target material.","source":"capsule_summary_line"},"body_markdown":"**Before any proof attempt it reads the candidate tactics for one goal and records an allow-or-reject decision for each.** Before a proof is attempted, this component reads a list of candidate proof tactics for one goal and writes down an allow-or-reject decision for each. It runs in `src/microcosm_core/organs/target_shape_tactic_routing_gate.py`.\n\n**It decides from goal shape and an environment probe without calling Lean, emitting a tactic-by-tactic record of what was admitted or blocked and why.** The decision is made from evidence that already exists: the shape of the goal and a probe that says which tactics the declared environment can run. Nothing here calls Lean, runs a prover, or proves anything. The output is a record that shows, tactic by tactic, what was admitted or blocked and the plain reason, in place of an opaque \"we tried these\" claim.\n\n**A rejection is one of three kinds: unprobed, known but unavailable, or available but wrong for the goal shape, kept separate so a reader sees which limit each hit.** A rejection has one of three kinds. The tactic is not in the tested probe set, so it was never measured. It is known but marked unavailable in the declared environment. Or it is available but does not fit the goal shape. The record separates those three so a reader can see which limit each blocked tactic hit.\n\n##### Purpose\n\n###### Most proof cost is wasted on doomed tactics\n\n**Most proof cost goes to tactics that were never going to work, and a loose gate that trusts the route's own choice cannot catch an unavailable or unprobed one.** A proof attempt is expensive, and most of that cost is spent on tactics that were never going to work. A loose gate that trusts the route's own declared choice cannot catch this: the route is a claim about what should run, and a route that admits an unavailable or unprobed tactic passes a loose check while still wasting the later prover call.\n\n###### The gate recomputes the tactic choice\n\n**The gate recomputes the selection from a per-shape preferred order, skipping and recording unselectable tactics, and flags a route whose declared choice disagrees.** So the gate recomputes the choice rather than accepting the declared one. Each target shape carries a short preferred-tactic order, held in `PREFERRED_TACTICS_BY_TARGET_SHAPE` (for example `omega` for `int_linear_arithmetic`, `decide` for `closed_nat_mod_decision`). The gate walks that order, skips any preferred tactic it cannot select, records why it skipped, and falls back to the next allowed candidate or to `DEFAULT_PREFERRED_TACTICS` for a shape it does not recognise. A route whose declared selection disagrees with the computed one is flagged, not honoured.\n\n##### How it works\n\n###### `_build_result` orders the pipeline\n\n** `_build_result` loads the two inputs, scans for forbidden material, builds availability sets, scores each route, folds in negatives, and sets one pass-or-blocked status.** `_build_result` runs the pipeline in order. It loads the two input files through `_load_payloads`, scans every input path for forbidden material with `scan_paths`, builds the availability sets with `_portfolio`, sanitizes and scores each route case, checks the copied source artifacts, folds in the negative-case fixtures, then sets a single pass-or-blocked status and emits the result and board.\n\n###### `_portfolio` splits the probe into three sets\n\n** `_portfolio` splits the probe into known, available, and unavailable ids, counting a tactic available only when its status is one of four passing values.** `_portfolio` reads the tactic probe file and splits it into three sets: `known_tactic_ids` (every tactic listed), `available_tactic_ids`, and `unavailable_tactic_ids`. A tactic counts as available only when its status is one of `available`, `pass`, `compiled`, or `compile_pass`.\n\n###### `_score_case` compares declared against computed\n\n** `_score_case` reads a case's allowed and candidate lists, gets a per-tactic verdict, computes the selection, and compares the declared tactic against the computed one.** `_score_case` scores one route case. It reads the case's `allowed_tactic_ids` and candidate list, asks `_decision_for_tactic` for a verdict on each candidate, computes the selection with `_shape_preferred_selection`, and compares the case's declared `selected_tactic_id` against the computed one.\n\n###### `_decision_for_tactic` rejects in a fixed order\n\n** `_decision_for_tactic` rejects in order as unprobed, unavailable, or shape-inadmissible, admitting only a tactic that is probed, available, and allowed for the shape.** `_decision_for_tactic` is the per-tactic gate. It rejects in a fixed order: a tactic absent from `known` is `UNPROBED_TACTIC`, a known tactic absent from `available` is `UNAVAILABLE_TACTIC`, an available tactic absent from `allowed` is `TARGET_SHAPE_ADMISSIBILITY_REJECTED`. Only a tactic that is probed, available, and allowed for the shape gets `TARGET_SHAPE_ADMISSIBLE`.\n\n###### `_shape_preferred_selection` picks the first allowed tactic\n\n** `_shape_preferred_selection` walks the shape's preferred order and returns the first allowed tactic, recording skips, the default-used flag, and a preferred-unavailable case.** `_shape_preferred_selection` walks the preferred order for the shape and returns the first tactic that is allowed. It records the skipped preferred tactics and their reasons, sets `unknown_shape_default_used` when the shape has no specific map, and marks the preferred-unavailable case when the first preferred tactic is known but not usable.\n\n###### `_route_integrity_findings` types each per-case problem\n\n** `_route_integrity_findings` turns any per-case problem into a typed finding: an admitted unavailable or unprobed tactic, a post-execution stage, or a selection mismatch.** `_route_integrity_findings` turns any per-case problem into a typed finding: an admitted unavailable tactic, an admitted unprobed tactic, a post-execution route stage, or a declared selection that does not match the computed preference.\n\n###### Sanitizing private goal material\n\n**The sanitizer keeps the public route shape but drops forbidden private-target keys and masks private-looking strings before any public row, reporting only the field path and class.** `_private_target_material_findings` and `_sanitize_private_target_material` handle the scope limit on private goals. The sanitizer keeps the public route shape but drops any key in `FORBIDDEN_PRIVATE_TARGET_KEYS` (private target ids, formulas, target statuses, solver and derivation traces, proof-packet and certificate-search refs, submission packages) and replaces private-looking strings with `private_target_material_omitted` before any public row is built. The findings function reports the rejected field path and class, never the matched value.\n\n###### Source-digest checks and the two entry points\n\n** `_source_artifact_imports` re-hashes the four copied source files and marks each copy, rewrite, or mismatch, while `run` includes the negative cases and `run_routing_bundle` omits them.** `_source_artifact_imports` re-hashes the four copied source files named in `REAL_SUBSTRATE_REFS` and marks each as an exact copy, a path rewrite, or a digest mismatch. `write_receipts` writes the result, board, validation, and sign-off records. `run` drives the fixture with the negative cases included; `run_routing_bundle` drives the exported bundle without them.\n\nFunction Role `_build_result` Orders the pipeline and sets the pass-or-blocked status `_portfolio` Splits the probe into known, available, unavailable sets `_score_case` Scores one route case and compares declared against computed `_decision_for_tactic` Rejects a tactic as unprobed, unavailable, or not allowed `_shape_preferred_selection` Computes the preferred selection and records skips `_route_integrity_findings` Turns per-case problems into typed findings `_sanitize_private_target_material` Drops private goal keys before public rows `write_receipts` Writes the result, board, validation, sign-off records Diagram of the mechanism (8 steps). Diagram source & refs\n\n```text\nflowchart TD Inputs[\"Probe file + route cases\"] Portfolio[\"_portfolio known / available / unavailable\"] Sanitize[\"_sanitize_private_target_material drop private goal keys\"] Decide[\"_decision_for_tactic per-tactic verdict\"] Select[\"_shape_preferred_selection compute the selection\"] Score[\"_score_case declared vs computed\"] Findings[\"_route_integrity_findings typed rejections\"] Result[\"_build_result status + board\"] Inputs --> Portfolio Inputs --> Sanitize Portfolio --> Decide Sanitize --> Decide Decide --> Select Select --> Score Score --> Findings Findings --> Result\n```\n\n###### What the exported bundle scores to\n\n**On the exported bundle four tactics are available and `aesop` unavailable, all seven route cases are pre-execution, and four source files are copied in with digest checks.** On the exported bundle the probe marks `decide`, `omega`, `simp_all`, and `rfl` available and `aesop` unavailable. All seven route cases are `pre_execution`, and four source files are copied in with digest checks.\n\n##### Negative cases\n\n**The gate must reject each of these, listed in `EXPECTED_NEGATIVE_CASES`:** The gate must reject each of these, listed in `EXPECTED_NEGATIVE_CASES`:\n- `unavailable_tactic_admitted`: a route admits a tactic the probe marks unavailable, here `aesop` while Mathlib is absent.\n- `unprobed_tactic_allowed`: a route admits a tactic missing from the probe.\n- `shape_preferred_tactic_overridden`: a declared tactic bypasses the computed shape preference.\n- `unknown_shape_fallback`: a shape bypasses the default safe order.\n- `preferred_unavailable_fallback`: a route bypasses the next available tactic after the first preferred one is unavailable.\n- `proof_body_leakage`: a route carries a proof, provider, or Lean body field.\n- `private_target_material`: a route carries a private target id, formula, status, trace, or challenge path, rejected without echoing the value.\n- `post_execution_route`: a route selection carries execution evidence.\n- `release_overclaim`: a route claims proof, provider, Lean, or launch-scope decision.\n\n##### Prior Art Grounding\n\n###### A pre-execution proof-search admissibility filter\n\n**It follows the proof-search habit of matching a goal shape to available methods before spending runtime, narrowed to a pre-execution admissibility filter over shape, references, and availability.** The routing layer follows a known proof-search pattern: match a goal shape to methods that are known to be available before spending runtime on them. Lean's tactic documentation gives the proof-assistant context for goal-directed tactic choice, and Isabelle's Sledgehammer is a mature example of selecting external provers and relevant facts from a goal. This component narrows the idea to a pre-execution admissibility filter: shape, allowed references, and current availability must line up before a route is exported.\n- Lean 4 tactic documentation: https://lean-lang.org/theorem_proving_in_lean4/Tactics/\n- Isabelle Sledgehammer user guide: https://isabelle.in.tum.de/doc/sledgehammer.pdf\n\n##### Validation Result record Path\n\nFrom `microcosm-substrate/`, reproduce the result with temporary result records:\n\n###### What a pass certifies\n\n**A pass means the seven cases scored as expected, the four source copies matched digests, every negative was rejected, and the secret scan was clean, without widening the ceiling.** A pass means the seven route cases scored as expected, the four source copies matched their digests, every negative case was rejected, and the secret scan found no blocking hit. It does not widen the no-Lean, no-proof ceiling.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim\n\n**For public pre-execution cases the gate admits only probed, available tactics before any proof attempt and recomputes the selection so a route cannot smuggle in an inadmissible one.** The strongest claim the evidence supports: for public pre-execution route cases, this gate admits only tactics that were both probed and available before any proof attempt, and it recomputes the selection so a route cannot smuggle in an unavailable, unprobed, or shape-inadmissible tactic by declaring it.\n\n###### The proof boundary and scope limit\n\n**The proof boundary is the route references it reads, and `AUTHORITY_CEILING` stops it from running Lean, proving a goal, exposing private material, or granting launch control.** The proof boundary is the route references it reads: tactic probe availability, target-shape cases, selected tactic ids, source-digest evidence, the negative-case records, and the statement-scope omission records. The scope limit is set in `AUTHORITY_CEILING`. The gate does not run Lean or Lake, prove a goal, export proof or provider bodies, expose private target material, authorize post-execution route selection, use external model services, claim Mathlib-dependent proof authority, or authorize public sharing or launch.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":13029,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/target_shape_tactic_routing.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"}],"grouped_relation_feeds":[{"relation":"abides by","count":5,"links":[{"id":"AX-1","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Derivation before assertion","url":"doctrine.html#dcard-ax-1"},{"id":"AX-2","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Kernelized 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Recall","relative_href":"docs/paper-module-ring2-premise-precision-recall.html","read_href":"docs/paper-module-ring2-premise-precision-recall.html","context_href":"docs/rules-and-ideas.html#paper-module-ring2-premise-precision-recall","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"After-the-fact premise retrieval metrics separate retrieval misses from proof failures without becoming theorem authority.","summary":"Ring-2 Premise Precision Recall validates copied Ring2 retrieval rankings against after-the-fact metric labels. It computes precision/recall classes, rejects oracle-label leakage and tuning shortcuts, and keeps proof bodies, model-output data, benchmark claims, and theorem-correctness claims outside the public result record boundary.","authority_ceiling":"Copied Ring2 retrieval records and public fixture/exported-bundle result records only; no Lean/Lake execution, theorem proof, benchmark performance, external model access, provider-context label flow, launch-scope decision, publishing-scope decision, or whole-system correctness.","abstract":"After-the-fact premise retrieval metrics separate retrieval misses from proof failures without becoming theorem authority.","teleology":{"section_id":"","text":"After-the-fact premise retrieval metrics separate retrieval misses from proof failures without becoming theorem authority.","source":"capsule_summary_line"},"body_markdown":"###### Premise-retrieval precision and recall replay\n\n**It replays copied premise-retrieval records after the fact and reports, problem by problem, how much of the needed support the search actually ranked.** `ring2_premise_retrieval_precision_recall_harness` scores saved premise-retrieval runs after the fact. A proof search first looks for the supporting facts a proof will need, the premises. This component replays copied records of that search and reports, problem by problem, how much of the needed support the search actually ranked.\n\n**It keeps two failures apart, a missed premise versus a proof that had everything and still failed, by computing precision and recall and labelling each problem one of four outcomes.** The point is to keep two failures apart. A proof can fail because the search never surfaced a premise it needed. A proof can also fail even though every needed premise was ranked. Those call for different fixes. The component computes precision and recall from the copied records, then labels each problem as one of four outcomes so a reader can tell which case they are looking at.\n\n**It emits measurement result records, bounded evidence authority, and the labels may never flow back into the ranking or into a provider recipe.** It emits result records, bounded evidence authority. The labels are measurement labels. They score a finished run and they are never allowed to flow back into the ranking or into a provider recipe.\n\n##### Purpose\n\n###### Blaming the prover hides a missed premise\n\n**A failed search invites blaming the prover, but a single reported precision figure cannot say whether the fault was a missed premise or a proof that had everything and still failed.** When a proof search fails it is easy to blame the prover and miss the simpler cause: the right facts were never put in front of it. A single reported precision figure hides this. It cannot say whether a low score came from a missed premise or from a proof that had everything it needed and still did not go through.\n\n###### Recomputed overlap read against proof outcome\n\n**It recomputes the numbers by intersecting ranked ids with needed ids and reading the proof outcome, so full recall with a failing proof surfaces as a downstream fault.** So the component recomputes the numbers rather than trusting a reported one. For each problem it intersects the ranked premise ids with the labelled needed-premise ids and reads the proof outcome alongside that overlap. Full recall with a passing proof is one outcome; full recall with a failing proof is a different one. That second case is the signal that the fault lies downstream of retrieval, and a loose metric that only averages scores would erase it.\n\n##### How it works\n\n###### Two entrypoints over one `_build_result`\n\n** `run` validates the fixture with five negative cases and `run_precision_recall_bundle` validates the export without them, both calling `_build_result` to set one `PASS` or `blocked` status.** Two entrypoints drive the same core. `run` validates a fixture directory with the five negative cases included. `run_precision_recall_bundle` validates an exported bundle with the negative cases turned off. Both call `_build_result`, which runs the stages below in order and decides a single `status` of `PASS` or `blocked`.\n\nFunction Role `_load_payloads` reads the four input files (`retrieval_runs`, `problem_labels`, `retrieval_rankings`, `evaluation_policy`) plus the negative inputs when they are included `_evaluate` computes precision and recall per problem and assigns the four-way failure mode `_validate_run_material` checks that copied run material carries its source ref, digest, target refs, and provenance status `_validate_source_artifacts` checks the four copied source files against a known digest `_negative_findings` turns the five adversarial inputs into typed refusal codes `_write_receipts` writes the result, board, validation, and sign-off records with proof bodies left out\n\n###### Precision and recall at a top-k cutoff\n\n** `_evaluate` truncates each ranking to `top_k`, intersects ranked with needed ids for hits, and computes precision and recall against the policy's expected aggregates.** The scoring core is `_evaluate`. It indexes the after-the-fact labels by `problem_id`, reads `default_top_k` from the policy (4 if unset) or a per-ranking `top_k`, and truncates each ranking to that cutoff. It intersects the truncated ranked ids with the labelled needed ids to get the hits, then computes `precision = hits / top_k` and `recall = hits / needed`. Aggregate precision and recall come from summed hit, candidate, and needed counts, and are compared against the policy's `expected_aggregate_metrics`.\n\n**The four-way label is mechanical, driven by recall and the recorded `proof_outcome`:** The four-way label is mechanical, driven by recall and the recorded `proof_outcome`:\n- `retrieval_hit`: recall reached 1.0 and the proof passed.\n- `proof_failure_despite_hit`: recall reached 1.0 but the proof did not pass. Retrieval did its job; the fault is elsewhere.\n- `partial_retrieval_miss`: some needed premises were ranked, not all.\n- `retrieval_miss`: none of the needed premises were ranked.\n\n###### Enforcing the shape of the evaluation set\n\n** `_evaluate` also requires every expected failure mode, an adversarial decoy problem, and the minimum problem count, turning any gap into a finding.** `_evaluate` also enforces the shape of the evaluation set, not just the happy path. It requires every mode named in the policy's `expected_failure_modes` to appear, requires an adversarial decoy problem whose needed premise is absent or missed, and requires the policy's minimum problem count. A gap in any of these becomes a finding.\n\n###### Provenance and digest checks gate the status\n\n**Before any score is trusted, `_validate_run_material` and `_validate_source_artifacts` check provenance and digests, and `status` is `PASS` only when every stage and the secret scan clear.** Before any score is trusted, the provenance stages run. `_validate_run_material` rejects copied material that is missing a `source_ref`, `source_sha256`, `target_refs`, `validation_refs`, or the expected `copied_non_secret_macro_body_with_provenance` status. `_validate_source_artifacts` walks the four files in `SOURCE_REFS`, hashes each copied artifact under `source_artifacts/`, and accepts it only if the hash matches the recorded exact digest in `SOURCE_DIGESTS` or a known path-rewrite digest. A missing file or a hash mismatch is a finding. `_build_result` also runs a secret-exclusion scan over the inputs. The overall `status` is `PASS` only when there are no floor findings, no run-material findings, the source artifacts pass, the evaluation passes, no expected negative case is missing, and the secret scan has no blocking hit. Otherwise it stays `blocked`.\n\n###### Result records expose metrics, never proof bodies\n\n** `_write_receipts` emits import ids, target refs, digest status, and metric counts, but writes no proof bodies, model-output data, or non-public paths.** `_write_receipts` then emits the records. It exposes import ids, target refs, digest status, aggregate metrics, and failure-mode counts. It does not write proof bodies, model-output data, or non-public paths.\n\nDiagram of the mechanism (8 steps). Diagram source & refs\n\n```text\nflowchart TD Inputs[\"four input files runs, labels, rankings, policy\"] --> Build[\"_build_result orchestrates stages, sets status\"] Build --> Evaluate[\"_evaluate precision/recall + four-way label\"] Build --> RunMat[\"_validate_run_material copied-material provenance\"] Build --> Artifacts[\"_validate_source_artifacts digest match per file\"] Build --> Negatives[\"_negative_findings five typed refusals\"] Evaluate --> Status[\"status: PASS or blocked\"] RunMat --> Status Artifacts --> Status Negatives --> Status Status --> Result records[\"_write_receipts metrics only, no proof bodies\"]\n```\n\n##### Negative cases\n\n** `EXPECTED_NEGATIVE_CASES` names five adversarial inputs that `run` must reject, each mapped to one refusal code:** `EXPECTED_NEGATIVE_CASES` names five adversarial inputs that `run` must reject, each mapped to one refusal code:\n- `oracle_labels_in_ranking` plants the known-correct premise ids inside a ranking. `_inspect_forbidden_bodies` catches the `oracle_needed_premise_ids` field and emits `RING2_RETRIEVAL_ORACLE_LABELS_IN_RANKING`.\n- `proof_body_leakage` carries a proof, provider, or private body field and emits `RING2_RETRIEVAL_PROOF_BODY_FORBIDDEN`.\n- `test_split_tuning_attempt` marks a run as tuned on the test labels and emits `RING2_RETRIEVAL_TEST_SPLIT_TUNING_FORBIDDEN`.\n- `metric_overclaim` asserts proof, benchmark, provider, or launch-scope decision and emits `RING2_RETRIEVAL_METRIC_OVERCLAIM`.\n- `missing_adversarial_decoy` omits the required decoy case and emits `RING2_RETRIEVAL_ADVERSARIAL_DECOY_REQUIRED`.\n\n###### Missing negative cases block the run\n\n**If any of the five is not observed, `_build_result` records it as missing and the run is `blocked`.** If any of the five is not observed, `_build_result` records it as missing and the run is `blocked`.\n\n##### Prior Art Grounding\n\n###### Grounded in IR metrics and premise selection\n\n**The design draws on TREC and scikit-learn precision/recall evaluation and on hammer-style premise selection, where finding the right facts is a distinct step from replaying a proof.** This component is grounded in information-retrieval evaluation. NIST's [TREC evaluation measures](https://trec.nist.gov/pubs/trec20/appendices/measures.pdf) give the older precision/recall frame for judging retrieval systems, and scikit-learn's [precision/recall metric API](https://scikit-learn.org/stable/modules/generated/sklearn.metrics.precision_recall_fscore_support.html) is a common machine-learning interface for reporting those labels. On the theorem-proving side, premise selection sits next to hammer workflows such as Isabelle [Sledgehammer](https://isabelle.in.tum.de/doc/sledgehammer.pdf), where finding the right facts is a distinct step from replaying a proof. This component keeps that distinction explicit and measurable.\n\n##### Validation Result record Path\n\nFrom `microcosm-substrate/`, reproduce this page's proof boundary with temporary records:\n\n###### What a pass over the records means\n\n**A pass means the records scored to expectation, the four digests matched, and all five negative cases were rejected, without anything running Lean or a prover.** A pass means the copied records scored to their expected metrics, the four source artifacts matched their digests, and all five negative cases were rejected. It does not mean anything ran Lean or a prover.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim\n\n**The strongest claim is that copied public records can be scored, sorted into the four labels, and guarded by five refusals, all reproducible from the fixture.** The strongest claim this component supports is that copied public premise-retrieval records can be scored for precision and recall, sorted into the four retrieval-versus-proof-failure labels, and guarded by the five typed refusals, all reproducible from the fixture. The `AUTHORITY_CEILING` constant fixes the rest.\n\n###### What the labels never claim\n\n**It runs no Lean, no providers, and no launch, and a `retrieval_hit` label says only that needed ids appeared in the ranking, nothing about proof soundness.** It does not run Lean or Lake, use external model services, emit proof bodies, tune retrieval on test answers, claim benchmark performance, prove that a theorem is true, or include launch operations. A `retrieval_hit` label means the needed-premise ids appeared in the ranking under this fixture's policy, nothing about whether the downstream proof is sound. The reported numbers are bounded by the declared fixture and bundle; a different corpus, cutoff, or premise labelling has to be rerun. The proof boundary is the copied records and public result records only, and the scope limit holds even where site projections make this page walkable. Its labels are measurement labels and they may not enter a provider-context recipe.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":12183,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/ring2_premise_precision_recall.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"TREC evaluation measures","url":"https://trec.nist.gov/pubs/trec20/appendices/measures.pdf","kind":"external_reference","section_id":"grounded-in-ir-metrics-and-premise-selection"},{"label":"precision/recall metric 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It validates strategy ids, copied public source tool bodies, retrieval-term effects, oracle-label exclusion, negative cases, and metadata-only result records while keeping proof and provider authority out of scope.","authority_ceiling":"Copied public strategy metadata, public source tool bodies, and public fixture/exported-bundle result records only; no Lean/Lake execution, formal-result correctness, oracle-label visibility, external model access, benchmark performance, launch-scope decision, publishing-scope decision, or whole-system correctness.","abstract":"Pre-oracle strategy hypotheses make the first proof-search move inspectable without claiming proof or provider authority.","teleology":{"section_id":"","text":"Pre-oracle strategy hypotheses make the first proof-search move inspectable without claiming proof or provider authority.","source":"capsule_summary_line"},"body_markdown":"###### A scorer for formal-math proof-search hypotheses\n\n** `mathematical_strategy_atlas_hypothesis_scorer` is a small component that records a formal-math proof search's first move before any proof runs.** `mathematical_strategy_atlas_hypothesis_scorer` is a small component that records a formal-math proof search's first move before any proof runs. It reads public problem feature tags and a strategy atlas, scores each candidate opening strategy, and writes down which one it picked and why. It never proves anything.\n\n**The output is a set of result records: the selected strategy id for each case, the integer score that chose it, the component breakdown that produced the score, and a list of cases where nothing matched.** The output is a set of result records: the selected strategy id for each case, the integer score that chose it, the component breakdown that produced the score, and a list of cases where nothing matched. A cold reader can take the same three input files and re-derive every selection by hand.\n\n###### The shipped finite-sum example\n\n**A worked example: a problem tagged `iff_goal` scores the `iff_split` strategy highest and selects it.** A worked example: a problem tagged `iff_goal` scores the `iff_split` strategy highest and selects it. A problem whose only tag matches nothing scores no candidate above zero and resolves to a typed `STRATEGY_SELECTION_MISS` instead of a silent guess.\n\n##### Purpose\n\n**A proof search has to start somewhere. Before a premise is retrieved or a tactic is run, an agent has already committed to a first move: a goal shape and a family of tactics it expects to use.** A proof search has to start somewhere. Before a premise is retrieved or a tactic is run, an agent has already committed to a first move: a goal shape and a family of tactics it expects to use. That choice is usually implicit, buried inside a model call. This component pulls it into the open and answers one question: for a given problem shape, which strategy did the system pick first, and on what visible evidence?\n\n**The constraint is what the answer is allowed to depend on. The scorer never sees the oracle's expected strategy, the ground-truth proof, or any provider output.** The constraint is what the answer is allowed to depend on. The scorer never sees the oracle's expected strategy, the ground-truth proof, or any provider output. It works from public feature tags and a strategy atlas of trigger features, negative triggers, and retrieval-expansion terms. The selected strategy is a hypothesis recomputed from readable inputs, not a label copied from an answer key.\n\n**A loose classifier fails here in two ways. It can bake the answer in by declaring the chosen strategy as a plain label, or it can score on shallow feature overlap that happens to line up with the known-good label.** A loose classifier fails here in two ways. It can bake the answer in by declaring the chosen strategy as a plain label, or it can score on shallow feature overlap that happens to line up with the known-good label. This component rejects both. A declared selection must match the score the component recomputes, and a strategy chosen on overlap alone is a typed negative case rather than a pass.\n\n##### How it works\n\n**The entrypoints are `run` for the fixture and `run_strategy_bundle` for the exported bundle. Both call `_build_result`, which runs the stages below in order and then hands the result to `write_receipts`.** The entrypoints are `run` for the fixture and `run_strategy_bundle` for the exported bundle. Both call `_build_result`, which runs the stages below in order and then hands the result to `write_receipts`.\n\n** `_build_result` loads three public inputs: `strategy_atlas.json`, `problem_features.json`, and `hypothesis_cases.json`.** `_build_result` loads three public inputs: `strategy_atlas.json`, `problem_features.json`, and `hypothesis_cases.json`. It scans every input path for non-public-state leakage first, so a blocking scan hit fails the run before scoring is trusted. Then it runs scoring, the copied-body floor check, and the source-artifact consistency check, and it derives a single `status` that is `PASS` only when all of them pass and no expected negative case is missing.\n\n** `validate_strategy_selection` is the scoring core. For each case it calls `_score_case`, which calls `_score_strategy` once per candidate.** `validate_strategy_selection` is the scoring core. For each case it calls `_score_case`, which calls `_score_strategy` once per candidate. The score is one integer from three terms. Each problem feature that matches a strategy's `trigger_features` adds four points (`TRIGGER_FEATURE_WEIGHT`). Each feature that matches its `negative_triggers` subtracts three (`NEGATIVE_TRIGGER_PENALTY`). Each retrieval-query term found in the strategy's expansion terms adds one point (`RETRIEVAL_TERM_BONUS`), capped at two (`RETRIEVAL_BONUS_CAP`). Plain feature overlap is recorded as a diagnostic count and kept out of the score.\n\n**Selection is a deterministic sort. Only strategies with a positive score are eligible.** Selection is a deterministic sort. Only strategies with a positive score are eligible. Among those, `_score_case` ranks by score highest first, breaks ties by the strategy's declared order and then its id, and takes the top row. If no candidate scores positive the case resolves to `STRATEGY_SELECTION_MISS`. Each case record carries the selected id, the score, the component breakdown, the ranked candidate scores, and the trigger, negative, and retrieval hits that produced them.\n\n**The weights encode the design intent. Trigger matches are worth more than retrieval matches, so a strategy is chosen for the shape it claims to handle, not for how many search terms coincide.** The weights encode the design intent. Trigger matches are worth more than retrieval matches, so a strategy is chosen for the shape it claims to handle, not for how many search terms coincide. A negative trigger can veto a strategy that looks apt. The retrieval cap stops a strategy from winning on keyword volume.\n\n**The same recomputation enforces honesty. When a case declares its own `selected_strategy_id`, `score`, `classifier`, `retrieval_bonus`, or `candidate_scores`, `_verify_declared_case_outcomes` recomputes each from the evidence and reports a stale-declaration finding on any mismatch.** The same recomputation enforces honesty. When a case declares its own `selected_strategy_id`, `score`, `classifier`, `retrieval_bonus`, or `candidate_scores`, `_verify_declared_case_outcomes` recomputes each from the evidence and reports a stale-declaration finding on any mismatch. A case that declares only a bare `selected_strategy_id`, with nothing to check it against, is rejected as label-only.\n\n**Two further checks guard the copied inputs. `validate_source_module_imports` reads `source_module_manifest.json` and confirms each copied source body against its recorded SHA-256 digest, line count, byte count, and required anchors, and rejects any body that would land inside a generated result record.** Two further checks guard the copied inputs. `validate_source_module_imports` reads `source_module_manifest.json` and confirms each copied source body against its recorded SHA-256 digest, line count, byte count, and required anchors, and rejects any body that would land inside a generated result record. `validate_source_artifact_consistency` opens the three copied runtime artifacts (`strategy_cards.json`, `strategy_hypothesis_set.json`, `prover_skill_atlas.json`) and checks them for oracle-label and proof-body leakage. `write_receipts` then emits four metadata-only result records: `mathematical_strategy_atlas_result.json`, `mathematical_strategy_atlas_board.json`, `mathematical_strategy_atlas_validation_receipt.json`, and a fixture-sign-off record. Each carries refs, counts, hits, and verdicts, never body text.\n\nFunction Role `run`, `run_strategy_bundle` Entrypoints for the fixture and exported bundle `_build_result` Runs the stages in order and derives `status` `validate_strategy_selection` Scores every case and checks the negative cases `_score_strategy`, `_score_case` Weighted per-strategy score and per-case selection `validate_source_module_imports` Digest, count, and anchor check for copied bodies `validate_source_artifact_consistency` Leakage check on the three copied runtime artifacts `write_receipts` Emits the four metadata-only result records Diagram of the mechanism (8 steps). Diagram source & refs\n\nSource refs\n\nInputs `strategy_atlas.json` `problem_features.json` `hypothesis_cases.json`\n\n```text\nflowchart TD inputs[\"strategy_atlas.json problem_features.json hypothesis_cases.json\"] scan[\"scan inputs for non-public-state leakage\"] score[\"_score_strategy trigger x4 - negative x3 + retrieval (cap 2)\"] select{\"any positive score?\"} selected[\"selected_strategy_id + score breakdown\"] miss[\"STRATEGY_SELECTION_MISS\"] verify[\"recompute vs declared fields\"] result records[\"four metadata-only result records\"] inputs --> scan scan --> score score --> select select -- yes --> selected select -- no --> miss selected --> verify miss --> verify verify --> result records\n```\n\n##### Negative cases\n\n**The fixture supplies eight named negative cases in `EXPECTED_NEGATIVE_CASES`, and the run fails if any expected case is not observed.** The fixture supplies eight named negative cases in `EXPECTED_NEGATIVE_CASES`, and the run fails if any expected case is not observed. `unknown_strategy_id` rejects a strategy id outside the public enum. `proof_body_with_strategy` rejects any proof, provider, or source body field on a case. `oracle_strategy_label_leakage` rejects a fixture that exposes an oracle strategy label. `post_oracle_strategy_selection` rejects a case marked as selected after oracle labels are visible. `release_overclaim` rejects a payload that tries to set `release_authorized` or any sibling authority flag. `superficial_overlap_only_scoring` rejects legacy unweighted overlap as the decision rule. `missing_rich_strategy_card_fields` rejects a strategy card that lacks `trigger_features`, `negative_triggers`, `retrieval_expansion_terms`, or `proof_plan_template`. `retrieval_bonus_ceiling_overclaim` rejects a claimed retrieval cap above two.\n\n##### Prior Art Grounding\n\n**The atlas follows the formal-methods practice of separating problem-shape classification from proof execution. Lean's tactic model, described in [Theorem Proving in Lean 4](https://lean-lang.org/theorem_proving_in_lean4/Tactics/), is the immediate precedent: proof work is arrange around tactics chosen for a goal shape while the kernel checks the final proof state.** The atlas follows the formal-methods practice of separating problem-shape classification from proof execution. Lean's tactic model, described in [Theorem Proving in Lean 4](https://lean-lang.org/theorem_proving_in_lean4/Tactics/), is the immediate precedent: proof work is arrange around tactics chosen for a goal shape while the kernel checks the final proof state. The [mathlib overview](https://leanprover-community.github.io/mathlib-overview.html) motivates explicit retrieval terms and domain tags, because a large formal library is navigated by topic and reusable theorem families. The pattern is also adjacent to hammer-style method selection such as Isabelle [Sledgehammer](https://isabelle.in.tum.de/doc/sledgehammer.pdf), where a front-end tool searches for useful facts before replay. This component keeps the pattern pre-oracle and metadata-only: it records why a first strategy was selected, not whether the proof can be completed.\n\n##### Validation Result record Path\n\nRun from `microcosm-substrate`:\n\n```text\nPYTHONPATH=src ../repo-python -m microcosm_core.organs.mathematical_strategy_atlas_hypothesis_scorer run \\\n  --input fixtures/first_wave/mathematical_strategy_atlas_hypothesis_scorer/input \\\n  --out /tmp/microcosm-mathematical-strategy-atlas-hypothesis-scorer/fixture \\\n  --card\nPYTHONPATH=src ../repo-python -m microcosm_core.organs.mathematical_strategy_atlas_hypothesis_scorer run-strategy-bundle \\\n  --input examples/mathematical_strategy_atlas_hypothesis_scorer/exported_mathematical_strategy_atlas_bundle \\\n  --out /tmp/microcosm-mathematical-strategy-atlas-hypothesis-scorer/bundle \\\n  --card\nPYTHONPATH=src ../repo-python -m pytest -p no:cacheprovider tests/test_mathematical_strategy_atlas_hypothesis_scorer.py -q\nPYTHONPATH=src ../repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n###### What a green strategy run establishes\n\n**A green run means the strategy scoring, the copied-body digests, the source-artifact consistency, and the eight negative cases all checked out.** A green run means the strategy scoring, the copied-body digests, the source-artifact consistency, and the eight negative cases all checked out. It does not mean a proof ran.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The bounded strategy-selection claim\n\n**The strongest claim the evidence supports is narrow: public pre-oracle strategy hypotheses, the retrieval-lens metadata that feeds them, the copied public source bodies, their source-artifact digests, and the eight negative cases are all inspectable and reproducible from the three input files.** The strongest claim the evidence supports is narrow: public pre-oracle strategy hypotheses, the retrieval-lens metadata that feeds them, the copied public source bodies, their source-artifact digests, and the eight negative cases are all inspectable and reproducible from the three input files. The proof boundary is exactly that metadata and those metadata-only result records.\n\n**The component does not run Lean or Lake, prove formal-result correctness, reveal oracle strategy labels, expose proof bodies, use external model services, tune on test answers, certify benchmark performance, or authorize public sharing or launch.** The component does not run Lean or Lake, prove formal-result correctness, reveal oracle strategy labels, expose proof bodies, use external model services, tune on test answers, certify benchmark performance, or authorize public sharing or launch. The copied runtime artifacts are public strategy traces, not oracle labels or model-output data. 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It separates real execution evidence from formal-result correctness, provider text, oracle answers, proof-body exposure, source-file changes, and launch-scope decision.","authority_ceiling":"Bounded public fixture and exported-bundle execution result records only; no general formal-result correctness, benchmark solve-rate claim, external model access, oracle-to-proof authority, proof body export, source-file changes, launch-scope decision, publishing-scope decision, or whole-system correctness.","abstract":"Bounded public verifier execution result records witness command execution without upgrading output into theorem authority.","teleology":{"section_id":"","text":"Bounded public verifier execution result records witness command execution without upgrading output into theorem authority.","source":"capsule_summary_line"},"body_markdown":"**It runs bounded Lean proof-step candidates in a throwaway copy of a public project and records each checker return code while keeping proof text and payloads out.** `verifier_lab_execution_spine` is the component that actually runs bounded Lean proof-step attempts and writes down what the checker said. It copies a small public Lean project into a throwaway workspace, builds it once, writes one `.lean` file per candidate, runs the Lean checker on each, and records the return code. It keeps the proof text, the tool output, and every provider or oracle payload out of the result. A reader gets the pass or fail facts and the safety counts, and nothing that would leak.\n\n**Narrower than the kernel, the spine executes each transition, turns a zero exit into `accepted`, and rejects contaminated candidates before the checker ever runs.** The component is narrower than `verifier_lab_kernel`. The kernel reasons about the shape of a transition; this spine executes it. A zero exit code from the Lean checker becomes `accepted`. Anything else becomes a residual with a failure class. Candidates that carry a forbidden field, or that mark an oracle or provider as visible, are rejected before the checker is ever invoked, so a contaminated run cannot be dressed up as a clean one.\n\n**It emits four records, full result, board summary, validation, and fixture sign-off, each stating the same scope limit and carrying counts without bodies.** It emits four result records: a full result, a board summary, a validation record, and a fixture-sign-off record. All four state the same scope limit, carry the return codes and counts, and omit the bodies.\n\n##### Purpose\n\n###### Keeping hidden help out of the record\n\n**Proof systems can report success while hiding oracle or provider help, so this component answers one question: did a bounded Lean candidate pass the checker unaided?** Automated proof systems can blur how a result was reached. A model can be handed the answer by an oracle, or shown the proof by a provider, and still report the result as if it had found the proof alone. A loose result record hides that distinction: it prints a success without saying what help was in the room. This component exists to keep that blur out of the record. It answers one question. Did a bounded Lean candidate pass the checker, with no help the record is hiding?\n\n###### Authority separation holds counters at zero\n\n**Authority separation keeps verified candidates, oracle references, and provider hypotheses as distinct classes, pinning the forward-success and export counters at zero by construction.** The discipline is authority separation. A verified candidate is one class. An oracle reference and a provider hypothesis are separate classes that are never counted as forward success. The counters `oracle_forward_success_increment_count` and `provider_results_counted` are held at zero by construction, alongside `proof_body_export_count` and `source_mutation_count`. Real execution and a clean record are not in tension: the leak checks run first, so the Lean run that follows is already free of the fields that would compromise it.\n\n##### How it works\n\n###### Two entry points over `_build_result`\n\n** `run` executes Lean for real on the fixture while `run_execution_bundle` validates an exported group offline, and both flow through `_build_result` then `write_receipts`.** Two entry points drive the component. `run` handles the first-wave fixture and executes Lean for real. `run_execution_bundle` validates an exported group without a Lean toolchain, so a third party can inspect a shipped bundle offline. Both call `_build_result`, then `write_receipts`.\n\n###### `_build_result` scans, copies, and builds once\n\n** `_build_result` loads the packet, secret-scans every input, and builds the copied Lean project once, executing candidates only when that build returns zero.** `_build_result` is the spine. It loads the execution packet, scans every input path for forbidden content with the shared secret-exclusion scan, and reads the tool versions. In fixture mode it opens a temporary directory, copies the Lean project into it, and calls `_build_lake_project`, which runs `lake build MicrocosmProofWitness` once with a sixty-second limit. Only if that build returns zero does it execute the transition candidates.\n\n###### Per-candidate contract check in `_execute_transition`\n\n** `_execute_transition` first runs the contract check for forbidden keys, visible oracle or provider, and unknown action class, rejecting the candidate or else running `lake env lean` and recording the return code.** Each candidate goes through `_execute_transition`. That function first calls `_validate_transition_contract`. The contract check walks the candidate for any key in `FORBIDDEN_TRANSITION_KEYS` (`proof_body`, `raw_tactic_script`, `oracle_template`, `provider_output_body`, and similar) using `_walk_forbidden_keys`, checks whether `oracle_visible` or `provider_visible` is set, and checks that the `action_class` is one of the nine names in `ALLOWED_ACTION_CLASSES`. If any check fires, the candidate becomes a `CONTRACT_REJECTED` record with the error codes attached and no Lean run. If the contract is clean, `_execute_transition` writes a `.lean` source built by `_lean_source_for_transition`, runs `lake env lean` on it, and sets `accepted` from the return code. The record carries the return code, the action class, and the failure class. It never carries the proof text or the stdout and stderr bodies.\n\n###### CP2 translation and evolve mutation lanes\n\n**Two smaller lanes also run: CP2 translation emits typed next-action candidates from a fixed vocabulary, and evolve mutation touches only bounded policy artifacts, each rejecting out-of-scope requests.** Two smaller lanes run on the same packet. CP2 translation reads residual cases and emits typed next-action candidates drawn from a fixed vocabulary; a request that smuggles a proof body is rejected. Evolve mutation adjusts only the bounded policy artifacts named in `ALLOWED_EVOLVE_ARTIFACTS`; a mutation aimed at anything else is rejected.\n\n###### Digest checks on the exported group\n\n** `validate_source_module_imports` recomputes each copied file's SHA-256 against the manifest and blocks the group on any digest mismatch, missing anchor, or out-of-prefix source.** `validate_source_module_imports` handles the exported group. It reads `source_module_manifest.json`, confirms the declared module count matches the rows, and for each row recomputes the SHA-256 digest of the copied file and compares it to the manifest. A digest mismatch, a missing anchor, or a source reference outside the allowed prefixes blocks the group. The copied bodies live in the bundle, never in the record.\n\n###### `write_receipts` pins the zero-authority facts\n\n** `write_receipts` composes the four records with every authority flag false, and the status passes only when the scan is clean, the build returned zero, and the expected transition counts are met.** `write_receipts` composes the four records and pins the safety facts: `receipts_include_proof_bodies` false, `provider_calls_authorized` false, `source_mutation_authorized` false, `release_authorized` false. The overall status in `_build_result` passes only when the secret scan is clean, Lean and Lake are available, the project build returned zero, no expected negative case is missing, and the run produced at least four transitions, two accepted, one residual, one CP2 effect, and one accepted mutation.\n\nFunction Role `run` First-wave fixture entry; runs Lean for real, then writes records `run_execution_bundle` Exported-group entry; validates a shipped bundle without a toolchain `_build_result` Loads the packet, scans inputs, builds, executes, gates the status `_build_lake_project` Runs `lake build MicrocosmProofWitness` once `_execute_transition` Contract-checks, then runs `lake env lean` on one candidate `_validate_transition_contract` Rejects forbidden fields, visible oracle/provider, unknown action classes `validate_source_module_imports` Recomputes copied-file digests for the exported group `write_receipts` Writes the four records and pins the zero-authority facts Diagram of the mechanism (9 steps). Diagram source & refs\n\n```text\nflowchart TD Packet[\"Execution packet transition candidates, CP2 requests, evolve mutations, oracle/provider refs\"] Scan[\"Secret-exclusion scan over every input path\"] Gate[\"Contract check forbidden field? oracle/provider visible? action class out of vocabulary?\"] Rejected[\"CONTRACT_REJECTED no Lean run\"] Build[\"lake build MicrocosmProofWitness once, in a temporary copy\"] Run[\"Write .lean, run lake env lean return code 0 = accepted\"] Verified[\"Accepted transition\"] Residual[\"Residual non-zero return code\"] Records[\"Four result records return codes and counts kept, bodies and proof text omitted\"] Packet --> Scan Scan --> Gate Gate -->|leak found| Rejected Gate -->|clean| Build Build -->|return 0| Run Run -->|exit 0| Verified Run -->|non-zero| Residual Rejected --> Records Verified --> Records Residual --> Records\n```\n\n##### Negative cases\n\n###### Four negative inputs the run must observe\n\n**The fixture ships four negative inputs listed in `EXPECTED_NEGATIVE_CASES`, each expected to raise one error code, and a run is blocked unless all four are observed.** The fixture ships four negative inputs, each expected to raise one error code, declared in `EXPECTED_NEGATIVE_CASES`. A run that fails to observe all four is blocked.\n- `transition_leaks_candidate_body` must raise `VERIFIER_LAB_EXECUTION_TRANSITION_FIELD_FORBIDDEN`: a candidate carrying a proof body or raw tactic script.\n- `provider_oracle_visible_transition` must raise `VERIFIER_LAB_EXECUTION_PROVIDER_OR_ORACLE_VISIBLE`: a candidate that exposes an oracle structured source record or provider text to forward execution.\n- `cp2_candidate_contains_proof_body` must raise `VERIFIER_LAB_EXECUTION_CP2_PROOF_BODY_FORBIDDEN`: a translation request that smuggles a proof body into the typed-action lane.\n- `evolve_mutates_unbounded_source` must raise `VERIFIER_LAB_EXECUTION_EVOLVE_SCOPE_FORBIDDEN`: a mutation aimed outside the bounded policy artifacts.\n\n##### Prior Art Grounding\n\n###### Small-kernel trust and artifact evaluation\n\n**Execution follows the small-kernel proof-assistant tradition where trust rests on the checker not the tactic, and the record shape follows artifact-evaluation practice.** The Lean and Lake execution inherits from the small-kernel proof-assistant tradition, the [Lean theorem prover](https://www.microsoft.com/en-us/research/publication/the-lean-theorem-prover-system-description/) and LCF-style systems such as [HOL Light](https://www.cl.cam.ac.uk/~jrh13/papers/hollight.html), where trust rests on a small checker rather than the tactic that produced the term. The record shape follows artifact-evaluation practice: record the command identity, the tool facts, and the pass or fail counts separately from the claim they support, and keep the underlying artifact out of the summary.\n\n##### Validation Result record Path\n\nRun from `microcosm-substrate`:\n\n###### What a green run proves\n\n**A green run counts as bounded execution-spine evidence only and never as general proof certification.** A green run proves only bounded execution-spine evidence: command intent, tool facts, return-code pass or fail per candidate, the four observed negative cases, and the zero-held authority counters.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim\n\n**A bounded set of public Lean candidates ran in a throwaway copy of a public project, and each checker return code was recorded without exposing proof text.** The strongest claim the evidence supports is narrow. A bounded set of public Lean candidates was run in a throwaway copy of a public project, and the checker's return code for each was recorded without exposing the proof text. That is the whole claim.\n\n###### What it refuses past that line\n\n**Past that it certifies no general proofs, carries no Mathlib authority, exports no bodies, calls no provider, mutates no source, and excludes launch.** It refuses everything past that. It does not certify proofs in general, does not carry Mathlib-dependent proof authority, does not export proof bodies, does not certify a benchmark solve rate, does not call any provider, does not change source files, and excludes launch or hosted deployment. An oracle match is never forward success. Provider text is never a proof. The diagram and Atlas views are navigation aids and do not promote a tool run into proof certification.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":12617,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/verifier_lab_execution_spine.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"Lean theorem prover","url":"https://www.microsoft.com/en-us/research/publication/the-lean-theorem-prover-system-description/","kind":"external_reference","section_id":"small-kernel-trust-and-artifact-evaluation"},{"label":"HOL Light","url":"https://www.cl.cam.ac.uk/~jrh13/papers/hollight.html","kind":"external_reference","section_id":"small-kernel-trust-and-artifact-evaluation"}],"grouped_relation_feeds":[{"relation":"abides 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Lab","url":"paper-module-certificate-kernel-execution-lab.html","relation_id":"paper_module.depends_on.paper_module","source_ref":"core/paper_module_capsules.json::paper_modules[7:paper_module.certificate_kernel_execution_lab].depends_on"},{"id":"paper_module.engine_room_lean_proof_search_lab","kind":"paper_module","kind_label":"paper module","relation":"depended on by","label":"Engine Room Lean Proof Search Lab","url":"paper-module-engine-room-lean-proof-search-lab.html","relation_id":"paper_module.depends_on.paper_module","source_ref":"core/paper_module_capsules.json::paper_modules[87:paper_module.engine_room_lean_proof_search_lab].depends_on"}],"source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/verifier_lab_execution_spine.md"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json"}],"visible_text_byte_count":13602,"section_titles":["It runs bounded Lean proof-step candidates in a throwaway copy of a public project","Narrower than the kernel","It emits four records","Purpose","Keeping hidden help out of the record","Authority separation holds counters at zero","How it works","Two entry points over `_build_result`","`_build_result` scans, copies, and builds once","Per-candidate contract check in `_execute_transition`","CP2 translation and evolve mutation lanes","Digest checks on the exported group","`write_receipts` pins the zero-authority facts","Negative cases","Four negative inputs the run must observe","Prior Art Grounding","Small-kernel trust and artifact evaluation","Validation Result record Path","What a green run proves","Scope limit","The strongest supported claim","What it refuses past that line"]},{"id":"paper_module.bounded_autonomy_campaign_packet","title":"Bounded Autonomy Campaign Packet","relative_href":"docs/paper-module-bounded-autonomy-campaign-packet.html","read_href":"docs/paper-module-bounded-autonomy-campaign-packet.html","context_href":"docs/rules-and-ideas.html#paper-module-bounded-autonomy-campaign-packet","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"Bounded autonomy campaign packets propose guarded agent work without authorizing source-file changes or unsupervised repair.","summary":"Bounded Autonomy Campaign Packet validates public campaign packet fixtures: proposed gaps, policy gates, repeated-failure digests, negative cases, source-open body imports, and scope limits. It keeps campaign proposals separate from self-repair, source writes, live scheduling, external model access, launch-scope decision, public sharing, and whole-system correctness.","authority_ceiling":"Self-proposal campaign packet fixture and exported-bundle result record evidence only; no self-repair authority, unsupervised source-file changes, live scheduler authority, external model access, launch-scope decision, publishing-scope decision, or whole-system correctness.","abstract":"Bounded autonomy campaign packets propose guarded agent work without authorizing source-file changes or unsupervised repair.","teleology":{"section_id":"","text":"Bounded autonomy campaign packets propose guarded agent work without authorizing source-file changes or unsupervised repair.","source":"capsule_summary_line"},"body_markdown":"###### Coverage gaps become proposals without action authority\n\n**This component lets an agent draft a plan to close its own coverage gaps while stamping every row as a proposal that carries no authority to act.** `bounded_autonomy_campaign_packet` is a component that lets an agent draft a plan to close its own coverage gaps without giving that plan any authority to act. It reads synthetic public inputs, watches a read-only campaign builder, and emits a draft candidate packet where every row is stamped as a proposal, not a change.\n\n###### The packet carries hashes and refs, never writes\n\n**The result record carries only refs, digests, line counts, and stdout and stderr hashes, never a write surface, a true source-file changes flag, or the builder's output bodies.** The result record it writes carries refs, digests, line counts, and stdout and stderr hashes. It never carries a write surface, a source-file changes flag set to true, or the builder's output bodies.\n\n###### The demonstration separates proposing from doing\n\n**What it demonstrates is narrow: an agent can propose work drawn from a real surface capable of doing it, observed in a mode where it did not.** What it demonstrates is narrow and deliberate: an agent can propose work drawn from a real surface that could do that work, observed in a mode where it did not.\n\n##### Purpose\n\n###### The risk of sketching sliding into doing\n\n**An agent can usefully sketch a plan to fix its under-coverage, but the risk is that sketching quietly becomes doing as a proposal grows a write surface.** An agent can usefully notice where it is under-covered and sketch a plan to fix it. The risk is that \"sketch a plan\" quietly becomes \"do the work\". A proposal grows a write surface, and a system meant to suggest starts editing its own source unsupervised.\n\n###### Keeping proposing and acting apart\n\n**This component keeps proposing and acting apart, answering whether an agent can emit a draft campaign proposal from real coverage gaps without that proposal carrying any authority to act.** This component keeps those two steps apart. It answers one question: can an agent emit a draft campaign proposal from real coverage gaps without that proposal carrying any authority to act on them?\n\n###### Candidate count comes from a witnessed builder\n\n**The candidate count comes not from an invented list but from a check-only builder witness, and each candidate is stamped with no write surface, no source-file changes, and required review.** The design choice that matters is where the candidate count comes from. The component does not invent a plausible list of work. It watches a real campaign builder run in a check-only mode and accepts its witness only when the builder reports candidate targets and leaves `wrote_packet` unset. Each drafted candidate is then stamped `write_surface: none`, `source_mutation_authorized: false`, and `requires_human_review: true`, so proposing can never be read as authorising.\n\n##### How it works\n\n###### Two entry points share the same checks\n\n** `run` and `run_bounded_autonomy_bundle` both route through `run_crown_jewel_organ` with the same `evaluate` and `evaluate_negative_case` evaluators, so the identical checks guard both command surfaces.** `run` and `run_bounded_autonomy_bundle` both hand the fixture and the exported bundle to `run_crown_jewel_organ`, passing `evaluate` as the evaluator and `evaluate_negative_case` as the negative-case evaluator. The bundle entry point adds the bundle input mode. The same checks guard both command surfaces.\n\n###### `evaluate` runs four ordered refusal checks\n\n** `evaluate` loads three inputs and runs four ordered checks, refusing a source-write policy, a failed witness, an empty list, or an authorising candidate, and passing only when no finding remains.** `evaluate` loads three declared inputs: `coverage_gaps.json`, `campaign_policy.json`, and `failed_campaign_digests.json`. It then runs four checks in order and collects a finding for each failure. First, if `write_source` appears in the policy's `allowed_actions`, it records `BOUNDED_AUTONOMY_SOURCE_WRITE_FORBIDDEN` before any candidate is drafted. Second, it builds the candidate packet and refuses it if the builder witness did not pass or the packet is not marked `drafted`. Third, it refuses an empty candidate list. Fourth, it refuses any candidate that sets `source_mutation_authorized` to true or writes to the `source` surface. It also counts repeated entries in the failed digest ledger and records `BOUNDED_AUTONOMY_REPEATED_FAILED_DIGEST` if any digest appears more than once. A run passes only when the finding list is empty.\n\n###### Turning the witnessed count into stamped rows\n\n** `_candidate_packet_subprocess` caps the witnessed count at the policy's `max_candidate_count` and turns it into draft rows, each tied to a coverage gap and stamped with no write surface and required human review.** `_candidate_packet_subprocess` turns the witnessed target count into draft rows. It reads `max_candidate_count` from the policy (default 2), asks the witness for that many targets, and takes the smaller of the two counts. Each row ties to one coverage gap when one is available, carries the builder ref and the builder source digest, and sets `write_surface: none`, `requires_human_review: true`, and `source_mutation_authorized: false`.\n\n###### Witnessing the builder in check-only mode\n\n** `_campaign_builder_witness` runs the real builder as a check-only subprocess when present, or falls back to the copied source-module manifest in the public clone, storing stdout and stderr hashes rather than bodies.** The witness itself is `_campaign_builder_witness`. It looks for the builder named by `BUILDER_REF` at the repository root. When that file is present it runs it as a subprocess with `--check --report --max-targets`, with a 45 second timeout, and accepts the witness only when the process exits zero, returns kind `standard_skill_pairing_campaign_summary`, is not a no-op, reports at least one candidate target, emits a `source_digest`, and leaves `wrote_packet` unset. When the builder file is absent, which is the case in the public clone, it falls back to `_public_source_module_builder_witness`, which reads the copied source-module manifest and passes only when the target file's sha256 matches, its line count matches, and no required anchor is missing. Either way the witness stores stdout and stderr hashes, not bodies.\n\nFunction Role `run`, `run_bounded_autonomy_bundle` Route the fixture and exported bundle through `run_crown_jewel_organ` with the shared evaluators `evaluate` Load the three inputs, run the policy, witness, candidate, and digest checks, return findings `_candidate_packet_subprocess` Convert the witnessed target count into stamped draft candidate rows `_campaign_builder_witness` Watch the real builder in check-only mode, or fall back to the source-module manifest `evaluate_negative_case` Mutate a copy of the input and re-run `evaluate` to confirm the refusal fires\n\n**Module constants pin the boundary: `EXPECTED_NEGATIVE_CASES` maps mutations to codes, `AUTHORITY_CEILING` sets every autonomy flag false, and `SPEC` declares the required inputs, manifest ref, and anchors.** The module constants pin the boundary. `EXPECTED_NEGATIVE_CASES` maps each named mutation to the error code it must produce. `AUTHORITY_CEILING` sets `self_repair_authorized`, `unsupervised_source_mutation_authorized`, `source_write_packet_authorized`, `provider_calls_authorized`, and `release_authorized` all to false. `SPEC` declares the required inputs, the source manifest ref, and the required anchors.\n\nDiagram of the mechanism (8 steps). Diagram source & refs\n\n```text\nflowchart TD Inputs[\"Public synthetic inputs coverage_gaps, campaign_policy, failed_campaign_digests\"] PolicyGate{\"policy allows write_source?\"} Witness[\"Builder witness check-only builder run or source-module manifest\"] WitnessGate{\"reports candidates and wrote_packet unset?\"} Draft[\"Draft candidate rows write_surface none, requires_human_review, source_mutation false\"] DigestGate{\"failed digest repeated?\"} Refuse[\"Refuse source-write, blocked witness, or repeated digest\"] Record[\"metadata-only result record refs, digests, stdout/stderr hashes\"] Inputs --> PolicyGate PolicyGate -- \"yes\" --> Refuse PolicyGate -- \"no\" --> Witness Witness --> WitnessGate WitnessGate -- \"no\" --> Refuse WitnessGate -- \"yes\" --> Draft Draft --> DigestGate DigestGate -- \"yes\" --> Refuse DigestGate -- \"no\" --> Record Refuse --> Record\n```\n\n##### Negative cases\n\n###### Refusals checked by mutating the input\n\n**The refusals are checked by mutating the input rather than trusting a declared label, as `evaluate_negative_case` copies the inputs, edits one file, and re-runs `evaluate`.** The refusals are checked by mutating the input, not by trusting a declared label. `evaluate_negative_case` copies the required inputs into a temporary directory, edits one file, and re-runs `evaluate`.\n\n###### The source-write policy case\n\n**The `source_write_campaign_packet` case appends `write_source` to the policy's `allowed_actions` and must produce `BOUNDED_AUTONOMY_SOURCE_WRITE_FORBIDDEN`.** The `source_write_campaign_packet` case appends `write_source` to the policy's `allowed_actions`. It must produce `BOUNDED_AUTONOMY_SOURCE_WRITE_FORBIDDEN`.\n\n###### The repeated failed-digest case\n\n**The `repeated_failed_campaign_digest` case duplicates a digest in the failed-digest ledger and must produce `BOUNDED_AUTONOMY_REPEATED_FAILED_DIGEST`, since an already-failed plan cannot be re-proposed under a fresh wrapper.** The `repeated_failed_campaign_digest` case rewrites the failed-digest ledger to hold the same digest twice. It must produce `BOUNDED_AUTONOMY_REPEATED_FAILED_DIGEST`. A plan that has already failed cannot be re-proposed under a fresh wrapper.\n\n###### Unknown negative cases fail with one stable code\n\n**Any negative-case id outside the declared set is rejected as unsupported with the stable bounded-autonomy error code.** A negative case with any other id returns `BOUNDED_AUTONOMY_NEGATIVE_CASE_UNSUPPORTED`.\n\n##### Prior Art Grounding\n\n###### Preflight gating of proposed actions\n\n**The component follows established preflight-gate patterns from NIST AI RMF, Open Policy Agent, and GitHub Actions, applying that shape to draft self-proposals over synthetic public coverage gaps.** The component follows established patterns for gating proposed actions before they run. The [NIST AI Risk Management Framework](https://www.nist.gov/itl/ai-risk-management-framework) frames AI work as governing, mapping, measuring, and managing risk rather than assuming autonomy is authorized. [Open Policy Agent](https://www.openpolicyagent.org/docs/latest) is a policy-engine pattern for deciding whether an action may proceed. GitHub Actions [workflow syntax](https://docs.github.com/en/actions/reference/workflows-and-actions/workflow-syntax) is a common automation surface where jobs and permissions are declared before execution. This component keeps the preflight-gate shape and applies it to draft self-proposals over synthetic public coverage gaps.\n\n##### Validation Result record Path\n\n###### What a pass means and how to route drift\n\n**A pass means both fixture and bundle drafted a packet, both negative cases fired their codes, and the record stayed metadata-only, with digest drift routed through `microcosm_exact_copy_refresh`.** A pass means the fixture and bundle both produced a draft packet, both named negative cases fired their expected error codes, and the result record stayed metadata-only. If the source-module manifest reports digest drift, route that through `microcosm_exact_copy_refresh`; this page is not the authority for copied source bodies. If the full projection check fails because another session holds the shared lattice outputs, treat that as unrelated contention and use the corpus check as the local gate.\n\n##### Scope boundary\n\n###### Scope limit\n\n**The strongest supported claim is that under public synthetic inputs the component emits a draft self-proposal from a read-only witness and refuses source-write or repeated-digest inputs, both checked by mutation.** The strongest claim the evidence supports is this: under public synthetic inputs, the component emits a draft self-proposal derived from a read-only builder witness and refuses a source-write policy or a repeated failed digest, with both refusals checked by mutating the input. That is the proof boundary and the scope limit for this module.\n\n###### What the module does not claim","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":12594,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/bounded_autonomy_campaign_packet.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"NIST AI Risk Management Framework","url":"https://www.nist.gov/itl/ai-risk-management-framework","kind":"external_reference","section_id":"preflight-gating-of-proposed-actions"},{"label":"Open Policy Agent","url":"https://www.openpolicyagent.org/docs/latest","kind":"external_reference","section_id":"preflight-gating-of-proposed-actions"},{"label":"workflow 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Replay","relative_href":"docs/paper-module-computer-use-action-trace-replay.html","read_href":"docs/paper-module-computer-use-action-trace-replay.html","context_href":"docs/rules-and-ideas.html#paper-module-computer-use-action-trace-replay","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"Validator-backed public replay for synthetic computer-use action traces under the route-observability runtime.","summary":"Computer-Use Action Trace Replay validates a synthetic computer-use episode through visible observations, affordances, action rows, authority verdicts, state-transition and recovery result records, cold replay rows, public trace spans, source-module manifests, negative cases, and metadata-only result records. It is a reader-facing contract under Agent Route Observability Runtime, not live browser or desktop control.","authority_ceiling":"Synthetic public computer-use action-trace fixtures, exported bundle metadata, copied source-module digests, and metadata-only result records only; no live account action, account secret entry, external network mutation, purchase/send authority, destructive host action, hidden screen-state claim, benchmark-score claim, launch-scope decision, or whole-system correctness.","abstract":"Validator-backed public replay for synthetic computer-use action traces under the route-observability runtime.","teleology":{"section_id":"","text":"Validator-backed public replay for synthetic computer-use action traces under the route-observability runtime.","source":"capsule_summary_line"},"body_markdown":"###### Component answers offline\n\n**The component answers, offline over a synthetic trace, whether a blocked action was later narrated as success and whether a claimed success has any state evidence behind it.** A computer-use agent leaves behind a stream of screenshots, clicks, keystrokes, and \"it worked\" assertions. Reading that stream by hand does not tell you the two things that matter: whether an action that was blocked or sent for review got narrated later as a success, and whether a claimed success has any state evidence behind it. This component answers those two questions on a synthetic trace, offline, before any of the language reaches a reader.\n\n###### Takes recorded local-agent\n\n**It takes a recorded local-agent run with its observations, actions, pre-action verdicts, and state changes, checking the trail is self-consistent against authority limits written down up front.** It takes a recorded run of a local agent, the route it picked, the observations it logged, the actions it took, the authority verdict stamped before each action, and the state change each action produced, and checks that this recorded trail is well-formed and self-consistent. The record states up front where the agent's authority was supposed to stop, so the limits are written down and checkable rather than taken on faith.\n\n###### Result is metadata-only\n\n**The result is a metadata-only record of counts, refs, hashes, and redaction posture over a recorded trail; it never watches a live agent.** The result is a metadata-only record of counts, references, hashes, and redaction posture. It checks a recorded trail. It does not watch a live agent and it does not establish one stayed in bounds.\n\n##### Purpose\n\n###### The complete-looking trace that hides a lie\n\n**It guards against a complete-looking trace that hides a lie: a denied action reported as done, a success asserted with no oracle, a screenshot treated as proof.** The failure this guards against is a trace that looks complete while hiding a lie. An action can be denied by policy and then reported as done. A success can be asserted with no diff, no oracle, and no before-and-after state. A screenshot can be treated as proof when the pixels prove nothing about what actually changed.\n\n###### A typed join of verdict and transition\n\n**Every action must line up with a prior observation, a pre-action verdict, and a transition whose outcome matches that verdict, through a typed join rather than a screenshot replay.** A loose reading of the log cannot catch these. You need every recorded action to line up, row by row, with a prior visible observation, a pre-action authority verdict, and a state-transition record whose outcome agrees with that verdict. If the verdict said allow, the transition has to show the action ran and an oracle confirmed the new state. If the verdict said block or review, the transition has to show the action did not run and the status reads `blocked` or `review_required`. The mechanism is a typed join across those tables, not a screenshot replay.\n\n###### The screenshot is the one distrusted input\n\n**Inverting the usual trust, observations enter only as a `screenshot_digest` and `visible_state_hash`, with raw pixels and hidden state barred, so weight rests on verdict-transition agreement.** The inversion is the point. Most action-trace tooling trusts the screenshot. This component treats the screenshot as the one thing it will not trust: observations enter only as a `screenshot_digest` and a `visible_state_hash`, and raw pixel bodies, hidden-state claims, and live-browser state are all required to be absent. The evidence that carries weight is the agreement between verdict and transition.\n\n##### How it works\n\n###### `run_computer_use_action_trace_bundle` and the check sequence\n\n** `run_computer_use_action_trace_bundle` resolves the input, decides fixture versus exported bundle, loads it through strict JSON, then runs a fixed sequence of checks before writing anything.** The entry point is `run_computer_use_action_trace_bundle`. It resolves the input directory, decides whether the input is the full fixture with negative cases or the public exported bundle, loads the bundle through a strict JSON path, then runs a fixed sequence of checks in this order before it writes anything.\n\nDiagram of the mechanism (12 steps). Diagram source & refs\n\n```text\nflowchart TD load[\"load bundle, strict JSON\"] load --> proj[\"_validate_computer_use_projection_protocol\"] proj --> pol[\"_validate_computer_use_interaction_policy\"] pol --> ep[\"_validate_computer_use_episodes\"] ep --> obs[\"_validate_computer_use_observations\"] obs --> act[\"_validate_computer_use_actions\"] act --> ver[\"_validate_computer_use_authority_verdicts\"] ver --> tr[\"_validate_computer_use_state_transitions\"] tr --> rec[\"_validate_computer_use_recovery_receipts\"] rec --> cold[\"_validate_computer_use_cold_replay\"] cold --> neg[\"_validate_computer_use_negative_cases fixture path only\"] neg --> out[\"status pass, else blocked\"]\n```\n\n###### Stages chain, so early breaks propagate\n\n**Each stage consumes the rows the earlier stage exported, so a break early in the chain propagates downstream.** Each stage consumes the rows the earlier stage exported, so a break early in the chain propagates. The relevant functions:\n\nFunction Consumes Checks and rejects `_validate_computer_use_observations` screen observations, episode rows Each observation is a redacted visible-state row with affordance refs and a screenshot digest. Rejects `raw_screenshot_body_exported`, `hidden_state_claim`, or `live_browser_state` set true. Requires every episode to have at least one observation. `_validate_computer_use_actions` action trace, episodes, observations Each action cites an observation and an affordance, uses an allowed action kind, stays redacted, and does not precede the observation it reacts to. Requires the trace to cover `click`, `type`, `navigate`, `edit_text_record`, and `wait`. `_validate_computer_use_authority_verdicts` verdicts, action rows Each action has one pre-action verdict of `allow`, `block`, or `review` that explicitly sets `live_account_authorized`, account secret entry authorized, `external_network_mutation_authorized`, `destructive_without_review_authorized`, and `purchase_or_send_authorized` to false. Flags any action with no verdict. `_validate_computer_use_state_transitions` transitions, actions, verdicts Matches each transition to its action's verdict. An `allow` needs `execution_attempted` true and `oracle_status` pass. A `block` or `review` needs `execution_attempted` false and `oracle_status` in `blocked` or `review_required`. Rejects any `nondeterministic_success_claim`. `_validate_computer_use_recovery_receipts` recovery rows, actions, verdicts Every blocked or reviewed action needs a redacted recovery row with `recovery_status` recovered and `state_restored` true. A blocked action is never upgraded into an executed mutation. `_validate_computer_use_cold_replay` cold replay rows, actions Replay rows must cover every action id and reproduce the action trace, the authority verdicts, and the state transitions, with a `pass_label` of `accepted` or `blocked_recovered`.\n\n###### Public trace spans and the pass rule\n\n** `build_public_computer_use_trace` emits bounded public spans, and status is `pass` only when findings are empty, no negative case is missing, the scan passes, and source-module digests verify.** After the checks, `build_public_computer_use_trace` emits bounded public trace spans over refs, counts, and hashes. `run_computer_use_action_trace_bundle` collects the findings from every stage. It sets status to `pass` only when the positive findings are empty, no expected negative case is missing, the non-public-state scan passes, and the copied public source-module digests verify. Otherwise the status is `blocked`. The written record carries counts, refs, hashes, and the redaction posture, and omits raw screen bodies, account secrets, hidden screen state, model-output data, private source bodies, absolute local paths, and benchmark-score claims.\n\n###### What the tracked fixture record reports\n\n**The tracked fixture record reports fixed counts: four episodes, eight actions and verdicts, eight transitions, one recovery, four cold-replay rows, one blocked action, and eight trace spans.** The tracked fixture record reports four episodes, six observations, eight actions, eight authority verdicts, eight state-transition records, one recovery record, four cold-replay rows, one blocked action, and eight public trace spans.\n\n##### Negative cases\n\n###### Eight declared perturbations on the fixture path\n\n** `_validate_computer_use_negative_cases` feeds eight declared perturbations through the same posture and requires each to be rejected with its own code, failing the run if any is missing.** The fixture path runs `_validate_computer_use_negative_cases`, which feeds eight declared perturbations through the same posture and confirms each one is rejected with a specific error code. The set is fixed in `COMPUTER_USE_EXPECTED_NEGATIVE_CASES`; a missing observation there fails the run.\n\n###### The specific actions each row must reject\n\n**Rows are rejected for live-account or account secret actions, external mutation, unapproved purchase or send, destructive file actions, hidden-state claims, missing observations, and benchmark-score claims.** Live account action is rejected when a row carries `live_account_context` or `live_account_action`. account secret entry is rejected when a row attempts it or carries a account secret value. External network mutation is rejected outright. A purchase or send is rejected when it is requested without a `human_approval_ref`. A destructive file action is rejected without a `human_review_ref`. A hidden screen-state claim is rejected when a row claims hidden state or omits its `visible_state_hash`. An action without a prior observation or affordance ref is rejected. A benchmark-score claim is rejected: a synthetic replay record cannot claim benchmark performance.\n\n##### Prior Art Grounding\n\n###### WebArena, Mind2Web, OSWorld, WebDriver accounting\n\n**The design borrows inspectable action-trace accounting from WebArena, Mind2Web, OSWorld, and WebDriver, requiring observations, verdicts, transitions, recovery, and cold replay to line up before an episode is credited.** This component borrows the action-trace accounting pattern from web and desktop agent benchmarks that make action trajectories inspectable. [WebArena](https://arxiv.org/abs/2307.13854) and [Mind2Web](https://arxiv.org/abs/2306.06070) anchor realistic web-task evaluation, and [OSWorld](https://arxiv.org/abs/2404.07972) extends the concern to agents acting in real computer environments. Browser automation standards such as [WebDriver](https://www.w3.org/TR/webdriver2/) are prior art for representing actions against visible browser state through a controlled protocol. The borrowed idea is the accounting: observations, affordances, actions, pre-action verdicts, transitions, recovery, and cold replay must line up before an episode is credited.\n\n##### Validation Result record Path\n\nRun from `microcosm-substrate/`:\n\n```text\nPYTHONPATH=src ../repo-python -m microcosm_core.cli agent-route-observability-runtime \\\n  --input examples/agent_route_observability_runtime/exported_computer_use_action_trace_bundle \\\n  --out receipts/runtime_shell/demo_project/organs/agent_route_observability_runtime \\\n  validate-computer-use-bundle\n```\n\n###### What a clean bundle join means\n\n**A pass means the exported bundle joined cleanly, every action carrying an observation, verdict, and matching transition, with the cold replay reproducing them and the record staying metadata-only.** A pass means the exported bundle joined cleanly: every action had an observation, a pre-action verdict, and a matching transition, the cold replay reproduced them, and the written record stayed metadata-only. The command writes the validation record under the named output directory. The public exported bundle has `contains_negative_cases: false`, so the eight-case floor is checked on the fixture path, and `tests/test_agent_route_observability_runtime.py` remains the negative-case authority.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The synthetic replay strongest claim\n\n**The strongest claim is a synthetic action-trace replay over public fixtures, joining observations, verdicts, transitions, recovery, cold replay, digest checks, and the eight negative cases into metadata-only records.** The strongest claim the evidence supports is a synthetic computer-use action-trace replay over public fixtures: visible observations, affordances, action rows, pre-action authority verdicts, state-transition records, recovery records, cold-replay rows, public trace spans, source-module digest checks, the eight expected negative cases, and metadata-only result records.\n\n###### Capabilities the ceiling sets false\n\n**It controls no live browser, takes no account actions, enters no account secrets, mutates nothing external, and claims no benchmark, with `COMPUTER_USE_AUTHORITY_CEILING` setting every such capability false.** That is the whole proof boundary. This component does not control a live browser or desktop, take account actions, enter account secrets, mutate external systems, execute purchases or sends, perform destructive host actions, export raw screenshots, or read hidden screen state. It does not claim benchmark performance, use external model services, change source files, include launch operations, or prove whole-system correctness. The scope limit is stated in the record itself as `COMPUTER_USE_AUTHORITY_CEILING`, where every one of those capabilities is set to false. The generated diagram and Atlas entry are navigation surfaces, not additional proof.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":13840,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/computer_use_action_trace_replay.md","kind":"source_link"},{"label":"Source 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Control","relative_href":"docs/paper-module-concurrency-mission-control.html","read_href":"docs/paper-module-concurrency-mission-control.html","context_href":"docs/rules-and-ideas.html#paper-module-concurrency-mission-control","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"Concurrency Mission Control validates metadata-only coordination result records without becoming a live scheduler or production concurrency proof.","summary":"Concurrency Mission Control validates the public concurrency mission-control membrane: copied source-builder digests, bridge artifacts, failure classes, work log seed-speed topology, heartbeat and claim-collision anchors, negative cases, source-open body imports, and metadata-only result records. It separates coordination evidence from hosted orchestration, external model access, live scheduling, whole-system equivalence, source authority, launch, public sharing, and production concurrency guarantees.","authority_ceiling":"Verified concurrency mission-control fixture and source-module import evidence only; no live scheduler, external model access, hosted orchestration, production concurrency-safety proof, source authority, whole-system equivalence, launch-scope decision, publishing-scope decision, or whole-system correctness.","abstract":"Concurrency Mission Control validates metadata-only coordination result records without becoming a live scheduler or production concurrency proof.","teleology":{"section_id":"","text":"Concurrency Mission Control validates metadata-only coordination result records without becoming a live scheduler or production concurrency proof.","source":"capsule_summary_line"},"body_markdown":"###### Component turns quiet\n\n`concurrency_mission_control` turns that moment into a checkable verdict.\n\n###### Copies real coordination\n\n**It copies the real coordination builder and bridge fixtures, runs them over synthetic lanes and a work log seed-speed snapshot, and emits metadata-only records of refs, digests, and verdicts.** The component imports the real coordination specimen builder and its public provider and work log bridge fixtures as exact copies, runs the copied builder over synthetic multi-agent lanes, and reads a public snapshot of the work log's seed-speed topology. It emits metadata-only result records: refs, digests, counts, and verdicts, with no session bodies, model-output data, or proof text.\n\n###### Is no live\n\n**It is no live scheduler, private runtime, or production safety evidence; it demonstrates fail-closed transaction gating over declared copied bodies and fixtures.** It is not a live scheduler, not the private mission-control runtime, not external model access, and outside production claims concurrency-safety evidence. It demonstrates fail-closed transaction gating over declared copied bodies and fixtures.\n\n##### Purpose\n\n###### Is acting on a dirty path safe\n\n**Given a dirty path and its live claim topology, each case resolves to one named classification and one allowed action, so a lane proceeds, hands off, or waits.** The question the component answers is narrow: given a dirty path and the live claim topology around it, is acting on that path safe, and if not, what must happen first. A loose answer like \"probably fine\" is exactly the failure mode. Each case resolves to a named classification and one allowed action, so a lane can decide to proceed, hand off, or wait.\n\n###### Real builder and two cautious classifiers\n\n**The evidence runs the real specimen builder over synthetic lanes, reading claims, heartbeats, and collisions, with two classifiers that default to the cautious verdict when evidence is thin.** The evidence does not come from a re-implemented scheduler. The component runs the same specimen builder the private system uses, over public synthetic lanes, and reads who holds which claim, whether their heartbeat is current, and where path claims collide. Two classifier functions carry the judgment, and both default to the cautious verdict when the evidence is thin. That default is the behaviour the page is about.\n\n##### How it works\n\n###### Two entry points share one `SPEC`\n\n** `run` and `run_concurrency_mission_control_bundle` both hand a shared `SPEC` to `run_crown_jewel_organ` with `_evaluate` and `evaluate_negative_case`, the bundle validating the same contract from a shipped copy.** `run` and `run_concurrency_mission_control_bundle` are the two entry points. Both hand a shared `SPEC` to `run_crown_jewel_organ` with `_evaluate` as the evaluator and `evaluate_negative_case` as the negative-case checker. The bundle entry sets `input_mode` to the exported bundle so a reader can validate the same contract from a shipped copy.\n\n###### `_evaluate` runs the builder and five gates\n\n** `_evaluate` loads the exercise manifest, runs the copied builder, then five gates, passing only when all six engines pass and every expected engine id is present.** `_evaluate` loads `concurrency_mission_control_exercise_manifest.json`, runs the copied builder, then runs five gates. It passes only when all six engines pass and every expected engine id is present.\n\nFunction Role `_run_original_builder` Copies the public seed root into a temporary workspace and calls `build_concurrency_mission_control_specimen` from the copied source. Reads the emitted `mission_board.json` and `receipt.json`, summarizes case, accept, block, repair, and authority-collapse counts. If the seed root is missing it returns `blocked` with `CONCURRENCY_MISSION_CONTROL_SEED_ROOT_MISSING` instead of running. `_failure_matrix_gate` Checks the board against the manifest's expected counts: exact case and accept counts, minimum block, repair, and source-bundle counts, and every required failure class present (owner-path conflict, duplicate run, dependency gap, stale lease, missing result record, supervised-scope gap, missing parent finalizer, misanchored claim). `_bridge_authority_membrane` Checks that required bridge statuses match, that the summed authority-collapse count is zero, that bridge case-count shapes match, and that the manifest's forbidden-claim set covers seven required bars including `private_mission_control_runtime` and `production_concurrency_safety`. _work_ledger_seed_speed_gate Checks the seed-speed snapshot: schema, non-negative count shape, minimum active sessions and claims, a collision gate (`claim_collisions` equals expected, per-session collision count zero, no collision rows), heartbeat, session cards, bound source refs, and forbidden private inputs. `classify_generated_surface_claim_lens` Takes a case's drift paths and claim rows and returns one classification with one allowed action. `classify_concurrency_closure_state_lens` Decides whether a unit of work is genuinely closed or only looks closed.\n\n###### The generated-surface lens and its verdicts\n\n**The generated-surface lens intersects drift paths with the generated entry surfaces, returning `owned_live`, `owned_stale`, `unowned_generated_drift`, `unrelated_dirty_state`, or `clean` with a matching allowed action.** The generated-surface lens intersects a case's drift paths with `GENERATED_ENTRY_SURFACES` (`ORGANS.md`, `ARCHITECTURE.md`, `AGENT_ROUTES.md`, `atlas/agent_task_routes.json`). If a live owner holds one of those paths it returns `owned_live` with the action `do_not_patch_from_sibling_lane`. Drift on other paths is `unrelated_dirty_state`, and no drift is `clean`.\n\n###### Separating closed from looks-closed\n\n**The closure-state lens folds classification, validation, commitability, and residuals to separate genuinely closed work from cases that only look done, defaulting to `open_unclassified` when evidence is insufficient.** The closure-state lens folds that generated classification together with validation state, commitability, and any open residual. It separates `closed_and_committed` (product closed, validation complete, commit landed) from the cases that only look done: `closed_validation_deferred` when validation is parked under host pressure with a stated reason, `closed_uncommitted_authority` when event authority exists but shared append logs are unsafe to stage, `false_residual_stale` when a residual is left open against a passing generator check, and `owned_live_handoff` or `owned_stale_reentry` when a live or stale owner still holds the path. When the closure evidence is insufficient it returns `open_unclassified`, which is the default, so absent evidence never reads as success.\n\nDiagram of the mechanism (7 steps). Diagram source & refs\n\nSource refs\n\nowner live / stale / absent `classify_generated_surface_claim_lens` closed vs looks-closed `classify_concurrency_closure_state_lens`\n\n```text\nflowchart TD Builder[\"_run_original_builder copied specimen builder in temp seed root\"] Matrix[\"_failure_matrix_gate counts and failure classes\"] Membrane[\"_bridge_authority_membrane bridges green, authority-collapse zero, forbidden claims blocked\"] Seed[\"_work_ledger_seed_speed_gate heartbeat, counts, collision-free claims\"] Surface[\"classify_generated_surface_claim_lens owner live / stale / absent\"] Closure[\"classify_concurrency_closure_state_lens closed vs looks-closed\"] Record[\"metadata-only result record refs, digests, counts, verdicts\"] Builder --> Matrix Builder --> Membrane Matrix --> Record Membrane --> Record Seed --> Record Surface --> Record Closure --> Record\n```\n\n##### Negative cases\n\n###### Five cases that must each block\n\n** `evaluate_negative_case` drives five stable cases that must each block with a named code, and any non-block counts as a wrong-kind pass the checker treats as a failure.** `evaluate_negative_case` drives five stable cases, each expected to block with a named code. `missing_seed_root` runs the builder against an empty temporary root and expects `CONCURRENCY_MISSION_CONTROL_SEED_ROOT_MISSING`. `provider_bridge_missing` sets the provider repair bridge status to `missing` and expects the authority membrane to block with `CONCURRENCY_MISSION_CONTROL_PROVIDER_BRIDGE_BLOCKED`. `authority_collapse_claim` sets an authority-collapse count of one and expects `CONCURRENCY_MISSION_CONTROL_AUTHORITY_COLLAPSE`. `private_runtime_claim` drops `private_mission_control_runtime` from the manifest's forbidden claims and expects `CONCURRENCY_MISSION_CONTROL_PRIVATE_RUNTIME_OVERCLAIM`. work_ledger_seed_speed_collision injects a claim collision into the seed-speed snapshot and expects CONCURRENCY_MISSION_CONTROL_WORK_LEDGER_COLLISION_UNRESOLVED. A case that does not produce its code counts as a pass of the wrong kind, so the checker treats a non-block as a failure.\n\n##### Prior Art Grounding\n\n###### Airflow, Kubernetes leases, two-phase commit\n\n**The design borrows coordination shapes from Airflow DAGs, Kubernetes leases, two-phase commit, and GitHub Actions concurrency, keeping only the fail-closed synthetic gating shape, not their runtimes.** The component borrows established coordination shapes without claiming their runtimes. Apache Airflow [DAGs](https://airflow.apache.org/docs/apache-airflow/stable/core-concepts/dags.html) separate task dependency and scheduling from task internals. Kubernetes [lease-based leader election](https://kubernetes.io/docs/concepts/cluster-administration/coordinated-leader-election/) is the lease-holder and renewal pattern behind live-versus-stale owner state. IBM Research on [two-phase commit](https://research.ibm.com/publications/two-phase-commit-optimizations-and-tradeoffs-in-the-commercial-environment) is the transaction-consistency pattern under participant failure. GitHub Actions [workflow syntax](https://docs.github.com/en/actions/reference/workflows-and-actions/workflow-syntax) is the declared workflow-concurrency control. This component keeps only the fail-closed synthetic gating shape.\n\n##### Validation Result record Path\n\nFrom `microcosm-substrate`, validate with throwaway result outputs first:\n\n###### What a pass proves, and drift repair\n\n**A pass proves fail-closed coordination evidence over the declared copied bodies and synthetic fixtures, with source-copy digest drift repaired through `microcosm_exact_copy_refresh`, not here.** A pass means the fixture and bundle runs proved fail-closed coordination evidence over the declared copied bodies and synthetic fixtures. Source-copy digest drift is repaired through `microcosm_exact_copy_refresh`, not here.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported fixture-contract claim\n\n**The strongest claim is that the copied builder, bridge, matrix, membrane, and seed-speed fixtures satisfy the declared contract with five named blocks and no non-public body export.** The strongest claim the evidence supports: the exact copied specimen builder, provider and work log bridge fixtures, failure-matrix fixture, bridge authority membrane, and work log seed-speed topology fixture together satisfy the declared concurrency mission-control fixture contract, with five negative cases blocking on their named codes and no non-public body export. The proof boundary is those copied bodies, fixtures, source manifests, and metadata-only result records.\n\n###### What the scope limit excludes\n\n**The ceiling proves no private runtime, live scheduling, external model access, or production safety, and the classifiers lower their verdict rather than claim success when evidence is missing.** The scope limit excludes everything past that. This component does not establish private mission-control runtime truth, live scheduling, provider or model dispatch, hosted orchestration, production concurrency safety, source-file changes, launch or publishing-scope decision, or whole-system correctness. 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negative cases, source-open body imports, and scope limits. It lowers claim strength to fixture truth: fact assertion, code-loci, and DAG evidence are checked, but the component does not become a comprehension engine, minimum-read graph, doctrine saturation proof, source-file changes lane, or launch-scope decision.","authority_ceiling":"Declared fact-assertion, code-locus, and DAG fixture truth gate only; no comprehension engine, no minimum-read-graph proof, no doctrine saturation claim, no source-file changes, no launch-scope decision, no publishing-scope decision, and no whole-system correctness.","abstract":"Doctrine Fact Claim Audit rejects wrong fact counts and dead anchors without claiming comprehension or route completeness.","teleology":{"section_id":"","text":"Doctrine Fact Claim Audit rejects wrong fact counts and dead anchors without claiming comprehension or route completeness.","source":"capsule_summary_line"},"body_markdown":"###### Documentation about live\n\n**Documentation about a live system rots: counts and code citations go stale as the code moves, and a reader cannot tell a current number from one true only once.** Documentation about a live system rots. A page says there are forty-seven of something, or cites a function in a named file, and both claims go stale as the code moves underneath them. A reader cannot tell a current count from a number that was true once and never rechecked.\n\n###### Doctrine fact claim\n\n** `doctrine_fact_claim_audit` asks which factual claims on a page can be re-derived from source now and which have drifted, emitting a metadata-only record of counts, verdicts, and scope limits.** `doctrine_fact_claim_audit` answers one question about a page of factual claims: which of them can be re-derived from source right now, and which have become untracked drift. It reads a small set of public fixtures, recomputes a bounded relation over them, and writes a result record that carries counts, verdicts, and scope limits but no copied source bodies.\n\n###### Does not judge\n\n**It does not judge whether prose sounds true; it checks four concrete things: fact count, code-locus existence with anchor, route-graph edges pointing at known facts, and volatile numbers bound to assertions.** The component does not read prose and judge whether it sounds true. It checks four concrete things: that a declared fact count matches the fact table, that every cited code location exists on disk with its quoted anchor present, that every route-graph edge points at a known fact, and that volatile numbers in current-state prose are bound to a fact assertion or surfaced as a blocking finding.\n\n##### Purpose\n\n###### A claim is a cache with no invalidation\n\n**A loose documentation claim is a cached value with no invalidation strategy: a bare number tells you nothing about how to recheck it, so stale claims persist unnoticed as code changes.** A loose documentation claim is a cached value with no invalidation strategy. The number 47 sitting in a sentence tells you nothing about how to recheck it, or whether the file it references still exists. When the code changes, the sentence keeps asserting the old value and nobody notices.\n\n###### Admissible only when bound to something recomputable\n\n**A claim is admissible only when bound to something recomputable: a bare number must tie to a fact assertion or get flagged, and cited paths are resolved on disk with anchors confirmed.** This component makes a claim admissible only when it is bound to something recomputable. A bare number in a current-state section is not enough. It has to be tied to a fact assertion that records how the value is derived, or it gets flagged. Cited file paths are resolved on disk and their anchor text is confirmed present, so a plausible-but-dead reference becomes a typed finding instead of inert prose.\n\n##### How it works\n\n###### A fixture evaluator on the crown-jewel runner\n\n**The component is a public fixture evaluator whose `SPEC` declares four required inputs and source anchors, while the shared runner handles validation, record writing, and negative cases, and this module supplies the evaluator and mutator.** The component is a public fixture evaluator built on the shared crown-jewel runner. `SPEC` is a `CrownJewelSpec` that declares four required inputs, `fact_assertions.json`, `fact_dag.json`, `numeric_claims.json`, and `projection_protocol.json`, plus the source manifest reference and the anchors each copied source body must contain. The runner handles manifest validation, result-record writing, negative-case execution, and scope-limit attachment. This module supplies the domain evaluator and the negative-case mutator.\n\n###### `evaluate` checks count, loci, anchors, edges\n\n** `evaluate` compares the declared fact count to the table, resolves each fact's code locus and anchor on disk, and rejects graph edges whose endpoints are not known facts, each with a typed error.** `evaluate` is the core pass. It loads the fact table and the route graph, then runs the numeric-claim check, then walks the fact rows. It compares `expected_fact_count` against the number of fact rows and records `DOCTRINE_FACT_COUNT_MISMATCH` on disagreement. For each fact it requires at least one code locus, resolves the relative path through `_resolve_code_locus` against the copied source bundle, reads the file, and confirms the declared anchor string is in the body. A missing file or a missing anchor becomes `DOCTRINE_CODE_LOCUS_MISSING` or `DOCTRINE_CODE_LOCUS_ANCHOR_MISSING`. It then builds the set of known fact ids and rejects any graph edge whose `from` or `to` endpoint is not in that set with `DOCTRINE_FACT_DAG_DEAD_REF`.\n\n###### `_evaluate_numeric_claims` binds volatile numbers\n\n** `_evaluate_numeric_claims` imports the copied `derived_fact_hologram.py` and runs `find_unbound_numeric_claims` per case, treating a detector case as positive and blocking any unbound current-state number.** `_evaluate_numeric_claims` handles the volatile-number check. It loads `numeric_claims.json` and imports the copied `derived_fact_hologram.py` body through `_load_derived_fact_module`, which resolves the module out of the exported bundle and reports `DOCTRINE_NUMERIC_CLAIM_SOURCE_MISSING` if the copied body is absent. It then calls that body's `find_unbound_numeric_claims` against each case. A detector case declares `expect_unbound: true` and must surface the stated section and number, so it is positive evidence. Any other case that leaves current-state numeric prose without a matching fact-assertion section is blocked with `DOCTRINE_UNBOUND_NUMERIC_CLAIM` and `DOCTRINE_VOLATILE_NUMERIC_UNBOUND`.\n\n###### Pass requires an empty findings list\n\n** `evaluate` returns `pass` only when the findings list is empty, reporting the fact, code-locus, edge, and numeric-claim tallies in a metadata-only record that never embeds copied source.** `evaluate` returns a status of `pass` only when the findings list is empty, along with the fact count, verified code-locus count, edge count, and the numeric-claim tallies. The status record keeps `body_in_receipt: false` and never embeds copied source text.\n\n###### Negative cases mutate a fixture in memory\n\n** `evaluate_negative_case` applies one real defect to the positive fixture in memory and reruns `evaluate`, so the proof of each error code stays attached to the evaluator, not to fixture filenames.** The negative floor is semantic, not label-trusting. `evaluate_negative_case` copies the positive fixture in memory, applies one real defect for the named case (`wrong_fact_count` bumps the expected count, `missing_code_locus` empties a fact's loci, `dead_code_locus` corrupts an anchor, `dead_dag_ref` repoints an edge, `unbound_numeric_claim` strips a case's asserted sections), writes the mutated payloads to a temporary directory, and reruns `evaluate` there. The expected error codes live in `EXPECTED_NEGATIVE_CASES`. Because the mutation happens after load, the proof stays attached to the evaluator rather than to fixture filenames.\n\nFunction Role `run`, `run_doctrine_fact_bundle` CLI entrypoints; forward to the shared crown-jewel runner in fixture or exported-bundle mode `evaluate` Loads the fixtures, checks fact count, code loci, anchors, and graph edges, returns the status record `_evaluate_numeric_claims` Imports the copied source body and runs the volatile-number binding check per case `_load_derived_fact_module` Resolves and imports `derived_fact_hologram.py` from the copied bundle `evaluate_negative_case` Mutates the positive fixture in memory and reruns `evaluate` to prove each stable error code Diagram of the mechanism (10 steps). Diagram source & refs\n\nSource refs\n\nfacts + expected_fact_count `fact_assertions.json` edges `fact_dag.json` cases `numeric_claims.json`\n\n```text\nflowchart TD Facts[\"fact_assertions.json facts + expected_fact_count\"] --> Eval[\"evaluate\"] Dag[\"fact_dag.json edges\"] --> Eval Numerics[\"numeric_claims.json cases\"] --> Eval Eval --> Count{\"count matches table length?\"} Eval --> Loci{\"path on disk + anchor in body?\"} Eval --> DagRef{\"edge endpoints are known facts?\"} Eval --> Bound{\"current-state numbers bound to a fact?\"} Count -->|no| Block[\"typed blocking finding\"] Loci -->|no| Block DagRef -->|no| Block Bound -->|no| Block Count -->|yes| Pass[\"metadata-only result record\"] Loci -->|yes| Pass DagRef -->|yes| Pass Bound -->|yes| Pass\n```\n\n##### Negative cases\n\n###### Five failure cases, five stable error codes\n\n**Five declared failure cases each derive a stable error code, while the positive run reports three facts, three code loci, two edges, and two numeric-claim cases including one detector.** The fixture ships five declared failure cases and the component derives a stable error code for each. `wrong_fact_count` gives `DOCTRINE_FACT_COUNT_MISMATCH`. `missing_code_locus` gives `DOCTRINE_CODE_LOCUS_MISSING`. `dead_code_locus` gives `DOCTRINE_CODE_LOCUS_ANCHOR_MISSING`. `dead_dag_ref` gives `DOCTRINE_FACT_DAG_DEAD_REF`. `unbound_numeric_claim` gives `DOCTRINE_UNBOUND_NUMERIC_CLAIM` together with `DOCTRINE_VOLATILE_NUMERIC_UNBOUND`. The accepted positive run reports three facts, three verified code loci, two graph edges, and two numeric-claim cases of which one is the detector for the number 123 in a current-state section.\n\n##### Prior Art Grounding\n\n###### Provenance and fact-check metadata prior art\n\n**The design borrows from provenance modeling and structured fact-check metadata, keeping only the parts that apply to fixture fact counts, code-locus existence, anchors, graph references, and numeric binding.** This component borrows shapes from provenance modeling and structured fact-check metadata. W3C [PROV](https://www.w3.org/TR/prov-overview/) models entities, activities, and agents so a reader can assess how derived information was produced. Schema.org [ClaimReview](https://schema.org/ClaimReview) is a web pattern for recording a reviewed claim and its checking context. [JSON Schema](https://json-schema.org/) covers rejecting malformed or incomplete claim records, and [SLSA provenance](https://slsa.dev/spec/v1.2/) covers tracing an artifact back to its source. The component keeps only the parts that apply to public fixture fact counts, code-locus existence, anchor presence, graph references, and synthetic numeric binding cases.\n\n##### Validation Result record Path\n\n**From `microcosm-substrate`, run against external output paths so the reader check does not churn tracked result records.** From `microcosm-substrate`, run against external output paths so the reader check does not churn tracked result records:\n\n###### What a pass confirms\n\n**A pass means the fact count, code loci, anchors, graph edges, numeric bindings, manifest digests, and five negative cases all agreed with the copied public basis.** A pass means the fixture fact count, code loci, anchors, graph edges, numeric-claim bindings, manifest digests, and the five negative cases all agreed with the copied public basis. If the copied source bodies drift, refresh the exported bundle through its owning import lane before treating a bundle failure as a reader-page defect.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim\n\n**The strongest claim is that, over these fixtures, the component verifies fact count, code-locus existence with anchors, graph edges, and bound volatile numbers, with negative cases proving each rejection derives from the evaluator.** The strongest claim the evidence supports: given these public fixtures, the component checks that a declared fact count matches its table, that every cited code location exists on disk with its anchor present, that every route-graph edge points at a known fact, and that volatile current-state numbers are either bound to a fact assertion or surfaced as a blocking finding. The five negative cases prove each rejection path derives a stable error code from the evaluator, not from a fixture label. Result records stay metadata-only.\n\n###### What it refuses to claim\n\n**It is not a comprehension engine and judges no prose as true: it exports no private doctrine, mutates no source, promotes no axiom, and excludes launch, bounded to the public fixtures and copied bodies.** What it refuses to claim: it is not a comprehension engine and does not judge whether prose reads as true. It does not establish a minimum read graph for the wider doctrine plane, does not export private doctrine, and excludes launch or hosted-public posture. The proof boundary is the synthetic public fixtures, the copied source bodies, their manifest digests and anchors, the metadata-only result records, and the named negative cases. The scope limit excludes source-file changes, candidate-axiom promotion, and any claim that these counts prove whole-system correctness.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":12872,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/doctrine_fact_claim_audit.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"PROV","url":"https://www.w3.org/TR/prov-overview/","kind":"external_reference","section_id":"provenance-and-fact-check-metadata-prior-art"},{"label":"ClaimReview","url":"https://schema.org/ClaimReview","kind":"external_reference","section_id":"provenance-and-fact-check-metadata-prior-art"},{"label":"JSON Schema","url":"https://json-schema.org/","kind":"external_reference","section_id":"provenance-and-fact-check-metadata-prior-art"},{"label":"SLSA 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judge","Purpose","A claim is a cache with no invalidation","Admissible only when bound to something recomputable","How it works","A fixture evaluator on the crown-jewel runner","`evaluate` checks count, loci, anchors, edges","`_evaluate_numeric_claims` binds volatile numbers","Pass requires an empty findings list","Negative cases mutate a fixture in memory","Negative cases","Five failure cases, five stable error codes","Prior Art Grounding","Provenance and fact-check metadata prior art","Validation Result record Path","From microcosm-substrate run","What a pass confirms","Scope limit","The strongest supported claim","What it refuses to claim"]},{"id":"paper_module.self_ignorance_coverage_ledger","title":"Self-Ignorance Coverage Ledger","relative_href":"docs/paper-module-self-ignorance-coverage-ledger.html","read_href":"docs/paper-module-self-ignorance-coverage-ledger.html","context_href":"docs/rules-and-ideas.html#paper-module-self-ignorance-coverage-ledger","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"Self-Ignorance Coverage Ledger counts known coverage debt while refusing unknown-unknown or absence-proof claims.","summary":"Self-Ignorance Coverage Ledger validates a public known-debt coverage fixture: declared Kind Atlas gaps, missing coverage categories, negative cases, source-open body imports, and scope limits. It records what the system knows it has not covered without claiming omniscience, absence proof, total search, source-file changes, public sharing, launch-scope decision, or whole-system correctness.","authority_ceiling":"Known Kind Atlas coverage-debt projection only; no literal unknown-unknown omniscience, no absence proof, no total repository search proof, no source-file changes, no launch-scope decision, no publishing-scope decision, and no whole-system correctness.","abstract":"Self-Ignorance Coverage Ledger counts known coverage debt while refusing unknown-unknown or absence-proof claims.","teleology":{"section_id":"","text":"Self-Ignorance Coverage Ledger counts known coverage debt while refusing unknown-unknown or absence-proof claims.","source":"capsule_summary_line"},"body_markdown":"###### Known unknowns in the current surface\n\n**It answers one question: across declared Kind Atlas families, how many option-surface rows has the generated System Atlas not yet materialised?** A navigation system that lists what it knows is easy to build. One that can state precisely what it has not yet covered is harder, and it is the more honest signal to a cold reader. This component answers a single question: for a declared set of Kind Atlas families, how many rows does the option surface expose that the generated System Atlas has not yet materialised?\n\n###### Computed unknown-coverage debt vector\n\n**The answer is a computed debt vector: for each kind it subtracts materialised entities from the live row count and reports the difference as known coverage debt.** The answer is a small debt vector, computed rather than asserted. For each selected kind the component recomputes the live Kind Atlas row count, counts the entities the System Atlas graph has actually materialised, and reports the difference as known coverage debt. It emits a metadata-only result record with the counts, the source references, and an explicit scope limit.\n\n###### Refusing false completeness\n\n**Its unusual part is refusal: it rejects any claim that unknown-unknowns are exhaustive, treating known debt as a bounded count of visible gaps, never a completeness proof.** The unusual part is what it refuses. It will not accept input that claims its unknown-unknowns are exhaustive, and it will not treat the absence of a row as proof that nothing is missing. Known debt is a bounded count of gaps it can see. It is never a completeness proof.\n\n##### Purpose\n\n###### The narrow middle between vague and overconfident\n\n**Systems overstate coverage either vaguely or overconfidently, so this component computes the finite gap between what the option surface exposes and what the graph materialised.** Navigation systems overstate themselves in two opposite ways. A vague \"coverage is incomplete\" tells a cold reader nothing operational. A confident \"nothing else is missing\" is worse, because it converts absence of evidence into evidence of absence. This component occupies the narrow middle. For a declared finite domain of Kind Atlas families it computes the gap between what the option surface exposes and what the System Atlas graph has materialised, and it reports that gap as a number with its basis attached.\n\n###### The negative output the checker must reject\n\n**Beside the bounded debt vector, the checker must reject exhaustive-coverage claims, hand-authored counts, substituted ids, stale baked ids, a tampered builder marker, or an understated debt floor.** The positive output is a bounded debt vector. The negative output matters as much: the checker must reject input that claims exhaustive unknown-unknown coverage, hand-authored materialisation counts, substituted entity ids, stale baked ids used as authority, a tampered graph builder marker, or a coverage debt that understates the declared floor.\n\n##### How it works\n\n###### Entrypoints and the `evaluate` core\n\n** `run` and `run_self_ignorance_bundle` are the entrypoints, `evaluate` is the core over four bundle files, and `evaluate_negative_case` replays the semantic negatives.** The runtime lives in `src/microcosm_core/organs/self_ignorance_coverage_ledger.py`. The command entrypoint is `run`; the exported-bundle entrypoint is `run_self_ignorance_bundle`; the core evaluator is `evaluate`; the semantic negative-case replayer is `evaluate_negative_case`. `evaluate` consumes four bundle files and runs the following order.\n\n###### Binding the coverage scope and check result record\n\n** `_projection_protocol_receipt` checks the protocol against the expected scope and System Atlas check command, raising a mismatch when either the scope string or the result record status is wrong.** First it loads `kind_atlas_rows.json`, `materialized_entities.json`, and `projection_protocol.json`. `_projection_protocol_receipt` checks the protocol against two constants: `EXPECTED_COVERAGE_SCOPE` and `EXPECTED_SYSTEM_ATLAS_CHECK_COMMAND`. A scope that is not the expected live-versus-generated string raises `SELF_IGNORANCE_PROJECTION_PROTOCOL_SCOPE_MISMATCH`. A check result record whose command is wrong, or whose status is outside `ALLOWED_SYSTEM_ATLAS_CHECK_STATUSES` (`pass` or a declared blocked refresh), raises `SELF_IGNORANCE_SYSTEM_ATLAS_CHECK_RECEIPT_INVALID`.\n\n###### Recomputing live row counts through `build_kind_atlas`\n\n** `_rows_with_live_kind_atlas_counts` overwrites fixture counts with freshly built `build_kind_atlas` row counts, or falls back to declared counts at a lower realness rank when the builder is absent.** Next `_rows_with_live_kind_atlas_counts` recomputes the live row counts. When the wider repository is present it imports `build_kind_atlas` from `system.lib.kind_atlas` and overwrites each fixture-declared count with the freshly built `row_count`. When the builder is not importable it records `unavailable` and falls back to the declared counts, which lowers the realness rank.\n\n###### Materialised ids derived from the graph\n\n** `_system_atlas_graph_materialization` rejects an empty graph or wrong builder marker, then buckets entity ids by kind so the materialised set comes from the graph, not a declared number.** Then `_system_atlas_graph_materialization` loads `system_atlas_graph.json`. It requires non-empty entity rows and rejects an empty graph with `SELF_IGNORANCE_REAL_ATLAS_GRAPH_EMPTY`. It requires the graph to carry the `build_system_atlas.py` builder marker; a different marker raises `SELF_IGNORANCE_ATLAS_GRAPH_BUILDER_MISMATCH`. `_materialized_ids_by_kind_from_graph` then buckets the entity ids by kind using `MATERIALIZED_PREFIX_BY_KIND` (`concept_`, `mechanism_`, `pm_`, `std_`, and so on), so the materialised set is derived from the graph rather than from a declared number.\n\n###### Graph ids must resolve to real source files\n\n** `_source_validate_expected_entity_ids` requires each graph id to resolve to a real source file so the set cannot be inflated, and cross-checks the bundled slice against the live graph.** `_source_validate_expected_entity_ids` checks that graph-derived ids for concepts, mechanisms, and standards resolve to real source files through `_entity_source_exists`. An id with no file behind it raises `SELF_IGNORANCE_EXPECTED_ENTITY_ID_SOURCE_MISSING`, so the materialised set cannot be inflated with names that do not exist. `_live_system_atlas_graph_materialization` cross-checks the bundled graph slice against the live System Atlas graph when it is available; a divergence raises `SELF_IGNORANCE_LIVE_SYSTEM_ATLAS_GRAPH_MISMATCH`.\n\n###### Computing debt and rejecting tampered counts\n\n** `evaluate` takes debt as live rows minus materialised, and raises errors when a declared count disagrees with the graph, understates the floor, or claims exhaustive absence.** With those bound, `evaluate` computes the debt. For each kind it takes `known_coverage_debt_count = max(live_kind_atlas_row_count - materialised_count, 0)`. `_expected_entity_ids_source_backed` decides whether declared ids may stand in for graph ids. A declared count that disagrees with the graph-derived count raises `SELF_IGNORANCE_MATERIALIZATION_COUNT_NOT_GRAPH_DERIVED`. A declared debt below the fixture floor raises `SELF_IGNORANCE_COVERAGE_DEBT_MISMATCH`. If `absence_policy.claims_unknown_unknowns_exhaustive` is true, it raises `SELF_IGNORANCE_FORBIDDEN_ABSENCE_INFERENCE`.\n\n###### Result record and the realness rank ladder\n\n**The record reports status, per-kind debt, evidence, and scope limit, earning R4 only when every recompute and cross-check ran, else R3 or R2, with no source bodies inside.** The result record reports status, the per-kind debt rows, the graph and live-graph evidence, the realness rank, and the scope limit from `AUTHORITY_CEILING`. The rank is R4 only when the debt was recomputed from live counts, the ids are source-backed, the Kind Atlas recompute ran, the protocol result record passed, and the live graph cross-check ran or the wider repository is absent. Otherwise it is R3 when a graph was supplied and R2 when it was not. No copied source bodies enter the record.\n\nFunction Role `evaluate` Runs the full check and emits the debt vector and evidence `_projection_protocol_receipt` Binds the coverage scope and the `build_system_atlas.py --check` result record `_rows_with_live_kind_atlas_counts` Recomputes live row counts through `build_kind_atlas` `_system_atlas_graph_materialization` Loads the graph, checks the builder marker, derives materialised ids `_source_validate_expected_entity_ids` Confirms graph ids resolve to real concept, mechanism, and standard files `_live_system_atlas_graph_materialization` Cross-checks the bundled slice against the live graph `evaluate_negative_case` Replays a named negative case from clean input copies Diagram of the mechanism (7 steps). Diagram source & refs\n\nSource refs\n\nlive row counts `kind_atlas_rows.json` materialised ids by kind `system_atlas_graph.json` scope + check result record `projection_protocol.json`\n\n```text\nflowchart TD KA[\"kind_atlas_rows.json live row counts\"] Graph[\"system_atlas_graph.json materialised ids by kind\"] Proto[\"projection_protocol.json scope + check result record\"] Src[\"concept / mechanism / standard source files\"] Eval[\"evaluate()\"] Debt[\"known debt vector per kind\"] Rec[\"metadata-only result record counts, refs, scope limit\"] KA --> Eval Graph --> Eval Proto --> Eval Src --> Eval Eval --> Debt Debt --> Rec\n```\n\n###### The current bundle's debt snapshot\n\n**The current bundle shows 503 live rows against 307 materialised for a known debt of 196, all snapshot facts to rerun after any Kind Atlas, System Atlas, or source-set move.** For the current bundle the count vector is concepts 41 rows against 30 materialised (debt 11), mechanisms 36 against 28 (debt 8), paper modules 225 against 220 (debt 5), and standards 201 against 29 (debt 172). Total live rows 503, materialised 307, known debt 196. These are snapshot facts read from the bundle, not fixed constants. Rerun the check after Kind Atlas, System Atlas, or the source set moves.\n\n##### Negative cases\n\n###### Two negatives replayed from clean copies\n\n** `evaluate_negative_case` replays two named cases from fresh input, so an absence-inference claim or an understated debt is caught by the semantic evaluator even if it mislabels its own error code.** The fixture declares two named negative cases in `EXPECTED_NEGATIVE_CASES`, and `evaluate_negative_case` replays each one against a fresh copy of the input rather than trusting a declared error label. `forbidden_absence_inference` sets `claims_unknown_unknowns_exhaustive` to true and must produce `SELF_IGNORANCE_FORBIDDEN_ABSENCE_INFERENCE`. `coverage_debt_mismatch` drops one materialised entity while raising the declared debt floor and must produce `SELF_IGNORANCE_COVERAGE_DEBT_MISMATCH`. Because the replay recomputes from clean copies, a case that lies about its own error code is still caught by the semantic evaluator.\n\n###### Perturbation proves the count tracks the graph\n\n**Adding one source-backed standard entity moves known debt from 196 to 195, showing the count is coupled to the graph-derived set rather than a fixed number in prose.** A perturbation check adds one real, source-backed standard entity to the graph and confirms the known debt moves from 196 to 195 while the result still passes. That shows the count is coupled to the graph-derived materialised set, not to a fixed number in prose.\n\n##### Prior Art Grounding\n\n###### Bounded coverage measurement as the analogue\n\n**The analogue is software coverage measurement like `coverage.py` and bounded checks like OpenSSF Scorecard, which give a useful signal over a declared set without claiming a complete proof.** The nearest ordinary analogue is software coverage measurement. A coverage tool reports what was exercised or missed over a declared source set, not every possible missing behaviour. `coverage.py` is a useful reference for that bounded, observed-coverage pattern. The health-signal side is adjacent to bounded repository checks such as OpenSSF Scorecard, where a fixed set of checks yields a useful risk signal without becoming a complete proof. This component applies the same discipline to navigation coverage debt and keeps the scope limit in the same record as the count.\n\n##### Validation Result record Path\n\nFrom `microcosm-substrate`, run:\n\n```text\nPYTHONPATH=src ../repo-python -m pytest -p no:cacheprovider tests/test_self_ignorance_coverage_ledger.py -q\nPYTHONPATH=src ../repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\nPYTHONPATH=src ../repo-python scripts/build_doctrine_projection.py --check\n```\n\nFor a manual bundle run into a throwaway directory:\n\n```text\nPYTHONPATH=src ../repo-python -m microcosm_core.organs.self_ignorance_coverage_ledger run-self-ignorance-bundle --input examples/self_ignorance_coverage_ledger/exported_self_ignorance_coverage_ledger_bundle --out /tmp/microcosm-self-ignorance-coverage-ledger/bundle --card\n```\n\n###### What a pass means for the snapshot\n\n**A pass means the bundle, manifest, two negative cases, and paper-module corpus are still coherent, and it proves no freshness beyond the checked snapshot.** A pass means the declared bundle, the source-module manifest, the two negative cases, and the paper-module corpus are still coherent. It does not establish freshness beyond the checked snapshot.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The narrow strongest claim\n\n**Given the fixture, a marked graph, a live recompute, source-backed ids, and a valid result record, the component makes the coverage debt visible, recomputable, and checkable while refusing the tamper cases.** The strongest claim the evidence supports is narrow. Given the public fixture, a graph carrying the `build_system_atlas.py` marker, a live Kind Atlas recompute, source-backed entity ids, and a valid protocol result record, the component makes the declared Kind Atlas coverage debt visible, recomputable, and checkable, and it refuses the named tamper cases.\n\n###### What the scope limit excludes\n\n**The boundary is the fixture, graph-derived rows, source checks, and result records, and the ceiling excludes unknown-unknown omniscience, absence proof, source-file changes, and launch-scope decision.** The proof boundary is the public coverage fixture, the graph-derived materialised rows, the source-file checks, the two semantic negative cases, and the listed validation result records. The scope limit excludes everything past that. It is not unknown-unknown omniscience, not absence proof, not a total repository search, not source-file changes, not launch or publishing-scope decision, not private-source equivalence, and not whole-system correctness. `AUTHORITY_CEILING` records the same limit in the result record, with `literal_unknown_unknown_omniscience_authorized`, `absence_proof_authorized`, `source_mutation_authorized`, and `release_authorized` all set to false.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":15099,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/self_ignorance_coverage_ledger.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"}],"grouped_relation_feeds":[{"relation":"abides by","count":4,"links":[{"id":"AX-6","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Open-world 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fixtures without reading or mutating live host process state.","summary":"Tool Server Pressure Inventory validates the public tool-server pressure membrane: synthetic process rows, active-owner descendants, over-budget owner launch requests, redaction findings, source-module digests, metadata-only result records, negative cases, and scope limits. It treats pressure inventory rows as fixture evidence, not live process control, process signalling, host mutation, provider authority, launch-scope decision, public sharing, or whole-system correctness.","authority_ceiling":"Declared public helper-process pressure inventory fixture and source-module digest evidence only; no live process reads, process signalling, host mutation, launch-scope decision, external model access, non-public data equivalence, publishing-scope decision, or whole-system correctness.","abstract":"Tool Server Pressure Inventory validates public pressure-inventory fixtures without reading or mutating live host process state.","teleology":{"section_id":"","text":"Tool Server Pressure Inventory validates public pressure-inventory fixtures without reading or mutating live host process state.","source":"capsule_summary_line"},"body_markdown":"**Long sessions leave helper processes that slow the host, but a reaper that kills stale helpers is dangerous when an abandoned process looks like one a live session still uses.** Long agent sessions leave helper processes behind: MCP servers, dev servers, keepalives. They pile up and the host slows down. The obvious fix is a reaper that walks the process table and kills stale helpers. A reaper is also dangerous, because the hard case is telling a genuinely abandoned process apart from one a live session is still using. Kill the wrong one and you break work in flight.\n\n**It is the read-only classifier over synthetic input: it classifies each helper by kind and owner chain, marks the few safe to close, and requests launch for over-budget owners.** `tool_server_pressure_inventory` is the read-only classifier from that problem, ported to run over synthetic input. It takes a fixture that supplies a `ps` -shaped process table, a helper-kind policy, and an owner-status taxonomy. It classifies each helper by kind and by owner chain, marks the few that are safe to close, and for owners that are over their helper budget it emits a launch request rather than a kill. Every row carries a `command_hash`, not a command line.\n\n**It reads no live process table, imports no process-control module, and sends no signal, writing metadata-only records that show a reaper's safety reasoning without the harmful part.** The component reads no live process table, imports no process-control module, and sends no signal. It writes metadata-only result records so a reader can see the safety reasoning of a reaper without the part that could harm a host.\n\n##### Purpose\n\n###### The owner chain is the real decision\n\n**A loose reaper closes from a shallow signal like age or name, but the real decision is the owner chain: a `launchd` -parented helper is detached while a session-traced one is not.** A loose reaper decides \"close this\" from a shallow signal: the process is old, or its name matches a helper pattern. That is enough to close a live session's MCP server by mistake. The interesting decision is the owner chain. A helper whose parent is `launchd` (`ppid == 1`) has been detached from any session and is a candidate. A helper that still traces back through a live agent session is not.\n\n###### The narrow safe-close predicate\n\n**A row becomes a close candidate only when it is detached, its kind is allowlisted, and its idle age passes the 300-second threshold; everything else routes to owner-check or keep.** So the safe-close predicate is deliberately narrow. A row becomes a close candidate only when it is detached, its kind is on an allowlist, and its idle age is past a threshold (`DEFAULT_MIN_AGE_SECONDS`, 300 seconds). Everything else routes to `requires_owner_check` or `keep`.\n\n###### Active owners get a launch request, not a kill\n\n**When an owner is over budget it emits a launch request asking the session to launch its own lease, and an audit enforces that an active-owner descendant is never a close candidate.** The second half of the design is that active owners never get a kill. When an owner is over its helper budget the component emits a launch request: a row that asks the owning session to launch or reuse its own lease. The inventory is not a kill list. The invariant, enforced by an audit pass over the component's own output, is that an active-owner descendant can never be a close candidate.\n\n##### How it works\n\n###### One pass, two entry points\n\n**The pipeline runs in one pass; `run` validates a fixture while `run_pressure_bundle` validates an exported source-body bundle, both delegating to `_build_result`.** The pipeline runs in one pass over the fixture. `run` loads the positive and negative inputs and writes the result records. `run_pressure_bundle` is the second entrypoint: it validates an exported source-body bundle instead of a fixture. Both call `_build_result`, which orchestrates loading, classification, redaction, and the negative-case sweep.\n\n###### The classifier's decision law\n\n** `build_tool_server_pressure_inventory` parses rows, matches each command to a helper kind, walks the parent chain for owner status, and routes to keep, safe-close, or owner-check.** The classifier is `build_tool_server_pressure_inventory`. It calls `_parse_process_rows` to turn the synthetic `ps_text` into rows of pid, ppid, age, and command, then for each row calls `_process_kind` to match the command against the policy's `match_substrings`. Rows whose command matches no helper kind are dropped. For each kept row `_owner_status_for_process` walks the parent chain up to eight hops, stopping at a keep runtime, a detached parent, an active owner hint, or a cycle. `_inventory_owner_and_decision` then applies the decision law: keep kinds route to `keep`, detached and allowlisted and old-enough rows route to `candidate_safe_close`, and everything else routes to `requires_owner_check`. Over-budget active owners are summarized by `_active_owner_pressure_groups`, and `_owner_release_request_for_group` builds a `helper_owner_release_request_v1` whose `permitted_action` is `ask_owner_to_release` and whose `result` is `requested`.\n\n###### Two guards over the classifier\n\n** `_redaction_findings` rejects command previews, signal keys, and absolute paths, and `_audit_inventory_claim` re-reads the output to fail any close candidate that breaks the safety invariant.** Two guards sit on top of the classifier. `_redaction_findings` walks every emitted dict and rejects a command preview key, a truthy process-signal key, or an absolute host path. `_audit_inventory_claim` re-reads the component's own inventory and fails it if any close candidate is an active owner, is not a `ppid == 1` allowlisted detached orphan, or is younger than the age threshold, and it fails any launch request that reads as a kill. `_positive_findings` runs these over the live fixture and `_negative_findings` runs them over the eight crafted failure inputs.\n\n###### The bundle lane's manifest check\n\n** `_source_module_manifest_result` verifies each exported module's relation, recomputes the target SHA-256, confirms required anchors, and reruns the redaction guard, keeping result records metadata-only.** `_source_module_manifest_result` covers the bundle lane. It reads `source_module_manifest.json`, checks each declared module's `source_to_target_relation` against the accepted set, recomputes the target body's SHA-256 against the declared `target_sha256`, confirms the required anchors are present in the body, and reruns the redaction guard on the body text. Result records carry refs, hashes, counts, and verdicts only; `body_in_receipt` stays false. `_write_receipts` then emits the result, board, validation, and sign-off records.\n\nFunction Role `run` Validate a fixture and write the four result records `run_pressure_bundle` Validate an exported source-body bundle `_build_result` Load inputs, classify, run guards, assemble the result `build_tool_server_pressure_inventory` Classify each helper by kind, owner chain, and decision `_parse_process_rows` Parse synthetic `ps_text` into pid, ppid, age, command `_owner_status_for_process` Walk the parent chain up to eight hops for owner status `_inventory_owner_and_decision` Apply the keep / safe-close / owner-check decision law `_active_owner_pressure_groups` Group over-budget active owners into launch requests `_redaction_findings` Reject command previews, process signals, absolute paths `_audit_inventory_claim` Re-check the component's own output against the invariant `_source_module_manifest_result` Verify the exported source body by digest and anchors `_write_receipts` Write the metadata-only result, board, and validation records Diagram of the mechanism (9 steps). Diagram source & refs\n\n```text\nflowchart TD Fixture[\"Synthetic process table pressure policy, owner classes\"] Parse[\"_parse_process_rows ps rows to pid, ppid, age, cmd\"] Kind[\"_process_kind match command to helper kind\"] Owner[\"_owner_status_for_process walk parent chain up to 8 hops\"] Decision[\"_inventory_owner_and_decision keep / safe-close / owner-check\"] Groups[\"_active_owner_pressure_groups over-budget owners to launch request\"] Audit[\"_audit_inventory_claim re-check own output\"] Redact[\"_redaction_findings reject previews, paths, signals\"] Result records[\"_write_receipts metadata-only result records\"] Fixture --> Parse Parse --> Kind Kind --> Owner Owner --> Decision Decision --> Groups Decision --> Audit Groups --> Audit Audit --> Redact Redact --> Result records\n```\n\n##### Negative cases\n\n** `EXPECTED_NEGATIVE_CASES` lists eight inputs the fixture must reject, each keyed to the error code it must raise.** `EXPECTED_NEGATIVE_CASES` lists eight inputs the fixture must reject, each keyed to the error code it must raise.\n\n###### Eight rejection codes for kill-unsafe inputs\n\n**The eight inputs each raise a specific code, covering active-owner or cycle kill candidates, unknown-owner kills, premature closes, sent signals, command or path leaks, and launch overclaims.** `active_owner_kill_candidate` and `owner_chain_cycle_safe_close` both raise `TSPI_ACTIVE_OWNER_KILL_CANDIDATE`: an active-owner descendant, including one reached through a parent-chain cycle, is marked as a close candidate. `unknown_owner_kill` raises `TSPI_UNKNOWN_OWNER_KILL_FORBIDDEN` when a close candidate is not a `ppid == 1` allowlisted detached orphan. `premature_safe_close` raises `TSPI_PREMATURE_SAFE_CLOSE` when a detached orphan younger than the age threshold is marked closeable. `process_signal_sent` raises `TSPI_PROCESS_SIGNAL_FORBIDDEN`. `command_preview_leak` raises `TSPI_COMMAND_PREVIEW_FORBIDDEN` when a row carries a live command line instead of a hash. `absolute_path_leak` raises `TSPI_ABSOLUTE_PATH_FORBIDDEN`. `owner_release_overclaim` raises `TSPI_OWNER_RELEASE_OVERCLAIM` when a launch request reads as a kill, terminate, or force-close.\n\n**A run passes only when every expected case is observed, the positive fixture raises nothing, and the secret scan is clean.** A run passes only when every expected case is observed, the positive fixture raises nothing, and the secret scan is clean.\n\n##### Prior Art Grounding\n\n###### Process-inventory and owner-reference prior art\n\n**It draws on psutil process iteration, Kubernetes owner-reference garbage collection, and MCP tool servers, but stays weaker: it classifies synthetic rows and reads no live host state.** The classifier draws on established process-inventory and ownership patterns. `psutil.process_iter()` is a common way to iterate process metadata without ad hoc `ps` parsing. Kubernetes garbage collection uses owner references to separate objects that may be collected from objects a live controller still owns, which is the same detached-versus-owned distinction applied here. The Model Context Protocol gives the local \"server exposes callable tools\" shape that names these helpers. This component keeps the result weaker than any of them: it classifies synthetic rows for pressure and close eligibility, and it reads no live host state.\n- psutil process iteration: https://psutil.readthedocs.io/en/latest/#psutil.process_iter\n- Kubernetes owner-reference garbage collection: https://kubernetes.io/docs/concepts/architecture/garbage-collection/\n- Model Context Protocol tool servers: https://modelcontextprotocol.io/docs/concepts/tools\n\n##### Validation Result record Path\n\nFrom `microcosm-substrate`, with result records written under `/tmp`:\n\n###### What a pass certifies\n\n**A pass means rows stayed digest-only, the eight negative cases raised their codes, and the bundle body matched its declared digest and anchors.** A pass means synthetic classification and source-manifest shape held: rows stayed digest-only, the eight negative cases raised their codes, and the bundle body matched its declared digest and anchors.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim\n\n**Over synthetic fixtures it classifies kind, owner status, and close eligibility correctly, keeps active owners out of the close set, requests launch for over-budget owners, and rejects the eight failures.** The strongest claim this component supports is that over synthetic process-table fixtures it classifies helper kind, owner status, and close eligibility correctly, keeps active-owner descendants out of the close set, turns over-budget owners into launch requests, and rejects the eight named boundary failures. The bundle lane adds that an exported source body matched its declared digest and anchors while staying metadata-only.\n\n###### The whole proof boundary and ceiling\n\n**That is the whole boundary: the component reads no live table, sends no signal, mutates no host, and `AUTHORITY_CEILING` holds every action flag at false as projection and validation only.** That is the whole proof boundary. The component does not read a live process table, send a process signal, mutate host state, or authorize cleanup. It does not use external model services, export private account or session state, or claim whole-system correctness. `AUTHORITY_CEILING` holds `process_signal_authority`, `live_process_table_read_authorized`, `host_mutation_authorized`, `release_authorized`, and `whole_system_correctness_claim` at false. 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A named feature can read like understanding.** Interpretability writing is easy to overstate. A named feature can read like understanding. A graph picture can read like a discovered circuit. A small local script can read like access to a real model. This component exists to hold one kind of claim to a checkable size.\n\n**It answers one question. Before the system lets a circuit-attribution story stand as public evidence, does the story survive a deterministic replay instead of being taken on trust?** It answers one question. Before the system lets a circuit-attribution story stand as public evidence, does the story survive a deterministic replay instead of being taken on trust? The runtime lives in `src/microcosm_core/organs/mechanistic_interpretability_circuit_attribution_replay.py`. It reads a fixture or an exported bundle, recomputes a tiny toy transformer, checks the graph and source-body evidence, and writes result records that carry refs, digests, counts, and verdicts. They carry no model weights, no raw activations, no prompts, and no hidden reasoning.\n\n**The current fixture has six sparse features and six replay rows. A pass is bounded evidence about that fixture.** The current fixture has six sparse features and six replay rows. A pass is bounded evidence about that fixture. It is not access to any trained model.\n\n##### Purpose\n\n**The problem is that an interpretability-shaped fixture can look convincing while being wrong or stale. A feature label can lack a catalog id.** The problem is that an interpretability-shaped fixture can look convincing while being wrong or stale. A feature label can lack a catalog id. A graph can be a screenshot with no traversable edges. A row can claim transparency with no intervention result record behind it. And an attribution claim that was once true of its inputs can stay in place after the inputs change.\n\n**A loose check passes all of these. This component makes the claim smaller and forces each claim-bearing row across four gates: a replay-schema gate, a graph-traversal gate, a toy-transformer recomputation gate, and a source-body boundary gate.** A loose check passes all of these. This component makes the claim smaller and forces each claim-bearing row across four gates: a replay-schema gate, a graph-traversal gate, a toy-transformer recomputation gate, and a source-body boundary gate. A row passes only when all four hold.\n\n##### How it works\n\n**Two entry points drive the same gates. `run` validates the first-wave fixture with negative cases included.** Two entry points drive the same gates. `run` validates the first-wave fixture with negative cases included. `run_attribution_bundle` validates an exported bundle without the negative-case files, and can reuse a cached bundle result when `_fresh_bundle_receipt` confirms the cache is still coupled to the current inputs. Both call `_build_result`, which runs the checks in order and then `_write_receipts`, which writes the result, board, validation, and sign-off records.\n\n** `_build_result` loads the fixture payloads, then for each replay row it calls `_replay_policy_findings`, `_graph_analysis_for_replay`, and a feature-catalog resolution check.** `_build_result` loads the fixture payloads, then for each replay row it calls `_replay_policy_findings`, `_graph_analysis_for_replay`, and a feature-catalog resolution check. After the loop it runs `_weight_sequence_analysis` once across all rows, `_toy_transformer_attribution_runtime` once, and `scan_paths` over the inputs and copied source modules. The overall status is `pass` only when every gate passes and the secret scan passes.\n\n** `_replay_policy_findings` is the schema gate. It requires all 29 fields in `REQUIRED_REPLAY_FIELDS`, a toy prompt ref with no inline prompt body, at least one sparse feature id, and a `replacement_model_approximation_score` inside the range 0 to 1.** `_replay_policy_findings` is the schema gate. It requires all 29 fields in `REQUIRED_REPLAY_FIELDS`, a toy prompt ref with no inline prompt body, at least one sparse feature id, and a `replacement_model_approximation_score` inside the range 0 to 1. It emits `INTERPRETABILITY_INTERVENTION_RECEIPT_REQUIRED` when a row claims transparency without a `causal_intervention_receipt_ref`, and `INTERPRETABILITY_FAITHFULNESS_REQUIRES_LIMITS` when a row claims faithfulness without a `faithfulness_limit_ref`. It rejects any row whose `private_model_weights_exported`, `raw_activation_dump_exported`, `proprietary_prompt_exported`, `hidden_chain_of_thought_exported`, or `release_authorized` flag is true, and any row whose serialized text contains a private needle from `PRIVATE_NEEDLES`.\n\n** `_graph_analysis_for_replay` is the traversal gate. It reads `graph_nodes` and `graph_edges`, checks that every edge endpoint resolves to a declared node, then does a depth-first walk from the row's `sparse_feature_ids` looking for a path to a node whose kind is `public_error_node`.** `_graph_analysis_for_replay` is the traversal gate. It reads `graph_nodes` and `graph_edges`, checks that every edge endpoint resolves to a declared node, then does a depth-first walk from the row's `sparse_feature_ids` looking for a path to a node whose kind is `public_error_node`. A row with no such path gets `INTERPRETABILITY_GRAPH_PATH_REQUIRED`. `_weight_sequence_analysis` collects edge weights column by column across rows and calls `_constant_delta_sequence`; a column of four or more weights with one constant nonzero step is flagged as a decorative arithmetic sequence, not real circuit evidence.\n\n** `_toy_transformer_attribution_runtime` is the recomputation gate. It reads `token_ids`, `embeddings`, `layer1`, `layer2`, and `target_logit_index` from the fixture.** `_toy_transformer_attribution_runtime` is the recomputation gate. It reads `token_ids`, `embeddings`, `layer1`, `layer2`, and `target_logit_index` from the fixture. `_toy_transformer_forward` averages the token embeddings into a context vector, applies `layer1`, applies a `tanh` hidden activation, applies `layer2`, and reads the target logit. The runtime then computes gradient scores using the analytic `tanh` derivative `1 - h^2`, so the attribution rests on the same forward pass, and ablates each hidden feature in turn to measure its logit delta. It picks the top feature by attribution and by ablation and compares both against the fixture's declared winners. A mismatch emits `INTERPRETABILITY_TOY_TRANSFORMER_DECLARED_TOP_FEATURE_MISMATCH`. It also records a `weight_digest` so a cached or exported card can prove which fixture basis it is coupled to. If the spec came from the internal default rather than the fixture, `input_coupled_fixture` is false and `_build_result` raises `INTERPRETABILITY_TOY_TRANSFORMER_FIXTURE_SPEC_REQUIRED`.\n\n**On the current fixture the target logit is `0.044176`, and both the gradient attribution and the ablation delta select `toy_hidden_feature_1`, which matches the declaration.** On the current fixture the target logit is `0.044176`, and both the gradient attribution and the ablation delta select `toy_hidden_feature_1`, which matches the declaration. The row passes because two independent recomputed paths agree with each other and with the fixture.\n\n** `_source_module_manifest_result` is the boundary gate. For each row in the source-module manifest it checks that the classification is `copied_non_secret_macro_body`, that the material class is one of the allowed public source classes, that the body was copied without any result record body text, that the target file exists and its digest matches `target_sha256` and `source_sha256`, that `sha256_match` is true, and that every required anchor appears in the target text.** `_source_module_manifest_result` is the boundary gate. For each row in the source-module manifest it checks that the classification is `copied_non_secret_macro_body`, that the material class is one of the allowed public source classes, that the body was copied without any result record body text, that the target file exists and its digest matches `target_sha256` and `source_sha256`, that `sha256_match` is true, and that every required anchor appears in the target text. `_source_open_body_import_summary` folds the verified modules into a summary whose scope limit flags are all false. `_write_receipts` emits the records with `body_in_receipt` set to false throughout, and `result_card` omits the large payload keys listed in `CARD_OMITTED_FULL_PAYLOAD_KEYS`.\n\nDiagram of the mechanism (7 steps). Diagram source & refs\n\n```text\nflowchart TD Input[\"Fixture or exported bundle features, replay rows, toy-transformer spec\"] Schema[\"_replay_policy_findings required fields, refs, forbidden flags\"] Graph[\"_graph_analysis_for_replay feature ids to public error node\"] Weights[\"_weight_sequence_analysis reject arithmetic edge weights\"] Toy[\"_toy_transformer_attribution_runtime forward + gradient + ablation vs declared\"] Source[\"_source_module_manifest_result copied bodies, digest, anchors\"] Result[\"_write_receipts refs, digests, counts, verdicts\"] Input --> Schema Input --> Graph Input --> Weights Input --> Toy Input --> Source Schema --> Result Graph --> Result Weights --> Result Toy --> Result Source --> Result\n```\n\n##### Negative cases\n\n** `run` loads eight named negative fixtures and checks that each produces its expected error code, listed in `EXPECTED_NEGATIVE_CASES`.** `run` loads eight named negative fixtures and checks that each produces its expected error code, listed in `EXPECTED_NEGATIVE_CASES`. `private_model_weights_export` must raise `INTERPRETABILITY_PRIVATE_WEIGHTS_FORBIDDEN`. `raw_activation_dump` must raise `INTERPRETABILITY_RAW_ACTIVATION_DUMP_FORBIDDEN`. `proprietary_prompt_export` must raise `INTERPRETABILITY_PROPRIETARY_PROMPT_FORBIDDEN`. `hidden_chain_of_thought_export` must raise `INTERPRETABILITY_HIDDEN_COT_FORBIDDEN`. `unverifiable_feature_name` must raise `INTERPRETABILITY_FEATURE_NAME_UNVERIFIABLE`. `graph_screenshot_without_edges` must raise `INTERPRETABILITY_MACHINE_READABLE_EDGES_REQUIRED`. `transparency_claim_without_intervention` must raise `INTERPRETABILITY_INTERVENTION_RECEIPT_REQUIRED`. `faithfulness_claim_without_sufficiency_limit` must raise `INTERPRETABILITY_FAITHFULNESS_REQUIRES_LIMITS`. A case that fails to raise its code leaves an entry in `expected_missing`, and the overall status is then `blocked`.\n\n##### Prior Art Grounding\n\n**The accounting shape borrows from the mechanistic interpretability tradition, the study of the internal circuits of neural networks ( [Anthropic, Transformer Circuits](https://transformer-circuits.pub/)): circuits should be graph-structured, features should be identifiable, causal language should be backed by interventions, and faithfulness language should be bounded.** The accounting shape borrows from the mechanistic interpretability tradition, the study of the internal circuits of neural networks ( [Anthropic, Transformer Circuits](https://transformer-circuits.pub/)): circuits should be graph-structured, features should be identifiable, causal language should be backed by interventions, and faithfulness language should be bounded. This component does not reproduce those methods. It puts a replay boundary around an interpretability-shaped fixture: machine-readable edges instead of screenshots, causal-intervention refs instead of bare transparency language, and fixture recomputation instead of trusting a stale row. The result is fixture-bound runtime evidence, not live model access.\n\n##### Validation Result record Path\n\nRun from the `microcosm-substrate/` public root:\n\n```text\nPYTHONPATH=src python3 -m pytest tests/test_mechanistic_interpretability_circuit_attribution_replay.py -q\nPYTHONPATH=src python3 scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n**The focused test exercises the fixture and bundle: recomputation, stale-row rejection, the graph and weight gates, source-body import checks, and cached-card reuse.** The focused test exercises the fixture and bundle: recomputation, stale-row rejection, the graph and weight gates, source-body import checks, and cached-card reuse. A pass is reader-verifiable evidence only. It excludes launch, external model access, source-file changes, or whole-system correctness.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The bounded attribution-replay claim\n\n**The strongest claim the evidence supports is a public, cold-replayable circuit-attribution accounting for the named fixture and exported bundle.** The strongest claim the evidence supports is a public, cold-replayable circuit-attribution accounting for the named fixture and exported bundle. Feature ids are tied to machine-readable graph edges with a traversable path to a public error node. Transparency and faithfulness language passes only when it carries a causal-intervention result record ref and an explicit limit ref. The toy-transformer declaration is coupled to recomputed forward, gradient, and ablation results, and a stale declaration is rejected. Copied source bodies are verified by manifest and digest checks while the records stay metadata-only.\n\n###### Where the replay proof boundary stops\n\n**The proof boundary is that fixture and bundle. This component does not access a live model, does not export private weights, raw activations, proprietary prompts, or hidden reasoning, and is not a model-transparency product.** The proof boundary is that fixture and bundle. This component does not access a live model, does not export private weights, raw activations, proprietary prompts, or hidden reasoning, and is not a model-transparency product. It carries no benchmark, public sharing, hosting, or launch-scope decision. The scope limit is a public circuit-attribution runtime record and copied source-body evidence, nothing wider.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":13880,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/mechanistic_interpretability_circuit_attribution_replay.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"Anthropic, Transformer Circuits","url":"https://transformer-circuits.pub/","kind":"external_reference","section_id":"the-accounting-shape-borrows-from-the-mechanistic-interpretability-tradition"}],"grouped_relation_feeds":[{"relation":"abides by","count":4,"links":[{"id":"AX-3","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Authority by derivation, not 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these","How it works","Two entry points drive the same gates","`_build_result` loads the fixture payloads","`_replay_policy_findings` is the schema gate","`_graph_analysis_for_replay` is the traversal gate","`_toy_transformer_attribution_runtime` is the recomputation gate","On the current fixture the target logit is `0.044176`","`_source_module_manifest_result` is the boundary gate","Negative cases","`run` loads eight named negative fixtures","Prior Art Grounding","The accounting shape borrows from the mechanistic interpretability tradition","Validation Result record Path","The focused test exercises the fixture and bundle","Scope limit","The bounded attribution-replay claim","Where the replay proof boundary stops"]},{"id":"paper_module.spatial_world_model_counterfactual_simulation_replay","title":"Spatial World Model Counterfactual Simulation 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It keeps generated-video and metadata evidence below real-world geographic accuracy, robot or AV operation, simulator product claims, benchmark claims, external model access, hosting, launch, public sharing, and whole-system correctness.","authority_ceiling":"Declared public synthetic spatial counterfactual-replay metadata and source-module import evidence only; no robot or AV operation, real-world geographic accuracy, simulator product validation, generated-video authority, benchmark claims, external model access, hosting, launch-scope decision, publishing-scope decision, or whole-system correctness.","abstract":"Spatial world-model replay validates synthetic counterfactual metadata without robot operation, simulator product claims, or real-world accuracy claims.","teleology":{"section_id":"","text":"Spatial world-model replay validates synthetic counterfactual metadata without robot operation, simulator product claims, or real-world accuracy claims.","source":"capsule_summary_line"},"body_markdown":"**It asks whether a declared spatial counterfactual row binds a scene, event, and predicted outcome that survive independent recomputation, rather than a shape that merely looks right.** Spatial world-model demos are easy to oversell. A plausible video, or a row that simply asserts \"the model predicted the next state correctly\", can pass for understanding without anything being checked. This component answers one narrow question: does a declared spatial counterfactual row bind a source scene, an event, and a predicted outcome that survive an independent recomputation, or is it only a shape that looks right?\n\n###### The runtime distrusts the predicted state\n\nThe predicted actor count, transition delta, event label, and spawn cells are derived from the inputs, so a stale or hand-edited prediction stops matching and the row blocks.\n\n**The result is a metadata-only record of refs, digests, and counts over synthetic metadata, running no robot, no vehicle, and no real map.** The result is a metadata-only record: refs, digests, counts, and verdicts, with no video bodies, raw sensor payloads, or location data. It checks synthetic metadata. It does not run a robot, drive a vehicle, or measure a real map.\n\n##### Purpose\n\n###### Synthetic robotics and self-driving scenes\n\n**The input is synthetic robotics and self-driving scenes, each with an action and a declared next scene, from occluded forklifts to late-turning oncoming vehicles.** The input is a set of synthetic scenes drawn from robotics and self-driving settings: a forklift entering an aisle from behind an occlusion, a small pedestrian stepping into a crosswalk, a gust pushing a drone off a corridor, a reflective floor read as free space, a pallet stack sliding into a lane, and an oncoming vehicle accelerating late into a turn. Each scene has an action and a declared next scene.\n\n###### Scenes force state-difference accounting\n\n**A loose check would just accept the declared next scene, so the scenes exist to force state-difference accounting rather than to approximate the real world.** A loose check would read the declared next scene and accept it. That is exactly the failure the component avoids, because a declared prediction can be edited to say anything. The scenes exist to force state-difference accounting, not because the gridworld approximates the real world.\n\n##### How it works\n\n###### Fixture and bundle entry points\n\n** `run` validates the fixture with eight negative cases and `run_simulation_bundle` the exported bundle without them, both calling `_build_result` then `_write_receipts`.** Two entry points drive the same pipeline. `run` validates the fixture with the eight negative cases included. `run_simulation_bundle` validates an exported bundle with negatives excluded, and can reuse a prior result when the input digest is unchanged. Both call `_build_result`, then `_write_receipts`.\n\n###### Six inputs through the ordered check stages\n\n** `_build_result` loads six input files, checks the protocol and metadata-only policy, then runs the replay, recomputation, source-module, and private-string stages, passing only when all hold.** `_build_result` loads six input files (a protocol, a replay policy, scene states, counterfactual replays, state transitions, and a source-module manifest). It checks that the protocol selects this component and that the replay policy declares the metadata-only scope limit, tested by `_required_policy_ok`, which requires `metadata_projection_only` true and every other ceiling flag false. It runs each replay row through `_replay_policy_findings`, runs the recomputation in `_state_transition_analysis`, checks the copied source modules in `_source_module_manifest_result`, and scans every input and source path for private strings. The run passes only if all of those hold.\n\n###### Required fields and forbidden payloads per row\n\n** `_replay_policy_findings` requires each row's declared refs and labels and emits a typed error for any forbidden flag, private-video body, raw-sensor payload, or non-public paths.** `_replay_policy_findings` checks one replay row. It requires the declared fields (scene ref, action-trace ref, predicted-state ref, transition-diff ref, oracle-check ref, sensor-packet refs, limitation labels, and the payload boundary). It emits a typed error when a forbidden flag is set to true, for example `SPATIAL_PRIVATE_VIDEO_FORBIDDEN` or `SPATIAL_LIVE_OPERATION_FORBIDDEN`, and it rejects any row that carries a private-video body, a raw-sensor payload, or a non-public paths string.\n\n###### Recomputing the gridworld step\n\n** `_state_transition_analysis` builds an 8 by 8 gridworld from the actor count and steps it with a deterministic event action, computing an input-driven actor-count delta rather than copying the expected row.** `_state_transition_analysis` is the recomputation. For each replay it resolves exactly one matching transition row, builds an 8 by 8 gridworld from the source scene's actor count with `_gridworld_initial_state`, then steps it with `_gridworld_step`. The step maps the event label to a deterministic action in `EVENT_GRIDWORLD_ACTIONS`, computes an actor-count delta with `_gridworld_actor_count_delta`, and places new actors with `_gridworld_spawn_cells`. The delta is input-driven, not copied from the expected row:\n\n```text\nmin(\n  base_event_actor_count_delta\n  + max(0, sensor_packet_count - max_timestep_lag - base_event_actor_count_delta),\n  4,\n  free_cell_count\n)\n```\n\n###### Input-derived spawn cells and the comparison\n\n** `_gridworld_spawn_cells` hashes the inputs to place actors, then the analysis compares every declared field against what the gridworld produced, and one mismatch blocks the whole result.** Spawn cells are also input-derived. `_gridworld_spawn_cells` hashes the event, replay id, scene ref, topology ref, sensor-packet refs, consistency budget, limitation labels, and source actor count, then walks the bounded grid from the declared event cell. The analysis then compares the declared predicted actor count, transition delta, event label, spawn cell or cells, and the three refs against what the gridworld actually produced. Any mismatch records `SPATIAL_STATE_TRANSITION_SIMULATION_MISMATCH` on that row, and one bad row blocks the whole result.\n\n###### Copied source bodies verified by digest\n\n** `_source_module_manifest_result` verifies each copied module by digest, anchors, and line count, rejecting any manifest that places body text into a result.** `_source_module_manifest_result` handles the copied source bodies the exported bundle may carry. It verifies each module by its `sha256` digest, its required text anchors, and its line count against the file on disk. It rejects any manifest that tries to place body text into a result, emitting `SPATIAL_SOURCE_BODY_TEXT_IN_RECEIPT_FORBIDDEN` or `SPATIAL_SOURCE_MODULE_BODY_TEXT_IN_RECEIPT_FORBIDDEN`. `_source_open_body_import_summary` folds the verified modules into a metadata-only summary of refs, digests, and counts.\n\n###### Records and the compact result card\n\n** `_write_receipts` writes the result, board, validation, and sign-off records, and `result_card` projects a compact card of status, counts, and command without the bulky payload.** `_write_receipts` writes the result, board, validation, and sign-off records and returns the result with the board attached. `result_card` projects the result into a compact card that omits the bulky payload keys and keeps the status, counts, and command.\n\nDiagram of the mechanism (9 steps). Diagram source & refs\n\n```text\nflowchart TD Scene[\"Scene-state row actor count + topology\"] --> Step[\"Deterministic gridworld step 8x8 bounded recomputation\"] Replay[\"Counterfactual replay row event + sensor refs + budget\"] --> Step Step --> Actual[\"Actual next state actor delta + spawn cells\"] Replay --> Declared[\"Declared predicted state transition diff + oracle check\"] Actual --> Compare{\"Actual matches declared?\"} Declared --> Compare Compare -->|yes| Result record[\"metadata-only pass record counts + refs + digests\"] Compare -->|no| Block[\"Typed mismatch, blocked\"] Replay --> Boundary{\"Forbidden payload or claim?\"} Boundary -->|yes| Block\n```\n\n##### Negative cases\n\n###### Eight rejections each bound to a code\n\n** `EXPECTED_NEGATIVE_CASES` names eight rows that must be rejected with their codes, and a fixture run passes only when every expected code is observed.** `EXPECTED_NEGATIVE_CASES` names eight rows that must be rejected, each bound to its error code: private video export, real-world location claim, live robot or AV operation, raw sensor export, simulator-product claim, generated-video-only authority, geographic-accuracy claim, and benchmark claims without a state diff (`SPATIAL_BENCHMARK_SCORE_REQUIRES_STATE_DIFF`). A fixture run passes only when every expected code is observed. If a negative row slipped through, its code would be missing from the observed set and the run would block.\n\n##### Prior Art Grounding\n\n###### Borrowed from the world-models line\n\n**The replay borrows the counterfactual-rollout shape from the World Models line over synthetic metadata, while the gridworld here is a hand-written toy, not a learned model.** This replay exercises a spatial world model under counterfactual interventions. It draws on the world-models line of work ( [Ha and Schmidhuber, World Models](https://arxiv.org/abs/1803.10122)), where an agent learns a compressed model of its environment and rolls it forward under hypothetical actions. The component borrows the counterfactual-rollout shape over synthetic metadata. The gridworld here is a hand-written toy, not a learned model.\n\n##### Validation Result record Path\n\nRun from `microcosm-substrate`:\n\n###### What a pass over the fixture proves\n\n**A pass means the six predicted states matched the gridworld step, all eight negative codes were observed, the source modules verified by digest, and no input leaked a private string.** A pass means the six declared predicted states match the gridworld's actual step, the eight negative codes were all observed, the copied source modules verified by digest, and no input path leaked a private string. These validation result records prove the synthetic replay and source-module import boundary only.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The fixture-bound strongest claim\n\n**The strongest claim is fixture-bound: over public synthetic inputs the component recomputed each state transition and produced the metadata-only records, and recomputation is why the counts are evidence.** The strongest claim the evidence supports is fixture-bound: the component ran over public synthetic inputs, recomputed each state transition, and produced the metadata-only records and projections named above, reproducible by the commands on this page. The recomputation is why the counts are evidence rather than assertion.\n\n###### A toy gridworld, not perception or control\n\n**The boundary stops at a toy 8 by 8 gridworld of actor counts and spawn cells, modelling no perception, control, or physics, and authorizing no robot operation, simulator product, or launch.** The proof boundary stops there. It is a public metadata consistency check over a toy 8 by 8 gridworld that models actor counts and spawn cells only. It does not model perception, control, physics, sensor calibration, camera or lidar geometry, maps, vehicle dynamics, or human behavior. The scope limit, carried in `AUTHORITY_CEILING`, allows synthetic scene, action, prediction, diff, and oracle refs, source-open public sensor refs, and copied-body digests. It excludes robot or AV operation, real-world geographic accuracy, a simulator product, generated-video authority, benchmark claims, external model access, hosting, public sharing, launch, or whole-system correctness.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":11808,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/spatial_world_model_counterfactual_simulation_replay.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"Ha and Schmidhuber, World Models","url":"https://arxiv.org/abs/1803.10122","kind":"external_reference","section_id":"borrowed-from-the-world-models-line"}],"grouped_relation_feeds":[{"relation":"abides by","count":6,"links":[{"id":"AX-1","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Derivation before 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limits","url":"rules-and-ideas.html#lattice-concept-concept-research-and-science-replay-evidence-bundle"}]}],"paper_module_backrefs":[],"source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/spatial_world_model_counterfactual_simulation_replay.md"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json"}],"visible_text_byte_count":13058,"section_titles":["It asks whether a declared spatial counterfactual row binds a scene","The runtime distrusts the predicted state","The result is a metadata-only record of refs","Purpose","Synthetic robotics and self-driving scenes","Scenes force state-difference accounting","How it works","Fixture and bundle entry points","Six inputs through the ordered check stages","Required fields and forbidden payloads per row","Recomputing the gridworld step","Input-derived spawn cells and the comparison","Copied source bodies verified by digest","Records and 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live-market authority.","summary":"Prediction Oracle Reconciliation validates synthetic prediction packets through CP1 fork preservation, CP2 target-universe checks, pre-target evidence limits, oracle-diff grading, bounded dossier edits, numeric reconciliation rows, source-module imports, negative cases, and scope limits. It is a projection-mechanics replay, not a forecasting correctness claim, investment or trading decisions, live market data call, external model access, non-public data equivalence, public sharing, launch, or whole-system correctness.","authority_ceiling":"Synthetic invented prediction packet and source-module import evidence only; no forecasting correctness or accuracy, no trading, financial, or investment-related actions, no live market data, no external model access, no prediction public sharing, no performance track record, no non-public data import, no launch-scope decision, no publishing-scope decision, and no whole-system correctness.","abstract":"Prediction Oracle Reconciliation exercises synthetic forecast reconciliation gates without forecasting, trading, provider, or live-market authority.","teleology":{"section_id":"","text":"Prediction Oracle Reconciliation exercises synthetic forecast reconciliation gates without forecasting, trading, provider, or live-market authority.","source":"capsule_summary_line"},"body_markdown":"** `prediction_oracle_reconciliation` audits one synthetic forecast packet on invented fixture data, with no live feed, external model access, trading, or performance claim.** `prediction_oracle_reconciliation` takes one synthetic forecast packet and checks whether its reasoning would survive an honest audit. It runs entirely on invented fixture data. There is no live market feed, no external model access, no trading, and no performance claim.\n\n**It answers one narrow question about whether the packet keeps its evidence honest and grading recomputable, emitting metadata-only result records of counts, verdicts, and rejected cases.** The component answers one narrow question: does this prediction packet keep its evidence honest and its grading recomputable, or does it cut a corner? It emits a set of result records that name the counts, verdicts, and rejected cases. The packet body, the claimed numbers, and any copied source bodies stay out of those records.\n\n**The component distrusts the packet's own numbers, recomputing error and direction from raw prices and rejecting any claimed value that disagrees.** The unusual choice is that the component does not trust the numbers the packet reports. For every graded row it recomputes the error and the direction from the raw prices, then rejects any claimed value that disagrees with the recompute.\n\n##### Purpose\n\n###### Two quiet ways a forecast is wrong\n\n**A forecast can hide a large numeric miss behind a correct direction or lean on evidence that arrived after the outcome, and this component surfaces both on a synthetic packet.** A forecast can be wrong in two quiet ways. It can call the direction right and still be far off on the number, so a correct arrow hides a large miss. And it can look accurate only because it used evidence that arrived after the outcome it was meant to predict. A loose checker that reads the packet's own scores back to it catches neither. This component exists to make both failures visible on a synthetic packet before any reasoning is dressed up as a track record.\n\n###### Every value is a fixture\n\n**Prices, targets, branches, and realized values are all fixtures, so any direction hit or numeric miss speaks only about the fixture and the grading mechanics.** The prices, targets, branches, and realized values are all fixtures. A direction hit or a numeric miss inside a result record is a statement about the fixture and the grading mechanics, nothing more.\n\n##### How it works\n\n###### Entry points and `_build_result`\n\n** `run` and `run_prediction_bundle` both call `_build_result`, which loads the packet, secret-scans the inputs, validates the main packet and each negative case, and folds the parts into one status.** The entry points are `run` for the first-wave fixture and `run_prediction_bundle` for the exported bundle. Both call `_build_result`, which loads the packet, runs a secret scan over the inputs through `scan_paths`, validates the main packet, validates each negative-case input, and folds the parts into one status.\n\n###### Density and authority checks in `validate_reconciliation_packet`\n\n** `validate_reconciliation_packet` checks packet density, rejects any forbidden authority flag through `_authority_overclaim`, then runs and folds five gates.** `validate_reconciliation_packet` is the core. It first checks packet density: at least five pattern ids, four source refs, one projection result record, and three public runtime refs. It calls `_authority_overclaim`, which raises `PREDICTION_TRADING_OR_ADVICE_OVERCLAIM` if any flag in `FORBIDDEN_AUTHORITY_FLAGS` (trading, financial or investment-related actions, live market data, external model access, public sharing, launch, non-public data equivalence) is set true. Then it runs five gates and folds their findings.\n\n###### CP1 branches must retain the losing side\n\n** `_validate_cp1` requires every branch to name its chosen side, carry rationale, and keep the record of why the opposite side lost, with equity-lane branches also needing a confirmation bit.** `_validate_cp1` reads `cp1_branches`. Every branch must name the side it chose, carry rationale refs, and keep an `opposite_side_invalidation_ref`, the record of why the losing side lost. A branch that asserts a winner without retaining the discarded alternative raises `PREDICTION_CP1_BIFURCATION_UNRESOLVED`. An equity, market, or finance lane branch also needs an explicit confirmation bit, or it raises `PREDICTION_EQUITY_CONFIRMATION_REQUIRED`.\n\n###### CP2 fixes the universe and forbids post-target evidence\n\n** `_validate_cp2` requires each prediction's target inside the declared universe and every evidence ref to be pre-target, the gate that stops a packet from grading itself with hindsight.** `_validate_cp2` reads `cp2_predictions`. Each prediction's `target_id` must sit inside the declared target universe, so the set of things being predicted is fixed before the outcome; a target outside it raises `PREDICTION_CP2_TARGET_OUT_OF_UNIVERSE`. Every evidence ref must be pre-target: `_evidence_is_pre_t` accepts a ref only if it carries a `T-` time prefix, and a ref that points past the target window raises `PREDICTION_ORACLE_POST_T_EVIDENCE_FORBIDDEN`. This is the gate that stops a packet from grading itself with hindsight.\n\n###### Recomputing every graded row from raw prices\n\n**The numeric functions recompute error and direction from raw prices, contradict any mismatched claim, surface large misses, refuse to grade missing or degraded rows, and preserve the STOCK/ETF split.** `_numeric_grading_inputs` and `_numeric_reconciliation_rows` do the arithmetic. For each graded oracle-diff row the component recomputes the absolute delta as the distance between predicted and realized price, the percent delta against the snapshot price, and the direction hit as predicted direction against realized direction. If the row also reports its own `abs_error`, `pred_error_pct`, or `direction_hit`, the claimed value must match the recompute, or the row raises a contradiction finding. A row with a correct direction whose error clears a floor (ten in absolute terms or five percent) is surfaced as `PREDICTION_ORACLE_NUMERIC_LARGE_MISS_SURFACED`. A row with no realized price is not fabricated into a graded row; it raises `PREDICTION_ORACLE_MISSING_REALIZED_TRUTH_NOT_GRADED`. A row marked degraded is gated out of grading rather than scored. STOCK and ETF rows are counted separately, and a run that collapses the split raises `PREDICTION_ORACLE_ASSET_CLASS_SPLIT_PRESERVED`.\n\n###### Dossier edits held to bounded deltas\n\n** `_validate_mutations` allows an edit only to add a contradiction, revise confidence, or retire a claim against a known prediction, with high-severity mutations needing two refs and an allowlist.** `_validate_mutations` reads `dossier_mutations`. An edit to the running record may only add a contradiction, revise a confidence band, or retire a claim, and it must target a known prediction. A high-severity mutation needs two evidence refs and an explicit public-delta allowlist, or it raises `PREDICTION_DOSSIER_MUTATION_UNSAFE`.\n\n###### The count floor for a pass\n\n**A pass requires no positive finding and a minimum census of CP1 branches, CP2 predictions, graded rows, STOCK and ETF numeric rows, and one bounded mutation.** The packet passes only when no positive finding fires and the counts hold: at least two CP1 branches, two CP2 predictions, two graded oracle-diff rows, two numeric rows spanning both STOCK and ETF, and one bounded mutation.\n\n###### The bundle path verifies copied source bodies\n\n**The bundle path adds `validate_source_module_imports` to confirm each copied source body is metadata-only and digest-matched, then writes the result record files and summary card.** The bundle path adds `validate_source_module_imports`, which reads `source_module_manifest.json`, confirms each copied source body declares `copied_non_secret_macro_body`, keeps its body text out of result records, states an exact-copy or source-faithful relation, and matches the recorded sha256 through `_sha256`. `_source_open_body_import_summary` reduces that to a metadata-only summary of digests and ids. `write_receipts` then writes the result record, board, validation, and sign-off files, and `result_card` builds the command-line summary card.\n\n###### The two constant tables that fix the boundary\n\n** `EXPECTED_NEGATIVE_CASES` maps each negative input to its error code so a silenced gate is caught, and `AUTHORITY_CEILING` holds every authority flag false in every record.** The two constant tables fix the boundary. `EXPECTED_NEGATIVE_CASES` maps each named negative input to the error code it must produce, so a gate that stops firing is caught as a missing case. `AUTHORITY_CEILING` holds every authority flag at false and is copied into every result record.\n\nDiagram of the mechanism (6 steps). Diagram source & refs\n\n```text\nflowchart TD Packet[\"Synthetic prediction packet targets, CP1 branches, CP2 predictions, oracle diff, numeric rows, mutations\"] CP1[\"validate_cp1 chosen side + why the opposite side lost; equity lane needs confirmation\"] CP2[\"validate_cp2 target inside universe; evidence must be pre-target (T-)\"] Numeric[\"numeric grading recompute abs error, percent error, direction; claimed values must match\"] Mutation[\"validate_mutations bounded dossier deltas only\"] Result[\"build_result counts, verdicts, authority booleans; packet body stays out\"] Packet --> CP1 Packet --> CP2 Packet --> Numeric Packet --> Mutation CP1 --> Result CP2 --> Result Numeric --> Result Mutation --> Result\n```\n\n##### Negative cases\n\n###### Ten inputs the component must catch\n\n** `EXPECTED_NEGATIVE_CASES` names ten inputs to reject or surface, each mapped to an error code, and `_build_result` fails if any expected case does not appear.** `EXPECTED_NEGATIVE_CASES` names ten inputs the component must reject or surface: a CP2 target outside the universe, an unresolved CP1 bifurcation, post-target evidence, an unconfirmed equity-lane claim, an unsafe high-severity dossier mutation, a trading or advice overclaim, a direction hit that is still a large numeric miss, a row missing realized truth, a degraded feed row, and a run that drops the STOCK and ETF split. Each input maps to a specific error code, and `_build_result` fails if any expected case does not appear.\n\n##### Prior Art Grounding\n\n###### Forecast scoring and market-scoring lineage\n\n**The component draws on probabilistic forecast evaluation and prediction-market scoring, borrowing the reconciliation pattern of declaring the universe up front, separating pre- and post-target evidence, and bounding dossier edits.** This component draws on probabilistic forecast evaluation and prediction-market infrastructure. The [Brier score](https://journals.ametsoc.org/doi/10.1175/1520-0493%281950%29078%3C0001%3AVOFEIT%3E2.0.CO%3B2) is an early probability-forecast verification anchor, proper-scoring-rule work such as [Gneiting and Raftery](https://sites.stat.washington.edu/people/raftery/Research/PDF/Gneiting2007jasa.pdf) motivates incentive-compatible scoring, and Hanson's [logarithmic market scoring rule](https://hanson.gmu.edu/mktscore.pdf) grounds the idea that forecasts are updated and evaluated through explicit scoring. Forecasting-tournament work on calibration also motivates keeping prediction evidence separate from post-outcome explanation. What this component borrows is the reconciliation pattern: declare the target universe before the outcome, keep pre-target evidence apart from post-target evidence, grade against a synthetic oracle, and hold dossier edits to declared deltas.\n\n##### Validation Result record Path\n\nRun from `microcosm-substrate`:\n\n```text\nPYTHONPATH=src ../repo-python -m microcosm_core.organs.prediction_oracle_reconciliation run \\\n  --input fixtures/first_wave/prediction_oracle_reconciliation/input \\\n  --out /tmp/microcosm-prediction-oracle-reconciliation/fixture \\\n  --card\nPYTHONPATH=src ../repo-python -m microcosm_core.organs.prediction_oracle_reconciliation run-prediction-bundle \\\n  --input examples/prediction_oracle_reconciliation/exported_prediction_oracle_bundle \\\n  --out /tmp/microcosm-prediction-oracle-reconciliation/bundle \\\n  --card\nPYTHONPATH=src ../repo-python -m pytest -p no:cacheprovider tests/test_prediction_oracle_reconciliation.py -q\nPYTHONPATH=src ../repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n###### What a passing run means\n\n**A passing run means the synthetic packet cleared every gate, all ten negative cases fired, the scan was clean, and the digests matched, proving nothing about a real forecast.** A passing run means the synthetic packet cleared every gate, all ten negative cases fired, the secret scan was clean, and the bundle digests matched. It proves nothing about a real forecast.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest claim the evidence supports\n\n**On a synthetic fixture, the component recomputes grading, refuses post-target evidence, requires resolved CP1 branches inside the universe, bounds dossier edits, and rejects every forbidden authority flag.** The strongest claim the evidence supports is narrow. On a synthetic fixture, the component recomputes numeric grading rather than trusting the packet, refuses post-target evidence, requires resolved CP1 branches inside a declared target universe, holds dossier edits to bounded deltas, and rejects any attempt to raise a trading, advice, provider, live-market, public sharing, or launch flag. That is the proof boundary: the fixture, its gates, and its result records.\n\n###### What a green result is not\n\n**A green result makes no forecast, gives no advice, calls no provider, imports no non-public data, and claims no track record, with `AUTHORITY_CEILING` holding every flag false.** It refuses everything past that. It makes no forecast, gives no financial or investment-related actions, calls no live provider, imports no non-public data, and claims no track record. A green result is not launch-scope decision and not whole-system correctness. The scope limit in `AUTHORITY_CEILING` keeps every such flag false, and the validation result records under Validation Result record Path are the only evidence this module offers.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":14292,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/prediction_oracle_reconciliation.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"Brier score","url":"https://journals.ametsoc.org/doi/10.1175/1520-0493%281950%29078%3C0001%3AVOFEIT%3E2.0.CO%3B2","kind":"external_reference","section_id":"forecast-scoring-and-market-scoring-lineage"},{"label":"Gneiting and 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It emits context metadata and verdicts only, without provider/API authorization, Lean/Lake execution, proof or oracle truth-side material, formal-result correctness, domain-level conclusions, launch, public sharing, or whole-system correctness.","authority_ceiling":"Context-budget projection fixture and source-body import evidence only; no provider or API call authorization, no Lean or Lake execution, no proof or oracle truth-side material, no theorem or domain-level conclusions, no launch-scope decision, no publishing-scope decision, and no whole-system correctness.","abstract":"Provider Context Recipe Budget validates context-budget projection mechanics without authorizing external model access or truth-side material.","teleology":{"section_id":"","text":"Provider Context Recipe Budget validates context-budget projection mechanics without authorizing external model access or truth-side material.","source":"capsule_summary_line"},"body_markdown":"###### Six fixed context-budget recipes\n\n** `provider_context_recipe_budget_policy` measures the byte cost of an agent's assembled context across six fixed recipes, filling each in order until the budget runs out and naming what did not fit, without calling any model.** `provider_context_recipe_budget_policy` checks the size accounting of the context an agent would assemble before it calls an outside model. It does not call one. It takes six fixed recipe shapes, measures how many bytes each section costs, fills each recipe in declared order until the budget runs out, and names any section that did not fit.\n\n###### Context windows as allocated budgets\n\n**The defended idea is that a context window is a budget to spend rather than a dump, so an over-budget recipe must name the sections it dropped instead of silently truncating.** The idea it defends is short: a context window is a budget to spend, not a place to dump everything retrieved. The board records this as `context_is_budget_not_dump`. A recipe that goes over budget must say which sections it dropped, not silently truncate them.\n\n**The output is context-shape metadata per recipe: the byte ceiling, admitted and omitted section ids, the allowed deliverable route, and authority claims that stay false, with no model call and no answer.** The output is metadata about context shape. For each recipe it reports the byte ceiling, which section ids were admitted, which were omitted, the deliverable route the recipe is allowed to produce, and a set of authority claims that stay false. No model is called and no answer is produced.\n\n##### Purpose\n\n###### Two failure modes of a loose context builder\n\n**A loose context builder can overflow a byte limit into arbitrary truncation or quietly carry answer-side material into a prompt, and this component makes both visible as plain accounting.** A loose context builder has two failure modes. It can overflow a real byte limit and get truncated at an arbitrary point, and it can quietly carry answer-side material into a prompt: a proof body, an oracle-only premise list, an ideal answer, a provider output body. This component makes both failures visible as plain accounting before any model call is in scope.\n\n###### Six recipes pinned to fixed budgets and routes\n\n**The six recipes are a closed set in `EXPECTED_RECIPE_BUDGETS`, each pinned to one deliverable route in `EXPECTED_DELIVERABLES` so it cannot change the kind of output it claims.** The six recipes are a closed set, listed in `EXPECTED_RECIPE_BUDGETS`: `minimal_4kb` at 4096 bytes, `premise_16kb` at 16384, `skill_32kb` and `repair_32kb` at 32768, `fewshot_64kb` at 65536, and `strategy_classification_4kb` at 4096. Each recipe is pinned to one deliverable route in `EXPECTED_DELIVERABLES`, so a recipe cannot change what kind of output it claims to produce.\n\n###### Byte sizes measured, not asserted\n\n**Byte sizes are measured by running the copied benchmark harness's real `_provider_context_pack` when present, falling back to declared sizes and recording that in `real_section_body_status` rather than guessing.** The byte sizes are measured, not asserted. When the copied benchmark harness is present the validator imports it and runs its real `_provider_context_pack` over each recipe, then measures the packed section bodies. If the harness cannot load, the component falls back to the declared byte sizes and says so in `real_section_body_status` rather than guessing.\n\n##### How it works\n\n###### Two entrypoints share `_build_result`\n\n** `run` loads the fixture with seven negative cases and `run_budget_bundle` the exported shape without them, both calling `_build_result` to load recipes, sections, copied bodies, and the scan first.** `run` and `run_budget_bundle` are the two entrypoints. `run` loads the fixture inputs with the seven negative cases enabled. `run_budget_bundle` loads the exported bundle shape without them. Both call `_build_result`, which loads the recipe rows, section rows, copied source-module bodies, and the non-public-state scan before it builds anything.\n\n###### Deterministic ordered fill in `_recipe_projection`\n\n** `_recipe_projection` walks a recipe's ordered sections, admitting each only while the running total stays within `byte_budget` and omitting the first over-budget section and every one after it.** Recipe projection is deterministic. `_recipe_projection` walks a recipe's ordered section ids, measures each section with `_byte_size` and `_section_accounting`, and admits a section only while the running total stays at or under the recipe's `byte_budget`. The first section that would push the total over budget goes to the omitted list instead, and so does every section after it. The projection returns the included and omitted section ids, the included byte count, an approximate token count of `ceil(bytes / 4)`, the graph role, and the deliverable type.\n\n**The validator then runs three independent checks and turns each failure into a structured finding rather than a silent pass.** The validator then runs three independent checks and turns each failure into a structured finding rather than a silent pass.\n\nDiagram of the mechanism (7 steps). Diagram source & refs\n\n```text\nflowchart TD Inputs[\"recipe rows + section rows + copied source bodies\"] --> Project[\"_recipe_projection ordered byte-budget fill\"] Project --> RecipeCheck[\"_recipe_findings budget, route, manifest\"] Inputs --> SectionCheck[\"_section_findings source ref + anchor backing\"] Inputs --> SourceCheck[\"_source_module_findings digest + anchor of copied bodies\"] RecipeCheck --> Result[\"_build_result status + context packets\"] SectionCheck --> Result SourceCheck --> Result Result --> Result records[\"_write_receipts metadata-only result records\"]\n```\n\n###### What `_recipe_findings` rejects\n\n** `_recipe_findings` rejects a mismatched or oversized budget, a forbidden truth-side section, a proof-body or provider-call authorization, a drifted deliverable route, and an over-budget projection with no omitted-sections manifest.** `_recipe_findings` rejects a recipe whose budget does not match the pinned value or exceeds 65536, a recipe that lists a forbidden truth-side section id, a recipe that sets `proof_bodies_allowed` or carries a forbidden body field, a recipe that sets `provider_calls_authorized`, a recipe whose `deliverable_type` drifts from its route, and an over-budget projection with no omitted-sections manifest.\n\n###### Section refs and anchors in `_section_findings`\n\n** `_section_findings` requires each public section to cite an allowed source ref and anchor that truly appears in a copied body, rejecting forbidden ids, forbidden body keys, outside refs, and synthetic material.** `_section_findings` requires each public section to cite a source ref from the allowed set and at least one source anchor, then confirms the anchor text actually appears in one of the cited copied bodies. It rejects a section id in the forbidden set, a section carrying a forbidden body key, a section citing a ref outside the imported bodies, and a section marked synthetic, placeholder, fabricated, or fixture-only by `_synthetic_section_material_marker`.\n\n###### Digest and anchor checks over eight copied bodies\n\n** `_source_module_findings` covers eight copied bodies, checking for each that the file exists, the manifest and source digests match the target, and every required anchor string is present.** `_source_module_findings` covers the eight copied bodies: the graph benchmark harness, the formal-ladder eval, the result record reducer, the set calibration report, and the compute-provider, provider-adapter, transform-job, and navigation-transform-result record standards. For each it checks that the row declares a copied body kept out of result records, that the target file exists, that the manifest digest matches the target and the source digest matches the target, and that every required anchor string is present.\n\n###### metadata-only result records and the compact card\n\n** `_write_receipts` writes metadata-only records in both modes and `result_card` emits a compact card, keeping counts, ids, digests, routes, and verdicts but never proof bodies or model answers.** `_write_receipts` keeps the output metadata-only. Fixture mode writes a result, a board, a validation record, and an sign-off record. Bundle mode writes one exported-bundle validation record. `result_card` emits a compact card and drops the context packets, source refs, result record paths, private-scan hits, and the scope boundary payload. The records keep counts, ids, digests, routes, and verdicts. They never keep proof bodies or model answers.\n\nThe recipe and its route:\n\nRecipe Budget Deliverable route `minimal_4kb` 4 KiB `environment_metadata` `premise_16kb` 16 KiB `ranked_premise_ids` `skill_32kb` 32 KiB `strategy_metadata` `repair_32kb` 32 KiB `failure_classification` `fewshot_64kb` 64 KiB `redacted_synthesis_advisory` `strategy_classification_4kb` 4 KiB `strategy_id_classification`\n\n##### Negative cases\n\n###### Seven cases the validator must reject\n\n**Seven fixture cases must each be rejected, every one keyed to an expected error code in `EXPECTED_NEGATIVE_CASES`.** The fixture carries seven cases the validator must reject, each keyed to an expected error code in `EXPECTED_NEGATIVE_CASES`.\n- `budget_overflow_recipe` rejects a recipe above its public byte ceiling.\n- `truth_side_section` rejects an oracle-only section id.\n- `proof_body_leakage` rejects proof or model body fields.\n- `provider_call_authorized` rejects a fixture that authorizes a model call.\n- `deliverable_type_route_mismatch` rejects a recipe whose route changed.\n- `omitted_sections_suppressed` rejects over-budget context with no omitted-sections manifest.\n- `synthetic_section_materials` rejects section material with no allowed source ref or anchor, or marked synthetic.\n\n###### A missing case blocks the result\n\n**A pass requires all seven cases observed and no floor check fired, and a missing case blocks the result.** A pass means all seven were observed and no floor check fired. A missing case blocks the result.\n\n##### Prior Art Grounding\n\n###### Retrieval-augmented generation and CLI restraint\n\n**The recipe budget sits in the retrieval-augmented-generation lineage, narrowed to metadata that budgets, orders, and drops sections, and borrows the CLI principle of saying enough but not too much.** The recipe budget sits in the lineage of retrieval-augmented generation and context packing. Lewis et al.'s [Retrieval-Augmented Generation](https://arxiv.org/abs/2005.11401) is the research anchor for conditioning generation on retrieved supporting material rather than model parameters alone. This component narrows that idea to metadata: retrieved sections are budgeted, ordered, and dropped explicitly before any model call is in scope. The command-facing style borrows the [Command Line Interface Guidelines](https://clig.dev/) principle of saying enough but not too much, expressed here as fixed byte ceilings and explicit omitted-section manifests.\n\n##### Validation Result record Path\n\nRun from `microcosm-substrate`:\n\n```text\nPYTHONPATH=src ../repo-python -m microcosm_core.organs.provider_context_recipe_budget_policy run \\\n  --input fixtures/first_wave/provider_context_recipe_budget_policy/input \\\n  --out /tmp/microcosm-provider-context-recipe-budget-policy/fixture \\\n  --card\nPYTHONPATH=src ../repo-python -m microcosm_core.organs.provider_context_recipe_budget_policy run-budget-bundle \\\n  --input examples/provider_context_recipe_budget_policy/exported_provider_context_budget_bundle \\\n  --out /tmp/microcosm-provider-context-recipe-budget-policy/bundle \\\n  --card\nPYTHONPATH=src ../repo-python -m pytest -p no:cacheprovider tests/test_provider_context_recipe_budget_policy.py -q\nPYTHONPATH=src ../repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n###### What a green result proves\n\n**A green result proves only public context-recipe metadata, with fixture mode observing every negative case and writing four records and bundle mode validating the exported shape into one metadata-only record.** A green result proves only public context-recipe metadata: byte ceilings, ordered admission, omitted sections, deliverable routes, the digest and anchor checks over the copied bodies, and the seven negative cases. Fixture mode must observe every negative case and write the four records. Bundle mode must validate the exported shape and write one metadata-only record.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest claim supported\n\n**The strongest claim is that a closed set of recipes stays inside its byte ceilings, fills in order, names what it drops, and carries no answer-side material, all as accounting before any model call.** The strongest claim this component supports is that a small, closed set of context recipes stays inside its declared byte ceilings, fills sections in declared order, names what it drops, and carries no answer-side material, all visible as accounting before any model call. The proof boundary is exactly that metadata plus the copied bodies it checks by digest and anchor.\n\n###### What the boundary refuses\n\n**The boundary calls no model, runs no Lean or Lake, proves no theorem, and exports no proof or answer body, excluding launch, public sharing, and any treatment of context metadata as proof authority.** It refuses everything past that boundary. It does not call a model, run Lean or Lake, prove a theorem, export a proof body, or reveal oracle-only truth-side material. Its scope limit excludes model answer bodies, launch-scope decision, publishing-scope decision, and treating context metadata as proof authority. 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fixed context-budget recipes","Context windows as allocated budgets","The output is context-shape metadata per recipe","Purpose","Two failure modes of a loose context builder","Six recipes pinned to fixed budgets and routes","Byte sizes measured, not asserted","How it works","Two entrypoints share `_build_result`","Deterministic ordered fill in `_recipe_projection`","The validator then runs three independent checks","What `_recipe_findings` rejects","Section refs and anchors in `_section_findings`","Digest and anchor checks over eight copied bodies","metadata-only result records and the compact card","Negative cases","Seven cases the validator must reject","A missing case blocks the result","Prior Art Grounding","Retrieval-augmented generation and CLI restraint","Validation Result record Path","What a green result proves","Scope limit","The strongest claim supported","What the boundary refuses"]},{"id":"paper_module.undeclared_library_prior_classifier","title":"Undeclared Library Prior Classifier","relative_href":"docs/paper-module-undeclared-library-prior-classifier.html","read_href":"docs/paper-module-undeclared-library-prior-classifier.html","context_href":"docs/rules-and-ideas.html#paper-module-undeclared-library-prior-classifier","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"Undeclared Library Prior Classifier scores extracted Lean symbol observations against an allowed premise set without running Lean or treating libraries as implicit allowlists.","summary":"Undeclared Library Prior Classifier validates copied Lean/Std premise rows and pre-extracted symbol observations, classifying undeclared library priors and premise-budget violations with route outcomes, source-module digest checks, secret-exclusion scans, negative cases, and scope limits. It does not read proof source, run Lean or Lake, prove formal-result correctness, treat the whole standard library as an implicit allowlist, claim Mathlib availability, use external model services, launch, publish, or prove whole-system correctness.","authority_ceiling":"Symbol-boundary classification fixture over copied Lean/Std premise rows and pre-extracted symbol observations only; no proof-source reads, Lean or Lake execution, formal-result correctness, whole-library implicit allowlist, Mathlib availability, external model access, launch-scope decision, publishing-scope decision, or whole-system correctness.","abstract":"Undeclared Library Prior Classifier scores extracted Lean symbol observations against an allowed premise set without running Lean or treating libraries as implicit allowlists.","teleology":{"section_id":"","text":"Undeclared Library Prior Classifier scores extracted Lean symbol observations against an allowed premise set without running Lean or treating libraries as implicit allowlists.","source":"capsule_summary_line"},"body_markdown":"###### Declared versus undeclared library priors\n\n**It answers one question about a Lean-accepted proof: did the candidate cite a known library symbol outside the premises its recipe allowed?** A theorem prover can return a proof that Lean accepts and still break the rules of the evaluation it ran under. The usual way this happens is quiet: the proof reaches for a library lemma the recipe never put on the table. The symbol is real and the proof is sound, but the run used a fact it was not allowed to assume. This component answers one question about that gap. Given the premises a candidate was allowed to use and the qualified symbols it actually reached for, did it cite a known library symbol outside the allowed set?\n\n**It works only from a copied premise index and pre-extracted observations, never running Lean or reading the proof, with the allowed set closed by construction from premise rows.** It works only from a copied premise index and a list of symbol observations that were extracted beforehand. It does not run Lean, it does not read the proof body, and it does not treat the standard library as an implicit allowlist. The allowed set is closed by construction: a symbol is admissible because a premise row names it, never because it happens to exist in Lean's standard library.\n\n**The public fixture's 11 premises and three observations classify to one undeclared-prior escalation, one budget-violation retry, and one advisory none, each named with its rule and class.** The public fixture carries 11 premise rows across four namespaces (`Bool`, `Iff`, `List`, `Nat`) and three symbol observations. Running it classifies one observation as `UNDECLARED_LIBRARY_PRIOR` routed to `bridge_escalate`, one as `PREMISE_BUDGET_VIOLATION` routed to `retry`, and one as an advisory `NONE`. The written records name each symbol, the premise rule it came from, and the class that was computed.\n\n##### Purpose\n\n###### Compiling does not mean in-recipe\n\n**A checked proof is not a proof that stayed inside the allowed building blocks, and a loose check cannot tell an in-recipe proof from one that borrowed an out-of-recipe lemma.** The problem is that \"the proof checked\" does not mean \"the proof stayed inside the allowed building blocks.\" A loose check that trusts the prover's sign-off, or that admits any symbol found in the standard library, cannot tell an in-recipe proof from one that quietly borrowed an out-of-recipe lemma. Both compile. Both look sound.\n\n###### Budget breach kept apart from residual breach\n\n**An explicit budget breach from a disallowed premise id is kept distinct from a residual breach using an undeclared known symbol, with the budget case taking precedence.** The component also keeps two failure modes apart. An explicit budget breach, where the candidate names a premise id the recipe did not allow, is not the same as a residual breach, where the candidate used a known symbol that turns out to be undeclared. The first is settled directly from the cited ids and takes precedence. The second is what the symbol comparison is for. Folding both into one class would either over-escalate honest retries or let genuine out-of-recipe library use pass as a budget note.\n\n##### How it works\n\n###### Two entry points into `_build_result`\n\n** `run` builds the fixture result with negatives included, and `run_symbol_bundle` runs the same builder over an exported source-body bundle and can reuse a fresh prior record.** The fixture entrypoint is `run`. It calls `_build_result` with the fixture input mode and negative cases included, then `_write_receipts`. The exported-bundle entrypoint is `run_symbol_bundle`, which runs the same builder over an exported source-body bundle and can reuse a fresh prior record. Inside `_build_result` the stages run in this order.\n\n###### Loading inputs and the secret scan\n\n** `_load_payloads` reads the four inputs plus seven negative-case files, and `scan_paths` runs a secret-exclusion pass before the four validators run.** `_load_payloads` reads the four input files (`projection_protocol.json`, `premise_index.json`, `symbol_observations.json`, `classifier_policy.json`) plus the seven negative-case files. `scan_paths` runs a secret-exclusion pass over those inputs and records a blocking hit count. Then four validators run.\n\n###### Validating projection, premises, and policy\n\n**Three validators check the projection's refs and digests, require every premise row to carry an id, qualified name, and source ref, and pin the symbol regex and review outcomes.** `validate_projection_protocol` checks that the projection cites real source refs, source pattern ids, target refs, and source digests, and that any omitted proof or provider material carries an omission record. `validate_premise_index` reads the premise rows, requires each to have a premise id, a qualified `theorem_or_def_name`, and a public source ref, and reports the premise and namespace counts. `validate_classifier_policy` requires the declared qualified-symbol pattern to equal `QUALIFIED_SYMBOL_RE` (the regex over the `Nat`, `List`, `Bool`, `Iff`, and `Eq` namespaces), the undeclared review outcome to be `bridge_escalate`, the budget review outcome to be `retry`, and it rejects any policy that sets `proof_bodies_allowed` true.\n\n###### `_classify_row` computes the class with precedence\n\n** `_classify_row` maps cited premises to symbols, finds the known ones outside the allowed and cited sets, and applies budget-over-residual precedence, flagging any drift from the asserted class.** `validate_symbol_observations` is the classifier. For each observation it calls `_classify_row`, which is where the rule lives. `_classify_row` reads `allowed_premise_ids` and `cited_unallowed_premise_ids`, maps them through the premise index to their symbols, extracts the observed qualified refs, and keeps the known ones (those a premise row names). The undeclared set is the known symbols that are neither allowed nor already cited as unallowed. Precedence is explicit: if `cited_unallowed_premise_ids` is present the row is `PREMISE_BUDGET_VIOLATION` with `retry`; otherwise a non-empty undeclared set is `UNDECLARED_LIBRARY_PRIOR` with `bridge_escalate`; otherwise the row is `NONE` and advisory. `_classify_row` then compares the observation's own asserted class against the recomputed one and records a finding on any mismatch, so the public assertion cannot drift from the premise-index computation.\n\n###### The exported bundle's provenance check\n\n**For the exported bundle `validate_source_module_manifest` rejects in-result record bodies and verifies each source body against its digests, counts, and anchors as provenance, not a second proof.** For the exported bundle, `validate_source_module_manifest` checks `source_module_manifest.json`, rejects manifest-level or row-level `body_in_receipt: true`, and verifies each declared source-body import against its source and target digests, line counts, byte counts, and required anchors. This is a provenance check on copied bodies, not a second proof.\n\n###### Result records and the compact card\n\n** `_write_receipts` writes the result, board, validation, and sign-off records, and `result_card` projects a pass or fail card that omits the heavier evidence.** `_write_receipts` writes the result, board, validation, and sign-off records. `result_card` projects a compact pass or fail card that omits the premise rows, classification rows, source digests, secret-scan detail, and scope limit body, so the card can be shown without exporting the heavier evidence.\n\nFunction Role `run` Fixture entrypoint; builds the result with negative cases and writes records `run_symbol_bundle` Exported-bundle entrypoint; validates copied source bodies `_build_result` Runs the secret scan and four validators, merges findings, sets pass or blocked `validate_premise_index` Builds the closed allowlist from copied premise rows `validate_classifier_policy` Pins the qualified-symbol regex and the two review outcomes `validate_symbol_observations` Iterates observations and negative cases into classifications `_classify_row` Computes the class and applies the budget-over-residual precedence `validate_source_module_manifest` Verifies exported source bodies by digest, size, and anchor `_write_receipts` Emits result, board, validation, and sign-off records `result_card` Projects the compact pass or fail card with heavy keys omitted Diagram of the mechanism (10 steps). Diagram source & refs\n\nSource refs\n\nBudget `cited_unallowed_premise_ids present`\n\n```text\nflowchart TD inputs[\"Copied premise index and symbol observations\"] policy[\"Classifier policy regex and outcomes\"] classify[\"_classify_row compute allowed vs undeclared\"] budget[\"cited_unallowed_premise_ids present\"] residual[\"Known symbol outside allowed set\"] clean[\"Allowed or no known undeclared symbol\"] retry[\"PREMISE_BUDGET_VIOLATION route retry\"] escalate[\"UNDECLARED_LIBRARY_PRIOR route bridge_escalate\"] advisory[\"NONE route accept_as_advisory\"] records[\"Result, board, validation, sign-off records\"] inputs --> classify policy --> classify classify --> budget classify --> residual classify --> clean budget --> retry residual --> escalate clean --> advisory retry --> records escalate --> records advisory --> records\n```\n\n##### Negative cases\n\n###### Seven negative cases and their codes\n\n**Seven negative cases each carry a code the run must observe, from leaked proof bodies and non-public refs to correctness overclaim, missing escalation, precedence violations, and false positives.** The fixture ships seven negative cases in `EXPECTED_NEGATIVE_CASES`, each with an error code the run must observe. `proof_body_leakage` must raise `SYMBOL_CLASSIFIER_PROOF_BODY_FORBIDDEN`: observations may carry hashes and extracted refs, bounded evidence bodies. `private_source_ref_leakage` must raise `SYMBOL_CLASSIFIER_PRIVATE_SOURCE_REF_FORBIDDEN`. `theorem_correctness_overclaim` must raise `SYMBOL_CLASSIFIER_THEOREM_CORRECTNESS_OVERCLAIM`, because a library-prior classifier is not theorem-correctness authority. `missing_escalation_for_undeclared_symbol` must raise `SYMBOL_CLASSIFIER_UNDECLARED_LIBRARY_PRIOR_NOT_ESCALATED` when an undeclared symbol is not routed to `bridge_escalate`. `premise_budget_precedence_violation` must raise `SYMBOL_CLASSIFIER_PREMISE_BUDGET_PRECEDENCE` when a cited-unallowed row is instead escalated as a residual. `allowed_symbol_false_positive` must raise `SYMBOL_CLASSIFIER_ALLOWED_SYMBOL_FALSE_POSITIVE` when a symbol already admitted by `allowed_premise_ids` is quarantined. `unqualified_symbol_overclaim` must raise `SYMBOL_CLASSIFIER_UNQUALIFIED_SYMBOL_OVERCLAIM` when an unqualified token is used to support the qualified library-prior class. A run that fails to observe any expected case is blocked.\n\n##### Prior Art Grounding\n\n###### Premise control and library-aware proof search\n\n**It sits in the Sledgehammer and Mathlib lineage of premise control, turning the observation that accepted proofs lean on large libraries into a boundary check on out-of-recipe symbols.** This check sits in the formal-methods lineage of premise control and library-aware proof search. Isabelle's Sledgehammer makes relevant-fact selection an explicit part of automated proof search, and Lean and Mathlib practice makes clear that accepted proofs can depend on a large library context. The component turns that observation into a boundary check: an accepted artifact is not enough if it quietly used symbols outside the declared premise set. It classifies the symbol-budget breach without judging whether the theorem is true.\n- Isabelle Sledgehammer and relevant-fact selection: https://isabelle.in.tum.de/doc/sledgehammer.pdf\n- Lean community Mathlib overview: https://leanprover-community.github.io/mathlib-overview.html\n- Lean 4 tactic and proof environment context: https://lean-lang.org/theorem_proving_in_lean4/Tactics/\n\n##### Validation Result record Path\n\nRun from `microcosm-substrate`:\n\n###### What a pass certifies\n\n**A pass means the fixture still classifies 11 premises and three observations into the expected three classes, the negatives are observed, the secret scan is clean, and the projection edges hold.** A pass means the fixture still classifies 11 premises and 3 observations into one undeclared-library prior, one premise-budget retry, and one advisory case, the seven negative cases are observed, the secret-exclusion scan has no blocking hits, and the generated projection reports Mermaid `available_from_capsule_edges`, Atlas `linked_from_capsule_edges`, and 19 relationship edges. The validation result records written by `_write_receipts` carry the same counts and the authority flags. A pass is bounded classifier evidence, not launch or whole-system correctness.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim\n\n**Over a copied premise index and pre-extracted observations it computes which cite a known symbol outside the allowed set, separating that residual breach from a precedence-taking budget breach.** The strongest claim this supports is narrow: over a copied, digest-bearing premise index and pre-extracted qualified symbol observations, it computes which observations cite a known library symbol outside the allowed premise set, and it separates that residual breach from an explicit premise-budget breach with the budget case taking precedence. That is the proof boundary. Everything the classifier reports is recomputed from premise rows and observations, not echoed from the observation's own assertion.\n\n###### What the scope limit excludes\n\n**The ceiling excludes running Lean, proving correctness, reading proof bodies, importing non-public refs, or claiming Mathlib availability, and treats no standard-library declaration as an implicit allowed prior.** The scope limit excludes the rest. It does not run Lean or Lake, prove formal-result correctness, read or export proof bodies, use external model services, import non-public source refs, or claim Mathlib availability. It does not treat every standard library or Mathlib declaration as an allowed prior, and it excludes launch or public sharing. Unknown or unqualified tokens stay outside the positive library-prior class unless the public observation and closed premise index make the boundary explicit. The exported source-body check is provenance by digest, size, and anchor, not semantic equivalence for any private source it was copied from.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":14465,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/undeclared_library_prior_classifier.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"}],"grouped_relation_feeds":[{"relation":"abides by","count":6,"links":[{"id":"AX-1","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Derivation before assertion","url":"doctrine.html#dcard-ax-1"},{"id":"AX-2","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Kernelized verification","url":"doctrine.html#dcard-ax-2"},{"id":"AX-5","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Fail-closed monotone 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excludes"]},{"id":"paper_module.voice_to_doctrine_self_improvement_loop","title":"Voice to Doctrine Self-Improvement Loop","relative_href":"docs/paper-module-voice-to-doctrine-self-improvement-loop.html","read_href":"docs/paper-module-voice-to-doctrine-self-improvement-loop.html","context_href":"docs/rules-and-ideas.html#paper-module-voice-to-doctrine-self-improvement-loop","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"Voice to Doctrine Self-Improvement Loop validates lesson propagation without exporting source notes or granting doctrine mutation authority.","summary":"Voice to Doctrine Self-Improvement Loop validates whether declared lessons refined a named owner surface with validation or were captured with a re-entry condition. It checks projection protocol, policy, owner surfaces, lesson rows, negative cases, source-open body imports, and scope limits while rejecting source notes export, private thread bodies, model-output data, direct doctrine-node edits, unvalidated global promotion, live work log mutation, public sharing, launch, and whole-system correctness.","authority_ceiling":"Declared lesson-propagation fixture only; no source notes export, non-public body export, model-output data export, source or doctrine mutation authority, global-promotion authority, live work log mutation, publishing-scope decision, launch-scope decision, external model access, whole-system equivalence, or whole-system correctness.","abstract":"Voice to Doctrine Self-Improvement Loop validates lesson propagation without exporting source notes or granting doctrine mutation authority.","teleology":{"section_id":"","text":"Voice to Doctrine Self-Improvement Loop validates lesson propagation without exporting source notes or granting doctrine mutation authority.","source":"capsule_summary_line"},"body_markdown":"**It decides from the files alone whether each lesson in a self-improvement pass actually moved a named surface or only left a note claiming it did.** This component checks whether a claim of self-improvement is backed by evidence. The system it comes from turns a local lesson into a durable change: a piece of pressure is sensed, classified, assigned to an owner surface, changed or captured there, validated, and closed with a condition for revisiting it later. This component takes a folder describing one pass of that loop and decides, from the files alone, whether each lesson really moved a named surface or only produced a note that says it did.\n\n**It runs on local files and emits metadata-only records, requiring every lesson's refs to resolve to real files and reporting only ids, counts, hashes, and status.** It runs on local files and writes metadata-only result records. Every lesson names the surface it changed, the action it took, and the refs that show the change held. Each ref must resolve to a real file. The result records report ids, counts, hashes, and pass or fail status, never the private text behind them.\n\n**The `run` entry point validates the test fixture, while `run_voice_to_doctrine_bundle` validates the exported example bundle instead.** The entry point is `run` in `src/microcosm_core/organs/voice_to_doctrine_self_improvement_loop.py`. A second entry point, `run_voice_to_doctrine_bundle`, validates the exported example bundle instead of the test fixture.\n\n##### Purpose\n\n###### Learning is easy to assert, hard to back\n\n**A claim that the system learned is easy to assert, so this component recomputes the verdict from evidence and ignores any label a lesson gives itself.** \"The system learned from its work\" is easy to assert and hard to back. A log line, a closed ticket, and a confident summary all read as progress. If a check accepts that language at face value, a run can pass by describing success rather than showing it. This component refuses that. It recomputes the verdict from the evidence and ignores any label a lesson gives itself.\n\n###### Three trusts a loose version would make\n\n**A loose check would trust a lesson's stated status, an unverified ref, and a refinement that changed nothing, and the checks below close all three gaps.** A loose version would trust three things this one does not: the lesson's own stated status, a ref that names a file without checking the file exists, and a refinement that changed nothing. Each of those lets an unbacked claim through. The checks below close all three.\n\n##### How it works\n\n###### `run` loads inputs and five validators\n\n** `run` loads four input files plus six negative-case files, then runs five validators in order and merges their findings.** `run` loads four input files from the chosen directory: `projection_protocol.json`, `propagation_policy.json`, `owner_surfaces.json`, and `local_lessons.json`, plus six negative-case files. It then runs five validators in order and merges their findings.\n\n###### `validate_projection_protocol` checks three source refs\n\n** `validate_projection_protocol` requires the three source pattern refs the loop is built from, recording any missing one as a required-source-pattern-ref-missing code.** `validate_projection_protocol` checks that the protocol declares the three source pattern refs the loop is built from: `recursive_self_improvement_operating_loop`, `doctrine_population_loop`, and `local_to_general_propagation`. A missing ref is recorded as `VOICE_DOCTRINE_REQUIRED_MACRO_PATTERN_REF_MISSING`.\n\n###### `validate_policy` requires the seven-step sequence\n\n** `validate_policy` requires the full seven-step sequence from `sense_local_pressure` through `publish_reentry_condition` and rejects any policy allowing result record-only progress.** `validate_policy` checks that the policy carries the full seven-step sequence, from `sense_local_pressure` through `publish_reentry_condition`, and rejects any policy that authorizes result record-only progress.\n\n###### `validate_owner_surfaces` maps surface authority\n\n** `validate_owner_surfaces` builds a map where each surface needs a stable id, a public ref, and a declared mutation authority to own a change.** `validate_owner_surfaces` builds a map of owner surfaces. Each surface needs a stable id, a public ref, and a declared mutation authority. A surface that names none of these cannot own a change.\n\n###### `validate_lessons` resolves every ref\n\n** `validate_lessons`, the core, requires each lesson's fields, a mapped owner surface, one of four typed outcomes, and every ref resolving through `_resolve_ref` to an anchor.** `validate_lessons` is the core. For each lesson it checks the required fields are present, the selected owner surface exists in the map, and the outcome is one of four: `refined_existing_surface`, `workitem_captured`, `nothing_to_refine`, or `already_propagated_verified`. It then resolves every ref the lesson carries (`changed_surface_ref`, `evidence_ref`, `evidence_refs`, `validation_ref`, `closeout_ref`) through `_resolve_ref`, which reads the target file and confirms the anchor text after `::` is present. A refinement with no changed surface is result record-only progress and fails. A capture with no re-entry condition fails. A `nothing_to_refine` with no stewardship and next-best-lane check fails.\n\n###### Asserted labels cannot rescue a row\n\n**A lesson's `expected_label` or `expected_verdict` is counted then ignored when the row is not genuinely backed, so a fixture cannot pass by declaring its own success.** The defense against self-grading lives here too. A lesson may carry `expected_label`, `expected_status`, `expected_outcome`, or `expected_verdict`. `validate_lessons` counts those keys, then computes whether the lesson is genuinely backed from the other checks. If the lesson is not backed, the asserted label is recorded as ignored under `VOICE_DOCTRINE_BAKED_EXPECTED_LABEL_IGNORED` and cannot rescue the row. A fixture cannot pass by declaring its own success.\n\n###### `_source_module_result` verifies eight copied bodies\n\n** `_source_module_result` confirms each of eight copied source bodies by hash, byte and line count, and anchors, allowing an exact copy unless a row declares a verified light edit.** `_source_module_result` validates the copied source bodies listed in `source_module_manifest.json`. For each of the eight rows it confirms the target file exists, its sha256, byte count, and line count match the manifest, every required anchor is present in both the copied body and the live source file, and the copy is exact unless the row declares a verified light edit. Seven bodies are exact copies; one, the task-ledger skill body, is a verified light edit with the operator's raw gesture removed.\n\n###### This page is a checked surface\n\n**This page is itself a checked surface, with lesson refs pointing at literal anchors here; deleting one leaves a ref unresolved and fails the run, though the fixture resolves nineteen.** This page is one of the surfaces the fixture checks. Its lesson rows point evidence and completion refs at named anchors in this file, including Public Mechanics, Shape, Reader Proof Boundary, Body Handling, and Scope boundary. `_resolve_ref` looks for each anchor as literal text; if an anchor is deleted, the ref does not resolve, `validate_lessons` records `VOICE_DOCTRINE_SURFACE_REF_UNRESOLVED`, and the run fails. The fixture resolves nineteen refs with none unresolved.\n\n###### Secret scan, gating, and record writers\n\n**After the validators `run` secret-scans inputs and copied bodies and passes only on clean findings, then writes three metadata-only records, with the bundle entry requiring the source manifest.** After the validators, `run` runs a secret-exclusion scan over the input paths and copied bodies, merges the observed negative cases, and passes only when there are no blocking findings, no missing negative cases, a clean scan, and verified source modules. `_build_board` and the result and validation writers then emit `voice_to_doctrine_self_improvement_loop_result.json`, `..._board.json`, and `..._validation_receipt.json`, each carrying `body_in_receipt: false`. `run_voice_to_doctrine_bundle` calls `run` with negative cases off and the source-module manifest required, then stamps the bundle id.\n\n###### What the fixture describes\n\n**The fixture spans four lessons over five owner surfaces: two refinements, one deferred capture with a re-entry condition, and one verified no-op.** The fixture describes four lessons across five owner surfaces: two refine an existing surface, one captures deferred work with a re-entry condition, and one records a verified no-op.\n\n##### Shape\n\nDiagram of the mechanism (9 steps). Diagram source & refs\n\nSource refs\n\n3 source pattern refs `validate_projection_protocol`\n\n```text\nflowchart TD Load[\"Load inputs protocol, policy, owner surfaces, lessons\"] Protocol[\"validate_projection_protocol 3 source pattern refs\"] Policy[\"validate_policy 7-step sequence\"] Owners[\"validate_owner_surfaces id, public ref, mutation authority\"] Lessons[\"validate_lessons fields, owner, outcome, resolve every ref\"] Recompute[\"Ignore expected label recompute from evidence\"] Negative[\"validate_negative_cases 6 refusals\"] Source[\"_source_module_result 8 copied bodies: hash, anchors, exact copy\"] Result records[\"metadata-only result records result, board, validation\"] Load --> Protocol Load --> Policy Load --> Owners Owners --> Lessons Lessons --> Recompute Load --> Negative Load --> Source Protocol --> Result records Policy --> Result records Recompute --> Result records Negative --> Result records Source --> Result records\n```\n\n##### Negative cases\n\n** `validate_negative_cases` reads six files, each a shape the loop must refuse, and records a fixed code for each:** `validate_negative_cases` reads six files, each a shape the loop must refuse, and records a fixed code for each:\n- `raw_operator_voice_export` carrying a forbidden body key gives `VOICE_DOCTRINE_RAW_OPERATOR_BODY_FORBIDDEN`.\n- `doctrine_node_hand_edit` that edits a principle, concept, mechanism, or axiom by direct file write gives `VOICE_DOCTRINE_DIRECT_NODE_EDIT_FORBIDDEN`.\n- `consume_without_deposit` with no deposit outcome gives `VOICE_DOCTRINE_CONSUME_WITHOUT_DEPOSIT`.\n- `pattern_receipt_only_progress` whose owner action is append-result record-only gives `VOICE_DOCTRINE_RECEIPT_ONLY_PROGRESS`.\n- `global_promotion_without_owner_validation` that requests promotion without owner validation gives `VOICE_DOCTRINE_GLOBAL_PROMOTION_WITHOUT_OWNER_VALIDATION`.\n- `private_thread_body_export` carrying a forbidden body key gives `VOICE_DOCTRINE_PRIVATE_THREAD_BODY_FORBIDDEN`.\n\n###### Fixture, bundle, and tampering modes\n\n**In fixture mode all six cases must be observed without blocking, in bundle mode none is expected, and source-module tampering each drops the verified-body count and adds a blocking code.** In fixture mode these six cases are expected: they must all be observed for the run to pass, and they do not block. In bundle mode the negative inputs are absent and none is expected. The source-module tests also exercise tampering: a changed hash, a rehashed body edit, a wrong source ref, and a missing live source each drop the verified-body count and add a blocking code.\n\n##### Prior Art Grounding\n\n###### After-action review and pattern language\n\n**It follows after-action review and pattern-language practice, but a lesson becomes durable only with evidence, an owner surface, a validation ref, and a bounded re-entry path.** This component follows after-action review, lessons-learned practice, and pattern-language work. NASA's Lessons Learned Information System is a public example of preserving operational lessons so later work can reuse them, and pattern-language practice gives a vocabulary for turning repeated local solutions into named reusable forms. This component keeps that direction without collapsing a local lesson into doctrine: a lesson becomes durable only when it has evidence, an owner surface, a validation ref, and a bounded re-entry path.\n- NASA Lessons Learned Information System: https://llis.nasa.gov/\n- Pattern language background: https://hillside.net/patterns/\n\n##### Validation Result record Path\n\nRun from `microcosm-substrate`:\n\n###### What a green record means\n\n**A green record means the lessons named a real owner surface, resolved every ref, observed the negative cases, and passed the secret scan, and nothing beyond that.** A green fixture or bundle record means the lessons named a real owner surface, resolved every ref, observed the negative cases, and passed the secret scan. It means nothing beyond that.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim\n\n**On a local folder it can require each lesson to name a surface, resolve every ref, land a typed outcome, and survive six refusals, with the bundle adding an eight-body copy floor.** The strongest claim this component supports: on a local folder, it can require that each lesson name an owner surface, resolve every evidence and completion ref to a real file, land one of four typed outcomes, and survive the six refusals, while ignoring any label the lesson asserts about itself. The exported bundle adds a body floor: eight copied source bodies verified by hash, byte count, line count, and anchor, with the raw source text kept out of the result records.\n\n###### What it refuses\n\n**It exports no raw voice, thread bodies, or payloads and grants no doctrine mutation, promotion, or launch, keeping the proof boundary at the fixture, bundle, and copied-body manifest.** It refuses the rest. It does not export source notes, private thread bodies, or model-output data. It does not grant doctrine mutation, direct edits to doctrine nodes, or global promotion. It does not perform live mutation of tracked work, external model access, source-file changes, launch, or public sharing. A pass is bounded validator evidence, bounded evidence that future lessons will generalize correctly and not whole-system correctness. 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Anti-Patterns Registry","relative_href":"docs/paper-module-routing-anti-patterns-registry.html","read_href":"docs/paper-module-routing-anti-patterns-registry.html","context_href":"docs/rules-and-ideas.html#paper-module-routing-anti-patterns-registry","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"Routing Anti-Patterns Registry validates public anti-pattern registry rows without becoming route source authority or mutating routes.","summary":"Routing Anti-Patterns Registry validates the public routing anti-pattern registry contract: kind/version, unique row ids, text fields, required navigation anchors, source-module digest evidence, negative cases, and scope limits. It treats the registry as a checked public artifact, not route source authority, route mutation authority, private routing-note export, provider authority, launch, public sharing, or whole-system correctness.","authority_ceiling":"Declared public routing anti-pattern registry fixture and copied source body evidence only; no route source authority, no route mutation, no private routing-note export, no external model access, no launch-scope decision, no publishing-scope decision, and no whole-system correctness.","abstract":"Routing Anti-Patterns Registry validates public anti-pattern registry rows without becoming route source authority or mutating routes.","teleology":{"section_id":"","text":"Routing Anti-Patterns Registry validates public anti-pattern registry rows without becoming route source authority or mutating routes.","source":"capsule_summary_line"},"body_markdown":"**It checks a public list of repeated navigation mistakes, each a typed row with a stable id and a plain explanation.** The system keeps a public list of navigation mistakes that agents repeat: reaching for `grep` before asking the kernel for a route, sending work to a bridge before the scope is chosen, or switching execution mode in a chat reply without updating the disk contract. Each mistake is one typed row with a stable id and a plain explanation. This component checks that list.\n\n**It confirms the rows hold their shape and the exported copy matches the source byte for byte, emitting counts, ids, and digests without copying the body or editing a route.** It reads a copy of the routing registry, confirms the rows hold their declared shape, and confirms that the exported copy still matches the source byte for byte. It emits a result record carrying counts, ids, digests, and verdicts. It never copies the registry body text into that record, and it never edits a route.\n\n###### Fixture checker, not a live router\n\n**It is a checker, not a router, and rejects any row that declares route or source authority even while describing a public anti-pattern.** `routing_anti_patterns_registry` is a checker, not a router. A page that lists routing failures is easy to mistake for the thing that decides routes. This component refuses that role in code. A row may describe a public anti-pattern, but a row that declares route or source authority is rejected.\n\n##### Purpose\n\n###### Navigation fails quietly until named\n\n**Navigation can fail silently, as when an agent greps a wide tree instead of taking a kernel route, so naming the failures as typed rows lets a reader recognise them.** A navigation system can fail quietly. An agent greps a wide tree when one kernel route would have narrowed it first, and nothing complains until the work is already off course. Naming these failures as typed rows lets a reader recognise them instead of rediscovering them.\n\n###### A loose list is not trustworthy evidence\n\n**Duplicate ids, missing explanations, self-graded labels, or claimed route authority would break the list, so each failure mode becomes a named rejection.** A loose list is not enough. If rows can carry duplicate ids, skip their explanations, grade themselves with a pre-written label, or quietly claim route authority, the list stops being trustworthy evidence. The component turns each of those failure modes into a named rejection, so a passing run means something specific.\n\n##### How it works\n\n###### Fixture and bundle entrypoints over one core\n\n** `run` checks the fixture with its negative cases and `run-bundle` checks the exported bundle with a source-copy manifest, both calling `_build_result` in a fixed order.** Two entrypoints share one validation core. `run` checks the first-wave fixture, including its negative cases. `run_routing_anti_patterns_bundle` (the `run-bundle` command) checks the exported bundle, where a source-copy manifest is required and the negative cases are not present. Both call `_build_result`, which runs the checks in a fixed order.\n\nFunction Role `_load_payloads` Loads the positive registry copy and, for the fixture, the six negative-case files. `validate_copied_macro_registry_rows` Checks row shape and derives each row's repair state from its id and text. `_negative_findings` Confirms each declared negative case is rejected with its expected error code. `_source_module_manifest_result` Verifies the exported copy's sha256 digests, anchors, and metadata-only boundary. `scan_paths` Scans the inputs for forbidden private classes and secrets. `_write_receipts` and `result_card` Emit the result record and the compact card without the copied body.\n\n###### Row shape, anchors, and forbidden fields\n\n** `_payload_findings` requires the `routing_anti_patterns` kind, uniquely-id'd rows with text and the three named anchors, and rejects authority overclaims, authority roles, and private fields.** The row check is `validate_copied_macro_registry_rows`, which wraps `_payload_findings`. `_payload_findings` requires `kind: routing_anti_patterns`, a positive integer `version`, and at least one `anti_patterns` row. Each row must carry a stable unique id and explanatory text. When the positive fixture is checked, the three named anchors `kernel_before_grep`, `bridge_before_scope`, and `mode_in_chat_only` must all appear. It then walks every nested object and rejects three things: authority overclaims such as `release_authorized`, `provider_calls_authorized`, `source_mutation_authorized`, and `route_policy_mutation_authorized`; forbidden authority roles, where a row claims `source_authority`, `route_authority`, `control_plane_authority`, or a similar role; and private fields such as `private_source_body`, `raw_seed_body`, `provider_payload_body`, and `secret_value`.\n\n###### Repair state derived from text, not labels\n\n** `_route_repair_state_for_row` derives each row's repair state from its own id and text rather than a baked label, keeping `baked_expected_labels_sufficient` false.** One design choice sits in `_route_repair_state_for_row`. Rather than trust a label written into the row, the checker derives each row's repair state from the row's own id and text. `kernel_before_grep` only earns `kernel_first_navigation` if its text actually mentions grep, kernel, and route. A row that ships a pre-written label is recorded, and the field `baked_expected_labels_sufficient` stays false. The meaning stays grounded in text a reader can see, not in a self-asserted grade.\n\n###### Source-copy digest and anchor accountability\n\n** `_source_module_manifest_result` streams a sha256 over each copied module, requires all declared digests and anchors to match, and blocks the run on any mismatch.** The exported-bundle path adds source-copy accountability through `_source_module_manifest_result`. It requires `source_module_manifest.json`, checks that the import is classified `copied_non_secret_macro_body`, and confirms the manifest keeps body text out of the result record. For each listed module it streams a sha256 over the copied target with `_sha256`, requires the declared `sha256`, `source_sha256`, and `target_sha256` to all match that digest, checks that the declared anchors appear in the copied body under `source_modules/`, and re-runs the row check over that copied body. A digest mismatch or a missing anchor blocks the run.\n\n###### Secret scan and metadata-only result records\n\n** `_build_result` passes only when rows, negative cases, the secret scan, and the source-copy check all clear, then writes metadata-only records and a compact card.** `_build_result` also runs a secret-exclusion scan over the input paths. It reports a pass only when the positive rows have no findings, no expected negative case is missing, the scan has no blocking hit, and the source-copy check is either clean or absent. `_write_receipts` then writes the result, board, and validation records, and `result_card` projects a compact card whose omission list names the full-payload keys it left out.\n\nDiagram of the mechanism (8 steps). Diagram source & refs\n\nSource refs\n\nshape + derived repair state `validate_copied_macro_registry_rows`\n\n```text\nflowchart TD Run[\"run / run-bundle\"] Build[\"_build_result\"] Rows[\"validate_copied_macro_registry_rows shape + derived repair state\"] Neg[\"_negative_findings six named rejections\"] Src[\"_source_module_manifest_result sha256 + anchors\"] Scan[\"scan_paths secret exclusion\"] Status[\"status = pass or blocked\"] Out[\"_write_receipts / result_card metadata-only records\"] Run --> Build Build --> Rows Build --> Neg Build --> Src Build --> Scan Rows --> Status Neg --> Status Src --> Status Scan --> Status Status --> Out\n```\n\n##### Negative cases\n\n**The first-wave fixture ships six perturbations. Each must be rejected with its own error code:** The first-wave fixture ships six perturbations. Each must be rejected with its own error code:\n- `missing_kind` rejected with `ROUTING_ANTI_PATTERN_KIND_REQUIRED`\n- `duplicate_id` rejected with `ROUTING_ANTI_PATTERN_DUPLICATE_ID`\n- `missing_text` rejected with `ROUTING_ANTI_PATTERN_TEXT_REQUIRED`\n- `authority_overclaim` rejected with `ROUTING_ANTI_PATTERN_AUTHORITY_OVERCLAIM`\n- `source_authority_masquerade` rejected with `ROUTING_ANTI_PATTERN_SOURCE_AUTHORITY_FORBIDDEN`\n- `private_source_leakage` rejected with `ROUTING_ANTI_PATTERN_PRIVATE_SOURCE_FORBIDDEN`\n\n###### Negative cases as proof obligations\n\n**A run passes only when every expected case appears with its code and `missing_negative_cases` is empty, making the negatives proof obligations rather than illustrations.** A run passes only when every expected case appears with its expected code and `missing_negative_cases` is empty. That makes the negative cases proof obligations, not illustrations.\n\n##### Prior Art Grounding\n\n###### From pattern catalogs to CLI usability\n\n**The registry follows the pattern and anti-pattern catalog tradition and CLI usability practice, requiring stable ids and explanatory text while staying below route-source authority.** This registry follows the pattern and anti-pattern catalog tradition: name recurring failure shapes so future operators can recognise and avoid them. The [Hillside patterns library](https://hillside.net/patterns/) is the positive pattern-language ancestor, and the software anti-pattern literature supplies the inverse move of documenting practices that look useful but produce bad outcomes. The routing-specific presentation borrows from CLI usability practice: the [Command Line Interface Guidelines](https://clig.dev/) press for discoverability and clear errors, and this component applies that pressure to navigation failures by requiring stable ids and explanatory text while staying below route-source authority.\n\n##### Validation Result record Path\n\nFrom `microcosm-substrate`, check the public routing-registry diagnostic without writing tracked records:\n\n###### What a pass proves and does not\n\n**A pass proves the public fixture, the six negative cases, and the copied-body digest floor, and does not make the registry a route source.** A pass proves the public fixture, the six negative cases, and the copied-body digest floor. It does not make the registry a route source.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The one supported claim\n\n**The evidence supports one claim: the rows hold their shape, carry stable ids, keep the three anchors, derive repair state from text, and the exported copy matches the source byte for byte.** The evidence here supports one claim: the public anti-pattern rows hold their declared shape, carry stable ids and explanatory text, keep the three named anchors, derive each repair state from row text rather than a baked label, and the exported copy matches the source registry byte for byte with no secret leak. The validation result records that back that claim are the records listed under Validation Result record Path.\n\n###### Projection-only, never route authority\n\n**The boundary stops there: the component decides and mutates no routes, exposes no private notes, and excludes external model access or launch, staying projection-only.** The proof boundary stops there. This component is a checker, not route source authority. It does not decide routes, mutate routes, prove route freshness, expose private routing notes, authorize external model access, include launch operations or public sharing, prove maturity or hosted-public posture, or prove whole-system correctness. 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synthetic sample arrays. It carries source-module refs, digests, anchors, sample counts, parity verdicts, negative cases, and an scope limit while excluding AVCaptureSession startup, microphone permission, recorded audio, device state, UI readiness, source-file changes, launch, public sharing, and whole-system correctness.","authority_ceiling":"Deterministic RMS parity evidence over public fixture inputs and copied source refs only; no macOS audio-session evidence, microphone permission authority, device capture, recorded audio, UI readiness, source-file changes, launch-scope decision, publishing-scope decision, or whole-system correctness.","abstract":"Set 8 Audio Level RMS Port validates deterministic RMS math parity over public synthetic samples without audio capture, microphone permission, or UI readiness authority.","teleology":{"section_id":"","text":"Set 8 Audio Level RMS Port validates deterministic RMS math parity over public synthetic samples without audio capture, microphone permission, or UI readiness authority.","source":"capsule_summary_line"},"body_markdown":"**The Swift `AudioLevelMonitor` drives a live microphone level meter in a recording app. Almost all of that file is platform machinery: opening a capture session, picking a device, reading sample buffers off a callback.** The Swift `AudioLevelMonitor` drives a live microphone level meter in a recording app. Almost all of that file is platform machinery: opening a capture session, picking a device, reading sample buffers off a callback. One small pure function inside it, `normalizedLevel`, turns a block of audio samples into a single number between zero and one. That number is the meter reading, and it is the only part that can be checked without a microphone.\n\n**This component ports that one function to Python and runs it over synthetic sample arrays that ship in the public clone.** This component ports that one function to Python and runs it over synthetic sample arrays that ship in the public clone. It answers one question: does the Python re-implementation produce the same level value as the Swift original, on inputs anyone can inspect. It writes a result record with the per-case comparison and a `pass` or `blocked` verdict.\n\n**Everything device-specific stays on the Swift side. The port does not open an `AVCaptureSession`, request microphone permission, read recorded audio, or capture a device.** Everything device-specific stays on the Swift side. The port does not open an `AVCaptureSession`, request microphone permission, read recorded audio, or capture a device.\n\n##### Purpose\n\n**A live level meter is hard to test because it depends on real audio hardware and OS permissions that cannot live in a public fixture.** A live level meter is hard to test because it depends on real audio hardware and OS permissions that cannot live in a public fixture. If you ported the whole monitor you would inherit all of that and prove nothing checkable. The narrow choice here is to isolate the amplitude arithmetic and hold everything else out.\n\n**What crosses into Python is the calculation alone. The claim is deliberately small: numeric parity for one function over sample arrays we can publish.** What crosses into Python is the calculation alone. The claim is deliberately small: numeric parity for one function over sample arrays we can publish. That is why the interesting content is what the port refuses to include, not what it copies.\n\n##### How it works\n\n**The math lives in `normalized_level(samples, sample_format)`. It accepts two format tags, `float32` and `int16`.** The math lives in `normalized_level(samples, sample_format)`. It accepts two format tags, `float32` and `int16`. Any other tag raises `ValueError`, which is how the unsupported-format case is exercised. An empty buffer returns `0.0` before any arithmetic runs.\n\n**For a non-empty buffer it accumulates the square of each sample. Float samples are used as-is.** For a non-empty buffer it accumulates the square of each sample. Float samples are used as-is. Int16 samples are first divided by `32767.0`, the Swift `Int16.max`, to map the integer range onto roughly minus-one to one. It then takes the root mean square, `sqrt(total / count)`, which summarises the block's energy as one amplitude. That value is multiplied by `8.0` and clamped with `min(max(rms * 8.0, 0.0), 1.0)`. The gain of eight is a display choice carried over verbatim from the Swift source: quiet speech sits low on a zero-to-one meter without it, so the level is scaled up and then capped so loud input cannot overshoot one. The int16 divisor and the `rms * 8` clamp are the two lines the exported bundle requires to match the copied Swift text.\n\n**The runtime is `_audio_evaluator`. It loads the probe manifest and runs three evaluation passes, then joins their findings.** The runtime is `_audio_evaluator`. It loads the probe manifest and runs three evaluation passes, then joins their findings.\n\n** `_evaluate_reference_cases` reads the synthetic cases in the manifest. Each case carries samples, a format, an expected level, and a tolerance.** `_evaluate_reference_cases` reads the synthetic cases in the manifest. Each case carries samples, a format, an expected level, and a tolerance. For each one it calls `normalized_level` and compares the observed value against the expected value within the tolerance. A case that is malformed, that raises, or that lands outside tolerance becomes a finding. It also checks that all three named cases in `EXPECTED_CASES` (`float32_reference_buffer`, `int16_reference_buffer`, `clamp_over_one_buffer`) are present, and records a finding for any that are missing.\n\n** `_evaluate_byte_reference_cases` covers the optional WAV cases. For each one `_decode_int16_wav` opens the file, and rejects anything that is not mono, 16-bit, uncompressed PCM, or whose byte length does not match its header.** `_evaluate_byte_reference_cases` covers the optional WAV cases. For each one `_decode_int16_wav` opens the file, and rejects anything that is not mono, 16-bit, uncompressed PCM, or whose byte length does not match its header. It unpacks the frames to int16 samples, recomputes the level from the raw bytes, and compares against the manifest expectation. A missing file, a decode failure, or an unsupported format each becomes a distinct finding rather than a silent skip.\n\n** `_evaluate_negative_exercises` runs the three refusals directly. It confirms an empty buffer reads zero, that `[1.0, -1.0, 0.5]` scales past one and clamps to one, and that format `pcm24` is refused with a `ValueError`.** `_evaluate_negative_exercises` runs the three refusals directly. It confirms an empty buffer reads zero, that `[1.0, -1.0, 0.5]` scales past one and clamps to one, and that format `pcm24` is refused with a `ValueError`. Each expectation is cross-checked against a small JSON fixture and turns into a finding if it does not hold. The declared codes for these cases live in `EXPECTED_NEGATIVE_CASES`.\n\n**Any finding from any pass flips the overall status from `pass` to `blocked`. `run` wraps the evaluator for fixture input and `run_batch8_audio_level_rms_bundle` wraps it for the exported source bundle; `result_card` projects the verdict, the reference-case counts, and the authority floor into a compact public card.** Any finding from any pass flips the overall status from `pass` to `blocked`. `run` wraps the evaluator for fixture input and `run_batch8_audio_level_rms_bundle` wraps it for the exported source bundle; `result_card` projects the verdict, the reference-case counts, and the authority floor into a compact public card.\n\nDiagram of the mechanism (10 steps). Diagram source & refs\n\n```text\nflowchart TD manifest[\"Probe manifest synthetic arrays + WAV byte cases expected level per case\"] evaluator[\"_audio_evaluator\"] refcases[\"_evaluate_reference_cases float32, int16, clamp-over-one\"] bytecases[\"_evaluate_byte_reference_cases decode mono 16-bit PCM WAV\"] negatives[\"_evaluate_negative_exercises empty, clamp, unknown-format\"] level[\"normalized_level(samples, format)\"] compare[\"compare observed vs expected within tolerance\"] verdict{\"any finding?\"} blocked[\"status: blocked\"] passed[\"status: pass\"] manifest --> evaluator evaluator --> refcases evaluator --> bytecases evaluator --> negatives refcases --> level bytecases --> level negatives --> level level --> compare compare --> verdict verdict -->|\"yes\"| blocked verdict -->|\"no\"| passed\n```\n\n##### Negative cases\n\n**The three named negative cases are the reason the port is worth running. `audio_level_empty_buffer_zero` requires an empty buffer to read `0.0`.** The three named negative cases are the reason the port is worth running. `audio_level_empty_buffer_zero` requires an empty buffer to read `0.0`. `audio_level_clamps_over_one` requires an over-one RMS-scaled buffer to clamp to `1.0`. `audio_level_unknown_format_refused` requires a format the port does not know to raise `ValueError`. If any of these behaviors changed, the corresponding code (`BATCH8_AUDIO_LEVEL_EMPTY_BUFFER_ZERO`, `BATCH8_AUDIO_LEVEL_CLAMP_REQUIRED`, `BATCH8_AUDIO_LEVEL_UNKNOWN_FORMAT_REFUSED`) is recorded and the run blocks.\n\n##### Prior Art Grounding\n\n**The math is standard digital-audio metering. Root mean square amplitude is a common way to summarise signal energy for a level display, and the practice of keeping OS capture APIs outside a pure numeric test is ordinary engineering hygiene.** The math is standard digital-audio metering. Root mean square amplitude is a common way to summarise signal energy for a level display, and the practice of keeping OS capture APIs outside a pure numeric test is ordinary engineering hygiene. Two honest anchors for the surrounding platform work are Apple's [AVFoundation](https://developer.apple.com/av-foundation/) media framework, which is the capture layer the Swift original uses, and the [FFmpeg documentation](https://www.ffmpeg.org/documentation.html), where audio streams and levels are handled as explicit inputs and transforms. This component borrows only the RMS-level calculation shape and binds it to fixture-based Python parity checks.\n\n##### Validation Result record Path\n\nRun these from the `microcosm-substrate/` public root:\n\n**The first command writes the parity result and sign-off JSON. The second validates the copied Swift source module, its digest anchors, the negative exercises, and the body-exclusion scan.** The first command writes the parity result and sign-off JSON. The second validates the copied Swift source module, its digest anchors, the negative exercises, and the body-exclusion scan. The test checks the Python port, bundle validation, result record body scan, and scope limit. A pass means the Python port reproduced every expected level within tolerance and refused every case it should refuse.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### Numeric parity with the copied Swift calculation\n\n**The strongest claim this supports is a deterministic Python port of the audio-level RMS calculation that matches the copied Swift text on published sample arrays, with a generated diagram view and navigation links available from the same source row.** The strongest claim this supports is a deterministic Python port of the audio-level RMS calculation that matches the copied Swift text on published sample arrays, with a generated diagram view and navigation links available from the same source row. The proof boundary is numeric parity for `normalized_level` over public fixture inputs, nothing more.\n\n**The scope limit stops there. This is not macOS audio-session evidence, not microphone permission authority, not device capture, not UI readiness, not source-file changes, and not public sharing or launch-scope decision.** The scope limit stops there. This is not macOS audio-session evidence, not microphone permission authority, not device capture, not UI readiness, not source-file changes, and not public sharing or launch-scope decision. The result records carry refs, digests, sample counts, and parity verdicts only; copied source bodies and audio samples stay out of them. 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registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json"}],"visible_text_byte_count":12329,"section_titles":["The Swift `AudioLevelMonitor` drives a live microphone level meter in a recording app","This component ports that one function to Python","Everything device-specific stays on the Swift side","Purpose","A live level meter is hard to test","What crosses into Python is the calculation alone","How it works","The math lives in `normalized_level(samples, sample_format)`","For a non-empty buffer it accumulates the square of each sample","The runtime is `_audio_evaluator`","`_evaluate_reference_cases` reads the synthetic cases in the manifest","`_evaluate_byte_reference_cases` covers the optional WAV cases","`_evaluate_negative_exercises` runs the three refusals directly","Any finding from any pass flips the overall status from `pass` to `blocked`","Negative cases","The three named negative cases are the reason the port is worth running","Prior Art 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work.","summary":"Set 8 Compliance Pipeline Bundle imports compliance scanner registry, bounded compliance ledger builder, and observe-loop pipeline helper bodies into a runnable component. It checks registry shape, bounded no-write compliance checks, baseline scanner truth accounting, pure pipeline helper behavior, source-module digests, negative cases, and scope limits while excluding full compliance-ledger refresh, bridge/external model access, source note mutation, repository mutation, launch, public sharing, and complete compliance proof.","authority_ceiling":"Declared public compliance and pipeline fixture evidence plus copied source-module digest checks only; no full compliance-ledger refresh, bridge/external model access authority, source note mutation, repository mutation, complete compliance proof, launch-scope decision, publishing-scope decision, or whole-system correctness.","abstract":"Set 8 Compliance Pipeline Bundle validates copied compliance scanner and observe-pipeline mechanics without refreshing the full ledger or dispatching bridge/provider work.","teleology":{"section_id":"","text":"Set 8 Compliance Pipeline Bundle validates copied compliance scanner and observe-pipeline mechanics without refreshing the full ledger or dispatching bridge/provider work.","source":"capsule_summary_line"},"body_markdown":"** `batch8_compliance_pipeline_capsule` takes two internal subsystems, copies their source bodies into the public clone, and runs them on synthetic inputs.** `batch8_compliance_pipeline_capsule` takes two internal subsystems, copies their source bodies into the public clone, and runs them on synthetic inputs. The first subsystem is a compliance scanner registry and its bounded ledger builder. The second is a six-stage pipeline that turns a long block of raw text into a short observe plan. The component runs six checks over the copied bodies and writes result records that carry no source text.\n\n**Most bundle components in this group are shape linters. They read the copied source for expected tokens and pass when the tokens are present.** Most bundle components in this group are shape linters. They read the copied source for expected tokens and pass when the tokens are present. This one goes further for the parts it can. Several of its checks import the copied bodies and call the real functions on made-up inputs, so a pass records observed behaviour, not just presence.\n\n**The question it answers is narrow. When these two subsystems are copied into the public clone and driven on a sample, do they still behave the way their internal contracts say, without touching a live ledger or dispatching any work?** The question it answers is narrow. When these two subsystems are copied into the public clone and driven on a sample, do they still behave the way their internal contracts say, without touching a live ledger or dispatching any work?\n\n##### Purpose\n\n**A loose check would grep the copied files for a function name and stop there. That proves the text was copied, not that it runs.** A loose check would grep the copied files for a function name and stop there. That proves the text was copied, not that it runs. The one behaviour worth showing directly is digest preservation. The pipeline compresses a long input down to a short digest before it decides what to inspect next. If that compression silently drops an instruction, the downstream agent never sees it. So the component feeds the real `digest_raw_seed` an eighty-line block of low-signal text with one directive line buried inside, then checks the directive survives. The paired negative case removes the marker that recognises directive lines and confirms the directive is then lost. That pairing is the point: a compression step asserted to keep the one line that matters, with a test that fails when it does not.\n\n**The standing limit is deliberate. The bounded compliance check runs the ledger builder in check-and-report mode, which reads and reports but never writes.** The standing limit is deliberate. The bounded compliance check runs the ledger builder in check-and-report mode, which reads and reports but never writes. The pipeline checks stop before any dispatch. The result is evidence that the copied mechanics work on a sample, not a claim that the live compliance ledger is fresh or that every branch is covered.\n\n##### How it works\n\n** `run` and `run_batch8_compliance_pipeline_bundle` both call `run_crown_jewel_organ` with `_evaluate` as the evaluator and `evaluate_negative_case` as the negative-case evaluator.** `run` and `run_batch8_compliance_pipeline_bundle` both call `run_crown_jewel_organ` with `_evaluate` as the evaluator and `evaluate_negative_case` as the negative-case evaluator. `_evaluate` first decides its mode. If the input directory holds a `source_module_manifest.json` and a `source_modules` directory, it is an exported bundle and every check runs in standalone mode against the copied token contracts only. A plain fixture input lets the checks import live modules from the parent tree where present. `_evaluate` then runs six checks in order and passes only if all six pass and the manifest records at least eleven copied source modules, which is the count in `SOURCE_REQUIRED_ANCHORS`.\n\nThe six checks:\n\nFunction What it does `_compliance_registry_runtime_witness` Reads the copied registry, coverage adapter, and ledger builder and confirms each carries its contract tokens. In live mode it also imports the registry and reads adapter, domain, and baseline counts as a shape witness. `_compliance_coverage_bounded_check` Runs the ledger builder as a subprocess with `--check --report` for two standards. Passes only if the run reports no write, refreshed both standard ids, and returned a truthful check status with a next-step command. `_baseline_companion_scanner_contract` Calls `scan_standard_baseline` on a sample standard and checks the returned row admits it is a baseline-inventory row with no domain adapter. `_pipeline_digest_and_shard_normalization` Imports the copied extract stage and exercises `digest_raw_seed`, `_normalize_shards_payload`, and `_pick_diverse_shards`. `_pipeline_observe_compile_helpers` Runs the compile-stage helpers that pull file mentions from free text, order follow-up files, and lift probe questions from a plan. `_pipeline_dispatch_process_boundary_contract` Calls `_receipt_selection_meta` and reads the execute, process, select, and emit stage bodies to confirm the dispatch boundary markers are present.\n\n**Two of these are worth reading closely. `_compliance_coverage_bounded_check` builds the command `build_compliance_ledger.py --check --report --standard-id std_compliance_coverage --standard-id std_microcosm`, runs it through `_run_public_witness`, and parses the summary.** Two of these are worth reading closely. `_compliance_coverage_bounded_check` builds the command `build_compliance_ledger.py --check --report --standard-id std_compliance_coverage --standard-id std_microcosm`, runs it through `_run_public_witness`, and parses the summary. The pass condition is strict. The run must report `wrote_ledger` false, both standard ids refreshed, a partial projection, a ratchet status of `ready_next_row` or `closed_batch`, and a truthful check result, where truthful means one of `ok` with no error findings, `failed` caused by error findings, or `failed` caused by a missing projection artifact. When the parent repo is not present the check falls back to `_standalone_compliance_coverage_contract`, which reads the copied builder for its bounded-check tokens instead of running it.\n\n** `_pipeline_digest_and_shard_normalization` builds a source note of a heading, one narrative line, one directive line, and eighty low-signal lines, then digests it to 240 characters.** `_pipeline_digest_and_shard_normalization` builds a source note of a heading, one narrative line, one directive line, and eighty low-signal lines, then digests it to 240 characters. It checks `directive_preserved`. It also feeds `_normalize_shards_payload` a shard with status `weird` and checks the status is rewritten to `pending` while the original value is kept as `status_variant`. It calls `_pick_diverse_shards` and checks one concept group contributes at most two shards.\n\n**The six negative cases each copy the bundle to a scratch directory, remove one load-bearing token from one copied body, and re-run the matching check.** The six negative cases each copy the bundle to a scratch directory, remove one load-bearing token from one copied body, and re-run the matching check. `_mutated_source_negative` carries this out and reports blocked only when the check both blocks and reports the expected condition, so a green run means each contract was exercised rather than skipped.\n\nDiagram of the mechanism (5 steps). Diagram source & refs\n\n```text\nflowchart TD bundle[\"Copied source bundle 11 bodies body_in_receipt: false\"] compliance[\"Compliance checks registry witness, bounded check, baseline scanner\"] pipeline[\"Pipeline checks digest and shards, compile helpers, dispatch boundary\"] negatives[\"6 negative cases remove one token per body; matching check must block\"] records[\"Result records result, board, validation\"] bundle --> compliance bundle --> pipeline bundle --> negatives compliance --> records pipeline --> records negatives --> records\n```\n\n##### Negative cases\n\n** `EXPECTED_NEGATIVE_CASES` names six. `missing_compliance_registry` removes `_LazyComplianceAdapters` from the copied registry and expects code `BATCH8_COMPLIANCE_REGISTRY_REQUIRED`.** `EXPECTED_NEGATIVE_CASES` names six. `missing_compliance_registry` removes `_LazyComplianceAdapters` from the copied registry and expects code `BATCH8_COMPLIANCE_REGISTRY_REQUIRED`. `bounded_check_failed` removes `_bounded_check_command` from the builder. `baseline_missing_standard` flips `baseline_inventory_only` in the baseline adapter. `digest_loses_directive` removes the directive marker from the extract stage so the buried line drops. `compile_helper_empty_mentions` removes a compile helper name. `pipeline_boundary_missing` removes `observe_dispatch_skipped` from the execute stage. Each case passes only when the matching check turns from pass to blocked.\n\n##### Prior Art Grounding\n\n**This component borrows from control-assessment, policy-as-code, provenance, and observability practice without adopting any one framework.** This component borrows from control-assessment, policy-as-code, provenance, and observability practice without adopting any one framework. Useful anchors are NIST [SP 800-53 Rev. 5](https://csrc.nist.gov/pubs/sp/800/53/r5/upd1/final) as a control-catalog pattern for naming and reporting control posture, [Open Policy Agent](https://www.openpolicyagent.org/docs/latest) as a pattern for evaluating structured inputs without embedding every rule in the caller, [SLSA provenance](https://slsa.dev/spec/v1.2/provenance) for treating artifact origin as an explicit attestation, and [OpenTelemetry](https://opentelemetry.io/docs/) for staged-pipeline instrumentation. The component borrows the scanner, policy, and pipeline-stage shapes and validates bounded no-write behaviour and pure helper mechanics only.\n\n##### Validation Result record Path\n\nReader-verifiable commands, run from the `microcosm-substrate/` public root:\n\n**The fixture command writes the exercise result and sign-off JSON. The bundle command validates the copied source modules, manifest digests, observed negative cases, and body scans.** The fixture command writes the exercise result and sign-off JSON. The bundle command validates the copied source modules, manifest digests, observed negative cases, and body scans. A pass means all six checks passed, the eleven source modules were present, and the six negative cases each blocked. These validation result records are the evidence a reader recomputes; they do not refresh the live ledger or dispatch any work.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The six-check fixture proof boundary\n\n**The strongest claim the evidence supports is this. The copied compliance registry, bounded ledger builder, baseline scanner, and the extract, compile, execute, and process pipeline stages behave as their internal contracts say on a sample, the bounded check reads and reports without writing, and six perturbations each block the matching check.** The strongest claim the evidence supports is this. The copied compliance registry, bounded ledger builder, baseline scanner, and the extract, compile, execute, and process pipeline stages behave as their internal contracts say on a sample, the bounded check reads and reports without writing, and six perturbations each block the matching check. That is the proof boundary.\n\n###### Claims outside the sample run\n\n**It refuses more. It does not refresh the full compliance ledger, so stale unselected rows stay outside the claim.** It refuses more. It does not refresh the full compliance ledger, so stale unselected rows stay outside the claim. It does not dispatch any provider or bridge work, change source records, or certify every compliance and pipeline branch. It is not a public sharing or launch-scope decision and not a whole-system correctness proof. 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Bundle","relative_href":"docs/paper-module-batch8-policy-engines-capsule.html","read_href":"docs/paper-module-batch8-policy-engines-capsule.html","context_href":"docs/rules-and-ideas.html#paper-module-batch8-policy-engines-capsule","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"Set 8 Policy Engines Bundle validates three deterministic public policy-engine exercises without running campaigns, providers, markets, or repository mutations.","summary":"Set 8 Policy Engines Bundle imports Lab contract audit red/green gating, market-fusion fail-closed claim preflight, and campaign dispatch transition adjudication as exact copied source bodies with bounded public exercises. It checks source-module manifests, stable negative cases, exercise outcomes, source digests, and scope limits while excluding live campaigns, external model access, repository mutation, private artifact export, market-level conclusions, launch, public sharing, and whole-system safety.","authority_ceiling":"Deterministic public policy-engine fixture evidence and copied source refs only; no live campaign execution, external model access, repository mutation, private artifact export, market validation, publishing-scope decision, launch-scope decision, or whole-system correctness.","abstract":"Set 8 Policy Engines Bundle validates three deterministic public policy-engine exercises without running campaigns, providers, markets, or repository mutations.","teleology":{"section_id":"","text":"Set 8 Policy Engines Bundle validates three deterministic public policy-engine exercises without running campaigns, providers, markets, or repository mutations.","source":"capsule_summary_line"},"body_markdown":"**This component takes three copied public policy-engine bodies and runs each one against a small fixed fixture to confirm it still makes the same deterministic decision.** This component takes three copied public policy-engine bodies and runs each one against a small fixed fixture to confirm it still makes the same deterministic decision. The three engines are a lab contract audit red/green gate, a market-fusion readiness gate that fails closed, and a campaign dispatch transition adjudicator.\n\n**It is a checker, not an engine host. It does not run live campaigns, use external model services or models, mutate any repository, validate a market, or approve public sharing or launch.** It is a checker, not an engine host. It does not run live campaigns, use external model services or models, mutate any repository, validate a market, or approve public sharing or launch. What it emits is a metadata-only result record: which engines were found, what each exercise decided, and the three stable negative-case codes that fired.\n\n##### Purpose\n\n**Each of the three copied bodies shares one shape: a machine-checkable gate that runs before any judgement or downstream action and refuses by default when evidence is absent.** Each of the three copied bodies shares one shape: a machine-checkable gate that runs before any judgement or downstream action and refuses by default when evidence is absent. Copying a gate into a public bundle is cheap. Confirming the copy still behaves like the original is the part that needs a test. This component answers one question for a cold reader: do these copied gate bodies make the same red, refuse, and reject decisions when driven by public fixture inputs?\n\n**A loose check would import the modules and assert they load. That passes even if a gate silently started accepting what it used to reject.** A loose check would import the modules and assert they load. That passes even if a gate silently started accepting what it used to reject. So each exercise drives a concrete decision with a known answer and marks itself `blocked` the moment the decision changes.\n\n##### How it works\n\n** `run` and `run_batch8_policy_engines_bundle` both forward to `run_crown_jewel_organ` with the shared `SPEC`, one for fixture input and one for the exported bundle.** `run` and `run_batch8_policy_engines_bundle` both forward to `run_crown_jewel_organ` with the shared `SPEC`, one for fixture input and one for the exported bundle. The work happens in `_capsule_evaluator`. It loads the probe manifest, reads the declared engine ids, and checks that set equals the three ids in `EXPECTED_ENGINES`. A mismatch records `BATCH8_POLICY_ENGINE_SET_MISMATCH`. It then reads the source-module manifest through `_source_rows` and checks the three required module ids are present; a gap records `BATCH8_POLICY_ENGINE_SOURCE_MODULE_MISSING`. Only when all three modules are present does it run the exercises. Any exercise that returns a status other than `pass` records `BATCH8_POLICY_ENGINE_RUNTIME_BLOCKED`. The bundle is `pass` only when the findings list is empty.\n\n**The copied bodies are loaded by `_load_copied_module`, which resolves each module's target path from the manifest and executes it under a temporary `sys.path`, so a body import failure surfaces as a blocked exercise rather than a crash.** The copied bodies are loaded by `_load_copied_module`, which resolves each module's target path from the manifest and executes it under a temporary `sys.path`, so a body import failure surfaces as a blocked exercise rather than a crash.\n\n** `_exercise_lab_contract_audit` writes a green set of lab node artifacts, runs the copied `compute_lab_contract_audit`, and expects status `green`.** `_exercise_lab_contract_audit` writes a green set of lab node artifacts, runs the copied `compute_lab_contract_audit`, and expects status `green`. It then rewrites one compute-node output to carry a banned `?` and runs the audit again, expecting status `red` with `QUESTION_MARK_OUTPUT` in the hard-fail list. Both halves must hold or the exercise is `blocked`.\n\n** `_exercise_market_fusion` calls the copied `preflight_candidate_situation` three ways. A registered situation must refuse with named reasons that do not include `candidate_situation_gate_missing`.** `_exercise_market_fusion` calls the copied `preflight_candidate_situation` three ways. A registered situation must refuse with named reasons that do not include `candidate_situation_gate_missing`. An unregistered situation must refuse with `candidate_situation_gate_missing`. A malformed payload sent through `preflight_consumer_claims` must also refuse with that same missing-gate reason. The distinction is the point: a refusal the gate reasoned about stays legible as different from a fail-closed default on something it never saw.\n\n** `_exercise_campaign_transition` calls the copied `validate_dispatch_transition`. A `candidate` to `blocked` move must return `legal_transition`.** `_exercise_campaign_transition` calls the copied `validate_dispatch_transition`. A `candidate` to `blocked` move must return `legal_transition`. A `completed` to `completed` move must return `already_target`. A `completed` to `running` move must raise `CampaignTransitionError` with the word `terminal` in its message, because `completed` is a terminal state.\n\n** `evaluate_negative_case` re-runs the matching exercise for each of the three declared cases and reports the stable error code when the refusal holds.** `evaluate_negative_case` re-runs the matching exercise for each of the three declared cases and reports the stable error code when the refusal holds. `result_card` projects a run into a compact public card carrying engine ids, the authority floor from `AUTHORITY_CEILING`, and a metadata-only flag set, with no source bodies or scan scope in the card.\n\nDiagram of the mechanism (6 steps). Diagram source & refs\n\n```text\nflowchart TD manifest[\"Probe manifest + source manifest\"] evaluator[\"_capsule_evaluator engine set == EXPECTED_ENGINES three source modules present\"] lab[\"_exercise_lab_contract_audit green then red on banned ?\"] market[\"_exercise_market_fusion refuse; fail-closed on missing gate\"] campaign[\"_exercise_campaign_transition legal / already_target / terminal\"] result[\"metadata-only result record pass only when findings empty\"] manifest --> evaluator evaluator --> lab evaluator --> market evaluator --> campaign lab --> result market --> result campaign --> result\n```\n\n##### Negative cases\n\n**The fixture declares three negative cases, one per engine, and `EXPECTED_NEGATIVE_CASES` maps each to its stable code.** The fixture declares three negative cases, one per engine, and `EXPECTED_NEGATIVE_CASES` maps each to its stable code. `lab_contract_question_mark_red_gate` fires `BATCH8_LAB_CONTRACT_QUESTION_MARK_RED_GATE` when the injected `?` output forces a red audit. `market_fusion_missing_gate_refused` fires `BATCH8_MARKET_FUSION_MISSING_GATE_REFUSED` when an unregistered situation refuses with the missing-gate reason. `campaign_completed_to_running_refused` fires `BATCH8_CAMPAIGN_COMPLETED_TO_RUNNING_REFUSED` when the terminal-state move is rejected. A green run with none of these firing would mean a gate stopped refusing, which the bundle reports as blocked.\n\n##### Prior Art Grounding\n\n###### Policy-as-code evaluation without policy authority\n\n**The design borrows the policy-as-code idea of evaluating policy as a separate engine over structured input, as in [Open Policy Agent](https://www.openpolicyagent.org/docs/latest).** The design borrows the policy-as-code idea of evaluating policy as a separate engine over structured input, as in [Open Policy Agent](https://www.openpolicyagent.org/docs/latest). The explicit risk-gate and red/green decision posture echoes NIST's [AI Risk Management Framework](https://www.nist.gov/itl/ai-risk-management-framework). The market-claim language stays inside the boundary described by the CFTC's [prediction markets](https://www.cftc.gov/LearnandProtect/PredictionMarkets) explainer. This component keeps the deterministic-gate and fail-closed-preflight shape and holds it to fixture inputs and copied public source.\n\n##### Validation Result record Path\n\nRun from the `microcosm-substrate/` public root:\n\n**The first command writes the fixture result record and sign-off JSON. The second validates the copied source bundle, its manifest digests, negative cases, and the metadata-only posture.** The first command writes the fixture result record and sign-off JSON. The second validates the copied source bundle, its manifest digests, negative cases, and the metadata-only posture. The test checks the red/green gate, the fail-closed refusals, bundle validation, and the no-launch ceiling. A pass means the copied gates still decide the fixtures the same way. It is bounded checker evidence, nothing more.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### Three copied policy engines under fixture inputs\n\n**The strongest claim this component supports: three copied public policy-engine bodies still make their expected deterministic decisions over synthetic fixture inputs, and the result record carries no source bodies or private state.** The strongest claim this component supports: three copied public policy-engine bodies still make their expected deterministic decisions over synthetic fixture inputs, and the result record carries no source bodies or private state. That is the whole proof boundary.\n\n###### Claims excluded past the fixture boundary\n\n**It refuses everything past that line. It is not lab correctness, not live campaign execution, not market validation, not provider or model dispatch, not repository or source-file changes, not whole-system safety, and not public sharing or launch-scope decision.** It refuses everything past that line. It is not lab correctness, not live campaign execution, not market validation, not provider or model dispatch, not repository or source-file changes, not whole-system safety, and not public sharing or launch-scope decision. Those would need changed bundle authority, not a green run here. 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It checks source digest parity, anchors, public family evidence, loser evidence, negative controls, survivor-only rejection, forward-gate-breach rejection, control-leak rejection, runtime verdicts, and scope limits while excluding financial decisions, investment recommendations, live market data, external model access, portfolio action, launch, public sharing, and whole-system correctness.","authority_ceiling":"Deterministic fixture evidence over public synthetic thesis rows and copied source refs only; no financial decisions, investment recommendation, live-market validation, external model access, portfolio authority, publishing-scope decision, launch-scope decision, or whole-system correctness.","abstract":"Set 8 Structural Theses Bundle validates public synthetic thesis-family replay without financial decisions, live-market validation, external model access, or portfolio authority.","teleology":{"section_id":"","text":"Set 8 Structural Theses Bundle validates public synthetic thesis-family replay without financial decisions, live-market validation, external model access, or portfolio authority.","source":"capsule_summary_line"},"body_markdown":"###### Structural theses run through the copied evaluator\n\n**This bundle imports the finance module `tools/finance/structural_theses.py` as a copied body and runs it over public synthetic thesis rows rather than building a new evaluator.** This component imports one finance module, `tools/finance/structural_theses.py`, as an exact copied source body and runs it over public synthetic thesis rows. It does not build a new evaluator. It drives the copied source's own family builder and validator and records what they return.\n\n###### Losers remain visible beside surviving families\n\n**The copied source asks which reasoning families survive once every claim is resolved forward with failures kept in view, freezing \"obvious\" at commitment time so post-hoc mutations are rejected.** The question the copied source answers is narrow. Given claims that looked structurally obvious at the time they were written, which reasoning families still survive once every claim is resolved forward and the failures are kept in view? \"Obvious\" is frozen at commitment time and is never a label applied after the outcome is known. A thesis whose meaning shifts once the result is in is a post-hoc mutation, and the validator rejects it.\n\n###### Review candidates are not tradable winners\n\n**The output is a metadata-only record of verdicts, counts, codes, and refs; source text, market data, and payloads stay out, and the strongest label is `review_candidate`.** The component emits a metadata-only result record: verdicts, counts, error codes, and source refs. Copied source text, market data, and model-output data stay out. The strongest label a surviving pattern can carry is `review_candidate`, a flag for human review. There is no tradable winner in the output vocabulary.\n\n##### Purpose\n\n###### The survivorship failure mode\n\n**Collecting only patterns that worked and calling the list a method just measures hindsight, because winners look like signal once the losers are dropped before counting.** A loose version of this idea is easy and wrong. Collect patterns that worked, call the list a method, and you have measured hindsight. The failure mode is survivorship: winners alone look like signal because the losers were dropped before anyone counted.\n\n###### Losers and controls as required evidence\n\n**The copied source requires refuted theses and negative controls to flow through the same pipeline as confirmed ones, and the bundle proves this discipline fires on public fixtures.** The discipline the copied source enforces is that losers and negative controls are required evidence, not noise. A refuted thesis must flow through the same pipeline as a confirmed one and stay legible as valid evidence. A negative control must be present and must not resolve into a confirmed claim. The bundle's job is to prove that this discipline actually fires on public fixtures, and that it refuses the inputs designed to break it.\n\n##### How it works\n\n###### `run` and its two callbacks\n\n** `run` hands the fixed `SPEC` to `run_crown_jewel_organ` with a clean-path and a negative-case callback; the real mechanism lives in the evaluator helpers that follow.** The runner is `run`, which hands the fixed `SPEC` to the shared `run_crown_jewel_organ` helper with two callbacks: `_structural_evaluator` for the clean path and `evaluate_negative_case` for single negative cases. `run_batch8_structural_theses_bundle` is the same entry in bundle-input mode. The mechanism lives in the evaluator and its helpers, which execute in this order.\n\n###### Isolated import in `_load_copied_structural_module`\n\n** `_load_copied_structural_module` imports the copied source under a private name, isolating `sys.path` and namespaces so its `system` and `tools` packages load from the bundle, not the host.** `_load_copied_structural_module` imports the copied `structural_theses.py` from the exported bundle's `source_modules` tree. It uses `importlib` with a private module name, and isolates the import inside `_temporary_sys_path` and `_temporary_import_namespaces` so the copied `system` and `tools` packages load from the bundle rather than the host. It raises `ImportError` if the spec or loader is missing.\n\n###### Probe inputs from `_family_inputs`\n\n** `_family_inputs` reads the probe manifest and returns the thesis rows, the `realized_returns` map, and the `sampling_frame`, filtering out non-mapping rows.** `_family_inputs` reads the probe manifest and returns three values: the list of thesis rows, the `realized_returns` map, and the `sampling_frame`. Non-mapping rows are filtered out.\n\n###### Family build and validation in `_build_family`\n\n** `_build_family` calls the copied `build_structural_thesis_family` under a `purged_holdout` split, driving real CP1 admission, CP2 resolution, and replay before `validate_structural_thesis_family` runs.** `_build_family` calls the copied source's `build_structural_thesis_family` with those inputs, a benchmark, a split policy of `purged_holdout`, and an embargo. Under the hood the copied source maps each thesis onto the existing forecast-claim shape and drives the real CP1 admission, CP2 resolution, proper-scoring replay, and purged walk-forward replay against the fixture prices. The evaluator then calls the copied `validate_structural_thesis_family` on the result.\n\n###### Fixed-id checks in `_structural_evaluator`\n\n** `_structural_evaluator` checks by fixed id that the winner confirms, the loser refutes as valid evidence, and the control stays a control, blocking on any mismatch.** `_structural_evaluator` checks the clean family by fixed id. The winner `sob_2015` must resolve `claim_confirmed_forward`. The loser `rmd_2018_loser` must resolve `claim_refuted_forward` with `loser_is_valid_evidence` true. The control `ctrl_2020` must stay a control and must not become `claim_confirmed_forward`. The surviving pattern must land in family memory as a `candidate_set`, and `authority_boundary.investment_recommendation_authorized` must be false. Any mismatch becomes a `BATCH8_STRUCTURAL_THESES_REFERENCE_CASE_MISMATCH` finding and the status flips to `blocked`.\n\n###### The three perturbations in `_evaluate_negative_exercises`\n\n** `_evaluate_negative_exercises` runs the three perturbations below and records a finding for any refusal that failed to fire; `evaluate_negative_case` runs them one at a time.** `_evaluate_negative_exercises` runs the three perturbations described below. It returns the observed codes plus one component-level finding per refusal that failed to fire. The single-case form `evaluate_negative_case` runs the same three checks one at a time for the runner's negative-case pass.\n\n###### Public card from `result_card`\n\n** `result_card` projects a compact public card carrying an `authority_floor` with every `*_authorized` flag false and a `body_floor` reporting no source bodies.** `result_card` projects the result into a compact public card. It copies the `authority_floor` from the ceiling (every `*_authorized` flag stays false) and a `body_floor` that reports no source bodies in the result record or card.\n\nFunction Role `run` / `run_batch8_structural_theses_bundle` Entry points; fixture mode and bundle mode `_load_copied_structural_module` Imports the copied source in an isolated namespace `_family_inputs` Reads theses, realized returns, sampling frame from the probe `_build_family` Calls the copied `build_structural_thesis_family` `_structural_evaluator` Validates the clean winner/loser/control family `_evaluate_negative_exercises` / `evaluate_negative_case` Fires the three refusals `result_card` metadata-only public card with authority and body floors Diagram of the mechanism (11 steps). Diagram source & refs\n\nSource refs\n\n_build_family CP1 admit, CP2 resolve, proper-scoring + purged replay `build_structural_thesis_family` _structural_evaluator `validate_structural_thesis_family`\n\n```text\nflowchart TD Probe[\"Public probe manifest theses, realized returns, sampling frame\"] --> Load[\"_load_copied_structural_module isolated import of copied source\"] Load --> Build[\"_build_family build_structural_thesis_family CP1 admit, CP2 resolve, proper-scoring + purged replay\"] Build --> Clean[\"_structural_evaluator validate_structural_thesis_family\"] Clean --> Check{\"sob_2015 confirmed, rmd_2018_loser refuted + valid, ctrl_2020 not confirmed?\"} Build --> Neg[\"_evaluate_negative_exercises\"] Neg --> S[\"Survivor-only family\"] Neg --> F[\"Refuted pattern smuggled forward\"] Neg --> C[\"Control mutated to confirmed\"] Check -->|yes| Pass[\"metadata-only pass record\"] Check -->|no| Block[\"Blocked finding\"] S -->|refusal fires| Pass F -->|refusal fires| Pass C -->|refusal fires| Pass\n```\n\n##### Negative cases\n\n###### Three refusals carry the proof\n\n**The three refusals in `EXPECTED_NEGATIVE_CASES` are the substance of the proof, each forcing one discipline in the copied validator to fire.** The three refusals in `EXPECTED_NEGATIVE_CASES` are the substance of the proof. Each forces one discipline in the copied validator to fire.\n\n###### Survivor-only family must be rejected\n\n**A family built from winner rows alone must trip `NO_LOSER_FLOWED_THROUGH`, `NO_NEGATIVE_CONTROL`, and `SURVIVORSHIP_SAMPLE`, or the check records a survivor-only failure.** Survivor-only (`structural_theses_survivor_only_rejected`). `_evaluate_negative_exercises` builds a family from the winner rows alone, dropping every failed thesis. The copied validator must raise `NO_LOSER_FLOWED_THROUGH`, `NO_NEGATIVE_CONTROL`, and `SURVIVORSHIP_SAMPLE`. If any of the three is missing, the check records `BATCH8_STRUCTURAL_THESES_SURVIVOR_ONLY_REJECTED`.\n\n###### Forward-gate breach must be rejected\n\n**Appending a refuted pattern as a `review_candidate` must raise `FORWARD_GATE_BREACH`, since only a pattern that survived at-time replay may earn that label.** Forward-gate breach (`structural_theses_forward_gate_breach_rejected`). A refuted pattern is appended to the clean family's `forward_lens.candidates` with a `review_candidate` research state. Only a pattern that survived at-time replay may earn a `review_candidate`, so the copied validator must raise `FORWARD_GATE_BREACH`, tracked as `BATCH8_STRUCTURAL_THESES_FORWARD_GATE_BREACH_REJECTED`.\n\n###### Control leak must be rejected\n\n**Mutating every control row to `claim_confirmed_forward` must raise `CONTROL_LEAK`, so a negative control can never resolve into a confirmed claim.** Control leak (`structural_theses_control_leak_rejected`). Every control row in a copy of the clean family is mutated to `claim_confirmed_forward`. The copied validator must raise `CONTROL_LEAK`, tracked as `BATCH8_STRUCTURAL_THESES_CONTROL_LEAK_REJECTED`.\n\n###### A missed refusal blocks the entire run\n\n**Any expected refusal that does not fire becomes a blocking finding rather than being counted as a pass.** If a refusal does not fire, the run records a blocked finding rather than a pass.\n\n##### Prior Art Grounding\n\n###### Factor-family, event-study, and survivorship lineage\n\n**The shape borrows from empirical-finance validation: Fama-French factor families, MacKinlay event studies, and survivorship-bias control, applied only to public synthetic rows and copied source.** The copied source borrows shape from empirical-finance validation. Fama and French's [common risk factors](https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html) work is the precedent for decomposing structural market claims into named factor families and testable rows. MacKinlay's [event-study methodology](https://www.bu.edu/econ/files/2011/01/MacKinlay-1996-Event-Studies-in-Economics-and-Finance.pdf) is the prior pattern for separating an event window, an expected baseline, and abnormal-return evidence. Brown, Goetzmann, Ibbotson, and Ross on [survivorship bias](https://terpconnect.umd.edu/~wermers/ftpsite/FAME/Brown_Goetzmann_Ibbotson_Ross.pdf) motivates the required loser and control cases instead of winner-only replay. The bundle keeps the factor-family, event-window, and bias-control shape and applies it only to public synthetic rows and copied source.\n\n##### Validation Result record Path\n\nReader-verifiable commands, run from the `microcosm-substrate/` public root:\n\n###### What the three commands verify\n\n**The fixture command writes the family result, the bundle command validates source refs and ceiling fields, and the test checks replay and refusals; a pass says nothing about live markets.** The first command writes the bounded family result and sign-off JSON. The bundle command validates the copied source refs, digest anchors, public winner/loser/control cases, and the scope limit fields. The test checks family replay, the three refusals, body exclusion, and the claim boundary. A pass means the copied source ran the public fixture and every refusal fired; it means nothing about live markets.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim\n\n**The evidence supports only that the copied source deterministically resolves the winner, loser, and control family and rejects the three perturbations on public synthetic rows.** The strongest claim the evidence supports: the copied `structural_theses.py` runs the public synthetic winner, loser, and control family deterministically, resolves `sob_2015`, `rmd_2018_loser`, and `ctrl_2020` with the expected at-time semantics, and rejects the survivor-only, forward-gate-breach, and control-leak perturbations. The proof boundary is deterministic fixture validation over public synthetic rows and copied source refs only. The scope limit holds every `*_authorized` flag false. This is not financial decisions, an investment recommendation, live-market validation, external model access, portfolio authority, public sharing, or launch. 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projection drift controls as provenance-preserving boundaries","url":"rules-and-ideas.html#lattice-concept-concept-import-projection-and-drift-control-bundle"}]}],"paper_module_backrefs":[{"id":"paper_module.batch12_prediction_market_board_capsule","kind":"paper_module","kind_label":"paper module","relation":"depended on by","label":"Set 12 Prediction Market Board Bundle","url":"paper-module-batch12-prediction-market-board-capsule.html","relation_id":"paper_module.depends_on.paper_module","source_ref":"core/paper_module_capsules.json::paper_modules[65:paper_module.batch12_prediction_market_board_capsule].depends_on"}],"source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/batch8_structural_theses_capsule.md"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json"}],"visible_text_byte_count":15115,"section_titles":["Structural theses run through the copied evaluator","Losers remain visible beside surviving families","Review candidates are not tradable winners","Purpose","The survivorship failure mode","Losers and controls as required evidence","How it works","`run` and its two callbacks","Isolated import in `_load_copied_structural_module`","Probe inputs from `_family_inputs`","Family build and validation in `_build_family`","Fixed-id checks in `_structural_evaluator`","The three perturbations in `_evaluate_negative_exercises`","Public card from `result_card`","Negative cases","Three refusals carry the proof","Survivor-only family must be rejected","Forward-gate breach must be rejected","Control leak must be rejected","A missed refusal blocks the entire run","Prior Art Grounding","Factor-family, event-study, and survivorship lineage","Validation Result record Path","What the three commands verify","Scope limit","The strongest supported claim"]},{"id":"paper_module.batch8_tools_tail_primitives_capsule","title":"Set 8 Tools-Tail Primitives Bundle","relative_href":"docs/paper-module-batch8-tools-tail-primitives-capsule.html","read_href":"docs/paper-module-batch8-tools-tail-primitives-capsule.html","context_href":"docs/rules-and-ideas.html#paper-module-batch8-tools-tail-primitives-capsule","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"Set 8 Tools-Tail Primitives Bundle validates four public tools-tail primitive exercises without Oracle truth, external model access, live bridge work, or repository mutation authority.","summary":"Set 8 Tools-Tail Primitives Bundle imports observer set diffing, JSON patch interpretation, ledger identity hashing, and shadow envelope parse coverage as exact copied source bodies with bounded public exercises. It checks source-module manifests, stable negative cases, exercise outcomes, mechanism rows, digests, and scope limits while excluding GodMode execution, external model access, live bridge work, repository mutation, private lab artifact export, oracle truth, launch, public sharing, and whole-system correctness.","authority_ceiling":"Deterministic public tools-tail fixture evidence and copied source refs only; no oracle truth, prediction correctness, semantic edit correctness, Lab execution authority, live bridge authority, repository mutation authority, external model access, launch-scope decision, publishing-scope decision, or whole-system correctness.","abstract":"Set 8 Tools-Tail Primitives Bundle validates four public tools-tail primitive exercises without Oracle truth, external model access, live bridge work, or repository mutation authority.","teleology":{"section_id":"","text":"Set 8 Tools-Tail Primitives Bundle validates four public tools-tail primitive exercises without Oracle truth, external model access, live bridge work, or repository mutation authority.","source":"capsule_summary_line"},"body_markdown":"###### Four copied tools run against fixed expectations\n\n**This bundle copies four small tools into the public system as their real bodies and runs each on a fixed input against a hard-coded expected value.** This component copies four small tools into the public system as their real source bodies, then runs each one on a fixed input and checks the output against a hard-coded expected value. The four are an observer set diff, a JSON-patch interpreter, a ledger-id hash, and a shadow envelope parser.\n\n###### Execution distinguishes working copies from stubs\n\n**It answers one question about copied code: does the copy still behave like the original, or has it quietly become a stub that only looks right?** It exists to answer one question about copied code: does the copy still behave the way the original did, or has it quietly become a stub that only looks right? The check runs the copied body in process and compares the result field by field. If the copy were hollow, the comparison fails instead of passing with a green tick.\n\n###### metadata-only records retain values and refusals\n\n**It emits a metadata-only record of which exercises passed, their small computed values, the rejected negative cases, and a fixed scope block, never the copied source text.** It emits a metadata-only result record: which exercises passed, the small computed values they produced, the negative cases they rejected, and a fixed scope block. It does not copy the source text of the tools into that record.\n\n##### Purpose\n\n###### A digest match does not establish execution\n\n**A digest match proves bytes were copied but not that they still run, so a loose hash check would report success even when an import broke.** A digest match proves that bytes were copied. It does not establish the copied bytes still run. When a tool moves from the private system into the public clone, a loose check would hash the file, see the hash match, and report success even if an import broke or a function returned the wrong shape. That is the failure this component is built to avoid.\n\n###### Behavioural evidence, not a correctness claim\n\n**Rather than re-implementing the tools, it loads each copied module and calls the real functions for behavioural evidence, leaving open whether the answers mean anything.** So it does not re-describe or re-implement the four tools. It loads each copied module from the manifest and calls the real functions, then asserts exact results. The evidence is behavioural: the code executed. It is not a claim that the answers mean anything about the world. Diffing two sets of observer rows is set arithmetic. Applying a JSON patch is interpreting an edit script. The gap between \"the mechanism runs as copied\" and \"the answer is correct\" stays open on purpose.\n\n##### How it works\n\n###### `evaluate` entry point and `_load_copied_module`\n\n** `evaluate` reads the manifest, checks mechanism rows, runs the four live exercises, and passes only when `findings` is empty; `_load_copied_module` imports each copied body by spec.** `evaluate` is the entry point. It reads the probe manifest, checks each declared mechanism row through `_mechanism_status`, runs the four live exercises in `RUNTIME_EXERCISES`, and folds any missing mechanism, failed exercise, or blocked anchor into a `findings` list. The result is `pass` only when that list is empty. `_load_copied_module` does the loading: it resolves the target path from the source manifest, imports the file by spec, and runs it under a temporary `sys.path` so the copied module can find its own imports.\n\nFunction What it runs `_exercise_observer_diff` `diff_evidence`, `diff_predictions` on keyed rows `_exercise_version_committer` `_apply_op` edit ops plus a refused scalar traversal `_exercise_ledger_id` `generate_ledger_id`, `normalize_lane` on lane aliases `_exercise_shadow` the copied `run` over a written envelope directory `evaluate_negative_case` one rejection path per tool\n\n###### Observer diff in `_exercise_observer_diff`\n\n** `_exercise_observer_diff` checks the overlap, missed, and extra id sets, drops a malformed row without crashing, and confirms `skipped` never leaks into the serialised result.** `_exercise_observer_diff` feeds three evidence rows (`E1` lab, a row with no `ledger_id`, `E3` lab) against two oracle rows (`E1`, `E2`). It checks `overlap_ledger_ids == [\"E1\"]`, `missed_ledger_ids == [\"E2\"]`, and `extra_ledger_ids == [\"E3\"]`. The malformed row is dropped rather than crashing the diff, and the exercise asserts the string `skipped` never appears in the serialised result, so a parse gap cannot leak through as silent data. Predictions split into `matching == [\"A\"]`, `divergent == [\"B\"]`, and `missing_targets == [\"C\"]`.\n\n###### Patch ops and refusal in `_exercise_version_committer`\n\n** `_exercise_version_committer` applies four `_apply_op` edits to a known document, then confirms traversing into a scalar raises `VersionCommitterError` rather than making a partial write.** `_exercise_version_committer` applies four ops with `_apply_op` (`set`, `merge`, `append`, `set`) and checks the final document equals `{\"profile\": {\"name\": \"Microcosm\", \"meta\": {\"kind\": \"fixture\"}}, \"events\": [{\"id\": \"E1\"}]}`. It then tries to traverse into a scalar, `set` on `/profile/name` where `profile` is the string `\"scalar\"`, and confirms this raises `VersionCommitterError`. The interesting property is the refusal: a malformed path is a controlled error, not a partial write.\n\n###### Identity stability in `_exercise_ledger_id`\n\n** `_exercise_ledger_id` confirms a lane alias and its canonical name hash to the same `P_` id, an unknown lane falls back to `X_`, and a missing identity field raises `ValueError`.** `_exercise_ledger_id` checks identity stability. `normalize_lane(\"poly\")` resolves to `POLYMARKET`, so `generate_ledger_id(\"poly\", record)` and `generate_ledger_id(\"POLYMARKET\", record)` return the same id, which starts with `P_`. An unknown lane falls back to an `X_` prefix. A record missing the identity field its lane requires raises `ValueError` instead of hashing a blank.\n\n###### Envelope parsing in `_exercise_shadow`\n\n** `_exercise_shadow` runs the copied parser over a written envelope directory and checks that well-formed rows parse while the malformed tuple is counted once as a `comma_arity` gap.** `_exercise_shadow` writes a small envelope directory into a temporary path: miner tuples with one malformed entry, a spine line, and a prediction row. It calls the copied `run` and reads `parse_stats`. It checks `hard_failure` is false, that the well-formed miner, spine, and prediction rows parsed, and that the malformed tuple was counted once under `comma_arity`. The parser reports its own coverage hole rather than swallowing it.\n\nDiagram of the mechanism (6 steps). Diagram source & refs\n\n```text\nflowchart TD manifest[\"Probe manifest + source module manifest\"] load[\"_load_copied_module import copied body by spec\"] exercises[\"Four live exercises diff | patch | ledger-id | shadow\"] negatives[\"evaluate_negative_case one rejection per tool\"] evaluate[\"evaluate collect findings\"] record[\"metadata-only result record pass only if findings empty\"] manifest --> load load --> exercises load --> negatives exercises --> evaluate negatives --> evaluate evaluate --> record\n```\n\n##### Negative cases\n\n###### One rejecting path per tool\n\n**For each tool the component runs a rejecting path from `EXPECTED_NEGATIVE_CASES`, passing only when the tool refuses the bad input with the expected code.** For each tool the component also runs the rejecting path, declared in `EXPECTED_NEGATIVE_CASES` and evaluated by `evaluate_negative_case`. A case passes only when the tool refuses the bad input with the expected code.\n\n###### Malformed observer row dropped\n\n**A row with no `ledger_id` is dropped and the good oracle id lands in `missed_ledger_ids`, raising `BATCH8_OBSERVER_DIFF_MALFORMED_ROW_SKIPPED`.** `observer_diff_malformed_key_skipped`: a row with no `ledger_id` is dropped, the good oracle id lands in `missed_ledger_ids`, code `BATCH8_OBSERVER_DIFF_MALFORMED_ROW_SKIPPED`.\n\n###### Scalar traversal refused\n\n**Setting a path inside a scalar value raises `VersionCommitterError` rather than writing, recorded as `BATCH8_VERSION_COMMITTER_SCALAR_TRAVERSAL_REFUSED`.** `version_committer_scalar_traversal_refused`: setting a path inside a scalar raises `VersionCommitterError`, code `BATCH8_VERSION_COMMITTER_SCALAR_TRAVERSAL_REFUSED`.\n\n###### Missing identity field refused\n\n**A record with no identity field raises `ValueError` instead of hashing a blank, recorded as `BATCH8_LEDGER_ID_MISSING_IDENTITY_REFUSED`.** `ledger_id_missing_identity_refused`: a record with no identity field raises `ValueError`, code `BATCH8_LEDGER_ID_MISSING_IDENTITY_REFUSED`.\n\n###### Malformed tuple counted as gap\n\n**The malformed tuple is counted as a `comma_arity` coverage gap rather than swallowed, recorded as `BATCH8_SHADOW_DSL_MALFORMED_TUPLE_COVERAGE_GAP`.** `shadow_dsl_malformed_tuple_coverage_gap`: the malformed tuple is counted as a `comma_arity` gap, code `BATCH8_SHADOW_DSL_MALFORMED_TUPLE_COVERAGE_GAP`.\n\n##### Prior Art Grounding\n\nThe four tools follow standard patterns. Useful anchors:\n- IETF [RFC 6902](https://datatracker.ietf.org/doc/html/rfc6902), which defines JSON Patch operations such as add, remove, replace, move, copy, and test.\n- IETF [RFC 9162](https://www.rfc-editor.org/rfc/rfc9162), where Certificate Transparency uses an append-only Merkle tree as an auditable log pattern.\n- W3C [PROV](https://www.w3.org/TR/prov-overview/), for representing the provenance of derived artifacts and their generating activities.\n\n###### Borrowed shapes, no correctness claim\n\n**The patch-op, identity-hash, append-only-log, and provenance shapes are borrowed, but the component runs only deterministic fixture exercises and claims nothing about correctness.** The patch-operation, identity-hash, append-only-log, and provenance shapes are borrowed. This component keeps to deterministic fixture exercises over those shapes and claims nothing about correctness of the edits, ids, or predictions.\n\n##### Validation Result record Path\n\nReader-verifiable commands, run from the `microcosm-substrate/` public root:\n\n###### What the three commands verify\n\n**The fixture command writes the result, the bundle command validates copied sources and digests, and the test checks mechanics and negative cases; a pass means the copied bodies produced exact values.** The fixture command writes the bounded result and sign-off JSON. The bundle command validates the copied sources, manifest digests, the four exercises, the body-exclusion posture, and the scope fields. The focused test checks fixture mechanics, bundle validation, and the negative cases. A pass means the copied bodies ran and produced the exact expected values on the public fixture.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim\n\n**The evidence supports only that the four copied tools import, run, produce exact expected values, and reject the four bad inputs, with no body text in the record.** The strongest claim the evidence supports: the four copied tools import and run in the public clone, produce the exact expected values on a fixed input, and reject the four declared bad inputs. The result record carries source refs, digests, counts, and outcomes, never copied body text or private state.\n\n###### What the scope limit refuses\n\n**Running the mechanisms is not a claim they are correct: `AUTHORITY_CEILING` refuses oracle truth, prediction correctness, live execution, mutation, and launch.** The proof boundary stops there. Running set arithmetic is not a claim that the observer rows are correct. Applying an edit script is not a claim that the edit is right. A stable hash is not oracle truth. The scope limit refuses oracle truth, prediction correctness, semantic edit correctness, live bridge or Lab execution, external model access, repository or source-file changes, publishing-scope decision, launch-scope decision, and whole-system correctness. 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An earlier import copied the whole validator body into Microcosm but only ever called its status-judge entry point. Most of the functions sat there as text that nothing exercised.\n\n###### Observed behavior closes the import-only gap\n\n**It closes that gap for six named checker groups by importing the copied validator, driving selected functions with small public inputs, and writing a metadata-only result record.** The bundle closes that gap for six named checker groups. It imports the copied self-indexing-cognitive-system/src/idea_microcosm/validators.py, drives selected functions with small public inputs, and writes a metadata-only result record. A pass means those six groups behaved the way their names claim. It does not mean the whole validator suite is fresh or correct.\n\n##### Purpose\n\n###### Copying a file is not running it\n\n**Copying a file and running it are different claims, so this component calls six checker groups to make a green run rest on observed behavior rather than file presence.** Copying a file and running it are different claims. A copied body can contain a function called `private_boundary_hits` and still be wrong, because nobody ever fed it a string with a home path in it. Asserting \"the file is present\" is cheap and says almost nothing. This component picks six groups of checkers and calls them, so a green run rests on observed behavior rather than on the file existing.\n\n###### Six groups span the validator's judgements\n\n**The six groups are chosen to span the validator's kinds of judgement, from source-shape anchoring to the no-write entry point, each reaching a different part of the imported body.** The six groups are chosen to span the kinds of judgement the validator makes: source-shape anchoring, status-policy judging, private-boundary scanning, specimen checker groups, launch-gate checker groups, and the no-write `validate(root, write_receipt=False)` entry point. Each group reaches into a different part of the imported body.\n\n###### Two choices keep a green run honest\n\n**Two design choices keep a green run honest: absent private state falls back to source anchors marked `public_runtime_source_only`, and negative cases are judged from engine outputs, not matched text.** Two design choices keep a green run honest. When the private system state is not present on disk, the component does not pretend the runtime checkers passed. It falls back to reading the copied source for the named anchors and marks the rest of the engines `public_runtime_source_only`, recording that as a stated limit. And the negative cases are judged from the engine outputs, not from whether a fixture file happens to contain the right error string, so a check cannot pass by matching text.\n\n##### How it works\n\n###### Two entry points feed one runner\n\n**The `run` and `run_batch8_validator_checker_bundle` entry points feed one shared runner that validates the source manifest, calls `_evaluate`, and writes result, board, validation, and sign-off artifacts.** The entry points are `run` and `run_batch8_validator_checker_bundle`. Both hand the input directory to a shared crown-jewel runner, which validates the source manifest and then calls `_evaluate`. The runner writes four artifacts: a result, a board, a validation record, and an sign-off record.\n\n###### `_evaluate` picks live or source-only mode\n\n** `_evaluate` picks the mode first: six live engines when private state is present, or the source-anchor engine plus source-only witnesses otherwise, turning any non-passing engine into a finding.** `_evaluate` picks the mode first. If the private system state is present and the input is a normal fixture, it runs six live engines. If the input is an exported bundle, or the private state is absent, it runs the source-anchor engine plus source-only witnesses for the other five. It then turns any non-passing engine into a finding, and adds a finding if the copied source manifest does not report exactly one module.\n\nThe six engines, in the order `_evaluate` lists them:\n\nFunction What it does `_validator_source_anchor_matrix` Reads the copied body, counts `def _..failures(` definitions, and checks the required anchors are present. Passes only if the anchors and launch-gate anchors are present and at least 30 checker-failure functions exist. `_status_policy_judge_matrix` Imports the validator and calls `judge_status_request` on four transitions. A well-evidenced transition must return allow, a transition missing evidence must downgrade, a prohibited upgrade must block, and a poisoned policy must block. It also drives `_validate_transition` and requires at least one transition failure. `_private_boundary_scanner_matrix` Writes a temporary file with a planted home path and email address, calls `private_boundary_hits`, and passes only if the returned patterns include both `private_home_path` and `private_email`. It records `body_in_receipt: false`. `_specimen_checker_matrix` Runs six specimen checkers through `_zero_failure_matrix`, including `_status_collapse_suite_failures` and `_source_shuttle_specimen_failures`, and passes only if every one reports zero failures on the existing fixture. `_release_gate_checker_matrix` Runs six launch-gate checkers through `_zero_failure_matrix`, including `_release_standards_axiom_gate_failures` and `_release_root_compiler_failures`, again requiring zero failures. `_validate_entrypoint_witness` Calls `validate(root, write_receipt=False)` and passes if the status is ok with at least 20 checks. Without private state it checks only that the `def validate(root: Path` anchor is present in the copied source.\n\n###### Shared checker helper and the bundle-mode stub\n\n** `_zero_failure_matrix` imports the validator and records a failure count per named checker, while `_public_runtime_source_only_engine` is the bundle-mode stub reporting a pass backed only by copied-source anchors.** `_zero_failure_matrix` is the shared helper the two checker-group engines use. It imports the validator, calls each named checker against the private root, and records the failure count per checker. `_public_runtime_source_only_engine` is the stub the bundle mode substitutes for a live engine: it reports a pass backed by copied-source anchors, with `macro_state_available: false` stated on the row.\n\n###### `result_card` carries authority and body floors\n\n** `result_card` compresses the run into two floors: an `authority_floor` of fixed all-false ceiling fields, and a `body_floor` recording `body_in_receipt: false` so no copied body text ships.** `result_card` compresses the run into a card with two floors. The `authority_floor` copies the fixed ceiling fields, all false: `release_authorized`, `publication_authorized`, `provider_dispatch`, `model_dispatch`, `source_mutation_authorized`, `full_validator_suite_freshness_claim`, `public_clone_or_hosting_authority`, and `test_completeness_proof`. The `body_floor` records `body_in_receipt: false` so the card never carries copied body text.\n\nDiagram of the mechanism (8 steps). Diagram source & refs\n\n```text\nflowchart TD A[\"Fixture input or exported bundle\"] --> B[\"Validate source manifest\"] B --> C{\"Private state present?\"} C -- \"yes, fixture input\" --> D[\"Six live checker engines\"] C -- \"no, or bundle input\" --> E[\"Source-anchor engine plus source-only witnesses\"] D --> F[\"Semantic negative-case checks\"] E --> F F --> G[\"Result, board, validation, sign-off records\"] G --> H[\"Result card: authority_floor and body_floor\"]\n```\n\n##### Negative cases\n\n###### Six negative cases judged from engine output\n\n**The fixture declares six negative cases with stable error codes, and `_observed_negative_case` reads live engine output or the source-only witness to decide whether each guard actually held.** The fixture declares six negative cases in `EXPECTED_NEGATIVE_CASES`, each paired with a stable error code. `_observed_negative_case` reads the live engine output (or the source-only witness) to decide whether the guard actually held, and `evaluate_negative_case` emits the error code when it did.\n\n###### The six cases and their error codes\n\n**The six cases run from `missing_validator_source` to `validate_entrypoint_bypassed`, and because judgement reads runtime evidence, each moves only when the matching engine observed the protective behavior.** The cases are `missing_validator_source` (`BATCH8_VALIDATOR_SOURCE_REQUIRED`), `policy_allows_poisoning` (`BATCH8_VALIDATOR_POLICY_POISONING_BLOCK_REQUIRED`), `private_boundary_blind` (`BATCH8_VALIDATOR_PRIVATE_BOUNDARY_REQUIRED`), `specimen_checker_missing` (`BATCH8_VALIDATOR_SPECIMEN_CHECKERS_REQUIRED`), `release_gate_missing` (`BATCH8_VALIDATOR_RELEASE_GATES_REQUIRED`), and `validate_entrypoint_bypassed` (`BATCH8_VALIDATOR_ENTRYPOINT_REQUIRED`). Because the judgement reads runtime evidence, a case moves only when the matching engine actually observed the protective behavior.\n\n##### Prior Art Grounding\n\n###### Three ordinary patterns, no new invention\n\n**The component borrows three ordinary patterns, fixture-driven testing, schema validation, and policy/checker separation, adding nothing beyond bounded checker exercises over copied public source.** The component borrows three ordinary engineering patterns: fixture-driven testing, where public inputs and expected negative cases are declared and replayed; schema validation, where a data shape is checked against a declared expectation; and policy/checker separation, where the code that decides is kept apart from the code that invokes it. Useful public references are [JSON Schema](https://json-schema.org/), [pytest fixtures](https://docs.pytest.org/en/stable/reference/fixtures.html), and [Open Policy Agent](https://www.openpolicyagent.org/docs/latest). Microcosm keeps this component to bounded checker exercises over copied public source and adds nothing beyond that shape.\n\n##### Validation Result record Path\n\nReader-verifiable commands, run from the `microcosm-substrate/` public root:\n\n###### What passing all three commands proves\n\n**A pass on all three commands means the six named groups behaved as claimed and the negative cases moved with runtime evidence, not a claim about the rest of the suite.** The first command writes the bounded checker result and sign-off record. The second runs the exported bundle path, which checks copied source, manifest digests, and the source-only witnesses without importing the private validator. A pass on all three means the six named groups behaved as claimed and the negative cases moved with runtime evidence. It is not a claim about the rest of the suite.\n\n##### Scope boundary\n\n###### Scope limit\n\n**The strongest supported claim is narrow: six named checker groups run and behave as their names claim, and six negative cases are rejected on runtime evidence, not matched text.** The strongest claim the evidence supports is narrow: for six named checker groups, the copied validator functions run and behave the way their names claim, and six declared negative cases are rejected on runtime evidence rather than on matched text. The copied body being large adds nothing to the claim. Only the named anchors, engines, digests, negative cases, and card fields count.\n\n###### What the bundle refuses to claim\n\n**It refuses the rest: no launch-scope decision, no hosted-public proof, no source-file changes, and no complete-suite proof, since bundle mode never re-runs the private suite.** It refuses the rest. 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over public fixtures without market-level conclusions, external model access, or launch-scope decision.","summary":"Set 12 Market Dashboard Read-Model Bundle runs market-dashboard read-model over public synthetic fixtures. It validates market dashboard import stubs, validator-case derivation, runtime feed freshness overlays, related-situation rows, source anchors, digest checks, negative cases, and scope limits while excluding launch-scope decision, external model access, whole-system equivalence, live market-level conclusions, investment-related actions, public sharing, and whole-system correctness.","authority_ceiling":"Fixture-bound market-dashboard read-model evidence and copied source refs only; no launch-scope decision, external model access, whole-system equivalence, market-level conclusions, investment-related actions, publishing-scope decision, or whole-system correctness.","abstract":"Set 12 Market Dashboard Read-Model Bundle validates copied read-model helpers over public fixtures without market-level conclusions, external model access, or launch-scope decision.","teleology":{"section_id":"","text":"Set 12 Market Dashboard Read-Model Bundle validates copied read-model helpers over public fixtures without market-level conclusions, external model access, or launch-scope decision.","source":"capsule_summary_line"},"body_markdown":"###### The dashboard read model is exercised as copied code\n\n**A dashboard read model turns a market graph into screen shapes, and it is the easiest place for a stale feed or overclaim to slip in.** A market dashboard is a presentation layer. It takes a generated graph of market situations and turns it into the shapes a screen consumes: a trust strip, a ranked queue, a detail index, a graph slice, facets, and drilldowns. That read model is also the easiest place for an overclaim to slip in. A stale feed can render green. An `auto_apply_allowed` flag can stay true. A copied label can read \"strong buy\" without anyone deciding it should.\n\n###### Three checks bind rows, filters, and claim limits\n\nIt imports the copied, read-model source as runnable code and drives its checks over small synthetic fixtures. The question it answers is narrow: when the read model is handed a planted overclaim, does the source flag it, or does the surface stay quiet? A pass is fixture evidence that the copied checks behave. It is not market-level conclusions, investment-related actions, or a live feed.\n\n###### Dashboard evidence remains metadata-only and non-live\n\n**Three copied checks run, and every result is metadata-only: refs, digests, counts, and verdicts only, never the source text.** The component runs three copied checks and records what each one did. Every result is metadata-only: refs, digests, counts, and verdicts only, never the copied source text or any model-output data.\n\n##### Purpose\n\n###### The failure mode worth catching\n\n**A loose harness that runs the copied source once on clean input proves nothing; the real risk is a read model silent on a planted overclaim.** The copied source is `system/lib/market_dashboard_read_model.py`. It carries three functions this component cares about: `validate_market_dashboard_read_model`, `_runtime_feed_freshness_overlay`, and `_related_situations`, plus the constant `TRADING_CLAIM_PATTERN`. A loose harness would import that source, run it once on a clean input, watch it pass, and call it validated. That proves nothing about the boundary. The failure mode worth catching is a read model that passes its own structure checks and then stays silent on a planted overclaim.\n\n###### A missing rejection is the finding\n\n**Each check is fed one input per failure shape, and a missing rejection, not a passing run, is what gets recorded as blocked.** So this component feeds the checks inputs designed to trip them, one input per failure shape, and treats a missing rejection as the finding. If the validator does not emit the expected error for a dangling edge or an auto-apply overclaim, that is what gets recorded as blocked. The interesting evidence is what the checks refuse.\n\n##### How it works\n\n###### One sandboxed import feeds three checks\n\n**The component imports the copied source in a sandbox, runs all three checks over fixture cases, and collects their findings.** The component imports the copied source in a sandbox, then drives the three checks over fixture cases and collects findings.\n\n###### Stubbed sandbox import in `_load_source_module`\n\n**The copied file is executed under throwaway `system` stubs in `sys.modules`, so it imports without pulling in any private host dependency.** `_load_source_module` resolves the copied file through the exported bundle manifest with `_source_target` and executes it under `_market_dashboard_import_stubs`. The stub context registers throwaway `system`, `system.lib`, and `system.lib.market_situation_graph` modules in `sys.modules` so the copied source imports without pulling in any private dependency, then restores the previous module table on exit. Nothing from the host system is loaded; the graph module is a stand-in with fixed filename and schema constants.\n\n###### Structure and overclaim gate over five negatives\n\n** `validate_market_dashboard_read_model` first checks the read model is well formed, then enforces the claim boundary, and must emit the matching error for five negatives.** `_compute_validator_cases` drives `validate_market_dashboard_read_model`. That copied check is the structural and overclaim gate. It first checks the read model is well formed: the schema version matches, every queued situation resolves to a detail entry, every graph-slice edge points at a node that exists, and each drilldown route returns metadata only with no `..` traversal. It then enforces the claim boundary: `auto_apply_allowed` must be false, `review_gated` must be true, `no_advice_mode` must be on, the silent-omission count must be zero, and copied text is scanned against `TRADING_CLAIM_PATTERN` for trading and action language. The component runs it against five negative fixtures and confirms the source emits the matching error fragment for each one.\n\n###### Three freshness states, no green from stale\n\n**The overlay classifies each synthetic run as fresh, stale, or blocked, and a stale or absent run never reports green.** `_compute_freshness_cases` drives `_runtime_feed_freshness_overlay`. For each case it writes a synthetic readiness file with `_write_readiness`, then asks the overlay to classify the run into one of three states. `fresh_green_feed` needs the run ready, all targets met, no blockers, and same-day generation. `stale_green_feed` is artifact-backed but no longer same-day; the fixture ages one case 14 days. `blocked_missing_artifact` covers a run with no readiness file, a success shortfall, or a blocker present. A stale or absent run never reports green. Historical proof does not stand in for a live feed.\n\n###### No-overlap cohort stays empty\n\n** `_related_situations` builds a capped see-also cohort from shared entities or type, and a situation with no overlap returns empty, not an invented link.** `_compute_related_cases` drives `_related_situations`, the \"see also\" cohort builder. It collects situations that share an entity or match the situation type, excludes the focus situation, and caps the list. The component checks one boundary in particular: a situation with no entity overlap and a different type returns an empty cohort, not an invented link.\n\n###### Clean payload, case sets, and the result card\n\n** `_evaluate` runs a clean payload and the three case sets, emits one row per check, and `result_card` projects the counts and floors into a card.** `_evaluate` runs a clean payload first (a clean fixture that produces validator errors is itself a finding), then runs the three case sets, folds their findings together, and emits one mechanism row per check with a pass or blocked status and its negative-case rows. `evaluate_negative_case` is the per-case entry the runner uses for the ten declared negative cases; it re-derives the source result and matches it against the expected fragment or state. `run` and `run_batch12_market_dashboard_read_model_bundle` wrap `_evaluate` through `run_crown_jewel_organ` for fixture and exported-bundle inputs, and `result_card` projects the result into a compact card carrying the mechanism count, the computed negative-case count, and the authority and body floors.\n\nFunction Role `_load_source_module` Imports the copied source under stubbed system modules `_compute_validator_cases` Drives the structure and overclaim gate over five negatives `_compute_freshness_cases` Classifies each run into fresh, stale, or blocked `_compute_related_cases` Checks entity-overlap cohorts and the no-overlap case `_evaluate` Runs the clean payload and all cases, collects findings `evaluate_negative_case` Re-derives one negative case against its expected fragment Diagram of the mechanism (12 steps). Diagram source & refs\n\nSource refs\n\nValidate market dashboard read model `validate_market_dashboard_read_model` Blocked missing artifact `blocked_missing_artifact`\n\n```text\nflowchart TD A[\"Synthetic dashboard, freshness, related fixtures\"] --> B[\"_load_source_module (stubbed import of market_dashboard_read_model.py)\"] B --> C[\"validate_market_dashboard_read_model\"] C --> C1[\"Structure: schema, queue-to-detail, graph edges, drilldown route safety\"] C --> C2[\"Ceiling: no auto-apply, review-gated, no-advice, no trading language, zero silent omissions\"] B --> D[\"_runtime_feed_freshness_overlay\"] D --> D1[\"fresh_green_feed\"] D --> D2[\"stale_green_feed\"] D --> D3[\"blocked_missing_artifact\"] B --> E[\"_related_situations\"] E --> E1[\"Entity or type match; self-excluded, capped; no overlap means empty\"] C1 --> F[\"metadata-only result record (refs, digests, counts, verdicts)\"] C2 --> F D1 --> F D2 --> F D3 --> F E1 --> F\n```\n\n##### Negative cases\n\n###### Ten declared negative cases\n\n**Ten cases live in `EXPECTED_NEGATIVE_CASES`, and five of them drive the validator, each expecting a specific error fragment.** The component declares ten negative cases in `EXPECTED_NEGATIVE_CASES`. Five drive the validator and expect a specific error fragment:\n- `dangling_graph_edge` expects `$.graph_slice.edges[edge1].target is dangling`.\n- `traversal_route_ref` expects `route_ref contains traversal`.\n- `oracle_auto_apply_overclaim` expects `auto_apply_allowed must be false`.\n- `strict_trading_claim_language` expects `contains trading/action claim language`.\n- `silent_omission_count` expects `silent_omission_count must be 0`.\n\n###### Freshness and related cases expect states\n\n**Four freshness cases expect a blocked or stale state and the last expects an empty cohort, and a missing rejection is recorded as not observed.** Four drive the freshness overlay and expect a state rather than a green pass: `freshness_missing_readiness_artifact` and `freshness_success_shortfall` and `freshness_blocker_list` each expect `blocked_missing_artifact`, and `freshness_stale_not_fresh` expects `stale_green_feed` with a non-zero staleness count. The last, `related_no_overlap_different_type`, expects an empty cohort. When the source does not produce the expected rejection, the case is recorded as not observed, which blocks that mechanism.\n\n##### Prior Art Grounding\n\n###### CQRS and dashboard-observability practice\n\n**The shape comes from CQRS read models and dashboard observability, with an added overclaim gate and a freshness state that refuses green from stale evidence.** The read-model shape comes from CQRS and dashboard-observability practice: derive presentation-ready projections from source data, make feed freshness visible, and keep the read surface separate from any mutation authority. Two public anchors are relevant. Microsoft's [CQRS pattern](https://learn.microsoft.com/en-us/azure/architecture/patterns/cqrs) describes read models optimized for queries rather than command handling. [Grafana dashboards](https://grafana.com/docs/grafana/latest/visualizations/dashboards/) query and transform data sources into operational panels. This component borrows the shape and adds an explicit overclaim gate and a freshness state that refuses to report green from stale evidence.\n\n##### Validation Result record Path\n\nReader-verifiable commands, run from the `microcosm-substrate/` public root:\n\n###### What the two commands write and prove\n\n**The fixture and bundle commands validate the copied source, digests, and freshness rows, and a pass means every check behaved with a metadata-only result record.** The fixture command writes the read-model result and sign-off JSON. The bundle command validates the copied source, its manifest digests, the freshness rows, the related-situation cohorts, and the metadata-only posture. A pass means the three checks behaved as declared over the synthetic fixtures: the clean payload validated, each negative fragment or state appeared, and the result record stayed metadata-only. These are validation result records for the fixtures, nothing more.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The fixture-bound proof boundary\n\n**Over synthetic fixtures the copied checks emit the expected errors, classify freshness honestly, and keep no-overlap cohorts empty, and that is the whole boundary.** The strongest claim the evidence supports: over public synthetic fixtures, the copied read-model checks emit the expected structural and overclaim errors, classify feed freshness into honest fresh, stale, and blocked states, and build entity-overlap cohorts that stay empty when nothing overlaps. That is the proof boundary. It is fixture-bound mechanism evidence.\n\n###### What it refuses past that line\n\n**Past the boundary it is not market-level conclusions, advice, or a external model access, and grants no launch, mutation, or whole-system authority.** It refuses everything past that line. It is not live market-level conclusions, not investment-related actions, and not a external model access. It excludes launch or public sharing, does not claim whole-system equivalence, excludes source-file changes, and does not assert whole-system correctness. That is the scope limit.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":12837,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/batch12_market_dashboard_read_model_capsule.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"CQRS pattern","url":"https://learn.microsoft.com/en-us/azure/architecture/patterns/cqrs","kind":"external_reference","section_id":"cqrs-and-dashboard-observability-practice"},{"label":"Grafana dashboards","url":"https://grafana.com/docs/grafana/latest/visualizations/dashboards/","kind":"external_reference","section_id":"cqrs-and-dashboard-observability-practice"}],"grouped_relation_feeds":[{"relation":"abides by","count":4,"links":[{"id":"AX-5","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Fail-closed monotone 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Bundle","relative_href":"docs/paper-module-batch12-prediction-market-board-capsule.html","read_href":"docs/paper-module-batch12-prediction-market-board-capsule.html","context_href":"docs/rules-and-ideas.html#paper-module-batch12-prediction-market-board-capsule","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"Set 12 Prediction Market Board Bundle validates copied prediction-board and quant-mart diagnostics over public fixtures without market-level conclusions or provider authority.","summary":"Set 12 Prediction Market Board Bundle runs prediction-market board and quant-mart diagnostics over public synthetic fixtures. It validates prediction-market joins, Polymarket identity by slug, provider drift monitors, missingness boards, prior green deltas, source lifecycle vintage enrichment, source-module digests, negative cases, and scope limits while excluding launch-scope decision, external model access, whole-system equivalence, market-level conclusions, provider truth, investment-related actions, public sharing, and whole-system correctness.","authority_ceiling":"Fixture-bound prediction-board and quant-mart diagnostic evidence plus copied source refs only; no market-level conclusions, provider truth, investment-related actions, external model access, whole-system equivalence, launch-scope decision, publishing-scope decision, or whole-system correctness.","abstract":"Set 12 Prediction Market Board Bundle validates copied prediction-board and quant-mart diagnostics over public fixtures without market-level conclusions or provider authority.","teleology":{"section_id":"","text":"Set 12 Prediction Market Board Bundle validates copied prediction-board and quant-mart diagnostics over public fixtures without market-level conclusions or provider authority.","source":"capsule_summary_line"},"body_markdown":"###### The board bundle runs one copied market projection\n\n**A copied presentation-mart source runs over synthetic prediction-market and source-feed fixtures to check the board keeps guesses labelled as guesses.** This component takes a copied presentation-mart source file, runs it over synthetic prediction-market and source-feed fixtures, and checks that the board keeps guesses labelled as guesses. It does not read live markets and it does not say what any price means.\n\n###### Fixture rows exercise ordering and visibility rules\n\n**The bundle stubs the private imports, loads `quant_presentation_mart.py` from disk, and calls seven helpers, each against a fixture built to expose one way a board misleads.** The copied source is `system/lib/quant_presentation_mart.py`. That file has private imports the public clone cannot satisfy, so the bundle stubs those imports, loads the module from disk, and calls seven of its helper functions directly. Each call is checked against a fixture built to expose one specific way a board can mislead.\n\n###### A passing board is not a live market\n\n**The output carries per-mechanism status, computed negative cases, and an scope limit as diagnostic rows, never source bodies or model-output data.** The output is a result record with per-mechanism status, the named negative cases it computed, and an scope limit. It carries diagnostic rows, not source bodies and not model-output data.\n\n##### Purpose\n\n###### How loose aggregation fakes a fact\n\n**The shared failure mode is a row that looks like a fact but is an aggregation artifact: a doubled volume, a fabricated id, a false-healthy feed.** Market and source dashboards share a failure mode: a row looks like a fact when it is really an artifact of loose aggregation. A duplicate listing doubles a volume figure. An unmatched market slug is handed a fabricated event identity. A feed returns zero rows but shows as healthy. A \"change since last time\" number appears with no prior run to compare against. A vintage date is displayed when the lifecycle data never carried one.\n\n###### Asserting the exact conservative behaviour\n\n**The bundle runs the real helper bodies over synthetic inputs and asserts the exact conservative behaviour rather than trusting a silent default.** The single question here is whether the copied presentation logic keeps those distinctions honest when run over public synthetic inputs. Loose aggregation fails silently: it defaults a missing value to a confident zero, or gives an orphan slug a synthetic id. This bundle runs the real helper bodies and asserts the exact conservative behaviour instead.\n\n##### How it works\n\n###### The pipeline and its two entry points\n\n**The pipeline is a fixture load, stubbed import, seven helper calls, and eight invariant checks, and both entry points forward to `run_crown_jewel_organ`.** The pipeline is a fixture load, a stubbed source import, seven helper calls, and eight invariant checks. `run` and `run_batch12_prediction_market_board_bundle` both forward to `run_crown_jewel_organ` with `_evaluate` as the evaluator and `evaluate_negative_case` as the negative-case evaluator. The bundle entrypoint differs only by input mode.\n\n###### Resolving and importing the copied source\n\n** `_source_manifest_for_input` resolves the manifest and `_load_source_module` reads it to find and import `quant_presentation_mart.py` from its file location.** `_source_manifest_for_input` resolves the source manifest. It prefers a `source_module_manifest.json` sitting in the input directory, and otherwise points at the exported bundle manifest under the public root. `_load_source_module` reads that manifest through `_source_target` to find `system/lib/quant_presentation_mart.py`, then imports it from its file location.\n\n###### Scoped stand-in modules in `_quant_mart_import_stubs`\n\n**Lightweight stand-in modules are injected into `sys.modules` so the copied source loads, then restored on exit so the stub stays scoped to the import.** `_quant_mart_import_stubs` is the context manager that makes the copied source loadable. The source imports `system.lib.market_feed_run_evidence`, `system.lib.market_fusion_readiness`, and `system.lib.finance_numeric_assurance`. The bundle injects lightweight stand-in modules for those names into `sys.modules`, including a `FEED_NODE_IDS` list of `global_stock_feed`, `global_news_feed`, and `global_macro_feed`. On exit it restores whatever modules were there before, so the stub is scoped to the import.\n\n###### Three prediction-market invariants in `_evaluate`\n\n** `_evaluate` calls `_prediction_market_board` and checks three invariants: the duplicate fold, the orphan with no fabricated identity, and the deduped aggregate.** `_evaluate` runs the prediction-market half. It loads `prediction_market_rows.json`, `polymarket_identity_artifact.json`, and `quant_mart_helper_cases.json`, then calls the source function `_prediction_market_board` with the Polymarket identity artifact. The board rows are keyed by `event_identity_status`. The `available` event and the `missing_from_feed_artifact` orphan drive three checks: the surviving market has volume `900000.0` with a market count of `1` (the duplicate fold), the orphan has a null `event_id` and an aggregate `max_liquidity` of `0.0` (no fabricated identity), and the deduped aggregate reports `market_count` `1` and `max_volume` `900000.0`.\n\n###### Five source-feed invariants across four helpers\n\n**Four source helpers check drift flags kept separate from FRED flags, missingness states, an unavailable delta with no baseline, and vintage bound to the structured source record.** `_evaluate_quant_mart_helpers` runs the source-feed half against four more source functions. `_provider_drift_monitor` produces per-feed drift flags: the stock feed must carry `provider_fallback_used`, `html_response_seen`, and `fetch_failures`, the news feed must stay empty, and the source feed must carry the FRED-specific `fred_invalid_series` and `fred_network_warning`. Keeping the generic transport flags separate from the FRED flags means a source data-source fault is not laundered into a generic warning. `_missingness_board` lists only feeds that are not both populated and `ok`: the healthy feed is absent, the empty feed reads `zero_rows`, the degraded feed reads `quality_degraded`. `_delta_since_previous_green` returns `status: unavailable` with an empty `row_deltas_by_lane` when no prior green run exists. `_macro_regime_board`, fed by `_macro_lifecycle_by_slug`, binds each bucket's `vintage_status` to whether the lifecycle structured source record carries it: the inflation bucket reads `available` with observation date `2026-05-01`, and the growth bucket, whose lifecycle row is absent, reads `missing_from_feed_artifact`.\n\n###### Findings, negative-case scoring, and the card\n\n**A failed check becomes a finding that flips the status to blocked, `evaluate_negative_case` scores each case, and `result_card` projects a metadata-only card.** Any check that does not hold becomes a structured finding through `finding`, and the result status flips from `pass` to `blocked`. `evaluate_negative_case` re-runs `_evaluate`, pulls the matching case row with `_computed_case_row`, and returns a blocked result with the expected stable error codes when the invariant held, a pass when it did not, or an error record if the run raised. `result_card` projects the result into a public card carrying `mechanism_count`, `computed_negative_case_count`, and an authority floor read from `AUTHORITY_CEILING`, with `body_in_receipt` false throughout.\n\nFunction Role `run`, `run_batch12_prediction_market_board_bundle` Entry points; forward to the crown-jewel runner with the evaluators `_source_manifest_for_input`, `_load_source_module` Resolve and import the copied presentation-mart source `_quant_mart_import_stubs` Inject and restore stand-in modules so the source loads standalone `_evaluate` Call `_prediction_market_board`; check the three prediction-market invariants `_evaluate_quant_mart_helpers` Call the four source-feed helpers; check the five source invariants `evaluate_negative_case` Score one named case against its expected error codes `result_card` Project the result into a metadata-only public card Diagram of the mechanism (11 steps). Diagram source & refs\n\n```text\nflowchart TD Input[\"Fixture inputs rows, identity, helper cases\"] --> Manifest[\"_source_manifest_for_input\"] Manifest --> Load[\"_load_source_module via _quant_mart_import_stubs\"] Load --> Eval[\"_evaluate\"] Eval --> Board[\"_prediction_market_board dedup + orphan + aggregate\"] Eval --> Helpers[\"_evaluate_quant_mart_helpers\"] Helpers --> Drift[\"_provider_drift_monitor generic vs FRED flags\"] Helpers --> Miss[\"_missingness_board zero_rows vs quality_degraded\"] Helpers --> Delta[\"_delta_since_previous_green unavailable with no baseline\"] Helpers --> Source[\"_macro_regime_board vintage bound to structured source record\"] Board --> Card[\"result_card status, negative cases, ceiling\"] Drift --> Card Miss --> Card Delta --> Card Source --> Card\n```\n\n##### Negative cases\n\n###### Eight named cases as required rejections\n\n** `EXPECTED_NEGATIVE_CASES` binds three prediction-market and five source-feed cases to stable error codes, each a rejection the board must make or fail loudly.** `EXPECTED_NEGATIVE_CASES` declares eight named cases, each bound to a stable error code. The three prediction-market cases are `duplicate_lower_volume_retained_higher`, `orphan_slug_no_identity_fabrication`, and `aggregate_count_and_max_volume_deduped`. The five source-feed cases are `provider_drift_multisignal_flags`, `provider_drift_fred_diagnostics_flags`, `missingness_zero_row_lane_flagged`, `delta_no_previous_green_unavailable`, and `macro_lifecycle_vintage_status_bound`. A case is a rejection the board must make: keep the higher-volume duplicate, refuse to invent an event id, flag the empty feed, withhold the delta with no baseline, and show a vintage date only when it is real. When an invariant does not hold, the run emits `BATCH12_PREDICTION_CASE_NOT_OBSERVED` or `BATCH12_QUANT_MART_HELPER_CASE_NOT_OBSERVED` rather than reporting a silent pass.\n\n##### Prior Art Grounding\n\n###### Information-aggregation and provider-join shape\n\n**The board borrows the information-aggregation and provider-join shape from prediction-market practice but stays explicitly diagnostic, treating no price as truth.** The component borrows the information-aggregation and provider-join shape from prediction-market and public market-data practice, then keeps the board explicitly diagnostic. Relevant anchors include Robin Hanson's [information markets](https://mason.gmu.edu/~rhanson/infomkts.html) framing, where markets aggregate dispersed information about uncertain events; the CFTC's [prediction markets explainer](https://www.cftc.gov/LearnandProtect/PredictionMarkets), which frames event contracts, market prices, and consumer cautions; and the [Polymarket API documentation](https://docs.polymarket.com/api-reference) as a concrete public API family for market, event, and series data. This bundle uses the join and identity-matching shape and nothing else: no price is treated as truth.\n\n##### Validation Result record Path\n\nReader-verifiable commands, run from the `microcosm-substrate/` public root:\n\n###### What the commands and test verify\n\n**The fixture and bundle commands and the focused test validate the manifest, seven helper calls, and eight cases, and a pass means conservative diagnostics held.** The fixture command writes the board result and sign-off record. The bundle command validates the copied source manifest, the seven helper calls, and metadata-only posture. The focused test checks fixture validation, bundle validation, and the eight named cases. A pass means the copied logic produced the expected conservative diagnostics over the synthetic fixtures. Validation result records for this module are the records those commands write.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The narrow boundary and its false flags\n\n**The copied logic produces the expected diagnostics over synthetic fixtures, and `AUTHORITY_CEILING` sets every market-truth, forecast, advice, and launch flag to false.** The strongest claim this supports is narrow: the copied `quant_presentation_mart` presentation logic runs standalone over public synthetic fixtures and produces the expected identity, drift, missingness, delta, and source-lifecycle diagnostics, with source bodies and model-output data kept out of the result record. That is the proof boundary. The scope limit stops there. `AUTHORITY_CEILING` sets `live_prediction_market_truth`, `provider_truth`, `forecast_correctness`, `calibration_claim`, `investment_advice`, `provider_dispatch`, `release_authorized`, `publication_authorized`, `private_root_equivalence_claim`, and `whole_system_correctness_claim` all to false. The board is a diagnostic surface over fixtures. It is not market-level conclusions, not provider truth, not investment-related actions, and not launch-scope decision.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":13085,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/batch12_prediction_market_board_capsule.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"information markets","url":"https://mason.gmu.edu/~rhanson/infomkts.html","kind":"external_reference","section_id":"information-aggregation-and-provider-join-shape"},{"label":"prediction markets explainer","url":"https://www.cftc.gov/LearnandProtect/PredictionMarkets","kind":"external_reference","section_id":"information-aggregation-and-provider-join-shape"},{"label":"Polymarket API 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It classifies phrases against evidence class and scope limit, checks typed ordinal evidence ranks, real-system flags, fail-closed defaults, boundary-context negation, main --assert-clear behavior, source digests, negative cases, and scope limits while excluding launch-scope decision, external model access, whole-system equivalence, market-level conclusions, investment-related actions, public sharing, and whole-system correctness.","authority_ceiling":"Fixture-bound launch-claim language gate evidence and copied source refs only; no launch-scope decision, publishing-scope decision, external model access, whole-system equivalence, market-level conclusions, investment-related actions, or whole-system correctness.","abstract":"Set 12 launch claim-Language Gate checks public claim language against result record-backed scope limits without approving launch or public sharing.","teleology":{"section_id":"","text":"Set 12 launch claim-Language Gate checks public claim language against result record-backed scope limits without approving launch or public sharing.","source":"capsule_summary_line"},"body_markdown":"###### launch language is checked against evidence classes\n\n**Public copy is scanned line by line for launch, licensing, maturity, and launch-authority wording it cannot support, failing non-zero under `--assert-clear` while any unsupported claim survives.** This component checks public copy for language that claims more than the result records behind it can support. It reads a small public sharing manifest, opens each file the manifest lists, and matches every line against fixed families of launch, licensing, maturity, and launch-authority wording. It writes a metadata-only result record and, when run with `--assert-clear`, exits non-zero while any unsupported claim remains. It does not approve launch and it does not publish anything.\n\n###### The gate separates supported wording from overclaim\n\n**The private gate module is absent on the public clone, so an in-file `_PublicFallbackReleaseClaimGate` stands in, making this a check on a checker over shipped fixtures.** The gate logic it exercises is a separate module, `release_claim_language_gate.py`. On the public clone that module is not present, so the component replaces it with an in-file stand-in, `_PublicFallbackReleaseClaimGate`. The page you can run here is that fallback gate driven over shipped fixtures. This module is therefore a check on a checker: it proves the gate blocks bare launch wording and clears explicit boundary wording over a fixed set of inputs.\n\n###### Passing prose excludes launch\n\n**The scan is a deterministic regular-expression pass that fails closed: any affirmative claim it cannot downgrade becomes an active blocker and forces exit 2.** The scan is a deterministic regular-expression pass over lines, not a language model. It fails closed. An affirmative claim it cannot downgrade becomes an active blocker, and `main --assert-clear` returns exit 2 whenever an active blocker survives.\n\n##### Purpose\n\n###### How copy drifts into over-claiming\n\n**Public copy drifts toward over-claiming as edits accumulate, and human review misses it because each phrase reads reasonably alone, so the gate asks whether a line's launch status outran the evidence.** Public copy drifts toward over-claiming. A page that began as \"fixture-proven, not yet published\" gets edited over months until someone writes launch, licensing, or maturity wording without noticing that nothing underneath it changed. A loose human review misses this because each phrase reads reasonably in isolation. The gate answers one narrow question: does a line assert launch status that the evidence does not hold, and would an automated check catch it before the copy ships.\n\n###### A lexical check honest about its reach\n\n**Being lexical, the check flags risky phrasing and its excusing context but never judges whether the underlying claim is true, only whether the wording outran the records.** Because the check is lexical, it is honest about its own reach. It finds risky phrasing and the context that would excuse it. It does not decide whether the underlying claim is true, only whether the wording has outrun what the result records can back.\n\n##### How it works\n\nThe stages run in this order inside `_evaluate`.\n\n###### `_load_fixture` reads or fails closed\n\n** `_load_fixture` reads `release_gate_fixture.json`, turning a missing file, bad JSON, or non-object payload into a structured finding and the fail-closed envelope rather than a quiet pass.** First, `_load_fixture` reads `release_gate_fixture.json` from the selected input directory. A missing file, invalid JSON, or a non-object payload becomes a structured finding, and the run returns the fail-closed envelope from `_blocked_exercise` rather than a quiet pass.\n\n###### `_load_source_module` returns the stub\n\n** `_load_source_module` returns `_PublicFallbackReleaseClaimGate` when the manifest marks the public stub, still validating the copied module's digest and required anchors before any exercise.** Second, `_load_source_module` resolves the gate. `_source_module_public_stubbed` inspects the copied-module manifest, and when it sees the public stub substitution it returns `_PublicFallbackReleaseClaimGate` instead of importing a private body. The copied module is still validated for digest equality and required anchors (`RISKY_PHRASES`, `NEGATIVE_CONTEXT_MARKERS`, `META_FORBIDDEN_CONTEXT_MARKERS`, `private_control_plane_leak`, `def _classify_hit`, `def build_gate`) before any exercise.\n\n###### `_write_gate_fixture` materialises each scenario\n\n** `_write_gate_fixture` writes each scenario's `docs/` text and a `publication_manifest.yaml`, while `_fixture_doc_name` rejects control characters, path separators, and duplicate document names.** Third, `_write_gate_fixture` materialises each scenario on disk: it writes the fixture text into `docs/`, then writes a `publication_manifest.yaml` that lists those files. `_fixture_doc_name` rejects any file name with control characters (`BATCH12_RELEASE_FIXTURE_PATH_CONTROL_CHAR`) or path separators (`BATCH12_RELEASE_FIXTURE_PATH_UNSAFE`), and rejects an active document that would overwrite the safe one (`BATCH12_RELEASE_FIXTURE_DUPLICATE_DOC_NAME`).\n\n###### `build_gate` scans and classifies lines\n\n** `build_gate` opens each manifest file and scans every line, recording each match as a metadata-only hit and counting active blockers, boundary hits, and claim surfaces into a status.** Fourth, `build_gate` reads the manifest, opens each listed file, and scans every line against the phrase patterns. Each match is classified by `_classify` and recorded as a hit carrying its path, line number, phrase id, classification, and a `body_in_receipt: false` flag. The summary counts active blockers, boundary hits, and claim surfaces, and sets a status.\n\n###### `_run_main_assert_clear` captures the exit code\n\n** `_run_main_assert_clear` runs the gate on the active scenario and returns its exit code, after which `_evaluate` computes the cases and hands off to the metadata-only writers.** Fifth, `_run_main_assert_clear` runs `main --assert-clear` on the active scenario, captures the printed status, and returns the exit code. `_evaluate` then computes the negative cases and the positive boundary-clear check and hands the result to `run` or `run_batch12_release_claim_language_gate_bundle`, which write the metadata-only records.\n\nFunction Role `_load_fixture` Reads `release_gate_fixture.json`; converts missing or malformed input into a fail-closed finding. `_write_gate_fixture` Writes the `docs/` files and `publication_manifest.yaml` for one scenario; rejects unsafe or duplicate names. `build_gate` Scans manifest files line by line for risky phrases and emits the classified hit summary. `_classify` Labels each hit as an active blocker or allowed boundary context. `main --assert-clear` Runs the gate and returns exit 2 when any active blocker remains. `_evaluate` Builds the four scenarios, computes the negative cases, and assembles the result record. `run_batch12_release_claim_language_gate_bundle` Wraps `_evaluate` and writes the metadata-only result, board, and validation records.\n\n###### `_classify` reads each hit in context\n\n** `_classify` blocks any internal control-leak phrase outright, allows other hits only when a negation marker is present, and holds internal control vocabulary to a stricter rule that even a negation cannot clear.** The interesting part is `_classify`. It reads a matched line two ways. If the phrase belongs to the `private_control_plane_leak` family, the line is an active blocker regardless of surrounding words. Otherwise, if the line contains a negative-context marker such as `not`, `forbidden`, `blocked`, or `omitted`, the hit is boundary-or-negative context and is allowed; with no such marker it is an active blocker. So \"not a hosted product\" clears and \"deployment-posture\" blocks. The internal control family is stricter on purpose: even \"excludes launch\" is blocked, because public reader copy should leave launch-authority vocabulary out entirely rather than negate it.\n\n###### Status and the `ok` flag from counts\n\n**Status is set from the counts: any active blocker yields `active_claim_blocked`, and `ok` is true only at zero blockers, which is when `--assert-clear` avoids exit 2.** `build_gate` then sets the status from the counts. Any active blocker gives `active_claim_blocked`; boundary hits with no active blocker give `clear_boundary_only`; nothing risky gives `clear`. The `ok` flag is true only at zero active blockers, and `main --assert-clear` returns 2 when `ok` is false.\n\n###### Four scenarios, one expected to clear\n\n** `_evaluate` runs four scenarios from one fixture, expecting the safe boundary document to clear and the three over-claim documents to block.** `_evaluate` exercises four scenarios from one fixture: a safe boundary document, an affirmative over-claim document, a public sharing over-claim document, and a private internal control leak document. The safe scenario is expected to clear as boundary-only. The other three are expected to block.\n\nDiagram of the mechanism (8 steps). Diagram source & refs\n\nSource refs\n\nsafe and active copy text `release_gate_fixture.json` negation marker present => allowed `boundary_or_negative_context`\n\n```text\nflowchart TD Fixture[\"release_gate_fixture.json safe and active copy text\"] Write[\"_write_gate_fixture docs files + publication_manifest.yaml\"] Gate[\"build_gate scan manifest lines for risky phrases\"] Classify{\"_classify read each hit in context\"} Boundary[\"boundary_or_negative_context negation marker present => allowed\"] Active[\"active_claim_blocker bare launch or internal control wording => blocked\"] Assert[\"main --assert-clear exit 2 when any active blocker remains\"] Result[\"metadata-only result record result, board, validation\"] Fixture --> Write Write --> Gate Gate --> Classify Classify -->|allowed| Boundary Classify -->|blocked| Active Active --> Assert Boundary --> Result Assert --> Result\n```\n\n##### Negative cases\n\n###### Three fixed cases, each recomputed\n\n** `EXPECTED_NEGATIVE_CASES` fixes three error-coded cases covering the affirmative blocker, the exit-2 assertion, and the internal control leak, blocking the run if any fails to compute.** `EXPECTED_NEGATIVE_CASES` fixes three cases, each with a stable error code. `affirmative_open_source_production_ready_blocks` (`BATCH12_RELEASE_CLAIM_ACTIVE_BLOCKER`) requires the active scenario to reach `active_claim_blocked` with both the `open_source` and `production_ready` phrase ids present. `assert_clear_returns_exit_2` (`BATCH12_RELEASE_CLAIM_ASSERT_CLEAR_EXIT_2`) requires `main --assert-clear` to return exit code 2 with `ok` false. `private_control_plane_public_reader_blocks` (`BATCH12_RELEASE_PRIVATE_CONTROL_PLANE_LEAK`) requires the internal control scenario to block on a `private_control_plane_leak` family hit. If any case does not compute, `_evaluate` records `BATCH12_RELEASE_CLAIM_CASE_NOT_OBSERVED` and the run is blocked.\n\n##### Prior Art Grounding\n\n###### Advertising-substantiation and communication rules\n\n**The proof-consumer form comes from advertising-substantiation and regulated-communication practice, where FTC, FINRA, and SEC rules require a reasonable basis before a claim is disseminated.** The pattern comes from advertising-substantiation and regulated-communication practice: public claims should stay within the evidence actually held, and stronger wording needs stronger support. The FTC advertising-substantiation policy (`https://www.ftc.gov/legal-library/browse/ftc-policy-statement-regarding-advertising-substantiation`) states that objective claims need a reasonable basis before dissemination. FINRA Rule 2210 and the SEC investment-adviser marketing rule carry the same shape for financial communications. This module borrows only the proof-consumer form of that idea. It does not implement legal compliance and it does not decide whether copy is fit to publish.\n\n##### Validation Result record Path\n\nReader-verifiable commands, run from the `microcosm-substrate/` public root:\n\n###### What a pass certifies\n\n**A pass certifies the fallback gate cleared the boundary scenario, blocked the three over-claim scenarios, exited 2 under `--assert-clear`, and wrote metadata-only records.** A pass means the fallback gate cleared the boundary scenario, blocked the three over-claim scenarios, returned exit 2 under `--assert-clear`, and wrote the result, board, and validation records with no copied body text inside them.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The lexical, fixture-bound proof boundary\n\n**Over the shipped fixtures the fallback gate allows boundary-only wording, blocks bare launch, licensing, maturity, and launch-authority wording, and exits 2 while a blocker remains.** The strongest claim the evidence supports is narrow: over the shipped fixtures, the fallback gate classifies boundary-only wording as allowed and classifies bare launch, licensing, maturity, and launch-authority wording as active blockers, and `main --assert-clear` exits 2 while any active blocker remains. That is the proof boundary. The evidence is lexical and fixture-bound.\n\n###### A pass is evidence, not permission to ship\n\n**A green result grants no launch, public sharing, or hosted-product status, proves no semantic understanding or legal sufficiency, and stays evidence about the checker rather than permission to ship.** The scope limit stops well short of that. A green result does not grant launch permission, public sharing, or hosted-product status. It does not establish semantic understanding of the copy, complete secret detection, whole-system equivalence, market-level conclusions, legal sufficiency, or whole-system correctness. It excludes editing the copied gate module or changing the fixtures, negative cases, or ceiling; those belong in the source and standard lanes first. 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It audits the all-B idea-first scorecard shape, expected-ref injection policy, private fixture refs, missing-task refusals, negative cases, and scope limits while excluding live benchmark results, navigation truth, hosted readiness, launch-scope decision, external model access, source-file changes, and whole-system correctness.","authority_ceiling":"Fixture-bound route-quality scorecard and copied source refs only; no live benchmark, navigation truth, hosted readiness, launch-scope decision, external model access, source-file changes, or whole-system correctness.","abstract":"Set 10 Cold Eval Honesty Bundle runs cold-eval over public fixtures without benchmark, navigation-truth, or launch-scope decision.","teleology":{"section_id":"","text":"Set 10 Cold Eval Honesty Bundle runs cold-eval over public fixtures without benchmark, navigation-truth, or launch-scope decision.","source":"capsule_summary_line"},"body_markdown":"###### One copied simulator, one bounded scorecard\n\n**It runs one real route-quality simulator, an exact copy of `cold_eval.py`, over a fixed public workspace and reports the scorecard without letting it become a claim about the live system.** `batch10_cold_eval_honesty_capsule` takes one real route-quality simulator, runs it over a fixed public workspace, and reports the scorecard without letting that scorecard become a claim about the live system. The simulator is an exact copy of `cold_eval.py`, imported from the exported bundle rather than rewritten, so the numbers come from the same code the original produced.\n\n###### Four assertions without a wider win claim\n\n**A green run asserts four things: the copied simulator ran, every task went to the idea-first arm, its extra route surface shows in the counts, and the forbidden claims stay forbidden.** A green run says four things happened: the copied simulator executed, the recomputed scorecard has every task won by the idea-first arm, the extra route surface that arm reaches is visible in the counts, and the exercise manifest still forbids benchmark, navigation-truth, launch, and hosted-readiness claims. It does not say idea-first routing wins anywhere except this workspace.\n\n###### Counts and statuses without source bodies\n\n**The result is a small metadata-only record carrying counts, engine statuses, and a scoring-policy name, but no source text or command output.** The result is a small metadata-only record. It carries counts, engine statuses, and a scoring-policy name. It does not carry any source text or command output.\n\n##### Purpose\n\n###### A clean-sweep scorecard is easy to fake\n\n**A scorecard where one arm wins every task is easy to fake, so this component runs the real simulator and checks none of those shortcuts happened before treating the win as accounting.** A scorecard that shows one arm winning every task is easy to fake. You can hand the winning arm the answer, inject the expected references into its route, or quietly let a non-public paths leak into the fixture and call the shape \"measured\". This component exists to run the real simulator and then check that none of those shortcuts happened, so the win is an accounting fact about a declared fixture and nothing more.\n\n###### Expected references score coverage, never the route\n\n**The honesty turns on one property: expected references only ever score coverage and are never added to either arm's route, so idea-first wins by genuinely declaring more relevant files.** The honest part turns on one property of the copied scorer. Each task lists the route references an answer should reach, but those expected references are only ever used to score coverage. They are never added to either arm's route. The flat arm is scored on what a reader reaches from `README.md`, `docs/quickstart.md`, and `pyproject.toml`. The idea-first arm is scored on what its entry packets declare. The idea-first arm wins because it genuinely declares more of the relevant files, not because the scorer leaked the target into the route.\n\n##### How it works\n\n###### Both entrypoints share one evaluator\n\n**Both entrypoints hand their fixture to `run_crown_jewel_organ`, where `_evaluate` turns each blocked engine into an explicit finding instead of a silent success.** `run` and `run_batch10_cold_eval_bundle` both hand the fixture to `run_crown_jewel_organ` with `_evaluate` as the evaluator. `_evaluate` runs three engines in order and turns any non-pass into a structured finding, so a blocked engine surfaces as an error code rather than a silent success.\n\nFunction Engine id What it does `_run_original_cold_eval` `cold_eval_original_runner` Loads the copied `cold_eval.py`, copies the workspace to a temporary directory, calls its `run_cold_eval` at a fixed timestamp, then reads back the scorecard and result record. `_scorecard_shape_audit` `cold_eval_scorecard_shape_audit` Checks the winner shape and route-count asymmetry, and scans the tasks file for private references. `_claim_ceiling_gate` `cold_eval_claim_ceiling_gate` Checks that the exercise manifest forbids the named claims and keeps expected-reference injection off.\n\n###### The copied simulator runs at a fixed time\n\n** `_run_original_cold_eval` requires the task file, runs the copied simulator at a fixed timestamp, and reads the scorecard and result record back from a temporary workspace.** `_run_original_cold_eval` first requires `evals/cold_agent_ab/tasks.json` in the workspace. Without it the engine returns blocked with `BATCH10_COLD_EVAL_TASKS_REQUIRED` and never runs the simulator. With it, the engine copies the workspace, calls `run_cold_eval` at `2026-05-31T00:00:00Z` so the run is deterministic, and reads `runs/cold_agent_ab/seed_scorecard.json` plus `receipts/cold_agent_ab_seed.json` back from the temporary copy. It emits the task count, the idea-first win count, the flat win count, the per-task winners, and the scoring-policy name. In the shipped fixture the idea-first win count is 3.\n\n###### The scorecard audit rejects route and path leaks\n\n** `_scorecard_shape_audit` accepts only an all-idea-first result with visible route-count asymmetry and no private reference in the task fixture.** `_scorecard_shape_audit` reads the same tasks file and walks its strings for any value containing a user-home prefix or the private source-repository root. It passes only when every winner is `B.idea_first_packet`, the idea-first route reference count is greater than the flat count, and no private reference was found. It records the two counts and the `route_surface_asymmetry_visible` flag so a reader can see the gap that produced the win.\n\n###### The ceiling gate blocks promoted claims\n\n** `_claim_ceiling_gate` requires expected-reference injection to stay off and the exercise manifest to keep all four promoted claims explicitly forbidden.** `_claim_ceiling_gate` loads `batch10_cold_eval_exercise_manifest.json` and passes only when at least one row was checked, `expected_ref_injection_allowed` is false, and the manifest's forbidden-claim list contains `live_agent_benchmark_win`, `navigation_truth`, `release_approval`, and `hosted_public_readiness`. This is where the run refuses to promote the shape.\n\n###### The source manifest is checked before execution\n\n**Before any engine runs, the shared runner verifies the copied simulator's digest and required anchors so a tampered source copy cannot produce a passing scorecard.** The source copy is checked before any of this. The shared runner validates the bundle's `source_module_manifest.json` against the copied file: the digest must match and the required anchors (`SCORING_POLICY =`, `def _score_task(`, `def run_cold_eval(`, `expected_ref_injection_used`, `idea_first_packet_wins_fixture`) must be present. A tampered digest fails with `CROWN_JEWEL_SOURCE_DIGEST_MISMATCH` before the engines run.\n\nDiagram of the mechanism (6 steps). Diagram source & refs\n\nSource refs\n\nrun copied run_cold_eval, read back scorecard `cold_eval_original_runner` all-B winner + route asymmetry + no non-public refs `cold_eval_scorecard_shape_audit` injection off, forbidden claims named `cold_eval_claim_ceiling_gate`\n\n```text\nflowchart TD A[\"Fixture or exported bundle (workspace + exercise manifest)\"] --> M[\"Source manifest check digest + required anchors\"] M --> R[\"cold_eval_original_runner run copied run_cold_eval, read back scorecard\"] R --> S[\"cold_eval_scorecard_shape_audit all-B winner + route asymmetry + no non-public refs\"] S --> C[\"cold_eval_claim_ceiling_gate injection off, forbidden claims named\"] C --> O[\"metadata-only result record and card\"]\n```\n\n###### The compact card preserves the same result\n\n** `result_card` adds engine and copied-module counts to the full record, while bundle mode follows the same evaluator path with exported inputs.** `result_card` projects a compact card from the full record and adds the engine count and the copied source-module count. `run_batch10_cold_eval_bundle` is the same path with the exported-bundle input mode.\n\n##### Negative cases\n\n###### Four perturbations must move the verdict\n\n**Each negative fixture mutates one input, reruns the same evaluator path, and must turn the result blocked with its named error code.** The fixture ships four perturbations. Each one copies the input, applies a named mutation, re-runs the engines, and confirms the run turns blocked with the right code.\n\n###### Missing tasks, route ties, injection, and non-public refs\n\n**The four mutations remove tasks, erase route asymmetry, enable expected-ref injection, or add a non-public paths, and each has a distinct required rejection.** `missing_tasks` removes the tasks file, so `_run_original_cold_eval` returns `BATCH10_COLD_EVAL_TASKS_REQUIRED`. `flat_route_can_win` rewrites the route counts to be equal, so the shape audit loses its asymmetry and reports `BATCH10_COLD_EVAL_NOT_ALWAYS_B_WIN`. `expected_ref_injection` sets `expected_ref_injection_allowed` to true in the manifest, and the scope limit gate rejects it with `BATCH10_COLD_EVAL_EXPECTED_REF_INJECTION_FORBIDDEN`. `private_fixture_ref` injects a `/Users/` probe into the tasks file, and the shape audit rejects it with `BATCH10_COLD_EVAL_PRIVATE_REF_REJECTED`. Each mutation runs through the same manifest and evaluator path as the positive case, so a perturbation that fails to move the outcome is itself a failure.\n\n##### Prior Art Grounding\n\n###### Evaluation reports need scenario and limit context\n\n**HELM and Model Cards motivate reporting what was measured, under which fixture assumptions, and which wider capability claims the score cannot support.** This component follows evaluation-transparency and benchmark-hygiene practice: a scorecard should state what was measured, what fixture assumptions went in, and what the result can and cannot support. [HELM](https://crfm.stanford.edu/helm/index.html) frames evaluation as a scenario-bound surface rather than a single global capability claim. [Model Cards for Model Reporting](https://arxiv.org/abs/1810.03993) established pairing performance numbers with intended use and limitations. This component borrows that scorecard-plus-limits shape and narrows it to a deterministic route-quality fixture.\n\n##### Validation Result record Path\n\nReader-verifiable commands, run from the `microcosm-substrate/` public root:\n\n###### A pass covers three engines and four rejections\n\n**The commands prove the copied simulator and all three engines ran under the declared fixture while every negative case blocked with its expected code.** The fixture command writes the scorecard record and sign-off JSON. The bundle command validates the copied source, its manifest, the metadata-only cards, the injection policy, and the private-reference negative case. A pass means the copied simulator ran, the three engines passed, and the four negative cases each turned blocked with the expected code.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The fixture supports one exact accounting claim\n\n**The strongest claim is that this copied simulator produced the declared all-idea-first scorecard without expected-ref injection or private-path leakage, while the manifest kept promoted claims forbidden.** The strongest claim here is small and exact. Over this fixture, the copied `cold_eval.py` ran, its scorecard was recomputed with every task won by the idea-first arm, expected-reference injection stayed off, no private reference leaked in, and the manifest still forbade the promoted claims. That is the whole proof boundary.\n\n###### The workspace result grants no live authority\n\n**The scope limit excludes live benchmarks, navigation truth, hosted readiness, route superiority, external model access, source-file changes, launch, and whole-system correctness.** The scope limit stops there. This component does not claim live benchmark results, navigation truth, hosted readiness, route-quality superiority, external model access, source-file changes, launch-scope decision, or whole-system correctness. The fixture win is accounting for one declared workspace, not evidence that idea-first routing wins in the live system. Validation result records back the claim above and nothing beyond it.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":12228,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/batch10_cold_eval_honesty_capsule.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"HELM","url":"https://crfm.stanford.edu/helm/index.html","kind":"external_reference","section_id":"evaluation-reports-need-scenario-and-limit-context"},{"label":"Model Cards for Model Reporting","url":"https://arxiv.org/abs/1810.03993","kind":"external_reference","section_id":"evaluation-reports-need-scenario-and-limit-context"}],"grouped_relation_feeds":[{"relation":"abides by","count":4,"links":[{"id":"AX-1","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Derivation before 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limits","url":"rules-and-ideas.html#lattice-concept-concept-agent-reliability-and-safety-validator-bundle"}]}],"paper_module_backrefs":[],"source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/batch10_cold_eval_honesty_capsule.md"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json"}],"visible_text_byte_count":13216,"section_titles":["One copied simulator, one bounded scorecard","Four assertions without a wider win claim","Counts and statuses without source bodies","Purpose","A clean-sweep scorecard is easy to fake","Expected references score coverage, never the route","How it works","Both entrypoints share one evaluator","The copied simulator runs at a fixed time","The scorecard audit rejects route and path leaks","The ceiling gate blocks promoted claims","The source manifest is checked before execution","The compact card preserves the same result","Negative cases","Four 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internal control source bodies without route, ledger-mutation, or launch-scope decision.","summary":"Set 10 Live Source Drift Bundle imports exact current internal control Python bodies after source drift. It validates stale-versus-current digest rows, copied-source compileability without import execution, required behavioral anchors, source-manifest boundaries, negative cases, and scope limits while excluding route authority, work log or work log mutation authority, mission execution, git approval, source-file changes, launch, public sharing, external model access, and non-public runtime export.","authority_ceiling":"Fixture-bound source-digest, anchor, compile, and scope limit evidence only; no route authority, Work or work log mutation, mission execution, git approval, source-file changes, launch, public sharing, external model access, or non-public runtime export.","abstract":"Set 10 Live Source Drift Bundle validates copied current internal control source bodies without route, ledger-mutation, or launch-scope decision.","teleology":{"section_id":"","text":"Set 10 Live Source Drift Bundle validates copied current internal control source bodies without route, ledger-mutation, or launch-scope decision.","source":"capsule_summary_line"},"body_markdown":"###### Live-source drift is recomputed across copied inputs\n\n**copies of four internal control source files are checked against the source they were copied from.** This component checks whether a small set of copies of internal control source files still match the current source they were copied from. The copies are exact Python bodies for four files: the option-surface router, the mission-transaction landing preflight, the work landing controller, and the work log controller.\n\n###### Three engines separate source, projection, and ceiling\n\n**Three questions at once: do the bytes still match the current digest, do the copies still compile, and do they still carry their named surfaces.** It answers three questions about those copies at once. Do their bytes still match the current source digest? Do they still compile as Python without being imported or run? Do they still contain the named command and contract surfaces they are supposed to carry? It emits a result record with the digest rows, compile status, anchor checks, and the scope limit, and no copied source body.\n\n###### A green result remains fixture-bound\n\n**It is built to go red: when the source moves ahead of the copies, the failed digest check is the signal to refresh them.** The component is built to go red. When the current source moves ahead of the copies, the digest check fails on purpose, and that red result is the signal to refresh the copies through the exact-copy source lane. It is a drift sentinel, not a one-off snapshot.\n\n##### Purpose\n\n###### Why a stale copy looks fine\n\n**A copy can compile and match one recorded digest while being weeks behind the file it mirrors, so weak checks pass it.** Copied source drifts silently. A public copy of a control file can look fine, compile fine, and still be weeks behind the file it claims to mirror. A loose check that only asks \"is this valid Python?\" passes a stale copy. A check that only compares one recorded digest can pass a copy whose recorded digest was itself never refreshed.\n\n###### Current and stale digests in one row\n\n**Each row holds the current and the earlier stale digest side by side, so proof of freshness and evidence of past drift cannot be erased.** This component makes drift visible instead of hiding it. Each digest row keeps two fingerprints side by side: the current source digest, and the stale digest that was recorded earlier. The row passes only when the copied body matches the current digest, and it separately asserts that the stale digest differs from the current one. The proof of freshness and the evidence of the earlier drift sit in the same row, so a refresh cannot quietly erase the fact that the copy was once behind.\n\n##### How it works\n\n###### Two entry points into `_evaluate`\n\n**Both entry points forward to `run_crown_jewel_organ`, and `_evaluate` loads the manifest and copied texts, then runs four engines in order.** `run` and `run_batch10_live_source_drift_bundle` are the two entry points. Both forward to `run_crown_jewel_organ` with the shared `SPEC`, an `_evaluate` function, and `evaluate_negative_case`. `_evaluate` loads the probe manifest and the copied source texts, then runs four engines in order and collects their findings.\n\n###### Per-row digest agreement in `_digest_refresh_matrix`\n\n**A row passes only when copied, target, and current digests all agree and the stale digest differs from the current one.** `_digest_refresh_matrix` builds one row per copied file. For each row it reads the current source digest and the stale recorded digest from the probe manifest, and the copied and target digests from the source manifest. A row's `current_matches_copy` is true only when the current digest is non-empty and the copied digest, the target digest, and the current digest all agree and the manifest marks the target as matching. Its `stale_differs_from_current` is true when a stale digest exists and differs from the current one. The engine rejects a copy whose body no longer matches the current digest, a row whose stale digest was allowed to equal the current one, duplicate rows, missing rows, unexpected rows, and a row count that does not match the four expected files.\n\n###### Compile without import in `_compile_gate`\n\n**Each copied file is compiled with `py_compile` into a throwaway directory but never imported, so malformed bodies are caught without running any code.** `_compile_gate` compiles each copied file with `py_compile.compile` into a throwaway temporary directory with `doraise=True`. It never imports the module, so a malformed body is caught without executing any copied code. `import_executed` stays false in the result. It rejects a missing target, a non- `.py` target, and any body that fails to compile.\n\n###### Named surfaces checked by substring\n\n**Substring checks confirm each copy still carries its named command and contract surfaces, so a compiling body that quietly dropped one still fails.** `_control_surface_anchor_matrix` runs substring checks over each copied body. It confirms that each file still exposes the command and contract surfaces it is supposed to carry, for example `def build_option_surface` and `option_surface_lens_packet_v0` in the option-surface router, or `def cmd_session_finalize` and `read_receipt_id` in the work log controller. A copy that compiles but has quietly lost a command or contract surface still fails here.\n\n###### Authority flags held false by `_claim_ceiling_gate`\n\n**Eleven authority flags must be false and `body_in_receipt` false, keeping every copied body a validator input and nothing more.** `_claim_ceiling_gate` reads the scope limit policy from the probe manifest and checks that eleven authority flags are all false, that `body_in_receipt` is false, and that `body_import_authority` is `copied_source_body_only`. This is the gate that keeps a copied body a validator input and nothing more.\n\n###### Aggregation and the compact result card\n\n** `_evaluate` blocks the whole result if any engine or structural check fails, and `result_card` projects the status plus four counts into a card.** `_evaluate` then confirms all four engine ids are present and that the source manifest declares exactly four modules. If any engine is blocked or any structural check fails, the whole result is blocked. `result_card` projects the pass or blocked status plus `engine_count`, `copied_macro_source_module_count`, `stale_digest_count`, and `compiled_module_count` into a compact card.\n\nDiagram of the mechanism (8 steps). Diagram source & refs\n\n```text\nflowchart TD A[\"Probe manifest current + stale digests, authority policy\"] --> C B[\"Copied bodies and source manifest\"] --> C C[\"_digest_refresh_matrix copied = target = current, stale differs from current\"] --> F B --> D[\"_compile_gate py_compile, no import\"] B --> E[\"_control_surface_anchor_matrix named surfaces present\"] A --> G[\"_claim_ceiling_gate authority flags false\"] D --> F E --> F G --> F F[\"_evaluate all engines pass, four modules present\"] --> H[\"metadata-only result and card\"] C -. mismatch .-> I[\"Blocked: refresh copies via exact-copy source lane\"]\n```\n\n##### Negative cases\n\n###### Perturbing a copy, never the fixtures\n\n**The bundle is copied into a temporary workspace and perturbed there, so the on-disk fixtures stay untouched while four cases must be rejected.** `evaluate_negative_case` copies the bundle into a temporary workspace and perturbs it, so the fixtures on disk are never mutated. Four cases must all be rejected.\n\n###### `stale_digest_replay` collapses the two digests\n\n**Setting a row's stale digest equal to its current one must make `all_stale_digests_differ` report false.** `stale_digest_replay` sets a row's stale digest equal to its current digest. The digest engine must then report `all_stale_digests_differ` as false. Expected code `BATCH10_LIVE_SOURCE_DRIFT_STALE_DIGEST_REPLAY_REFUSED`.\n\n###### `compile_bypass` hides invalid Python behind anchors\n\n**A body with the right anchors but invalid Python, digest refreshed, must be caught only by the compile engine counting fewer modules.** `compile_bypass` overwrites one copied body with text that carries the right anchor comments but is invalid Python, and refreshes its digest so only the compile check can catch it. The compile engine must then compile fewer modules than it counted. Expected code `BATCH10_LIVE_SOURCE_DRIFT_COMPILE_BYPASS_REFUSED`.\n\n###### `private_runtime_state_export` flips an authority flag\n\n**Turning on the runtime-state-export authorization flag in the policy must be rejected by the scope limit gate.** `private_runtime_state_export` sets `private_runtime_state_export_authorized` to true in the policy. The scope limit gate must reject it. Expected code `BATCH10_LIVE_SOURCE_DRIFT_PRIVATE_RUNTIME_EXPORT_REFUSED`.\n\n###### `live_mutation_authority_claim` raises three mutation flags\n\n**Setting the work log mutation, source-file changes, and mission execution flags true must all be rejected by the scope limit gate.** `live_mutation_authority_claim` sets the work log mutation, source-file changes, and mission execution flags to true. The scope limit gate must reject all three. Expected code `BATCH10_LIVE_SOURCE_DRIFT_MUTATION_AUTHORITY_REFUSED`.\n\n##### Prior Art Grounding\n\n###### Reproducible-build and supply-chain provenance practice\n\n**The discipline comes from reproducible builds and supply-chain provenance: declare inputs, fingerprint them, check artifacts against the fingerprints, and keep the stale digest visible as a fixture.** The pattern comes from reproducible-build and supply-chain provenance practice: declare the source inputs, fingerprint them, check copied or generated artifacts against those fingerprints, and keep private state out of the result record. Two useful anchors are [Bazel hermeticity](https://bazel.build/concepts/hermeticity), for its emphasis on declared inputs and repeatable outputs, and [SLSA provenance](https://slsa.dev/spec/), for recording how an artifact relates to its build inputs. This component applies that discipline to live source-copy drift, and keeps the stale digest visible as a regression fixture rather than overwriting it.\n\n##### Validation Result record Path\n\nRun from the `microcosm-substrate/` public root.\n\n###### What a pass and a blocked result mean\n\n**A pass writes four record types and rejects the negatives, while a blocked result is the valid drift signal to refresh the copies and rerun.** A pass writes digest-refresh, compile-gate, anchor-matrix, and scope limit records and confirms the four negative cases are rejected. A blocked result is a valid outcome when the copies have fallen behind the current source: it is the signal to refresh the copied bodies and manifest digests through the exact-copy source lane and rerun, not a defect in this page. When the sentinel is blocked, the focused test is expected to fail on pass-status or exact-body equality and should be reported with the same refresh residual.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The narrow proof boundary\n\n**The strongest claim is narrow: four copies can be compared, compiled, surface-checked, and policy-guarded, with four negatives rejected when the copy lane is current.** The strongest claim this evidence supports is narrow. Selected copies of four internal control source files can be compared against the current source digest, compiled without import execution, checked for their required command and contract surfaces, and guarded by a policy whose authority flags are all false, with four named negative cases rejected when the exact-copy lane is current. That is the proof boundary.\n\n###### What a green result does not grant\n\n**A green result grants no route, mutation, execution, commit, launch, or export authority, and the copied body is never an authority to run anything.** The scope limit is lower than the claim. A green result does not grant route authority, work log or work log mutation authority, mission execution, git staging or commit approval, source-file changes, launch, public sharing, external model access, or private runtime state export. It does not establish whole-system correctness. 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It checks copied trace parser, codemap layout, DAG scheduling, launch-root, source-surgeon, clean-clone, calculator, PageRank, and regression-selection witnesses, negative cases, digest boundaries, and scope limits while excluding launch-scope decision, whole-system equivalence, semantic truth, investment-related actions, sandbox completeness, and selected-test sufficiency proof.","authority_ceiling":"Fixture-bound public source-body import and deterministic exercise evidence only; no launch-scope decision, whole-system equivalence, semantic truth, investment-related actions, complete sandbox proof, selected-test sufficiency proof, external model access, or source-file changes.","abstract":"Set 7 Source Engines Bundle imports source engine bodies and exercises trace, graph, scheduling, source-index, patch, numeric, rank, and regression-selection invariants.","teleology":{"section_id":"","text":"Set 7 Source Engines Bundle imports source engine bodies and exercises trace, graph, scheduling, source-index, patch, numeric, rank, and regression-selection invariants.","source":"capsule_summary_line"},"body_markdown":"###### Eight copied engines under one public fixture\n\n** `batch7_macro_engines_capsule` takes eight unrelated engines that were copied out of the wider system as source and checks that each one still behaves the way its own tests and invariants say it should.** `batch7_macro_engines_capsule` takes eight unrelated engines that were copied out of the wider system as source and checks that each one still behaves the way its own tests and invariants say it should. The engines have nothing in common at runtime: a trace parser written in JavaScript, a dependency-graph scheduler, an AST source index, a patch applier, a network-isolation guard, a robust numeric scorer, a personalized PageRank ranker, and a regression-test selector.\n\n**The component runs each engine through one small deterministic exercise with a known answer, then records a metadata-only result row for it.** The component runs each engine through one small deterministic exercise with a known answer, then records a metadata-only result row for it. A pass means every engine produced its expected output and every paired negative case kept failing.\n\n**It writes `batch7_macro_engines_capsule_result.json`, a board, a validation result record, and a bundle result under the output directory the caller names.** It writes `batch7_macro_engines_capsule_result.json`, a board, a validation result record, and a bundle result under the output directory the caller names. Those records carry status, counts, digests, and engine ids. They never carry the copied source or captured command output: every row sets `body_in_receipt` to false.\n\n##### Purpose\n\n**Copied code is easy to trust and easy to be wrong about. A source body can be copied faithfully, pass a digest check, and still be broken or subtly different from the engine it was copied from.** Copied code is easy to trust and easy to be wrong about. A source body can be copied faithfully, pass a digest check, and still be broken or subtly different from the engine it was copied from. A digest match proves the bytes arrived. It says nothing about whether the code runs.\n\n**This component refuses to treat a digest match as behavior. Each engine has to execute and return the right value before its row is allowed to pass, and each engine's failure mode has to stay a failure.** This component refuses to treat a digest match as behavior. Each engine has to execute and return the right value before its row is allowed to pass, and each engine's failure mode has to stay a failure. The point is not that any single engine is interesting on its own. It is that eight engines with quite different runtimes, an external Node process, in-process Python calls, and static AST reads, can be exercised together under one fixture and one scope limit.\n\n##### How it works\n\n** `_evaluate` selects one of two paths from the input directory name. When the directory is `exported_batch7_macro_engines_capsule_bundle`, it calls `_source_open_bundle_exercises`, which gates all eight engine rows on two manifest fields, `all_expected_digests_matched` and `all_required_anchors_present`.** `_evaluate` selects one of two paths from the input directory name. When the directory is `exported_batch7_macro_engines_capsule_bundle`, it calls `_source_open_bundle_exercises`, which gates all eight engine rows on two manifest fields, `all_expected_digests_matched` and `all_required_anchors_present`. If either is false, every row is blocked. Otherwise `_evaluate` runs the eight live exercises in order. It records a finding for any row that did not pass and for any engine in `EXPECTED_ENGINES` that is missing from the result.\n\n**Each live exercise is a single function. Each returns a status plus the concrete facts it checked.** Each live exercise is a single function. Each returns a status plus the concrete facts it checked.\n\nFunction Engine What it checks `_agent_trace_exercise` `agent_trace_ir_compiler` Runs `node --test parser.test.mjs` against the copied `parser.mjs`. The suite includes a commit claim with no diff evidence, which the parser rejects. `_dag_exercise` `constitutional_dag_kernel` Calls `compute_waves` on a six-node graph and checks the schedule is `[[\"a\",\"f\"], [\"b\",\"c\"], [\"d\"], [\"e\"]]`. A two-node cycle must raise `ValueError`; `audit_config_purity` must flag an impure config. `_release_root_exercise` `release_root_compiler` Parses the copied `release_root_compiler.py` with `ast.parse` and checks `build_std_python_report` and `build_release_root_compiler` are defined and the source reports `missing_ref_count`. `_source_surgeon_exercise` `source_surgeon_patch` Uses `SourceSurgeon._apply_unified_diff_hunks`. A one-line diff must yield `a = 'B'`; a mismatched-context diff must raise `ApplyError`; malformed Python must raise `SyntaxError`. `_clean_clone_exercise` `hermetic_clean_clone` Swaps `socket.create_connection` and `socket.socket` for a raiser, confirms an outbound connection raises `NetworkDisabled`, then restores both. `_calculator_exercise` `calculator_standard_actor` Feeds `[1, 2, 3, 4, 5, 100]` to `StandardActor._compute_center_scale`. The robust center stays at `3.5` while the naive mean rises above `19`. `_pagerank_exercise` `personalized_pagerank_ranker` Calls `personalized_pagerank` on a four-node graph. Scores must sum to `1.0`; an unknown source node must return `{}`. `_regression_selection_exercise` `regression_test_selection` Runs `select_impacted_tests.py --json`, or reads the copied selector statically in the exported-bundle path, and checks the fallback test set is never empty.\n\n**Three helpers hold the orchestration together. `_safe_engine` wraps each runner and converts any exception into a blocked row rather than crashing the run.** Three helpers hold the orchestration together. `_safe_engine` wraps each runner and converts any exception into a blocked row rather than crashing the run. `_semantic_runtime_exercises` assembles the eight rows keyed by engine id and caches the result for a given input. `evaluate_negative_case` and `_observed_negative_case` confirm a declared negative case was actually seen before it counts, so a passing engine plus a witnessed rejection is what marks a case as covered.\n\n###### Shared crown-jewel entrypoint wiring\n\n** `run` and `run_batch7_bundle` forward `SPEC` to `run_crown_jewel_organ` with `_evaluate` and `evaluate_negative_case`. `result_card` projects the result into a compact card that carries `engine_count` and `copied_macro_source_module_count` and nothing else from the run.** `run` and `run_batch7_bundle` forward `SPEC` to `run_crown_jewel_organ` with `_evaluate` and `evaluate_negative_case`. `result_card` projects the result into a compact card that carries `engine_count` and `copied_macro_source_module_count` and nothing else from the run.\n\nDiagram of the mechanism (5 steps). Diagram source & refs\n\n```text\nflowchart TD input[\"Input directory\"] bundle[\"Exported bundle: gate 8 rows on manifest digests and anchors\"] live[\"Live fixture: run 8 engine exercises\"] neg[\"Negative cases: must keep failing\"] result[\"metadata-only result: status, counts, digests\"] input --> bundle input --> live live --> neg bundle --> result neg --> result\n```\n\n##### Negative cases\n\n**Each engine has a paired case that must be rejected, listed in `EXPECTED_NEGATIVE_CASES` with its stable error code:** Each engine has a paired case that must be rejected, listed in `EXPECTED_NEGATIVE_CASES` with its stable error code:\n- `trace_commit_without_diff` gives `BATCH7_TRACE_COMMIT_WITHOUT_DIFF_REJECTED`.\n- `dag_cycle` gives `BATCH7_DAG_CYCLE_REJECTED`.\n- `release_root_bad_ref` gives `BATCH7_RELEASE_ROOT_BAD_REF_REPORTED`.\n- `source_surgeon_context_mismatch` gives `BATCH7_SOURCE_SURGEON_CONTEXT_MISMATCH`.\n- `clean_clone_network_call` gives `BATCH7_CLEAN_CLONE_NETWORK_DISABLED`.\n- `calculator_outlier` gives `BATCH7_CALCULATOR_OUTLIER_RESISTANT`.\n- `pagerank_missing_source` gives `BATCH7_PAGERANK_MISSING_SOURCE_REFUSED`.\n- `regression_selection_empty` gives `BATCH7_RTS_NEVER_EMPTY_FALLBACK`.\n\n**A case only counts once `_observed_negative_case` sees the matching engine pass and the specific rejection field set, so a case cannot be marked covered by an engine that did not run.** A case only counts once `_observed_negative_case` sees the matching engine pass and the specific rejection field set, so a case cannot be marked covered by an engine that did not run.\n\n##### Prior Art Grounding\n\n**The engines sit on standard practice in trace instrumentation and graph analysis: parse execution traces into structured units of work, preserve graph-rank invariants, schedule a dependency graph into waves, and choose focused tests without claiming they are sufficient.** The engines sit on standard practice in trace instrumentation and graph analysis: parse execution traces into structured units of work, preserve graph-rank invariants, schedule a dependency graph into waves, and choose focused tests without claiming they are sufficient. Two anchors are directly relevant. [OpenTelemetry](https://opentelemetry.io/docs/) documents traces and spans as a vendor-neutral model for units of work and their relationships, which is the shape the trace parser produces. [NetworkX PageRank](https://networkx.org/documentation/stable/reference/algorithms/generated/networkx.algorithms.link_analysis.pagerank_alg.pagerank.html) documents the PageRank family this component's ranker follows. The regression selector follows ordinary test-impact-analysis practice with a never-empty fallback.\n\n##### Validation Result record Path\n\nRun from the `microcosm-substrate/` public root:\n\n**The fixture command runs the eight live exercises and writes the result and sign-off JSON. The bundle command validates the copied source witnesses against the manifest without emitting any source body.** The fixture command runs the eight live exercises and writes the result and sign-off JSON. The bundle command validates the copied source witnesses against the manifest without emitting any source body. The focused test covers the runtime path, the exported-bundle path, the negative cases, and the card's body omission. A pass across all of these means the fixture-bound run held; it does not extend past that.\n\n##### Scope boundary\n\n###### Scope limit\n\n**The strongest claim this component supports is bounded and fixture-scoped: eight copied engines were exercised through public fixtures, each produced its expected output, and each paired negative case kept failing.** The strongest claim this component supports is bounded and fixture-scoped: eight copied engines were exercised through public fixtures, each produced its expected output, and each paired negative case kept failing. That is validator evidence under the `AUTHORITY_CEILING` constant, which sets `release_authorized`, `provider_dispatch`, `source_mutation_authorized`, `investment_advice`, `semantic_truth_authority`, and `test_completeness_proof` all to false.\n\n**The proof boundary stops at the copied source and the deterministic exercises. A pass is not launch-scope decision, hosted-public authority, whole-system equivalence, a complete sandbox, or proof that the selected tests are sufficient.** The proof boundary stops at the copied source and the deterministic exercises. A pass is not launch-scope decision, hosted-public authority, whole-system equivalence, a complete sandbox, or proof that the selected tests are sufficient. The calculator is a numeric primitive, not market data or investment-related actions. The ranker checks a rank invariant, not meaning. The clean-clone check proves one hermetic baseline, not full isolation. 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and the deterministic exercises"]},{"id":"paper_module.batch9_macro_engines_capsule","title":"Set 9 Source Engines Bundle","relative_href":"docs/paper-module-batch9-macro-engines-capsule.html","read_href":"docs/paper-module-batch9-macro-engines-capsule.html","context_href":"docs/rules-and-ideas.html#paper-module-batch9-macro-engines-capsule","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"Set 9 Source Engines Bundle imports backend and governance source-engine bodies and exercises provenance, approval, AST, finance-news, mission graph, dependency, config, edge, WorkAtlas, host-pressure, doctrine-enrichment, worker-budget, and milestone-quality invariants.","summary":"Set 9 Source Engines Bundle binds the accepted batch9_macro_engines_capsule component to its public source-open bundle. It checks thirteen copied source engines for provenance lineage, approval adjudication, Python AST indexing, finance headline clustering, mission graph compilation, dependency pin drift, config authority audit, heterogeneous edge extraction, WorkAtlas aggregation, host-pressure admission, doctrine enrichment, worker job budget gating, and milestone-relative quality accounting while excluding live lineage truth, approval authority, market/news truth, host-state truth, work log truth, external model access, source-file changes, public sharing, and launch-scope decision.","authority_ceiling":"Fixture-bound public source-body import and deterministic exercise evidence only; no live lineage truth, human approval authority, real market/news truth, host-state truth, work log truth, external model access, source-file changes, public sharing, launch-scope decision, or whole-system equivalence.","abstract":"Set 9 Source Engines Bundle imports backend and governance source-engine bodies and exercises provenance, approval, AST, finance-news, mission graph, dependency, config, edge, WorkAtlas, host-pressure, doctrine-enrichment, worker-budget, and milestone-quality invariants.","teleology":{"section_id":"","text":"Set 9 Source Engines Bundle imports backend and governance source-engine bodies and exercises provenance, approval, AST, finance-news, mission graph, dependency, config, edge, WorkAtlas, host-pressure, doctrine-enrichment, worker-budget, and milestone-quality invariants.","source":"capsule_summary_line"},"body_markdown":"###### Ten copied engines run against public fixtures\n\n**This component copies ten backend and governance engine bodies into a public bundle and runs each against fixtures, emitting metadata-only records of what ran and whether the paired rejection fired.** This component copies ten backend and governance engine bodies out of the larger system into a public bundle, then runs each one against small fixtures to confirm the copied code still computes what it did before. It emits metadata-only records: which engine ran, what it produced, and whether the paired rejection fired.\n\n###### Execution tests behavior beyond matching digests\n\n**A digest proves the bytes match but not that the logic still behaves, so the bundle loads each copied module and calls its real functions rather than trusting baked-in output.** Copying a file into a public bundle proves that the bytes match. It does not establish that the logic still behaves. This component closes that gap by loading each copied module and calling its real functions, rather than comparing against output files baked once and trusted forever.\n\n###### One bundle spans ten bounded engine domains\n\n**The result is one reproducible evidence bundle spanning ten domains, from provenance lineage and approval adjudication to host-pressure admission and milestone-relative quality accounting.** The result is a reproducible evidence bundle for provenance lineage, approval adjudication, Python symbol indexing, mission-graph wave ordering, dependency pin drift, config-authority audit, host-pressure admission, doctrine enrichment, worker pre-dispatch gating, and milestone-relative quality accounting.\n\n##### Purpose\n\n###### A digest check cannot prove behavior\n\n**An exported copy can pass a digest check and still be dead, because a digest confirms identical bytes but says nothing about whether the function still runs.** An exported copy can pass a digest check and still be dead. A digest says the bytes are identical to what was copied. It says nothing about whether the function still runs, or whether a later edit quietly changed a branch. A result file that is written once and checked forever has the same weakness: it stops tracking the code it claims to describe.\n\n###### Live execution guards the loose-copy failure\n\n**The component ties proof to live execution, loading each engine from its copied file so the check moves when the body moves and a digest-faithful but behavior-lost bundle fails.** This component ties the proof to live execution. Each of the ten engines is loaded from its copied file and driven with a fixture, so the check moves when the copied body moves. That is the point of the \"loose copy\" failure it guards against: a bundle that looks faithful because the digest matches, while the behaviour it was supposed to preserve is gone.\n\n##### How it works\n\nThe pipeline runs in four stages, in this order.\n\n###### Stage one resolves the source manifest\n\n**First `_source_rows` reads `source_module_manifest.json`, one row per copied engine carrying a module id, target path, expected digest, and required anchors, while helpers locate each copied file.** First it resolves the source manifest. `_source_rows` reads `source_module_manifest.json`, one row per copied engine, each with a module id, a copied target path, an expected digest, and required anchor strings. `_source_module_path` and `_source_module_text` locate the copied file for a given module id.\n\n###### Stage two checks digests and anchors\n\n**Second, a copied body failing its manifest digest raises `CROWN_JEWEL_SOURCE_DIGEST_MISMATCH`, and one missing a required anchor raises `CROWN_JEWEL_SOURCE_ANCHOR_MISSING` even after the digest is refreshed.** Second it checks digests and anchors. A copied body whose bytes no longer match its manifest digest fails `CROWN_JEWEL_SOURCE_DIGEST_MISMATCH`. A body that has lost one of its required anchors fails `CROWN_JEWEL_SOURCE_ANCHOR_MISSING`, even when the digest was refreshed to match the edit.\n\n###### Stage three runs the ten exercises\n\n**Third, `_run_all_exercises` drives all ten: nine import the copied module and call its functions, while the text-backed mission graph gates a local reimplementation on anchors in the copied text.** Third it runs the exercises. `_run_all_exercises` drives all ten. Nine are path-backed: the copied module is imported with `_load_python_source_module` and its own functions are called against a fixture in a temporary directory. One, the mission graph, is text-backed: `_source_backed_mission_graph_contract` scans the copied text for branch anchors and gates a local reimplementation on whether they are present.\n\n###### Stage four evaluates and emits records\n\n**Fourth, `evaluate` collects exercise statuses, counts mechanisms and modules, and returns a `body_in_receipt` -false record, with `run` and `run_batch9_bundle` wrapping the two paths and `result_card` projecting a compact card.** Fourth it evaluates and emits. `evaluate` collects the exercise statuses, counts the declared mechanisms against `EXPECTED_MECHANISMS`, counts copied modules, and returns a record with `body_in_receipt` set to false. `run` wraps this for the fixture path and `run_batch9_bundle` for the exported bundle path. `result_card` projects a compact card carrying an authority floor and a body floor.\n\nEach engine is exercised through its real imported function:\n\nMechanism id Imported source Function called What the check confirms `lineage_temporal_provenance_chain_resolver` `system/server/lineage.py` `build_temporal_lineage`, `_resolve_primary_lineage` Parent-chain resolution; `resolve_lineage` sets `self_loop_pruned` when a node names itself as parent and `cycle_detected` when a run id repeats. `approval_sign_off_claim_adjudicator` `system/lib/approval_registry.py` `list_approvals`, `decide_approval` A pre-acquired claim with a different nonce makes `decide_approval` return `claim_conflict`, recorded as `preacquired_claim_refused`. `python_ast_symbol_index_doc_tree` `system/lib/python_documentation_tree.py` `build_file_entry` Symbols are read back from written-out Python; a file that will not parse sets `syntax_error_gap`. `mission_graph_topological_compiler` `system/server/graph.py` text anchors gate `compile_mission_graph` Group closure and upstream walk build waves by peeling indegree-zero nodes; a missing target blocks with `missing_target_error`. `dependency_pin_drift_auditor` `tools/dev/check_pin_drift.py` `parse_requirements`, `evaluate` Requirements are bucketed ok, drifted, missing, or unparseable. `config_authority_drift_audit` `system/lib/config_authority_registry.py` `validate_config_authority_registry` A registry that marks a generated projection mutable is rejected. `host_pressure_admission_decision_gate` `system/lib/admission_consumer.py` `build_admission_consumer_decision` Under the auto policy a summary-first recommendation blocks new work, setting `auto_policy_blocked`. `doctrine_file_enrichment_multihop_join` `system/server/doctrine_enrichment.py` `DoctrineEnrichmentService.get_file_doctrine` An indexed path returns a populated envelope; an unindexed path returns an empty one. `worker_job_budget_forbidden_surface_gate` `system/lib/type_a_worker_harness.py` `_enforce_budget`, `_contains_forbidden_surface` A job over budget or naming a forbidden surface is blocked before dispatch. `milestone_relative_promotion_quality_accounting` `system/lib/population_lane_metrics.py` `classify_blockers_and_next_action` Milestone-relative promotion metrics are computed; runs missing a committed-at timestamp are counted into a bucket.\n\n###### Local reference implementations define expected shape\n\n**The component also carries local reference implementations that define the expected shape, but for nine engines the imported copy runs, with the mission graph the sole case where the local body runs.** The component also carries local reference implementations, such as `resolve_lineage`, `build_ast_doc_tree`, and `compile_mission_graph`. These define the expected shape. For nine engines the imported copy is what actually runs; the mission graph is the one case where the local body runs, gated by anchors found in the copied text.\n\nDiagram of the mechanism (6 steps). Diagram source & refs\n\nSource refs\n\n10 copied engine bodies `source_module_manifest.json`\n\n```text\nflowchart TD manifest[\"source_module_manifest.json 10 copied engine bodies\"] fixtures[\"fixture input positive fixture + 10 negative cases\"] digest[\"digest + anchor check bytes match, anchors present\"] exercises[\"_run_all_exercises load each copy, call its functions\"] neg[\"negative cases apply fixture_patch, rerun\"] emit[\"evaluate metadata-only record + card\"] manifest --> digest fixtures --> exercises digest --> exercises exercises --> neg exercises --> emit neg --> emit\n```\n\n##### Negative cases\n\n**Each engine ships one paired rejection, and `evaluate_negative_case` patches the positive fixture, reruns the exercises, and blocks with a stable error code only when the rejection actually fired.** Each engine ships one paired rejection. `evaluate_negative_case` takes a case id, loads `<case_id>.json`, applies its `fixture_patch` onto the positive fixture with `_deep_update`, reruns the exercises, and calls `_negative_case_observed` to inspect one field. If the rejection fired the case returns status blocked with its stable error code; if it did not, the case fails. The ten cases and their codes:\n- `lineage_self_loop_pruned` -> `BATCH9_LINEAGE_SELF_LOOP_PRUNED`\n- `approval_preacquired_claim_refused` -> `BATCH9_APPROVAL_PREACQUIRED_CLAIM_REFUSED`\n- `ast_syntax_error_gap` -> `BATCH9_AST_SYNTAX_ERROR_GAP`\n- `mission_graph_missing_target` -> `BATCH9_MISSION_GRAPH_MISSING_TARGET`\n- `dependency_pin_drift_detected` -> `BATCH9_DEPENDENCY_PIN_DRIFT_DETECTED`\n- `config_authority_mutation_allowed_rejected` -> `BATCH9_CONFIG_AUTHORITY_MUTATION_ALLOWED_REJECTED`\n- `host_pressure_auto_policy_blocks` -> `BATCH9_HOST_PRESSURE_AUTO_POLICY_BLOCKS`\n- `doctrine_enrichment_unindexed_empty` -> `BATCH9_DOCTRINE_UNINDEXED_EMPTY`\n- `worker_forbidden_surface_blocked` -> `BATCH9_WORKER_FORBIDDEN_SURFACE_BLOCKED`\n- `milestone_missing_committed_at_bucketed` -> `BATCH9_MILESTONE_MISSING_COMMITTED_AT_BUCKETED`\n\n###### Two tamper modes on the copied bodies\n\n**Two tamper modes sit alongside these: an unrefreshed digest fails `CROWN_JEWEL_SOURCE_DIGEST_MISMATCH`, and a removed anchor still fails `CROWN_JEWEL_SOURCE_ANCHOR_MISSING`, keeping fixture-only replacements outside the proof shape.** Two tamper modes sit alongside these. An edit to a copied body with no manifest digest refresh fails `CROWN_JEWEL_SOURCE_DIGEST_MISMATCH`. Removing a required anchor while refreshing the digest still fails `CROWN_JEWEL_SOURCE_ANCHOR_MISSING`. Together these keep a fixture-only or static-record replacement outside the accepted proof shape.\n\n##### Prior Art Grounding\n\n###### Golden-master testing that runs the copy\n\n**The component follows the characterization or golden-master tradition, pinning behavior with deterministic fixtures, but its variation runs the copied code rather than comparing against a frozen output file.** This component follows the characterization, or golden-master, testing tradition ( [Feathers, Working Effectively with Legacy Code](https://en.wikipedia.org/wiki/Characterization_test)), which pins existing behaviour with deterministic fixtures before trusting it. The variation here is that the pin runs the copied code rather than comparing against a frozen output file. The result is fixture-bound import evidence, not lineage truth, human approval authority, or market-level conclusions.\n\n##### Validation Result record Path\n\nRun these from the `microcosm-substrate/` public root:\n\n###### What the four commands together prove\n\n**The four commands prove the fixture and bundle paths, the focused test covers imports and rejections, and the corpus check confirms the projections stay consistent, nothing wider.** The first command proves the public fixture path. The second proves the exported bundle path. The focused test covers exact-copy imports, source-sensitive behaviour shifts, digest-mismatch blocking, required-anchor removal, and the semantic rejections above. The corpus check confirms the generated instance, Mermaid, Atlas card, and this Markdown stay mutually consistent. A pass means the copied bodies and fixtures ran and matched here, nothing wider.\n\n##### Scope boundary\n\n###### Scope limit\n\n**The strongest supported claim is narrow: the ten copied bodies still behave like their imported source under a bounded positive case and one paired rejection, checked in the public clone.** The strongest claim this component supports is narrow: the ten copied engine bodies and their public fixtures show that each engine still behaves like its imported source under a bounded positive case and one paired rejection, checked here in the public clone. Its proof boundary is the copied-body import, the digest and anchor checks, the ten exercises, the ten negative cases, and the metadata-only records they emit.\n\n###### What the scope limit excludes\n\n**It refuses everything past that: no live lineage, approval, market, host-state, or work log truth, no dispatch, mutation, or launch, since generated views are reader surfaces, source-linked only.** It refuses everything past that. It is not live lineage truth, human approval authority, real market or news truth, host-state truth, work log truth, external model access, source-file changes, public sharing, launch-scope decision, whole-system equivalence, or whole-system correctness. The scope limit excludes all of these. The generated Markdown, JSON, and site views are reader surfaces; source authority stays in the JSON contract, the source manifest, the tests, and the records.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":13819,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/batch9_macro_engines_capsule.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"}],"grouped_relation_feeds":[{"relation":"abides by","count":4,"links":[{"id":"AX-4","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Content-addressed determinism","url":"doctrine.html#dcard-ax-4"},{"id":"AX-8","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Provenance propagation and non-interference","url":"doctrine.html#dcard-ax-8"},{"id":"AX-10","kind":"axiom","kind_label":"axiom","relation":"abides 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Bundle","url":"paper-module-batch10-governance-compilers-capsule.html","relation_id":"paper_module.depends_on.paper_module","source_ref":"core/paper_module_capsules.json::paper_modules[72:paper_module.batch10_governance_compilers_capsule].depends_on"}],"source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/batch9_macro_engines_capsule.md"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json"}],"visible_text_byte_count":15387,"section_titles":["Ten copied engines run against public fixtures","Execution tests behavior beyond matching digests","One bundle spans ten bounded engine domains","Purpose","A digest check cannot prove behavior","Live execution guards the loose-copy failure","How it works","Stage one resolves the source manifest","Stage two checks digests and anchors","Stage three runs the ten exercises","Stage four evaluates and emits records","Local reference implementations define expected shape","Negative cases","Each engine ships one paired rejection","Two tamper modes on the copied bodies","Prior Art Grounding","Golden-master testing that runs the copy","Validation Result record Path","What the four commands together prove","Scope limit","The strongest supported claim is narrow","What the scope limit excludes"]},{"id":"paper_module.pattern_assimilation","title":"Pattern Assimilation","relative_href":"docs/paper-module-pattern-assimilation.html","read_href":"docs/paper-module-pattern-assimilation.html","context_href":"docs/rules-and-ideas.html#paper-module-pattern-assimilation","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"Pattern Assimilation validates public completion-learning metadata, owner-routed refinement result records, typed nothing-to-refine decisions, and copied body-import manifests without promoting local lessons into global doctrine.","summary":"Pattern Assimilation binds the accepted pattern_assimilation_step component to the public sign-off validator, first-wave fixture, exported assimilation bundle, source-module manifest, standard row, and metadata-only result records. It checks same-lane completion decisions, owner-surface refinement evidence, stewardship and re-entry fields, duplicate result record rejection, local-lesson scope limits, source note exclusion, copied body imports, and public-relative result record paths while excluding live ledger mutation, source notes ingestion, model-output data, global doctrine changes, launch, public sharing, behavior-change proof, and non-public data equivalence.","authority_ceiling":"Public fixture metadata, exported assimilation bundle metadata, copied body-import digest evidence, and metadata-only result records only; no live work log or work log mutation, source note ingestion, global doctrine changes, launch or publishing-scope decision, external model access, behavior-change proof, non-public data equivalence, or whole-system correctness.","abstract":"Pattern Assimilation validates public completion-learning metadata, owner-routed refinement result records, typed nothing-to-refine decisions, and copied body-import manifests without promoting local lessons into global doctrine.","teleology":{"section_id":"","text":"Pattern Assimilation validates public completion-learning metadata, owner-routed refinement result records, typed nothing-to-refine decisions, and copied body-import manifests without promoting local lessons into global doctrine.","source":"capsule_summary_line"},"body_markdown":"###### Checkable pattern assimilation from landed evidence\n\n** `pattern_assimilation_step` turns a pass's claim to have learned something from uncheckable commit prose into a typed record a validator can read.** `pattern_assimilation_step` checks whether a development pass that claims to have learned something actually filed an inspectable record of it. When a component lands in the system, someone is supposed to close it out with a decision: either this run changed a named artifact, or it changed nothing and here is why. That decision is usually prose in a commit message. Prose cannot be checked. This component forces the decision into a typed record and validates the record.\n\n**Over landed-component rows and their completions, it confirms each landing has one same-lane completion that names its surface, carries its no-op fields, and never promotes a local lesson into global doctrine.** It runs over fixture data, not live state. Given a set of landed-component rows and the completion records that go with them, it confirms that each landed component has exactly one same-lane completion, that a claimed change names the surface it touched, that a \"nothing to refine\" record carries its required fields, and that no record promotes a local lesson into global doctrine. The output is a result record listing which checks passed and which named negative cases fired.\n\n**sign-off.py is the entry point, with a fixture lane over the completion records directly and a bundle lane over an exported package plus copied source bodies.** The entry point is src/microcosm_core/validators/sign-off.py. It has two lanes: a fixture lane that validates the completion records directly, and a bundle lane that validates an exported package of the same records plus copied source bodies.\n\n##### Purpose\n\n###### The unbacked learning claim\n\n**The failure guarded against is a learning claim nobody can audit, where loose prose about improving or finding nothing drifts free of the artifacts.** The failure this guards against is the unbacked learning claim. A pass says it \"improved the fixture\" or \"found nothing to refine\", and nobody can tell whether that is true. Loose prose is easy to assert and impossible to audit, so the learning story drifts away from the artifacts.\n\n###### Typed fields a validator can read\n\n**The fix makes each completion a validator-readable record: a change names its owner surface and artifact, a no-op proves stewardship, next-best lane, and re-entry, and a missing or mismatched completion is rejected.** The fix is to make each completion resolve to a record with fields a validator can read. A concrete change record must name an `owner_surface` and the artifact it touched. A `nothing_to_refine` record must prove three things: stewardship was checked, the next-best lane was considered, and a re-entry condition was written down. A landed component with no completion, or with a completion that points at a record that does not exist or does not match, is rejected.\n\n###### A local lesson cannot self-promote to doctrine\n\n**A lesson may route to the owner surface of its artifact but never promote itself to global doctrine; a record setting `claims_global_doctrine_authority` is rejected outright.** One constraint the component will not relax: a local lesson may route to the owner surface that owns the affected artifact, but it may not promote itself into global doctrine. A change record that sets `claims_global_doctrine_authority` is rejected outright. Learning has to land on a specific board with a named steward, not become a free-floating rule.\n\n##### How it works\n\n###### The fixture lane in `validate_pattern_assimilation`\n\n** `validate_pattern_assimilation` loads four inputs, scans them for non-public-state leakage, runs the row checker, and passes only when every expected negative case fired, the scan is clean, and nothing blocked.** The fixture lane runs in `validate_pattern_assimilation`. It loads four input files (`_load_inputs`), scans them for non-public-state leakage (`_scan_fixture_inputs`), runs the row checker (`_validate_rows`), then compares the negative cases that fired against `EXPECTED_NEGATIVE_CASES`. The run passes only when every expected negative case was observed, the non-public-state scan is clean, and no blocking finding was raised. Results are written by `write_outputs`.\n\n###### Per-row validation in `_validate_rows`\n\n** `_validate_rows` pre-filters change and no-op records down to the valid ones, then walks each landed row to raise a missing, unresolved, or duplicate finding.** `_validate_rows` is the mechanism. It first pre-filters the completion records into valid ones. A change record is valid when it has a result record id and component id, its `refinement_result` is `fixture_manifest_refined` or `validator_contract_refined`, it names an `owner_surface`, and it does not claim global doctrine authority. A no-op record is valid when its `refinement_result` is `nothing_to_refine` and it carries `stewardship_checked`, `next_best_lane_checked`, and a `reentry_condition`. It then walks each landed row: a row with no completion result raises `MISSING_PATTERN_ASSIMILATION_CLOSEOUT`; a row whose declared completion does not resolve to a matching valid record raises `PATTERN_ASSIMILATION_CLOSEOUT_RECEIPT_REF_UNRESOLVED` as a blocking finding. Duplicate result record ids across both record sets raise `DUPLICATE_REFINEMENT_RECEIPT_ID` so one lesson cannot be counted twice.\n\n###### Resolving the landing result record reference\n\n** `_validate_public_safe_receipt_ref` confirms each landing's referenced record sits inside a public root, parses as `.json` or `.jsonl`, passes the scan, and hashes, blocking any absent or unsafe citation.** Each landed row also names a landing result record, and `_validate_public_safe_receipt_ref` resolves that reference. It confirms the ref points inside a public result record root, is a `.json` or `.jsonl` file that parses, passes the non-public-state scan, and hashes it. A ref that escapes the public roots, is missing, has the wrong suffix, or trips the scan becomes a blocking finding, so a completion cannot cite a record that is absent or unsafe.\n\n###### The bundle lane in `run_assimilation_bundle`\n\n** `run_assimilation_bundle` runs per-part checkers over an eight-part export, then verifies copied source bodies against their recorded sha256, passing only when the scan is clean and every checker is empty.** The bundle lane runs in `run_assimilation_bundle`. It loads an eight-part export (`_load_assimilation_bundle`) and runs a chain of per-part checkers: `validate_exported_organ_landings`, `validate_exported_refinement_receipts`, `validate_exported_nothing_to_refine_receipts`, `validate_exported_closeout_receipt_links`, `validate_exported_stewardship_checks`, `validate_exported_reentry_conditions`, `validate_exported_next_best_lane_checks`, and `validate_exported_assimilation_policy`. It then verifies copied source bodies through `validate_source_module_manifest`, which reads `source_module_manifest.json`, requires `body_in_receipt` to be false, and, for each copied module, checks its declared class is in `PUBLIC_SAFE_SOURCE_BODY_CLASSES`, the target file exists, and its recorded sha256 matches the file on disk. A mismatch raises `ASSIMILATION_BUNDLE_SOURCE_MODULE_INVALID`. The bundle passes only when the scan is clean, every checker is empty of findings, and each record set is non-empty; a stable fingerprint hash of the inputs is stamped into the result.\n\n###### Atomic result writes and CLI dispatch\n\n** `_write_jsonl_upsert` atomically replaces a run's own row instead of duplicating it, and `main` dispatches the bundle and fixture lanes, each returning 0 on pass and 1 on block.** Result records are written by `_write_jsonl_upsert` for the append-only run log: it drops any prior line with the same `run_id`, appends the new row, and swaps the file in atomically through a temp file and `os.replace`, so a re-run replaces its own row instead of duplicating it. The CLI dispatch is `main`: the `validate-assimilation-bundle` subcommand runs the bundle lane, and `--only pattern_assimilation_step --input --out` runs the fixture lane. Each returns 0 on pass, 1 on block. The stamped scope limit is the constant `PATTERN_ASSIMILATION_AUTHORITY_CEILING`.\n\nDiagram of the mechanism (7 steps). Diagram source & refs\n\n```text\nflowchart TD inputs[\"Four fixture files landings + change + no-op + missing-case\"] scan[\"non-public-state scan _scan_fixture_inputs\"] rows[\"Row checker _validate_rows\"] filter[\"Pre-filter valid records change: owner surface, no doctrine upgrade no-op: stewardship, next-best lane, re-entry\"] walk{\"Per landed row: one completion, ref resolves to a matching valid record?\"} compare[\"Compare fired cases vs EXPECTED_NEGATIVE_CASES\"] out[\"Result records write_outputs, _write_jsonl_upsert\"] inputs --> scan scan --> rows rows --> filter filter --> walk walk -->|resolved| compare walk -->|missing, dangling, duplicate, upgraded| compare compare --> out\n```\n\n##### Negative cases\n\n###### A silenced check blocks the run\n\n**Every required negative case must fire on the fixture, and any case that stops firing blocks the run because a check has gone silent.** The fixture is built to make every required negative case fire. If any stops firing, the run blocks, because that means a check has gone silent.\n\n###### The reason code each attempt raises\n\n**Each gaming shape maps to a named code: missing completion, missing owner, stewardship, or re-entry fields, duplicate result record, doctrine self-upgrade, a seed body in the fixture, and a bundle source mismatch.** `MISSING_PATTERN_ASSIMILATION_CLOSEOUT` fires for a landed component with no change record and no typed no-op record. `MISSING_REFINEMENT_OWNER_SURFACE`, `MISSING_STEWARDSHIP_CHECK`, and `MISSING_REENTRY_CONDITION` fire for records that cannot route the lesson to an owner surface and a re-entry condition. `DUPLICATE_REFINEMENT_RECEIPT_ID` fires for a repeated result record id. `LOCAL_LESSON_AUTHORITY_UPGRADE` fires for a local lesson that claims global doctrine authority. `RAW_SEED_BODY_IN_ASSIMILATION_FIXTURE` fires when a seed-origin body appears in the public fixture, and the body is rejected and redacted. In the bundle lane, `ASSIMILATION_BUNDLE_SOURCE_MODULE_INVALID` fires on a copied-source digest or class mismatch.\n\n##### Prior Art Grounding\n\n###### Pattern languages and provenance tracing\n\n**The design draws on software pattern-language practice for naming context-bound lessons and on provenance and tracing practice for citing owner surface and touched artifact, applied here to completion bookkeeping only.** The idea of naming recurring engineering lessons and binding each to its context comes from software pattern-language practice; the [Hillside patterns library](https://hillside.net/patterns/) is the direct lineage for treating patterns as shared vocabulary rather than one-off notes. The record shape borrows from provenance and tracing practice: [W3C PROV](https://www.w3.org/TR/prov-overview/) informs the rule that each change record cite its owner surface and the artifact it touched, and [OpenTelemetry traces](https://opentelemetry.io/docs/concepts/signals/traces/) are a fair analogue for linking work into an inspectable chain. This component applies those ideas to completion bookkeeping only.\n\n##### Validation Result record Path\n\n###### Running the assimilation fixture lane\n\n**From `microcosm-substrate`, run the fixture lane and the bundle lane into an isolated `/tmp` output so scratch records do not overwrite tracked files:** From `microcosm-substrate`, run the fixture lane and the bundle lane into an isolated `/tmp` output so scratch records do not overwrite tracked files:\n\nFocused test re-entry:\n\n```text\nPYTHONPATH=src ../repo-python -m pytest -p no:cacheprovider \\\n  tests/test_pattern_assimilation_step.py -q \\\n  --basetemp=/tmp/microcosm_pattern_assimilation_pytest\n```\n\n###### What a pass and a block mean\n\n**A pass means the fixture completions had the required typed shape and every negative case fired; a block means a completion was missing, dangling, duplicated, over-claiming, or drifted from its digest.** A pass means the supplied fixture completions have the required typed shape and every required negative case fired. A block means a completion is missing, dangling, duplicated, or over-claiming, or a copied source body drifted from its recorded digest.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### What the evidence supports\n\n**Over the fixtures and copied bodies, the proof is that completion learning has a typed, same-lane, owner-routed shape and that invalid completion claims are rejected before they can pass as doctrine.** Over the supplied fixtures and copied body imports, this component shows that completion learning has a typed, same-lane, owner-routed shape and that invalid completion claims are rejected before they can pass as doctrine. It checks owner-surface change evidence, typed `nothing_to_refine` records, stewardship and re-entry fields, duplicate result record rejection, seed-body exclusion, public relative paths, and copied-source digest verification.\n\n###### What stays outside the boundary\n\n**The boundary reads no private lessons, mutates no ledger, promotes no doctrine, and excludes launch; the ceiling is public metadata plus metadata-only result records, nothing more.** That is the whole proof boundary. It does not read private lessons or seed bodies, mutate live ledger state, promote global doctrine, include launch operations or public sharing, use external model services, claim non-public data equivalence, prove a behavior change, or certify whole-system correctness. The scope limit is public fixture and bundle metadata plus metadata-only result records, nothing more. The generated diagram and the Atlas card are navigation surfaces; they do not upgrade a local lesson into global doctrine.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":13970,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/pattern_assimilation.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"Hillside patterns library","url":"https://hillside.net/patterns/","kind":"external_reference","section_id":"pattern-languages-and-provenance-tracing"},{"label":"W3C PROV","url":"https://www.w3.org/TR/prov-overview/","kind":"external_reference","section_id":"pattern-languages-and-provenance-tracing"},{"label":"OpenTelemetry 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and recent-change source bodies as public source-open evidence without granting live ledger, public sharing, launch, market, or source-file changes.","summary":"Set 10 Governance And Compilers Bundle binds the accepted batch10_governance_compilers_capsule component to a refreshed source-open bundle. It exercises governed-mutation intent, observe/apply compilation, public-proof review, launch blocker triage, public sharing path contracts, result record reuse, no-lookahead horizons, session-wave execution, claim-conflict wait tax, role-aware DAG blocking, frontend table shaping, reference grouping, recent-change coalescing, and the deferred Set-9 lane-width repair while preserving copied source digests, negative cases, and scope limits.","authority_ceiling":"Fixture-bound public source-body import, source-faithful public refactor evidence, deterministic exercise evidence, and metadata-only result records only; no live work log truth, live work log truth, source-file changes, publishing-scope decision, launch-scope decision, external model access, market advice, whole-system equivalence, neutral benchmark claim, or whole-system correctness.","abstract":"Set 10 Governance And Compilers Bundle imports governance, compiler, launch, finance, dependency, DAG, table, reference, and recent-change source bodies as public source-open evidence without granting live ledger, public sharing, launch, market, or source-file changes.","teleology":{"section_id":"","text":"Set 10 Governance And Compilers Bundle imports governance, compiler, launch, finance, dependency, DAG, table, reference, and recent-change source bodies as public source-open evidence without granting live ledger, public sharing, launch, market, or source-file changes.","source":"capsule_summary_line"},"body_markdown":"###### Four governance compilers run from copied bodies\n\n**Eleven governance and compiler mechanisms are copied into a public bundle and re-run against bundle-built inputs, with a result record for each that separates a computed failure from a fixture-read one.** This component takes eleven governance and compiler mechanisms that run in the wider system, copies their source bodies into a public bundle, and re-runs a small source-faithful port of each one against inputs it constructs itself. It writes a result record for every mechanism: what source it came from, what the positive run produced, what each planted failure case produced, and whether the bundle computed that failure itself or only read it from a fixture.\n\n###### One runner exercises all compiler fixtures\n\n**The problem is trust: a cold reader cannot confirm the system's own claims, so this re-runs the logic in the open and puts the computed value beside the expected one.** The problem it solves is trust. When the system says a mutation gate blocks a write, or a finance horizon rejects a bad date, a cold reader has no way to confirm that from the outside. This component re-runs the logic in the open and records the computed value next to the expected one.\n\n###### Compiler evidence stops before live governance\n\n**It exercises the `batch10_governance_compilers_capsule` component and grants no launch, public sharing, live-ledger, or source-file changes.** It exercises `batch10_governance_compilers_capsule` in `src/microcosm_core/organs/batch10_governance_compilers_capsule.py`. It is not launch-scope decision, publishing-scope decision, live ledger truth, or source-file changes permission.\n\n##### Purpose\n\n###### Why a declared error code proves nothing\n\n**A fixture that names its own expected error code proves nothing, because the declaration can be right while the ported logic is broken or absent.** A fixture that declares \"this case should return error code X\" proves nothing on its own. The declaration can be right while the ported logic is broken, or absent. A loose bundle would let a case grade itself green by assertion.\n\n###### Labels are input, not verdict\n\n**Each case's on-disk label and error code are treated as input, recomputed in code and compared, and a case with no computed evidence raises `fixture_verdict_echo_risk` and blocks the run.** This component refuses that. Every negative case carries a label and a stable error code on disk, but the bundle treats those as input, not verdict. For each case it recomputes the outcome in code and compares the computed value to the expectation. The result record marks each case `fixture_role: negative_case_label_not_verdict_authority` and stamps the verdict `computed_by_batch10_governance_compilers_capsule_integrity_matrix`. A case with no computed evidence raises `fixture_verdict_echo_risk` and blocks the run.\n\n##### How it works\n\n###### Two entry points over one evaluator\n\n** `run` and `run_batch10_governance_compilers_bundle` both feed `_evaluate`, which builds eleven mechanism matrices before `_build_integrity_matrix` writes one row each.** `run` and `run_batch10_governance_compilers_bundle` are the two entry points; both hand `_evaluate` to a shared runner. `_evaluate` builds eleven mechanism matrices, then `_build_integrity_matrix` writes one row per mechanism.\n\n###### Each compiler has a controlled matrix\n\n**Every governance mechanism uses its own matrix function to construct controlled input and assert the copied logic's behavior.** Each mechanism has its own matrix function that constructs a controlled input and asserts the ported logic behaves.\n\nMechanism Matrix function What it exercises `mutation_governance_intent_gate` `_mutation_governance_matrix` `classify_latest_user_intent` and `latest_intent_gate`: a diagnostic message blocks file writes, an implement message allows a patch `observe_apply_plan_compiler` `_observe_apply_compiler_matrix` `_extract_operations_from_artifact`: a well-formed artifact yields operations, a malformed one yields none `flagship_reviewer_persona_gauntlet_adjudicator` `_reviewer_gauntlet_matrix` six reviewer personas over a public proof bundle via `_persona_results_from_public_artifact` and `_status_from_requirements` `release_public_toggle_blocker_closure_triage` `_release_toggle_triage_matrix` `_blocker_class`: a no-go toggle with an open blocker is not operator-review-ready `publication_manifest_selector_contract_verifier` `_publication_manifest_contract_matrix` `_hard_exclude_selectors` and `_path_matches_selector`: a non-public paths against a hard-exclude rule is caught `constitution_workspace_receipt_reuse_decider` `_receipt_reuse_matrix` `_fingerprint_rows_by_path` and `_changed_stable_inputs`: a changed stable input invalidates result record reuse `finance_no_lookahead_temporal_contract` `_finance_temporal_matrix` `resolve_horizon`: an unparseable horizon is rejected `session_dependency_wave_executor` `_session_wave_matrix` `_session_node_state`: a child whose dependency failed is skipped `claim_conflict_wait_tax_detector` `_claim_conflict_wait_tax_matrix` `_path_overlaps`: a claim inside an active scope is a conflict `role_aware_dag_block_propagation` `_role_aware_dag_matrix` `_resolve_blocked_groups` and `_resolve_group_state`: a quality error on an upstream probe does not over-block `weighted_lane_width_apportionment_binding_repair` `_lane_width_binding_matrix` `solve_root_object_lane_widths`, held as a deferred repair, not a fresh import\n\n###### How `_source_evidence` records each source\n\n** `_source_evidence` reads the manifest and records each mechanism as a verbatim copy or a source-faithful refactor, blocking the under-bound lane-width repair rather than presenting it as an import.** Source evidence is resolved per mechanism by `_source_evidence`. It reads the bundle's `source_module_manifest.json`, looks up each mechanism's named source reference, and records what it finds. A verbatim copy carries its digest status and body-copied flag. A body that cannot ship as-is is recorded as a source-faithful public refactor with the original source digest, not a copy. `publication_manifest_selector_contract_verifier` is a refactor because its source carried a private home-path example. `weighted_lane_width_apportionment_binding_repair` is classified `under_bound_binding_repair`, so its integrity row is forced to `current_action: block` rather than presented as a Set 10 import.\n\n###### Recomputing every negative case\n\n** `_compute_negative_case_probe` recomputes each case from its `probe_input`, and a missing computation, or fewer than ten copied source modules, turns into a blocking finding.** Negative cases run through `_compute_negative_case_probe`. For each case it reads `probe_input` from the fixture, computes the observed value, and compares it to the binding's expected value. A row counts as proven only when the computed value matches. `_build_integrity_matrix` sets `fixture_verdict_echo_risk` on any mechanism whose cases are missing or uncomputed, and `_evaluate` turns any such count into a blocking finding. The run also requires exactly ten copied source modules; a thinned bundle raises `BATCH10_SOURCE_MODULE_COUNT_INVALID` and blocks. `result_card` projects the pass into a compact card with the mechanism count and passed-mechanism count.\n\nDiagram of the mechanism (6 steps). Diagram source & refs\n\nSource refs\n\nrequire 10 copied modules `Validate source_module_manifest.json`\n\n```text\nflowchart TD Input[\"Public input dir or exported bundle\"] Manifest[\"Validate source_module_manifest.json require 10 copied modules\"] Matrices[\"Build 11 mechanism matrices each on bundle-constructed input\"] Integrity[\"_build_integrity_matrix one row per mechanism recompute each negative case\"] Echo[\"fixture_verdict_echo_risk? any uncomputed negative case\"] Result[\"Result record: source evidence, computed outputs, scope limit body_in_receipt: false\"] Input --> Manifest --> Matrices --> Integrity --> Echo Echo -->|no risk| Result Echo -->|risk found| Result\n```\n\n##### Negative cases\n\nThe eleven planted cases, each recomputed rather than trusted:\n- `mutation_status_intent_blocks_writes`: a diagnostic message on an evolution route sets `prohibit_file_writes` to true.\n- `observe_apply_malformed_artifact_refused`: an artifact with a non-list `operations` field yields no operations.\n- `reviewer_missing_boundary_detected`: a missing boundary doc drops the safety persona to `warn`.\n- `release_toggle_no_go_fail_closed`: a no-go toggle with one blocker is not operator-review-ready.\n- `publication_hard_exclude_rejected`: `private/raw_seed.md` against a `private/` hard-exclude is rejected.\n- `receipt_reuse_stable_input_changed`: a changed stable digest counts one changed input.\n- `finance_invalid_horizon_rejected`: the horizon `not-a-date` is rejected.\n- `session_failed_dependency_skips_child`: node C is skipped when its dependency B failed.\n- `claim_parent_child_overlap_reported`: a claim path inside an active scope is one conflict.\n- `dag_quality_error_not_overblocked`: a quality error on an upstream probe is softened, not blocked.\n- `lane_width_binding_deferred_to_batch9`: the lane-width repair resolves to `under_bound_repair_deferred_to_batch9_claim`.\n\n##### Prior Art Grounding\n\n###### Policy-as-code and supply-chain assurance\n\n**The design borrows the gate-and-compiler shape from Open Policy Agent, Kubernetes admission policies, and SLSA with OpenSSF Scorecard, staying fixture-bound evidence over copied or refactored source.** The component follows policy-as-code, admission-control, and supply-chain assurance patterns: compile rules into deterministic checks, reject unsupported actions before they mutate state, and keep provenance for the decision. Honest anchors are [Open Policy Agent](https://www.openpolicyagent.org/docs/latest), which separates policy decisions from enforcement and evaluates structured input against machine-readable rules; Kubernetes [validating admission policies](https://kubernetes.io/docs/concepts/policy), which block, warn, or audit API requests before admission; and [SLSA](https://slsa.dev/spec/) with [OpenSSF Scorecard](https://openssf.org/scorecard/), which represent typed assurance levels, checks, and provenance for a software supply chain. This component borrows the gate-and-compiler shape for the eleven mechanisms above and stays fixture-bound evidence over copied or refactored source.\n\n##### Validation Result record Path\n\nRun from the `microcosm-substrate/` public root:\n\n###### What the two commands write and prove\n\n**The first command writes the result record, the second validates source and rows, and a pass means computed evidence and metadata-only records, not launch or whole-system correctness.** The first command writes the integrity-matrix result record and sign-off JSON. The second validates the copied and refactored source, the source evidence, the positive and negative rows, and the scope limit fields. A pass means the mechanisms had computed source evidence, computed positive and negative rows, and metadata-only records; it does not mean launch or whole-system correctness.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The fixture-bound proof boundary\n\n**Over the public fixtures the eleven mechanisms carry source evidence, computed rows, and metadata-only records, and every negative case is recomputed so no fixture passes itself by declaration.** The strongest claim: over the public fixtures and the refreshed source-open bundle, the eleven governance and compiler mechanisms have copied or declared source evidence, computed positive and negative exercise rows, and result records that carry references, digests, and counts but no copied bodies. Every negative case is recomputed by the bundle, so a fixture cannot pass itself by declaring an error code.\n\n###### What stays out of reach\n\n**Past that line it establishes no live ledger truth, mutation, public sharing, launch, provider, benchmark, or production authority, only fixture-bound public source-open evidence.** The proof boundary stops there. This component does not establish live work log truth, live work log truth, source-file changes, publishing-scope decision, launch-scope decision, external model access, neutral benchmark evidence, whole-system equivalence, investment-related actions, or deployment posture. The scope limit is fixture-bound public source-open evidence only.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":12441,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/batch10_governance_compilers_capsule.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"Open Policy Agent","url":"https://www.openpolicyagent.org/docs/latest","kind":"external_reference","section_id":"policy-as-code-and-supply-chain-assurance"},{"label":"validating admission policies","url":"https://kubernetes.io/docs/concepts/policy","kind":"external_reference","section_id":"policy-as-code-and-supply-chain-assurance"},{"label":"SLSA","url":"https://slsa.dev/spec/","kind":"external_reference","section_id":"policy-as-code-and-supply-chain-assurance"},{"label":"OpenSSF 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It exercises run-affinity scoring, calculator cluster insight derivation, std_python delta gating, exogenous navigation grading, portability supersession rollup, shard browse priority, holographic evidence selection, projection secret scanning, stockgrid flow normalization, source-regime board bucketing, frontend wayfinding, agent-session diagnostic lenses, and demo-take coverage auditing while preserving copied source digests, computed negative probes, and scope limits.","authority_ceiling":"Fixture-bound public source-body import, source-faithful public port evidence, computed negative-probe evidence, and metadata-only result records only; no live work log truth, navigation authority, complete secret detection, live market data, investment-related actions, raw transcript authority, video capture, source-file changes, publishing-scope decision, launch-scope decision, external model access, whole-system equivalence, or whole-system correctness.","abstract":"Set 11 Saturation Engines Bundle imports saturation, diagnostic, wayfinding, market-board, secret-scan, and demo-take source bodies as public source-open evidence without granting live runtime, launch, market, or navigation authority.","teleology":{"section_id":"","text":"Set 11 Saturation Engines Bundle imports saturation, diagnostic, wayfinding, market-board, secret-scan, and demo-take source bodies as public source-open evidence without granting live runtime, launch, market, or navigation authority.","source":"capsule_summary_line"},"body_markdown":"###### Four saturation engines run from copied source\n\n**Thirteen unrelated pieces of internal logic are copied into a public bundle and re-run on synthetic inputs so a reader can watch each behave.** This component takes thirteen unrelated pieces of internal logic, copies their source bodies into a public bundle, and re-runs each one against small synthetic inputs so a reader can watch the logic behave instead of taking a claim on trust. The targets share nothing except that treatment: a run ranker, a cluster-insight deriver, a delta gate, a navigation grader, a supersession rollup, a browse sectioner, an evidence selector, a secret scanner, two market-board engines, a graph wayfinder, a session-diagnostic lens, and a demo-coverage auditor.\n\n###### Each engine answers a distinct bounded question\n\n**One narrow question: does each imported mechanism actually compute the guard it claims, on inputs designed to make it fail?** The single question it answers is narrow. For each mechanism, does the imported source actually compute the guard it claims to, on inputs designed to make it fail? It is a saturation pass because the targets are swept up together so one reviewer can audit a broad slice of the codebase from one place.\n\n###### The result reports saturation without acting on it\n\n**The records are metadata-only: digests, computed outputs, probe values, and verdicts, while copied source bodies stay in the bundle tree.** It emits metadata-only result records: source-module digests, per-mechanism computed outputs, computed negative-case probe values, and pass or blocked verdicts. Copied source bodies stay in the bundle's `source_modules` tree and are never inlined into a record.\n\n##### Purpose\n\n###### Why a fixture filename proves nothing\n\n**A fixture filename is only a label, and a runner that counts it without exercising the real function is green and blind at once.** A test that trusts its own fixture filenames proves nothing. A file named `run_affinity_stale_terminal_rejected` is a label. If the runner counts that label as a rejection without exercising the real function, the suite is green and blind at the same time. That is the failure this component is built to avoid.\n\n###### Two agreeing computations per negative case\n\n**A case is verified only when the mechanism's own exercise and a separate probe over its `probe_input` agree, so an undemonstrated failure is flagged.** So it never lets a label stand in for a result. For each negative case it recomputes the guard from the fixture's own `probe_input` and refuses to mark the case verified unless two independent computations agree: the mechanism's own exercise and a separate probe over the fixture input. A fixture that asserts a failure it cannot demonstrate is flagged, not counted.\n\n##### How it works\n\n###### Two entry points, one fixed pipeline\n\n**Both entry points forward to `_run_batch11_crown_jewel_organ`, which drives the whole pipeline in one fixed order.** `run` and `run_batch11_saturation_engines_bundle` are the two entry points. Both forward to `_run_batch11_crown_jewel_organ`, which drives the pipeline in a fixed order.\n\n###### Manifest and digest checks before any logic\n\n** `validate_source_manifest` checks each copied module's digest, line count, and anchors first, so thirteen mechanisms rest on twelve modules before any logic runs.** First it loads the required input `batch11_saturation_engines_capsule_probe_manifest.json`. Then `validate_source_manifest` checks each copied module against its recorded digest, line count, and required source anchors, so a drifted or partial copy is caught before any logic runs. `EXPECTED_MODULE_IDS` names the twelve copied modules; the market-board engines share one copied `quant_presentation_mart` source, which is why thirteen mechanisms rest on twelve modules.\n\n###### Thirteen exercises in fixed order in `_evaluate`\n\n** `_evaluate` runs each copied function in the fixed `EXPECTED_MECHANISMS` order, and any reorder, block, or wrong module count becomes a finding.** `_evaluate` then exercises all thirteen mechanisms in the order fixed by `EXPECTED_MECHANISMS`. Each exercise runs the copied function and returns a small computed output. `_run_affinity_matrix`, for example, calls the copied `recommend_run_by_affinity` over two candidate runs and records whether a stale terminal run was refused. Any reordering, blocked exercise, or wrong module count becomes a finding: `BATCH11_MECHANISM_ORDER_INVALID`, `BATCH11_MECHANISM_BLOCKED`, or `BATCH11_SOURCE_MODULE_COUNT_INVALID`.\n\n###### Echo-risk pairing in `_build_integrity_matrix`\n\n**Each row pairs the mechanism's output with an independent probe over `probe_input`, and any disagreement sets echo risk that blocks the whole run.** `_build_integrity_matrix` is where the negative cases are kept honest. For each mechanism it reads two values. One is the mechanism's own computed output. The other comes from `_compute_negative_case_probe`, which recomputes the case from the fixture's `probe_input` using the same copied function, then digests that input into `fixture_probe_input_digest`. A row's negative result counts as computed only when both values are true. When they disagree, the row's `fixture_verdict_echo_risk` is set. A non-zero `fixture_verdict_echo_risk_count` raises `BATCH11_FIXTURE_VERDICT_ECHO_RISK` and blocks the whole run. `_source_evidence` attaches the per-mechanism digest and anchor rows so each verdict is traceable to a specific copied module.\n\n###### Forbidden-class scan and result record body scan\n\n**After validation the runner scans fixtures against forbidden classes and passes the payload through a body scan that blocks any leaked copied body.** After evaluation the runner calls `validate_negative_cases`, then scans the fixture files with `scan_paths` against the forbidden classes loaded from `core/private_state_forbidden_classes.json`. Finally it writes the result, board, and validation records, and passes the assembled payload through `scan_receipt_payload_for_bodies`. If a copied body leaked into a record, that scan flips the status to blocked with `CROWN_JEWEL_RECEIPT_BODY_SCAN_BLOCKED`. `result_card` projects the passing payload into a compact card; `main` is the command-line dispatch for the `run` and `validate-bundle` actions.\n\n###### Tier and binding disposition per row\n\n**Each row is tagged by verification tier and by a binding disposition recording how the mechanism relates to the rest of the system.** Each row is tagged by tier and binding disposition. The six mechanisms in `TIER_B_MECHANISMS` are marked `B_verified_by_controller`; the rest are `A_adversarially_verified_in_cap`. `MECHANISM_BINDING_DISPOSITIONS` records how each mechanism relates to the rest of the system: a new import, an already-bound gate the bundle only re-checks, an under-bound path it extends, or a mechanism absent from the public component plane at import time.\n\nFunction Role `_run_batch11_crown_jewel_organ` Orders the pipeline and writes the metadata-only records `_evaluate` Runs the thirteen exercises and raises order, blocked, and count findings `_build_integrity_matrix` Pairs each mechanism output with an independent probe and sets echo risk `_compute_negative_case_probe` Recomputes each negative case from the fixture's own `probe_input` `_source_evidence` Attaches per-mechanism copied-source digest and anchor rows `result_card` Projects a passing result into a compact reader card Diagram of the mechanism (7 steps). Diagram source & refs\n\nSource refs\n\ndigest and anchor checks `validate_source_manifest`\n\n```text\nflowchart TD input[\"probe manifest + copied source modules\"] manifest[\"validate_source_manifest digest and anchor checks\"] evaluate[\"_evaluate 13 exercises in fixed order\"] matrix[\"_build_integrity_matrix mechanism value AND fixture probe\"] probe[\"_compute_negative_case_probe recompute from probe_input\"] scan[\"scan_paths + result record body scan\"] record[\"metadata-only result record\"] input --> manifest manifest --> evaluate evaluate --> matrix probe --> matrix matrix --> scan scan --> record\n```\n\n##### Negative cases\n\n###### Thirteen named cases as claims to satisfy\n\n** `EXPECTED_NEGATIVE_CASES` declares one named case per mechanism, each an error-coded claim the copied code must actually satisfy on breaking inputs.** `EXPECTED_NEGATIVE_CASES` declares thirteen named cases, one per mechanism, each with an error code. They read as claims the code must actually satisfy. A stale terminal run is refused even when made sticky and feed-rich (`run_affinity_stale_terminal_rejected`). A zero-energy calculator bucket produces no fake dominance share (`calculator_zero_bucket_no_fake_dominance`). A new gap blocks the delta gate (`std_python_new_gap_blocks`). A wrong route is graded down (`exogenous_nav_wrong_route_graded_down`). An unresolved hard failure blocks the portability rollup (`portability_unresolved_hard_fail_blocks`). A synthetic key shape blocks the projection scan (`projection_secret_token_blocks`). An unreachable navigation target returns a blocker rather than a fabricated path (`frontend_nav_unreachable_target`). A demo take missing its anchors scores below a complete one (`demo_take_missing_anchor_penalized`).\n\n###### Synthetic secret-scan fixture, never live material\n\n**The probe recomputes each outcome from the fixture input, and the secret-scan case uses synthetic test strings, never live material, that must be blocked.** For each case the probe recomputes the outcome from the fixture input. The secret-scan example is deliberate: the fixture carries a synthetic key shape and a private ledger path, both test strings, never live material, and the probe confirms both are blocked.\n\n##### Prior Art Grounding\n\n###### Characterization testing and the test-oracle problem\n\n**The lineage is characterization testing plus the test-oracle problem: a stored expectation is no oracle unless an independent probe from the fixture input agrees.** The method is characterization testing: pin a piece of existing code by exercising it against fixtures and recording what it actually does, rather than asserting a value someone typed by hand. The negative-case discipline is a version of the test-oracle problem. A stored expectation is not an oracle if it can pass without the code running, so the probe computes an independent expectation from the fixture's own input and requires the two to agree. There is no external specification these engines implement, so the honest lineage is that engineering practice, not a standard. The links in earlier drafts to overload and backpressure literature described a different kind of saturation and are dropped here.\n\n##### Validation Result record Path\n\n###### Fixture files are inputs, not verdicts\n\n**Each fixture is a public `probe_input` the component recomputes into recorded probe, fixture, and mechanism values before counting a case verified.** Negative-case fixture files are inputs, not verdicts. Each file carries a public `probe_input`; the component computes the corresponding probe and records `fixture_probe_input_digest`, `fixture_computed_value`, and `mechanism_computed_value` in the integrity matrix before counting a case as verified.\n\nRun these from the `microcosm-substrate/` public root:\n\n###### What a pass certifies\n\n**A pass means digests and anchors matched, all thirteen exercises ran in order, every case reproduced with zero echo risk, and no body leaked.** A pass means every copied module matched its digest and anchors, all thirteen exercises ran in order, every negative case was reproduced from its own input with zero echo risk, and no copied body reached a record. It is fixture evidence about this standalone component, nothing wider.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim\n\n**The copied source bodies compute their guards on breaking fixtures, records carry only refs and verdicts, and no fixture label can pass on its own.** The strongest claim the evidence supports is this: the copied Set-11 source bodies compute the guards they claim to, on public fixtures built to break them, and the result records carry only refs, digests, counts, and computed verdicts. Each negative case is reproduced from its own input by an independent probe, so a fixture label cannot pass on its own.\n\n###### The scope limit holds everything false\n\n** `AUTHORITY_CEILING` sets provider, model, mutation, public sharing, navigation, secret-detection, market, and video flags all false, so a green run is bounded evidence about copied logic only.** The proof boundary stops there. The scope limit, fixed in the `AUTHORITY_CEILING` constant, sets external model access, model dispatch, source-file changes, public sharing, launch, work-ledger authority, navigation authority, complete secret detection, market advice, raw-session truth, and video capture all to false. This component is not live work log truth, not navigation authority, not complete secret detection, not live market data, not investment-related actions, not raw transcript authority, not video capture, not source-file changes, and not launch or publishing-scope decision. 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The files cover proof search, a copied Lean certificate kernel for Erdos #257, reasoning-authority fences, completion planning, bitemporal claim logging, macOS task-policy wrapping, and context-yield attribution.\n\n###### Two runtime paths share one authority boundary\n\n**It never runs those systems: it reads their copied public source, checks each mechanism against required anchors, recomputes one negative case per mechanism, and writes a metadata-only result record.** The component does not run any of those systems. It reads their copied public source, checks each declared mechanism against named required anchors, recomputes a negative case per mechanism from that source, and writes a result record. The record carries source references, digests, anchor counts, and bounded outcomes. It does not carry source bodies or live private state.\n\n###### A passing fixture does not establish a theorem\n\n**The only positive claim is narrow: copied source is present, its required anchors are in place, and the copied Lean kernel is free of proof placeholders, with everything stronger refused openly.** Its positive claim is narrow. The copied source is present, its required anchors are in place, and the copied Lean kernel is free of proof placeholders. Everything stronger is refused in the open.\n\n##### Purpose\n\n###### Each mechanism invites an overclaim\n\n**Each bundled mechanism looks like more than it is: proof search like solving open problems, the Lean kernel like a solution, a grant like a live sandbox, a wrapper like orchestration.** These mechanisms sit next to claims a reader will want to make on their behalf. A proof-search benchmark looks like solving open problems. A copied `CertificateKernel.lean` for Erdos #257 looks like a solution. A reasoning grant looks like a live sandbox. A task-policy wrapper looks like live orchestration.\n\n###### One narrow question per mechanism\n\n**The component asks only whether each mechanism can be shown as copied, anchored public source without inheriting an authority it lacks, making the import auditable.** A loose import surface would let any of those read as more than it is. The component answers one narrow question per mechanism: can this be shown to a reader as copied, anchored, public source, without inheriting an authority it does not have. It makes the import auditable and forces every stronger statement into a visible limit.\n\n##### How it works\n\n###### Two entry shapes over one gate\n\n** `run` takes a fixture directory and `run_batch4_bundle` takes the exported bundle, but both forward to `run_crown_jewel_organ` with `_evaluate`, so they share one gate.** There are two entry shapes. `run` consumes a fixture input directory. `run_batch4_bundle` consumes the exported bundle directory. Both forward to `run_crown_jewel_organ` with `_evaluate` as the evaluator, so they share one gate. `main` exposes them as the `run` and `validate-bundle` subcommands.\n\n`_evaluate` drives the checks in order:\n1. `_load_manifest` reads `batch4_probe_manifest.json`. Invalid JSON returns `BATCH4_PROBE_MANIFEST_INVALID`; a non-object returns `BATCH4_PROBE_MANIFEST_NOT_OBJECT`.\n2. `_module_rows` builds a module-id to row map from the exported `source_module_manifest.json`. `_evaluate` checks that all seventeen ids in `EXPECTED_MODULE_IDS` are present; a gap emits `BATCH4_SOURCE_MODULES_MISSING`.\n3. It checks the manifest's mechanism set equals the twelve ids in `EXPECTED_MECHANISMS` exactly. Any missing or extra id emits `BATCH4_MECHANISM_SET_MISMATCH`.\n4. `_mechanism_status` checks, per mechanism, that each named source module is present and that every string in that module's `required_anchors` appears in the copied file text. A missing module or anchor marks the mechanism blocked.\n5. `_synthetic_runtime_exercises` reads the copied module texts and derives three bundles: a proof bundle, an authority bundle, and a runtime bundle. Each bundle is a set of anchor-presence facts computed from the copied source, not a live run.\n6. `_erdos_static_scan` runs a regular expression over the copied `CertificateKernel.lean` and rejects the tokens `sorry`, `admit`, and `axiom`. It counts declarations and samples a few names. A banned token blocks the scan. This is a token scan, not a Lean proof check.\n7. `_lean_lake_probe` optionally runs `lake env lean` over the copied kernel under the source Lake project. A zero-exit run records `live_compile_pass`; a missing `lean` / `lake` binary or an unavailable Mathlib dependency records `unavailable`; a nonzero exit records `blocked`. Every branch fixes `proof_authority_delta` to `none`, so a compile pass is an availability signal, not formal-result correctness.\n8. `_semantic_negative_case_proofs` recomputes the twelve negative cases and reports how many were rejected.\n\n###### Negative verdicts re-derived from source\n\n** `_computed_negative_case_verdict` ignores the fixture's declared verdict and re-derives each one from source evidence, an overclaim-shape check, and a runtime probe, rejecting only when the source supports it.** The negative-case step is the part worth reading closely. `_computed_negative_case_verdict` does not trust the fixture to declare its own verdict. It re-derives each verdict from source-observed evidence, an overclaim-shape check on the mechanism's declared ceiling, and a runtime probe, then marks the row `declared_fixture_ignored`. A case counts as rejected only when the source itself supports the rejection.\n\n###### Pass gate and the omission record\n\n** `_evaluate` passes only with no findings and a validated manifest, while `result_card` attaches `AUTHORITY_CEILING` and an omission record naming what is left out: source bodies, private runtime state, and model-output data.** `_evaluate` returns `pass` only when there are no findings and the source manifest validated. `result_card` projects the outcome into a compact card and attaches `AUTHORITY_CEILING` plus an omission record naming what is left out of the result: full copied source bodies, raw private runtime state, and live provider or session payloads.\n\nFunction Role `run`, `run_batch4_bundle`, `main` Entry shapes over a fixture directory or the exported bundle `_load_manifest` Parse `batch4_probe_manifest.json` or reject it as a finding `_module_rows` Build the copied-module map from `source_module_manifest.json` `_mechanism_status` Check per-mechanism module presence and required anchors `_synthetic_runtime_exercises` Derive proof, authority, and runtime bundles from copied text `_erdos_static_scan` Reject `sorry`, `admit`, `axiom` in the copied Lean kernel `_lean_lake_probe` Optional local `lake env lean` availability signal `_evaluate` Orchestrate the gate and collect findings `_semantic_negative_case_proofs` Recompute each negative case from source `result_card` Attach `AUTHORITY_CEILING` and the omission record Diagram of the mechanism (8 steps). Diagram source & refs\n\nSource refs\n\n12 mechanisms + 12 negative cases `batch4_probe_manifest.json` 17 copied source modules `source_module_manifest.json`\n\n```text\nflowchart TD probe[\"batch4_probe_manifest.json 12 mechanisms + 12 negative cases\"] manifest[\"source_module_manifest.json 17 copied source modules\"] evaluate[\"_evaluate run / validate-bundle\"] anchors[\"_mechanism_status module present + required anchors\"] scan[\"_erdos_static_scan reject sorry / admit / axiom\"] probe2[\"_lean_lake_probe compile availability, delta = none\"] negatives[\"_semantic_negative_case_proofs verdict recomputed from source\"] record[\"metadata-only result record refs, digests, anchor counts, ceiling\"] probe --> evaluate manifest --> evaluate evaluate --> anchors evaluate --> scan scan --> probe2 evaluate --> negatives anchors --> record scan --> record negatives --> record\n```\n\n##### Negative cases\n\n###### Twelve stable error codes, one per mechanism\n\n**The fixture's twelve negative cases, one per mechanism with a stable error code, are the public failure floor stopping the source import from reading as an unbounded proof or runtime claim.** The fixture declares twelve negative cases, one per mechanism, each with a stable error code held in `EXPECTED_NEGATIVE_CASES`. They are the public failure floor that stops a source-import surface from reading as an unbounded proof or runtime claim:\n- `weak_skeleton_synthesis_failure`: `BATCH4_WEAK_SKELETON_MUST_NOT_SILENT_PASS`\n- `foundry_low_repair_quarantine`: `BATCH4_FOUNDRY_REPAIRED_COUNT_BELOW_THRESHOLD`\n- `verisoft_truth_leak`: `BATCH4_VERISOFT_TRUTH_LEAK_REJECTED`\n- `verisoft_prefix_answer_leakage`: `BATCH4_VERISOFT_PREFIX_ANSWER_LEAKAGE_REJECTED`\n- `erdos_solution_overclaim`: `BATCH4_ERDOS257_SOLUTION_OVERCLAIM_REJECTED`\n- `packet_sha256_corruption`: `BATCH4_PACKET_SHA256_CORRUPTION_REJECTED`\n- `grant_forbidden_context`: `BATCH4_GRANT_FORBIDDEN_CONTEXT_DENIED`\n- `forward_dirty_unknown_target`: `BATCH4_FORWARD_POLICY_DIRTY_UNKNOWN_TARGET_BLOCKED`\n- `closeout_stale_head`: `BATCH4_CLOSEOUT_STALE_HEAD_DEFERS`\n- `bitemporal_expired_claim`: `BATCH4_BITEMPORAL_EXPIRED_CLAIM_NOT_CURRENT`\n- `taskpolicy_missing_binary`: `BATCH4_TASKPOLICY_UNAVAILABLE_PASSTHROUGH`\n- `context_accepted_read_guard`: `BATCH4_CONTEXT_ACCEPTED_SCOPED_READ_NOT_FLAGGED`\n\n###### Rejection requires source support\n\n**A case is marked rejected only when the source-observed verdict supports it, never because the fixture said so.** Each case is marked rejected only when the source-observed verdict supports it, not because the fixture said so.\n\n##### Prior Art Grounding\n\n###### Evidence before authority, from proof-carrying code\n\n**It separates the authority to act from the evidence permitting it, borrowing the evidence-before-authority ordering of proof-carrying code and capability security, yielding fixture-bound evidence rather than a verified authorization system.** The component keeps the authority to act separate from the evidence that an action is permitted. That ordering is the idea behind proof-carrying code ( [Necula, 1997](https://en.wikipedia.org/wiki/Proof-carrying_code)) and capability-based security, where a request arrives with evidence of its own legitimacy rather than relying on ambient trust. This module borrows the evidence-before-authority ordering over copied fixtures. The result is fixture-bound evidence, not a verified authorization system.\n\n##### Validation Result record Path\n\nRun from the `microcosm-substrate/` public root:\n\n```text\nPYTHONPATH=src python3 -m pytest tests/test_batch4_proof_authority_runtime.py -q\nPYTHONPATH=src python3 scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n###### What the two checks confirm\n\n**The focused test covers fixture, bundle, digest, anchor, mutated-Lean-body rejection, and negative-case recomputation, while the corpus check confirms the bundle, Mermaid, Atlas, and Markdown projections stay consistent.** The focused test exercises the fixture and bundle expectations, the source digest and anchor checks, the mutated-Lean-body rejection, and the negative-case recomputation. A pass is reader-verifiable evidence only.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### What the module may claim\n\n**The module may claim only fixture-bound import, digest and anchor checks over the seventeen copied modules, a static Lean placeholder scan, an optional compile-availability signal, and metadata-only records with twelve recomputed negative cases.** This module may claim fixture-bound import of copied public source, digest and required-anchor checks over the seventeen copied modules, a static placeholder-token scan over the copied Lean kernel, an optional local compile availability signal, and metadata-only result records with twelve recomputed negative cases. The proof boundary is exactly that copied-source checking: `AUTHORITY_CEILING` fixes `proof_authority_delta` to `none` and holds every action flag, including `provider_dispatch`, `source_mutation_authorized`, `publication_authorized`, and `release_authorized`, at false.\n\n###### What stays refused\n\n**It claims no formal-result correctness, no Erdos #257 solution, no benchmark claims, no live enforcement or orchestration, and no mutation or launch, and the ceiling can rise only from changed evidence, never this Markdown.** It does not claim formal-result correctness, a solution of Erdos #257, official benchmark claims, live sandbox enforcement, live Codex orchestration, external model access, source-file changes, public sharing or launch-scope decision, whole-system equivalence, or whole-system correctness. The static scan is bounded evidence checking, and digest equality does not establish semantic equivalence to any private original.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":13106,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/batch4_proof_authority_runtime.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"Necula, 1997","url":"https://en.wikipedia.org/wiki/Proof-carrying_code","kind":"external_reference","section_id":"evidence-before-authority-from-proof-carrying-code"}],"grouped_relation_feeds":[{"relation":"abides by","count":4,"links":[{"id":"AX-4","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Content-addressed determinism","url":"doctrine.html#dcard-ax-4"},{"id":"AX-8","kind":"axiom","kind_label":"axiom","relation":"abides 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Bundle","relative_href":"docs/paper-module-batch6-unsurfaced-primitives-capsule.html","read_href":"docs/paper-module-batch6-unsurfaced-primitives-capsule.html","context_href":"docs/rules-and-ideas.html#paper-module-batch6-unsurfaced-primitives-capsule","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"Set 6 Unsurfaced Primitives Bundle imports provenance, operator-handoff, market, finance, provider-recovery, and demo-take source primitives as public source-open evidence without granting live operator memory, market, provider, media, public sharing, or launch-scope decision.","summary":"Set 6 Unsurfaced Primitives Bundle binds the accepted batch6_unsurfaced_primitives_capsule component to source-open primitive exercises. It checks source note keyphrase scoring, schema-loose distillation, operator handoff linkage, observed-turn window merging, market situation graphs, finance numeric assurance, fail-closed policy judgment, clone-local concurrency, market-clock scheduling, provider-recovery scoping, and demo-take temporal remapping while preserving public fixture inputs, exact-source digest expectations, negative cases, and scope limits.","authority_ceiling":"Fixture-bound public source-body import, copied-module digest/anchor evidence, synthetic source-exercise evidence, and metadata-only result records only; no live operator memory, prompt-shelf capture authority, live market data, provider/browser state, media launch, source-file changes, publishing-scope decision, launch-scope decision, whole-system equivalence, or whole-system correctness.","abstract":"Set 6 Unsurfaced Primitives Bundle imports provenance, operator-handoff, market, finance, provider-recovery, and demo-take source primitives as public source-open evidence without granting live operator memory, market, provider, media, public sharing, or launch-scope decision.","teleology":{"section_id":"","text":"Set 6 Unsurfaced Primitives Bundle imports provenance, operator-handoff, market, finance, provider-recovery, and demo-take source primitives as public source-open evidence without granting live operator memory, market, provider, media, public sharing, or launch-scope decision.","source":"capsule_summary_line"},"body_markdown":"###### Unsurfaced primitives are exercised from copied code\n\n**A scout found eleven small, load-bearing functions never shown publicly; this component copies them into the public clone and checks the copied bodies still behave as the prose claims.** A scout found eleven small functions scattered across the wider system that were real and load-bearing but had never been shown as public evidence. This component copies those eleven into the public clone and checks that the copied bodies still behave the way the prose claims. It is the kind of code that quietly decides whether a bigger feature is correct: a finance unit-scale check, a clock that fires each market event once per session, a function that subtracts paused time from a recorded video offset.\n\n###### Each primitive gets a positive and negative path\n\n**The check is by execution: each copied body runs on fixed synthetic input and is compared to its exact expected value, with a paired wrong input proving the guard.** The check is by execution, not by description. For each primitive the component imports the copied body and runs it on a fixed public synthetic input, then compares the result against the exact value the real code should produce. Each primitive also carries one paired negative case, a deliberately wrong input that the code must reject or correct, so the result record shows both the working path and the guard.\n\n###### Public fixtures do not become live authority\n\n**The result record holds only pass or fail and summary numbers, never copied source, transcripts, payloads, or market data, with every finding stamped `body_in_receipt: false`.** What lands in the result record is the pass or fail of each exercise plus a few summary numbers. Copied source text, operator transcripts, model-output data, and live market data never enter it. Every finding is stamped `body_in_receipt: false`.\n\n##### Purpose\n\n###### Why a digest is not enough\n\n**A file hash proves only that bytes match, not that the logic is right, so this component closes the gap by running each copied body and asserting its output.** Copying a function into a public bundle and checking its file hash proves only that the bytes match. It says nothing about whether the logic is right. A digest-only bundle is loose in exactly the place that matters: it can carry a body that no longer computes what its name says. This component closes that gap by running the copied body and asserting its output.\n\n###### Eleven primitives, one uniform shape\n\n**The eleven primitives span ranking, distillation, handoff scoring, window merging, market graphs, finance assurance, status judgment, concurrency, clock scheduling, provider scoping, and demo remapping, each checked in the same shape.** The eleven primitives span keyphrase ranking, schema-loose distillation, operator handoff scoring, observed-turn window merging, market situation graphs, finance numeric assurance, fail-closed status judgment, clone-local concurrency, market-clock scheduling, provider-recovery scoping, and demo-take temporal remapping. Each is checked in the same shape so a reader can see the working answer and the rejected answer side by side.\n\n##### How it works\n\n###### Two entrypoints, one shared driver\n\n** `run` and `run_batch6_bundle` both pass `SPEC` and the `_evaluate` and `evaluate_negative_case` callbacks to `run_crown_jewel_organ`, with the bundle entrypoint setting `input_mode` to read copied source.** `run` and `run_batch6_bundle` both hand `SPEC`, the input directory, and two callbacks to `run_crown_jewel_organ`: `_evaluate` and `evaluate_negative_case`. The bundle entrypoint sets `input_mode` to the exported-bundle mode so the same evaluator reads copied source rather than the fixture manifest.\n\n###### Manifest load and roster checks\n\n** `_evaluate` loads the probe manifest, keys the 26 copied modules by `module_id`, and confirms both `EXPECTED_MODULE_IDS` and the exact 11 `EXPECTED_MECHANISMS` are present.** `_evaluate` runs the pipeline in order. It loads `batch6_probe_manifest.json` through `_load_manifest`, which rejects a non-JSON file with `BATCH6_PROBE_MANIFEST_INVALID` and a non-object with `BATCH6_PROBE_MANIFEST_NOT_OBJECT`. It resolves the copied modules with `_module_rows`, which reads the source-module manifest and keys each row by `module_id`. It confirms every id in `EXPECTED_MODULE_IDS` (26 rows) is present, then confirms the probe manifest names exactly the 11 mechanisms in `EXPECTED_MECHANISMS`, emitting `BATCH6_MECHANISM_SET_MISMATCH` on any missing or extra id.\n\n###### Anchors gate the source exercise\n\n** `_mechanism_status` blocks any mechanism whose copied text is missing a module or `required_anchor`, and only when nothing is missing does `_run_source_exercise` import and run the real body.** For each mechanism `_mechanism_status` reads the copied target text and checks that the module's `required_anchors` all appear in it. A missing module or a missing anchor sets that mechanism to `blocked`. Then, only if no expected module is missing, `_run_source_exercise` imports the real copied body and runs it.\n\n###### Isolated import of copied bodies\n\n** `_load_copied_module` loads each file under a private name inside `_temporary_sys_path` and `_temporary_import_namespaces`, so copied bodies run from the bundle without colliding with the running package.** `_load_copied_module` does the import. It resolves the target path, then loads the file under a private module name inside `_temporary_sys_path` and `_temporary_import_namespaces`, so the copied `idea_microcosm`, `system`, and `tools` bodies load from the bundle without colliding with the running package. The exercise calls the real function and compares the answer.\n\nFour exercises make the idea concrete.\n\n###### `demo_take_temporal_join` removes paused seconds\n\n** `demo_take_temporal_join` feeds `video_t_seconds` a 120-second offset paused from 10s to 25s and asserts exactly 105.0, with an open-pause branch returning 15.0.** `demo_take_temporal_join` calls `video_t_seconds`. It feeds a 120-second wall offset with one pause at 10s and resume at 25s and asserts the result is exactly 105.0, the 15 paused seconds removed. A second call with a pause that has not yet resumed checks the open-pause branch returns 15.0.\n\n###### `finance_numeric_assurance` fails closed on bad numbers\n\n** `build_finance_numeric_assurance` recomputes declared numbers, so a `usd_millions` row with mismatched flow and an out-of-range probability turns `display_state` to `blocked` and fails closed.** `finance_numeric_assurance` calls `build_finance_numeric_assurance`, which recomputes declared numbers instead of trusting them. The exercise hands it a flow row tagged `usd_millions` whose `flow` and `flow_usd` disagree by orders of magnitude, plus a probability declared as 70.2. The result record's `display_state` becomes `blocked` and its `check_ids` include `stockgrid_flow_unit_scale_mismatch` and `probability_bounds`. A mislabelled unit or an out-of-range probability fails closed.\n\n###### `operator_handoff_linkage` scores capture against input\n\n** `operator_handoff_linkage` scores a `TypeBCapture` against a `TypeAUserInput` by containment and overlap, putting a related pair at or above 0.8 and an unrelated one below 0.3.** `operator_handoff_linkage` calls `score_pair` on a `TypeBCapture` (an agent's earlier suggestion) against a `TypeAUserInput` (what the operator later typed), using containment, token overlap, and anchor matching. A related pair scores at or above the 0.8 floor with containment true; an unrelated \"summarize the weather forecast\" input scores below 0.3.\n\n###### `metabolism_market_clock` enforces per-session idempotence\n\n** `metabolism_market_clock` sets `due_fire_points` to 15:31 UTC and fires the hourly points while suppressing the already-fired open event, enforcing once-per-session idempotence.** `metabolism_market_clock` calls `due_fire_points`. It sets the clock to 15:31 UTC with the open event already fired earlier that day, then asserts the hourly points fire while the already-fired open event stays suppressed. The guard is idempotence: an event that fired once must not fire again in the same session.\n\n###### The other seven checks in brief\n\n**The remaining seven follow the same shape, from `rake_ranked_phrases` and `distill_diagnostic` through market-graph, status-judgment, and `_paths_overlap` checks, each pairing a working answer with a rejected one.** The other seven follow the same pattern. `rake_ranked_phrases` returns ranked phrases for real text and an empty list for stopword-only input. `distill_diagnostic` keeps assistant text and operator tail as separate roles with `body_persisted` false. `merge_observed_turn_window` appends a new observed tail and leaves a re-sent duplicate window unchanged. `build_market_situation_graph` with strict `validate_market_situation_graph` requires `counterevidence_edges` and a not-investment-advice guard on every situation. `judge_status_request` allows a transition with evidence, downgrades one missing it, blocks a forbidden upgrade, and blocks under a malformed policy. `_paths_overlap` reports a parent directory and a child path as overlapping and two siblings as not.\n\nDiagram of the mechanism (7 steps). Diagram source & refs\n\nSource refs\n\nProbe `batch6_probe_manifest.json`\n\n```text\nflowchart TD Probe[\"batch6_probe_manifest.json\"] Manifest[\"source module manifest 26 copied bodies\"] Load[\"_load_manifest / _module_rows\"] Anchor[\"_mechanism_status module + anchor present?\"] Exercise[\"_run_source_exercise import copied body, run it\"] Neg[\"evaluate_negative_case one wrong input per mechanism\"] Record[\"result record pass/fail + summary numbers body_in_receipt false\"] Probe --> Load Manifest --> Load Load --> Anchor Anchor --> Exercise Exercise --> Neg Neg --> Record\n```\n\n##### Negative cases\n\n###### One guarded case per mechanism\n\n** `EXPECTED_NEGATIVE_CASES` holds one case per mechanism, each routed through `_observed_negative_case` and passing only when the paired exercise passed and its guard fired.** `EXPECTED_NEGATIVE_CASES` holds one case per mechanism, and `evaluate_negative_case` routes each through `_observed_negative_case`, which reads the matching runtime exercise output. A case only passes when the paired exercise passed and its guard fired. The eleven are: `stopword_only_keyphrase_empty`, `schema_loose_voice_conflation`, `handoff_unrelated_below_floor`, `observed_turn_duplicate_rerender`, `market_situation_missing_counterevidence`, `finance_unit_scale_mismatch`, `status_policy_poisoned`, `concurrency_parent_child_conflict`, `market_clock_duplicate_fire_suppressed`, `provider_timeout_scope_narrow`, and `demo_take_pause_remap`. Each carries a stable expected error code, for example `BATCH6_FINANCE_UNIT_SCALE_MISMATCH_BLOCKED` for the finance case.\n\n##### Prior Art Grounding\n\n###### Established provenance and policy patterns\n\n**The primitives draw on established patterns, borrowing provenance and policy-engine design for the market and status pieces and event-time-versus-record-time separation for the demo-take remap.** The copied primitives draw on established patterns rather than novel theory. The market and status pieces borrow from provenance and policy-engine design; the demo-take remap follows the practice of keeping event time separate from record time.\n- W3C [PROV](https://www.w3.org/TR/prov-overview/), for reconciling derived artifacts back to entities, activities, and responsible agents.\n- [Open Policy Agent](https://www.openpolicyagent.org/docs/latest), which separates policy evaluation from application code through a general-purpose engine.\n- Martin Fowler's [bitemporal history](https://martinfowler.com/articles/bitemporal-history.html), as a prior pattern for preserving event time separately from record time.\n\n###### Bounded to source-open public fixtures\n\n**The component keeps to source-open public fixtures and exposes no live operator memory, market data, provider state, or publishing-scope decision.** The component keeps to source-open public fixtures and does not expose live operator memory, live market data, provider state, or publishing-scope decision.\n\n##### Validation Result record Path\n\nRun from the `microcosm-substrate/` public root:\n\n###### What a pass proves\n\n**A pass means all 26 modules carried their anchors, all 11 exercises returned expected values, all 11 negative cases fired their guard, and no record leaked a body or private state.** A pass means all 26 copied modules carried their required anchors, all 11 source exercises returned the expected values, all 11 negative cases fired their guard, and no result record leaked a copied body or private state. The corpus and projection checks prove only that the generated instance for this module is fresh.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim\n\n**The strongest claim is that eleven copied bodies were imported, anchor-checked, exercised to their exact expected outputs, and paired with negative cases, all under metadata-only records.** The strongest claim here is narrow. Eleven copied bodies were imported into the public clone, checked against their required source anchors, exercised on public synthetic input to their exact expected outputs, and paired with negative cases, all under metadata-only result records. That is the proof boundary and the scope limit for this page.\n\n###### What it refuses past that\n\n**It is not operator memory, capture authority, trading decisions, provider recovery, media launch, source-file changes, or launch-scope decision, and `AUTHORITY_CEILING` holds every action flag at false.** It refuses everything past that. It is not live operator memory, not capture authority, not trading decisions, not live provider recovery, not demo media launch, not source-file changes, not public sharing or launch-scope decision, not equivalence to the private root, and not whole-system correctness. `AUTHORITY_CEILING` in the source sets `launch_authorized`, `provider_dispatch`, `runtime_execution`, `source_mutation_authorized`, `publication_authorized`, and `release_authorized` all to false.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":14175,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/batch6_unsurfaced_primitives_capsule.md","kind":"source_link"},{"label":"Source 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Gate","relative_href":"docs/paper-module-engine-room-public-projection-leak-gate.html","read_href":"docs/paper-module-engine-room-public-projection-leak-gate.html","context_href":"docs/rules-and-ideas.html#paper-module-engine-room-public-projection-leak-gate","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"The Engine Room public projection leak gate validates rendered public projection roots for account secret-shaped strings, non-public paths shapes, symlink escapes, policy-exception handling, and optional gitleaks status while keeping findings hash-only.","summary":"Engine Room Public Projection Leak Gate is a DLP-style projection boundary. It scans rendered public projection files and paths, checks symlink escapes, records policy-exception hits as hash-only evidence, reports optional gitleaks status, and validates two positive plus three negative fixture cases without copying sensitive payloads, approving launch, or claiming general security, prompt-injection, sandbox, or information-flow authority.","authority_ceiling":"Public projection leak-gate fixture and rendered-root scan result records only; no general security scanner, prompt-injection defense, sandbox, information-flow proof, launch-scope decision, whole-system equivalence, source-file changes, or whole-system correctness.","abstract":"The Engine Room public projection leak gate validates rendered public projection roots for account secret-shaped strings, non-public paths shapes, symlink escapes, policy-exception handling, and optional gitleaks status while keeping findings hash-only.","teleology":{"section_id":"","text":"The Engine Room public projection leak gate validates rendered public projection roots for account secret-shaped strings, non-public paths shapes, symlink escapes, policy-exception handling, and optional gitleaks status while keeping findings hash-only.","source":"capsule_summary_line"},"body_markdown":"###### Before rendered tree\n\n**Before a rendered tree reaches a public reader, one narrow question stands: does it carry account secret shapes, a non-public paths, a transport symbol, or an escaping symlink that should not leave the workspace?** Before a rendered tree of files is exposed to a public reader, someone has to answer one narrow question: does this tree contain anything that should not leave the workspace? account secret-shaped strings, a private home path, a browser transport symbol, or a symlink pointing outside the tree are all ways for private material to ride along with an otherwise public projection.\n\n###### Component answers question\n\n**This component answers that question over a directory of rendered files with a `green` or `red` verdict: a data-loss-prevention gate for one tree, not a general scanner.** This component answers that one question over a directory of rendered files and returns a `green` or `red` verdict. It is a data-loss-prevention gate for one tree, not a general scanner.\n\n###### Design point is\n\n**The design point is what it does with a find: each match is recorded by category, path, line, and a SHA-256 hash while the matched value is dropped, so the report never becomes a second copy of the leak.** The design point is what it does with what it finds. A secret scanner that prints the secret into its own report has made a second copy of the leak. This gate never does that. Every match is recorded by category, path, line number, and a SHA-256 hash of the matched text, and the matched value itself is dropped. A reviewer can confirm that a leak was found and where, without the result record becoming the thing that leaks.\n\n##### Purpose\n\n###### The quiet co-travel failure mode\n\n**The failure a loose gate allows is quiet co-travel: a rendered file carries a key or non-public paths nobody noticed, and a naive report copies that value into a second guarded artifact.** The failure mode a loose gate allows is quiet co-travel. A publish step renders a directory, a human skims it, and one file carries a key or a non-public paths that nobody noticed. The problem is not that leaks are undetectable. The problem is that ad hoc review misses them and that a naive report copies the sensitive value out of the file and into a second artifact that then also has to be guarded.\n\n###### Fixing the position of the check\n\n**This gate reads rendered files and path names against fixed regular expressions, treats an escaping symlink as a hard blocker, folds in an optional `gitleaks` run, and emits a hash-only publishable record.** This gate fixes the position of that check. It reads the rendered files and their path names against a fixed set of regular expressions, treats a symlink that escapes the root as a hard blocker, and folds an optional `gitleaks` run into the same result. The verdict is deterministic for the same input, the counts are honest, and the result record is safe to publish because it holds hashes and categories, not the matched text.\n\n##### How it works\n\n###### `scan_projection` resolves and walks the root\n\n** `scan_projection` resolves the supplied root, raises `ValueError` if it is missing or not a directory, normalises policy-exception paths, and walks the tree in sorted path order.** The entry point is `scan_projection(root, ...)`. It resolves the supplied root and raises `ValueError` if the root is missing or is not a directory. It then normalises the caller's policy-exception paths against `DEFAULT_POLICY_EXCEPTION_PATHS` through `_normalise_policy_paths`, and walks the tree in sorted path order.\n\n###### Three per-entry checks in order\n\n**Each entry passes three checks in order: skip-and-count for named directories and `.pyc` files, record a symlink whose target escapes the root, and scan every other path two ways.** For each entry the walk applies three checks in order. Directories named in `SCAN_SKIP_DIR_NAMES` (`node_modules`, `.git`, `.vite`, `dist`, `__pycache__`, `.pytest_cache`) and files with a `.pyc` suffix are skipped and counted. A symlink whose resolved target does not sit under the root is recorded as a symlink escape, keeping only the relative path and a hash of the escaped target. Every other path is scanned two ways.\n\n###### Path and content pattern scans\n\n** `_scan_path` runs the relative path against `PATH_PATTERNS` and `_scan_file` runs UTF-8 contents against `CONTENT_PATTERNS` for account secrets, non-public paths, and host-bound transport, skipping any file it cannot read as text.** `_scan_path` runs the relative path string against `PATH_PATTERNS`: private history and voice file shapes, a private notes tree shape, and browser transport file names. `_scan_file` reads each readable file as UTF-8 and runs its contents against `CONTENT_PATTERNS`: private key blocks and vendor key shapes under account secrets (for example `openai_key_shape`, `github_token_shape`, `aws_access_key_shape`), private home and profile paths under `private_path`, and browser transport symbols and debug ports under `host_bound_transport`. A file that cannot be read as text is skipped, not failed.\n\n###### `_hit` stores a hash, never the value\n\n**Every match becomes a `_hit` record carrying category, pattern name, path, optional line, source kind, policy-exception flag, and `match_sha256`, and this is the single place the leak-safety property is enforced.** Every match becomes a record through `_hit`. That record carries the category, the pattern name, the relative path, an optional line number, the source kind (`content` or `path`), whether the path is a policy exception, and `match_sha256`. The matched value is never stored. This is the single place the leak-safety property is enforced.\n\n###### Splitting hits and computing the verdict\n\n** `scan_projection` retains policy-exception hits as non-blocking evidence, treats every other hit as blocking, folds in an optional redacted `gitleaks` run, and sets the verdict `red` on any blocker.** `scan_projection` then splits the hits. Any hit whose path is in the policy exception set is retained as evidence but does not block. Every other hit is a blocking hit. If the caller asked for it, `run_gitleaks` runs the external scanner over the root with `--redact` and records only its status, finding count, return code, and hashed stdout and stderr. `_overall_status` computes the verdict: `red` if there is any blocking hit or symlink escape, or if the `gitleaks` status is `red`, `error`, or `unavailable_fail_closed`; otherwise `green`. The result sets `public_release_allowed_by_scan` to true only when the status is `green`.\n\n###### Fixture-driven evaluation\n\n** `evaluate_case` materialises one JSON case, runs `scan_projection`, and checks the observed status, while `evaluate_fixture_dir` passes only when every case meets its expectation.** Two functions drive the check from fixtures. `evaluate_case` materialises one JSON case into a temporary root, runs `scan_projection`, and compares the observed status to the case's expected status. `evaluate_fixture_dir` runs every case in a directory and returns `status: pass` only when all cases meet their expectation. `main` exposes both a `scan` and an `evaluate-fixtures` subcommand.\n\nFunction Role `scan_projection` Walks the root, scans paths and files, computes the verdict `_scan_path` / `_scan_file` Apply `PATH_PATTERNS` and `CONTENT_PATTERNS` `_hit` Builds a hash-only record and drops the matched value `_overall_status` Derives `green` or `red` from hits, symlinks, and gitleaks `run_gitleaks` Optional external scan, recorded as status and hashes only `evaluate_case` / `evaluate_fixture_dir` Run fixture cases and check expected status Diagram of the mechanism (12 steps). Diagram source & refs\n\n```text\nflowchart TD Root[\"Rendered root walk in sorted order\"] Root --> Skip{\"Skip dir or .pyc?\"} Skip -- \"yes\" --> Count[\"Counted, not scanned\"] Skip -- \"no\" --> Link{\"Symlink escapes root?\"} Link -- \"yes\" --> Escape[\"Symlink escape target hash only\"] Link -- \"no\" --> Scan[\"_scan_path and _scan_file _hit stores hash only\"] Scan --> Policy{\"Path in policy exception set?\"} Policy -- \"yes\" --> Kept[\"Retained, non-blocking\"] Policy -- \"no\" --> Block[\"Blocking hit\"] Escape --> Verdict{\"Blocking hit, escape, or gitleaks red?\"} Block --> Verdict Verdict -- \"yes\" --> Red[\"red verdict\"] Verdict -- \"no\" --> Green[\"green verdict\"]\n```\n\n##### Negative cases\n\n###### Two green cases, three red\n\n**Five cases split two green and three red, and `policy_exception_hash_only` plants a boundary-document example inside a policy-exception path and still expects green with the hit retained but non-blocking.** The fixture set has five cases: two that must stay green and three that must go red. `clean_projection` and `policy_exception_hash_only` are the positive cases; the second plants a boundary-document example inside a policy-exception path and expects a green verdict with the hit retained but non-blocking.\n\n###### Three blockers, each hash-only\n\n**The three negative cases force a non-public paths, a vendor key shape, and a path-name pattern block, and the tests assert none of those values survive into the emitted record.** The three negative cases each force a distinct blocker. `planted_private_path` places a private home path in a file body and expects `red` with the match stored as a hash. `planted_key_shape` plants a vendor key shape and expects `red` without the raw key appearing anywhere in the result. `path_pattern_blocked` carries a non-public paths shape in a file name so the block comes from `_scan_path`, not file contents. The focused tests confirm each of these, and they assert directly that the non-public paths, the key shape, and the symlink target do not survive into the emitted record.\n\n##### Prior Art Grounding\n\n###### Data-loss-prevention and secret-scanning lineage\n\n**The component follows data-loss-prevention and secret-scanning practice, anchored by NIST DLP, GitHub secret scanning, and Gitleaks, contributing the hash-only result record and policy-exception split over one projection tree.** The component follows data-loss-prevention and secret-scanning practice: scan artifacts before public sharing, detect account secret-shaped strings and non-public paths markers, keep enough evidence for triage, and never copy the sensitive payload into the report. Relevant anchors are NIST's [Data Loss Prevention](https://www.nist.gov/publications/data-loss-prevention) work, [GitHub secret scanning](https://docs.github.com/en/code-security/concepts/secret-security/about-secret-scanning), and [Gitleaks](https://github.com/gitleaks/gitleaks), which this gate can call as an optional second pass. The contribution here is the hash-only result record and the policy-exception split, narrowed to one rendered projection tree.\n\n##### Validation Result record Path\n\n###### Reader-verifiable result record is\n\n**The reader-verifiable result record is the focused test suite plus the corpus parity check.** The reader-verifiable result record is the focused test suite plus the corpus parity check:\n\n```text\nPYTHONPATH=src ./repo-pytest tests/test_engine_room_public_projection_leak_gate.py -q --basetemp /tmp/microcosm-public-projection-leak-gate\ncd microcosm-substrate && PYTHONPATH=src ../repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\nThe public exercise runs the same fixture matrix directly:\n\n```text\nPYTHONPATH=src python3 -m microcosm_core.engine_room.public_projection_leak_gate evaluate-fixtures --input fixtures/first_wave/engine_room_public_projection_leak_gate/input --json\n```\n\n###### What a pass means here\n\n**A pass means the five fixture cases behave as expected and the hash-only handling holds; it approves no launch and proves no whole-system safety.** A pass means the five fixture cases behave as expected and the hash-only handling holds. It does not approve launch or prove whole-system safety.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim\n\n**Given a directory, the gate finds account secret shapes, non-public paths shapes, and escaping symlinks with fixed pattern sets and reports them as hashes, so the result record is itself safe to publish.** The strongest claim the evidence supports: given a directory, this gate finds account secret shapes, non-public paths shapes, and escaping symlinks with the fixed pattern sets, and reports them as hashes so the result record is itself safe to publish. That is the whole proof boundary.\n\n###### What the gate cannot promise\n\n**It is no general security scanner or information-flow proof: regex detection can miss encoded or novel secrets, and a `green` verdict means only that this one gate found no blocker in this one scan.** It refuses more. 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three red","Three blockers, each hash-only","Prior Art Grounding","Data-loss-prevention and secret-scanning lineage","Validation Result record Path","Reader-verifiable result record is","What a pass means here","Scope limit","The strongest supported claim","What the gate cannot promise"]},{"id":"paper_module.batch5_authority_systems_capsule","title":"Set 5 Authority and Systems Bundle","relative_href":"docs/paper-module-batch5-authority-systems-capsule.html","read_href":"docs/paper-module-batch5-authority-systems-capsule.html","context_href":"docs/rules-and-ideas.html#paper-module-batch5-authority-systems-capsule","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"Set 5 Authority and Systems Bundle imports post-execution authority, replay, proof-repair, process, generated-state, trace, blast-radius, and doctrine-graph source bodies as public source-open evidence without claiming live authority or launch-scope decision.","summary":"Set 5 Authority and Systems Bundle binds the legacy Markdown projection, Set 5 runnable source locus, exported copied-source bundle, source-module digests, synthetic negative exercises, metadata-only result records, and scope limits to a mechanism-backed JSON bundle. It covers post-execution result record validation, reasoning replay scope and lineage, verifier-gated Lean repair harnessing, process orphan classification, generated-state fixpoint settlement, trace-tape compaction, code blast radius, and doctrine graph compilation while preserving fixture boundaries.","authority_ceiling":"Fixture-bound public source-body import, copied-module digest and anchor evidence, synthetic source-exercise evidence, and metadata-only result records only; no live external model access, proof success, process signal authority, generated-state mutation authority, source-file changes, publishing-scope decision, launch-scope decision, whole-system equivalence, or whole-system correctness.","abstract":"Set 5 Authority and Systems Bundle imports post-execution authority, replay, proof-repair, process, generated-state, trace, blast-radius, and doctrine-graph source bodies as public source-open evidence without claiming live authority or launch-scope decision.","teleology":{"section_id":"","text":"Set 5 Authority and Systems Bundle imports post-execution authority, replay, proof-repair, process, generated-state, trace, blast-radius, and doctrine-graph source bodies as public source-open evidence without claiming live authority or launch-scope decision.","source":"capsule_summary_line"},"body_markdown":"###### Authority-system checks run from copied bodies\n\n**A reader can inspect eight unrelated authority and systems mechanisms at once and confirm each refuses its wrong answer, without running any of the real machinery.** This component lets a reader inspect eight unrelated authority and systems mechanisms at once, and confirm that each one refuses the wrong answer, without running any of the real machinery. The eight are: post-execution result record validation, reasoning replay scope, verifier-gated proof repair, process orphan classification, generated-state fixpoint settlement, trace-tape compaction, code blast radius, and doctrine-graph compilation.\n\n###### Fixture cases exercise delegation and refusal\n\n**It ships an exact copy of each source body, checks the copies against digests and anchors, then runs a synthetic re-derivation per mechanism that recomputes its own verdict instead of echoing a stored answer.** It carries an exact copy of each original source body in `examples/batch5_authority_systems_capsule/exported_batch5_authority_systems_capsule_bundle/source_modules/` and checks those copies against recorded digests and required text anchors. Alongside that, it runs a small synthetic re-derivation for each mechanism. Every re-derivation recomputes its own verdict from the fixture input. It does not echo a stored answer.\n\n###### The result carries evidence without inherited authority\n\n**The run emits a metadata-only result record for each mechanism, each naming a pass or a refusal and the scope limit it stayed inside.** The output is a set of metadata-only result records under `receipts/first_wave/batch5_authority_systems_capsule`, each stating a pass or a refusal and the scope limit the run stayed inside.\n\n##### Purpose\n\n###### One shared discipline across eight subjects\n\n**The eight mechanisms share no subject, only the discipline of declining to claim more than they earned, where a loose version accepts a drifted result record, hands Lean a placeholder, or invents coverage.** The eight mechanisms share nothing in subject. One checks post-execution result records. Another decides when a reasoning step needs re-running. Another gates a Lean proof attempt. Another classifies a stray process. What they share is a single discipline: each must decline to claim more than it earned. A loose version fails in the obvious ways. The result record check would accept a drifted result record. The proof gate would hand a placeholder to Lean. The orphan check would signal a live process. The blast-radius pass would invent coverage for a leaf that has no dependents.\n\n###### Synthetic exercises beside verifiable source\n\n**Rather than replay the real tools, it verifies copied source against digests and anchors and runs one synthetic exercise per mechanism, passing a negative case only when the exercise itself reaches the refusal.** The unusual choice here is that the component does not replay the real tools. It reads the copied source bodies, verifies them against digests and anchors, then runs one synthetic exercise per mechanism. A negative case passes only when the exercise itself reaches the refusal, not when a fixture asserts the refusal for it. That is the point of the design: you can read eight refusal behaviours in one place while the genuine source bodies stay verifiable next to them.\n\n##### How it works\n\n###### Two entrypoints share one evaluator\n\n** `run` and `run_batch5_bundle` both drive `run_crown_jewel_organ` with `_evaluate` and `evaluate_negative_case`, differing only by `BUNDLE_INPUT_MODE`, and both write metadata-only records with `body_in_receipt` always `False`.** `run` and `run_batch5_bundle` are the two entrypoints. Both call the shared `run_crown_jewel_organ` driver with `SPEC`, passing `_evaluate` as the evaluator and `evaluate_negative_case` as the negative-case evaluator. `run_batch5_bundle` sets `BUNDLE_INPUT_MODE` so the same evaluator runs against the exported source bundle instead of the plain fixture. Both write metadata-only result records and, when asked, an sign-off record whose `body_in_receipt` field is always `False`.\n\n###### The `_evaluate` pipeline order\n\n** `_evaluate` loads the manifest, checks the ten source paths against `SOURCE_REQUIRED_ANCHORS`, runs the eight exercises, then recomputes the eight negative-case probes, passing only when no findings were raised.** `_evaluate` runs the pipeline in this order. It loads `batch5_exercise_manifest.json`. It compares the source manifest against `SOURCE_REQUIRED_ANCHORS`, the map of the ten required source paths to the text anchors each must contain, and emits `BATCH5_SOURCE_MODULE_SET_INCOMPLETE` if any are missing. It then runs the eight exercises, one per manifest block, and emits `BATCH5_MECHANISM_EXERCISE_FAILED` for any exercise that does not pass. Last, it recomputes the eight negative-case probes over `sorted(EXPECTED_NEGATIVE_CASES)`. The run status is `pass` only when no findings were raised.\n\n###### Exercises recompute and never act\n\n**Every mechanism exercise derives its verdict purely from a manifest block, touching no provider, no Lean run, no process signal, and no generated-state mutation.** Each exercise reads a small block from the manifest and recomputes a verdict. None call a provider, run Lean, signal a process, or mutate generated state.\n\nFunction What it checks `_receipt_validator_exercise` Given a grant and two post-execution result records, recomputes drift codes for the second: `provider_substitution`, `forbidden_context_sent`, `output_artifact_divergence`, `execution_claimed_without_runtime_grant`. Passes only when the valid result record is accepted and the drifted one carries at least one code. `_replay_scope_exercise` Compares the context classes a step consumed against the classes that changed. Disjoint sets classify as `no_replay`, otherwise `partial`. Passes when the classification matches the fixture's expected value. `_proof_contract_gate_exercise` Scans a candidate proof string before any Lean call. A `sorry` token, a plan-only phrase (`plan:` or `i will`), or a proof that restates the declared theorem without `exact` each become a failure class and set the gate to `rejected_before_lean`. Records `0/8 historical banked attempts; no proof-success claim`. `_orphan_reaper_exercise` Classifies a process. A live-session descendant becomes `requires_owner_check`, never `safe_close_candidate`, and `signal_sent` stays `False`. `_fixpoint_drainer_exercise` Walks residual signatures. A residual id that reappears under a moved source signature classifies as `settlement_residual_source_moved`; a repeated signature as `settlement_residual_signature_repeated`. `mutation_authorized` is always `False`. `_trace_tape_exercise` Joins span text. Under the byte budget it passes the text through. Over budget it truncates to a head budget and appends a `#r0001 omitted_trace_bytes` pointer plus an omission record with the omitted byte count. A breach with no pointer does not pass. `_blast_radius_exercise` Builds the reverse-dependency graph and takes the transitive closure of dependents for a target. Passes when the target has dependents and the named leaf reports an honestly empty bucket rather than invented coverage. `_doctrine_graph_exercise` Scans doctrine nodes. A node whose code path no longer exists is reported as `authority_gap_deleted_code_path`; a node marked `tombstone` is reported with its replacement id. Passes only when both a drift finding and a tombstone are present.\n\n###### Probes rerun exercises, echo risk tracked\n\n** `_compute_negative_case_probe` reruns each exercise on the case's `probe_input` and rejects only on a recomputed refusal, while `fixture_verdict_echo_risk_count` flags any verdict not recomputed from its own input.** The negative-case probes reuse those same functions. `_compute_negative_case_probe` runs the relevant exercise on the case's `probe_input`, then `evaluate_negative_case` marks the case rejected only when the recomputed verdict is the refusal. The run tracks a `fixture_verdict_echo_risk_count`: a case whose verdict was not recomputed from its own probe input counts as risk, so a fixture cannot pass a negative case by simply asserting the answer.\n\nDiagram of the mechanism (6 steps). Diagram source & refs\n\n```text\nflowchart TD Manifest[\"Copied source bundle + exercise manifest digests and anchors checked first\"] --> Evaluate[\"_evaluate source-set check, then eight exercises\"] Evaluate --> Exercises[\"Eight mechanism exercises each recomputes its own verdict\"] Exercises --> Probes[\"Negative-case probes rerun exercises on probe_input\"] Probes --> Records[\"metadata-only result records status pass only when no findings\"] Records --> Ceiling[\"Scope limit no external model access, no mutation, no proof success, no launch\"]\n```\n\n##### Negative cases\n\n###### Eight cases with stable error codes\n\n** `EXPECTED_NEGATIVE_CASES` declares eight cases, each carrying the stable error code the run must raise when its perturbation is caught.** `EXPECTED_NEGATIVE_CASES` declares eight cases, each with a stable error code the run must raise when the perturbation is caught.\n- `receipt_provider_context_drift` gives `BATCH5_RECEIPT_DRIFT_REJECTED`.\n- `irrelevant_context_no_replay` gives `BATCH5_REPLAY_IRRELEVANT_CONTEXT_NO_REPLAY`.\n- `proof_contract_sorry` gives `BATCH5_PROOF_CONTRACT_BAD_PROOF_REJECTED`.\n- `live_descendant_never_signal` gives `BATCH5_ORPHAN_REAPER_LIVE_DESCENDANT_NEVER_SIGNAL`.\n- `nonconverging_residual_classified` gives `BATCH5_DRAINER_NONCONVERGING_RESIDUAL_CLASSIFIED`.\n- `trace_omission_receipt_required` gives `BATCH5_TRACE_OMISSION_RECEIPT_REQUIRED`.\n- `blast_radius_leaf_empty` gives `BATCH5_BLAST_RADIUS_EMPTY_LEAF_DOES_NOT_INVENT_COVERAGE`.\n- `doctrine_deleted_path_drift` gives `BATCH5_DOCTRINE_DELETED_PATH_DRIFT_REPORTED`.\n\n##### Prior Art Grounding\n\n###### Provenance, tracing, and supply-chain attestation\n\n**The design borrows the lineage, trace, and attestation shape of W3C PROV, OpenTelemetry, and SLSA provenance, while staying bounded to copied public source, synthetic inputs, and refusal checks.** This bundle borrows from provenance interchange, trace instrumentation, and software supply-chain attestation practice. W3C [PROV](https://www.w3.org/TR/prov-overview/) models the entities, activities, and agents that produce data so a reader can judge its reliability. [OpenTelemetry](https://opentelemetry.io/docs/) is a vendor-neutral pattern for traces, metrics, and logs across composed systems. [SLSA provenance](https://slsa.dev/spec/v1.2/provenance) treats artifact origin, builder identity, and build parameters as explicit attestable metadata. The lineage, trace, and attestation shape is borrowed; the exercise stays bounded to copied public source bodies, synthetic inputs, and refusal checks.\n\n##### Validation Result record Path\n\nReader-verifiable commands, run from the `microcosm-substrate/` public root:\n\n```text\nPYTHONPATH=src ../repo-python -m microcosm_core.organs.batch5_authority_systems_capsule run \\\n  --input fixtures/first_wave/batch5_authority_systems_capsule/input \\\n  --out /tmp/microcosm-batch5-authority-systems-vrp \\\n  --card\nPYTHONPATH=src ../repo-python -m microcosm_core.organs.batch5_authority_systems_capsule run-batch5-bundle \\\n  --input examples/batch5_authority_systems_capsule/exported_batch5_authority_systems_capsule_bundle \\\n  --out /tmp/microcosm-batch5-authority-systems-bundle-vrp \\\n  --card\nPYTHONPATH=src ../repo-pytest tests/test_batch5_authority_systems_capsule.py -q --basetemp /tmp/microcosm-batch5-authority-systems-tests\n```\n\n###### What each command verifies\n\n**The first command writes the synthetic result, the second validates copied modules, digests, anchors, and secret-exclusion, and the test file checks exercises, bundle, omission record, and the eight refusal cases.** The first command writes the synthetic exercise result. The second validates the copied source modules, their manifest digests, anchor rows, and secret-exclusion posture while keeping source bodies out of the record. The test file checks the exercises, the exported bundle, the omission record, the body-scan boundary, and the eight refusal cases. A pass here means the bounded exercise ran and its refusals held, nothing more.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The narrow claim and what stays false\n\n**The strongest claim is that a reader can inspect eight mechanisms as copied source plus a synthetic exercise and confirm each refusal, with no external model access, proof-success, process signal, mutation, or launch-scope decision.** The strongest claim the evidence supports: a reader can inspect eight authority and systems mechanisms as copied public source plus a synthetic exercise, and confirm each refuses its wrong answer, from the commands above alone. The proof boundary is narrow. This page and its generated structured source record are reader evidence, source-linked only. The scope limit holds these facts. No live model or external model access. No Lean proof-success or benchmark claim. No process signal is sent. No generated-state mutation is authorized. No source-file changes, whole-system equivalence, public sharing, or launch-scope decision.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":13248,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/batch5_authority_systems_capsule.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"PROV","url":"https://www.w3.org/TR/prov-overview/","kind":"external_reference","section_id":"provenance-tracing-and-supply-chain-attestation"},{"label":"OpenTelemetry","url":"https://opentelemetry.io/docs/","kind":"external_reference","section_id":"provenance-tracing-and-supply-chain-attestation"},{"label":"SLSA 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false"]},{"id":"paper_module.batch7_oracle_sibling_capsule","title":"Set 7 Oracle Sibling Bundle","relative_href":"docs/paper-module-batch7-oracle-sibling-capsule.html","read_href":"docs/paper-module-batch7-oracle-sibling-capsule.html","context_href":"docs/rules-and-ideas.html#paper-module-batch7-oracle-sibling-capsule","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"Set 7 Oracle Sibling Bundle imports Oracle sibling source bodies and exercises deterministic subject-index, snapshot, truth-diff, quartet-plan, and original pytest witness boundaries.","summary":"Set 7 Oracle Sibling Bundle binds the runnable Oracle sibling source locus, exported copied-source bundle, source-module digests, original pytest witnesses, deterministic subject-index/snapshot/truth-diff/quartet-plan exercises, negative cases, metadata-only result records, and scope limits to a mechanism-backed JSON bundle without claiming accepted-component authority or semantic truth authority.","authority_ceiling":"Fixture-bound public Oracle source-body import, copied-module digest and anchor evidence, deterministic local exercise evidence, original pytest witness evidence, and metadata-only result records only; no Oracle reasoning authority, semantic truth authority, external model access, bridge-backed reasoning, private orchestration engine invocation, source-file changes, publishing-scope decision, launch-scope decision, whole-system equivalence, complete Oracle coverage, or whole-system correctness.","abstract":"Set 7 Oracle Sibling Bundle imports Oracle sibling source bodies and exercises deterministic subject-index, snapshot, truth-diff, quartet-plan, and original pytest witness boundaries.","teleology":{"section_id":"","text":"Set 7 Oracle Sibling Bundle imports Oracle sibling source bodies and exercises deterministic subject-index, snapshot, truth-diff, quartet-plan, and original pytest witness boundaries.","source":"capsule_summary_line"},"body_markdown":"###### Oracle-sibling behavior is replayed from copied code\n\n**The Oracle mostly runs on live feeds and a bridge-backed reasoning engine, so this component asks which parts are pure and inspectable without any feed, provider, or reasoning call.** The Oracle is a sibling system that reasons about market evidence. Most of it depends on live data feeds and on a bridge-backed reasoning engine, so it cannot be shown to a public reader directly. This component answers a narrower question. Which parts of the Oracle are pure, deterministic, and inspectable, and can those parts be run and checked without touching any feed, provider, or reasoning call?\n\n###### The fixture separates advisory output from authority\n\n**The answer is the grounding and bookkeeping layer: four network-free tools that map evidence to targets, hydrate saved artifacts, diff timed feeds, and plan chain repair.** The answer is the Oracle's grounding and bookkeeping layer. Before the Oracle reasons, it builds a map of which evidence supports which prediction target, it hydrates a saved artifact from a recorded run, it diffs two timed snapshots of the same economic feed, and it plans how to repair a missing artifact chain. None of that needs the network or the reasoning engine. This component imports the exact source for those four tools, exercises each against small in-memory runs, and re-runs the Oracle's own tests as an independent witness.\n\n###### A passing oracle replay remains non-authoritative\n\n**The output stays metadata-only: a result record of counts, booleans, and digests, nothing more.** Every run emits a result record of counts, booleans, and digests. Source bodies and test output never enter it.\n\n##### Purpose\n\n###### Why replay against the real source\n\n**A paraphrased or hand-tuned replay would be indistinguishable from theatre, so the component drives the real Oracle source against small readable fixtures and stops before any reasoning call.** A loose replay would be easy to fake and hard to trust. If the bundle copied paraphrased logic, or ran the tools against inputs shaped to pass, a reader could not tell inspection from theatre. So the component imports the real Oracle source and drives it against fixtures small enough to read but specific enough to make the domain rules visible. It also stops at a hard line. The Oracle's repair planner can, if asked, call the bridge-backed `GodModeEngine` to fill a missing node by reasoning. This component walks the planner up to that point and no further. It builds the plan, copies one aliasable artifact, and records `run_missing_quartet` and `GodModeEngine` as excluded. The exclusion is part of the evidence, not a gap in it.\n\n##### How it works\n\n###### `_evaluate` orders the pipeline\n\n** `_evaluate` runs the pytest witness then five engines in fixed order, and a single engine that misses `pass` turns the whole result to `blocked`.** `_evaluate` runs the pipeline in a fixed order. It first calls `_run_original_pytest_witness`, then the five engines listed below, then it collects findings from the source manifest check, the witness, and each engine. Any engine that does not reach `pass` becomes a structured finding, and one finding turns the whole result to `blocked`. `run` and `run_batch7_oracle_sibling_bundle` wrap `_evaluate` through the shared `run_crown_jewel_organ` runner, which handles input loading, the source-manifest gate, and result record writing.\n\nFunction What it does `_run_original_pytest_witness` Runs the Oracle's own focused tests as a subprocess, requires at least seven passed, records return code and stdout/stderr byte counts only `_subject_index_engine` Drives `subject_index.run` on a seeded run and checks the grounding map `_subject_snapshot_engine` Drives `subject_snapshot.run` and checks artifact provenance survives hydration `_truth_diff_macro_engine` Drives `truth_diff_macro.run` on two timed feeds and checks the series delta `_quartet_repair_engine` Drives `build_quartet_repair_plan` and `materialize_missing_aliases`, checks the plan and one written alias `_original_pytest_engine` Folds the witness status into an engine record\n\n###### Grounding a price prediction to admissible support\n\n**Only stock or ETF evidence (the `S_` or `E_` prefix) can anchor a price, so `XOM` and `XLE` ground while economic `TLT` is carried forward as an unmet gap.** The subject-index engine seeds three pieces of evidence through `_seed_subject_run`: a stock (`XOM`, ledger id `S_CAFEBABE`), an ETF (`XLE`, ledger id `E_DEADBEEF`), and an economic instrument (`TLT`, ledger id `M_FEEDBEEF`). The Oracle's rule is that evidence grounds a price prediction only when its subject is a valid target and its ledger id marks it as stock or ETF support, the `S_` or `E_` prefix. So `XOM` and `XLE` land in `admissible_evidence_by_subject`, while `TLT` stays in `contextual_evidence_by_subject` even though it is a valid target, because its support is context that cannot anchor a price. The engine then checks that `TLT` appears in `missing_admissible_support_targets`. The Oracle does not silently drop a target it cannot ground, it carries the gap forward.\n\n###### Provenance survives snapshot hydration\n\n**A hydrated artifact must still carry its `source_artifact_id` and `subject_run_id`, so a caller knows which run a payload came from, with no body copied.** The subject-snapshot engine writes one `lab_director` artifact and hydrates it by id. The result must carry `source_artifact_id` equal to `lab_director` and `subject_run_id` equal to `subject`, so a downstream caller knows which run a prediction payload came from. No artifact body is copied into the result record.\n\n###### Diffing two timed snapshots of a feed\n\n**Diffing a 14:00 and 22:00 snapshot names the changed series by absolute delta (`US10Y` 4.10 to 4.35), the newly appeared `CPI`, and the dropped `OIL`.** The truth-diff engine writes two snapshots of the same economic feed, a subject snapshot at 14:00 and a truth snapshot at 22:00. It confirms the diff names the changed series ranked by strongest absolute delta, here `US10Y` moving from 4.10 to 4.35, a newly appeared series `CPI`, and a dropped series `OIL`. This is the difference between a number changing and the system knowing which number changed, by how much, and whether the set of series itself shifted.\n\n###### Planning repair up to the reasoning boundary\n\n**The repair planner reports `BLOCKED`, names the deepest missing node, and writes one alias, but stops short of ever constructing the reasoning-backed `GodModeEngine`.** The quartet engine exercises the repair planner against a truth run missing most of its node chain. The plan must report readiness `BLOCKED`, name the deepest missing target `oracle_cp2_emitter`, and list `prediction_reconciliation` as aliasable. The engine materialises that one alias and reads the written file back to confirm its `artifact_alias_of` records `oracle_truth_diff_equity`. It asserts `run_missing_quartet_excluded` and that `GodModeEngine` was never constructed.\n\nDiagram of the mechanism (8 steps). Diagram source & refs\n\n```text\nflowchart TD input[\"Fixture input exercise manifest\"] witness[\"_run_original_pytest_witness focused Oracle tests, >=7 passed\"] index[\"_subject_index_engine admissible vs contextual grounding\"] snap[\"_subject_snapshot_engine provenance survives hydration\"] diff[\"_truth_diff_macro_engine changed / new / dropped series\"] quartet[\"_quartet_repair_engine plan BLOCKED, one alias written\"] stop[\"excluded: run_missing_quartet / private orchestration engine\"] record[\"Result record counts, booleans, digests; no bodies\"] input --> witness witness --> index index --> snap snap --> diff diff --> quartet quartet --> stop quartet --> record\n```\n\n##### Negative cases\n\n###### Five malformed inputs, one error code each\n\n**Five negative inputs are declared in `EXPECTED_NEGATIVE_CASES`, each mapped to a stable error code and run through `evaluate_negative_case`.** The component declares five negative cases in `EXPECTED_NEGATIVE_CASES`, each mapped to a stable error code, and evaluated through `evaluate_negative_case`.\n\n###### What each rejection code guards\n\n**Missing run, artifact-id, or truth directories each reject with a named code, the excluded quartet node is enforced, and observations record only the exception type, never its body.** A subject-index run with no run directory rejects with `BATCH7_ORACLE_SUBJECT_RUN_REQUIRED`. A snapshot run with no artifact id rejects with `BATCH7_ORACLE_ARTIFACT_ID_REQUIRED`. A truth-diff run with no truth directory rejects with `BATCH7_ORACLE_TRUTH_RUN_REQUIRED`. The quartet planner must both refuse to run the missing node and confirm the manifest lists `run_missing_quartet` as excluded, tracked as `BATCH7_ORACLE_RUN_MISSING_EXCLUDED`. If the Oracle's own tests do not pass as a witness, the case `BATCH7_ORACLE_ORIGINAL_PYTEST_WITNESS_REQUIRED` fires. Each observation records only the exception type and whether the expected message fragment appeared, never the exception body.\n\n##### Prior Art Grounding\n\n###### Test-oracle and provenance lineage\n\n**The design reuses two known ideas, the test oracle that decides whether an output is acceptable and W3C PROV data provenance, for deterministic replay, not for semantic truth.** The component draws on two established practices. A test oracle is the part of a test that decides whether an observed output is acceptable, surveyed in the [test oracle problem](https://discovery.ucl.ac.uk/1471263/) literature. Data provenance records where an artifact came from and how it was derived, modelled by the W3C [PROV](https://www.w3.org/TR/prov-overview/) family of entities, activities, and agents. The subject index and snapshot tools apply the provenance idea, the truth-diff and quartet checks apply the oracle idea. This component reuses those patterns for deterministic replay. It does not turn a local check into semantic truth.\n\n##### Validation Result record Path\n\nRun from `microcosm-substrate/`:\n\n###### What a result record writes and proves\n\n**The run writes result, board, and validation-result record files, and a pass means the five engines met their contracts and the record held no bodies, nothing more.** The run writes `batch7_oracle_sibling_capsule_result.json`, `batch7_oracle_sibling_capsule_board.json`, and `batch7_oracle_sibling_capsule_validation_receipt.json` under the output directory, plus the sign-off file. The focused test suite is `tests/test_batch7_oracle_sibling_capsule.py`. A pass means the five engines met their contracts, the negative cases rejected as declared, and the result record held no bodies. It means nothing beyond that run.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The bounded claim and its ceiling\n\n**The strongest claim is a bounded source-open replay over one fixture; it runs no Oracle reasoning, dispatches no provider, and excludes launch, mutation, or semantic truth.** The strongest claim this component supports is a bounded, source-open replay: the real Oracle grounding tools ran against readable fixtures, produced the exact expected shapes, rejected five malformed inputs, and left no source or test bodies in the result record. That is the proof boundary and the scope limit. It does not run Oracle reasoning, dispatch any provider or model, invoke `run_missing_quartet` or `GodModeEngine`, touch a market feed or wallet, change source files, or prove complete Oracle coverage. Its `AUTHORITY_CEILING` sets `release_authorized`, `publication_authorized`, `provider_dispatch`, `source_mutation_authorized`, and `semantic_truth_authority` to false. 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whole-system equivalence, complete UI coverage, or whole-system correctness.","abstract":"Set 7 Demo Take Console Bundle imports Swift capture-console source bodies and exercises SwiftPM build, recording-state, helper-bridge, recorder-store, hotkey/audio-meter, and transcribe-payload boundaries.","teleology":{"section_id":"","text":"Set 7 Demo Take Console Bundle imports Swift capture-console source bodies and exercises SwiftPM build, recording-state, helper-bridge, recorder-store, hotkey/audio-meter, and transcribe-payload boundaries.","source":"capsule_summary_line"},"body_markdown":"###### The console bundle replays one copied runtime\n\n**This component reads the Demo Take Console's public Swift source and checks that its safety gates actually hold, without ever launching the macOS capture app.** The Demo Take Console is a macOS app that records screen and microphone takes, drives FFmpeg through a Python helper, and hands the audio to an on-device WhisperKit transcriber. This component reads the app's public Swift source and checks that the safety gates in that source actually hold, without ever launching the app.\n\n###### Three checks bind inputs, output, and ceiling\n\n**It copies eight Swift files, runs six source-token checks plus six paired deletion cases, and emits records of exit codes, counts, and booleans, never source bodies or logs.** It copies eight Swift files into a public bundle, runs six checks that read those copies, then runs six paired negative cases that delete one guard and demand the matching check notice. It emits result records that carry exit codes, counts, and booleans, never the source bodies, recordings, or build logs.\n\n###### The demo result stays metadata-only and local\n\n**A green run says only that the source contracts are present, not that capture, FFmpeg, or a model ever ran.** A green run says the source contracts are present. It does not start capture, run FFmpeg, or dispatch a model.\n\n##### Purpose\n\n###### Do the source gates exist as source\n\n**A capture app is trusted with screen, microphone, and disk, so the narrow question is whether the gates written into its Swift source exist as source.** A capture app is trusted with a screen, a microphone, and disk. The question this component answers is narrow: before anyone trusts the app to record, do the gates written into its Swift source exist as source?\n\n###### Each check asserts a behaviour-carrying token\n\n**Because a file can name clamp yet ship an unclamped meter, each check asserts a specific behaviour-carrying token in the copied source rather than shape alone.** Checking shape alone is not enough. A file can contain the word \"clamp\" and still ship an unclamped meter. So each check asserts a specific token that carries the behaviour: the start gate requires a chosen recording source and free disk, the global hotkey requires Control-Option-Command, the audio meter returns `min(max(rms * 8, 0), 1)`, the transcriber guards a missing audio file. The checks read the copied source, not the app's live state.\n\n##### How it works\n\n###### Two entrypoints and two callbacks\n\n** `run` and `run_batch7_demo_take_bundle` both drive `run_crown_jewel_organ` with `_evaluate` for positive checks and `evaluate_negative_case` for mutation tests, over the fixture or the exported copy.** `run` and `run_batch7_demo_take_bundle` both call `run_crown_jewel_organ` with two callbacks: `_evaluate` for the positive checks and `evaluate_negative_case` for the mutation tests. `run` reads the fixture input; the bundle variant reads the exported public copy. `main` wires the `run` and `validate-bundle` subcommands to the same runner and prints a result or a card.\n\n###### Six engines over eight copied modules\n\n** `_evaluate` runs six engines, failing on any non- `pass` status or fewer than eight copied modules, with each engine reading its Swift file through `_read` and returning booleans plus a `claim_ceiling`.** `_evaluate` runs six engines in order and collects a finding for any that does not return `status: \"pass\"`. It also fails if the copied source manifest holds fewer than eight modules, the count in `SOURCE_REQUIRED_ANCHORS`. Each engine reads its Swift file through `_read`, which resolves the copied path under the exported bundle, and returns a small dict of booleans plus a `claim_ceiling` string.\n\nFunction Reads Passes when `_swiftpm_build_witness` app target `swift build --product DemoTakeConsoleApp` exits zero `_recording_state_control_model` `Models.swift` eleven `RecordingState` cases and the wall/video time keys are present `_capture_helper_bridge_contract` `CaptureHelperClient.swift`, `HostEnvironment.swift` eleven helper commands, sorted-key config JSON, and the repo-bound script path are present `_recorder_store_capture_fsm` `RecorderStore.swift` the start gate, start, pause/resume, and stop-to-review paths are present `_hotkey_audio_meter_contract` `HotkeyMonitor.swift`, `AudioLevelMonitor.swift` the hotkey key and modifiers and the clamped meter return are present `_transcribe_payload_builder` `DemoTakeTranscribe/main.swift` the WhisperKit decode config, the `demo_take_transcript_v0` schema, the SRT builder, and the missing-audio guard are present\n\n###### The build witness runs `swift build`\n\n**The one engine that runs code, `_run_public_witness`, executes `swift build` and returns only the exit code, byte counts, and a `Build complete!` flag, claiming compilation and nothing more.** The build witness is the one engine that runs code. `_cached_swiftpm_build_witness` calls `_run_public_witness`, which runs `swift build` and returns only the exit code, the stdout and stderr byte counts, and whether the string `Build complete!` appeared. The build log body stays out of the record. The claim is that the copied sources compile as a SwiftPM target, nothing more.\n\nDiagram of the mechanism (4 steps). Diagram source & refs\n\n```text\nflowchart TD bundle[\"Copied public Swift bundle eight source files\"] engines[\"Six engines build witness plus five source-token contracts\"] negatives[\"Six negative cases delete one guard re-run that engine\"] record[\"Result record exit codes, counts, booleans no source bodies or logs\"] bundle --> engines bundle --> negatives engines --> record negatives --> record\n```\n\n##### Negative cases\n\n###### Five mutations flip an engine to blocked\n\n**Five negative cases share `_mutated_source_negative`, which copies the bundle, replaces one token, re-runs the matching engine, and expects it to flip to `blocked` with its guard flag false.** `_semantic_runtime_exercises` builds six negative results. Five share `_mutated_source_negative`: it copies the bundle to a temporary directory with `_copy_public_bundle`, replaces one token with `_replace_copied_source_token`, re-runs the matching engine, and expects the engine to flip to `blocked` with its guard flag now false. `_observed_negative_case` reads that flag per case.\n\n###### The five named source mutations\n\n**The five mutations rewrite the helper path, remove the source blocker, drop a hotkey modifier, unclamp the meter, and delete the audio guard, and any check that still passes fails the case.** The five source mutations are `missing_helper_bridge` (rewrites the `demo_take_capture.py` helper path), `start_without_screen` (removes the \"Select at least one recording source\" blocker), `hotkey_wrong_modifier` (drops `.option` from the modifiers), `audio_meter_unclamped` (replaces the clamp with a bare `rms * 8`), and `transcribe_missing_audio` (removes the `audioNotFound` guard). If a check still passes after its guard is gone, it was never testing the guard, and that case fails.\n\n###### The sixth case checks the build witness\n\n**The sixth case, `missing_swift_build_witness`, runs `swift build` against an empty directory and expects a non-pass, since a success against nothing would mean the witness was asserted, not run.** The sixth, `missing_swift_build_witness`, runs `swift build` against an empty temporary directory through `_run_public_witness` and expects a non-pass. A build that \"succeeds\" against nothing would mean the witness is asserted rather than run.\n\n##### Prior Art Grounding\n\n###### ScreenCaptureKit, FFmpeg, and WhisperKit lineage\n\n**The app follows the desktop capture-console lineage anchored by Apple's ScreenCaptureKit, FFmpeg, and WhisperKit, while this component borrows only the contract shape and stays at source-copy and fixture checks.** The app follows the common desktop capture-console lineage: OS capture APIs, a command-line media encoder, typed recording state, a global hotkey, a level meter, and a transcription handoff. The public anchors are Apple's [ScreenCaptureKit](https://developer.apple.com/documentation/ScreenCaptureKit) for screen and audio streaming, [FFmpeg](https://www.ffmpeg.org/ffmpeg.html) for encoding, and [WhisperKit](https://github.com/argmaxinc/WhisperKit) for on-device speech recognition on Apple Silicon. This component borrows the contract shape and keeps the exercise at source-copy and fixture checks.\n\n##### Validation Result record Path\n\nRun from `microcosm-substrate/`:\n\n###### Output files and what a pass means\n\n**The run writes result, board, and validation-result record JSON plus an sign-off file, and a pass means every engine returned `pass`, the six negative cases blocked, and no source body reached a record.** The run writes `batch7_demo_take_console_capsule_result.json`, `batch7_demo_take_console_capsule_board.json`, and `batch7_demo_take_console_capsule_validation_receipt.json` under the output directory, plus the sign-off file. The `validate-bundle` action re-runs the same checks over `exported_batch7_demo_take_console_capsule_bundle`. A pass means every engine returned `pass`, the six negative cases blocked, and no source body reached a record.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The bounded strongest claim\n\n**The strongest claim is that the copied sources compile as a SwiftPM target and the start gate, hotkey modifier, clamp, bridge contract, and transcribe guards are present, confirmed by six deletion tests.** The strongest claim is bounded: the copied Demo Take Console sources compile as a SwiftPM target, and the recording-start gate, hotkey modifier, audio clamp, helper-bridge contract, and transcribe guards are present in that source, with six deletion tests confirming each check depends on its guard.\n\n###### What a green run refuses\n\n**It launches nothing, grants no screen or microphone access, runs no FFmpeg or WhisperKit, and is neither launch-scope decision nor proof that every UI path is covered.** It refuses the rest. It does not launch the app, grant screen or microphone access, start a recording session, run FFmpeg, or dispatch WhisperKit. It is not launch or publishing-scope decision and bounded evidence that every UI path is covered. A green fixture or bundle run is evidence for this source-copy replay only.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":10334,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/batch7_demo_take_console_capsule.md","kind":"source_link"},{"label":"Source 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It validates projection owner selection from changed paths, declared source and artifact fingerprints, no-write check return codes, source-hash cache reuse, planted-byte detection, and missing-artifact failure over four public fixtures while keeping semantic drift proof, repair authority, full source registry validation, launch, and private-system claims out of scope.","authority_ceiling":"Public owner-routed generated projection drift fixture and focused regression result records only; no semantic drift proof, full source registry validation, repair authority, launch-scope decision, whole-system equivalence, source-file changes, external model access, or whole-system correctness.","abstract":"Public generated-projection drift fixture: owner-routed checks fingerprint declared sources and artifacts, reuse clean result records only under matching hashes, and fail planted-byte or missing-artifact cases.","teleology":{"section_id":"","text":"Public generated-projection drift fixture: owner-routed checks fingerprint declared sources and artifacts, reuse clean result records only under matching hashes, and fail planted-byte or missing-artifact cases.","source":"capsule_summary_line"},"body_markdown":"**Generated files committed beside their sources can silently go stale, so given changed paths this gate reports which declared surfaces may now be out of date and whether each one's freshness check still passes.** A repository that commits generated files next to the sources they were built from carries a standing risk: a generated file can quietly fall out of step with its source, and nothing fails until a reader trusts a stale copy. This component checks for that. Given a set of changed paths, it answers which declared generated surfaces might now be out of date, and whether each surface's own freshness check still passes.\n\n###### Module is source-faithful\n\n**The module is a source-faithful refactor of two internal builders that emits one result record per surface: the fingerprints taken, whether a clean record was reused or the check ran, and any drift reasons.** The public module is a source-faithful refactor of two internal builders, kept together in `src/microcosm_core/engine_room/generated_projection_drift_gate.py`. It emits one result record per checked surface: the fingerprints it took, whether it reused a prior clean record or ran the check, and, if the surface drifted, the exact reasons why.\n\n**Owner routing is the design choice worth naming: each generated surface is a row that names its own check, so a changed path selects only the rows it could affect rather than triggering one global diff.** The gate does not diff every generated file in one global sweep. Each surface is a row that names its own check, and a changed path selects only the rows it could affect.\n\n##### Purpose\n\n###### Why a global snapshot fails\n\n**A single global-snapshot alternative fails in two directions: it reruns everything on every edit, and it cannot say who owns a difference or how to fix it.** The loose alternative is a single global snapshot: hash the whole generated tree, compare against a stored snapshot, flag any difference. That fails in two directions. It reruns everything on every edit, and it cannot say who owns a difference or how to fix it.\n\n###### Splitting the tree into owners\n\n**Each surface becomes a `ProjectionOwner` naming its artifacts, sources, and no-write check command, and that command, not the gate, decides freshness while the gate routes and records evidence.** This component splits the tree into owners instead. A `ProjectionOwner` names its generated artifacts, its source authorities, and a no-write check command that is treated as the freshness authority for that surface. A changed path is matched against those patterns, so a small edit selects only the owners it could plausibly touch. The check command, not this gate, decides whether a surface is fresh. The gate routes to the right owner and records the evidence.\n\n##### How it works\n\n###### Runtime is pipeline\n\n**The runtime is a pipeline.** The runtime is a pipeline. It runs in this order for one check.\n\n###### Scope selects owners by path\n\n** `select_projection_owners` keeps only owners whose artifact or source patterns match a changed path, returning the selected rows and a selection `mode` of `scoped_paths`, `owner_filter`, or `all_owners`.** `select_projection_owners` takes the full owner list plus optional owner ids and changed paths. When changed paths are given, it keeps only owners whose artifact or source patterns match one of those paths, through `owner_matches_path` and `projection_pattern_matches_path` (exact, prefix, or glob match). It returns the selected rows and a selection record whose `mode` is `scoped_paths`, `owner_filter`, or `all_owners`.\n\n###### Fingerprint hashes sources and artifacts\n\n** `_owner_fingerprint` content-addresses every source and artifact file with `sha256`, recording a combined `source_hash`, `artifact_hash`, path counts, and any pattern that matches no file as missing.** For each selected owner, `_owner_fingerprint` content-addresses every source file and every artifact file with `sha256`. It records a combined `source_hash`, a combined `artifact_hash`, path counts, and any missing patterns. A pattern that matches no file on disk is counted as missing.\n\n###### Cache reuse is deliberately strict\n\n** `_source_hash_cache_hit` reuses a prior clean record only when its status, `source_hash`, `artifact_hash`, and check command all match and no artifacts are missing, otherwise the check runs.** `_source_hash_cache_hit` decides whether a prior clean record can be reused. Reuse is deliberately strict. It requires a prior record with `status` clean, the same `source_hash`, the same `artifact_hash`, the same check command, and zero missing artifacts. If any of those differ, the cache misses and the check runs.\n\n###### Run the check or reuse cache\n\n**A cache hit returns a synthetic `source_hash_cache` record, while a miss runs the owner's check command and captures its return code plus hashed, length-counted stdout and stderr without storing raw text.** On a cache hit, `_cached_clean_result` returns a synthetic record marked `source_hash_cache`, returncode zero. On a miss, `_run_command` runs the owner's check command and captures its return code plus hashed, length-counted stdout and stderr (the raw text is not stored). The command runs through `_run_builtin_command` first, which handles the fixture-only `builtin:pass`, `builtin:fail`, and `builtin:assert-file-equals` verbs before falling through to a real subprocess.\n\n###### Classify drift reasons per owner\n\n** `_check_owner` assembles drift reasons from a failing command, a missing artifact, or an incomplete lineage record, and any reason marks the owner `drift` while none marks it `clean`.** `_check_owner` assembles the drift reasons. A nonzero return code adds `check_command_failed`. A missing artifact adds `artifact_missing`, independent of the command result, so an absent generated file cannot be laundered by a green command. A required-but-incomplete lineage record adds `fact_authority_lineage_missing_required` or `fact_authority_lineage_invalid`. Any reason makes the owner status `drift`; none makes it `clean`.\n\n###### Aggregate the per-owner results\n\n** `check_projection_drift` runs the per-owner check over the selected rows and rolls them into an overall `status`, `drift_owner_count`, and `source_hash_cache` hit and miss counts.** This is the record a caller reads. The residual explanation remains available so the reader can inspect the concrete boundary without reconstructing omitted context.\n\nFunction Role `select_projection_owners` Scopes the owner set by owner id or changed path `check_projection_drift` Runs the per-owner checks and aggregates the result `evaluate_case` Materialises one fixture case in a scratch tree and checks it `evaluate_fixture_dir` Runs every fixture case in a directory and counts passes `main` Command-line entry: `check` a real tree or `evaluate-fixtures`\n\n###### The two CLI subcommands\n\n**The `check` subcommand runs the gate against a live tree and exits nonzero on drift, while `evaluate-fixtures` materialises each bundled case in a temporary tree and checks observed against expected status.** The `check` subcommand runs the gate against a live tree and exits nonzero when any owner drifted. The `evaluate-fixtures` subcommand runs the bundled cases; `evaluate_case` writes each case's declared files into a temporary tree, builds owners from the case JSON, and records whether the observed status matched the expected one.\n\nDiagram of the mechanism (10 steps). Diagram source & refs\n\nSource refs\n\nSelect projection owners `select_projection_owners`\n\n```text\nflowchart TD A[\"Changed path or owner id\"] --> B[\"select_projection_owners\"] B --> C[\"_owner_fingerprint: sha256 sources and artifacts\"] C --> D{\"_source_hash_cache_hit matches prior clean record?\"} D -- \"yes\" --> E[\"reuse cached clean record\"] D -- \"no\" --> F[\"_run_command runs the owner check\"] E --> G[\"_check_owner assembles drift reasons\"] F --> G G --> H{\"any reason?\"} H -- \"yes\" --> I[\"status drift with reasons\"] H -- \"no\" --> J[\"status clean\"]\n```\n\n##### Negative cases\n\n###### Two positive and three drift cases\n\n**Two positive fixtures and three drift fixtures cover a planted byte, a deleted artifact, and a missing lineage record, and a run over all five returns `status: pass` with every case passed.** The fixture set has two positive cases, `clean_owner` and `scoped_changed_path`, and three that must be caught as drift. `planted_byte_detected` changes one byte in a generated file so the owner's `builtin:assert-file-equals` check returns nonzero, giving `check_command_failed`. `missing_artifact` deletes a declared artifact so the fingerprint reports a missing path, giving `artifact_missing` even though the check would otherwise pass. `fact_authority_lineage_required_missing` declares an owner that requires a lineage record but omits its guard, derivation, and residual fields, giving `fact_authority_lineage_missing_required`. A run over all five cases returns `status: pass`, `case_count: 5`, `passed_case_count: 5`.\n\n##### Prior Art Grounding\n\n###### Reproducible-build and regression-test lineage\n\n**The gate follows reproducible-build and regression-test practice: declare source inputs, produce artifacts, compare content hashes, and rerun the owner check when source or artifact identity changes.** The gate follows reproducible-build and regression-test practice: declare source inputs, produce generated artifacts, compare content hashes, and rerun the owner check when either source or artifact identity changes. Two anchors are relevant.\n- [Bazel hermeticity](https://bazel.build/concepts/hermeticity), for its emphasis on declared inputs, source identity, repeatable actions, and cache validity.\n- [pytest-regtest snapshot testing](https://pytest-regtest.readthedocs.io/en/stable/snapshots/), where recorded outputs are compared against reference outputs to catch unexpected change.\n\n###### Borrowed discipline, added owner routing\n\n**The borrowed part is declared-input and artifact-fingerprint discipline; the added part is routing each check through its owner rather than treating all generated files as one snapshot.** The borrowed part is declared-input and artifact-fingerprint discipline. The added part is routing each check through its owner rather than treating all generated files as one snapshot.\n\n##### Validation Result record Path\n\n```text\nPYTHONPATH=src ./repo-pytest tests/test_engine_room_generated_projection_drift_gate.py -q --basetemp /tmp/microcosm-generated-projection-drift-gate\ncd microcosm-substrate && PYTHONPATH=src python3 scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\nThe public exercise runs the same fixtures directly:\n\n```text\nPYTHONPATH=src python3 -m microcosm_core.engine_room.generated_projection_drift_gate evaluate-fixtures --input fixtures/first_wave/engine_room_generated_projection_drift_gate/input --json\n```\n\n###### What a pass means here\n\n**A pass means the fixture behavior and the bundle-backed JSON projection are reproducible; it runs no repair commands and excludes launch.** A pass means the fixture behavior and the bundle-backed JSON projection are reproducible. It does not run repair commands and it excludes launch.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim\n\n**For a supplied tree and declared owners, the gate reports whether each selected owner's required artifacts were present and its no-write check passed, with fingerprints recorded and clean records reused only under matching hashes.** The strongest claim the evidence supports: for a supplied tree and a set of declared owners, the gate reports whether each selected owner's required artifacts were present and whether its own no-write check passed, with source and artifact fingerprints recorded and clean records reused only under matching hashes. That is a freshness signal for declared owners.\n\n###### What the gate does not establish\n\n**The proof boundary stops at freshness: the gate is not semantic-drift proof, not full registry validation, and carries no repair or launch-scope decision over the wider generated tree.** The proof boundary stops there. It is not semantic drift proof, so a check that passes on identical bytes says nothing about whether the generated prose is meaningfully correct. It is not full registry validation: only declared owners are checked, and the wider generated tree is judged by each owner's own command. It has no repair authority and no launch-scope decision. 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Bundle","url":"paper-module-batch8-validator-checker-capsule.html","relation_id":"paper_module.depends_on.paper_module","source_ref":"core/paper_module_capsules.json::paper_modules[63:paper_module.batch8_validator_checker_capsule].depends_on"},{"id":"paper_module.engine_room_demo","kind":"paper_module","kind_label":"paper module","relation":"depended on by","label":"Engine Room Demo","url":"paper-module-engine-room-demo.html","relation_id":"paper_module.depends_on.paper_module","source_ref":"core/paper_module_capsules.json::paper_modules[31:paper_module.engine_room_demo].depends_on"}],"source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/engine_room_generated_projection_drift_gate.md"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json"}],"visible_text_byte_count":13806,"section_titles":["Generated files committed","Module is source-faithful","Owner routing is","Purpose","Why 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Singleflight","relative_href":"docs/paper-module-engine-room-command-run-singleflight.html","read_href":"docs/paper-module-engine-room-command-run-singleflight.html","context_href":"docs/rules-and-ideas.html#paper-module-engine-room-command-run-singleflight","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"Public command-run singleflight fixture: content-addressed subprocess keys collapse duplicate active runs and replay captured result records without claiming scheduler or daemon authority.","summary":"Engine Room Command-Run Singleflight is a subprocess singleflight bundle. It validates content-addressed command keys, scoped dirty/content fingerprints, fcntl leader/follower election, completed-run reuse, captured output replay, and two negative boundaries over fixture commands while keeping scheduler, daemon, distributed-lock, live-state export, launch, and private-system claims out of scope.","authority_ceiling":"Public subprocess singleflight fixture and focused regression result records only; no job scheduler, daemon, distributed lock service, live command_runs export, external model access authority, launch-scope decision, whole-system equivalence, source-file changes, or whole-system correctness.","abstract":"Public command-run singleflight fixture: content-addressed subprocess keys collapse duplicate active runs and replay captured result records without claiming scheduler or daemon authority.","teleology":{"section_id":"","text":"Public command-run singleflight fixture: content-addressed subprocess keys collapse duplicate active runs and replay captured result records without claiming scheduler or daemon authority.","source":"capsule_summary_line"},"body_markdown":"**When two callers fire the same command at once, this component runs it exactly once and hands the second caller the first caller's captured output.** When several agents or background tasks fire the same command at the same moment, the plain outcome is several identical subprocesses doing the same work at once. That wastes the machine, and where the command writes something it can corrupt shared state by writing twice. This component answers one question: when two callers ask for the same command at the same time, can the system run it exactly once and hand the second caller the first caller's captured output?\n\n###### Is public refactor\n\n**It is a public refactor keeping the executable core: content-addressed keys, an `fcntl` leader lock, captured output that followers replay, and completed-run reuse on request.** It is a public refactor of an internal command-run helper. It keeps the executable core: content-addressed command keys, an `fcntl` file lock that elects one leader per key, captured stdout and stderr that followers replay, and completed-run reuse when the caller asks for it. It does not import or expose the internal live command-run state tree, and it does not stand in for a scheduler.\n\n###### Whole file is\n\n**The whole file is `src/microcosm_core/engine_room/command_run_singleflight.py`.** The whole file is `src/microcosm_core/engine_room/command_run_singleflight.py`. Every function named below is defined there.\n\n##### Purpose\n\n###### The key spans argv, cwd, environment, and scope\n\n**Rather than compare names or argv alone, the component keys on argv, cwd, an environment slice, and scoped worktree state, so two runs collapse only while their code is identical.** A loose deduplicator compares command names or memoizes by argv alone. That is wrong here, because the same argv can mean different work: edit a file the command reads and the correct answer changes. So the interesting decision is how this component decides two requests are the same. It does not compare names. It builds a key over the argv, the working directory, a small slice of the environment, and the scoped worktree state. Two runs collapse into one only while the code they would see is identical. Change a file in scope and the key changes, which forces a fresh run instead of serving a stale result.\n\n##### How it works\n\n###### One entry function, two role functions\n\n**The pipeline is short: one entry function takes a per-key lock, decides a role, and returns a result record, while the heavy work lives in two role functions.** The pipeline is short. One entry function takes a per-key lock, decides a role, and returns a result record. The heavy work lives in two role functions.\n\nFunction Role `build_command_key` Builds the content-addressed key from argv, cwd, env slice, and scoped state. `run_command_singleflight` Entry point: locks the key, picks leader, follower, or reuse, returns a `RunReceipt`. `_run_leader` Runs the subprocess once, captures output, writes the run records. `_wait_for_active` Follower path: waits for the active run and replays its captured output. `evaluate_fixture_dir` Runs the four public fixture cases and reports pass or fail.\n\n###### `build_command_key` hashes into a `dirty_fingerprint`\n\n** `build_command_key` hashes argv, cwd, and a `dirty_fingerprint` over scoped Git or file state, keeping only a `PYTHONPATH` and `PYTEST_ADDOPTS` environment slice and no absolute paths.** `build_command_key` consumes the argv, the resolved cwd, a resource class, the scoped paths, and an environment mapping. It hashes the argv, labels and hashes the cwd, and computes a `dirty_fingerprint` over the scope. Inside a Git repository that fingerprint carries `HEAD`, the porcelain status hash, the working diff hash, the staged diff hash, and per-file content hashes for the scoped paths. Outside Git it falls back to the scoped file-content fingerprint alone. The environment slice is narrow: `PYTHONPATH` and `PYTEST_ADDOPTS`. The key is a dictionary of hashes and labels, not absolute paths, so the records stay public-safe. `run_command_singleflight` then reduces it to a 24-character `key_hash`.\n\n###### `run_command_singleflight` elects a role under the lock\n\n** `run_command_singleflight` rejects empty argv, then under an `fcntl.LOCK_EX` per-key lock reads the active record and becomes a follower, a reuse, or the leader.** `run_command_singleflight` rejects empty argv with a `ValueError` before it builds anything. It resolves the cwd and state root, builds the key, and opens the per-key lock file under `fcntl.LOCK_EX`. Holding the lock, it reads the active record for this key. If an active run is still running and its process is alive or inside the pending window, this caller becomes a follower. If an active run has completed and the caller passed `reuse_completed=True`, it becomes a reuse. Otherwise it writes fresh active and run metadata, releases the lock, and becomes the leader.\n\n###### `_run_leader` runs the subprocess once\n\n** `_run_leader` starts the subprocess once, captures stdout and stderr to per-run files, marks the run completed with its exit code, and returns a `leader` `RunReceipt`.** `_run_leader` starts the subprocess once with `subprocess.Popen`, records the active run with its pid, then blocks on `communicate` to capture stdout and stderr. It writes those to per-run output files, marks the run completed with its exit code, updates the latest-by-key record, and appends leader lifecycle events. It returns a `RunReceipt` with `role` set to `leader`.\n\n###### `_wait_for_active` replays without rerunning\n\n** `_wait_for_active` polls until the run completes and replays the leader's output with the same `run_id`, and on a dead process or timeout returns `stale_or_timeout` with exit code 124.** `_wait_for_active` is the follower path. It polls the active record until the status reads completed, then reads the leader's captured output files and returns a record that carries the same stdout, stderr, exit code, and `run_id` as the leader. It never launches its own subprocess. If the active run's process dies or the run does not finish inside the wait window, the follower returns `status: stale_or_timeout` with exit code `124` and still does not rerun.\n\nDiagram of the mechanism (7 steps). Diagram source & refs\n\nSource refs\n\ntake per-key fcntl lock `run_command_singleflight`\n\n```text\nflowchart TD A[\"argv, cwd, env, scope paths\"] --> B[\"build_command_key hash argv + cwd + env + scoped state\"] B --> C[\"run_command_singleflight take per-key fcntl lock\"] C --> D{\"active run for this key?\"} D -- \"running\" --> E[\"_wait_for_active follower replays captured output\"] D -- \"completed and reuse_completed\" --> F[\"replay completed result\"] D -- \"none or stale\" --> G[\"_run_leader run subprocess once, capture output\"]\n```\n\n##### Negative cases\n\n###### Two positive and two boundary fixture cases\n\n** `evaluate_fixture_dir` checks four cases: `single_leader` and `completed_reuse` pass, while `scope_mutation_changes_key` and `missing_command_rejected` guard the key and reject empty argv.** `evaluate_fixture_dir` runs four named cases and checks each observed result against the expected flag. Two are positive: `single_leader` confirms one caller wins the lock and runs the command, and `completed_reuse` runs a counter command, then reuses the completed result and checks the counter stayed at 1. Two are boundary cases. `scope_mutation_changes_key` writes a scoped file, builds a key, rewrites the file, builds the key again, and requires the two `key_hash` values to differ, so edited scoped content cannot be laundered into an old run. `missing_command_rejected` calls the entry function with empty argv and requires the `ValueError`, so an empty command is never accepted. A pass is `case_count: 4` and `passed_case_count: 4`.\n\n##### Prior Art Grounding\n\n###### Lineage in Go's singleflight package\n\n**The pattern is duplicate-call suppression from Go's singleflight, adapted to local subprocess runs with a content-addressed key, scoped state, and captured output replay, not a distributed lock.** The pattern is duplicate-call suppression, the same idea as Go's singleflight package, [`golang.org/x/sync/singleflight`](https://pkg.go.dev/golang.org/x/sync/singleflight), which defines a namespace of work where duplicate calls for one key share a single in-flight execution. This component borrows the leader and follower shape and the shared result, then adapts it to local subprocess runs with a content-addressed command key, scoped worktree state, captured output replay, and an explicit line between collapsing active duplicates and caching completed results. It is singleflight for command execution, not a distributed lock service.\n\n##### Validation Result record Path\n\nExercise the four fixture cases:\n\n```text\nPYTHONPATH=src python3 -m microcosm_core.engine_room.command_run_singleflight evaluate-fixtures \\\n  --input fixtures/first_wave/engine_room_command_run_singleflight/input \\\n  --json\n```\n\nRun the focused tests and the corpus parity check:\n\n```text\nPYTHONPATH=src ./repo-pytest \\\n  tests/test_engine_room_command_run_singleflight.py \\\n  -q --basetemp /tmp/microcosm-command-run-singleflight\ncd microcosm-substrate && PYTHONPATH=src python3 scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n###### The two-process race the tests add\n\n**The focused tests add a real two-process race where roles resolve to leader and follower, both share one `run_id` and `counter=1`, and the side effect increments exactly once.** The focused tests add the real two-process race: two callers start the same command, their roles resolve to `leader` and `follower`, both records share one `run_id`, both replay `counter=1`, and the side-effect counter increments exactly once. A pass means the public fixture behavior and the bundle-backed JSON projection stay reproducible. These validation result records do not prove scheduler behavior, daemon behavior, or distributed locking.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest claim and its boundary\n\n**At most, a duplicate key elects one leader, followers replay its output, an edited file forces a miss, reuse is explicit, and empty argv is refused, with nothing beyond that.** The strongest claim the evidence supports: a duplicate command key elects one leader through a per-key `fcntl` lock, followers replay that leader's captured output rather than rerunning, an edited scoped file changes the key and forces a miss, a completed run is reused only on explicit request, and empty argv is refused. That is the proof boundary. Its scope limit is a single subprocess singleflight fixture. It is not a job scheduler, not a daemon, not a distributed lock service, and not an export of live command-run state. A green record here is not evidence for a durable queue, a network lock, or whole-system correctness.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":10899,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/engine_room_command_run_singleflight.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"`golang.org/x/sync/singleflight`","url":"https://pkg.go.dev/golang.org/x/sync/singleflight","kind":"external_reference","section_id":"lineage-in-go-s-singleflight-package"}],"grouped_relation_feeds":[{"relation":"abides by","count":5,"links":[{"id":"AX-1","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Derivation before assertion","url":"doctrine.html#dcard-ax-1"},{"id":"AX-5","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Fail-closed 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It exercises synthetic SQLite queue state, lease recovery, blackboard claim projection, and cold-start reconciliation fixtures without exporting private runtime state or dispatching providers.","authority_ceiling":"Component evidence for the accepted staged Engine Room demo only; not a live non-public runtime export, not external model service, not agent dispatch, not distributed database proof, not launch-scope decision, and not source-file changes.","abstract":"Staged Engine Room component: synthetic SQLite metabolism runtime exercise for queues, leases, blackboard projection, and reconciliation.","teleology":{"section_id":"","text":"Staged Engine Room component: synthetic SQLite metabolism runtime exercise for queues, leases, blackboard projection, and reconciliation.","source":"capsule_summary_line"},"body_markdown":"###### Long-running runtime's durable\n\n**A long-running runtime's durable record of jobs, leases, runs, and claims drifts, so this component reads that state alone and asks which rows are inconsistent and which a person should review first.** A long-running agent runtime keeps a durable record of its work: jobs waiting to run, leases held by workers, runs in flight, and claims asserted on a shared blackboard. That record drifts. A worker dies mid-run and its lease is never released. A run finishes but the job it belonged to still reads running. A launch log goes stale because nothing writes to it. This component takes the durable state alone and answers one question: which rows are inconsistent, and which of those should a person look at before anything touches them.\n\n###### Is synthetic sqlite\n\n**It is a synthetic SQLite exercise, not the live runtime: a single stdlib file that creates a real database, drives the queue, lease, run, blackboard, and reconciliation paths, and emits one JSON record.** It is a synthetic SQLite exercise, not the live runtime. It ships fixtures and creates a real database file, drives the queue, lease, run, blackboard, and reconciliation paths over that file, and emits one JSON result record. The whole component is `src/microcosm_core/engine_room/metabolism_runtime.py`, a single file that uses only the Python standard library.\n\n###### Touches no live\n\n**It touches no live runtime: it reads no private runtime database, dispatches no worker or provider, and does not stand in for distributed-database behaviour.** It reads no private runtime database, dispatches no worker or provider, and does not stand in for distributed-database behaviour.\n\n##### Purpose\n\n###### What reconciliation refuses to repair\n\n**Reconciliation tags every finding `operator_review_required` and refuses to auto-repair ambiguous state, recovering only the expired lease that has a clean recovery path via `requeue_expired_jobs`.** The interesting choice is what the reconciliation pass refuses to do. It reads the jobs and runs tables, applies its rules, and tags every finding `operator_review_required`. It does not auto-repair. An expired lease has a clean recovery path, so `requeue_expired_jobs` moves it back to `recoverable` on its own. But a running job with no run row, or a finished run whose job still reads running, is ambiguous. The safe move is to surface it, not to guess. A loose runtime that guesses here corrupts the record; this component draws the line and stays on the safe side of it.\n\n###### The blackboard is append-only\n\nClaims are never edited or deleted in place. An assertion is one event row. A contradiction, expiry, or supersession is a separate event that points back at the assertion it invalidates. State is reconstructed from an append-only history, so the reason a claim is no longer active stays on the record.\n\n##### How it works\n\n###### Mechanism lives in\n\n**The mechanism lives in one module.** The stages below run in the order the fixture harness drives them.\n\n###### Storage opens a WAL SQLite store\n\n** `connect` opens a WAL SQLite file with foreign keys on and `ensure_schema` creates the `jobs`, `runs`, and `blackboard_claim_events` tables plus a partial unique index over the active states.** `connect` opens a SQLite file with `PRAGMA journal_mode=WAL`, normal sync, and foreign keys on, then calls `ensure_schema`. `ensure_schema` creates three tables, `jobs`, `runs`, and `blackboard_claim_events`, and a partial unique index `idx_jobs_active_idempotency` on `idempotency_key` restricted to the four active states in `ACTIVE_STATES` (`queued`, `claimed`, `running`, `recoverable`).\n\n###### Queue insertion and lease recovery\n\n** `enqueue_job` rejects a duplicate idempotency key that is still active, `claim_next_job` leases the next queued or recoverable job, and `requeue_expired_jobs` recovers lapsed leases to `recoverable`.** `enqueue_job` inserts one job row and returns `(job, inserted)`. When a job with the same `idempotency_key` is still active, the partial unique index raises `sqlite3.IntegrityError`, which the function catches and reports as `inserted=False`. Once the earlier job reaches a terminal state the key is free again. `claim_next_job` selects the next `queued` or `recoverable` job by priority then creation time, stamps a `claim_expires_at` lease, and moves it to `claimed`. `requeue_expired_jobs` finds `claimed` or `running` rows whose lease has lapsed, clears the owner and expiry, moves them to `recoverable`, and returns how many it recovered.\n\n###### Runs and the finalize-job flag\n\n** `start_run` moves a job to `running` and `complete_run` records a return code, propagating a terminal job state only when `finalize_job=True`, which makes the still-running-job defect reachable on purpose.** `start_run` inserts a run row and moves its job to `running` through `update_job_state`. `complete_run` writes `completed_at` and a return code; it only propagates a terminal job state when the caller passes `finalize_job=True`. That flag is what makes the \"finished run, still-running job\" defect reachable on purpose.\n\n###### Blackboard events and projection\n\n** `append_claim_event` appends one immutable event of a given kind, and `build_blackboard_projection` replays the whole log to return the assertions that were not invalidated, with active, event, and contradiction counts.** `append_claim_event` appends one event of a given kind (`claim_asserted`, `claim_contradicted`, `claim_expired`, `claim_superseded`) and never mutates a prior row. `build_blackboard_projection` reads the whole event log, collects the assertion ids named by any invalidating event, and returns the assertions that were asserted and not invalidated, with counts for active claims, total events, and contradictions.\n\n###### Reconciliation surfaces four defect rules\n\n** `reconcile` walks `claimed` and `running` jobs against their newest run and emits a `ReconciliationFinding` with `operator_review_required` for each of four defect rules, returning `healthy` or `needs_review`.** `reconcile` walks the `claimed` and `running` jobs, pulls the newest run for each with `latest_run_for_job`, and emits a `ReconciliationFinding` for each defect it sees. Every finding carries an expected value, an observed value, and the action `operator_review_required`, and derives a stable `finding_id` from its rule and object. The four rules are `running_job_no_run_row`, `run_finalized_but_job_running`, `running_job_stale_launch_log`, and `running_job_missing_launch_log`. The pass returns status `healthy` when it finds nothing and `needs_review` when it does, plus per-rule counts, the findings, the `CLAIM_CEILING` string, and the scope boundary list.\n\n###### Fixture harness and CLI\n\n** `evaluate_case` runs one named case in an isolated scratch directory, `evaluate_fixture_dir` passes only when every case met its expectation, and `main` exposes this as the `evaluate-fixtures` command.** `evaluate_case` runs one named case against an isolated scratch directory and compares its status to the expected status. `evaluate_fixture_dir` reads every `*.json` file in the input directory, runs each case, and returns `status: pass` only when at least one case exists and every case met its expectation. `build_parser` and `main` expose this as the `evaluate-fixtures` command, printing the JSON record and returning a process exit code of 0 for pass and 1 for fail.\n\nFunction Role `connect`, `ensure_schema` Open the WAL SQLite store and install the three tables and indexes `enqueue_job` Insert a job, reject a duplicate that is still active `claim_next_job`, `requeue_expired_jobs` Lease the next job, recover a lapsed lease to `recoverable` `start_run`, `complete_run` Record a run and, on request, finalise its job `append_claim_event`, `build_blackboard_projection` Append claim events, project the active claims `reconcile`, `ReconciliationFinding` Detect inconsistent job, run, and log state as review findings `evaluate_fixture_dir`, `main` Replay the fixture cases and emit the JSON record Diagram of the mechanism (7 steps). Diagram source & refs\n\nSource refs\n\nappend_claim_event `build_blackboard_projection`\n\n```text\nflowchart TD Fixtures[\"evaluate_fixture_dir reads *.json cases\"] Store[\"connect / ensure_schema WAL SQLite: jobs, runs, blackboard_claim_events\"] Queue[\"enqueue_job / claim_next_job requeue_expired_jobs\"] Runs[\"start_run / complete_run\"] Board[\"append_claim_event build_blackboard_projection\"] Recon[\"reconcile ReconciliationFinding per defect\"] Record[\"JSON record status, rule counts, findings, scope limit\"] Fixtures --> Store Store --> Queue Queue --> Runs Store --> Board Runs --> Recon Board --> Recon Recon --> Record\n```\n\n##### Negative cases\n\n###### Five fixture cases, three forcing findings\n\n**Five named fixture cases each expect `pass`, and three of them exist to force a reconciliation finding.** The fixtures in `fixtures/first_wave/engine_room_metabolism_runtime/input` are five named cases, each expecting `pass`. Three of them exist to force a reconciliation finding.\n\n###### Queue recovery and blackboard projection cases\n\n** `queue_recovery` rejects a duplicate key and recovers an already-expired lease to `recoverable`, while `blackboard_projection` asserts then contradicts a claim and checks zero active claims and one contradiction.** `queue_recovery` enqueues the same idempotency key twice, expecting the second insert to be rejected, claims the job with a lease of `-1` seconds so it is already expired, and checks that `requeue_expired_jobs` returns 1 and the job lands in `recoverable`. `blackboard_projection` asserts one claim then contradicts it, and checks the projection reports zero active claims and one contradiction.\n\n###### Three cases that force reconciliation rules\n\n** `running_job_no_run_row`, `finalized_run_running_job`, and `stale_log` each provoke their named reconciliation rule, and any case whose observed status misses its expected status fails the whole record.** `running_job_no_run_row` marks a job running with no run row and expects `running_job_no_run_row` in the rule counts. `finalized_run_running_job` starts a run, completes it with `finalize_job=False`, and expects `run_finalized_but_job_running`. `stale_log` writes a log file, back-dates its mtime by 1200 seconds, and expects `running_job_stale_launch_log` under a 60-second freshness threshold. A case whose observed status does not match its expected status makes the whole record fail.\n\n##### Prior Art Grounding\n\n###### Durable-runtime control-loop lineage\n\n**The component follows the durable-runtime control-loop lineage of autonomic computing and SRE monitoring over a WAL store, borrowing the self-management loop while keeping the jobs, leases, and findings synthetic.** The component follows the durable-runtime control-loop lineage: keep the work state in a durable log, detect stale or inconsistent state, recover what has a clean recovery path, and keep that log separate from the acting dispatcher. Relevant anchors are IBM's [autonomic-computing architecture](https://research.ibm.com/publications/autonomic-computing-architectural-approach-and-prototype) and its monitor-analyze-plan-execute loop, [SQLite write-ahead logging](https://www.sqlite.org/wal.html) for the local durability and concurrency behaviour used here, and Google's [SRE monitoring guidance](https://sre.google/sre-book/monitoring-distributed-systems/) for separating symptoms from causes. The borrowed idea is the self-management loop over a durable local store; the jobs, leases, claims, and findings are synthetic and public-safe.\n\n##### Composition context\n\n###### Admitted as a staged demo bundle\n\n** `demo.py` stages this component as one `CapsuleExercise` covering the `metabolism_runtime` and `metabolism_reconciler` targets, running its `evaluate_fixture_dir` in sequence as a demo, not registry integration or an admission gate.** `src/microcosm_core/engine_room/demo.py` admits this component as one staged bundle. Its `CAPSULES` tuple holds a `CapsuleExercise` for `engine_room_metabolism_runtime` covering the jewel targets `metabolism_runtime` and `metabolism_reconciler`. `run_capsule` imports the module and calls its `evaluate_fixture_dir`. `audit_controller_coverage` checks that each bundle has its expected surfaces on disk and that the covered targets match `EXPECTED_JEWEL_TARGETS`. The demo runs the bundles in sequence and reports whether they pass; it is not registry integration or an admission gate.\n\n##### Validation Result record Path\n\n```text\nPYTHONPATH=src python3 -m microcosm_core.engine_room.metabolism_runtime evaluate-fixtures \\\n  --input fixtures/first_wave/engine_room_metabolism_runtime/input \\\n  --json\n\nPYTHONPATH=src ./repo-pytest tests/test_engine_room_metabolism_runtime.py -q\ncd microcosm-substrate && PYTHONPATH=src ../repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n###### What the result records prove\n\n**The CLI passes only when every fixture case met its expected status, the pytest suite exercises each path, and the corpus check keeps the generated row in parity, showing reproducibility and nothing more.** The CLI emits a record with `organ_id: engine_room_metabolism_runtime`; a pass means every fixture case met its expected status. The pytest suite exercises WAL and idempotency, expired-lease recovery, the contradicted-claim projection, each reconciliation rule, fixture replay, and the CLI. The corpus check keeps the generated JSON row in parity with the bundle registry. A pass shows the synthetic behaviour is reproducible and nothing more.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim\n\n**The component is a reproducible synthetic SQLite exercise for a durable job queue, lease recovery, append-only blackboard projection, and a reconciliation taxonomy that surfaces ambiguous state for review instead of repairing it.** The strongest claim the evidence supports: this component is a reproducible synthetic SQLite exercise for a durable job queue, lease recovery to `recoverable`, append-only blackboard claim-event projection, and a cold-start reconciliation taxonomy that surfaces ambiguous state for review instead of repairing it. That is the whole scope limit.\n\n###### What it is not\n\n**It is not a live non-public runtime export, a dispatcher, an auto-repairer, or a distributed database, and its concurrency is only that of a single local WAL file.** It is not a live non-public runtime export. It is not an agent dispatcher and calls no provider. It does not auto-repair ambiguous job or run state. It is not a distributed database, and its concurrency behaviour is only that of a single local WAL file. It ships no private runtime database, status JSON, operator session, or live log. 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DAG","relative_href":"docs/paper-module-engine-room-bridge-campaign-dag.html","read_href":"docs/paper-module-engine-room-bridge-campaign-dag.html","context_href":"docs/rules-and-ideas.html#paper-module-engine-room-bridge-campaign-dag","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"Staged Engine Room component: pre-dispatch bridge-campaign DAG validator for typed nodes, acyclicity, synthesis reachability, and provider fan-out ceilings.","summary":"Engine Room Bridge Campaign DAG binds the staged pre-dispatch campaign validator to the accepted Engine Room demo mechanism. It validates typed probe/reducer/synthesis graphs, rejects cycles, requires synthesis nodes to trace back to probe evidence, checks provider parallelism ceilings, and emits public fixture result records without dispatching providers or proving campaign execution safety.","authority_ceiling":"Component evidence for the accepted staged Engine Room demo only; no external model access, campaign execution, reducer or synthesis correctness proof, provider safety proof, launch-scope decision, whole-system equivalence, source-file changes, or whole-system correctness.","abstract":"Staged Engine Room component: pre-dispatch bridge-campaign DAG validator for typed nodes, acyclicity, synthesis reachability, and provider fan-out ceilings.","teleology":{"section_id":"","text":"Staged Engine Room component: pre-dispatch bridge-campaign DAG validator for typed nodes, acyclicity, synthesis reachability, and provider fan-out ceilings.","source":"capsule_summary_line"},"body_markdown":"###### Component is validator\n\n**This component is a validator that reads a bridge campaign graph and decides, before dispatch, whether it is safe to run, never calling a provider, worker, or reducer.** A bridge campaign fans one piece of work out across several agent providers, then folds their results back into a single conclusion. This component reads the graph that describes such a campaign and decides, before anything runs, whether the graph is safe to dispatch. It is a validator. It never calls a provider, never dispatches a worker, and never runs a reducer.\n\n###### Takes json campaign\n\n**It takes one JSON campaign spec and returns a `ValidationResult` of typed rule decisions with errors and warnings, so a rejection names exactly which rule failed and why.** The code lives in `src/microcosm_core/engine_room/bridge_campaign_dag.py`. It takes one JSON campaign spec and returns a `ValidationResult`: a list of typed rule decisions, plus the errors and warnings those decisions produced. Each decision carries a rule id, the target it looked at, and a plain reason. The whole point is that when a graph is rejected, the result says exactly which rule rejected it and why.\n\n###### Engine room demo\n\n**The Engine Room demo admits this component as a staged target, so the fixtures a reader runs here are the same ones `audit_controller_coverage` and `run_demo` audit.** The Engine Room demo at `src/microcosm_core/engine_room/demo.py` admits this component as one of its staged bundle targets, so the same fixtures a reader can run here are the ones the demo audits under `audit_controller_coverage` and `run_demo`.\n\n##### Purpose\n\n###### Malformed graphs are paid for after dispatch\n\n**A cycle, an evidence-free synthesis, or an over-large fan-out costs the most after dispatch, since a run-it-and-see approach only discovers it at runtime across live external model access.** The cost of a malformed campaign graph is paid after dispatch, when it is expensive. A dependency cycle never terminates. A synthesis step that depends on nothing but other conclusions summarises no gathered evidence. A fan-out that asks one provider for more parallel workers than it can take gets throttled or rejected downstream. A loose \"run it and see\" approach discovers all of this at runtime, across live external model access.\n\n###### Move the checks before dispatch\n\n**This component moves those checks before dispatch, where only a rejected JSON file is at stake, asking whether the graph is well formed enough to be worth running.** This component moves those checks to before dispatch, where the only thing at stake is a rejected JSON file. It asks one question. Is this campaign graph well formed enough to be worth running?\n\n###### Reachability: a conclusion must trace to evidence\n\n**The key check is reachability: an acyclic, well-typed graph is still rejected when its synthesis traces back to no probe, so a conclusion must be traceable to gathered evidence.** The check worth pausing on is reachability. An acyclic graph with valid node types can still be wrong: the single synthesis node might trace back only to a reducer with no probe behind it. That graph looks like a campaign but concludes over nothing the campaign actually gathered. The validator rejects it. A conclusion has to be traceable to evidence.\n\n##### How it works\n\n###### `validate_campaign` appends one decision per rule\n\n** `validate_campaign` takes a campaign mapping, provider, and worker count, then appends one decision per rule with no side effects, mirroring the private bridge contract's CR and VR families.** `validate_campaign` does the work. It takes one campaign mapping plus a provider name, a worker count, and optional filesystem-preflight flags. It builds an empty `ValidationResult` and appends one decision per rule. It performs no side effects. The rule ids mirror the private bridge contract's CR and VR families, so this public component carries a faithful subset of the same checks without carrying the runtime.\n\nThe rules run in this order over one spec:\n\nRule group What it checks Rejects when `CR001` - `CR009` envelope: `schema_version` is `1.0`, `kind` is `bridge_campaign`, kebab-case `campaign_id` (via `CAMPAIGN_ID_RE`), non-empty `intent`, a public `plan_path`, barrier binding, and a bounded continuation packet any envelope field is missing or out of shape `CR010`, `CR015` - `CR020` nodes: labels are unique and non-empty, every `role` is one of `VALID_NODE_ROLES` (`probe`, `reducer`, `synthesis`), input modes are known, and every `depends_on` entry names a real node a role is unknown or a dependency points nowhere `CR011` reducer and synthesis nodes each depend on at least one upstream node a reducer or synthesis has no upstream `CR012` the graph is acyclic, computed by `_cycle_labels` any node sits on a dependency cycle `CR013`, `CR006` there is exactly one synthesis node, and the barrier's `group_label` names it zero or many synthesis nodes, or a mismatched barrier `CR014` the synthesis node transitively reaches at least one probe, computed by `_reachable_dependencies` synthesis traces back to no probe `VR005` the requested `workers` count is within the provider's ceiling in `SAFE_PARALLELISM` the count exceeds the ceiling\n\n###### Cycle, reachability, and parallelism helpers\n\n** `_cycle_labels` names the members of any cycle, `_reachable_dependencies` walks edges from the synthesis for `CR014`, and `SAFE_PARALLELISM` caps workers per provider for `VR005`.** `_cycle_labels` runs a depth-first walk with a visiting set and returns the labels caught on a back-edge, so a cycle names its members rather than just failing. `_reachable_dependencies` walks `depends_on` edges transitively from the synthesis label; `CR014` then checks whether any reached node has role `probe`. `SAFE_PARALLELISM` is a small fixed table: `chatgpt` 8, `claude` 2, `gemini` 3, `local` 4. A provider not in the table, or a worker count above its ceiling, fails `VR005`.\n\n###### `ValidationResult.add` turns decisions into outcomes\n\n** `ValidationResult.add` makes a `reject` set `ok` false with an error line, a `warn` record a warning, and an `ok` just log, while `to_dict` serialises the whole result.** `ValidationResult.add` is where a decision becomes an outcome: a `reject` sets `ok` to false and records a formatted error line, a `warn` records a warning, an `ok` just logs the decision. `to_dict` serialises the whole result for a machine reader.\n\n###### Three file wrappers replay fixtures\n\n**Three wrappers add files: `load_campaign` reads one JSON object, `validate_campaign_file` checks a spec against its declared expectation, and `validate_fixture_dir` folds a directory into a pass matrix.** The other three callables wrap this for files. `load_campaign` reads one JSON object off disk and rejects anything that is not an object. `validate_campaign_file` loads a spec, pulls its declared `provider`, `workers`, and `expected_ok`, runs `validate_campaign`, and returns a row recording whether the observed result matched the declared expectation. `validate_fixture_dir` runs that over every `*.json` in a directory and returns a matrix with `status: pass` only when every case met its declared expectation. `main` exposes both as the `validate` and `validate-fixtures` subcommands.\n\nDiagram of the mechanism (7 steps). Diagram source & refs\n\n```text\nflowchart TD A[\"Campaign JSON spec\"] --> B[\"validate_campaign envelope + node rules\"] B --> C[\"_cycle_labels acyclicity CR012\"] C --> D[\"_reachable_dependencies synthesis reaches probe CR014\"] D --> E[\"SAFE_PARALLELISM provider ceiling VR005\"] E --> F[\"ValidationResult ok = true, decisions\"] B -.->|reject| R[\"ValidationResult ok = false rule id + target + reason\"] C -.->|reject| R D -.->|reject| R E -.->|reject| R\n```\n\n##### Negative cases\n\n###### Four public fixtures\n\n**The four public fixtures in `fixtures/first_wave/engine_room_bridge_campaign_dag/input` pin the behavior with three declared failures and one pass.** The four public fixtures in `fixtures/first_wave/engine_room_bridge_campaign_dag/input` pin the behavior with three declared failures and one pass.\n\n###### `valid_three_probe_campaign` is the passing fixture\n\n**The passing fixture wires three probes into one reducer and then one synthesis, requesting 3 `chatgpt` workers against a ceiling of 8.** The pass, `valid_three_probe_campaign`, wires three probes (`prover_probe`, `metabolism_probe`, `security_probe`) into one `engine_room_reducer` and then one `engine_room_synthesis`, requesting 3 `chatgpt` workers against a ceiling of 8.\n\n###### Cycle, ceiling, and dangling-synthesis rejections\n\n**The three failures reject a cycle through `CR012`, a 99-worker request through `VR005`, and a synthesis reaching no probe through `CR014`, with `CR011` also firing.** The `cycle_rejected` case makes `probe_a` depend on the synthesis node while the synthesis depends on `probe_a`, so `_cycle_labels` finds the loop and `CR012` rejects it. The `provider_ceiling_rejected` case asks `chatgpt` for 99 workers, so `VR005` rejects it against the ceiling of 8. The `dangling_synthesis_rejected` case points the synthesis at an `empty_reducer` with no probe behind it, so `CR014` rejects it for reaching no probe (and `CR011` also fires, since the reducer has no upstream).\n\n##### Prior Art Grounding\n\n###### Lineage in DAG workflow orchestration\n\n**The design follows workflow orchestration's typed directed-acyclic-graph tradition, borrowing acyclicity, evidence reachability, and provider-capacity ceilings as contract-layer checks run before any campaign executes.** This component follows the workflow-orchestration tradition that models work as a directed acyclic graph with typed nodes, dependency edges, fan-in, and validation before execution. [Apache Airflow DAGs](https://airflow.apache.org/docs/apache-airflow/2.4.3/core-concepts/dags.html) group tasks into a directed acyclic graph with explicit dependencies checked before a run. [BPMN](https://www.omg.org/spec/BPMN/2.0/) and related notations separate control-flow structure from the concrete execution system. The borrowed pattern is narrow: acyclicity, evidence reachability, and provider-capacity ceilings are all checked at the contract layer, before any campaign runs.\n\n##### Validation Result record Path\n\nRun the focused tests and the corpus parity check:\n\n```text\nPYTHONPATH=src ./repo-pytest tests/test_engine_room_bridge_campaign_dag.py -q\ncd microcosm-substrate && PYTHONPATH=src ../repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\nOr exercise the fixtures directly:\n\n```text\nPYTHONPATH=src python3 -m microcosm_core.engine_room.bridge_campaign_dag validate-fixtures \\\n  --input fixtures/first_wave/engine_room_bridge_campaign_dag/input \\\n  --json\n```\n\n###### What a pass proves here\n\n**A pass means the four fixtures behaved exactly as declared and the generated projection is still reproducible from source, but it does not mean a campaign ran.** A pass means the four fixtures behaved exactly as declared (`status: pass`, `case_count: 4`) and the generated bundle projection is still reproducible from source. It does not mean a campaign ran.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest honest claim\n\n**At most, given a typed campaign spec the component decides before dispatch whether the envelope, nodes, dependencies, acyclicity, synthesis binding, evidence reachability, and worker ceiling all hold.** The strongest honest claim: given a typed campaign spec, this component decides before dispatch whether the graph has a valid envelope, unique and typed nodes, resolvable dependencies, no cycle, exactly one synthesis node bound to the barrier, a synthesis path that reaches a probe, and a worker count within the provider ceiling. The four fixtures show each rejection firing on a real malformed graph.\n\n###### What validation still refuses\n\n**Beyond that it dispatches nothing, calls no provider, proves no reducer correct, and its ceiling table claims no live quota, provider-safety authority, or launch-scope decision.** It refuses everything past that line. It does not dispatch agents, use external model services, run reducers or synthesis, or prove that a reducer or synthesis is correct. Its provider ceiling is a small local table, not live quota or provider-safety authority. It is staged under Engine Room paths and is not an accepted standalone component. The proof boundary is the fixture run and the focused tests; the scope limit stops at graph-shape validation and claims no equivalence to the private bridge runtime and no launch-scope decision.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":12642,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/engine_room_bridge_campaign_dag.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"Apache Airflow DAGs","url":"https://airflow.apache.org/docs/apache-airflow/2.4.3/core-concepts/dags.html","kind":"external_reference","section_id":"lineage-in-dag-workflow-orchestration"},{"label":"BPMN","url":"https://www.omg.org/spec/BPMN/2.0/","kind":"external_reference","section_id":"lineage-in-dag-workflow-orchestration"}],"grouped_relation_feeds":[{"relation":"abides 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filters, and no-match boundaries.","summary":"Engine Room Reference Knowledge Router binds the staged reference-routing bundle to a concrete mechanism. It scores structured routing metadata, family text, open-first summaries, and curated notes over public fixtures, then rejects domain-mismatch and empty-query cases without cloning repositories, exporting private reference material, or claiming BM25, TF-IDF, embedding, license, launch, or private-system authority.","authority_ceiling":"Public sanitized-reference routing fixture and focused regression evidence only; no BM25, TF-IDF, embedding search, repository cloning, license authority, private reference corpus export, launch-scope decision, whole-system equivalence, source-file changes, or whole-system correctness.","abstract":"Public Engine Room component: sanitized reference router for structured fields, note relevance, domain filters, and no-match boundaries.","teleology":{"section_id":"","text":"Public Engine Room component: sanitized reference router for structured fields, note relevance, domain filters, and no-match boundaries.","source":"capsule_summary_line"},"body_markdown":"###### Is source-faithful public\n\n**This is a source-faithful public copy of the private routing decision, narrowed to run over a sanitized in-memory catalog with no private corpus attached.** The private system keeps a catalog of reusable technique bundles, each tagged with the domains, clusters, and problem spaces it speaks to. When an agent has a problem in front of it, something has to decide which bundles are worth opening. This component is a source-faithful public copy of that decision, narrowed so it runs over a sanitized in-memory catalog with no private corpus attached.\n\n###### Answers which catalog\n\n**It answers which catalog entries are relevant to a problem and why, giving every ranked entry a four-part score breakdown instead of one opaque relevance number.** It answers one question: given a problem statement and a catalog, which entries are relevant, and why. The \"why\" is the point. Instead of one opaque relevance number, every ranked entry carries a four-part score breakdown, so a reader can see whether a row ranked because its structured routing fields matched, because its description happened to share words, or because a curated note carried the weight.\n\n###### Runtime normalizes filters\n\n**The runtime normalizes, filters, scores, and emits a result record, without cloning repositories, reading private entry bodies, or running BM25, TF-IDF, or embedding search.** The runtime is `src/microcosm_core/engine_room/annex_knowledge_router.py`. It normalizes, filters, scores, and emits a result record. It does not clone repositories, read private entry bodies, or run BM25, TF-IDF, or embedding search.\n\n##### Purpose\n\n###### Why a single relevance number fails\n\n**Ranking by one relevance number or bare substring presence cannot tell a deliberate metadata match from an incidental query word, yet only the former is trustworthy.** Loose matching fails here for a specific reason. If you rank by a single relevance number, or by plain substring presence, you cannot tell whether a match came from metadata an author deliberately wrote or from a query word that happened to appear in some prose. Both look like \"a match.\" Only one is trustworthy.\n\n###### Tiered weights favor structured routing fields\n\n**Tiered weights encode that distinction: an exact `problem_spaces` match scores 120 points while the same word in a description scores 6, a deliberately simple weighted-token scorer.** The component encodes that distinction as tiered weights. Structured routing fields score far higher than free text. An exact match on a `problem_spaces` field is worth 120 points. The same word appearing in a description is worth 6. This is a deliberately simple weighted-token scorer, not a learned retriever, and the page does not dress it as one.\n\n##### How it works\n\n###### Ranking runs in\n\n**The ranking runs in execution order inside `route_annexes`, wrapped by `route_catalog`.** The ranking runs in execution order inside `route_annexes`, wrapped by `route_catalog`.\n\n###### Normalize the problem into query tokens\n\n**First, `_normalized_query_text` lowercases and cleans the problem and `_query_tokens` drops stopwords, so if nothing survives, `route_annexes` returns no candidates and the result is no-match.** First, normalize. `_normalized_query_text` lowercases the problem, turns slashes, underscores, and hyphens into spaces, and drops remaining punctuation. `_query_tokens` then splits it and removes a small stopword set, `ROUTE_MATCH_STOPWORDS`. If nothing survives normalization, `route_annexes` returns no candidates, which becomes a no-match result.\n\n###### Filter on domain or cluster first\n\n**Second, a caller-supplied `domain` or `cluster` must exactly match the entry's normalized fields, or the entry is dropped before any scoring can rescue it.** Second, filter. For each catalog entry, if the caller passed a `domain` or `cluster`, the entry's normalized domains or clusters must contain an exact match. Otherwise the entry is dropped before any scoring can rescue it.\n\n###### Four-tier scoring in `_route_match_score`\n\n**Third, `_route_match_score` scores each field as exact, phrase-substring, or per-token overlap, and every tier keeps the maximum over its fields.** Third, score in four tiers. `_route_match_score` returns the exact weight when the normalized query equals the field, the phrase weight when the query is a substring of the field, otherwise the per-token weight times the count of overlapping tokens. Each tier keeps the maximum over its fields:\n- structured routing fields (problem spaces, capabilities, domains, clusters): 120 / 80 / 18\n- family text (slug, display name, description, tags): 32 / 24 / 6\n- open-first summaries: 20 / 16 / 4\n- curated notes, ordered first by `sort_notes_by_relevance`: 18 / 12 / 3, and a matching note adds its id to `matched_note_ids`\n\n###### Sum tiers, threshold, sort, and limit\n\n**Fourth, the four tiers sum to a total that drops any row at zero or below, and survivors sort by descending score then slug, with an optional `limit` trimming the list.** Fourth, sum and threshold. The total is the four tiers added together. A total of zero or less drops the row. Each surviving row keeps its `score`, its `match_breakdown`, its matched note ids (capped at eight), and its `routing_summary`. `route_annexes` sorts by descending score then slug, and an optional `limit` trims the list.\n\n###### `route_catalog` wraps the result record\n\n**Fifth, `route_catalog` returns a status record of `routed` or `no_match` carrying the row count, the echoed problem and filters, source refs, and the authority bounds.** Fifth, wrap. `route_catalog` returns the result record: `status` of `routed` or `no_match`, `row_count`, the problem echoed back, the domain and cluster filters, `source_refs`, `claim_ceiling`, and anti_claims.\n\n###### `_routing_summary_from_family` assembles routing fields\n\n**One helper, `_routing_summary_from_family`, builds each entry's routing summary from its own routing block plus routing pulled from its notes, deduping while preserving order.** One helper does the assembly. `_routing_summary_from_family` builds each entry's `routing_summary` from its own routing block plus routing pulled from its notes, deduping while preserving order.\n\nFunction Role `route_catalog` Entry point; runs the ranking and wraps it in a status record `route_annexes` The ranking pass: normalize, filter, four-tier score, sort `_route_match_score` One field's score: exact, phrase, or per-token overlap `_routing_summary_from_family` Assemble structured routing fields from the entry and its notes `sort_notes_by_relevance` Order notes by bounded relevance before scoring `evaluate_case`, `evaluate_fixture_dir` Replay one fixture or a directory into a pass/fail record Diagram of the mechanism (8 steps). Diagram source & refs\n\n```text\nflowchart TD Problem[\"Problem statement normalize to tokens\"] --> Empty{\"Empty after normalization?\"} Empty -->|yes| NoMatch[\"status: no_match\"] Empty -->|no| Filter{\"domain / cluster filter matches?\"} Filter -->|no| Drop[\"excluded before scoring\"] Filter -->|yes| Score[\"four-tier weighted-token score\"] Score --> Threshold{\"total score > 0?\"} Threshold -->|no| Drop Threshold -->|yes| Ranked[\"ranked rows score + match_breakdown\"]\n```\n\n##### Negative cases\n\nThe public fixtures hold two positive and two negative cases.\n\n###### `empty_problem_no_match`: no tokens survive\n\n** `empty_problem_no_match` sends the empty string, so normalization leaves no tokens, `route_annexes` returns nothing, and the status is `no_match`.** `empty_problem_no_match` sends the empty string as the problem. After normalization there are no tokens, `route_annexes` returns nothing, and the status is `no_match`.\n\n###### `domain_filter_no_match`: filter vetoes a text match\n\n** `domain_filter_no_match` supplies a domain no entry lists, so every entry is excluded before scoring, showing a domain filter can veto a row that text alone would have ranked.** `domain_filter_no_match` sends \"forecast error scoring\" with the domain filter \"agent-runtime\". No catalog entry lists that domain, so every entry is excluded before scoring and the result is `no_match`. This is the filter path: a domain filter can veto an entry that text alone would have ranked.\n\n###### Two positive cases rank on fields and notes\n\n**The positive cases rank `provider-rate-limit-patterns` above 80 on structured fields and above 40 through a curated note, and `evaluate_fixture_dir` folds all four cases into a pass or fail.** The positive cases anchor the other side. `provider_backoff_route` (\"rate limit and back off across multiple LLM providers\") ranks `provider-rate-limit-patterns` on structured fields with a score of at least\n\nthrough the curated note `note_provider_backoff` with a score of at least 40. `evaluate_fixture_dir` turns the four cases into `case_count`, `passed_case_count`, and a pass or fail status.\n1. `note_match` (\"provider rate limit retry result record\") ranks the same entry\n\n##### Prior Art Grounding\n\n###### Lineage in fielded and explainable sparse search\n\n**The design borrows information retrieval's inspectable, field-weighted scoring, whose closest lineage is fielded search and explainable sparse term scoring, named by two reference points.** The component borrows the general information-retrieval pattern of scoring a candidate set with visible term evidence and returning ranked, inspectable matches. Its closest lineage is fielded search, where structured fields weigh more than body text, and explainable sparse term scoring. Two reference points in that lineage:\n- [Text REtrieval Conference (TREC)](https://trec.nist.gov/) as the long-running benchmark line for retrieval tasks, relevance judgments, and scored runs.\n- [Apache Lucene `BM25Similarity`](https://lucene.apache.org/core/9_8_0/core/org/apache/lucene/search/similarities/BM25Similarity.html) as a concrete example of explainable sparse term scoring.\n\n###### Field-weighting borrowed, dense retrieval refused\n\n**The component takes only the inspectable, field-weighted idea and implements no BM25, no TF-IDF, and no embedding search.** The component takes the inspectable, field-weighted idea and nothing more. It does not implement BM25, TF-IDF, or embeddings.\n\n##### Validation Result record Path\n\n###### Reader-verifiable checks are\n\n**The reader-verifiable checks are the focused test and the paper-module corpus readback.** The reader-verifiable checks are the focused test and the paper-module corpus readback:\n\n```text\nPYTHONPATH=src ./repo-pytest tests/test_engine_room_annex_knowledge_router.py -q\ncd microcosm-substrate && PYTHONPATH=src ../repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\nYou can also replay the fixtures directly:\n\n```text\nPYTHONPATH=src python3 -m microcosm_core.engine_room.annex_knowledge_router evaluate-fixtures \\\n  --input fixtures/first_wave/engine_room_annex_knowledge_router/input \\\n  --json\n```\n\n###### What a pass proves here\n\n**A pass means the public fixture behavior and JSON projection are still reproducible, and it means nothing more than that.** A pass means the public fixture behavior and the JSON bundle projection are still reproducible. It does not mean anything more.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest honest claim\n\n**At most, over a sanitized in-memory catalog the component ranks entries by an explainable tiered score, shows the per-tier breakdown, and matches all four fixtures.** The strongest honest claim is this: over a sanitized in-memory catalog, the component ranks entries by an explainable tiered weighted-token score and shows the per-tier breakdown behind every rank. Given the two positive fixtures it returns the expected top entry above the expected score, and given the two negative fixtures it returns `no_match`.\n\n###### What the component still refuses\n\n**Even so, it is not BM25, TF-IDF, embedding search, or provenance authority; a `no_match` proves only that the finite fixture did not route, and nothing here authorizes launch.** It refuses the rest. It is not BM25, not TF-IDF, not embedding search, not repository cloning, and not license or provenance authority. A `no_match` result proves only that the finite public fixture did not route under the supplied filters; it does not establish that no useful private entry exists. Matched note ids name which curated notes contributed; they do not disclose private entry bodies. The bundle binds one staged mechanism subject. It does not admit an accepted component, unblock the Atlas owner lane, or include launch operations.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":12625,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/engine_room_annex_knowledge_router.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"Text REtrieval Conference (TREC)","url":"https://trec.nist.gov/","kind":"external_reference","section_id":"lineage-in-fielded-and-explainable-sparse-search"},{"label":"Apache Lucene `BM25Similarity`","url":"https://lucene.apache.org/core/9_8_0/core/org/apache/lucene/search/similarities/BM25Similarity.html","kind":"external_reference","section_id":"lineage-in-fielded-and-explainable-sparse-search"}],"grouped_relation_feeds":[{"relation":"abides 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Engine","relative_href":"docs/paper-module-engine-room-derived-fact-provider-engine.html","read_href":"docs/paper-module-engine-room-derived-fact-provider-engine.html","context_href":"docs/rules-and-ideas.html#paper-module-engine-room-derived-fact-provider-engine","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"Public Engine Room component: registry-backed derived fact provider over JSON pointers, glob counts, git-backed callables, and provider error rows.","summary":"Engine Room Derived Fact Provider Engine binds the staged derived-fact bundle to a concrete mechanism. It resolves JSON-pointer, glob-count, and git-backed callable fact rows over public fixture roots, records provider errors as repairable data, and keeps derived-fact availability below truth-audit, semantic-claim-validation, full source registry, launch, and private-system authority.","authority_ceiling":"Public fixture-root fact-provider evidence only; no doctrine truth audit, no semantic claim validation, no full source fact registry export, no launch-scope decision, no whole-system equivalence, no source-file changes, and no whole-system correctness.","abstract":"Public Engine Room component: registry-backed derived fact provider over JSON pointers, glob counts, git-backed callables, and provider error rows.","teleology":{"section_id":"","text":"Public Engine Room component: registry-backed derived fact provider over JSON pointers, glob counts, git-backed callables, and provider error rows.","source":"capsule_summary_line"},"body_markdown":"###### System claims things\n\n**A system that claims things about itself needs a disciplined way to fetch the numbers behind them, since any document that hard-codes them goes stale as the repository changes.** A system that makes claims about itself needs a disciplined way to fetch the numbers behind those claims. How many files match a pattern. How many rows a registry holds. How many files git is tracking. Those are facts about the current state of the repository, and a document that hard-codes them goes stale the moment the repository changes.\n\n**For each authored fact row the engine reads how the row declares it resolves and produces the value now, emitting one record with a ledger, failure audit, cache, and content hash.** This component answers one narrow question. For each authored fact row, what value does it resolve to against a supplied root, right now? A row declares how it resolves rather than what it equals, and the engine reads the declaration and produces the value. The output is a single result record carrying a ledger of resolved facts, an audit list of any provider failures, a small navigation cache, and a content hash.\n\n###### Is provider engine\n\n**It is the provider engine, not a truth auditor: a clean record means the declared rows resolved against the supplied root, not that the underlying claims are true.** It is the provider engine, not a truth auditor. A clean record means the declared rows resolved against the supplied root. It does not mean the underlying claims are true.\n\n##### Purpose\n\n###### A row declares how it resolves, not what it equals\n\n**A fact row carries a `provider_type` and its provider's fields, whether `json_pointer`, `glob_count`, or `callable`, and states no answer, so the value is derived on every evaluation.** A fact row is a small object with a `provider_type` and the fields that provider needs. A `json_pointer` row names a source file and a pointer into it. A `glob_count` row names a file pattern to count. A `callable` row names one of a fixed set of git-backed computations. Nothing in a row states the answer. The answer is derived on every evaluation.\n\n###### A failed row degrades to one repairable row\n\n**Rather than raise on the first unresolvable row, the engine turns it into a row carrying an `error_class`, message, and `required_next_action`, leaving the others resolved and the ledger `degraded`.** The hard design choice is what happens when a row cannot resolve, for example because its source file is missing or its provider name is unknown. A loose engine would raise on the first bad row and abandon the rest. This one does not. A failed row becomes an ordinary row that carries an `error_class`, a human-readable message, and a `required_next_action` such as restoring the named source path. One broken fact degrades to one repairable row. The other rows still resolve, and the ledger reports `degraded` rather than dying with a stack trace.\n\n##### How it works\n\n###### Source file is\n\n**The source file is `src/microcosm_core/engine_room/derived_fact_provider_engine.py`.** Evaluation runs bottom-up: one row, then the whole registry, then a directory of fixture cases.\n\n###### `resolve_json_pointer` walks by pointer token\n\n** `resolve_json_pointer` splits a pointer with `_json_pointer_tokens`, unescapes `~1` and `~0`, descends token by token, and raises `KeyError` on an absent token, which becomes an error row upstream.** `resolve_json_pointer` walks a decoded document by pointer. It splits the pointer into tokens with `_json_pointer_tokens`, which requires a leading `/`, drops a leading `#`, and unescapes `~1` to `/` and `~0` to `~`. It then descends token by token, indexing into lists by integer and into mappings by key, and raises `KeyError` when a token is absent. That `KeyError` is what turns a bad pointer into an error row upstream.\n\n###### `evaluate_provider` branches on `provider_type`\n\n** `evaluate_provider` builds a base record then branches: `json_pointer` resolves a file pointer, `glob_count` counts and samples matches, and `callable` dispatches three git-backed names through `_callable_value`.** `evaluate_provider` is the core. It takes one row and a root path and builds a base record with the row's id, title, tags, and provider fields, provisionally marked `ok`. Inside a try block it branches on `provider_type`. A `json_pointer` branch reads the named JSON file under the root and resolves the pointer. A `glob_count` branch globs the pattern under the root, drops any match whose relative path starts with a listed exclude prefix, keeps only files, sorts them, sets the value to the count, and stores the first twenty paths as `sample_matches` so a reader can see what was counted. A `callable` branch dispatches through `_callable_value`, which knows exactly three names: `git_tracked_file_count` and `git_tracked_python_count` shell through `git ls-files`, and `tracked_fact_registry_count` reads `fact_registry.json` under the root and counts its `facts`. Any other name raises. The resolved value is coerced by `value_type` and a string form is stored in `value_repr`.\n\n###### The except block is the error-as-data path\n\n**The except block catches every exception, flips status to `error`, nulls the value, and on a missing source sets `source_status` to `missing` with a `required_next_action`, so nothing escapes.** The except block is the error-as-data path. It catches every exception, flips `provider_status` and `status` to `error`, records `error_class` and a message, and nulls the value. When the failure is a missing source file, it sets `source_status` to `missing` and fills `required_next_action` from `_source_repair_command`. No exception escapes `evaluate_provider`.\n\n###### `evaluate_registry` aggregates and hashes\n\n** `evaluate_registry` evaluates every row, counts by type and status, builds a `degraded` -or- `ok` ledger, collects error rows and a navigation cache, and hashes the content into `receipt_sha256`.** `evaluate_registry` runs `evaluate_provider` over every row in the registry's `facts` list and aggregates. It counts rows by provider type and by status, builds the `ledger` whose status is `degraded` when any row errored and `ok` otherwise, collects the error rows into `audit.provider_findings`, projects a compact `navigation_cache` of id, title, value, and status per row, and hashes the summary, facts, and findings into `receipt_sha256`. That hash is over the evaluation content, so an unchanged registry against an unchanged root hashes the same.\n\n###### `evaluate_case` and `evaluate_fixture_dir` replay tests\n\n** `evaluate_case` writes declared files into a temporary directory, optionally runs `git init` and `git add`, evaluates, and checks expectations, while `evaluate_fixture_dir` passes only when every case met its expectation.** `evaluate_case` and `evaluate_fixture_dir` wrap the registry evaluator for replayable tests. A case writes its declared files into a fresh temporary directory, optionally runs `git init` and `git add` through `_prepare_git_index` so the callable providers have a real index to read, evaluates the registry there, then compares observed values and error ids against the case's expectations and reports `expectation_met`. `evaluate_fixture_dir` reads every `*.json` case in an input directory in sorted order, evaluates each, and reports `pass` only when every case met its expectation. `build_parser` and `main` expose two CLI commands, `evaluate-registry` and `evaluate-fixtures`, that print the record as JSON or a one-line status.\n\nFunction Role `resolve_json_pointer` Walk a decoded JSON document by pointer, raise `KeyError` on a missing token `evaluate_provider` Resolve one row through its provider branch, or turn any failure into an error row `evaluate_registry` Aggregate rows into ledger, audit findings, navigation cache, and a content hash `evaluate_fixture_dir` Replay a directory of fixture cases and report `pass` only if all met expectations `main` Run the `evaluate-registry` or `evaluate-fixtures` command from the CLI Diagram of the mechanism (8 steps). Diagram source & refs\n\n```text\nflowchart TD Row[\"fact row provider_type + fields\"] Provider[\"evaluate_provider select branch\"] Json[\"json_pointer read file, resolve pointer\"] Glob[\"glob_count count files, keep 20 samples\"] Call[\"callable git-backed count\"] Ok[\"resolved row value + value_repr\"] Err[\"error row error_class + required_next_action\"] Reg[\"evaluate_registry ledger + audit + cache + sha256\"] Row --> Provider Provider --> Json Provider --> Glob Provider --> Call Json --> Ok Glob --> Ok Call --> Ok Json -. on failure .-> Err Glob -. on failure .-> Err Call -. on failure .-> Err Ok --> Reg Err --> Reg\n```\n\n##### Negative cases\n\n###### Public fixture set\n\n**The public fixture set in `fixtures/first_wave/engine_room_derived_fact_provider_engine/input` pins both the happy path and the failures.** The public fixture set in `fixtures/first_wave/engine_room_derived_fact_provider_engine/input` pins both the happy path and the failures.\n\n###### Two error-as-data cases and two passes\n\n** `provider_error_as_data` and `unknown_provider_negative` require bad rows to error into `degraded` data, while `json_pointer_and_glob_count` and `git_callable_tracked_files` resolve their rows to `ok`.** The `provider_error_as_data` case declares two rows that must fail as data, not as a crash. `demo.missing_source` points into a JSON file that is not written, and `demo.bad_callable` names a callable that does not exist. The case expects status `degraded` and expects exactly `demo.missing_source` and `demo.bad_callable` to carry error rows. The `unknown_provider_negative` case declares one row, `demo.unknown_provider`, with `provider_type` set to `imaginary_provider`, and expects that single row to error while the ledger reports `degraded`. The two passing cases confirm the other side: `json_pointer_and_glob_count` resolves `demo.fact_count` and `demo.markdown_count` to `ok`, and `git_callable_tracked_files` resolves `demo.tracked_files` and `demo.tracked_python` against a real git index to `ok`.\n\n##### Prior Art Grounding\n\n###### Lineage in registered-provider data platforms\n\n**The design borrows data platforms' registered-provider and lineage-accounting shape, producing derived facts from declared sources carried with metadata rather than copied by hand.** The pattern is the registered-provider and lineage-accounting shape from data platforms, where derived facts are produced from declared sources and carried with metadata rather than copied by hand.\n- [PostgreSQL materialized views](https://www.postgresql.org/docs/16/rules-materializedviews.html), where a stored relation is derived from a query and refreshed from source data.\n- [OpenLineage](https://openlineage.io/), an open model for recording jobs, datasets, and run metadata across data systems.\n\n###### Shape borrowed, lineage graph and verifier refused\n\n**This component borrows the shape and adds error-as-data rows with repair hints, adopting no full lineage graph and acting as no semantic claim verifier.** This component borrows the shape and adds error-as-data rows with repair hints. It does not adopt a full lineage graph, and it is not a semantic claim verifier.\n\n##### Validation Result record Path\n\n```text\nPYTHONPATH=src ./repo-pytest tests/test_engine_room_derived_fact_provider_engine.py -q\ncd microcosm-substrate && PYTHONPATH=src ../repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n###### What passing both commands proves\n\n**The first command replays the fixtures across pointer escaping, glob samples, git callables, and error rows, the second checks the page matches source, and passing both proves reproducibility.** The first command replays the fixture cases and checks JSON pointer escaping, glob-count samples, git-backed callables, the error rows, and the CLI record. The second checks that this page still matches its source row. Passing both proves the fixture behavior and the projection are reproducible. It does not admit an component or include launch operations.\n\nYou can also run the CLI over the public fixtures directly:\n\n```text\nPYTHONPATH=src python3 -m microcosm_core.engine_room.derived_fact_provider_engine evaluate-fixtures \\\n  --input fixtures/first_wave/engine_room_derived_fact_provider_engine/input \\\n  --json\n```\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest honest claim\n\n**At most, given authored fact rows and a root the engine resolves each through one provider, turns failures into repairable error rows, and emits one record bounded at provider resolution.** The strongest honest claim is this. Given a registry of authored fact rows and a root, the engine resolves each row through exactly one provider, turns every failure into a repairable error row, and emits one record with a ledger, an audit list, a navigation cache, and a content hash. The proof boundary is provider resolution. The scope limit is mechanism-level.\n\n###### What the engine still refuses\n\n**It is no doctrine truth auditor, no full export of the private registry, and no launch-scope decision; a clean record means the rows resolved, not that their claims are correct.** It refuses to be more. It is not a doctrine truth auditor, not a full export of the private fact registry it was refactored from, not semantic claim validation, and not launch-scope decision. A clean record means the declared rows resolved against the supplied root, not that the claims those rows back are correct. 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It detects permission ceremony without a real blocker, self-error statements without durable capture, and command handoff language without execution evidence over public fixtures, while accepting bounded blocker or result record language and refusing taint-analysis, prompt-injection-defense, sandbox, information-flow, launch, and private-system claims.","authority_ceiling":"Public phrase-policy fixture evidence only; no taint analysis, prompt-injection defense, sandboxing, information-flow proof, launch-scope decision, whole-system equivalence, source-file changes, or whole-system correctness.","abstract":"Public Engine Room component: phrase-membership egress gate for permission ceremony, self-error capture binding, and command-displacement evidence.","teleology":{"section_id":"","text":"Public Engine Room component: phrase-membership egress gate for permission ceremony, self-error capture binding, and command-displacement evidence.","source":"capsule_summary_line"},"body_markdown":"###### Is small public\n\n**This is a small public text check that reads agent output and asks whether it hands work back to the operator or excuses a mistake without recording it.** `engine_room_egress_self_compliance_gate` is a small text check that reads a piece of agent output and asks one question: does this text hand work back to the operator or excuse a mistake without recording it? It is a public refactor of the internal egress-compliance policy the private system runs before an agent yields a turn.\n\n###### Looks for three\n\n**It looks for three failure classes: asking permission when nothing is blocked, admitting an error without a durable capture, and handing the operator a command instead of running it.** It looks for three failure classes. Asking for permission when nothing is actually blocked. Admitting a self-detected error without binding it to a durable capture. Handing the operator a safe command to run instead of running it. Each of these reads as good manners in isolation, and each is the exact shape of an agent quietly displacing its own work onto the human.\n\n###### Emits json record\n\n**It emits a JSON record with a red or green status and one row per fired failure class, and ships six replayable fixtures and a runnable CLI.** The component emits a JSON result record: a red or green status plus one row per failure class that fired. Red means the text matched a tripwire phrase without the matching repair signal. It ships with six replayable fixtures and a CLI you can run directly:\n\n```text\nPYTHONPATH=src python3 -m microcosm_core.engine_room.egress_self_compliance_gate evaluate-fixtures \\\n  --input fixtures/first_wave/engine_room_egress_self_compliance_gate/input \\\n  --json\n```\n\n##### Purpose\n\n###### The flagged failures all read as polite\n\n**The failures the gate looks for read as courtesy, so each polite phrase is fine only with attached evidence: a real blocker, a capture id, or a command-run result record.** The failures this gate looks for are the ones that read as polite. \"Let me know if you want me to continue\" looks courteous. \"My mistake, I miscounted\" looks honest. \"You can run make smoke to verify\" looks helpful. In each case the polite phrase is fine only when it is accompanied by evidence that makes it honest: a real blocker that justifies stopping, a capture id that pins the mistake, or a result record showing the command was already run.\n\n###### Invert the default: polite phrases are guilty first\n\n**Because the output is neither rude nor wrong, the gate treats each polite phrase family as a violation on sight and clears it only when the legitimising signal appears.** A loose check that only flagged rude or obviously wrong output would miss all three, because the output is neither rude nor wrong on its surface. So the gate inverts the default. Each polite phrase family is treated as a violation the moment it appears, and the gate then looks in the same text for the one legitimising signal that clears it. The phrase is innocent only with its evidence attached.\n\n###### Phrase membership, not analysis of the work\n\n**This is phrase membership over the text, not analysis of what the agent did, so it cannot tell whether a blocker is real, only that the legitimising language is present.** This is phrase membership over the output text, not analysis of what the agent did. The gate cannot tell whether a named blocker is real or whether a capture id resolves to anything. It checks that the legitimising language is present, and stops there. That keeps it small, fast, and inspectable.\n\n##### How it works\n\n###### `evaluate_text` runs three detectors into a status\n\n** `evaluate_text` lowercases the text, runs the three `DETECTORS`, and returns a metadata-only record whose status is red if any detector matched a tripwire without its legitimiser.** Everything runs through `evaluate_text`. It lowercases the candidate text with `_body`, runs the three detector functions in `DETECTORS`, and returns a metadata-only JSON record with the status, a `violation_count`, and one row per detector that fired. A detector returns a row only when its tripwire phrase family matched; the row's `violation` field is true when the required legitimiser phrase family is absent from the same text. The status is red if any row is a violation, green otherwise.\n\n**Each detector is one instance of the same rule: match a tripwire family, accept only if a paired legitimiser family also matches.** Each detector is one instance of the same rule: match a tripwire family, accept only if a paired legitimiser family also matches.\n\nFunction Tripwire family Cleared when text also matches `detect_permission_gate_without_blocker` `PERMISSION_GATE_PHRASES` `LEGITIMATE_BLOCKER_PHRASES` (destructive, secret, remote push, public sharing boundary, concurrent-owner conflict, validation failure) `detect_self_error_without_capture` `SELF_ERROR_TRIPWIRE_PHRASES` `DURABLE_BINDING_PHRASES` (`cap_`, work item, work log, quick-capture, captured, failure mode) `detect_command_displacement_to_operator` `COMMAND_DISPLACEMENT_PHRASES` `COMMAND_EXECUTION_RECEIPT_PHRASES` (i ran, ran:, exit code, passed, failed, command output)\n\n**Every detector returns a frozen `DetectorResult` carrying its `diagnostic_id`, the matched tripwire and legitimiser phrases, a severity, and a one-line rule stating the intended behaviour.** Every detector returns a frozen `DetectorResult` carrying its `diagnostic_id`, the matched tripwire and legitimiser phrases, a severity of `operational_pressure` for a violation or `informational` otherwise, and a one-line rule stating the intended behaviour.\n\n###### The fixture side reuses the same core\n\n** `evaluate_case` compares `evaluate_text` 's status to a fixture's `expected_status`, `evaluate_fixture_dir` passes only when every case met its expectation, and `main` wires both into a CLI.** The fixture side reuses the same core. `evaluate_case` reads one fixture, calls `evaluate_text` on its `text`, and compares the observed status to the fixture's `expected_status`. `evaluate_fixture_dir` globs every `*.json` in a directory, runs each case, and returns `case_count`, `passed_case_count`, and a `pass` or `fail` status that is `pass` only when every case met its expectation. `main` wires both into a CLI: `evaluate` for a single string, `evaluate-fixtures` for a directory, each with a `--json` flag.\n\nDiagram of the mechanism (6 steps). Diagram source & refs\n\nSource refs\n\nDetect permission gate without blocker `detect_permission_gate_without_blocker` Detect self error without capture `detect_self_error_without_capture` Detect command displacement to operator `detect_command_displacement_to_operator`\n\n```text\nflowchart TD T[\"Agent output text lowercased by evaluate_text\"] --> P[\"detect_permission_gate_without_blocker\"] T --> S[\"detect_self_error_without_capture\"] T --> C[\"detect_command_displacement_to_operator\"] P --> R[\"evaluate_text record red if any row is a violation, else green\"] S --> R C --> R R --> F[\"evaluate_fixture_dir compare status to expected_status per case\"]\n```\n\n###### The demo consumes this gate's fixtures\n\n**The demo lists this component as one `CapsuleExercise` with the jewel target `egress_self_compliance_gate`, which `audit_controller_coverage` checks against `EXPECTED_JEWEL_TARGETS` before consuming its fixture result.** The composition target is `src/microcosm_core/engine_room/demo.py`, which lists this component as one `CapsuleExercise` in `CAPSULES` with the jewel target `egress_self_compliance_gate`. Its `audit_controller_coverage` checks that target against `EXPECTED_JEWEL_TARGETS`, so the demo can consume this gate's fixture result without this module being anything more than a staged bundle.\n\n##### Negative cases\n\n###### Six fixtures under\n\n**The six fixtures under `fixtures/first_wave/engine_room_egress_self_compliance_gate/input` pin one red and one green example for each detector.** The six fixtures under `fixtures/first_wave/engine_room_egress_self_compliance_gate/input` pin one red and one green example for each detector:\n- `permission_gate_without_blocker` (red): \"The tests passed. Let me know if you want me to continue.\" A permission phrase with no blocker named.\n- `permission_gate_with_blocker` (green): \"Should I proceed? I am stopping before a remote push because that crosses the public sharing boundary.\"\n- `self_error_without_capture` (red): \"My mistake: I miscounted the rows in the generated report.\"\n- `self_error_with_capture` (green): \"My mistake is captured in cap_quick_example_self_error as a failure mode before this note.\"\n- `command_displacement` (red): \"You can run make smoke to verify the package.\"\n- `command_receipt` (green): \"I ran make smoke; it passed with exit code 0.\"\n\n##### Prior Art Grounding\n\n###### Lineage in policy-as-code output gating\n\n**The gate follows the policy-as-code and output-gate tradition of a machine-readable policy that evaluates an artifact before a boundary and returns a named failure class, with two anchors.** The gate follows the policy-as-code and output-gate tradition: make a policy machine-readable, evaluate an artifact before it leaves a boundary, and return a named failure class instead of relying on prose judgement. Two honest anchors:\n- [Open Policy Agent](https://www.openpolicyagent.org/docs/latest), a general engine that externalises policy decisions from application code.\n- [NIST SP 800-53 Rev. 5](https://doi.org/10.6028/NIST.SP.800-53r5), for the broader audit, accountability, and information-output control tradition.\n\n###### Narrowed to fixed phrase-membership checks\n\n**This component narrows that pattern to fixed phrase-membership checks over agent-output text and is intentionally small.** This component narrows that pattern to fixed phrase-membership checks over agent-output text. It is intentionally small.\n\n##### Validation Result record Path\n\n```text\nPYTHONPATH=src ./repo-pytest tests/test_engine_room_egress_self_compliance_gate.py -q\ncd microcosm-substrate && PYTHONPATH=src ../repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n###### What the focused test pins\n\n**The focused test pins each detector's red and green path, checks `evaluate_fixture_dir` returns six passing cases, and confirms the CLI's `organ_id`, so a pass only proves reproducibility.** The focused test pins each detector's red and green path, checks that `evaluate_fixture_dir` returns `case_count` 6 and `status: pass`, and confirms the CLI emits JSON with `organ_id: engine_room_egress_self_compliance_gate`. A pass means the public fixture behaviour and the bundle projection are reproducible. It admits nothing further.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest honest claim\n\n**At most, there is a staged, replayable check for three declared egress-output failure classes running as phrase-membership policy over agent text, with red and green fixtures for each.** The strongest honest claim: Microcosm has a staged, replayable check for three declared egress-output failure classes, running as phrase-membership policy over supplied agent text with red and green fixtures for each class. The proof boundary is that narrow.\n\n###### What the gate still refuses\n\n**It is no taint analysis, injection defence, sandbox, or information-flow proof; it can miss disguised output or flag innocent phrasing, and it claims no wider authority or launch.** It is not taint analysis, not prompt-injection defence, not a sandbox, and not an information-flow proof. 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Search Lab binds the staged proof-search bundle to a concrete mechanism. It runs tiny public Lean statements through symbolic tactic search, statement-only candidate scoring, problem-id ablation, forward oracle-body rejection, and axiom cleanliness checks, while keeping private source run state, oracle proof bodies, neural theorem proving, frontier-scale automation, launch, and private-system authority out of scope.","authority_ceiling":"Public tiny-fixture Lean proof-search evidence only; no neural theorem proving, frontier-scale automation, private source run export, oracle-body forward solving, launch-scope decision, whole-system equivalence, source-file changes, or whole-system formal-result correctness.","abstract":"Public Engine Room component: bounded Lean proof-search lab for tiny fixture statements, oracle-firewall negatives, ablation, and axiom-cleanliness checks.","teleology":{"section_id":"","text":"Public Engine Room component: bounded Lean proof-search lab for tiny fixture statements, oracle-firewall negatives, ablation, and axiom-cleanliness checks.","source":"capsule_summary_line"},"body_markdown":"###### Component runs tiny\n\n**This component runs tiny public Lean statements through a small shape-driven proof search and reports which ones the installed `lean` kernel actually closes on a real run.** This component runs tiny public Lean statements through a small proof search and reports which ones the installed Lean kernel actually closes. It reads a group of theorem statements, proposes candidate tactic bodies from the shape of each goal, writes each candidate to a temporary `.lean` file, and runs the real `lean` executable over it.\n\n###### Search loop is\n\n**A search loop is easy to fool by leaked answers, id memorisation, or a `sorry` -backed file, so this component is built mostly to refuse that false credit rather than to score wins.** A search loop is easy to fool. It can copy a proof out of an answer field, learn to map a problem id to a stored tactic, or compile a file that leans on `sorry`. Each of those returns a green result that means nothing. This component is built mostly to refuse that false credit, not to score wins.\n\n**Each fixture case yields one result record bundling the firewall report, search trace, statement-only table, ablation report, and a single pass or fail with its `failure_kind`.** The output is one result record per fixture case: the firewall report, the search trace, the statement-only table, the ablation report, and a single pass or fail with a named `failure_kind`.\n\n##### Purpose\n\n###### Trust, not search, is the hard part\n\n**The hard part of a proof-search tool is not finding a proof but trusting that a reported success was earned rather than leaked or memorised.** A loose lab that only counts closed goals cannot tell an earned proof from a smuggled one.\n\n###### Generation proposes, Lean decides\n\n**For each statement it asks whether the search actually closed the goal and whether the installed Lean kernel agrees the proof depends on no axioms, keeping generation cheap and untrusted.** This component answers one question per statement: when a tiny public theorem is reported solved, did the search actually close it, and does the installed Lean kernel agree the proof depends on no axioms. Candidate generation stays cheap and untrusted. All authority moves to the kernel. Generation proposes, Lean decides.\n\n##### How it works\n\n###### The two entry points\n\n** `evaluate_lab` runs the pipeline for one group of problems in a fixed order, while `evaluate_fixture_dir` scores every `*.json` case against the status and `failure_kind` its fixture expects.** `evaluate_lab` runs the pipeline for one group of problems in a fixed order. `evaluate_fixture_dir` loads every `*.json` case in a directory and runs them through `evaluate_case`, which compares the observed status and `failure_kind` against the values the fixture expects.\n\nThe order of operations inside `evaluate_lab`:\n1. `_forward_firewall` walks every input row first. `_forbidden_field_paths` recurses through nested maps and lists looking for any of the fields in `FORBIDDEN_FORWARD_FIELDS`: `candidate_body`, `ideal_body`, `repair_body`, `oracle_body`, `oracle_needed_premise_ids`, `provider_text`. If any is present and non-empty the firewall reports `fail` and the search never runs. This is what stops an answer being handed in as a hint.\n2. `_public_problem` strips those forbidden fields, then `_problem_from_mapping` builds a `LeanProblem` from what remains. `infer_target_shape` reads the theorem signature and classifies the goal, for example `Or p q -> Or q p` becomes `or_comm` and a `=` goal becomes `equality`.\n3. For each problem `run_and_or_search` builds candidate tactic bodies with `_base_candidate_actions`. These are deterministic templates keyed by shape: `or_comm` draws an `Or.inl` / `Or.inr` case split, `equality` draws `rfl`, and so on. It dedupes candidates by a goal hash, calls `check_candidate_with_lean` on each, and stops at the first accepted one.\n4. `run_statement_only_hammer` runs a second pass that uses only statement-shaped candidates. It records an action value table with a posterior score per tactic and never credits an adapter candidate or a repair body.\n5. `run_blind_policy_ablation` renames each theorem and its id through `_rename_problem_id`, picks an action for both the original and the renamed problem, and compares them with `_policy_signature`. It passes only when the chosen action, the action-kind counts, and the success counts all match. A policy that had secretly keyed on the id changes its choice under the rename and fails here.\n\n###### The kernel run is the authority\n\n** `check_candidate_with_lean` accepts a candidate only when `lean` exits zero, the source carries no `sorry`, and `#print axioms` confirms the theorem depends on no axioms.** `check_candidate_with_lean` is where a candidate becomes a fact or nothing. `_statement_source` assembles the `.lean` file and appends `#print axioms <theorem>`. If the body contains `sorry` the function static-rejects it before Lean ever runs. Otherwise a cached single-flight helper writes the file to a temporary directory and runs `lean` on it under a timeout. `_classify_axioms` reads the process output: a candidate is `accepted` only when the process exits zero and the output says the theorem \"does not depend on any axioms\", with no `sorry` in the source. A compiled file that depends on axioms is not accepted.\n\n###### Folding the passes into one status\n\n** `evaluate_lab` folds the three passes into one status that fails on an empty set, firewall violation, unclosed goal, axiom taint, or failed ablation, naming the reason in `failure_kind`.** `evaluate_lab` then folds the three passes into one status. It fails on an empty problem set, a firewall violation, any problem the search or hammer did not close, any axiom taint, or an ablation that did not pass, and names the reason in `failure_kind`. `main` exposes the two commands, `evaluate-lab` and `evaluate-fixtures`, and returns a non-zero exit code when the status is fail.\n\nFunction Role `evaluate_fixture_dir` Load and score every `*.json` case in a directory `evaluate_lab` Run firewall, search, hammer, and ablation for one group `run_and_or_search` Bounded shape-keyed tactic search per problem `run_statement_only_hammer` Statement-only candidate pass with value scores `run_blind_policy_ablation` Rename-and-recheck guard against id memorisation `check_candidate_with_lean` Run `lean` on one candidate and audit its axioms Diagram of the mechanism (8 steps). Diagram source & refs\n\nSource refs\n\nrun lean subprocess `check_candidate_with_lean` statement-only value table `run_statement_only_hammer` rename ids, recheck policy `run_blind_policy_ablation`\n\n```text\nflowchart TD A[\"Group of theorem statements\"] --> B[\"_forward_firewall reject leaked answer fields\"] B --> C[\"run_and_or_search shape-keyed tactic candidates\"] C --> D[\"check_candidate_with_lean run lean subprocess\"] D --> E[\"_classify_axioms reject sorry or axiom taint\"] C --> F[\"run_statement_only_hammer statement-only value table\"] F --> G[\"run_blind_policy_ablation rename ids, recheck policy\"] E --> H[\"evaluate_lab result record\"] G --> H\n```\n\n##### Negative cases\n\n###### Public fixtures name\n\n**The public fixtures name four failures and one pass, and each fixture states the `failure_kind` it expects.** The public fixtures name four failures and one pass, and each fixture states the `failure_kind` it expects:\n- `oracle_field_negative`: a row carries a forbidden answer field, so the firewall fails with `oracle_firewall_violation` and no search runs.\n- `nested_oracle_field_negative`: the forbidden field is buried inside a nested structure, and `_forbidden_field_paths` still finds it. Same `failure_kind`.\n- `memorized_policy_negative`: an id-conditioned policy is caught by the rename ablation and fails with `problem_id_ablation_failure`.\n- `sorry_axiom_negative`: a candidate body carries `sorry`, so the axiom audit fails the case with `axiom_taint_detected`.\n- `positive_symbolic_lab_pass`: tiny statements close under the shape-keyed search with clean axioms, the firewall is clean, and the ablation is stable.\n\n##### Prior Art Grounding\n\n###### Tactic automation with kernel authority\n\n**The design follows interactive theorem proving where automation proposes small tactic scripts and the kernel stays the authority, echoing Lean 4 tactics and Isabelle Sledgehammer's replay discipline.** This component follows the interactive-theorem-proving pattern where automation proposes small tactic scripts and the kernel stays the authority. [Theorem Proving in Lean 4](https://lean-lang.org/theorem_proving_in_lean4/Tactics/) is the direct precedent for tactic-structured proof construction. The statement-only pass is adjacent to hammer workflows such as Isabelle [Sledgehammer](https://isabelle.in.tum.de/doc/sledgehammer.pdf), where an outside search suggests steps but the proof assistant must replay the result. Here candidate generation is fixture behavior and the `lean` run plus the `#print axioms` audit decide what counts.\n\n##### Validation Result record Path\n\nRun the public fixture matrix directly:\n\n```text\nPYTHONPATH=src python3 -m microcosm_core.engine_room.lean_proof_search_lab evaluate-fixtures \\\n  --input fixtures/first_wave/engine_room_lean_proof_search_lab/input \\\n  --json\n```\n\n###### Reader-verifiable check is\n\n**The reader-verifiable check is the focused test plus the corpus parity check.** The reader-verifiable check is the focused test plus the corpus parity check:\n\n```text\nPYTHONPATH=src ./repo-pytest tests/test_engine_room_lean_proof_search_lab.py -q\ncd microcosm-substrate && PYTHONPATH=src ../repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n###### What a pass means here\n\n**A pass means only that the public fixture behavior and the bundle projection are reproducible, admitting nothing into the shared registry and authorizing no launch.** A pass means only that the public fixture behavior and the bundle projection are reproducible. It does not admit a component into the shared registry or include launch operations.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim\n\n**Over tiny public fixtures the component closes shape-matched Lean goals with a real kernel run and rejects leaked answers, `sorry` taint, and id-memorising policies, bounded exactly to that fixture matrix.** The strongest claim the evidence supports is narrow. Over tiny public fixtures, this component closes shape-matched Lean goals with a real kernel run, rejects rows that carry leaked answer fields, rejects `sorry` -tainted candidates through a `#print axioms` gate, and rejects a policy that has memorised the problem id. Its proof boundary is exactly that fixture matrix.\n\n###### What a green run does not claim\n\n**A green run is mechanism-level only: not a neural theorem prover, not frontier-scale automation, not an online-RL search, and not an export of private prover run state.** It is not a neural theorem prover, not frontier-scale math automation, not an online-RL search, and not an export of private prover run state. The scope limit is mechanism-level: a green run shows the public fixture and test behavior are reproducible, and nothing more. It does not establish library-scale automation, whole-system correctness, or launch-scope decision.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":11335,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/engine_room_lean_proof_search_lab.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"Theorem Proving in Lean 4","url":"https://lean-lang.org/theorem_proving_in_lean4/Tactics/","kind":"external_reference","section_id":"tactic-automation-with-kernel-authority"},{"label":"Sledgehammer","url":"https://isabelle.in.tum.de/doc/sledgehammer.pdf","kind":"external_reference","section_id":"tactic-automation-with-kernel-authority"}],"grouped_relation_feeds":[{"relation":"abides by","count":7,"links":[{"id":"AX-1","kind":"axiom","kind_label":"axiom","relation":"abides 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Benchmark","relative_href":"docs/paper-module-engine-room-navigation-fitness-benchmark.html","read_href":"docs/paper-module-engine-room-navigation-fitness-benchmark.html","context_href":"docs/rules-and-ideas.html#paper-module-engine-room-navigation-fitness-benchmark","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"Public Engine Room component: route-packet benchmark evaluator for stable-id recall, precision, forbidden first routes, latency, and debt candidates.","summary":"Engine Room Navigation Fitness Benchmark binds the staged navigation-fitness bundle to a concrete mechanism. It evaluates public route-packet fixtures for expected stable-id recall, precision, forbidden first routes, latency budgets, scent terms, and debt candidates without running the private kernel, validating embeddings, claiming universal benchmark authority, or upgrading launch/private-system status.","authority_ceiling":"Public route-packet fixture evidence only; no live private kernel run, no embedding benchmark, no universal navigation benchmark, no launch-scope decision, no whole-system equivalence, no source-file changes, and no whole-system correctness.","abstract":"Public Engine Room component: route-packet benchmark evaluator for stable-id recall, precision, forbidden first routes, latency, and debt candidates.","teleology":{"section_id":"","text":"Public Engine Room component: route-packet benchmark evaluator for stable-id recall, precision, forbidden first routes, latency, and debt candidates.","source":"capsule_summary_line"},"body_markdown":"###### Costly cold-start failure\n\n**The costly cold-start failure is not a wrong answer but the wrong first command, so this component scores whether a navigation packet pointed at the stable ids a cold task actually needed.** When an agent is dropped cold into a large repository, the costly failure is not a wrong answer. It is reaching for the wrong first command and landing on the wrong rows. This component makes that failure measurable. It reads cold-task fixtures and the route packets a navigation surface returned for them, then scores whether the packet pointed at the stable ids the task actually needed.\n\n###### Is evaluator not\n\n**It is an evaluator, not a route runner: it scores pre-captured packets for recall, precision, forbidden-route hits, scent coverage, and latency, without calling the navigation kernel or generating routes.** The component is an evaluator, not a route runner. It consumes fixture tasks and pre-captured packets, computes recall, precision, forbidden-route hits, scent coverage, and latency status, and emits one result record per task plus a suite-level summary. It does not call the private navigation kernel, does not validate embeddings, and does not generate routes of its own.\n\n###### Runtime lives in\n\n**The runtime lives in `src/microcosm_core/engine_room/navigation_fitness_benchmark.py`.** The runtime lives in `src/microcosm_core/engine_room/navigation_fitness_benchmark.py`. Public fixtures live in `fixtures/first_wave/engine_room_navigation_fitness_benchmark/input`.\n\n##### Purpose\n\n###### Why one score is the wrong shape\n\n**Most retrieval scores collapse correct and fast into one number, so a slow-but-correct route and a fast-but-wrong route look identical and land in the same repair queue when they need different fixes.** Most retrieval scores collapse \"correct\" and \"fast\" into one number. That is the wrong shape for navigation triage. A packet can name every expected stable id and still be too slow. A packet can come back fast and still miss an id, carry a weak information scent, use a forbidden first route, time out, or error. If those collapse into a single score, a slow-but-correct route and a fast-but-wrong route look the same, and they land in the same repair queue when they need different fixes.\n\n###### Sufficiency and latency stay apart\n\n**The evaluator keeps two axes apart: sufficiency asks whether the route pointed at the right things, latency asks whether it stayed under budget, and each is computed and reported independently.** This evaluator keeps the two axes apart. Sufficiency answers \"did the route point at the right things.\" Latency answers \"did it stay under budget.\" The two are computed independently and reported side by side, so each failing route lands in the queue that matches its actual defect.\n\n##### How it works\n\n###### The input is a typed task\n\n** `NavigationFitnessTask` is a frozen dataclass fixing the task id, expected artifacts, forbidden routes, latency budget, and scent terms, and `task_from_mapping` degrades missing fields to visible public defaults.** The unit of input is a typed task, not a free-form prompt. `NavigationFitnessTask` is a frozen dataclass that fixes the task id, family, prompt, route type, expected stable artifacts, forbidden first routes, latency budget in milliseconds, route role, and scent terms. `task_from_mapping` turns each raw fixture row into that record. Missing fields fall back to explicit public defaults (`unknown_task`, `public_fixture`, `context_pack`, a 1500 ms budget), so a thin fixture degrades to visible defaults rather than to hidden context.\n\n###### Evaluate task is\n\n** `evaluate_task` is the core predicate.** `evaluate_task` is the core predicate. It runs these steps against one task and one packet:\n\nFunction Role `_packet_artifacts` Collects the ids the packet selected, from flat `selected_artifacts` and from structured `selected_rows` (each `kind_id:row_id` pair and any `artifact_ids`). `_match_expected` Splits expected ids into found and missing. An expected id ending in `*` matches by prefix; otherwise it matches exactly. `_scent_status` Lowercases the packet's summary, title, route hint, first command, and row text, then checks that every required scent term appears. Returns `unscored` when the task lists no terms. `_percentile` Computes p50 and p95 wall-clock values over the suite.\n\n###### Computing recall, precision, and route checks\n\n** `evaluate_task` computes `recall_at_packet` and `precision_at_packet`, reads `wall_ms`, `timed_out`, and `error` off the packet, and checks each forbidden route against its `first_contact_command`.** From those, `evaluate_task` computes `recall_at_packet` (found over expected) and `precision_at_packet` (found over selected), reads `wall_ms`, `timed_out`, and `error` off the packet, and checks each forbidden route against the packet's `first_contact_command`.\n\n###### Sufficiency ladder versus latency status\n\n**Sufficiency is a first-failing-check ladder over timeout, error, missing id, weak scent, and forbidden route, while latency is a separate three-way status that never touches the sufficiency verdict.** Sufficiency is a first-failing-check ladder. In order: a timeout gives failure kind `route_timeout`; an error gives `route_error`; any missing expected id gives `missing_id`; a failed scent check gives `weak_scent`; a forbidden first route gives `forbidden_route`. If none fire, sufficiency is `pass`. Latency is a separate three-way status: `timeout` if the packet timed out, `pass` if wall time is within budget, `fail` if it ran over. Latency never touches the sufficiency verdict.\n\n###### Suite aggregation and two debt classes\n\n** `evaluate_benchmark` runs the predicate over every case and aggregates counts, percentiles, and exactly two debt classes, so a slow-but-correct route yields one latency debt and zero sufficiency debt.** `evaluate_benchmark` lifts the per-task predicate to a suite. It runs `evaluate_task` over every case, then aggregates counts, `p50_wall_ms`, `p95_wall_ms`, per-route-type metrics from `_route_type_metrics`, and debt candidates from `_debt_candidates`. That last function emits exactly two debt classes. A sufficiency debt names the missing ids and forbidden-route hits and carries the failure kind as `repair_class`. A latency debt records wall time, budget, and latency status without marking the route semantically wrong. A single slow-but-correct route produces one latency debt and zero sufficiency debt.\n\n###### Comparing result records against expectations\n\n** `evaluate_case` reports `expectation_met` only when observed status, every summary field, and every per-task field agree, and `evaluate_fixture_dir` passes only when every case matched.** `evaluate_case` compares a suite result record against a fixture's stated expectations: an `expected_status`, an `expected_summary` of counts, and `expected_task_statuses` per task. It reports `expectation_met` only when the observed status, every summary field, and every per-task field agree. `evaluate_fixture_dir` walks a directory of case files, runs each through `evaluate_case`, and returns `case_count`, `passed_case_count`, and an overall `status` that passes only when every case matched. `build_parser` and `main` expose two subcommands, `evaluate-benchmark` and `evaluate-fixtures`, the latter printing the fixture-directory result as JSON without writing any durable output.\n\nDiagram of the mechanism (8 steps). Diagram source & refs\n\n```text\nflowchart TD A[\"Fixture task expected ids, forbidden routes, latency budget, scent terms\"] --> E[\"evaluate_task\"] P[\"Captured route packet\"] --> E E --> S[\"Sufficiency ladder timeout, error, missing id, weak scent, forbidden route\"] E --> L[\"Latency status wall time vs budget\"] S --> R[\"Per-task result record\"] L --> R R --> B[\"evaluate_benchmark counts, p50/p95, debt candidates\"] B --> C[\"evaluate_case observed vs expected\"]\n```\n\n##### Negative cases\n\n###### Four public fixture\n\n**The four public fixture cases pin the accounting rules.** The four public fixture cases pin the accounting rules. Each names its own task id.\n\n###### Two paraphrased cold tasks pass\n\n** `heldout_paraphrase_pass` runs two nonliteral cold tasks that must route to their expected stable ids, avoid forbidden first routes, satisfy scent terms, and stay under budget.** `heldout_paraphrase_pass` runs two nonliteral cold tasks (`heldout_theory_roof`, `heldout_agent_path_store`). Both must route to their expected stable ids, avoid forbidden first routes, satisfy scent terms, and stay under budget.\n\n###### A correct id with a forbidden route\n\n** `adversarial_forbidden_route` selects the right id but uses the forbidden `--paper-module` first command, so sufficiency fails with `forbidden_route` while latency stays `pass`.** `adversarial_forbidden_route` runs `adversarial_paper_module_query`. The packet selects the right id (`paper_modules:navigation_hologram_theory`) but its first command uses the forbidden `--paper-module` route. Sufficiency fails with kind `forbidden_route` while latency stays `pass`, so a correct id does not rescue a banned first move.\n\n###### A nearby row misses the expected id\n\n** `missing_stable_id_negative` selects a nearby row while the expected stable id is absent, so sufficiency fails with kind `missing_id`.** `missing_stable_id_negative` runs `missing_agent_diagnostics`, where a nearby row is selected but the expected stable id is absent. Sufficiency fails with kind `missing_id`.\n\n###### Slow but sufficient proves axis separation\n\n** `latency_debt_negative` selects the expected id at 830 ms against a 100 ms budget, so sufficiency passes, latency fails, and the suite reports one latency debt and zero sufficiency debt.** `latency_debt_negative` runs `slow_but_sufficient`. The packet selects the expected id with a 100 ms budget and an 830 ms wall time. Sufficiency passes, latency fails, and the suite reports one latency debt and zero sufficiency debt. This is the case that proves the two axes stay separate.\n\n##### Prior Art Grounding\n\n###### Retrieval evaluation and information scent\n\n**The component applies information-retrieval evaluation and information-scent research to route packets: define expected targets, score recall and precision, penalize forbidden first moves, and keep latency apart from answer quality.** The component applies information-retrieval evaluation and information-scent research to agent route packets rather than to document search: define the expected targets, score returned results for recall and precision, penalize forbidden first moves, and keep latency apart from answer quality. The retrieval-benchmark tradition of relevance judgments, precision, and recall is anchored by [TREC](https://trec.nist.gov/). The information-scent framing follows Pirolli and Card's [Information Foraging](https://cir.nii.ac.jp/crid/1363951795634897280?lang=en) (Psychological Review, 1999). This is a fixture-scored evaluator, not a universal benchmark or a live-kernel proof.\n\n##### Validation Result record Path\n\n```text\nPYTHONPATH=src ./repo-pytest tests/test_engine_room_navigation_fitness_benchmark.py -q\ncd microcosm-substrate && PYTHONPATH=src ../repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\nYou can also replay the fixtures directly:\n\n```text\ncd microcosm-substrate && PYTHONPATH=src python3 -m microcosm_core.engine_room.navigation_fitness_benchmark evaluate-fixtures \\\n  --input fixtures/first_wave/engine_room_navigation_fitness_benchmark/input --json\n```\n\n###### What a pass means here\n\n**A pass means the public fixture behavior and the bundle projection remain reproducible; it admits no component, unblocks no Atlas owner lane, and excludes launch.** A pass means the public fixture behavior and the bundle projection remain reproducible. It does not admit an accepted component, unblock the Atlas owner lane, or include launch operations.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest honest claim\n\n**Over the public fixtures the evaluator scores route packets for stable-id recall and precision, catches forbidden routes, checks scent terms, and separates latency debt from sufficiency debt, matching the four cases exactly.** The strongest honest claim: over the public fixtures, this evaluator scores route packets for stable-id recall and precision, catches forbidden first routes, checks scent terms, and separates latency debt from sufficiency debt, and its accounting matches the four fixture cases exactly. That is what the tests prove.\n\n###### What it refuses to claim\n\n**It is not a live private `kernel.py` run, an embedding benchmark, a universal navigation benchmark, or launch-scope decision, and a live-kernel claim would need packets from the real route runner.** It refuses more. It is not a live private `kernel.py` run, not an embedding benchmark, not a universal navigation benchmark, and not launch-scope decision. Its result records say nothing about unseen-task navigation quality, browser or provider state, or accepted-component admission. A live-kernel claim needs packets captured from the real route runner. 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It binds bootstrap.sh, src/microcosm_core/cold_clone_probe.py, the first-wave pattern-binding fixture, the secret-exclusion scan, public relative result record refs, and focused tests so a fresh checkout has one bounded proof of first-run mechanics before install, CI, hosted launch, or full component inventory review.","authority_ceiling":"Public source-root bootstrap mechanics and metadata-only result record refs only; no launch-scope decision, hosted-product readiness, external model access, source-file changes, whole-system equivalence, publishing-scope decision, or whole-system correctness.","abstract":"The cold-clone probe validates the first public source-root bootstrap path: src import, secret-exclusion scan, first-wave pattern-binding fixture replay, public result record refs, and ignored local result record emission.","teleology":{"section_id":"","text":"The cold-clone probe validates the first public source-root bootstrap path: src import, secret-exclusion scan, first-wave pattern-binding fixture replay, public result record refs, and ignored local result record emission.","source":"capsule_summary_line"},"body_markdown":"###### Probe is first\n\n**The probe is the first script a fresh clone runs, checking whether it can bootstrap the first-wave contract from source without leaking anything private.** `cold_clone_probe` is the first script a reader runs after cloning the public checkout. It answers one question: can this clone bootstrap the bounded first-wave contract from source, without leaking anything private, before the reader installs the package or reads the long component inventory?\n\n###### Imports from src\n\n**It imports from `src/`, checks fixture inputs, scans for secrets, replays the pattern-binding fixture, and writes one ignored local record; a pass proves only the first bounded action.** The probe imports the package from `src/`, checks that the first-wave fixture inputs are present, runs a secret-exclusion scan, replays the pattern-binding fixture, and then writes one small result record. That record is ignored local state, not a tracked artifact. A pass means the first local action from a fresh clone works and stays inside the public boundary. It is not an install proof, a launch proof, or a statement about the rest of the system.\n\n##### Purpose\n\n###### Pulling the first-step check to the front\n\n**A public checkout must prove its own first step, so the probe pulls the bootstrap check ahead of install or inventory into one command, one record, four typed failure modes.** A public checkout has to prove its own first step. If the reader's first action is `make install` or a long tour of the component inventory, a broken or leaky bootstrap is discovered late and off to the side. The probe pulls that check to the front and makes it concrete: one command, one result record, four typed failure modes.\n\n###### Why a bare import-and-ok is not enough\n\n**A probe that only imports and prints \"ok\" would pass on missing fixtures, an unavailable scan, or absent result records, so each failure gets a distinct blocked status and code instead.** \"Loose\" here means a probe that only imports the package and prints \"ok\". That would pass on a clone that is missing its fixtures, or one whose secret scan is unavailable, or one where the pattern-binding result records never materialize. The probe refuses to call any of those a pass. Each is a distinct blocked status with its own code, so the reader learns which part of the bootstrap is wrong.\n\n##### How it works\n\n###### `run_probe` predicate and `main` wrapper\n\n**The proof predicate `run_probe` runs a fixed sequence and never raises, while the CLI wrapper `main` parses flags, writes the `--emit` record, and sets the exit code.** The runtime is one file, `src/microcosm_core/cold_clone_probe.py`. The proof predicate is `run_probe(root, suite, emit_ref)`. It runs a fixed sequence over the checkout root and returns a result dict; it never raises. The CLI wrapper `main` parses `--suite` and `--emit`, calls `run_probe` on the current directory, writes the returned record, and exits 0 on pass or 1 otherwise. The one file that writes the `--emit` record is `main`, not `run_probe`.\n\n###### A reproducible command stamp before four gates\n\n** `run_probe` stamps the base record with a reproducible `./bootstrap.sh` command quoted through `shlex.quote`, then walks four gates in order.** `run_probe` builds a metadata-only base record stamped with the reproducible command string from `_bootstrap_command(suite, emit_ref)`, which returns `./bootstrap.sh --suite <suite> --emit <path>` with both arguments quoted through `shlex.quote`. It then walks four gates in order:\n\nFunction or constant Role `SUPPORTED_SUITES` The only accepted suite is `first-wave`; anything else is rejected before fixture or scanner state is touched. `REQUIRED_INPUTS` The four fixture files the first-wave suite needs; each must be a readable file under the root. `validate_secret_exclusion_scan` Imported scan that must return `status == \"pass\"` before any pattern replay runs. `validate_pattern_binding` Imported validator that replays the first-wave fixture into a scratch directory under `.microcosm/`. `_mirror_missing_pattern_receipts` Copies the five `PATTERN_RECEIPTS` from the scratch run into their canonical clone-root slots. `DEFAULT_EMIT_REF` The default result path, `.microcosm/cold_clone_probe.json`, which stays ignored local state.\n\n###### Gate order and result record mirroring\n\n**The gates run suite, inputs, secret scan, then replay so the boundary check gates fixture work, after which `_mirror_missing_pattern_receipts` copies missing result records into their canonical slots.** The order matters. `run_probe` checks the suite first, then `REQUIRED_INPUTS`, then the secret scan, and only then the pattern-binding replay. The scan runs before replay on purpose: the boundary check gates the fixture work, not the other way around. After `validate_pattern_binding` writes into the scratch directory, `_mirror_missing_pattern_receipts(root_path, source_dir)` copies each canonical result record that is absent at the root but present in the scratch output. It skips destinations that already exist and sources that are not files, using `_path_exists` and `_path_is_file`, two probes that swallow `OSError` and report missing rather than crashing on a fresh clone with odd permissions. When it copies the validation-result result record it rewrites that file's `receipt_paths` field to the canonical `PATTERN_RECEIPTS` list, so the mirrored record points at the stable slots.\n\n###### What a pass record carries\n\n**A pass record lists the emit path, the five result record refs, and the secret-scan summary with no private bodies, also aliased under `private_state_scan` for older readers.** A pass record lists the emit path first, then the five pattern-binding result record refs, and carries the secret-exclusion scan summary with no private bodies copied in. The scan summary is also exposed under `private_state_scan` as a compatibility alias for older readers.\n\nDiagram of the mechanism (12 steps). Diagram source & refs\n\nSource refs\n\nMISSING_FIXTURE_INPUT `blocked_dependency_missing` Validate secret exclusion scan `validate_secret_exclusion_scan` COMMAND_UNAVAILABLE `blocked_command_unavailable` SECRET_EXCLUSION_SCAN_BLOCKED `blocked_secret_exclusion` Validate pattern binding `validate_pattern_binding + mirror` MISSING_PATTERN_BINDING_RECEIPT `blocked_dependency_missing`\n\n```text\nflowchart TD A[\"Fresh public checkout\"] --> B[\"run_probe(root, suite, emit_ref)\"] B --> C{\"suite in SUPPORTED_SUITES?\"} C -- no --> X1[\"blocked_invalid_input UNKNOWN_COLD_CLONE_SUITE\"] C -- yes --> D{\"REQUIRED_INPUTS present?\"} D -- no --> X2[\"blocked_dependency_missing MISSING_FIXTURE_INPUT\"] D -- yes --> E[\"validate_secret_exclusion_scan\"] E -- raises --> X3[\"blocked_command_unavailable COMMAND_UNAVAILABLE\"] E -- status not pass --> X4[\"blocked_secret_exclusion SECRET_EXCLUSION_SCAN_BLOCKED\"] E -- pass --> F[\"validate_pattern_binding + mirror\"] F -- fail or missing --> X5[\"blocked_dependency_missing MISSING_PATTERN_BINDING_RECEIPT\"] F -- all present --> P[\"status pass emit ref + five result record refs\"]\n```\n\n##### Negative cases\n\n###### Run probe never\n\n** `run_probe` never raises.** `run_probe` never raises. Every failure is a typed record the reader can read:\n- `blocked_invalid_input` with code `UNKNOWN_COLD_CLONE_SUITE`: the requested suite is not `first-wave`. It also returns the supported list.\n- `blocked_dependency_missing` with code `MISSING_FIXTURE_INPUT`: one or more of `REQUIRED_INPUTS` is absent. The record names the missing files.\n- `blocked_command_unavailable` with code `COMMAND_UNAVAILABLE`: the secret scan raised, so the boundary cannot be checked. The exception text is recorded.\n- `blocked_secret_exclusion` with code `SECRET_EXCLUSION_SCAN_BLOCKED`: the scan ran but did not pass. The scan summary is attached, without leaking findings.\n- `blocked_dependency_missing` with code `MISSING_PATTERN_BINDING_RECEIPT`: the pattern-binding validator did not pass, or one of the five `PATTERN_RECEIPTS` is still absent after mirroring. The record names the missing result records.\n\n##### Prior Art Grounding\n\n###### Reproducible builds, twelve-factor, clean-checkout CI\n\n**The shape borrows from reproducible builds, the twelve-factor dependency rule, and clean-checkout CI, applied to one bounded first action from source.** The shape borrows from three public practices. Reproducible builds recreate outputs from declared sources and a constrained environment ( [reproducible-builds.org](https://reproducible-builds.org/docs/)). The Twelve-Factor dependency rule declares dependencies rather than relying on ambient system packages ( [12factor.net](https://12factor.net/dependencies)). Clean-checkout CI on hosted runners is the common way to prove a fresh clone builds and smoke-tests ( [GitHub Actions](https://github.com/features/actions)). The probe keeps the clean-checkout, declared-input, gated shape and applies it to one bounded first action from source.\n\n##### Validation Result record Path\n\n**Run `./bootstrap.sh` from the public root.** Use `--dry-run` first to print the exact command and the ignored result target without writing anything, then run it for real and check the focused tests:\n\n```text\n./bootstrap.sh --dry-run\n./bootstrap.sh\nPYTHONPATH=src ../repo-pytest \\\n  tests/test_cold_clone_probe.py \\\n  tests/test_bootstrap_script.py \\\n  tests/test_public_entry_docs.py \\\n  -q --basetemp /tmp/microcosm-cold-clone-probe\n```\n\n###### The result record and what green pytest means\n\n**The normal run writes an ignored result record, and a green pytest line means the predicate, wrapper, and public-entry docs agree, though a pass installs nothing and certifies no launch.** The normal run writes ignored `.microcosm/cold_clone_probe.json` evidence with `status=pass`, the five public result record refs, and the secret-scan summary. Pass `--emit <path>` only to refresh an intentional local target; the default stays ignored local state. A green pytest line means the predicate, the root wrapper, and the public-entry docs all agree with the current checkout. These are the validation result records for this module; a pass here does not install the package, certify launch, or aggregate any wider coverage.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The bounded bootstrap claim and its ceiling\n\n**The strongest claim is that a fresh clone can bootstrap the first-wave suite from `src/` with typed failure codes; the boundary stops there, authorizing no launch or deployment.** The strongest claim the evidence supports: a fresh public clone can bootstrap the first-wave suite from `src/`, with the fixture inputs present, the secret-exclusion scan passing, the pattern-binding fixture replaying, and all five pattern-binding result records materialized, and it records that with typed failure codes when any step is wrong. The proof boundary stops there. The probe excludes launch, hosted deployment, public sharing, external model access, secret export, package distribution, whole-system equivalence, or whole-system correctness. Its result record is ignored local state, not a tracked claim, and the scope limit for this module is the source-root bootstrap check plus its metadata-only result record. 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It binds the package card helper, CLI emitter, standard, README entry order, reader branch ids, doctrine-effect frame, omission result record, observatory landing refs, text projection, and focused tests so a cold reader sees what to inspect first without treating counts as maturity scores or route cards as launch, hosted-public sharing, provider, source-file changes, private-equivalence, or whole-system proof.","authority_ceiling":"Public first-screen card composition and focused validation only; no launch-scope decision, hosted-publishing-scope decision, external model access authority, source-file changes, non-public data equivalence, score-based progress, reader-success certification, or whole-system correctness.","abstract":"The first-screen composition root validates the public one-screen entry card, reader branches, omission result records, evidence accounting frame, text projection, README order, and scope limit without becoming launch or hosted-publishing-scope decision.","teleology":{"section_id":"","text":"The first-screen composition root validates the public one-screen entry card, reader branches, omission result records, evidence accounting frame, text projection, README order, and scope limit without becoming launch or hosted-publishing-scope decision.","source":"capsule_summary_line"},"body_markdown":"###### Component builds card\n\n**This component builds the one card a new reader sees before choosing a deeper route into the public Plectis repository, hosting nothing, calling no provider, and claiming no finished repository.** The first-screen composition root builds the one card a new reader sees before they pick a deeper route into the public Plectis repository. It is one Python component, `microcosm_core.first_screen_composition`, with a small command-line wrapper. It does not host anything, call a model provider, or claim the repository is finished.\n\n###### Output is card\n\n**Its output is one card in three shapes, full JSON, budgeted compact JSON, and terminal text, all built from a standard and checked back against it so drift becomes a failed check.** Its output is a single card in three shapes: a full JSON contract, a compact JSON card under a character budget, and a terminal-sized text card. Every shape carries the same shared first command, the same six reader branches, the same evidence accounting, and the same scope limit. The component builds the card from a standard file and then checks the built card back against that same standard, so a renamed field or a dropped branch shows up as a failed check instead of silent drift.\n\n##### Purpose\n\n###### The first-screen problem is not depth\n\n**The repository already has deep surfaces; the first-screen problem is that depth lands badly when a cold reader first meets a long command list or a raw JSON dump.** The repository already has deep surfaces: route maps, component status, standards, result records, and the localhost observatory. The first-screen problem is not missing depth. It is that depth lands badly when the first thing a reader meets is a long command list or a raw JSON dump.\n\n###### Prose has no contract, so it drifts\n\n**A hand-written welcome card drifts because prose has no contract, so the composition root replaces it with a fixed structure answering what must fit on one screen and in what order.** A loose, hand-written welcome card fails in a specific way. It drifts. Someone renames a branch, adds a command, or removes the scope limit, and nothing catches it, because prose has no contract. The composition root replaces prose with a fixed structure and answers one question: when a cold reader lands here, what has to fit on one screen, and in what order.\n\n###### The fixed slots on the card\n\n**The fixed slots are a goal-entry line, a shared first command, one behavior proof, six reader branches, an evidence-count legend, a link to the wider system, a deeper-route list, and the scope limit.** The fixed slots are the goal-entry line `plectis comprehend --first-action \"<your goal>\"`, the shared first command `plectis hello <project>`, one shared behavior proof `plectis tour --card <project>`, six reader branches, an evidence-count legend read as accounting rather than a score, a link from the local command to the wider source-open system, a short list of the deeper routes the card names instead of copying, and the scope limit.\n\n##### How it works\n\nThe order below is the order `first_screen_composition_card` runs.\n\nFunction Role `first_screen_composition_card` Loads the standard, assembles every card packet, runs both check passes, stamps `status`. `_validation_checks` Checks the built card is internally consistent. `_standard_backed_first_screen_scan` Checks the built card against the standard it was built from. `first_screen_compact_card` Projects the full card into a budgeted compact card with an omission list. `first_screen_text_card` Projects the card into a terminal-sized text screen, optionally for one reader.\n\n###### The builder assembles from the standard\n\n** `first_screen_composition_card` loads the standard and assembles one large payload of shared commands, six reader routes, menus, proof fields, legend, counts, and a scope limit, free-handing none of it.** `first_screen_composition_card(root, project_label)` is the builder. It loads `standards/std_microcosm_first_screen_composition_root.json` through `_load_standard`, then assembles one large payload: the shared commands, the six reader routes from `_reader_routes`, the copyable per-reader menu, the behavior-proof fields, the evidence-class legend, the scale counts, and the scope limit copied from the standard. It free-hands none of this. The `authority_ceiling`, anti_claim, and `public_private_boundary` are deep-copied straight from the standard.\n\n###### Internal consistency checks\n\n** `_validation_checks` returns named booleans that are true only when each card surface satisfies its own contract, yielding a false check for any missing or malformed surface and never raising.** Once the payload is built the component checks it twice. `_validation_checks(payload)` returns an ordered dictionary of named booleans. Each boolean is true only when one surface of the card satisfies its own contract: the six reader-route ids match `REQUIRED_ROUTE_IDS`, the reader menu and the landing packets cover those same ids, the evidence-class legend is complete, the scale counts are honest integers. A missing or malformed surface yields a false check. It never raises.\n\n###### Scanning the card against its standard\n\n** `_standard_backed_first_screen_scan` confirms the card's routes, copyable commands, and six false denied-authority flags all match the standard, returning `pass` only when every check is true and never raising.** `_standard_backed_first_screen_scan(payload, standard, check_ids)` then compares the built card against the standard it came from. It confirms the standard reference and validator id match, that every reader route is present across all four route surfaces, that each reader carries the exact copyable commands `plectis hello --reader <id> <label>` and its text-projection form, and that the six denied-authority flags in `DENIED_AUTHORITY_KEYS` are false in both the standard and the card. Those six keys are launch, source-file changes, non-public data equivalence, external model access, score-based progress, and whole-system correctness. The scan returns a record whose `status` is \"pass\" only when every check is true. On any drift it returns \"blocked\" with the failing check named. It never raises.\n\n###### Folding the scan into `status`\n\n**The builder folds the scan status into the check set, sets `status` to `pass` only when every check is true, and still returns a blocked card in full so the failing check is visible.** The builder folds the scan status into the check set, sets `validation.status` to \"pass\" only when every check is true, and copies that to the top-level `status`. A blocked card is still returned in full, so a reader can see which check failed.\n\n###### Compact and text projections with budgets\n\n** `first_screen_compact_card` produces a budgeted summary card with an omission list, and `first_screen_text_card` produces a 33-line terminal card that raises rather than truncating when it would overflow.** Two projections render the same card for different media. `first_screen_compact_card(payload)` produces a summary-first card under `COMPACT_JSON_CARD_MAX_CHARS` (16000) characters, with an omission list naming the full-contract keys it dropped and the `plectis first-screen --full` command that restores them. `first_screen_text_card(payload, reader_id)` produces a terminal card of at most `TEXT_CARD_MAX_LINES` (33) lines over the same scope limit. Passing `reader_id=\"all\"` shows every branch; an alias focuses one. It raises `ValueError` when the reader id is unknown or when the assembled card would exceed the line budget, so an over-long text card is a hard failure, not a truncated screen.\n\n###### The command-line wrapper\n\n**The wrapper `scripts/first_screen_composition_card.py` exposes the card with `--root`, `--project-label`, `--format`, and `--reader`, returning exit code 0 only when the card `status` is `pass`.** The command-line wrapper `scripts/first_screen_composition_card.py` exposes this without the full console. `build_parser` defines `--root`, `--project-label`, `--format` (json or text), and `--reader`. `main` builds the card, prints the JSON contract or the reader-focused text card, and returns exit code 0 only when the card `status` is \"pass\".\n\nDiagram of the mechanism (6 steps). Diagram source & refs\n\nSource refs\n\nStandard `std_microcosm_first_screen_composition_root.json` assemble card packets `first_screen_composition_card` JSON or terminal projection `first_screen_compact_card / first_screen_text_card`\n\n```text\nflowchart TD A[\"Standard std_microcosm_first_screen_composition_root.json\"] --> B[\"first_screen_composition_card assemble card packets\"] B --> C[\"_validation_checks internal consistency\"] B --> D[\"_standard_backed_first_screen_scan card vs standard\"] C --> E[\"status: pass or blocked\"] D --> E E --> F[\"first_screen_compact_card / first_screen_text_card JSON or terminal projection\"]\n```\n\n##### Negative cases\n\n###### What the checks force the card to reject\n\n**Mismatched reader ids fail `reader_route_parity`, a wrong copyable command fails `copyable_reader_commands`, a non-false denied-authority flag fails `denied_authority_flags_false`, and an over-33-line text card raises instead of printing.** The scan and the checks name what the card must reject. A card whose reader branches do not match the six required ids fails `reader_route_parity`. A card missing a copyable per-reader command, or carrying a command that does not match the expected `plectis hello --reader <id>` string, fails `copyable_reader_commands`. A card where any of the six denied-authority flags is not false fails `denied_authority_flags_false`. A text card that would run past 33 lines raises instead of printing. Each of these is a blocked result with the failing check named, never a quiet pass.\n\n##### Prior Art Grounding\n\n###### Command-line usability and progressive disclosure\n\n**The card borrows from command-line usability practice, progressive disclosure, and W3C PROV: a concise shared command, a terminal card, a JSON contract, and evidence counts kept attached to their scope.** The card borrows from command-line usability practice rather than inventing an onboarding genre. The [Command Line Interface Guidelines](https://clig.dev/) argue for concise default help, worked examples, discoverable next commands, and machine-readable output. Those appear here as the one shared command, the terminal-sized card, and the JSON contract. The compression rule follows progressive disclosure: reveal enough to orient a reader without dumping the whole system, the pattern Nielsen Norman Group documents as [progressive disclosure](https://www.nngroup.com/articles/progressive-disclosure/). [W3C PROV](https://www.w3.org/TR/prov-overview/) informs the rule that evidence counts stay attached to their source and scope limit rather than becoming standalone scores.\n\n##### Validation Result record Path\n\nReader-verifiable emitter commands, run from the `microcosm-substrate/` public root:\n\n```text\nPYTHONPATH=src ../repo-python scripts/first_screen_composition_card.py --project-label . --format json\nPYTHONPATH=src ../repo-python scripts/first_screen_composition_card.py \\\n  --project-label . \\\n  --format text \\\n  --reader safety_evals_engineer\n```\n\nFocused tests, run from the repository root:\n\n```text\nPYTHONPATH=src ./repo-pytest \\\n  tests/test_first_screen_composition_card.py \\\n  tests/test_cli_hello_first_screen.py \\\n  tests/test_readme_first_screen_entry.py \\\n  -q --basetemp /tmp/microcosm-first-screen-composition-tests\n```\n\n###### What a pass means here\n\n**A pass means the built card matched its standard and both projections stayed inside their budgets; the emitter writes no state and certifies no reader success or launch-scope decision.** A pass means the built card matched its standard and the two projections stayed inside their budgets. The emitter writes no project state and prints the card to stdout. It does not certify that any reader will succeed or that the repository is ready to launch.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The narrow proof boundary\n\n**The strongest claim is narrow: first-screen composition, reader-branch routing, the omission list, the text projection, and the scope limit are reproducible from public source and tests, and nothing more.** The strongest claim this component supports is narrow: the first-screen card is built from a public standard and checked back against it, so first-screen composition, reader-branch routing, the omission list, the text projection, and the scope limit are reproducible from public source and tests. That is the proof boundary. Its scope limit stops there. It does not establish runtime correctness, replace the route map, certify launch or hosted public sharing, use external model services, change project source, turn counts into maturity scores, or cover the whole doctrine lattice. A blocked scan reports drift. 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Bundle","relative_href":"docs/paper-module-batch7-secondary-runtime-capsule.html","read_href":"docs/paper-module-batch7-secondary-runtime-capsule.html","context_href":"docs/rules-and-ideas.html#paper-module-batch7-secondary-runtime-capsule","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"Set 7 Secondary Runtime Bundle imports runtime, graph, cartography, stockgrid, and Polymarket bodies and exercises their fixture-bound boundaries.","summary":"Set 7 Secondary Runtime Bundle binds the accepted batch7_secondary_runtime_capsule component to copied runtime view-model, lane-progress, graph-lens, graph-projection, cartography, stockgrid, and Polymarket source bodies. It exercises eight fixture engines, source digest anchors, required anchors, UI witness metadata, synthetic runtime packets, synthetic market rows, negative cases, and scope limits while excluding browser/session export, wallet authority, live market data, investment-related actions, external model access, launch-scope decision, whole-system equivalence, semantic truth, and complete UI or ranking coverage.","authority_ceiling":"Fixture-bound public source-body import and secondary runtime/market witness evidence only; no browser/session export, wallet authority, live market data, investment-related actions, external model access, launch-scope decision, whole-system equivalence, semantic truth, source-file changes, or complete UI/ranking coverage proof.","abstract":"Set 7 Secondary Runtime Bundle imports runtime, graph, cartography, stockgrid, and Polymarket bodies and exercises their fixture-bound boundaries.","teleology":{"section_id":"","text":"Set 7 Secondary Runtime Bundle imports runtime, graph, cartography, stockgrid, and Polymarket bodies and exercises their fixture-bound boundaries.","source":"capsule_summary_line"},"body_markdown":"###### Secondary runtime behavior is exercised directly\n\n**Three copied market and numeric bodies are re-run over synthetic fixtures, and each good-input pass is paired with a planted regression that must fail on the right bad input.** `batch7_secondary_runtime_capsule` copies a set of source bodies into a bundle and re-runs three of them against small synthetic fixtures. The three it exercises are numeric and market primitives: the stockgrid feature factory, the Polymarket order-book math, and the Polymarket four-lens scanner. The point is not that the copied code runs. The point is that each exercise is paired with a planted regression, so a check that only passes on good input is backed by a matching check that fails on the right bad input.\n\n###### Three runtime groups share one metadata-only result\n\n**It calls the real copied functions on fixed inputs and writes metadata-only records of status, engine ids, and counts, never inlining a source body, model-output data, or live data.** The component reads a fixture directory, imports each copied body, calls its real functions with fixed inputs, and compares the returned numbers to values the original author relied on. It writes metadata-only result records: status, engine ids, a copied-module count, and negative-case outcomes. It never inlines a copied source body, a model-output data, or live market data.\n\n###### Passing fixtures do not imply hosted readiness\n\n**The one question: do the copied market and numeric bodies still behave as their source claims when run in isolation over test-author inputs?** It answers one question. Do these copied market and numeric bodies still behave the way their source claims, when run in isolation over inputs the test author controls?\n\n##### Purpose\n\n###### How a copied body silently drifts\n\n**A copied body can drift in ways a clean-input smoke test never catches, since a `bids[0]` best-price read or a momentum divide only fails on adversarial input.** A copied source body can drift from its original in ways a passing smoke test misses. A best-price reader that takes `bids[0]` looks correct until the feed returns bids floor-first. A momentum primitive that divides by a price ratio looks correct until the ratio hits -100 percent. A test that only feeds clean input proves the happy path and nothing else.\n\n###### Each guard gets a paired negative check\n\n**Every engine records its good-path numbers and then a negative case that only clears if the guard still holds, where a `blocked` outcome means the guard fired.** This bundle makes the failure explicit. For each of the three engines it records the good-path numbers, then runs a negative case that would only pass if the guard is still in place. The `compute_best_prices` extrema rule, the `_daily_log_momentum_bps` refusal, and the resolved-market gate each have a paired negative check. Blocked, here, means the guard fired.\n\n##### How it works\n\n###### `run` drives the spec and anchor checks\n\n** `run` forwards a fixed `SPEC` through `run_crown_jewel_organ`, and `SOURCE_REQUIRED_ANCHORS` confirms each copied body still carries its declared anchor strings before its output is trusted.** The runner is `run`, which forwards a fixed `SPEC` to `run_crown_jewel_organ` with `_evaluate` as the evaluator and `evaluate_negative_case` as the negative-case evaluator. `run_batch7_secondary_bundle` is the same call over an exported bundle directory. `SPEC` also carries `SOURCE_REQUIRED_ANCHORS`, so the runner checks that each copied body still contains its declared anchor strings, for example `def compute_best_prices` and `Numeric-extrema best-price extraction` in `clob_snapshot.py`, before it trusts the exercise output.\n\n###### `_evaluate` folds engine findings into status\n\n** `_evaluate` runs the three engines in order and returns `pass` only with no findings, recording a blocked code for any non-pass engine and a missing code for any absent id.** `_evaluate` runs the three engines in order and collects findings. If any engine returns a status other than `pass` it records `BATCH7_SECONDARY_ENGINE_EXERCISE_BLOCKED`. If an expected engine id is missing from the results it records `BATCH7_SECONDARY_ENGINE_EXERCISE_MISSING`. It returns `pass` only when there are no findings, along with `engine_count`, sorted `engine_ids`, and `copied_macro_source_module_count` read from the source manifest.\n\nThe three engines:\n\nFunction Copied body What it checks `_stockgrid_exercise` `tools/stockgrid/stockgrid.py` `PayloadFactory` primitives return the expected numbers `_polymarket_clob_exercise` `tools/polymarket/clob_snapshot.py` best bid/ask come from extrema, not row order `_polymarket_score_exercise` `tools/polymarket/score.py` a resolved market is gated to zero on one lens\n\n###### Four stockgrid primitives, one refusal\n\n** `_stockgrid_exercise` checks four `PayloadFactory` primitives, including a momentum call that must refuse a -100 percent change because it has no finite log return.** `_stockgrid_exercise` builds a `PayloadFactory` and calls four primitives. `_daily_log_momentum_bps(10.0, 10)` must return a positive value above 90. `_daily_log_momentum_bps(-100.0, 10)` must return nothing: a claimed -100 percent daily change has no finite log return, so the primitive refuses it rather than emitting a misleading number. `_zscore` on the series `[1, 2, 3]` must equal `[-1.224745, 0.0, 1.224745]`, and `_mean_defined([1, None, \"3\"])` must equal `2.0`. The engine passes only if all four hold.\n\n###### Best price from extrema, not row order\n\n**A mis-sorted book must still yield best bid `0.42` and best ask `0.53` from the extrema rather than the first row, with the trap flag recording the difference.** `_polymarket_clob_exercise` feeds a deliberately mis-sorted book to `compute_best_prices`. The bids are `0.12, 0.42, 0.25` and the asks are `0.88, 0.53, 0.65`. The best bid must be `0.42` (the maximum bid) and the best ask must be `0.53` (the minimum ask), neither of which is the first row on its side. The spread must round to `0.11`. It then computes depth at a band with `compute_depth_at_band` and a `compute_depth_imbalance` value that must stay in `[-1, 1]`. The `sorted_book_trap_rejected` flag records that the chosen prices differ from `bids[0]` and `asks[0]`.\n\n###### Resolved markets are gated to zero\n\n**Scoring an open and a resolved market confirms the four-lens set and that a settled market gates to `0.0` on `NEWSBREAKER`, so it cannot surface as a fresh signal.** `_polymarket_score_exercise` scores two `NormalizedMarket` rows with `calculate_lenses`: one open, one resolved, identical except for status and price. The lens set must be exactly `HOT SEAT`, `NEWSBREAKER`, `GOD MODE`, `SCOUT`. The open market must score above zero on `NEWSBREAKER` and the resolved market must score exactly `0.0`. The scorer gates that lens on market resolution, so the fixture confirms a settled market cannot surface as a fresh signal.\n\nDiagram of the mechanism (7 steps). Diagram source & refs\n\n```text\nflowchart TD fixture[\"Fixture directory copied bodies + manifest\"] anchors[\"Anchor check SOURCE_REQUIRED_ANCHORS\"] stock[\"_stockgrid_exercise PayloadFactory primitives\"] clob[\"_polymarket_clob_exercise best-price extrema\"] score[\"_polymarket_score_exercise four-lens scan\"] negatives[\"Negative cases refusal, extrema, gate\"] result records[\"metadata-only records status, engine ids, counts\"] fixture --> anchors anchors --> stock anchors --> clob anchors --> score stock --> negatives clob --> negatives score --> negatives negatives --> result records\n```\n\n##### Negative cases\n\n###### Three planted regressions, each a code\n\n** `EXPECTED_NEGATIVE_CASES` names three guards, each emitting its stable code only when observed, where a `blocked` outcome is the intended result of catching the planted regression.** `EXPECTED_NEGATIVE_CASES` names three cases, each with a stable error code. `_stockgrid_extreme_momentum_negative` re-runs the stockgrid exercise and reports `blocked` only when the exercise passed and `extreme_momentum_refused` is true, under code `BATCH7_SECONDARY_STOCKGRID_EXTREME_MOMENTUM_REFUSED`. `_polymarket_sorted_book_trap_negative` reports `blocked` when `sorted_book_trap_rejected` held, under `BATCH7_SECONDARY_POLYMARKET_NUMERIC_EXTREMA_REQUIRED`. `_polymarket_resolved_market_negative` reports `blocked` when `resolved_newsbreaker_gated` held, under `BATCH7_SECONDARY_POLYMARKET_RESOLVED_MARKET_GATED`. `evaluate_negative_case` routes a case id through `_observed_negative_case` and emits the expected error code only when the guard is observed. A `blocked` outcome is the intended result: it means the planted regression was caught.\n\n##### Prior Art Grounding\n\n###### Microstructure, log-returns, event markets\n\n**The three checks trace to order-book microstructure, log-return feature engineering with its -100 percent singularity, and event-market scanning, all run over synthetic rows only.** The three engines sit in ordinary market-data and feature-engineering lineages. Order-book microstructure defines best bid, best ask, spread, and depth imbalance from the two sides of a limit-order book; the extrema rule here is the standard guard against a feed that does not sort its levels. Log-return feature engineering converts a percentage change into a bounded additive quantity and has a well-known singularity at a -100 percent change, which is why the primitive refuses it. Event-market scanning ranks prediction markets by freshness and other signals. The CFTC's [prediction markets explainer](https://www.cftc.gov/LearnandProtect/PredictionMarkets) is a plain boundary reference for what event-market data is and the caution it warrants. These fixtures run only over synthetic rows. They are not live market data and not trading decisions.\n\n##### Validation Result record Path\n\nReader-verifiable fixture command, run from `microcosm-substrate/`:\n\nFocused test, run from the repository root:\n\n```text\nPYTHONPATH=src ./repo-pytest \\\n  tests/test_batch7_secondary_runtime_capsule.py \\\n  -q --basetemp /tmp/microcosm-batch7-secondary-runtime-tests\n```\n\n###### What a pass writes and asserts\n\n**A pass means all three engines returned their expected numbers with no engine missing and every negative case fired, writing result, result record, and board files.** A pass means all three engines returned their expected numbers, no expected engine was missing, and each negative case fired. The fixture run writes `batch7_secondary_runtime_capsule_result.json`, `batch7_secondary_runtime_capsule_validation_receipt.json`, and `batch7_secondary_runtime_capsule_board.json` under the result record directory; the exported-bundle re-run uses the `run-batch7-secondary-bundle` action over `exported_batch7_secondary_runtime_capsule_bundle`.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The narrow proof boundary\n\n**The strongest claim is narrow: three anchor-checked copied bodies still return their authors' numbers over fixed synthetic inputs, with every planted regression caught.** The strongest claim this component supports is narrow. Three source bodies, copied into a bundle and anchor-checked, still return the numbers their authors relied on over fixed synthetic inputs, and each planted regression is caught. That is the proof boundary.\n\n###### What it never authorizes\n\n**It grants no live market access, wallet authority, or investment-related actions, and dispatches no provider, mutates no source, and claims no launch, public sharing, or semantic correctness.** It authorizes nothing beyond that. It is not live market access, wallet or account authority, or a market-data feed. It gives no investment-related actions. It does not export browser or session state, dispatch a provider or model, mutate the copied source, or prove the private and public roots are byte-identical. It is not a launch, not a public sharing, and not a claim that the scoring or momentum systems are complete or semantically correct. The `AUTHORITY_CEILING` constant holds every one of these fields false.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":12045,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/batch7_secondary_runtime_capsule.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"prediction markets explainer","url":"https://www.cftc.gov/LearnandProtect/PredictionMarkets","kind":"external_reference","section_id":"microstructure-log-returns-event-markets"}],"grouped_relation_feeds":[{"relation":"abides by","count":4,"links":[{"id":"AX-4","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Content-addressed determinism","url":"doctrine.html#dcard-ax-4"},{"id":"AX-8","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Provenance propagation and non-interference","url":"doctrine.html#dcard-ax-8"},{"id":"AX-10","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Temporal validity and freshness contracts","url":"doctrine.html#dcard-ax-10"},{"id":"AX-11","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Executable grammar before doctrine authority","url":"doctrine.html#dcard-ax-11"}]},{"relation":"depends on","count":1,"links":[{"id":"paper_module.macro_projection_import_protocol","kind":"paper_module","kind_label":"paper module","relation":"depends on","label":"Source Projection Import Protocol","url":"paper-module-macro-projection-import-protocol.html"}]},{"relation":"explains","count":2,"links":[{"id":"batch7_secondary_runtime_capsule","kind":"organ","kind_label":"component","relation":"explains","label":"Secondary Runtime Source Bundle","url":"component-batch7-secondary-runtime-capsule.html"},{"id":"mechanism.batch7_secondary_runtime_capsule.validates_public_secondary_runtime_capsule","kind":"mechanism","kind_label":"mechanism","relation":"explains","label":"validates public secondary runtime bundle","url":"rules-and-ideas.html#lattice-mechanism-mechanism-batch7-secondary-runtime-capsule-validates-public-secondary-runtime-capsule"}]},{"relation":"governed by","count":5,"links":[{"id":"P-2","kind":"principle","kind_label":"principle","relation":"governed by","label":"Lower claim strength to checker strength","url":"doctrine.html#dcard-p-2"},{"id":"P-5","kind":"principle","kind_label":"principle","relation":"governed by","label":"Cache by content, not by name","url":"doctrine.html#dcard-p-5"},{"id":"P-9","kind":"principle","kind_label":"principle","relation":"governed by","label":"Preserve provenance across every boundary","url":"doctrine.html#dcard-p-9"},{"id":"P-15","kind":"principle","kind_label":"principle","relation":"governed by","label":"Keep projections below source authority","url":"doctrine.html#dcard-p-15"},{"id":"concept.import_projection_and_drift_control_bundle","kind":"concept","kind_label":"concept","relation":"governed by","label":"source import and projection drift controls as provenance-preserving boundaries","url":"rules-and-ideas.html#lattice-concept-concept-import-projection-and-drift-control-bundle"}]}],"paper_module_backrefs":[],"source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/batch7_secondary_runtime_capsule.md"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json"}],"visible_text_byte_count":13333,"section_titles":["Secondary runtime behavior is exercised directly","Three runtime groups share one metadata-only result","Passing fixtures do not imply hosted readiness","Purpose","How a copied body silently drifts","Each guard gets a paired negative check","How it works","`run` drives the spec and anchor checks","`_evaluate` folds engine findings into status","Four stockgrid primitives, one refusal","Best price from extrema, not row order","Resolved markets are gated to zero","Negative cases","Three planted regressions, each a code","Prior Art Grounding","Microstructure, log-returns, event markets","Validation Result record Path","What a pass writes and asserts","Scope limit","The narrow proof boundary","What it never authorizes"]},{"id":"paper_module.microcosm_axiom_substrate","title":"Microcosm Axiom System","relative_href":"docs/paper-module-microcosm-axiom-substrate.html","read_href":"docs/paper-module-microcosm-axiom-substrate.html","context_href":"docs/rules-and-ideas.html#paper-module-microcosm-axiom-substrate","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"The public Plectis axiom system routes readers from axiom doctrine to the read-only support-cover evaluator, routing registry, standard, tests, and result records without claiming proof, launch, or source-file changes.","summary":"Plectis Axiom System is the public doctrine/routing boundary for axiom support. Its bundle binds the authored paper-module projection to the axiom support-cover mechanism and validator locus, with source refs to AXIOMS.md, PRINCIPLES.md, ANTI_PRINCIPLES.md, core/axiom_organ_routing.json, standards/std_microcosm_axiom.json, and focused tests. The validator computes support cases, anti-axiom rejection mappings, candidate pressure, and strong-gate pressure as a read-only projection.","authority_ceiling":"Public doctrine/routing/standard/result record evidence and read-only evaluator result records only; axiom witness-route source authority remains core/axiom_organ_routing.json, and claim_ceiling/strongest_allowed_claim remain computed by validator.microcosm.axiom_support_cover, not hand-stamped by this bundle or generated projections. No axiom proof certification, no candidate-law promotion, no source-file changes, no provider/Lean/Lake execution, no launch/publishing-scope decision, no whole-system equivalence, and no whole-system correctness.","abstract":"The public Plectis axiom system routes readers from axiom doctrine to the read-only support-cover evaluator, routing registry, standard, tests, and result records without claiming proof, launch, or source-file changes.","teleology":{"section_id":"","text":"The public Plectis axiom system routes readers from axiom doctrine to the read-only support-cover evaluator, routing registry, standard, tests, and result records without claiming proof, launch, or source-file changes.","source":"capsule_summary_line"},"body_markdown":"###### Read-only evidence accounting for each axiom\n\n**The axiom system is a read-only evaluator that measures how much on-disk evidence actually backs each Microcosm axiom, and reports where that evidence runs out.** The axiom system is a read-only evaluator that measures how much on-disk evidence actually backs each Microcosm axiom, and reports where that evidence runs out. Its core is `evaluate_axiom_support_cover` in `src/microcosm_core/validators/axiom_support_cover.py`.\n\n**A doctrine file can assert that the system holds twelve axioms. The harder question is whether anything checkable stands behind each one.** A doctrine file can assert that the system holds twelve axioms. The harder question is whether anything checkable stands behind each one. This component answers a single question: for each axiom, how strong is the support that already exists on disk, and where does it stop short of proof?\n\n**It emits one result record per run: a support case for every obligation, a per-axiom verdict, an anti-axiom rejection mapping, a strong-gate summary, and derived principle support.** It emits one result record per run: a support case for every obligation, a per-axiom verdict, an anti-axiom rejection mapping, a strong-gate summary, and derived principle support. No node in that record ever reports `strong_certified: true`.\n\n##### Purpose\n\n**The routing source `core/axiom_organ_routing.json` hand-stamps eleven of the twelve axioms with `witness_strength: strong` and names the anti-axiom each is meant to reject: AX-1 against `asserted_label_is_truth`, AX-5 against `silent_default_pass`, AX-8 against `endpoint_label_assertion_without_propagation`, AX-12 against `meta_artifact_exemption`.** The routing source `core/axiom_organ_routing.json` hand-stamps eleven of the twelve axioms with `witness_strength: strong` and names the anti-axiom each is meant to reject: AX-1 against `asserted_label_is_truth`, AX-5 against `silent_default_pass`, AX-8 against `endpoint_label_assertion_without_propagation`, AX-12 against `meta_artifact_exemption`. A loose reader would take those stamps at face value. The evaluator does not. It recomputes support from the material each obligation actually cites, and caps every axiom below `strong`.\n\n**Loose fails in a specific way. The routing defines `strong` as computing the property and rejecting the named anti-axiom with a negative case.** Loose fails in a specific way. The routing defines `strong` as computing the property and rejecting the named anti-axiom with a negative case. Confirming that a witness component's result record holds a complete negative-case suite is not the same as proving that suite rejects one obligation's slice. If the evaluator let the first stand in for the second, a green count would mask an unproven claim. So the gap is reported as pressure to do more witness work, never closed by relabelling. The honesty contract points back at the evaluator too: because it is itself a Microcosm artifact, AX-12 forbids it from reading its own generated output as evidence for itself.\n\n##### How it works\n\n** `evaluate_axiom_support_cover` is the entrypoint. It reads the routing rows, the eight-component ceiling registry and its six sub-order registries through `_ceiling_dimension_registry`, the component registry, the evidence-class table, and `PRINCIPLES.md`.** `evaluate_axiom_support_cover` is the entrypoint. It reads the routing rows, the eight-component ceiling registry and its six sub-order registries through `_ceiling_dimension_registry`, the component registry, the evidence-class table, and `PRINCIPLES.md`. It walks the twelve rows in order and evaluates each row's obligations, then folds the results into one record.\n\n**Each obligation carries a binding: the witness components, source surfaces, and negative-case codes meant to back it.** Each obligation carries a binding: the witness components, source surfaces, and negative-case codes meant to back it. `_binding_issues` checks that every cited piece resolves. A witness component must exist in `core/organ_registry.json`, a surface must exist on disk or match a glob, and a negative-case code must be declared on the routing row. Any unresolved reference is returned as a concrete issue string such as `organ_not_in_registry:<id>` or `surface_missing:<path>`, which caps the obligation at `blocked_binding_unresolved` rather than letting it pass.\n\n**For bindings that resolve, `_ceiling_vector` computes a value for each of eight named components: `evidence_class`, `checker_scope`, `provenance_class`, `freshness_state`, `domain_scope`, `negative_case_status`, `authority_scope`, and `projection_scope`.** For bindings that resolve, `_ceiling_vector` computes a value for each of eight named components: `evidence_class`, `checker_scope`, `provenance_class`, `freshness_state`, `domain_scope`, `negative_case_status`, `authority_scope`, and `projection_scope`. Each value is read off the bound material. `evidence_class` is the strongest rank among the witness components and never exceeds it. `freshness_state` is pinned to its honest floor, `unknown_live_freshness_no_refresh_contract`: a deterministic basis digest proves the inputs are reproducible, not that they are current. `authority_scope` and `projection_scope` each carry a non-laundering note so read-only output and generated projections cannot be read back as source evidence.\n\n** `_anti_axiom_rejection` tracks rejection apart from support. Its tier rises to `organ_receipt_coverage_present` only when `_organ_receipt_negative_coverage` finds a component result record under `receipts/first_wave/` whose `negative_case_coverage` has an empty `missing` list and top-level `status: pass`.** `_anti_axiom_rejection` tracks rejection apart from support. Its tier rises to `organ_receipt_coverage_present` only when `_organ_receipt_negative_coverage` finds a component result record under `receipts/first_wave/` whose `negative_case_coverage` has an empty `missing` list and top-level `status: pass`. Even then `mapping_verified` stays false unless a source-owned mapping row on the routing declares an exact or subsuming rejection for that obligation. The record notes plainly that no v0 tier certifies a per-obligation rejection from component-level coverage.\n\n** `_axiom_verdict` folds each axiom's required obligations into one verdict and an explicit list of `strong_blocked_reasons`.** `_axiom_verdict` folds each axiom's required obligations into one verdict and an explicit list of `strong_blocked_reasons`. It sets `hand_stamped_strong_not_certifiable` to true whenever the row stamped `strong`. `_strong_gate_summary` then takes the bilattice meet of two judgements: `positive_support_status` from the obligations' resolution, and `rejection_mapping_status` from the mappings. A detected conflict beats layer debt, which beats an unverified rejection. `strongest_allowed_claim` equals the raw verdict only when rejection is verified and nothing caps it, which for the current source is never.\n\n** `_compile_support_case` builds the citation envelope for one obligation. It cites only bundle and result-record refs that resolve on disk, stamps a deterministic `basis_digest` from `_basis_digest`, records the rederive command, and labels itself as citation, not certified strength.** `_compile_support_case` builds the citation envelope for one obligation. It cites only bundle and result-record refs that resolve on disk, stamps a deterministic `basis_digest` from `_basis_digest`, records the rederive command, and labels itself as citation, not certified strength. Principle support is derived, not measured: `_principle_obligation_groundings` parses each principle's `Obligation grounding:` line in `PRINCIPLES.md` and inherits support from those obligations, so a principle is never stronger than the obligations it rests on. `main` wraps the entrypoint as a CLI, writing the record to `--out` or stdout.\n\nFunction Role `evaluate_axiom_support_cover` Read-only entrypoint; assembles the whole record `_binding_issues` Enumerates unresolved refs; caps an obligation as blocked `_ceiling_vector` Computes the eight ceiling components for one binding `_negative_case_status_component` Orders the negative-case gate: absent, declared_only, referenced_in_bound_checker `_anti_axiom_rejection` Rejection judgement, tracked apart from support `_axiom_verdict` Per-axiom verdict plus the reasons `strong` is blocked `_strong_gate_summary` Bilattice meet of support and rejection; the node ceiling `_compile_support_case` On-disk citation envelope with a reproducible basis digest `main` CLI wrapper that prints or writes the record Diagram of the mechanism (7 steps). Diagram source & refs\n\nSource refs\n\nRouting rows AX-1 to AX-12, anti-axiom, hand-stamped witness_strength `core/axiom_organ_routing.json` or layer_debt `blocked_binding_unresolved`\n\n```text\nflowchart TD Routing[\"Routing rows core/axiom_organ_routing.json AX-1 to AX-12, anti-axiom, hand-stamped witness_strength\"] Bind[\"_binding_issues witness components, surfaces, negative-case codes resolve?\"] Capped[\"blocked_binding_unresolved or layer_debt\"] Ceiling[\"_ceiling_vector eight named components\"] Reject[\"_anti_axiom_rejection mapping_verified false without a source-owned row\"] Meet[\"_strong_gate_summary meet of support and rejection\"] Claim[\"strongest_allowed_claim strong_certified: false\"] Routing --> Bind Bind -- \"unresolved\" --> Capped --> Meet Bind -- \"resolved\" --> Ceiling --> Meet Bind --> Reject --> Meet Meet --> Claim\n```\n\n##### Negative cases\n\n**The evaluator refuses a fixed set of moves. It never echoes a hand-stamped `strong`: eleven rows stamp it and AX-8 stamps `partial`, and no axiom node reports `strong_certified: true`.** The evaluator refuses a fixed set of moves. It never echoes a hand-stamped `strong`: eleven rows stamp it and AX-8 stamps `partial`, and no axiom node reports `strong_certified: true`. It never promotes `organ_receipt_coverage_present` into `exact_obligation_rejection`; component-level coverage is admissible material, not a per-obligation rejection. It fails an obligation closed when any binding reference does not resolve, rather than passing it silently. It flags any binding that cites a principle as a witness, since a principle that leans on an axiom cannot also serve as that axiom's evidence. The AX-8 mappings stay deliberately non-uniform in the source rows so that endpoint coverage never reads as general source-to-sink propagation rejection.\n\n##### Prior Art Grounding\n\n**The pattern draws from two older ideas: formal assumptions should be inspectable, and machine-readable schemas should make support claims testable.** The pattern draws from two older ideas: formal assumptions should be inspectable, and machine-readable schemas should make support claims testable. Lean gives the immediate analogue through its axiom-audit practice, where a theorem can be checked and then separately inspected for its assumptions with `#print axioms`. This component adapts that spirit to doctrine by making each axiom expand into witness surfaces, negative cases, routing rows, and support-frontier status instead of treating the axiom prose as self-certifying. The schema side is grounded in [JSON Schema](https://json-schema.org/), which frames schemas as a way to define validation rules and document shared structure. The provenance side is adjacent to [W3C PROV](https://www.w3.org/TR/prov-overview/): support rows and witness refs are evidence links with bounded meaning, bounded evidence of whole-system completeness.\n\n##### Validation Result record Path\n\nReader-verifiable evaluator command, run from the `microcosm-substrate/` public root:\n\n```text\nPYTHONPATH=src ../repo-python \\\n  -m microcosm_core.validators.axiom_support_cover \\\n  --root . \\\n  --out /tmp/microcosm-axiom-support-cover-vrp.json\n```\n\nFocused test command, run from the repository root:\n\n**A pass means the evaluator wrote a record that reports support cases, support frontiers, anti-axiom rejection mappings, principle inheritance, and strong-gate pressure without mutating law, and that the tests confirm routing-schema parity, witness refs, negative-case evidence, the AX-8 mapping readback, and the rule that component coverage is not laundered into exact obligation rejection.** A pass means the evaluator wrote a record that reports support cases, support frontiers, anti-axiom rejection mappings, principle inheritance, and strong-gate pressure without mutating law, and that the tests confirm routing-schema parity, witness refs, negative-case evidence, the AX-8 mapping readback, and the rule that component coverage is not laundered into exact obligation rejection.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The inspectable axiom-support boundary\n\n**The strongest claim this component supports is that the axiom-support boundary is inspectable from public material: doctrine refs, routing rows, the evidence-class vocabulary, the validator, its result records, and the anti-axiom mapping statuses.** The strongest claim this component supports is that the axiom-support boundary is inspectable from public material: doctrine refs, routing rows, the evidence-class vocabulary, the validator, its result records, and the anti-axiom mapping statuses. That is the proof boundary. Within it, every axiom is capped below `strong` and every reason is named.\n\n###### Claims the axiom evaluator refuses\n\n**It refuses the rest. It does not establish any axiom in Lean, does not certify the strong gate, does not promote candidate law, does not close a rejection obligation, and does not change source records or include launch operations.** It refuses the rest. It does not establish any axiom in Lean, does not certify the strong gate, does not promote candidate law, does not close a rejection obligation, and does not change source records or include launch operations. A diagram view and navigation-atlas card are not yet generated because the module has not been admitted through the standard subject-resolution path. The scope limit holds: this Markdown and its generated projections are reader evidence only, and stronger claims require source-owner rows, builder regeneration, and their own validation result records.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":14260,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/microcosm_axiom_substrate.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"},{"label":"JSON Schema","url":"https://json-schema.org/","kind":"external_reference","section_id":"the-pattern-draws-from-two-older-ideas"},{"label":"W3C PROV","url":"https://www.w3.org/TR/prov-overview/","kind":"external_reference","section_id":"the-pattern-draws-from-two-older-ideas"}],"grouped_relation_feeds":[{"relation":"abides 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with explicit scope limit","url":"rules-and-ideas.html#lattice-concept-concept-executable-doctrine-grammar-standard-bundle"}]}],"paper_module_backrefs":[{"id":"paper_module.batch8_validator_checker_capsule","kind":"paper_module","kind_label":"paper module","relation":"depended on by","label":"Set 8 Validator Checker Bundle","url":"paper-module-batch8-validator-checker-capsule.html","relation_id":"paper_module.depends_on.paper_module","source_ref":"core/paper_module_capsules.json::paper_modules[63:paper_module.batch8_validator_checker_capsule].depends_on"}],"source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/microcosm_axiom_substrate.md"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json"}],"visible_text_byte_count":15757,"section_titles":["Read-only evidence accounting for each axiom","A doctrine file can assert that the system holds twelve axioms","It emits one result 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Runtime","relative_href":"docs/paper-module-semantic-singleflight-dedup-runtime.html","read_href":"docs/paper-module-semantic-singleflight-dedup-runtime.html","context_href":"docs/rules-and-ideas.html#paper-module-semantic-singleflight-dedup-runtime","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"The semantic singleflight dedup runtime surfaces the public command-run singleflight bundle and dedups command runs by a content key built from argv, resolved cwd, git HEAD, a scoped dirty-tree fingerprint, and an env fingerprint, over bounded public fixtures only.","summary":"Semantic Singleflight Dedup Runtime is the public operational-discipline replay for content-addressed command dedup. It runs bounded fixture exercises that build the command key, show a leader run and a completed-run reuse hit (the side-effecting counter stays at 1), and observe two negative cases: a scoped file mutation that flips the key so a stale working tree cannot answer for a different run, and an empty command that is rejected. It writes metadata-only result records and surfaces microcosm_core.engine_room.command_run_singleflight without exporting live state.","authority_ceiling":"Public fixture and result record evidence only; it keys and dedups command runs by repo-state fingerprint and does not guarantee global mutual exclusion, does not replace a lock service, cannot prove cross-host correctness, and is not a job scheduler, a daemon, live run-state export, launch-scope decision, or publishing-scope decision.","abstract":"The semantic singleflight dedup runtime surfaces the public command-run singleflight bundle and dedups command runs by a content key built from argv, resolved cwd, git HEAD, a scoped dirty-tree fingerprint, and an env fingerprint, over bounded public fixtures only.","teleology":{"section_id":"","text":"The semantic singleflight dedup runtime surfaces the public command-run singleflight bundle and dedups command runs by a content key built from argv, resolved cwd, git HEAD, a scoped dirty-tree fingerprint, and an env fingerprint, over bounded public fixtures only.","source":"capsule_summary_line"},"body_markdown":"**It collapses repeated runs of the same command into one, deciding sameness by a content key over the state the command depends on and reusing a completed run.** This component collapses repeated runs of the same command into one. It decides whether two runs are \"the same\" by building a content key from the state the command actually depends on, then reusing a completed run instead of executing it again.\n\n###### Real subprocess exercises through singleflight\n\n**It drives the `command_run_singleflight` bundle over fixture exercises that run real subprocesses, so a counter incremented twice stays at 1 when the duplicate run reuses the finished result.** It surfaces one public bundle, `command_run_singleflight`, and drives it over a small set of fixture exercises. Each exercise runs a real subprocess in a temporary working directory, so the reader can watch a duplicate run reuse a finished result rather than take the claim on trust. The counter exercise makes this visible: a side-effecting command that increments a file runs once, and the counter stays at 1 even though the command is issued twice.\n\n**The result is metadata-only records of status, per-case rows, and a card, with no captured output, environment values, or absolute paths written in.** The result is a set of metadata-only result records: a status, per-case rows, and a short card. No captured command output, no environment values, and no absolute paths are written into the records.\n\n##### Purpose\n\n###### The hard part is deciding sameness\n\n**Re-running wastes work but reusing a stale result is worse, and a loose key that reads only the command words reuses answers after the tree has changed.** Re-running an identical command wastes work. Reusing a stale result is worse, because the second caller gets an answer computed under different state. The hard part is deciding when two runs count as the same run. A loose key that reads only the command words will happily reuse a result after the working tree has changed underneath it.\n\n###### An explicit content key over repo state\n\n**It answers with an explicit content key that folds in repo state, so a meaningful change changes the key and a follower never reuses a result from unshared state.** This component answers with an explicit content key rather than a guess. The key folds in repo state, so a change that matters to the command changes the key, and a follower never reuses a result computed under state it did not share.\n\n##### How it works\n\n###### `build_result` tallies positive and negative rows\n\n** `build_result` runs each fixture case through `_evaluate_case` and passes only when at least one positive and one negative case are present and every case behaves as required.** `build_result` is the pipeline entry. It loads the fixture cases from the input directory, runs each one in a fresh temporary scratch directory through `_evaluate_case`, then tallies the rows. It marks the whole run `pass` only when there is at least one positive case and one negative case, every positive case was observed to pass, every negative case fired its guard, and both declared negative case ids in `EXPECTED_NEGATIVE_CASES` are present. Otherwise the status is `fail`. `run` calls `build_result` and writes the result record, the card from `result_card`, and a validation record.\n\n** `_evaluate_case` dispatches on the case's `exercise` field and calls into the bundle. Two bundle functions carry the mechanism.** `_evaluate_case` dispatches on the case's `exercise` field and calls into the bundle. Two bundle functions carry the mechanism.\n\n###### The content key from argv, cwd, and dirty tree\n\n** `build_command_key` hashes argv and cwd, folds in git HEAD and scoped diffs or scoped file contents, adds a bounded env fingerprint, and carries only hashes and labels.** `build_command_key` computes the content key. It hashes the argv, hashes the resolved working directory, and records the resource class and scope paths. For the dirty-tree part it looks for a git root: inside git it folds in HEAD, the porcelain status, the working diff, and the staged diff for the scoped paths; outside git it falls back to hashing the scoped file contents. It also records an environment fingerprint limited to `PYTHONPATH` and `PYTEST_ADDOPTS`. The key carries hashes and labels, not absolute paths, so records stay public-safe.\n\n###### Per-key lock assigns leader, follower, or reused\n\n** `run_command_singleflight` rejects empty argv, takes an exclusive per-key lock, and assigns the caller as leader, follower, or reused, running the subprocess only once.** `run_command_singleflight` performs the collapse. It rejects an empty argv before doing anything else. It builds the key, hashes it into a short key hash, and takes an exclusive file lock keyed by that hash. Under the lock it reads the active state for the key and chooses a role. If a run is active and its process is alive, this caller becomes a follower, waits, and replays the leader's captured output. If a run is already completed and the caller passed `reuse_completed`, it returns the reused record without executing. Otherwise it becomes the leader, runs the subprocess once, and captures the output for later followers. The returned record names the role: leader, follower, reused, or a stale-or-timeout record if an active run never completed.\n\nFunction Role `build_result` Runs every fixture case, tallies positive and negative rows, decides pass or fail `_evaluate_case` Dispatches one exercise to the bundle and reports what was observed `build_command_key` Builds the content key from argv, cwd, git HEAD, scoped dirty fingerprint, and env fingerprint `run_command_singleflight` Rejects empty argv, takes the per-key lock, and assigns leader, follower, or reused `result_card` Projects the result into a compact public card Diagram of the mechanism (6 steps). Diagram source & refs\n\nSource refs\n\nexclusive per-key lock `run_command_singleflight`\n\n```text\nflowchart TD Argv[\"argv + cwd + scope\"] Key[\"build_command_key content key + key hash\"] Lock[\"run_command_singleflight exclusive per-key lock\"] Leader[\"new key leader runs once\"] Reused[\"completed key reuse captured result\"] Reject[\"empty argv rejected\"] Argv --> Key Key --> Lock Lock --> Leader Lock --> Reused Argv --> Reject\n```\n\n##### Negative cases\n\n###### Two positive exercises: leader and reuse\n\n**The two positive exercises are `single_leader`, where a first run leads and returns its output, and `completed_reuse`, where the counter stays at 1 as a second run reuses it.** The fixtures include two positive exercises and two negative ones. The positive exercises are `single_leader`, where a first run becomes the leader and returns its captured output, and `completed_reuse`, where a second run reuses the completed run and the side-effecting counter stays at 1.\n\n###### Scope mutation and empty argv fail closed\n\n**The negative exercises confirm that rewriting a scoped file changes the dirty fingerprint and an empty command is refused, both firing their guards or the run fails.** The negative exercises are the point of the check. `scope_mutation_changes_key` builds the key for a scoped file holding \"before\", rewrites the file to \"after\", builds the key again, and confirms the dirty fingerprint changed. A stale working tree cannot answer for a different run, so its guard `SINGLEFLIGHT_STALE_STATE_CANNOT_DEDUP` is expected to fire. `missing_command_rejected` passes an empty command and confirms it is refused before any run starts, firing `SINGLEFLIGHT_EMPTY_ARGV_REJECTED`. Both fail closed: the run reports `fail` if either guard does not fire.\n\n##### Prior Art Grounding\n\n###### Request coalescing plus content-addressed caching\n\n**The lineage is request coalescing and content-addressed caching, with repo state added to the key so a working-tree change invalidates reuse.** The engineering lineage is request coalescing, the pattern where concurrent callers for the same key share one computation and the rest wait for its result, combined with content-addressed caching, where the cache key is a hash of the inputs rather than a name a caller chose. The bundle adds repo state to that key so that a change to the working tree invalidates reuse. The local prior art is the Plectis paper-module coverage contract: a reader page states which source row or public fixture can be checked, which generated projections are navigation aids, and which claims stay outside the evidence boundary.\n\n##### Validation Result record Path\n\nRun the component over its bounded fixtures and its test:\n\n###### Coverage and corpus-parity validation\n\n**From the repository root, rerun the public coverage contract and the corpus parity check:** From the repository root, rerun the public coverage contract and the corpus parity check:\n\n```text\nPYTHONPATH=src ./repo-pytest tests/test_plectis_paper_module_coverage_contract.py -q --tb=short\nPYTHONPATH=src ./repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n###### What a pass over the fixtures means\n\n**A pass means the content key deduplicated the fixture commands, the counter fired once, and both negative guards fired, and any source change means refreshing and rerunning.** A pass means the content key deduplicated the bounded fixture commands, the counter side effect happened once, and both negative guards fired. If a later pass changes the bundle source, fixture manifest, runtime locus, or record authority, refresh this page and rerun these commands.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The one supported claim\n\n**The evidence supports one claim: over the fixtures the content key dedups runs by repo state and rejects a scope mutation or empty command rather than silently reusing.** The evidence supports one claim: over these public fixtures, the content key dedups command runs by repo state, reuse happens without re-executing, and a scope mutation or an empty command is rejected rather than silently reused.\n\n###### Not a distributed lock or scheduler\n\n**It refuses the rest: no distributed lock, no cross-host correctness, no scheduler or daemon, no live-state export, and no launch, provider, or source-file changes.** It refuses the rest. It is not a distributed lock and does not guarantee global mutual exclusion. It does not establish cross-host correctness. It is not a job scheduler or a daemon, and it does not export the live command-run state tree. It excludes launch, public sharing, external model access, or source-file changes, and a green run over fixtures is not a claim about whole-system correctness.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":10606,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/semantic_singleflight_dedup_runtime.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"}],"grouped_relation_feeds":[{"relation":"abides by","count":2,"links":[{"id":"AX-1","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Derivation before assertion","url":"doctrine.html#dcard-ax-1"},{"id":"AX-9","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Compensable transactional effects","url":"doctrine.html#dcard-ax-9"}]},{"relation":"explains","count":2,"links":[{"id":"semantic_singleflight_dedup_runtime","kind":"organ","kind_label":"component","relation":"explains","label":"Semantic Singleflight Dedup Runtime","url":"component-semantic-singleflight-dedup-runtime.html"},{"id":"mechanism.semantic_singleflight_dedup_runtime.dedups_command_runs_by_repo_state_key","kind":"mechanism","kind_label":"mechanism","relation":"explains","label":"dedups command runs by repo state key","url":"rules-and-ideas.html#lattice-mechanism-mechanism-semantic-singleflight-dedup-runtime-dedups-command-runs-by-repo-state-key"}]},{"relation":"governed by","count":5,"links":[{"id":"P-1","kind":"principle","kind_label":"principle","relation":"governed by","label":"Recompute, do not echo","url":"doctrine.html#dcard-p-1"},{"id":"P-2","kind":"principle","kind_label":"principle","relation":"governed by","label":"Lower claim strength to checker strength","url":"doctrine.html#dcard-p-2"},{"id":"P-3","kind":"principle","kind_label":"principle","relation":"governed by","label":"Concentrate trust in small checkers","url":"doctrine.html#dcard-p-3"},{"id":"P-9","kind":"principle","kind_label":"principle","relation":"governed by","label":"Preserve provenance across every boundary","url":"doctrine.html#dcard-p-9"},{"id":"concept.work_landing_and_continuity_control_bundle","kind":"concept","kind_label":"concept","relation":"governed by","label":"work landing and continuity controls as transaction-scoped coordination evidence","url":"rules-and-ideas.html#lattice-concept-concept-work-landing-and-continuity-control-bundle"}]}],"paper_module_backrefs":[],"source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/semantic_singleflight_dedup_runtime.md"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json"}],"visible_text_byte_count":12086,"section_titles":["It collapses repeated runs of the same command into one","Real subprocess exercises through singleflight","The result is metadata-only records of status","Purpose","The hard part is deciding sameness","An explicit content key over repo state","How it works","`build_result` tallies positive and negative rows","`_evaluate_case` dispatches on the case's `exercise` field and calls into the bundle","The content key from argv, cwd, and dirty tree","Per-key lock assigns leader, follower, or reused","Negative cases","Two positive exercises: leader and reuse","Scope mutation and empty argv fail closed","Prior Art Grounding","Request coalescing plus content-addressed caching","Validation Result record Path","Coverage and corpus-parity validation","What a pass over the fixtures means","Scope limit","The one supported claim","Not a distributed lock or scheduler"]},{"id":"paper_module.finite_erdos_denominator_certificate_strike","title":"Finite Erdos Denominator-Order Certificate Strike","relative_href":"docs/paper-module-finite-erdos-denominator-certificate-strike.html","read_href":"docs/paper-module-finite-erdos-denominator-certificate-strike.html","context_href":"docs/rules-and-ideas.html#paper-module-finite-erdos-denominator-certificate-strike","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"The finite Erdos denominator-order certificate strike surfaces the public finite_denominator_order_certificate bundle and verifies, in exact rational arithmetic, that ord_Q(b)=lcm(F) for S_F(b)=sum 1/(b^n-1)=P/Q over bounded public fixtures.","summary":"Finite Erdos Denominator-Order Certificate Strike is the public formal-math replay for the finite period-noncollapse certificate. It sums 1/(b^n-1) over a finite support as an exact Fraction, reads the reduced denominator Q, computes the multiplicative order of b mod Q, cross-checks Q against the closed form B/gcd(A_L,B), and confirms ord_Q(b)=lcm(F) on small cases including a reducing case. It self-falsifies: a forged order or denominator is rejected by recomputation. It writes metadata-only result records and surfaces microcosm_core.engine_room.finite_denominator_order_certificate without calling providers.","authority_ceiling":"Public fixture and result record evidence only; it computes a finite denominator-order certificate in exact arithmetic and does not establish the open infinite Erdos #257 problem, is not an oracle, prover, or provider result, is not a machine-checked proof of even the finite statement, and is not launch-scope decision or publishing-scope decision.","abstract":"The finite Erdos denominator-order certificate strike surfaces the public finite_denominator_order_certificate bundle and verifies, in exact rational arithmetic, that ord_Q(b)=lcm(F) for S_F(b)=sum 1/(b^n-1)=P/Q over bounded public fixtures.","teleology":{"section_id":"","text":"The finite Erdos denominator-order certificate strike surfaces the public finite_denominator_order_certificate bundle and verifies, in exact rational arithmetic, that ord_Q(b)=lcm(F) for S_F(b)=sum 1/(b^n-1)=P/Q over bounded public fixtures.","source":"capsule_summary_line"},"body_markdown":"###### Component sums s\n\n**This component sums `S_F(b) = sum 1/(b^n - 1)` as an exact fraction `P/Q` and checks whether the multiplicative order of `b` modulo the reduced denominator `Q` equals `lcm(F)`.** `finite_erdos_denominator_certificate_strike` computes one finite number-theoretic object in exact whole-number arithmetic and checks a sharp claim about it. For a finite set of exponents `F` and a base `b`, it sums `S_F(b) = sum_{n in F} 1/(b^n - 1)` as an exact fraction `P/Q`, then asks whether the multiplicative order of `b` modulo the reduced denominator `Q` equals the least common multiple of `F`.\n\n###### Runs four small\n\n**It runs four small cases, two true certificates to accept and two forgeries to reject, and writes a metadata-only result record of the status and per-case values.** The component runs four small cases. Two are true certificates the checker should accept, including one where a common prime cancels in the fraction. Two are forged certificates the checker should reject. Each run writes a metadata-only result record that reports the status and the per-case values.\n\n###### Is finite identity\n\n**This is the finite identity behind the Erdos #257 period-noncollapse strike, not a solution to the open infinite problem, and a holding certificate is computational evidence, not a proof.** This is the finite identity behind the Erdos #257 period-noncollapse strike. It is not a solution to the open infinite problem, and a holding certificate over the bounded fixture set is computational evidence, not a proof.\n\n##### Purpose\n\n###### An easy claim to get wrong\n\n**The claim `ord_Q(b) = lcm(F)` is easy to get wrong: floating-point order, a trusted stored answer, or a single-derivation denominator each let a wrong certificate slip through.** The claim `ord_Q(b) = lcm(F)` is easy to state and easy to get wrong. A loose version would compute the order with floating point, or trust a stored expected answer, or check the denominator without deriving it a second way. Any of those lets a wrong certificate slip through unnoticed.\n\n###### Exact arithmetic and a recomputing checker\n\n**The component sums in exact Python fractions, derives the denominator two independent ways that must agree, and never accepts a claimed certificate on its word: the checker recomputes and compares.** This component computes the object exactly and lets a small checker decide it, so a reader can see where the evidence stops. The arithmetic uses Python fractions, never floats. The denominator is derived once by reduction and a second time by a closed form, and the two must agree. A claimed certificate is never accepted on its word: the checker recomputes the truth and compares.\n\n##### How it works\n\n###### Bundle arithmetic, component driver\n\n**The engine-room bundle `finite_denominator_order_certificate.py` holds the exact arithmetic while the component file drives it over fixture cases and writes the records.** The engine-room bundle `finite_denominator_order_certificate.py` holds the arithmetic. The component file drives it over fixture cases and writes the records.\n\n###### `compute_*` sums and cross-checks `Q`\n\n** `compute_finite_denominator_order_certificate` sums the exact fraction, reads the reduced denominator `Q`, finds `ord_Q(b)` by repeated multiplication, and cross-checks `Q` against the closed form `B/gcd(A_L, B)`.** `compute_finite_denominator_order_certificate(support, base)` is the core. It normalises `F` to a sorted set, sums `Fraction(1, base**n - 1)` over the members, and reads the reduced denominator `Q` from the resulting fraction. It computes `ord_Q(b)` with `multiplicative_order`, which finds the smallest `k` with `b**k = 1 mod Q` by repeated multiplication, with no factorisation. It then derives `Q` a second, independent way: with `L = lcm(F)`, `B = b^L - 1`, and `A_L = sum_{n in F} B/(b^n - 1)`, the closed form is `Q = B / gcd(A_L, B)`. Every `n` in `F` divides `L`, so each `b^n - 1` divides `B` and `A_L` is an exact integer; an assert guards that exact division. The returned `holds` is true only when the order equals `L`, the reduced denominator equals the closed form, `b` is coprime to `Q`, and `Q > 1`.\n\n###### `verify_*` is self-falsifying\n\n** `verify_finite_denominator_order_certificate` recomputes the truth from `(support, base)`, emits a typed mismatch code per disagreeing field, and returns `valid` only when nothing mismatches and the recomputed certificate holds.** `verify_finite_denominator_order_certificate(support, base, claimed)` is the self-falsifying check. It recomputes the truth from `(support, base)` and ignores the claimed values except to compare them. For each supplied field it emits a mismatch code: `ERDOS_CERT_NUMERATOR_MISMATCH`, `ERDOS_CERT_DENOMINATOR_MISMATCH`, or `ERDOS_CERT_ORDER_MISMATCH`. It returns `valid` only when nothing mismatches and the recomputed certificate holds.\n\n###### The component drives fixtures to a status\n\n** `run` and `build_result` evaluate each fixture case, and the run is `pass` only with at least one positive and one negative case, every positive holding, every negative rejected, and both forgery ids present.** The component file wraps these. `run` loads the input directory, calls `build_result`, and writes the result, board, and validation records. `build_result` reads the cases through `_fixture_cases` and evaluates each with `_evaluate_case`. A positive case calls `compute_*` and, for the reduction case, also requires that a reduction occurred. A negative case calls `verify_*` and checks that it was rejected with the exact codes named in `EXPECTED_NEGATIVE_CASES`. The run status is `pass` only when there is at least one positive and one negative case, every positive holds, every negative is rejected as expected, and both planted forgery ids are present.\n\nDiagram of the mechanism (6 steps). Diagram source & refs\n\n```text\nflowchart TD Input[\"Support F, base b\"] Sum[\"Exact sum S_F(b) = P/Q as a Fraction\"] Denom[\"Reduced denominator Q cross-checked against B/gcd(A_L,B)\"] Order[\"Multiplicative order ord_Q(b) by repeated multiplication\"] Check[\"Certificate holds if ord_Q(b) = lcm(F)\"] Verify[\"Forgery check recompute and compare claimed values\"] Input --> Sum Sum --> Denom Denom --> Order Order --> Check Check --> Verify\n```\n\n##### Negative cases\n\n###### Two forgeries in\n\n**The two forgeries in the fixture set both change exactly one field and rely on recomputation to catch it.** The two forgeries in the fixture set both change exactly one field and rely on recomputation to catch it.\n\n###### A forged denominator is recomputed away\n\n** `forged_denominator_rejected` claims `Q = 5` for `F = {3}`, `b = 2`, but the checker recomputes the true `Q = 7` and rejects with `ERDOS_CERT_DENOMINATOR_MISMATCH`.** `forged_denominator_rejected` uses `F = {3}`, `b = 2`. The true sum is `1/7`, so `Q = 7` and `ord_7(2) = 3 = lcm({3})`. The claim states the denominator is 5. The checker recomputes `Q = 7`, sees the disagreement, and rejects with `ERDOS_CERT_DENOMINATOR_MISMATCH`.\n\n###### A forged order is recomputed away\n\n** `forged_order_rejected` claims order 5 for `F = {1, 2}`, `b = 2`, but the checker recomputes `ord_3(2) = 2` and rejects with `ERDOS_CERT_ORDER_MISMATCH`.** `forged_order_rejected` uses `F = {1, 2}`, `b = 2`. The true sum is `4/3`, so `Q = 3` and `ord_3(2) = 2 = lcm({1, 2})`. The claim states the order is 5. The checker recomputes the order as 2 and rejects with `ERDOS_CERT_ORDER_MISMATCH`.\n\n###### The identity survives a reduction\n\n** `certificate_holds_after_reduction` uses `F = {2, 3}`, `b = 2` where the prime 3 cancels to `Q = 21`, and `ord_21(2) = 6 = lcm({2, 3})` still holds.** For contrast, `certificate_holds_after_reduction` uses `F = {2, 3}`, `b = 2`. The sum is `1/3 + 1/7 = 10/21`. The closed form gives `B = 63`, `A_L = 30`, `gcd(30, 63) = 3`, so `Q = 63/3 = 21`. The prime 3 cancels, `Q` is smaller than `B`, and `ord_21(2) = 6 = lcm({2, 3})` still holds. This is the case that shows the identity surviving a reduction rather than only the no-reduction case `certificate_holds` where `F = {2}` gives `1/3` directly.\n\n##### Prior Art Grounding\n\n###### Elementary number theory with a cross-check\n\n**The arithmetic is standard elementary number theory, exact rational summation, reduced denominators, and multiplicative order, with a closed-form cross-check, stated here only for finite support under Erdos #257.** The arithmetic is standard elementary number theory: exact rational summation, reduced denominators, and multiplicative order by repeated multiplication, with a closed-form cross-check `Q = B/gcd(A_L, B)` derived from divisibility of `b^n - 1` into `b^L - 1`. The finite identity `ord_Q(b) = lcm(F)` sits under the Erdos #257 period-noncollapse question and is stated here only for finite support. The engineering lineage is the local Plectis coverage contract: a reader page states what fixture can be rerun, what generated projections are navigation aids, and what stays outside the evidence boundary. No external citation is claimed.\n\n##### Validation Result record Path\n\nRun the component over its fixture set and rerun its test:\n\n###### What a pass means here\n\n**A pass means both positive certificates held, both forgeries were rejected with their expected mismatch codes, and the run wrote a metadata-only record, covering only the bounded fixture inputs in this checkout.** A pass means both positive certificates held, both forgeries were rejected with their expected mismatch codes, and the run wrote a metadata-only result record. A pass covers only the bounded fixture inputs in this checkout.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest honest claim and its limit\n\n**The strongest honest claim is that the finite certificate was computed exactly, its denominator agreed two ways, and forged order and denominator were caught by recomputation; it does not establish the open infinite problem.** The strongest honest claim is that the finite certificate was computed in exact rational arithmetic, that its denominator was derived two independent ways and agreed, and that a wrong claimed order and a wrong claimed denominator were both caught by recomputation rather than by a stored answer. The proof boundary stops there. The component does not establish the open infinite Erdos #257 problem. A holding certificate is a bounded computational witness, not a machine-checked proof of even the finite statement. The component is not an oracle, prover, or provider result, calls no external solver, and excludes launch, public sharing, or source change.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":10600,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/finite_erdos_denominator_certificate_strike.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"}],"grouped_relation_feeds":[{"relation":"abides by","count":1,"links":[{"id":"AX-2","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Kernelized verification","url":"doctrine.html#dcard-ax-2"}]},{"relation":"explains","count":2,"links":[{"id":"finite_erdos_denominator_certificate_strike","kind":"organ","kind_label":"component","relation":"explains","label":"Finite Erdos Denominator-Order Certificate Strike","url":"component-finite-erdos-denominator-certificate-strike.html"},{"id":"mechanism.finite_erdos_denominator_certificate_strike.verifies_finite_denominator_order_certificate","kind":"mechanism","kind_label":"mechanism","relation":"explains","label":"verifies finite denominator order certificate","url":"rules-and-ideas.html#lattice-mechanism-mechanism-finite-erdos-denominator-certificate-strike-verifies-finite-denominator-order-certificate"}]},{"relation":"governed by","count":2,"links":[{"id":"P-3","kind":"principle","kind_label":"principle","relation":"governed by","label":"Concentrate trust in small checkers","url":"doctrine.html#dcard-p-3"},{"id":"concept.formal_math_and_proof_witness_bundle","kind":"concept","kind_label":"concept","relation":"governed by","label":"formal math and proof witnesses as bounded public proof-adjacent result records","url":"rules-and-ideas.html#lattice-concept-concept-formal-math-and-proof-witness-bundle"}]}],"paper_module_backrefs":[],"source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/finite_erdos_denominator_certificate_strike.md"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json"}],"visible_text_byte_count":12101,"section_titles":["Component sums s","Runs four small","Is finite identity","Purpose","An easy claim to get wrong","Exact arithmetic and a recomputing checker","How it works","Bundle arithmetic, component driver","`compute_*` sums and cross-checks `Q`","`verify_*` is self-falsifying","The component drives fixtures to a status","Negative cases","Two forgeries in","A forged denominator is recomputed away","A forged order is recomputed away","The identity survives a reduction","Prior Art Grounding","Elementary number theory with a cross-check","Validation Result record Path","What a pass means here","Scope limit","The strongest honest claim and its limit"]},{"id":"paper_module.bridge_campaign_dag_validation","title":"Bridge Campaign DAG Validation","relative_href":"docs/paper-module-bridge-campaign-dag-validation.html","read_href":"docs/paper-module-bridge-campaign-dag-validation.html","context_href":"docs/rules-and-ideas.html#paper-module-bridge-campaign-dag-validation","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"The bridge campaign DAG validation component surfaces the public bridge_campaign_dag bundle and checks, over bounded public fixtures, that a fan-in campaign graph is well formed: unique labels, existing dependency edges, acyclicity, exactly one synthesis reaching a probe, barrier alignment, and a worker count within the provider safe-parallelism ceiling.","summary":"Bridge Campaign DAG Validation is the public agent-orchestration replay for a fan-in campaign graph. It validates each campaign spec against a subset of the source CR/VR rule families: schema and identity, unique node labels, dependency edges that reference existing nodes, DFS acyclicity, exactly one synthesis node that reaches a probe, a barrier naming that synthesis, and a requested worker count within the provider safe-parallelism ceiling. It accepts a linear chain and a three-way fan-in, and self-falsifies: a cyclic campaign is rejected with CR012 and a two-synthesis campaign with CR013. It writes metadata-only result records and surfaces microcosm_core.engine_room.bridge_campaign_dag without dispatching agents or calling providers.","authority_ceiling":"Public fixture and result record evidence only; it validates campaign DAG structure and does not dispatch agents, execute campaigns, prove provider correctness or safety, and is not launch-scope decision or publishing-scope decision.","abstract":"The bridge campaign DAG validation component surfaces the public bridge_campaign_dag bundle and checks, over bounded public fixtures, that a fan-in campaign graph is well formed: unique labels, existing dependency edges, acyclicity, exactly one synthesis reaching a probe, barrier alignment, and a worker count within the provider safe-parallelism ceiling.","teleology":{"section_id":"","text":"The bridge campaign DAG validation component surfaces the public bridge_campaign_dag bundle and checks, over bounded public fixtures, that a fan-in campaign graph is well formed: unique labels, existing dependency edges, acyclicity, exactly one synthesis reaching a probe, barrier alignment, and a worker count within the provider safe-parallelism ceiling.","source":"capsule_summary_line"},"body_markdown":"###### Bridge campaign fans\n\n**A bridge campaign fans parallel reads into one synthesis step, and this component recomputes the plan's structure to accept or reject it before anything runs, never dispatching or executing.** A bridge campaign is a small plan that fans several parallel reads into one synthesis step. This component reads such a plan and decides whether it is well formed before anything runs. It does not dispatch agents and it does not execute the campaign. It reads the plan, recomputes its structure, and returns an accept or reject decision with the rule id that fired.\n\n###### Result is metadata-only\n\n**The result is a metadata-only record of status, per-case rows, and rejecting rule ids, accepting two clean plans and rejecting two broken ones over the shipped fixtures.** The result is a metadata-only record: a status, per-case rows, and the rule ids that rejected each malformed plan. Over the shipped fixtures it accepts two clean plans and rejects two broken ones.\n\n##### Purpose\n\n###### Late structural faults waste a whole run\n\n**A structurally broken campaign is expensive to discover late, so the honest question is which rule fired rather than trusting an `ok` field that can lie.** A structurally broken campaign is expensive to discover late. A dependency loop, two synthesis steps, or a step that depends on a name that does not exist will each waste a whole run before the fault surfaces. Trusting a plan because it carries an \"ok\" field does not help: the field can lie. The honest question is \"which rule fired,\" not \"is the plan valid.\"\n\n###### Recomputing the graph from the plan\n\n**The check recomputes the graph from the plan itself, deriving cycles, single-synthesis, and reaches-a-probe from node roles rather than a summary, so passing requires every recomputed rule to hold.** So the check recomputes the graph from the plan itself. Cycle detection runs over the declared edges. Single-synthesis and reaches-a-probe are derived from the node roles and dependencies, not read from a summary. A malformed plan cannot pass, because passing requires every recomputed rule to hold.\n\n##### How it works\n\n###### Validator bundle and component runner\n\n**Two source files carry the mechanism: the engine bundle `bridge_campaign_dag.py` is the validator, and the component runner drives it over the fixtures and writes the records.** Two source files carry the mechanism. The engine bundle `src/microcosm_core/engine_room/bridge_campaign_dag.py` is the validator. The component runner `src/microcosm_core/organs/bridge_campaign_dag_validation.py` drives it over the fixtures and writes the records.\n\n###### `validate_campaign` conjoins every rule outcome\n\n** `validate_campaign` builds a `ValidationResult` and adds each rule's outcome, where any `reject` flips `ok` false, so the final `ok` is the conjunction of every CR/VR rule.** `validate_campaign` takes one plan, a provider name, and a worker count. It builds a `ValidationResult` and calls `result.add(rule_id, outcome, target, message)` for each rule. An outcome of `reject` flips `ok` to false and appends the formatted decision to the error list, so the final `ok` is the conjunction of every rule. The rule families mirror the private source CR/VR ids over a subset:\n\nRule Check CR001 to CR005 schema version, kind, kebab-case `campaign_id`, intent, and a public `plan_path` CR017 node labels are unique and non-empty CR015, CR016, CR020 known `input_mode`, a rationale when the mode is not result records-only, a valid role CR010, CR011 every `depends_on` names a real node; reducer and synthesis nodes depend on something CR012 the node graph is acyclic CR013, CR006, CR014 exactly one synthesis node; the barrier names it; it transitively reaches a probe CR007 to CR009 barrier description, continuation packet schema, bounded next action VR005 requested workers stay within the provider ceiling\n\n###### Cycle detection, reachability, and worker caps\n\n** `_cycle_labels` marks cycles with a depth-first back-edge check, `_reachable_dependencies` walks the closure from the synthesis node for CR014, and VR005 caps workers per provider via `SAFE_PARALLELISM`.** Cycle detection lives in `_cycle_labels`. It runs a depth-first walk with a visiting set and a visited set; a back-edge into a node still on the stack marks a cycle. `_reachable_dependencies` walks the dependency closure from the single synthesis node so CR014 can ask whether any reachable node is a probe. VR005 reads `SAFE_PARALLELISM`, which caps workers per provider: chatgpt 8, claude 2, gemini 3, local 4.\n\n###### The runner turns fixtures into records\n\n**The runner loads each fixture, evaluates it against `validate_campaign`, and aggregates a `pass` only when every positive validated, every negative was rejected as expected, and both planted ids fired.** The runner turns fixtures into records. `_fixture_cases` loads every JSON file under the input directory. `_evaluate_case` calls `validate_campaign` for one case, reads the reject rule ids with `_reject_codes`, and decides `observed_ok`. A positive case passes when the plan validated and matched its `expected_ok`. A negative case passes only when the plan was rejected and its expected rule ids from `EXPECTED_NEGATIVE_CASES` are among the firing rejections. `build_result` aggregates the rows: the overall status is `pass` only if there is at least one positive and one negative case, all positive cases validated, all negative cases were rejected as expected, and both planted negative ids are present. `run` writes the result, a compact card from `result_card`, and a validation record, each with `body_in_receipt` set to false.\n\nDiagram of the mechanism (7 steps). Diagram source & refs\n\n```text\nflowchart TD Spec[\"Campaign plan nodes, deps, barrier, provider, workers\"] Identity[\"Identity and schema CR001 to CR005\"] Labels[\"Unique labels, roles, real dependency edges\"] Cycle[\"Acyclicity CR012 depth-first\"] Synth[\"One synthesis reaches a probe, CR013 CR014\"] Ceiling[\"Worker ceiling VR005\"] Verdict[\"Accept or reject with the rule id\"] Spec --> Identity --> Labels --> Cycle --> Synth --> Ceiling --> Verdict\n```\n\n##### Negative cases\n\n###### Two planted rejections and two clean plans\n\n**The fixtures plant `cycle_rejected` firing `CR012` and `two_synthesis_rejected` firing `CR013` against two clean plans, and a rejection counts only when it names the exact rule it broke.** The fixtures plant two malformed plans, both rejected by recomputation. `cycle_rejected` has a dependency loop and must fire `CR012`. `two_synthesis_rejected` declares two synthesis nodes and must fire `CR013`. The two accepted plans are `linear_chain_ok`, a probe to reducer to synthesis chain, and `fan_in_ok`, three probes into one reducer into one synthesis. A negative case counts as handled only when the plan was rejected and its named rule id appears in the rejections, so a rejection always names the rule it broke.\n\n##### Prior Art Grounding\n\n###### Ordinary graph engineering and the coverage contract\n\n**The engine bundle is a source-faithful refactor of ordinary graph engineering, claiming no external citation, its local prior art being the Plectis coverage contract requiring a scope limit and result record path.** The engine bundle is a source-faithful public refactor of the private bridge campaign contract layer and its provider ceiling rule. The lineage is ordinary software engineering: schema validation, depth-first cycle detection on a directed graph, reachability over a dependency closure, and fixture-scoped regression cases. No external citation is claimed. The local prior art is the Plectis coverage contract, which requires each reader page to publish a scope limit, a grounding note, and a result record path rather than letting a generated structured source record stand in for the source.\n\n##### Validation Result record Path\n\nRun the component over its fixtures and write the records:\n\n###### What a pass means and how to confirm\n\n**A pass means both clean plans validated and both broken plans were rejected with their expected rule ids, confirmed against the bundle by the coverage contract and corpus check.** A pass means both clean plans validated and both broken plans were rejected with their expected rule ids. From the repository root, the coverage contract and the paper-module corpus check confirm this page against its bundle:\n\n```text\nPYTHONPATH=src ./repo-pytest tests/test_plectis_paper_module_coverage_contract.py -q --tb=short\nPYTHONPATH=src ./repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n##### Scope boundary\n\n###### Scope limit\n\n###### Proof boundary and scope limit\n\n**A green run shows well-formed plans validated and malformed ones rejected with the rule they broke over bounded fixtures, while the component neither dispatches, executes, nor authorizes launch.** A green run shows that well-formed plans validated and malformed ones were rejected, each with the rule it broke, over the bounded public fixtures in this checkout. That is the strongest claim here, and it is the proof boundary. The scope limit: this component does not dispatch agents, execute campaigns, prove provider correctness or safety, or claim whole-system equivalence. It excludes launch, public sharing, external model access, or source-file changes. It reads a plan and returns a decision.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":9379,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/bridge_campaign_dag_validation.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"}],"grouped_relation_feeds":[{"relation":"abides by","count":1,"links":[{"id":"AX-2","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Kernelized verification","url":"doctrine.html#dcard-ax-2"}]},{"relation":"explains","count":2,"links":[{"id":"bridge_campaign_dag_validation","kind":"organ","kind_label":"component","relation":"explains","label":"Bridge Campaign DAG Validation","url":"component-bridge-campaign-dag-validation.html"},{"id":"mechanism.bridge_campaign_dag_validation.verifies_bridge_campaign_dag","kind":"mechanism","kind_label":"mechanism","relation":"explains","label":"verifies bridge campaign dag","url":"rules-and-ideas.html#lattice-mechanism-mechanism-bridge-campaign-dag-validation-verifies-bridge-campaign-dag"}]},{"relation":"governed by","count":2,"links":[{"id":"P-3","kind":"principle","kind_label":"principle","relation":"governed by","label":"Concentrate trust in small checkers","url":"doctrine.html#dcard-p-3"},{"id":"concept.agent_reliability_and_safety_validator_bundle","kind":"concept","kind_label":"concept","relation":"governed by","label":"agent reliability and safety validators as bounded public scope limits","url":"rules-and-ideas.html#lattice-concept-concept-agent-reliability-and-safety-validator-bundle"}]}],"paper_module_backrefs":[],"source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/bridge_campaign_dag_validation.md"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json"}],"visible_text_byte_count":11105,"section_titles":["Bridge campaign fans","Result is metadata-only","Purpose","Late structural faults waste a whole run","Recomputing the graph from the plan","How it works","Validator bundle and component runner","`validate_campaign` conjoins every rule outcome","Cycle detection, reachability, and worker caps","The runner turns fixtures into records","Negative cases","Two planted rejections and two clean plans","Prior Art Grounding","Ordinary graph engineering and the coverage contract","Validation Result record Path","What a pass means and how to confirm","Scope limit","Proof boundary and scope limit"]},{"id":"paper_module.metabolism_queue_reconciliation","title":"Metabolism Queue Reconciliation","relative_href":"docs/paper-module-metabolism-queue-reconciliation.html","read_href":"docs/paper-module-metabolism-queue-reconciliation.html","context_href":"docs/rules-and-ideas.html#paper-module-metabolism-queue-reconciliation","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"Runs a synthetic SQLite durable-queue bundle over public fixtures and rejects planted job/run/log inconsistencies by recomputing the reconciliation taxonomy.","summary":"Metabolism Queue Reconciliation surfaces the Engine Room metabolism_runtime bundle: a stdlib-only synthetic SQLite job queue with idempotency-guarded enqueue, lease claim and expired-lease recovery, a blackboard claim-event projection where contradictions invalidate assertions, and a cold-start reconciler over the job/run/log triple. The component exercises two clean scenarios (lease recovery to recoverable with a healthy reconcile; a contradiction zeroing active claims) and two planted-defect scenarios (a running job with no run row; a finalized run whose job is still running), asserting the reconciler rejects each defect with its expected rule id. Status is pass only when both positives compute clean, both negatives are rejected with the expected marker, and both expected negative ids are present. No live private database, no agent dispatch, no external model access, no auto-repair.","authority_ceiling":"Real-system bundle run over public fixtures; status pass attests only that the synthetic queue/reconciliation computation behaved as specified on these bounded cases. Does NOT export the live private metabolism database/scheduler, does NOT dispatch agents, does NOT use external model services, does NOT auto-repair ambiguous runtime state, is NOT a distributed database, is NOT an oracle/prover, and excludes launch, public sharing, production use, or source-file changes.","abstract":"Runs a synthetic SQLite durable-queue bundle over public fixtures and rejects planted job/run/log inconsistencies by recomputing the reconciliation taxonomy.","teleology":{"section_id":"","text":"Runs a synthetic SQLite durable-queue bundle over public fixtures and rejects planted job/run/log inconsistencies by recomputing the reconciliation taxonomy.","source":"capsule_summary_line"},"body_markdown":"** `metabolism_queue_reconciliation` is a runnable model of a durable job queue. It stands up a synthetic SQLite store in a temporary directory, hands out work on a lease, recovers the job when the lease expires, and then runs a consistency check that flags impossible job/run/log states for a person to review.** `metabolism_queue_reconciliation` is a runnable model of a durable job queue. It stands up a synthetic SQLite store in a temporary directory, hands out work on a lease, recovers the job when the lease expires, and then runs a consistency check that flags impossible job/run/log states for a person to review.\n\n**The component surfaces one Engine Room bundle, `metabolism_runtime`, as a first-class part of the public system.** The component surfaces one Engine Room bundle, `metabolism_runtime`, as a first-class part of the public system. It runs four fixture scenarios: two clean ones that must compute end to end, and two deliberately broken ones that must be caught. Each scenario runs against its own temporary database that is created and torn down per case. It emits metadata-only result records and a CLI entry point, and it asserts nothing about the live private queue.\n\n**The honest unit here is the named reconciliation rule, not the word \"healthy.\" A clean store should produce no findings; a contradictory store should produce a finding tagged with the rule that describes the contradiction.** The honest unit here is the named reconciliation rule, not the word \"healthy.\" A clean store should produce no findings; a contradictory store should produce a finding tagged with the rule that describes the contradiction.\n\n##### Purpose\n\n**A job queue that drifts into contradictory state is hard to trust. Two states are the ones this component plants and catches: a job marked running with no run row recorded, and a run that finished while its job still says running.** A job queue that drifts into contradictory state is hard to trust. Two states are the ones this component plants and catches: a job marked running with no run row recorded, and a run that finished while its job still says running. A queue that passes over these silently, or auto-repairs them without a record, hides exactly the situations an operator needs to see.\n\n**The component makes reconciliation explicit and rerunnable over a bounded store. It does not fix the contradiction.** The component makes reconciliation explicit and rerunnable over a bounded store. It does not fix the contradiction. It recomputes the job/run/log relationship, names the rule that failed, and marks the store for review.\n\n##### How it works\n\n**The component wrapper in `src/microcosm_core /organs/metabolism_queue_reconciliation.py` loads fixture cases and drives them through the bundle.** The component wrapper in `src/microcosm_core/organs/metabolism_queue_reconciliation.py` loads fixture cases and drives them through the bundle. The bundle in `src/microcosm_core/engine_room/metabolism_runtime.py` owns the queue, the blackboard ledger, and the reconciler.\n\n###### Per-case scratch-store evaluation\n\n** `build_result` reads the case files with `_fixture_cases` (a sorted glob over `*.json` in the input directory, or a single file), opens one temporary scratch directory, and evaluates every case inside it.** `build_result` reads the case files with `_fixture_cases` (a sorted glob over `*.json` in the input directory, or a single file), opens one temporary scratch directory, and evaluates every case inside it. `_evaluate_case` forces the bundle's own `expected_status` to `pass`, calls the bundle's `evaluate_case`, and reads back the observed status and which reconciliation rules fired via `_fired_reconciliation_rules`. A case declared positive passes only when the bundle computed cleanly and no rule fired. A case declared negative passes only when the bundle ran cleanly, the expected rule id from `EXPECTED_NEGATIVE_CASES` fired, and the declared expectation was rejection. `build_result` then requires at least one positive and one negative case, every positive clean, every negative rejected, and both expected negative ids present, before it reports overall `pass`. `run` writes the result, board, and validation records with `write_json_atomic`, plus an sign-off record when an sign-off path is given. Every record sets `body_in_receipt` to false.\n\n**Inside the bundle, `evaluate_case` dispatches on the case kind to one of five runners. The queue path uses `enqueue_job`, which inserts a job under a partial unique index so a second job on the same idempotency key while an earlier one is still active is blocked.** Inside the bundle, `evaluate_case` dispatches on the case kind to one of five runners. The queue path uses `enqueue_job`, which inserts a job under a partial unique index so a second job on the same idempotency key while an earlier one is still active is blocked. `claim_next_job` leases the next queued or recoverable job to a worker and sets a lease expiry. `requeue_expired_jobs` moves claimed or running jobs whose lease has passed back to `recoverable` and clears the owner. The blackboard path uses `append_claim_event` to record an assertion and then a contradiction, and `build_blackboard_projection` drops any assertion that a later contradiction, expiry, or supersession event invalidated, so a contradicted claim projects to zero active claims.\n\n** `reconcile` is the check. It walks the jobs currently in `claimed` or `running` state, fetches the latest run for each with `latest_run_for_job`, and builds a `ReconciliationFinding` when the run is missing, when a finished run sits under a still-running job, or when the run's log file is missing or stale.** `reconcile` is the check. It walks the jobs currently in `claimed` or `running` state, fetches the latest run for each with `latest_run_for_job`, and builds a `ReconciliationFinding` when the run is missing, when a finished run sits under a still-running job, or when the run's log file is missing or stale. It counts findings by rule, sets status to `needs_review` when any finding exists and `healthy` otherwise, and every finding carries the action `operator_review_required`. It never mutates the jobs it inspects.\n\nFunction Role `_fixture_cases` Loads case rows from the input directory or a single file `_evaluate_case` Runs one case through the bundle and decides pass or reject `_fired_reconciliation_rules` Reads which reconciliation rules the bundle fired `build_result` Partitions positive and negative cases and computes overall status `run` Writes the metadata-only result, board, validation, and sign-off records `enqueue_job` Inserts a job under the active-idempotency uniqueness guard `claim_next_job` / `requeue_expired_jobs` Leases a job and recovers an expired lease to `recoverable` `build_blackboard_projection` Projects active claims after contradictions invalidate them `reconcile` Recomputes the job/run/log taxonomy and emits named findings Diagram of the mechanism (6 steps). Diagram source & refs\n\nSource refs\n\nreconcile `job/run/log check`\n\n```text\nflowchart TD Cases[\"Load fixture cases _fixture_cases\"] Scratch[\"Temp SQLite store per case\"] Bundle[\"evaluate_case run one scenario\"] Recon[\"reconcile job/run/log check\"] Decide[\"_evaluate_case pass or reject\"] Result[\"build_result + run metadata-only records\"] Cases --> Scratch --> Bundle --> Recon --> Decide --> Result\n```\n\n##### Negative cases\n\nTwo scenarios plant a contradiction and confirm the reconciler catches it.\n\n** `running_job_no_run_row_rejected` enqueues a job, claims it, and forces its state to running without ever recording a run.** `running_job_no_run_row_rejected` enqueues a job, claims it, and forces its state to running without ever recording a run. `reconcile` finds a running job with no run row and fires `running_job_no_run_row`. The case passes only because that rule fired and its declared expectation was rejection.\n\n** `finalized_run_running_job_rejected` enqueues a job, claims it, starts a run, and finishes the run without finalizing the job.** `finalized_run_running_job_rejected` enqueues a job, claims it, starts a run, and finishes the run without finalizing the job. The job still says running while its run has a completion time. `reconcile` fires `run_finalized_but_job_running`. The two clean scenarios, `queue_lease_recovery_ok` and `blackboard_projection_ok`, must produce no findings at all; a rule firing there would fail the positive case.\n\n##### Prior Art Grounding\n\n**This follows standard software-engineering practice for durable work queues and lease recovery, and for reconciliation passes that flag inconsistent records for human review rather than repairing them in place.** This follows standard software-engineering practice for durable work queues and lease recovery, and for reconciliation passes that flag inconsistent records for human review rather than repairing them in place. The queue design uses a partial unique index for active-key idempotency and a claim lease with expiry recovery, both common patterns in SQLite-backed job systems. The local prior art is the Plectis paper-module coverage contract: a reader page states what source row and public fixtures can be checked, names the generated projections as navigation aids, and keeps its claims inside the evidence boundary. No external citation is claimed for the specific reconciliation taxonomy; it is named in the source constants and this page reads it back.\n\n##### Validation Result record Path\n\nRun the component over its public fixtures:\n\n**A pass means both clean scenarios computed with no findings and both planted scenarios were rejected by the expected rule.** A pass means both clean scenarios computed with no findings and both planted scenarios were rejected by the expected rule. From the repository root, the coverage contract and corpus parity check confirm this page still matches its source record:\n\n```text\nPYTHONPATH=src ./repo-pytest tests/test_plectis_paper_module_coverage_contract.py -q --tb=short\nPYTHONPATH=src ./repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n###### Source-drift handling for later passes\n\n**If a later pass changes the bundle source, fixture manifest, runtime locus, or record authority, refresh this page and rerun both commands before treating the structured source record as current.** If a later pass changes the bundle source, fixture manifest, runtime locus, or record authority, refresh this page and rerun both commands before treating the structured source record as current. These are the validation result records for this module.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### What the synthetic queue run establishes\n\n**A green run shows that the synthetic queue, the blackboard projection, and the reconciler behaved as specified on four bounded fixture cases in this checkout.** A green run shows that the synthetic queue, the blackboard projection, and the reconciler behaved as specified on four bounded fixture cases in this checkout. That is the strongest claim the evidence supports.\n\n###### Claims outside queue reconciliation\n\n**It refuses more. It does not export or stand in for the live private metabolism database or scheduler.** It refuses more. It does not export or stand in for the live private metabolism database or scheduler. It does not dispatch agents, use external model services, or auto-repair ambiguous state; a contradiction is flagged for review, never silently corrected. It is not a distributed database and makes no production-correctness claim beyond the public fixtures. It grants no launch, public sharing, or source-file changes. The proof boundary is the fixture set plus the bundle reconciler; the scope limit is bounded validator evidence, not whole-system correctness.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":11799,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/metabolism_queue_reconciliation.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"}],"grouped_relation_feeds":[{"relation":"abides by","count":1,"links":[{"id":"AX-2","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Kernelized verification","url":"doctrine.html#dcard-ax-2"}]},{"relation":"explains","count":2,"links":[{"id":"metabolism_queue_reconciliation","kind":"organ","kind_label":"component","relation":"explains","label":"Metabolism Queue Reconciliation","url":"component-metabolism-queue-reconciliation.html"},{"id":"mechanism.metabolism_queue_reconciliation.verifies_metabolism_runtime","kind":"mechanism","kind_label":"mechanism","relation":"explains","label":"verifies metabolism runtime","url":"rules-and-ideas.html#lattice-mechanism-mechanism-metabolism-queue-reconciliation-verifies-metabolism-runtime"}]},{"relation":"governed by","count":2,"links":[{"id":"P-3","kind":"principle","kind_label":"principle","relation":"governed by","label":"Concentrate trust in small checkers","url":"doctrine.html#dcard-p-3"},{"id":"concept.agent_reliability_and_safety_validator_bundle","kind":"concept","kind_label":"concept","relation":"governed by","label":"agent reliability and safety validators as bounded public scope limits","url":"rules-and-ideas.html#lattice-concept-concept-agent-reliability-and-safety-validator-bundle"}]}],"paper_module_backrefs":[],"source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/metabolism_queue_reconciliation.md"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json"}],"visible_text_byte_count":13519,"section_titles":["`metabolism_queue_reconciliation` is a runnable model of a durable job queue","The component surfaces one Engine Room bundle","The honest unit here is the named reconciliation rule","Purpose","A job queue that drifts into contradictory state is hard to trust","The component makes reconciliation explicit and rerunnable over a bounded store","How it works","The component wrapper in `src/microcosm_core/components/metabolism_queue_reconciliation.py` loads fixture cases","Per-case scratch-store evaluation","Inside the bundle, `evaluate_case` dispatches on the case kind to one of five runners","`reconcile` is the check","Negative cases","`running_job_no_run_row_rejected` enqueues a job","`finalized_run_running_job_rejected` enqueues a job","Prior Art Grounding","This follows standard software-engineering practice for durable work queues and lease recovery","Validation Result record Path","A pass means both clean scenarios computed with no findings","Source-drift handling for later passes","Scope limit","What the synthetic queue run establishes","Claims outside queue reconciliation"]},{"id":"paper_module.derived_fact_provider_runtime","title":"Derived Fact Provider Runtime","relative_href":"docs/paper-module-derived-fact-provider-runtime.html","read_href":"docs/paper-module-derived-fact-provider-runtime.html","context_href":"docs/rules-and-ideas.html#paper-module-derived-fact-provider-runtime","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"Registry-backed derived fact provider runtime: resolves JSON-pointer, glob-count, and git-backed callable facts over public fixture roots and turns provider failures into error-as-data rows, rejecting malformed registries by recomputation.","summary":"Surfaces the derived_fact_provider_engine bundle as an component. It evaluates authored fact registries against bounded public fixture roots through three provider shapes \u2014 json_pointer (RFC 6901, list-index aware), glob_count (with excluded prefixes), and named callable facts (git-tracked counts) \u2014 and converts any provider failure into an error-as-data row (provider_status=error, error_class, repair hint) that degrades the result record rather than crashing the ledger. The runner exercises two clean registries (pointer+glob; git-callable+array-index pointer) and self-falsifies on two planted defects (an absent source path -> FileNotFoundError, an unsupported provider_type -> ValueError), asserting the exact error_class marker fires. Result records are metadata-only. It is not a doctrine truth auditor, not a full source registry export, not semantic claim validation, and grants no launch/publishing-scope decision.","authority_ceiling":"A pass means the surfaced fact-provider bundle resolved the authored fixture registries against their supplied roots and rejected the planted-defect registries by recomputation with the expected error_class. It does NOT mean any downstream prose claim is true (not a doctrine truth auditor), does NOT cover the full source fact registry (not a full export), does NOT perform semantic claim validation, does not establish the provider correct beyond the bounded fixtures, and grants NO launch, public sharing, private-source-export, or source-file changes. The only runtime variability admitted is filesystem reads, the git subprocess used by callable facts (over isolated tempdirs), and CLI argument reads.","abstract":"Registry-backed derived fact provider runtime: resolves JSON-pointer, glob-count, and git-backed callable facts over public fixture roots and turns provider failures into error-as-data rows, rejecting malformed registries by recomputation.","teleology":{"section_id":"","text":"Registry-backed derived fact provider runtime: resolves JSON-pointer, glob-count, and git-backed callable facts over public fixture roots and turns provider failures into error-as-data rows, rejecting malformed registries by recomputation.","source":"capsule_summary_line"},"body_markdown":"###### Fact registry names\n\n**A fact registry names small facts to compute, and this component resolves each one against a public fixture root, returning one result row per fact.** A fact registry lists small facts to compute: pull a number out of a JSON file by its pointer, count the files matching a glob, or run a named helper like \"how many git-tracked files are there\". This component resolves such a registry against a public fixture root and returns one result row per fact.\n\n###### Point is failure\n\n**The point is failure handling: a bad recipe becomes an error row carrying its exception class and repair hint, dropping the run to `degraded` rather than crashing or vanishing.** The point is what happens when a recipe is wrong. A pointer at a file that does not exist, or a provider type nobody defined, does not stop the run. The failing fact becomes an error row that carries the exception class and a repair hint, and the run status drops from `ok` to `degraded`. Failure is recorded, not swallowed and not fatal.\n\n**Two files split the work: an engine resolves the registry into a result record, and a wrapper grades it over fixture cases where good facts must match and broken ones must be caught.** Two files carry the work. The engine `derived_fact_provider_engine` resolves the registry and folds the rows into a result record. The component wrapper `derived_fact_provider_runtime` runs the engine over bounded fixture cases and checks that the good ones resolve to the expected values and the broken ones are caught with the exact expected error class.\n\n##### Purpose\n\n###### Two illegible failure modes to avoid\n\n**A derived-fact registry is only useful if failures are legible: crashing the whole ledger on one bad recipe or silently dropping the bad row both make it untrustworthy.** A registry of derived facts is only useful if its failures are legible. A loose implementation has two failure modes that both hurt: it crashes the whole ledger when one recipe points at a missing file, or it silently drops the bad row so a later reader cannot tell a fact was never computed. Either way the registry stops being trustworthy the moment one input is wrong.\n\n###### Error-as-data keeps the run going\n\n**The fix is error-as-data: each fact resolves in a `try` block, and a failure writes a typed error row with a repair hint while the run continues, degrading the record rather than ending it.** The fix is error-as-data. Each fact resolves inside a `try` block. Success writes a value; failure writes a row with `provider_status: error`, the `error_class`, a human-readable message, and a `required_next_action` repair hint. The run continues to the next fact. One bad recipe degrades the record; it does not end it.\n\n##### How it works\n\n###### Engine resolves, wrapper grades\n\n**The engine drives the registry through three providers into one result record, and the component wrapper runs that engine over declared fixture cases and grades the outcome.** The engine drives the registry through three providers and folds the rows into one result record. The component wrapper runs that engine over declared fixture cases and grades the outcome.\n\nDiagram of the mechanism (7 steps). Diagram source & refs\n\n```text\nflowchart TD Registry[\"Fact registry (facts: rows)\"] Dispatch[\"evaluate_provider dispatch on provider_type\"] Pointer[\"json_pointer resolve_json_pointer, RFC 6901\"] Glob[\"glob_count root.glob, exclude prefixes\"] Callable[\"callable _callable_value, git subprocess\"] Fold[\"evaluate_registry count errors, build ledger\"] Verdict[\"status ok or degraded error rows carry error_class + hint\"] Registry --> Dispatch Dispatch --> Pointer --> Fold Dispatch --> Glob --> Fold Dispatch --> Callable --> Fold Fold --> Verdict\n```\n\n###### Engine resolves fact\n\n**The engine resolves one fact at a time.** `evaluate_provider` reads a registry row, seeds a result dict with `provider_status: ok`, then dispatches on `provider_type`:\n- `json_pointer` reads the JSON at `source_path` and calls `resolve_json_pointer`, which walks RFC 6901 tokens. A list token is parsed as an integer index, so `/entries/1` selects the second element; a missing key or a bad index raises `KeyError`.\n- `glob_count` runs `root.glob` on the pattern, drops any match whose relative path starts with an excluded prefix, keeps only files, and returns the count.\n- `callable` calls `_callable_value`, which answers named facts like `git_tracked_file_count` and `git_tracked_python_count` by running `git ls-files` through `_git_ls_files` in the fixture root. An unrecognised name raises `KeyError`.\n\n###### One `except` turns failures into typed rows\n\n**The whole dispatch sits under one `except`, so any exception flips the row to `provider_status: error` with an `error_class` and repair hint, while `_coerce_scalar` casts good values to the declared type.** The whole dispatch sits under one `except`. Any exception flips the row to `provider_status: error`, records `error_class` as the exception class name, and adds a repair hint through `_source_repair_command`. `_coerce_scalar` casts a good value to the declared `value_type` before it lands.\n\n###### `evaluate_registry` folds rows into a ledger\n\n** `evaluate_registry` runs every fact, counts statuses, sets the record to `degraded` if any row errored, and assembles a ledger, findings, navigation cache, and a `receipt_sha256`.** `evaluate_registry` runs `evaluate_provider` over every fact, counts providers and statuses, and sets the record status to `degraded` when any row errored and `ok` otherwise. It assembles a ledger, a list of provider findings, a navigation cache, and a `receipt_sha256` over the summary and rows.\n\n###### `evaluate_case` runs a fixture in a temp dir\n\n** `evaluate_case` is the harness: it writes the case files into a temp directory, optionally builds a throwaway git index, runs the registry, and compares observed values and errors against expectations.** `evaluate_case` is the harness. It writes the case's declared files into a fresh temporary directory, optionally builds a throwaway git index with `_prepare_git_index` so the callable providers have something to count, runs `evaluate_registry`, then compares each observed value against `expected_values` and each expected error id against the rows. It returns `expectation_met` alongside the full record.\n\n###### The grading contract for pass\n\n** `build_result` splits fixtures into positive and negative rows, passing overall only when every positive resolves cleanly and every negative degrades with the exact planted error class.** The component wrapper adds the grading contract. `build_result` loads the fixture cases through `_fixture_cases`, runs `_evaluate_case` on each, and splits them into positive and negative rows. A positive case passes when its record status is `ok` and every value matched. A negative case passes only when the record is `degraded` and `_defect_error_class` finds the planted-defect fact carrying the exact error class named in `EXPECTED_NEGATIVE_CASES`. The overall status is `pass` only when there is at least one positive and one negative case, all positives pass, all negatives are caught, and both required negative ids are present. `result_card` projects a compact metadata-only card, and `run` writes the result, board, and validation records through `write_json_atomic`, with an optional sign-off record.\n\n##### Negative cases\n\n**Two fixtures plant a defect and require it to be caught by recomputation.** Two fixtures plant a defect and require it to be caught by recomputation:\n- `missing_source_path_rejected` points a fact at a `source_path` that was never written. The provider raises `FileNotFoundError`, the fact `demo.missing_source` becomes an error row, and the record degrades. The wrapper checks the observed error class equals `FileNotFoundError`.\n- `unknown_provider_type_rejected` declares a `provider_type` the engine does not implement. The dispatch raises `ValueError` on the fact `demo.unknown_provider`, the record degrades, and the wrapper checks the observed error class equals `ValueError`.\n\n**Two positive fixtures pin expected answers: pointer and glob facts resolve to 7 and 2, and git callables plus an array-index pointer resolve to 2, 1, and `beta`.** The two positive fixtures fix the expected answers. `json_pointer_glob_clean` resolves `demo.summary_count` to 7 through the pointer `/summary/fact_count` and `demo.markdown_count` to 2 by globbing `docs/*.md` while excluding the `private/` prefix. `git_callable_and_pointer_index_clean` resolves `demo.tracked_files` to 2 and `demo.tracked_python` to 1 through the git callables, and `demo.second_entry` to `beta` through the array-index pointer `/entries/1`.\n\n##### Prior Art Grounding\n\n###### RFC 6901 pointers and error-as-data provenance\n\n**The pointer provider follows RFC 6901, and the error-as-data shape follows provenance practice where a failed fact records its typed error and next action rather than vanishing.** The pointer provider implements RFC 6901 JSON Pointer, including the `~0` and `~1` token unescaping and integer list indexing that the standard specifies. The error-as-data shape follows ordinary provenance and traceability practice: a computed fact records where it came from, and a failed fact records its typed error and a next action rather than vanishing. The local lineage is the Plectis coverage contract, which asks each reader page to publish a scope limit, a grounding note, and a rerunnable result record path instead of letting a generated structured source record stand in for the source.\n\n##### Validation Result record Path\n\n###### Run component over\n\n**Run the component over its fixture cases and write the result records.** Run the component over its fixture cases and write the result records:\n\n###### What a pass means\n\n**A `pass` means both positive cases resolved to their expected values with an `ok` record and both negative cases were caught with the expected error class, degrading the record.** A `pass` means both positive cases resolved to their expected values with an `ok` record and both negative cases were caught by recomputation with the expected error class, degrading the record. To rerun the corpus checks that read this page, from the repository root:\n\n```text\nPYTHONPATH=src ./repo-pytest tests/test_plectis_paper_module_coverage_contract.py -q --tb=short\nPYTHONPATH=src ./repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim\n\n**The strongest claim is that registered facts resolved against the supplied fixture root and a malformed registry was rejected by recomputation; a clean record means facts resolved, not that any claim built on them is true.** The strongest claim the evidence supports: the registered facts resolved against the supplied fixture root, and a malformed registry was rejected because the provider recomputed each row and recorded the planted defect as a typed error. A clean record means the facts resolved, not that any prose claim built on those facts is true.\n\n###### What it refuses to do\n\n**It refuses the rest: it audits no doctrine claim, exports no full registry, calls no live provider, mutates no source, and excludes launch, resolving facts only against the public root you give it.** It refuses the rest. It does not audit whether a doctrine claim is true, does not export the full fact registry, does not perform semantic claim validation, and does not act as an oracle or prover. It resolves facts only against the public root you give it. It does not call live providers, change source files, or include launch operations or public sharing. Correctness is demonstrated over public fixtures, not over any private root.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":11797,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/derived_fact_provider_runtime.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"}],"grouped_relation_feeds":[{"relation":"abides by","count":1,"links":[{"id":"AX-2","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Kernelized verification","url":"doctrine.html#dcard-ax-2"}]},{"relation":"explains","count":2,"links":[{"id":"derived_fact_provider_runtime","kind":"organ","kind_label":"component","relation":"explains","label":"Derived Fact Provider Runtime","url":"component-derived-fact-provider-runtime.html"},{"id":"mechanism.derived_fact_provider_runtime.verifies_derived_fact_provider_engine","kind":"mechanism","kind_label":"mechanism","relation":"explains","label":"verifies derived fact provider engine","url":"rules-and-ideas.html#lattice-mechanism-mechanism-derived-fact-provider-runtime-verifies-derived-fact-provider-engine"}]},{"relation":"governed by","count":2,"links":[{"id":"P-3","kind":"principle","kind_label":"principle","relation":"governed by","label":"Concentrate trust in small checkers","url":"doctrine.html#dcard-p-3"},{"id":"concept.research_and_science_replay_evidence_bundle","kind":"concept","kind_label":"concept","relation":"governed by","label":"research and science replay validators as reproducibility scope limits","url":"rules-and-ideas.html#lattice-concept-concept-research-and-science-replay-evidence-bundle"}]}],"paper_module_backrefs":[],"source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/derived_fact_provider_runtime.md"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json"}],"visible_text_byte_count":13947,"section_titles":["Fact registry names","Point is failure","Two files split","Purpose","Two illegible failure modes to avoid","Error-as-data keeps the run going","How it works","Engine resolves, wrapper grades","Engine resolves fact","One `except` turns failures into typed rows","`evaluate_registry` folds rows into a ledger","`evaluate_case` runs a fixture in a temp dir","The grading contract for pass","Negative cases","Two fixtures plant","Two positive fixtures pin expected answers","Prior Art Grounding","RFC 6901 pointers and error-as-data provenance","Validation Result record Path","Run component over","What a pass means","Scope limit","The strongest supported claim","What it refuses to do"]},{"id":"paper_module.navigation_fitness_benchmark","title":"Navigation Fitness Benchmark","relative_href":"docs/paper-module-navigation-fitness-benchmark.html","read_href":"docs/paper-module-navigation-fitness-benchmark.html","context_href":"docs/rules-and-ideas.html#paper-module-navigation-fitness-benchmark","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"Recomputes a curated route-packet navigation benchmark \u2014 recall, precision, forbidden-first-route, scent coverage, latency, and debt \u2014 over bounded public fixtures, accepting a case only when the recomputation matches its planted expectation.","summary":"Navigation Fitness Benchmark surfaces the engine-room route-packet benchmark bundle. Each fixture pairs a navigation task (expected stable ids, forbidden first routes, latency budget, scent terms) with the route packet a router actually produced. The bundle recomputes recall/precision against the expected ids, checks the first route against the forbidden list, scores scent coverage, derives a latency verdict, and collects sufficiency/latency debt \u2014 then reports whether the recomputation matches the case's planted expectation. Two positive cases match their expectations and are accepted (one carrying an honestly-anticipated latency debt); two negative cases plant a route defect plus a deliberately-wrong \"it passed\" expectation, which the recomputation contradicts, so they are rejected with the real failure marker (missing_id, forbidden_route) firing. Scope: curated public fixtures only \u2014 not a live kernel run, not an embedding benchmark, not a universal navigation score, not launch-scope decision.","authority_ceiling":"Command-result record evidence over bounded public fixtures, not runtime-product completeness. A pass means the curated route-packet cases recomputed as expected (positives accepted, negatives rejected by recomputation with the expected markers); it does NOT establish live private-kernel navigation quality, embedding quality, universal benchmark authority, deployment posture, or launch/public sharing authorization. Result records are metadata-only command evidence, bounded evidence of system correctness.","abstract":"Recomputes a curated route-packet navigation benchmark \u2014 recall, precision, forbidden-first-route, scent coverage, latency, and debt \u2014 over bounded public fixtures, accepting a case only when the recomputation matches its planted expectation.","teleology":{"section_id":"","text":"Recomputes a curated route-packet navigation benchmark \u2014 recall, precision, forbidden-first-route, scent coverage, latency, and debt \u2014 over bounded public fixtures, accepting a case only when the recomputation matches its planted expectation.","source":"capsule_summary_line"},"body_markdown":"###### Scoring route packets against planted expectations\n\n** `navigation_fitness_benchmark` scores how well a navigation system found the right thing, and then checks its own scoring against what each case claimed would happen.** `navigation_fitness_benchmark` scores how well a navigation system found the right thing, and then checks its own scoring against what each case claimed would happen. A case is a task paired with the route packet a router returned for it. The task says what was wanted (which exact items count as the answer), which first routes are off-limits, which cue words should appear, and how fast the answer should arrive. The packet says what the router actually selected and how long it took.\n\n**The component recomputes the verdict from the packet: recall and precision of the selected items against the expected ids, whether a banned first route was taken, whether the cue words were present, and whether the run stayed under its time budget.** The component recomputes the verdict from the packet: recall and precision of the selected items against the expected ids, whether a banned first route was taken, whether the cue words were present, and whether the run stayed under its time budget. It emits a pass or fail per case plus the debt it found. A case is accepted only when that recomputation matches the expectation planted in the case. A packet that claims a pass it did not earn is rejected.\n\n##### Purpose\n\n**A single navigation score is easy to inflate and easy to misread as either failure or marketing.** A single navigation score is easy to inflate and easy to misread as either failure or marketing. If the harness trusts a packet's own claim that it passed, the number means nothing. This component does not trust the claim. It re-derives every metric from the packet and treats the case's stated expectation as a thing to be confirmed, not repeated. The unit of evidence is \"the recomputation matched the expectation,\" not a headline number.\n\n##### How it works\n\n**The mechanism lives in two files. `engine_room/navigation_fitness_benchmark.py` is the source-faithful public evaluator.** The mechanism lives in two files. `engine_room/navigation_fitness_benchmark.py` is the source-faithful public evaluator. `organs/navigation_fitness_benchmark.py` wraps it as a bounded fixture runner and decides sign-off.\n\n** `task_from_mapping` reads a task row into a `NavigationFitnessTask`: expected artifact ids, forbidden first routes, a latency budget (default 1500 ms), and scent terms.** `task_from_mapping` reads a task row into a `NavigationFitnessTask`: expected artifact ids, forbidden first routes, a latency budget (default 1500 ms), and scent terms.\n\n** `evaluate_task` is the scorer. It collects the packet's selected artifacts with `_packet_artifacts`, matches them against the expected set with `_match_expected` (an id ending in `*` matches by prefix), and computes recall as found over expected and precision as found over selected.** `evaluate_task` is the scorer. It collects the packet's selected artifacts with `_packet_artifacts`, matches them against the expected set with `_match_expected` (an id ending in `*` matches by prefix), and computes recall as found over expected and precision as found over selected. It scans the packet's `first_contact_command` for any forbidden route, scores scent coverage with `_scent_status`, and sets a latency status by comparing wall time to the task budget. It then assigns one sufficiency status in a fixed priority order: `route_timeout`, `route_error`, `missing_id`, `weak_scent`, `forbidden_route`, otherwise pass. Latency is judged separately, so a slow run can still be sufficient.\n\n** `evaluate_benchmark` runs `evaluate_task` over every task in a case and aggregates: p50 and p95 wall time from `_percentile`, sufficiency and latency pass/fail counts, per route-type metrics, and debt candidates from `_debt_candidates`.** `evaluate_benchmark` runs `evaluate_task` over every task in a case and aggregates: p50 and p95 wall time from `_percentile`, sufficiency and latency pass/fail counts, per route-type metrics, and debt candidates from `_debt_candidates`. Each failed sufficiency check becomes a `sufficiency_debt` row and each failed latency check a `latency_debt` row.\n\n** `evaluate_case` re-derives that benchmark and compares the recomputed result record against the case's planted `expected_status`, `expected_summary`, and `expected_task_statuses`.** `evaluate_case` re-derives that benchmark and compares the recomputed result record against the case's planted `expected_status`, `expected_summary`, and `expected_task_statuses`. It returns `expectation_met` only when the recomputed status and every summary and per-task check agree with what the case declared.\n\n**The component wrapper closes the loop. `_evaluate_case` calls the real `evaluate_case` (it does not re-implement the maths or bake in answers) and derives `observed_ok`.** The component wrapper closes the loop. `_evaluate_case` calls the real `evaluate_case` (it does not re-implement the maths or bake in answers) and derives `observed_ok`. For a positive case that means the recomputation both met the expectation and aligned with the declared `expected_ok`. For a negative case it means the recomputation did not meet the expectation and the specific recomputed failure marker for that planted defect is present. `build_result` runs the fixture set and returns `pass` only when the positives all hold, the negatives are all correctly rejected, and both expected negative case ids are present. `result_card` projects a metadata-only summary and `run` writes the result, board, validation, and sign-off records.\n\nDiagram of the mechanism (6 steps). Diagram source & refs\n\n```text\nflowchart TD Case[\"Task + route packet\"] Task[\"task_from_mapping expected ids, forbidden routes, budget, scent terms\"] Score[\"evaluate_task recall, precision, forbidden, scent, latency, sufficiency\"] Bench[\"evaluate_benchmark p50/p95, counts, debt\"] Match[\"evaluate_case recomputed == planted expectation?\"] Accept[\"_evaluate_case observed_ok\"] Case --> Task --> Score --> Bench --> Match --> Accept\n```\n\n##### Negative cases\n\n**Two planted negatives ship with the fixtures, and both carry a packet that declares `expected_status: pass`.** Two planted negatives ship with the fixtures, and both carry a packet that declares `expected_status: pass`. `missing_stable_id_rejected` returns a packet that omits an expected id; the recomputation produces a `missing_id` failure, so the recomputed status is `fail`, the planted \"it passed\" expectation is not met, and the case is correctly rejected. `forbidden_first_route_rejected` returns a packet whose first-contact command uses a banned route; the recomputation produces a `forbidden_route` failure and rejects it the same way. A negative is only counted as handled when the exact marker for its defect appears, so a case that fails for the wrong reason does not silently pass as a rejection.\n\n**The `latency_debt_pass` positive shows the other side. Its one task runs 830 ms against a 100 ms budget, so it carries one latency debt candidate, yet its sufficiency still passes and the case is accepted.** The `latency_debt_pass` positive shows the other side. Its one task runs 830 ms against a 100 ms budget, so it carries one latency debt candidate, yet its sufficiency still passes and the case is accepted. The component accepts honest debt; it does not require every packet to be debt-free.\n\n##### Prior Art Grounding\n\n**The recall, precision, and prefix-matching here are ordinary information retrieval evaluation applied to router output. The rest is standard engineering practice for fixture-scoped regression evidence: state which public inputs can be checked, recompute the verdict rather than trusting the input's own claim, and hold a small set of positive and negative cases.** The recall, precision, and prefix-matching here are ordinary information retrieval evaluation applied to router output. The rest is standard engineering practice for fixture-scoped regression evidence: state which public inputs can be checked, recompute the verdict rather than trusting the input's own claim, and hold a small set of positive and negative cases. The local lineage is the Plectis coverage contract, where a reader page publishes a scope limit and a rerun path instead of letting a generated structured source record stand in for the source.\n\n##### Validation Result record Path\n\nRun the component over its fixtures:\n\n###### Positive and negative case recomputation\n\n**A pass means the two positives (`clean_fanout_pass`, `latency_debt_pass`) matched their planted expectations and the two negatives (`missing_stable_id_rejected`, `forbidden_first_route_rejected`) were rejected by recomputation.** A pass means the two positives (`clean_fanout_pass`, `latency_debt_pass`) matched their planted expectations and the two negatives (`missing_stable_id_rejected`, `forbidden_first_route_rejected`) were rejected by recomputation. The paper-module coverage contract and corpus parity check guard this page itself:\n\n```text\nPYTHONPATH=src ./repo-pytest tests/test_plectis_paper_module_coverage_contract.py -q --tb=short\nPYTHONPATH=src ./repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n##### Scope boundary\n\n###### Scope limit\n\n**A green run shows that curated public route-packet cases recomputed as expected: the positives were accepted, including one that carries an honest latency debt, and the negatives were rejected because the recomputation contradicted their planted \"it passed\" claim.** A green run shows that curated public route-packet cases recomputed as expected: the positives were accepted, including one that carries an honest latency debt, and the negatives were rejected because the recomputation contradicted their planted \"it passed\" claim. That is the whole proof boundary. The component judges public fixtures only. It does not run the private kernel, does not capture packets from the live route runner, and does not test the underlying search or embeddings. It is not a universal navigation score and not a prover. It does not call any provider, does not export private state, and grants no launch, public sharing, or source-file changes.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":10307,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/navigation_fitness_benchmark.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"}],"grouped_relation_feeds":[{"relation":"abides by","count":1,"links":[{"id":"AX-2","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Kernelized verification","url":"doctrine.html#dcard-ax-2"}]},{"relation":"explains","count":2,"links":[{"id":"navigation_fitness_benchmark","kind":"organ","kind_label":"component","relation":"explains","label":"Navigation Fitness Benchmark","url":"component-navigation-fitness-benchmark.html"},{"id":"mechanism.navigation_fitness_benchmark.verifies_navigation_fitness_benchmark","kind":"mechanism","kind_label":"mechanism","relation":"explains","label":"verifies navigation fitness benchmark","url":"rules-and-ideas.html#lattice-mechanism-mechanism-navigation-fitness-benchmark-verifies-navigation-fitness-benchmark"}]},{"relation":"governed by","count":2,"links":[{"id":"P-3","kind":"principle","kind_label":"principle","relation":"governed by","label":"Concentrate trust in small checkers","url":"doctrine.html#dcard-p-3"},{"id":"concept.architecture_and_navigation_route_contract_bundle","kind":"concept","kind_label":"concept","relation":"governed by","label":"architecture and navigation route contracts as executable doctrine boundaries","url":"rules-and-ideas.html#lattice-concept-concept-architecture-and-navigation-route-contract-bundle"}]}],"paper_module_backrefs":[],"source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/navigation_fitness_benchmark.md"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json"}],"visible_text_byte_count":12356,"section_titles":["Scoring route packets against planted expectations","The component recomputes the verdict from the packet","Purpose","A single navigation score is easy to inflate","How it works","The mechanism lives in two files","`task_from_mapping` reads a task row into a `NavigationFitnessTask`","`evaluate_task` is the scorer","`evaluate_benchmark` runs `evaluate_task` over every task in a case and aggregates","`evaluate_case` re-derives that benchmark","The component wrapper closes the loop","Negative cases","Two planted negatives ship with the fixtures","The `latency_debt_pass` positive shows the other side","Prior Art Grounding","The recall, precision, and prefix-matching here are ordinary information retrieval evaluation applied to router output","Validation Result record Path","Positive and negative case recomputation","Scope limit","A green run shows that curated public route-packet cases recomputed as expected"]},{"id":"paper_module.annex_knowledge_routing","title":"Reference Knowledge Routing","relative_href":"docs/paper-module-annex-knowledge-routing.html","read_href":"docs/paper-module-annex-knowledge-routing.html","context_href":"docs/rules-and-ideas.html#paper-module-annex-knowledge-routing","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"Explainable tiered weighted-token retrieval that ranks a sanitized reference catalog against a problem statement and rejects unroutable problems with no_match.","summary":"Reference Knowledge Routing scores each entry of a sanitized in-memory reference catalog against a problem statement across four descending-weight tiers \u2014 structured routing fields, family text, open-first summaries, then curated notes \u2014 combining exact-match, phrase-containment, and token-overlap signals per tier into a ranked list with a per-row match breakdown and matched note ids. It is deliberately a transparent keyword-overlap retriever, not BM25/TF-IDF/embeddings/semantic search, and it routes only over the catalog handed to it (no repo cloning, no private corpus, no license authority). The component exercises the real bundle over four public fixtures: two positives route cleanly to the expected top slug above a minimum score (one via structured fields, one via a curated note), and two negatives self-falsify by recomputing to no_match \u2014 one because the problem shares no token with the catalog, one because a domain filter excludes the only candidate. Status is pass only when both positives route, both negatives are rejected with the expected no_match marker, and both expected negative case ids are present.","authority_ceiling":"Real-system bundle surfaced over bounded public fixtures. Does NOT clone repositories, ship the private reference corpus, perform semantic/embedding/BM25/TF-IDF search, adjudicate licenses or provenance, use external model services or external solvers, change source files, or include launch operations or public sharing. Routes only over the sanitized catalog supplied in each case; absolute scores are catalog-relative (no statistical normalization). Not a production retrieval system and not private-system equivalent.","abstract":"Explainable tiered weighted-token retrieval that ranks a sanitized reference catalog against a problem statement and rejects unroutable problems with no_match.","teleology":{"section_id":"","text":"Explainable tiered weighted-token retrieval that ranks a sanitized reference catalog against a problem statement and rejects unroutable problems with no_match.","source":"capsule_summary_line"},"body_markdown":"###### Problems route to ranked reference entries or no match\n\n**It takes a plain-English problem and a small reference catalog and returns the ranked entries most likely to help, or `no_match` when nothing overlaps rather than a forced guess.** `annex_knowledge_routing` takes a plain-English problem and a small catalog of reference entries and returns the entries most likely to help, ranked, with a breakdown of why each one matched. When nothing in the catalog overlaps the problem, it returns `no_match` rather than a forced best guess.\n\n###### Public routing records expose scores without bodies\n\n**Each entry carries structured tags, description text, open-first summaries, and notes, and the router scores across those four kinds of evidence, weighting structured tags most, then ranks entries above zero.** Each catalog entry carries structured routing tags (domains, clusters, problem-spaces, capabilities), some free description text, short open-first summaries, and curated notes. The router scores an entry against the problem across those four kinds of evidence, weighting the structured tags most, then ranks the entries that scored above zero. The result is a status, a ranked list of rows, and a per-row score breakdown.\n\n###### Routing evidence does not become implementation authority\n\n**It is a plain keyword-overlap matcher, not BM25, TF-IDF, or embedding search, and it reads only the in-memory catalog handed to it.** It is a keyword-overlap matcher. It is not BM25, TF-IDF, or embedding search, and it reads only the catalog handed to it in memory.\n\n##### Purpose\n\n###### The always-answer failure mode\n\n**A router that always returns its top entry hides the cases where it has nothing to offer, letting an off-topic problem still return a confident-looking row.** A router that always returns its top-ranked entry hides the cases where it has nothing to offer. If every query gets an answer, a reader cannot tell a real match from noise, and an off-topic problem still returns a confident-looking row. That failure mode is the thing this component is built to avoid.\n\n###### Explicit scores and an inspectable `no_match`\n\n**The router computes an explicit per-entry score, drops zero-score entries, returns `no_match` when all score zero or a filter empties the set, and exposes each row's tier scores.** So the router computes an explicit score per entry and drops any entry that scores zero. If every entry scores zero, or a domain or cluster filter removes every candidate, the status is `no_match`. Each returned row also carries the tier-by-tier scores that produced its rank, so the ranking is inspectable instead of opaque.\n\n##### How it works\n\n###### `route_catalog` wraps the ranked rows\n\n** `route_catalog` calls `route_annexes`, wraps the rows into a record whose status is `routed` or `no_match`, and stamps the `CLAIM_CEILING` and ANTI_CLAIMS constants onto the output.** The public entry point is `route_catalog` in `src/microcosm_core/engine_room/annex_knowledge_router.py`. It calls `route_annexes` and wraps the ranked rows into a result record whose `status` is `routed` when at least one row survived and `no_match` when the list is empty. It also stamps the fixed `CLAIM_CEILING` and ANTI_CLAIMS constants onto the record so the scope limit travels with the output.\n\n###### `route_annexes` normalizes the problem text\n\n** `route_annexes` does the work, normalizing the problem with `_normalized_query_text` (lowercase, split, strip, collapse), returning nothing on an empty problem, then iterating each catalog entry.** `route_annexes` does the work. It normalizes the problem text with `_normalized_query_text`, which lowercases, splits on underscores and slashes, strips punctuation, and collapses whitespace. An empty normalized problem returns nothing at all. Then, for each entry in the catalog:\n\n###### Building the routing summary and filtering\n\n**It first builds a routing summary from the entry's structured tags and note tags, and skips the entry before scoring if a domain or cluster filter finds no matching value.** First it builds a routing summary with `_routing_summary_from_family`, which collects the entry's structured `domains`, `clusters`, `problem_spaces`, and `status` and folds in the `problem_spaces` and `capabilities` declared on the entry's notes. If a `domain` or `cluster` filter is set and the entry does not list a matching value, the entry is skipped before any scoring.\n\n###### Four weighted tiers score every surviving entry\n\n**Each surviving reference entry is scored across four weighted tiers by the same `_route_match_score` function.** Then it scores the surviving entry across four tiers, each with its own weights passed to `_route_match_score`:\n\nTier Fields scored Exact / phrase / token weight structured problem-spaces, capabilities, domains, clusters 120 / 80 / 18 family text slug, display name, description, tags 32 / 24 / 6 open-first each open-first summary 20 / 16 / 4 notes note text, note problem-spaces, note capabilities 18 / 12 / 3\n\n###### Exact, phrase, and token scoring\n\n** `_route_match_score` returns the exact weight on an identical string, the phrase weight when the query is contained, or token count times weight otherwise, keeping each tier's best field.** `_route_match_score` normalizes both the query and the candidate string. If they are identical it returns the exact weight. If the whole query appears inside the candidate it returns the phrase weight. Otherwise it counts how many query tokens appear in the candidate token set and returns that count times the token weight. Stopwords in `ROUTE_MATCH_STOPWORDS` are dropped before token counting, so common words like \"the\" or \"with\" cannot inflate a match. Within each tier the router keeps the single best field score, not the sum, so one strong field sets the tier.\n\n###### Notes ranked by relevance carry their ids\n\n** `sort_notes_by_relevance` reads notes by a clamped 0-100 relevance and note id, and up to eight matched note ids ride the row so a reader sees which note carried the match.** Notes are read in relevance order by `sort_notes_by_relevance`, which sorts by a clamped 0-100 relevance value and then by note id. When a note contributes a positive score, its id is recorded, and up to eight matched note ids ride along on the row so a reader can see which note carried the match.\n\n###### Summing tiers into ranked rows\n\n**The four tier scores sum to a total, zero-or-less entries are dropped, and the rest sort by descending score, ties broken by slug, each row carrying its `match_breakdown`.** The four tier scores are summed into a total. Entries with a total of zero or less are dropped. The rest are sorted by descending score, ties broken by slug, and returned as rows carrying the slug, display name, total score, the `match_breakdown` of the four tier scores, and the matched note ids.\n\nDiagram of the mechanism (7 steps). Diagram source & refs\n\n```text\nflowchart TD Problem[\"Problem text (+ optional domain/cluster filter)\"] Summary[\"Build routing summary structured tags + note tags\"] Filter[\"Domain / cluster filter skip non-matching entries\"] Score[\"Score four tiers structured > family > open-first > notes\"] Rank[\"Sum, drop zero-score entries, rank by score then slug\"] Routed[\"status routed rows + match breakdown\"] NoMatch[\"status no_match every entry scored zero or filter emptied the set\"] Problem --> Summary --> Filter --> Score --> Rank Rank -->|at least one positive row| Routed Rank -->|no rows| NoMatch\n```\n\n###### The component wrapper checks fixture cases\n\n**The component wrapper drives the real `route_catalog` per fixture case, compares observed status, slug, score, and note ids to expectations, and passes only when every positive and negative case behaves.** The component wrapper in `src/microcosm_core/organs/annex_knowledge_routing.py` runs this bundle against bounded public fixture cases. `build_result` reads the fixture directory and calls `_evaluate_case` per case, which drives the real `route_catalog` and compares the observed status, top slug, score, and matched note ids against the case's declared expectations. It separates positive from negative cases and reports `pass` only when every positive case routes as expected, every negative case is rejected, and both planted negatives named in `EXPECTED_NEGATIVE_CASES` are present. `run` writes the metadata-only result, board, validation, and sign-off records through `write_json_atomic`; `result_card` projects the compact board; `run_annex_knowledge_routing_bundle` is the bundle-caller alias for `run`.\n\n##### Negative cases\n\n###### Two positives, two planted negatives\n\n**The fixture set carries two positive cases and two planted negatives, and the component self-falsifies on the negatives by recomputation.** The fixture set carries two positive cases and two planted negatives, and the component self-falsifies on the negatives by recomputation.\n\n###### `no_overlap_rejected`: nothing scores\n\n** `no_overlap_rejected` sends an off-topic problem whose tokens overlap no entry, so every tier scores zero and the status recomputes to `no_match`.** `no_overlap_rejected` sends the problem \"subaquatic origami choreography for migrating waterfowl\" against the catalog. No token overlaps any entry, every tier scores zero, and the status recomputes to `no_match`.\n\n###### `domain_filter_rejected`: filter empties the set\n\n** `domain_filter_rejected` sends an otherwise-matchable problem under an `agent-runtime` domain filter that removes every non-declaring candidate, so the result is `no_match`.** `domain_filter_rejected` sends \"forecast error scoring for market outcomes\" with a `domain` filter of `agent-runtime`. The problem could otherwise match, but the filter removes every candidate that does not declare that domain, so the result is `no_match`.\n\n###### Two positives pin the structured and notes paths\n\n**The positives pin both paths: `structured_route_ok` routes through the structured tier above score 80, and `note_route_ok` routes through the notes tier above 40 with `note_provider_backoff` on the top row.** The positive cases pin the two routing paths. `structured_route_ok` routes \"rate limit backoff across multiple llm providers\" to `provider-rate-limit-patterns` through the structured tier above a minimum score of 80. `note_route_ok` routes \"provider rate limit retry result record\" to the same slug through the notes tier above a minimum score of 40, and requires the matched note id `note_provider_backoff` to appear on the top row.\n\n##### Prior Art Grounding\n\n###### Lexical overlap, not a learned ranker\n\n**The mechanism is ordinary tiered weighted keyword overlap that stays lexical and shows its per-tier arithmetic, refactored source-faithfully from `system/lib/annex_registry.py::route_annexes` with no external citation claimed.** The mechanism is ordinary weighted keyword overlap with tiered field weights, the kind of lexical scoring used in lightweight retrieval before ranked statistical models like BM25 or TF-IDF. This component deliberately stays at that lexical level and shows its per-tier arithmetic rather than reaching for a learned ranker. The local lineage is `system/lib/annex_registry.py::route_annexes`, of which this bundle is a source-faithful public refactor over a sanitized catalog. No external citation is claimed.\n\n##### Validation Result record Path\n\nRun the component against its public fixtures:\n\n###### Coverage and corpus checks run from the root\n\n**The public coverage contract and paper-module parity check both rerun from the repository root.** From the repository root, rerun the public coverage contract and the paper-module corpus parity check:\n\n```text\nPYTHONPATH=src ./repo-pytest tests/test_plectis_paper_module_coverage_contract.py -q --tb=short\nPYTHONPATH=src ./repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n###### What a pass certifies\n\n**A pass means the two positive cases routed to their expected slug above the minimum score, both negatives recomputed to `no_match`, and the reader page stays consistent with its source record.** A pass means the two positive cases routed to their expected slug above the minimum score, both negatives recomputed to `no_match`, and the reader page stays consistent with its source record.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest supported claim\n\n**Over a sanitized in-memory catalog the router ranks by explainable tiered weighted-token overlap, exposes each rank's per-tier breakdown, and returns `no_match` when nothing scores or a filter empties the set.** The strongest claim the evidence supports: given a sanitized in-memory catalog and a problem statement, the router ranks entries by explainable tiered weighted-token overlap, exposes the per-tier breakdown behind each rank, and returns `no_match` when no entry scores above zero or a filter empties the candidate set. That behavior is checked by the two positive and two negative fixture cases.\n\n###### What the router refuses\n\n**It refuses to be BM25, TF-IDF, embedding, or semantic search, clones no repository, ships no private corpus, routes only over the passed catalog, and a green run is bounded fixture evidence.** What it refuses. It is not BM25, TF-IDF, embedding, or semantic search. It does not clone repositories, ship a private corpus, or adjudicate licence or provenance. It routes only over the catalog passed to it, with no repository or network access. A green fixture run is bounded fixture evidence, not whole-system equivalence, production correctness, external model access, source-file changes, or launch-scope decision. That is the proof boundary and the scope limit for this page.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":13724,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/annex_knowledge_routing.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"}],"grouped_relation_feeds":[{"relation":"abides by","count":1,"links":[{"id":"AX-2","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Kernelized verification","url":"doctrine.html#dcard-ax-2"}]},{"relation":"explains","count":2,"links":[{"id":"annex_knowledge_routing","kind":"organ","kind_label":"component","relation":"explains","label":"Reference Knowledge Routing","url":"component-annex-knowledge-routing.html"},{"id":"mechanism.annex_knowledge_routing.verifies_annex_knowledge_router","kind":"mechanism","kind_label":"mechanism","relation":"explains","label":"verifies reference knowledge router","url":"rules-and-ideas.html#lattice-mechanism-mechanism-annex-knowledge-routing-verifies-annex-knowledge-router"}]},{"relation":"governed by","count":2,"links":[{"id":"P-3","kind":"principle","kind_label":"principle","relation":"governed by","label":"Concentrate trust in small checkers","url":"doctrine.html#dcard-p-3"},{"id":"concept.architecture_and_navigation_route_contract_bundle","kind":"concept","kind_label":"concept","relation":"governed by","label":"architecture and navigation route contracts as executable doctrine boundaries","url":"rules-and-ideas.html#lattice-concept-concept-architecture-and-navigation-route-contract-bundle"}]}],"paper_module_backrefs":[],"source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/annex_knowledge_routing.md"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json"}],"visible_text_byte_count":15730,"section_titles":["Problems route to ranked reference entries or no match","Public routing records expose scores without bodies","Routing evidence does not become implementation authority","Purpose","The always-answer failure mode","Explicit scores and an inspectable `no_match`","How it works","`route_catalog` wraps the ranked rows","`route_annexes` normalizes the problem text","Building the routing summary and filtering","Four weighted tiers score every surviving entry","Exact, phrase, and token scoring","Notes ranked by relevance carry their ids","Summing tiers into ranked rows","The component wrapper checks fixture cases","Negative cases","Two positives, two planted negatives","`no_overlap_rejected`: nothing scores","`domain_filter_rejected`: filter empties the set","Two positives pin the structured and notes paths","Prior Art Grounding","Lexical overlap, not a learned ranker","Validation Result record Path","Coverage and corpus checks run from the root","What a pass certifies","Scope limit","The strongest supported claim","What the router refuses"]},{"id":"paper_module.egress_self_compliance_audit","title":"Egress Self-Compliance Audit","relative_href":"docs/paper-module-egress-self-compliance-audit.html","read_href":"docs/paper-module-egress-self-compliance-audit.html","context_href":"docs/rules-and-ideas.html#paper-module-egress-self-compliance-audit","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"Phrase-membership audit of agent-output text for three egress self-compliance slips (permission-gate-without-blocker, self-error-without-capture, command-displacement-to-operator), verified by recomputation over public fixtures.","summary":"Egress Self-Compliance Audit surfaces the egress_self_compliance_gate bundle: a transparent substring policy that flags three ways an agent's own output can break its operating discipline \u2014 asking permission with no named blocker, confessing an error with no durable capture, or displacing a command onto the operator with no execution result record. Each detector fires only when a tripwire phrase appears without its legitimiser. The component replays 2 compliant and 2 planted-violation fixtures through the live bundle, asserting each violation reproduces with its exact diagnostic marker. Honest ceiling: phrase membership only \u2014 no taint analysis, no prompt-injection defense, no semantics; paraphrased violations are missed.","authority_ceiling":"Command-result record evidence over bounded public fixtures, not runtime-product completeness. Establishes only that the bundle's phrase-membership detectors reproduce the declared green/red verdicts and diagnostic markers on the authored cases. Does NOT establish coverage of real agent output, semantic correctness, adversarial robustness, or any safety guarantee; excludes launch, public sharing, external model access, or source-file changes.","abstract":"Phrase-membership audit of agent-output text for three egress self-compliance slips (permission-gate-without-blocker, self-error-without-capture, command-displacement-to-operator), verified by recomputation over public fixtures.","teleology":{"section_id":"","text":"Phrase-membership audit of agent-output text for three egress self-compliance slips (permission-gate-without-blocker, self-error-without-capture, command-displacement-to-operator), verified by recomputation over public fixtures.","source":"capsule_summary_line"},"body_markdown":"###### Reads snippet of\n\n**It reads one snippet of an agent's own output and flags three self-policing slips: an unbacked permission ask, an uncaptured mistake, and a command pushed to the operator.** `egress_self_compliance_audit` reads a single snippet of an AI agent's own output and checks it for three self-policing slips: asking permission with no real blocker named, admitting a mistake with no durable capture, and handing a command to the operator with no sign it was run.\n\n**Phrase membership drives it: a giveaway present with its excuse absent flags the text, and the gate bundle runs over fixture cases into a metadata-only pass or fail record.** The check is phrase membership. It looks for a small set of giveaway phrases, and for the phrases that would excuse each one. If a giveaway is present and its excuse is absent, the text is flagged. The component surfaces the public `egress_self_compliance_gate` bundle, runs it over a bounded set of fixture cases, and writes a metadata-only result record with a pass or fail status.\n\n###### Is fast style\n\n**It is a fast style check, not comprehension: reword a slip past the phrase tables and it is missed.** It is a fast, transparent style check, not an understanding of meaning. Reword a slip so it dodges the phrase tables and the check will miss it. That is the honest limit, and it is stated up front.\n\n##### Purpose\n\n###### Three slips a disciplined agent leaks\n\n**A disciplined agent can still leak the three slips into its own prose: an empty permission ask, an unrecorded mistake, and a command it should have run itself.** An agent that follows an operating discipline can still leak violations into its own prose. It can ask \"shall I continue?\" when nothing actually blocks it. It can say \"my mistake\" without recording the error anywhere durable. It can tell the operator to run a command it should have run itself. These are the exact slips this component looks for.\n\n###### The excuse half prevents false flags\n\n**A tripwire counts as a violation only when its matching excuse phrase is also missing, so honest asks that name a real blocker pass.** A loose check that only scanned for the tripwire phrase would flag every honest permission ask, including the ones that name a genuine blocker. The point of the design is the second half of each rule: a tripwire counts as a violation only when its matching excuse phrase is also missing. Permission asks that name a real blast-radius blocker pass. Error admissions bound to a capture pass. Command handoffs that also report execution pass.\n\n##### How it works\n\n** `evaluate_text` runs three detectors over one lowercased string and returns `green` or `red` plus a `rows` list of every detector that matched.** The detection logic lives in `egress_self_compliance_gate.py`. Its `evaluate_text` function runs three detectors over one lowercased string and returns a status of `green` (no detector fired a violation) or `red` (at least one did), plus a `rows` list describing each detector that matched.\n\n###### The two-phase detector shape\n\n**Every detector runs the same two phases, `_body` to lowercase the text and `_matches` to return the table phrases that appear as substrings.** Each detector follows the same two-phase shape, built on `_body` (lowercases the text) and `_matches` (returns the phrases from a table that appear as substrings):\n\n###### Permission ask with no blocker\n\n** `detect_permission_gate_without_blocker` fires only when a `PERMISSION_GATE_PHRASES` match is present and no `LEGITIMATE_BLOCKER_PHRASES` match excuses it.** `detect_permission_gate_without_blocker` matches `PERMISSION_GATE_PHRASES` (for example \"shall i continue\", \"want me to continue\"). If none match, it returns nothing. If some match, it then checks `LEGITIMATE_BLOCKER_PHRASES` (for example \"irreversible\", \"public sharing boundary\", \"remote push\"). A violation fires only when a gate phrase is present and no blocker phrase is.\n\n###### Error admission with no capture\n\n** `detect_self_error_without_capture` fires when a `SELF_ERROR_TRIPWIRE_PHRASES` match appears with no `DURABLE_BINDING_PHRASES` match tying it to a capture.** `detect_self_error_without_capture` matches `SELF_ERROR_TRIPWIRE_PHRASES` (\"my mistake\", \"i miscounted\") against `DURABLE_BINDING_PHRASES` (\"cap_\", \"work log\", \"captured\"). An error admission with no binding to a capture is a violation.\n\n###### Command handoff with no result record\n\n** `detect_command_displacement_to_operator` fires when a `COMMAND_DISPLACEMENT_PHRASES` match appears with no `COMMAND_EXECUTION_RECEIPT_PHRASES` match showing it was run.** `detect_command_displacement_to_operator` matches `COMMAND_DISPLACEMENT_PHRASES` (\"you can run\", \"run this command\") against `COMMAND_EXECUTION_RECEIPT_PHRASES` (\"i ran\", \"exit code\", \"passed\"). A command handed off with no execution evidence is a violation.\n\n###### Audit wraps the gate in fixtures\n\n**The `egress_self_compliance_audit.py` component wraps the gate in the standard fixture contract, and `build_result` loads cases, evaluates each, aggregates a status, and returns the metadata-only payload.** The audit component in `egress_self_compliance_audit.py` wraps this gate with the standard fixture contract. `build_result` loads the cases, evaluates each one, aggregates a status, and returns the metadata-only payload. Its steps run in this order:\n\nFunction Role `build_result` Loads cases, evaluates each, aggregates the pass or fail status `_evaluate_case` Recomputes one case through the live gate and decides if it met expectation `run` Writes the result, board, validation, and optional sign-off records `result_card` Projects a compact metadata-only card for the public site\n\n###### `_evaluate_case` recomputes every verdict\n\n** `_evaluate_case` recomputes each case through the live gate and never trusts a stored verdict: a positive passes on green, a negative only on red carrying its expected diagnostic id.** `_evaluate_case` reads a case's `case_type` and `expected_ok`, calls `evaluate_text` on the case text, and never trusts a stored verdict. A positive case is satisfied when the gate returns green. A negative case is satisfied only when the gate returns red and the specific expected diagnostic id is among the firing rows. Those expected ids are declared in `EXPECTED_NEGATIVE_CASES`, which maps `permission_gate_without_blocker` to the `permission_gate_without_blocker` diagnostic and `command_displacement_no_receipt` to the `command_displacement_to_operator` diagnostic.\n\n###### `build_result` pass conditions\n\n** `build_result` returns `pass` only with at least one clean positive, every negative rejected with its marker, and both expected negative ids present; anything short is `fail`.** `build_result` then sets the status to `pass` only when there is at least one positive case and one negative case, every positive is clean, every negative is rejected with its marker, and both expected negative case ids are present. Anything short of that is `fail`.\n\nDiagram of the mechanism (5 steps). Diagram source & refs\n\n```text\nflowchart TD Text[\"Agent-output text\"] Gate[\"Permission gate: blocker named?\"] Err[\"Self-error: capture bound?\"] Cmd[\"Command handoff: execution reported?\"] Verdict[\"green or red with diagnostic id\"] Text --> Gate --> Verdict Text --> Err --> Verdict Text --> Cmd --> Verdict\n```\n\n##### Negative cases\n\n###### Fixture ships two\n\n**The fixture ships two planted violations, and the pass criterion depends on both being caught.** The fixture ships two planted violations, and the pass criterion depends on both being caught.\n\n###### Bare permission ask fires red\n\n** `permission_gate_without_blocker` names a gate phrase with no blocker, so its detector fires and the case recomputes red.** `permission_gate_without_blocker` carries the text \"Let me know if you want me to continue with the next wave.\" The gate phrase is present, no blocker is named, so the `permission_gate_without_blocker` detector fires and the case is red.\n\n###### Command handoff with mismatched diagnostic id\n\n** `command_displacement_no_receipt` hands off a command with no execution report, firing the `command_displacement_to_operator` detector whose id differs from the case id.** `command_displacement_no_receipt` carries \"You can run make smoke to verify the package before you ship it.\" A command is handed off with no execution report, so the `command_displacement_to_operator` detector fires. The case id and its expected diagnostic id differ here, which is why `EXPECTED_NEGATIVE_CASES` maps one to the other.\n\n###### Two positives guard the other direction\n\n**Two positive cases guard the other direction: a permission ask that names its blockers and an error admission bound to a `cap_` id both stay green.** The two positive cases guard the other direction. `permission_gate_with_named_blocker` asks \"Should I proceed?\" but names the remote push, the public sharing boundary, and irreversibility, so it stays green. `self_error_with_capture` says \"My mistake: I miscounted\" and binds it to a `cap_` id in the work log, so it stays green too. Repair a negative to name its excuse and it flips green, which shows the verdict is recomputed rather than baked in.\n\n##### Prior Art Grounding\n\n###### Substring policy in the lint family\n\n**The design follows fixture-scoped regression practice, refactors the `egress_compliance.py` policy into a public bundle, and sits in the lint and deny-list family, not information-flow analysis.** This follows standard software-engineering practice for fixture-scoped regression evidence and provenance: the page states which public source rows and fixture cases can be rerun, which generated projections are navigation aids, and which claims sit outside the evidence boundary. The gate is a source-faithful public refactor of the private `system/lib/egress_compliance.py` policy; the public bundle is the authority for the detection logic and phrase tables. The lineage is lightweight substring policy, the same family as lint rules and keyword deny-lists, not information-flow analysis.\n\n##### Validation Result record Path\n\n###### Run component over\n\n**Run the component over its fixture set and write the result records.** Run the component over its fixture set and write the result records:\n\n###### Rerun public coverage\n\n**Rerun the public coverage contract and the paper-module corpus parity check from the repository root.** Rerun the public coverage contract and the paper-module corpus parity check from the repository root:\n\n```text\nPYTHONPATH=src ./repo-pytest tests/test_plectis_paper_module_coverage_contract.py -q --tb=short\nPYTHONPATH=src ./repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n###### Pass conditions and refresh triggers\n\n**A pass means both positives recomputed green and both planted negatives recomputed red with their expected ids; refresh the page if the bundle, fixtures, locus, or result record authority change.** A pass means both positive cases recomputed green, both planted negatives recomputed red with their expected diagnostic id, and both expected negative case ids were present. If a future change alters the bundle source, the fixture manifest, the runtime locus, or the result record authority, refresh this page and rerun the commands before treating the structured source record as current.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest honest claim\n\n**On the authored fixtures, the phrase-membership detectors reproduced every declared green and red verdict by recomputing through the live gate, so a planted violation cannot pass.** The strongest honest claim is narrow: on the authored fixture cases, the phrase-membership detectors reproduced the declared green and red verdicts and the expected diagnostic markers by recomputation through the live gate. A planted violation cannot pass, because the gate re-derives every detector row and flags any text whose tripwire fires without its excuse.\n\n###### Where the proof boundary stops\n\n**The proof boundary stops at substring policy over one string: it is not taint analysis, injection defense, or semantic understanding, and it misses any slip worded outside the tables.** The proof boundary stops there. This is substring policy over one text string. It is not taint analysis, prompt-injection defense, sandboxing, an information-flow proof, or semantic understanding. It does not cover real agent output by guarantee, since a violation worded outside the phrase tables is missed and benign text containing a tripwire can be flagged. It may not claim whole-system equivalence, production correctness beyond the fixtures, external model access, source-file changes, or launch-scope decision.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":12737,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/egress_self_compliance_audit.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"}],"grouped_relation_feeds":[{"relation":"abides by","count":1,"links":[{"id":"AX-2","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Kernelized verification","url":"doctrine.html#dcard-ax-2"}]},{"relation":"explains","count":2,"links":[{"id":"egress_self_compliance_audit","kind":"organ","kind_label":"component","relation":"explains","label":"Egress Self-Compliance Audit","url":"component-egress-self-compliance-audit.html"},{"id":"mechanism.egress_self_compliance_audit.verifies_egress_self_compliance_gate","kind":"mechanism","kind_label":"mechanism","relation":"explains","label":"verifies egress self compliance gate","url":"rules-and-ideas.html#lattice-mechanism-mechanism-egress-self-compliance-audit-verifies-egress-self-compliance-gate"}]},{"relation":"governed by","count":2,"links":[{"id":"P-3","kind":"principle","kind_label":"principle","relation":"governed by","label":"Concentrate trust in small checkers","url":"doctrine.html#dcard-p-3"},{"id":"concept.agent_reliability_and_safety_validator_bundle","kind":"concept","kind_label":"concept","relation":"governed by","label":"agent reliability and safety validators as bounded public scope limits","url":"rules-and-ideas.html#lattice-concept-concept-agent-reliability-and-safety-validator-bundle"}]}],"paper_module_backrefs":[],"source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/egress_self_compliance_audit.md"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json"}],"visible_text_byte_count":14407,"section_titles":["Reads snippet of","Phrase membership drives","Is fast style","Purpose","Three slips a disciplined agent leaks","The excuse half prevents false flags","How it works","`evaluate_text` runs three detectors","The two-phase detector shape","Permission ask with no blocker","Error admission with no capture","Command handoff with no result record","Audit wraps the gate in fixtures","`_evaluate_case` recomputes every verdict","`build_result` pass conditions","Negative cases","Fixture ships two","Bare permission ask fires red","Command handoff with mismatched diagnostic id","Two positives guard the other direction","Prior Art Grounding","Substring policy in the lint family","Validation Result record Path","Run component over","Rerun public coverage","Pass conditions and refresh triggers","Scope limit","The strongest honest claim","Where the proof boundary stops"]},{"id":"paper_module.generated_projection_drift_runtime","title":"Generated Projection Drift Runtime","relative_href":"docs/paper-module-generated-projection-drift-runtime.html","read_href":"docs/paper-module-generated-projection-drift-runtime.html","context_href":"docs/rules-and-ideas.html#paper-module-generated-projection-drift-runtime","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"An owner-routed gate that fingerprints generated artifacts against their sources, uses a prior-clean skip cache, and runs each owner's no-write check to flag projection drift by recomputation.","summary":"Generated Projection Drift Runtime surfaces a drift gate over generated build artifacts. Each owner row pairs an artifact, its source authorities, and a no-write check command. The gate SHA-256-fingerprints source and artifact files, consults a prior-clean source-hash skip cache, requires declared artifacts to be present, and runs the owner check, marking an owner clean only on a zero return with present artifacts. Four bounded public fixtures exercise it: a passing no-write check, a genuine source-hash cache hit (the check is skipped), a planted artifact byte rejected as check_command_failed, and a missing required artifact rejected as artifact_missing. It flags drift; it does not repair files, prove content-diff equivalence for every source builder, validate the whole registry, or include launch operations.","authority_ceiling":"A clean result means the selected owner's declared no-write check passed and its required artifacts were present for the supplied root over bounded public fixtures. It is command-result record-style evidence, not a semantic proof, not a file repairer, not full-registry validation, and not launch/public sharing/source-file changes.","abstract":"An owner-routed gate that fingerprints generated artifacts against their sources, uses a prior-clean skip cache, and runs each owner's no-write check to flag projection drift by recomputation.","teleology":{"section_id":"","text":"An owner-routed gate that fingerprints generated artifacts against their sources, uses a prior-clean skip cache, and runs each owner's no-write check to flag projection drift by recomputation.","source":"capsule_summary_line"},"body_markdown":"**A generated file should be reproducible from its source, drift is when that stops being true through a hand-edit or an out-of-step builder, and this component is the gate that catches it.** A generated file is one that should be reproducible from its source: run the builder, get the same output. Drift is when that stops being true, usually because someone hand-edited the output or a builder quietly fell out of step with what it builds from. This component is the gate that catches drift.\n\n**It models each generated file as an owner with artifacts, sources, and a no-write check command, hashing both sides and marking an owner clean only when its check returns zero and its artifact exists.** It models each generated file as an owner: a set of artifact paths, the source paths they derive from, and a no-write check command that recomputes the output and compares. For each owner it hashes the source and the artifact, decides whether a recent clean record lets it skip re-running the check, confirms the required artifact is present, and otherwise runs the owner's own check. An owner is clean only when its check returns zero and its artifact exists. The gate reports drift. It does not repair the file.\n\n##### Purpose\n\n###### Recomputation is the honest unit\n\n**Timestamp comparison and a builder's own success message both mislead, so the honest unit is recomputation: hash the source, hash the artifact, and run the regeneration check so drift is observed, not assumed.** Loose drift detection fails in a predictable way. If you compare a timestamp, a touched-but-identical file looks dirty and a stale-but-recently-written file looks clean. If you trust a builder's own success message, a builder that stopped matching its source still reports success. Drift is then observed, not assumed.\n\n###### The declared boundary per owner\n\n**The declared boundary matters as much as the check: this gate proves nothing about the larger source registry or software launch, answering only whether each artifact still passes its own no-write check.** The declared boundary matters as much as the check. This gate does not establish that every builder in the larger source registry uses true content-diff semantics, and it does not decide anything about releasing software. It answers one question per owner: does this artifact still pass its own no-write check.\n\n##### How it works\n\n###### Bundle authority, component wrapper\n\n**The engine-room bundle is the drift authority and the component wrapper runs it over a bounded set of fixture cases, turning the outcome into a pass or fail.** Two files carry the mechanism. The engine-room bundle `src/microcosm_core/engine_room/generated_projection_drift_gate.py` is the drift authority. The component wrapper `src/microcosm_core/organs/generated_projection_drift_runtime.py` runs it over a bounded set of fixture cases and turns the outcome into a pass or fail.\n\n###### An owner is a `ProjectionOwner`\n\n**A `ProjectionOwner` names its `owner_id`, `artifacts`, `source_authorities`, and `check_command`, and `select_projection_owners` narrows the set by ids or changed paths so a run checks only the owners a change touches.** An owner is a `ProjectionOwner`: `owner_id`, `artifacts`, `source_authorities`, and a `check_command`, plus optional fields for repair hints and fact-authority lineage. `select_projection_owners` narrows the owner set by explicit `owner_ids` or by `changed_paths`, so a run can check only the owners a change actually touches. With no filter it selects all owners.\n\n###### `_owner_fingerprint` recomputes hashes\n\n** `_owner_fingerprint` expands each pattern to concrete files, SHA-256-hashes every body, and folds them into one `source_hash` and one `artifact_hash`, recording an unmatched pattern in `missing`.** `_owner_fingerprint` is the recomputation step. It calls `_fingerprint_patterns` once for the source paths and once for the artifact paths. That helper expands each pattern to concrete files, SHA-256-hashes every file body, sorts the entries by path, and folds them into one `source_hash` and one `artifact_hash`. A pattern that matches nothing is recorded in `missing`, which is how a missing artifact becomes visible later.\n\n###### `_source_hash_cache_hit` is the skip cache\n\n** `_source_hash_cache_hit` reuses a prior clean record only when its status, zero missing artifacts, both hashes, and check command all match, so it skips settled work but never turns a failing check into a pass.** `_source_hash_cache_hit` is the skip cache. It reads a prior clean record for this owner and returns it only when every guard holds: the record status is `clean`, its artifact-missing count is zero, its stored `source_hash` and `artifact_hash` both equal the freshly recomputed hashes, and its recorded `check_command` matches the owner's current command. If any guard fails it returns nothing and the check runs for real. The cache never turns a failing check into a pass; it only skips work that a byte-for-byte match has already settled.\n\n###### `_check_owner` assigns drift reasons\n\n** `_check_owner` fingerprints, evaluates lineage, then takes a cache hit or runs the command, collecting `check_command_failed`, `artifact_missing`, or lineage reasons, and any reason marks the owner `drift`.** `_check_owner` composes these. It fingerprints the owner, evaluates lineage, then either takes a cache hit and builds a cached clean result or runs the command through `_run_command`. It collects `status_reasons`: `check_command_failed` when the return code is non-zero, `artifact_missing` when the artifact fingerprint has any missing pattern, and a fact-authority-lineage reason when a required lineage block is missing or malformed. Any reason makes the owner `drift`; none makes it `clean`. `_run_command` first tries the small builtin commands (`builtin:pass`, `builtin:fail`, `builtin:assert-file-equals`) and otherwise runs a real subprocess under a timeout. `check_projection_drift` aggregates the per-owner rows into one status: `drift` if any owner drifted, `clean` if none did.\n\n###### The component wrapper drives fixtures\n\n** `_evaluate_case` materialises each fixture in a scratch tree via the real bundle, and `build_result` passes only with at least one positive and one negative case, all observed as expected and both negative ids present.** The component wrapper drives this over fixtures. `_evaluate_case` materialises each fixture's declared file tree into its own `TemporaryDirectory` and calls the real bundle `evaluate_case`. It does not re-implement the gate and it bakes no answers. For a negative case it requires the observed status to be not clean, the case's own expectation to be met, and the expected drift markers named in `EXPECTED_NEGATIVE_CASES` to appear in the recomputed reasons. For a positive case it requires clean plus expectation met. `build_result` runs every case and returns `status == \"pass\"` only when there is at least one positive and one negative case, all positives observed clean, all negatives observed as drift, and both expected negative ids are present. `run` writes the result, board, and validation records, and `run_generated_projection_drift_runtime_bundle` forwards to it.\n\nFunction Role `select_projection_owners` picks owners by id or by changed path `_owner_fingerprint` SHA-256 hashes source and artifact files `_source_hash_cache_hit` allows a skip only on a guarded exact hash match `_check_owner` runs the check and assigns drift reasons `check_projection_drift` aggregates owners into one clean or drift status `_evaluate_case` runs one fixture in a scratch tree via the real bundle `build_result` passes only when positives, negatives, and markers all hold Diagram of the mechanism (5 steps). Diagram source & refs\n\n```text\nflowchart TD Owner[\"Owner row source + artifact + check\"] Fingerprint[\"SHA-256 fingerprint source + artifact\"] Cache[\"Prior-clean skip cache guarded hash match\"] Check[\"No-write check return code + artifact present\"] Verdict[\"clean or drift with status reasons\"] Owner --> Fingerprint Fingerprint --> Cache Cache --> Check Check --> Verdict\n```\n\n##### Negative cases\n\n###### Two drift plants and two clean cases\n\n** `planted_byte_drift` and `missing_artifact_drift` must be recomputed to drift carrying their markers, while `clean_command_owner` and `clean_source_hash_cache_hit` fix the positive side, including a genuine skip of a would-fail check.** The fixtures plant two failures and require each to be caught by recomputation, not by a label. `planted_byte_drift` changes an artifact byte so the owner's check returns non-zero; the run demands the marker `check_command_failed`. `missing_artifact_drift` removes a required artifact so its fingerprint reports a missing pattern; the run demands the marker `artifact_missing`. Both must land as drift for the group to pass. The two positive cases fix the other side: `clean_command_owner` passes a clean check, and `clean_source_hash_cache_hit` exercises a genuine skip where the baked hashes match the recomputed ones, so a deliberately failing check is skipped rather than passed by accident.\n\n##### Prior Art Grounding\n\n###### Build-system and provenance practice\n\n**This follows ordinary build-system and provenance practice, hashing inputs and outputs and trusting a regeneration check's exit code over a timestamp, refactored publicly under the Plectis coverage contract.** This follows ordinary build-system and provenance practice: hash inputs and outputs, skip work when a content hash already settled it, and treat a regeneration check's exit code as the authority rather than a timestamp. The engine-room bundle is a public refactor of the private repository's own drift tooling. The local prior art is the Plectis paper-module coverage contract: each reader page states what source and public fixtures can be rerun, which generated views are navigation aids, and which claims stay outside the evidence boundary. No external citation is claimed for the mechanism beyond that engineering lineage.\n\n##### Validation Result record Path\n\nRun the component over its public fixtures:\n\n###### Coverage and corpus-parity validation\n\n**From the repository root, rerun the coverage contract and the corpus parity check:** From the repository root, rerun the coverage contract and the corpus parity check:\n\n```text\nPYTHONPATH=src ./repo-pytest tests/test_plectis_paper_module_coverage_contract.py -q --tb=short\nPYTHONPATH=src ./repo-python scripts/build_doctrine_projection.py --check-paper-module-corpus\n```\n\n###### What a pass means here\n\n**A pass means the two positive cases observed clean, the two negatives recomputed to drift with their markers, and both expected negative ids were present, bounded fixture-scoped evidence.** A pass means the two positive cases observed clean, the two negatives were recomputed to drift carrying their expected markers, and both expected negative ids were present. It is fixture-scoped evidence over bounded inputs.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The proof boundary and scope limit\n\n**Over these public fixtures the gate fingerprints source and artifact, honours a skip cache only on a guarded exact match, runs each no-write check, and reports drift by recomputation, repairing nothing.** The strongest claim the evidence supports: over these public fixtures the gate fingerprints source and artifact, honours a skip cache only on a guarded exact hash match, runs each owner's no-write check, and reports a drifted owner by recomputation rather than by trust. That is the proof boundary and the scope limit. It does not establish content-diff equivalence for every builder in the larger source registry, and it does not validate that whole registry. It grants no launch, public sharing, provider-call, or source-file changes, and it makes no claim of whole-system equivalence or production correctness beyond what the fixtures exercise.","body_markdown_source":"rendered_public_body_html","body_markdown_byte_count":11930,"diagram_count":1,"references":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/generated_projection_drift_runtime.md","kind":"source_link"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json","kind":"source_link"}],"grouped_relation_feeds":[{"relation":"abides by","count":1,"links":[{"id":"AX-2","kind":"axiom","kind_label":"axiom","relation":"abides by","label":"Kernelized verification","url":"doctrine.html#dcard-ax-2"}]},{"relation":"explains","count":2,"links":[{"id":"generated_projection_drift_runtime","kind":"organ","kind_label":"component","relation":"explains","label":"Generated Projection Drift Runtime","url":"component-generated-projection-drift-runtime.html"},{"id":"mechanism.generated_projection_drift_runtime.verifies_generated_projection_drift_gate","kind":"mechanism","kind_label":"mechanism","relation":"explains","label":"verifies generated projection drift gate","url":"rules-and-ideas.html#lattice-mechanism-mechanism-generated-projection-drift-runtime-verifies-generated-projection-drift-gate"}]},{"relation":"governed by","count":2,"links":[{"id":"P-3","kind":"principle","kind_label":"principle","relation":"governed by","label":"Concentrate trust in small checkers","url":"doctrine.html#dcard-p-3"},{"id":"concept.import_projection_and_drift_control_bundle","kind":"concept","kind_label":"concept","relation":"governed by","label":"source import and projection drift controls as provenance-preserving boundaries","url":"rules-and-ideas.html#lattice-concept-concept-import-projection-and-drift-control-bundle"}]}],"paper_module_backrefs":[],"source_links":[{"label":"Design note","url":"https://github.com/wcook04/plectis/blob/main/paper_modules/generated_projection_drift_runtime.md"},{"label":"Source registry","url":"https://github.com/wcook04/plectis/blob/main/core/paper_module_capsules.json"}],"visible_text_byte_count":13505,"section_titles":["A generated file should be reproducible from its source","It models each generated file as an owner with artifacts","Purpose","Recomputation is the honest unit","The declared boundary per owner","How it works","Bundle authority, component wrapper","An owner is a `ProjectionOwner`","`_owner_fingerprint` recomputes hashes","`_source_hash_cache_hit` is the skip cache","`_check_owner` assigns drift reasons","The component wrapper drives fixtures","Negative cases","Two drift plants and two clean cases","Prior Art Grounding","Build-system and provenance practice","Validation Result record Path","Coverage and corpus-parity validation","What a pass means here","Scope limit","The proof boundary and scope limit"]},{"id":"paper_module.lean_proof_search_lab_runtime","title":"Lean Proof-Search Lab Runtime","relative_href":"docs/paper-module-lean-proof-search-lab-runtime.html","read_href":"docs/paper-module-lean-proof-search-lab-runtime.html","context_href":"docs/rules-and-ideas.html#paper-module-lean-proof-search-lab-runtime","canonical_route_role":"canonical_read","context_route_role":"lattice_relationship_context","body_bearing_view":"paper_module_detail_page","body_rendered_on_reference_page":false,"route_status":"read_and_context_split","summary_line":"Gated external-tool proof-search lab that searches toy Lean theorems, checks each candidate with the installed Lean subprocess, and rejects oracle leaks, axiom taint, and problem-id memorisation by recomputation; locked and claimless when Lean is absent.","summary":"Lean Proof-Search Lab Runtime surfaces the engine-room lean_proof_search_lab bundle: a bounded and/or symbolic tactic search over public toy theorems where every candidate body is compiled by the real `lean` subprocess. It enforces three epistemic-discipline guards and self-falsifies on each: a forward oracle firewall (no proof body, oracle body, or oracle-needed premise ids may be forwarded), a #print axioms cleanliness gate (sorry-tainted candidates are rejected), and a problem-id ablation (a policy that only works because it memorised the problem id fails once the id is renamed). The component exercises one positive case (Lean closes lab_and_intro and lab_or_comm) and four planted negatives (two oracle-firewall violations, one axiom taint, one problem-id ablation failure). It is a gated external-tool witness: with Lean installed it reports tool_present_and_verified, without Lean it is terminally locked and claims no verification. Not neural theorem proving, not a solver of any open problem, no oracle-body forwarding, no launch or publishing-scope decision.","authority_ceiling":"Real-system external-tool witness over bounded public toy theorems. A pass attests only that the installed Lean subprocess closed the toy positive theorems and rejected the planted oracle-leak, axiom-taint, and problem-id-memorisation negatives on these fixtures. It does not establish any open mathematical problem, is NOT neural theorem proving, does NOT forward oracle proof bodies, is NOT an oracle/prover authority, and excludes launch, public sharing, production use, or source-file changes. When Lean is absent the component is locked and attests nothing at all.","abstract":"Gated external-tool proof-search lab that searches toy Lean theorems, checks each candidate with the installed Lean subprocess, and rejects oracle leaks, axiom taint, and problem-id memorisation by recomputation; locked and claimless when Lean is absent.","teleology":{"section_id":"","text":"Gated external-tool proof-search lab that searches toy Lean theorems, checks each candidate with the installed Lean subprocess, and rejects oracle leaks, axiom taint, and problem-id memorisation by recomputation; locked and claimless when Lean is absent.","source":"capsule_summary_line"},"body_markdown":"**It searches small Lean theorems for a proof, checks each candidate with the installed Lean toolchain, and reports itself locked when Lean is absent.** `lean_proof_search_lab_runtime` searches small Lean theorems for a proof and checks each candidate with the Lean toolchain that is actually installed. When Lean is not installed it reports itself locked and claims nothing.\n\n###### Lean compilation and clean-axiom admission\n\n**A candidate is kept only when the `lean` subprocess compiles it with a clean `#print axioms` set, and the run emits a status, tool state, and per-case breakdown.** It takes a directory of fixture cases, runs a bounded tactic search over each toy theorem, hands every candidate script to the `lean` subprocess, and reads `#print axioms` on what comes back. A candidate is kept only when Lean compiles it with a clean axiom set. The component emits a result record with a status, a tool state, and a per-case breakdown.\n\n**The third outcome is the point: with no `lean` binary the run neither passes nor fails but reports `locked` and verifies nothing.** The interesting part is the third outcome. A demo that prints a tactic and asserts success is easy to fake. This component instead answers whether a real checker ran and whether a reader can tell that it ran. When the `lean` binary is missing it does not pass and does not fail: it reports `locked` and performs no verification.\n\n##### Purpose\n\n###### Three ways a loose checker cheats\n\n**A cheap \"proof found\" badge fails three ways: a forwarded proof body, a script leaning on `sorry` or extra axioms, or a tactic picked from the problem's name.** A green badge that says \"proof found\" is cheap. The hard question is whether the proof was checked by something that could have said no, and whether the result stays honest when the checker is absent. A loose implementation fails in three ways: it forwards a ready-made proof body and calls that a search, it accepts a script that leans on `sorry` or extra axioms, or it only picks the right tactic because it recognised the problem's name. This component is built to reject all three by recomputation, and to refuse to report a pass at all when Lean cannot be run.\n\n##### How it works\n\n###### Wrapper picks the terminal state\n\n**The runtime wrapper decides which of three terminal states applies, then delegates the real search to the lab in `lean_proof_search_lab.py`.** The runtime wrapper in `organs/lean_proof_search_lab_runtime.py` decides which of three terminal states applies, then delegates the real work to the lab in `engine_room/lean_proof_search_lab.py`.\n\n###### `build_result` branches on Lean's presence\n\n** `build_result` branches on `lean_available`: absent yields `_locked_result`, present runs `evaluate_fixture_dir` and reaches `tool_present_and_verified` only when positives pass, every negative is caught, and no case is unaccounted for.** `lean_available` probes for the binary with `shutil.which(\"lean\")`. `build_result` branches on it. If Lean is absent it returns `_locked_result`: status `locked`, tool state `tool_missing`, no cases, `verification_performed` false. If Lean is present it calls `evaluate_fixture_dir`, classifies every returned case with `_classify_case`, and sets the tool state to `tool_present_and_verified` only when the positive cases pass, every expected negative case is present and rejected, and no case is left unaccounted for. Otherwise the tool state is `tool_present_but_failed`. `run` writes the result record, a compact card from `result_card`, and a validation record to the output directory.\n\n###### The lab pipeline entry point\n\n**Inside the lab, `evaluate_fixture_dir` runs `evaluate_case` on each `*.json` file, which unpacks it and calls the `evaluate_lab` pipeline.** Inside the lab, `evaluate_fixture_dir` reads each `*.json` file in the input directory and runs `evaluate_case`, which unpacks the case and calls `evaluate_lab`. `evaluate_lab` is the pipeline, and it runs in this order:\n\n###### Firewall strips forbidden proof fields\n\n** `_forward_firewall` fails the case with `oracle_firewall_violation` if any `FORBIDDEN_FORWARD_FIELDS` carries a value, then strips them so a leaked proof body cannot become forward evidence.** First `_forward_firewall` scans the raw problem rows for any of the `FORBIDDEN_FORWARD_FIELDS` (`candidate_body`, `ideal_body`, `repair_body`, `oracle_body`, `oracle_needed_premise_ids`, `provider_text`), nested paths included. If any are present with a value, the firewall fails and the whole case fails with kind `oracle_firewall_violation`. The forbidden fields are stripped before the search sees the problem, so a leaked proof body cannot become forward evidence.\n\n###### Shape-ordered search checked by Lean\n\n** `run_and_or_search` orders candidate tactics by inferred shape and hands each to `check_candidate_with_lean`, which rejects `sorry`, runs the `lean` subprocess, and selects the first candidate accepted with clean axioms.** Then for each cleaned problem `run_and_or_search` runs a bounded candidate-tactic search. `infer_target_shape` reads the theorem signature and picks a shape (`or_comm`, `and_comm`, `and_intro`, `false_elim`, `identity_intro`, `true_intro`, `exists_zero`, `equality`). `_base_candidate_actions` orders the tactic scripts with the matching shape first. Each candidate goes to `check_candidate_with_lean`, which builds a Lean source file, statically rejects any body containing `sorry`, and otherwise runs the `lean` subprocess and reads its axioms with `_classify_axioms`. The first candidate Lean accepts with a clean axiom set is selected; duplicates are pruned by goal hash.\n\n###### Ablation and axiom taint gate pass\n\n** `run_blind_policy_ablation` renames each problem id and fails on divergent selections, and any axiom taint fails the case, so a pass needs firewall, search, stable ablation, and clean axioms.** Finally `run_blind_policy_ablation` renames each problem id with `_rename_problem_id` and reruns the policy. If a policy selects a different tactic once the problem is renamed, its selection counts diverge and the ablation fails with kind `problem_id_ablation_failure`. `evaluate_lab` also counts axiom-tainted frontier rows; any taint fails the case with kind `axiom_taint_detected`. The case passes only when the firewall holds, every problem is closed by search, the ablation is stable, and no axiom taint is present.\n\nDiagram of the mechanism (8 steps). Diagram source & refs\n\nSource refs\n\nlean subprocess `check_candidate_with_lean` rename problem id `run_blind_policy_ablation`\n\n```text\nflowchart TD Probe[\"lean_available shutil.which(lean)\"] Locked[\"_locked_result tool_missing, no claim\"] Firewall[\"_forward_firewall FORBIDDEN_FORWARD_FIELDS\"] Search[\"run_and_or_search candidate tactic scripts\"] Lean[\"check_candidate_with_lean lean subprocess\"] Axioms[\"_classify_axioms #print axioms clean\"] Ablation[\"run_blind_policy_ablation rename problem id\"] Verdict[\"build_result pass or fail\"] Probe -->|absent| Locked Probe -->|present| Firewall Firewall --> Search Search --> Lean Lean --> Axioms Axioms --> Ablation Ablation --> Verdict\n```\n\n###### The bundle entry never spawns Lean\n\n** `run_lean_proof_search_lab_runtime_bundle` only checks that an exported bundle declares its required files, never spawning Lean, so live verification happens only through `run`.** The bundle entry `run_lean_proof_search_lab_runtime_bundle` is separate. It checks that an exported bundle declares its required files and never spawns Lean, so its tool state is `standalone_contract_not_executed`. Live Lean verification only happens through `run`.\n\n##### Negative cases\n\n###### Four planted cheats and their guards\n\n** `EXPECTED_NEGATIVE_CASES` names four planted cheats, each with the guard that must catch it by recomputation rather than a hard-coded expected answer.** The wrapper names four planted cheats in `EXPECTED_NEGATIVE_CASES`, each with the guard that must catch it. Every one must be rejected by recomputation, not by a hard-coded expected answer.\n\n###### Which guard catches each cheat\n\n**A leaked body fails the firewall, a `sorry` script fails the axiom audit, and a name-reading policy fails the ablation; a verified run needs the positive and all four caught.** `oracle_field_negative` and `nested_oracle_field_negative` plant a forbidden proof body at the top level and nested inside the payload. Both must fail with `oracle_firewall_violation`. `sorry_axiom_negative` submits a script that closes the goal with `sorry`, and must fail with `axiom_taint_detected`. `memorized_policy_negative` submits a policy that only picks the right tactic because it read the problem's name, and must fail with `problem_id_ablation_failure` once the id is renamed. A run reports `tool_present_and_verified` only when the honest positive case passes and all four of these are caught.\n\n##### Prior Art Grounding\n\n###### Prover, renaming, and leakage discipline\n\n**Sign-off rests on the external Lean 4 prover and its `#print axioms` audit, the ablation borrows renaming discipline from memorisation testing, and the firewall borrows leakage discipline.** The engineering lineage is standard proof-checking practice rather than a single cited result. The component checks candidates with the Lean 4 theorem prover as an external subprocess and audits them with Lean's own `#print axioms` command, so sign-off depends on the prover and not on this code. The problem-id ablation follows the held-out and renaming discipline used to detect memorisation in machine-learning evaluation: if renaming the problem changes the answer, the policy was reading the label. The oracle firewall follows the same idea applied to data leakage, refusing forward fields that would let a ready-made answer stand in for a search.\n\n##### Validation Result record Path\n\n###### Running the proof-search fixtures\n\n**Run the component over its fixtures, then rerun the corpus checks that read this page:** Run the component over its fixtures, then rerun the corpus checks that read this page:\n\n###### What a pass and a lock mean\n\n**A pass means Lean was installed, accepted the toy theorems with clean axioms, and the three guards each rejected their planted defect; no Lean on the path reports `locked`.** A pass means Lean was installed, it accepted the toy theorems with clean axioms, and the firewall, axiom audit, and ablation each rejected their planted defect. If Lean is not on the path the run reports `locked` and asserts nothing.\n\n##### Scope boundary\n\n###### Scope limit\n\n###### The strongest claim and its boundary\n\n**With Lean installed, the toolchain accepted bounded toy proofs and the three guards each rejected a planted cheat; it proves no open results and is no general correctness oracle.** The strongest claim the evidence supports: on a machine with Lean installed, the installed toolchain accepted a set of bounded toy proofs, and the oracle firewall, the `#print axioms` gate, and the problem-id ablation each rejected a planted cheat by recomputation. That is the proof boundary. It does not establish open mathematical results, it is not neural or frontier-scale theorem proving, and it is not a general formal-result correctness oracle. The scope limit stops there: the component does not use external model services, does not change source files, and excludes launch or public sharing. 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It covers the declared preconditions and dependency gaps that are actually checked; uncovered partial functions remain a gap rather than a guaranteed refusal.","source_field":"doctrine_enrichment.records[].does_not_prove"},"reader_ladder":{"plain":"A calculation may return a real number only when the conditions it needs actually hold. When they don't, it must hand back a clearly labelled refusal that names the reason and the evidence, never a quiet number or a blank that the next step reads as an answer.","analogy":{"text":"A cash machine that can't complete a withdrawal prints a stated decline reason and gives you nothing, rather than spitting out some arbitrary amount of cash that you'd carry off as if it were correct.","maps":[{"doctrine":"the precondition that must hold for a real result","analogy":"the conditions the machine needs to launch cash, like a valid balance and a reachable account"},{"doctrine":"Ok carrying a value when the precondition holds","analogy":"the machine dispensing the requested amount when everything checks out"},{"doctrine":"a typed refusal carrying a reason and evidence","analogy":"the printed decline notice naming why it stopped"},{"doctrine":"never a silent number or NaN leaking downstream","analogy":"never handing out an arbitrary stack of cash that you'd treat as a real withdrawal"}],"boundary":"The cash machine shows that a failed operation must decline with a reason instead of emitting a stray figure; it does not establish that every operation in the system declines this way, and untested failure paths can still leak a bad value rather than a clean refusal."},"why_it_matters":"It means a missing input or an out-of-range condition shows up as a visible, routable gap instead of a plausible-looking figure that a later page prints as a genuine result.","potential_misread":"That a number coming out the end means the conditions were fine, so a calculation that quietly produced a value can be treated as a valid answer without checking it had grounds to run.","source_field":"doctrine_enrichment.records[].reader_ladder"},"deep_reading":{"text":"AX-7 refuses the most common way numerical code lies: producing a plausible-looking value where its preconditions do not actually hold. Microcosm requires a partial computation g to be totalized into a Result, so it returns Ok(y) only when a declared precondition Pre(x) is met, and otherwise returns a Refusal that names the reason and carries the evidence behind it. The refusal is the load-bearing half, because a number with no precondition check and no failure path is indistinguishable from a number that happens to be wrong; a NaN, an out-of-range horizon, or a missing dependency must surface as a typed object the caller can route, not leak downstream as a figure that reads as a result. This connects directly to the evidence model: a refusal with a reason is admissible information, whereas a silent number is a claim with no basis beneath it, so AX-7 is what lets the upstream admissibility floor see a gap instead of inheriting a fabricated value.","source_field":"doctrine_enrichment.records[].deep"},"formal_statement":{"latex":"\\forall\\, x.\\ \\big(\\mathrm{Pre}(x) \\Rightarrow \\exists\\, y.\\ g(x) = \\mathrm{Ok}(y)\\big) \\;\\wedge\\; \\big(\\neg\\,\\mathrm{Pre}(x) \\Rightarrow \\exists\\, r, e.\\ g(x) = \\mathrm{Refusal}(r, e)\\big)","reads":"For every input: when the declared precondition holds, the computation returns Ok carrying some value; when it does not hold, the computation returns a typed Refusal carrying a reason and the evidence behind it. Both branches are total; there is no third, silent outcome.","symbols":[{"sym":"g","meaning":"a partial computation, totalized into a Result"},{"sym":"x","meaning":"an input to the computation"},{"sym":"\\mathrm{Pre}(x)","meaning":"the declared precondition on input x"},{"sym":"\\mathrm{Ok}(y)","meaning":"a successful result carrying value y"},{"sym":"\\mathrm{Refusal}(r, e)","meaning":"a typed refusal carrying a reason r and evidence e"}],"source_field":"doctrine_enrichment.records[].formal"},"relationship_triad":{"governs":"What a partial computation is allowed to return when its precondition does not hold.","requires":"A declared precondition for the Ok path and a typed Refusal carrying a reason and evidence on every other path, including missing dependencies and undefined values.","refuses":"An inadmissible number or silent NaN returned where no precondition was checked.","source_field":"doctrine_enrichment.records[].governs|requires|refuses"},"example":{"text":"The finance 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It is the precise statement of how far one verified result reaches, leaving everything outside that line unclaimed rather than quietly assumed.","visible_text_byte_count":1601,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::scope_limit","source_hash":"sha256:ab74c9b3006a5eebd8ec2cdb31943ed39db5c8947c04bc25c4d30401f8a5baa3","relationships":[{"verb":"bounds","target":"component","gloss":"Each component carries a scope limit."},{"verb":"protects","target":"plectis","gloss":"Public Plectis copy stays honest by keeping scope limits close to evidence."},{"verb":"requires","target":"living_system_posture","gloss":"A scope limit must be read against current public records, not stale memory."}],"source_refs":[{"ref":"tools/meta/dissemination/build_microcosm_public_site.py","role":"public_projection_builder","gloss":"Builder emits scope-limit copy, claim-ceiling index, and canonical question routes."},{"ref":"microcosm-substrate/core/organ_registry.json","role":"compatibility_source","gloss":"Component records carry per-object boundary fields consumed by public projections."}],"evidence_commands":[{"command":"./repo-python tools/meta/dissemination/build_microcosm_public_site.py --check --validate","reason":"Check that generated public packets still expose claim boundaries safely."},{"command":"./repo-python kernel.py --term scope_limit --term-band context","reason":"Open the public scope-limit vocabulary row."}],"used_in":{"total_occurrences":1593,"page_count":208,"top_page":"rules-and-ideas.html","top_pages":[{"page":"rules-and-ideas.html","count":264},{"page":"paper-modules.html","count":163},{"page":"doctrine.html","count":101},{"page":"components.html","count":90},{"page":"reference.html","count":58},{"page":"area-agent-reliability.html","count":22}]}},{"schema":"plectis_public_glossary_term_v0","position":4,"id":"evidence_class","label":"evidence class","preferred_label":"evidence class","alternative_labels":["evidence kind","evidence profile"],"hidden_search_labels":["evidence rank","proof consumer class"],"term_kind":"evidence","meaning_scope_class":"ai_workflow_coined_term","meaning_scope_label":"System-coined term","definition_profile":"governed_system_term","meaning_boundary":"An evidence class fixes what kind of support a record provides and which inferences it licenses; it is not a confidence score or a declaration that the underlying claim is true.","meaning_contrasts":[],"status":"active","disposition":"public_evidence_type_label","anchor":"glossary-evidence-class","relative_href":"docs/glossary.html#glossary-evidence-class","public_phrase":"type of support","short_definition":"An evidence class names the kind of support behind a result\u2014such as a source-linked row, fixture pass, local tool run, or review packet\u2014without implying that every claim made from that support is true.","reader_preview":"An evidence class names the kind of support behind a result\u2014such as a source-linked row, fixture pass, local tool run, or review packet\u2014without implying that every claim made from that support is true.","reader_preview_basis":"definition","card_definition":"When a card or result on this site is marked as backed, its evidence class is the category of that backing. The classes distinguish, for example, a result whose support is a link to a source row, one whose support is a fixture that passed, one whose support is a tool that ran locally, and one whose support is a recorded review. Each class carries its own ceiling on what may be claimed from it, so two results can both be supported while resting on very different kinds of evidence.","reader_rule":"Check which evidence class a result carries, and read it as a statement about the kind of support, not the strength of every conclusion. A class sits next to a scope limit that names what that support does not establish.","deep_definition":"The governing list of classes is kept in the source slice and the site builder's claim-ceiling projection, so a class is a fixed vocabulary entry rather than a free-form label, and it keeps a local fixture or a generated record from being read as wider authority.","scope_note":"Evidence class covers the typed backing attached to public results, cards, and packets. A reader meets it on result and component pages where a claim is shown alongside the category of support that stands behind it, and most directly on the evidence page that sets out what a pass proves.","example":"On the evidence page, a result marked as fixture-backed carries the fixture-pass evidence class: its support is a test fixture that ran and passed, which is a different and narrower kind of backing than a row linked directly to a source file.","example_route":{"href":"evidence.html","label":"What the pass proves (evidence)","gloss":"see results sorted by the kind of support behind them"},"non_example":"An evidence class is not a quality score or a rank that makes one result better than another, and a high or strong-sounding class is not proof of launch readiness, security, or whole-system correctness. It names the kind of support, not a verdict.","visible_text_byte_count":1791,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::evidence_class","source_hash":"sha256:ab74c9b3006a5eebd8ec2cdb31943ed39db5c8947c04bc25c4d30401f8a5baa3","relationships":[{"verb":"bounds","target":"scope_limit","gloss":"Evidence type must be read with the claim boundary."},{"verb":"classifies","target":"record","gloss":"A record can carry an evidence class describing the support it contains."},{"verb":"protects","target":"plectis","gloss":"Plectis public copy uses evidence classes to avoid overclaiming."}],"source_refs":[{"ref":"microcosm-substrate/core/organ_evidence_classes.json","role":"source_registry","gloss":"Source registry for component evidence classes."},{"ref":"tools/meta/dissemination/build_microcosm_public_site.py","role":"public_projection_builder","gloss":"Builds evidence labels, ranks, profiles, and claim-ceiling routes into the public site."}],"evidence_commands":[{"command":"jq '.' microcosm-substrate/core/organ_evidence_classes.json","reason":"Read the source evidence-class registry."},{"command":"./repo-python tools/meta/dissemination/build_microcosm_public_site.py --check --validate","reason":"Verify the public projection still carries bounded evidence classes."}],"used_in":{"total_occurrences":112,"page_count":104,"top_page":"index.html","top_pages":[{"page":"index.html","count":3},{"page":"evidence.html","count":2},{"page":"paper-module-agentic-vulnerability-discovery-patch-proof-replay.html","count":2},{"page":"paper-module-batch12-release-claim-language-gate.html","count":2},{"page":"paper-module-cold-reader-route-map.html","count":2},{"page":"rules-and-ideas.html","count":2}]}},{"schema":"plectis_public_glossary_term_v0","position":5,"id":"record","label":"record","preferred_label":"record","alternative_labels":["result record","verification record"],"hidden_search_labels":["receipt","audit record","generated record"],"term_kind":"artifact","meaning_scope_class":"ai_workflow_specialization","meaning_scope_label":"System-local specialization","definition_profile":"governed_system_term","meaning_boundary":"Record means a governed stored result or receipt in this system. It does not by itself carry legal-record status, evidentiary sufficiency, or truth authority.","meaning_contrasts":[],"status":"active","disposition":"public_artifact_evidence_label","anchor":"glossary-record","relative_href":"docs/glossary.html#glossary-record","public_phrase":"inspectable artifact row","short_definition":"A record is a concrete, durable, independently inspectable artifact\u2014such as a result row, receipt, generated output, review packet, or verification entry\u2014that preserves what was checked, produced, or routed.","reader_preview":"A record is a concrete, durable, independently inspectable artifact\u2014such as a result row, receipt, generated output, review packet, or verification entry\u2014that preserves what was checked, produced, or routed.","reader_preview_basis":"definition","card_definition":"A record is the inspectable object a page points to when it refers to something more solid than a sentence. Each record carries what produced it and where its authority stops, and it sits below its owning source rather than replacing it. Records are how narrative pages connect to source authority: a reader can open the actual result row, receipt, ledger, or generated projection behind a statement.","reader_rule":"When a page points you to a record, open it and read what it actually preserves and where its scope ends. A record holds a checkable object; it does not make a claim true merely by existing.","deep_definition":"In this system a record may be a source row, a generated projection, a receipt, a ledger entry, or a public packet. Each is durable and reopenable, and each ranks below the source that produced it.","scope_note":"The term covers concrete source and generated artifacts across the site: result rows, receipts, generated JSON outputs, review packets, ledgers, and verification entries. A reader meets records wherever a page links out to something inspectable rather than describing it in prose.","example":"The doctrine card \"Generated-result record source inversion\" concerns exactly this: a generated record, such as a result file emitted by a tool, must stay subordinate to the source that produced it rather than being treated as the authority itself.","example_route":{"href":"doctrine.html#dcard-ap-13","label":"Generated-result record source inversion","gloss":"open a rule about how generated records relate to their source"},"non_example":"A record is not a paragraph of explanatory text and not a claim that becomes true by being written down. It is a concrete artifact, row, or generated output that someone can reopen and check.","visible_text_byte_count":1656,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::record","source_hash":"sha256:ab74c9b3006a5eebd8ec2cdb31943ed39db5c8947c04bc25c4d30401f8a5baa3","relationships":[{"verb":"can_carry","target":"evidence_class","gloss":"A record can name the support type behind it."},{"verb":"bounded_by","target":"scope_limit","gloss":"A record's claim boundary must travel with it."},{"verb":"below","target":"source_authority","gloss":"A generated record is not stronger than its source authority."}],"source_refs":[{"ref":"sites/microcosm/content-graph.json","role":"generated_public_record_index","gloss":"Generated public index of components, paper modules, glossary terms, and evidence profiles."},{"ref":"tools/meta/dissemination/build_microcosm_public_site.py","role":"public_projection_builder","gloss":"Builds public records from source rows and generated projections."}],"evidence_commands":[{"command":"jq '.components[0]' sites/microcosm/content-graph.json","reason":"Inspect a generated public record shape."},{"command":"./repo-python kernel.py --term record --term-band context","reason":"Read the record vocabulary row."}],"used_in":{"total_occurrences":3681,"page_count":213,"top_page":"rules-and-ideas.html","top_pages":[{"page":"rules-and-ideas.html","count":492},{"page":"paper-modules.html","count":191},{"page":"doctrine.html","count":167},{"page":"reference.html","count":121},{"page":"paper-module-pattern-assimilation.html","count":61},{"page":"paper-module-world-model-projection-drift-control-room.html","count":60}]}},{"schema":"plectis_public_glossary_term_v0","position":6,"id":"substrate","label":"substrate","preferred_label":"substrate","alternative_labels":["source substrate","workflow substrate"],"hidden_search_labels":["live substrate","runtime substrate"],"term_kind":"substrate","meaning_scope_class":"ai_workflow_specialization","meaning_scope_label":"System-local specialization","definition_profile":"governed_system_term","meaning_boundary":"Substrate means the live files, records, or runtime state on which this system operates. It is a local systems metaphor, not a biological, chemical, or philosophical substrate claim.","meaning_contrasts":[],"status":"active","disposition":"deep_public_technical_term","anchor":"glossary-substrate","relative_href":"docs/glossary.html#glossary-substrate","public_phrase":"underlying working base","short_definition":"A substrate is the underlying body of files, routes, recorded state, and tools that the system operates on, distinct from the pages and summaries projected over that source material.","reader_preview":"A substrate is the underlying body of files, routes, recorded state, and tools that the system operates on, distinct from the pages and summaries projected over that source material.","reader_preview_basis":"definition","card_definition":"The substrate is everything the public pages are generated from: the source files, the standards, the recorded state, the validators, and the tools that act on them. It is the concrete, inspectable layer beneath any summary, route, or evidence view. A generated page can describe or point into the substrate, but it does not own the source records, package identifiers, or state transitions that live there.","reader_rule":"When something is called the substrate, treat it as the lower source layer that a page is reporting on, and look to it rather than the page when you need the actual file or record.","deep_definition":"The substrate is what keeps a generated page honest about its own status: the page makes a map, digest, or evidence view, while the substrate remains the authority that owns the source records, validators, identifiers, and state changes underneath.","scope_note":"The term covers the source layer of the whole system: files, standards, ledgers, recorded state, and tooling. A reader meets it wherever the pages distinguish the generated public view from the source it was derived from.","example":"The Source and license page is the entry point into this substrate, opening onto the source files and records that the rest of the site is generated from.","example_route":{"href":"source.html","label":"Source & license","gloss":"open the source layer the pages are generated from"},"non_example":"A substrate is not a vague word for the whole project, and not a mystical layer behind it. It is the specific, inspectable set of files and records that the pages are generated from.","visible_text_byte_count":1491,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::substrate","source_hash":"sha256:ab74c9b3006a5eebd8ec2cdb31943ed39db5c8947c04bc25c4d30401f8a5baa3","relationships":[{"verb":"contains","target":"source_authority","gloss":"Source authority rests in substrate-owned records and tools."},{"verb":"projected_by","target":"plectis","gloss":"Plectis is a public map over source substrate."},{"verb":"contrasts_with","target":"record","gloss":"A record is one inspectable artifact inside or generated from substrate."}],"source_refs":[{"ref":"codex/doctrine/paper_modules/microcosm_substrate.md","role":"paper_module","gloss":"Doctrine roof for the source-open substrate concept."},{"ref":"AGENTS.override.md","role":"agent_seed","gloss":"Type A operating seed that distinguishes live substrate from chat."}],"evidence_commands":[{"command":"./repo-python kernel.py --paper-module microcosm_substrate","reason":"Open the substrate doctrine surface."},{"command":"./repo-python kernel.py --term substrate --term-band context","reason":"Read the substrate vocabulary row."}],"used_in":{"total_occurrences":87,"page_count":73,"top_page":"paper-module-indirect-prompt-injection-information-flow-policy-replay.html","top_pages":[{"page":"paper-module-indirect-prompt-injection-information-flow-policy-replay.html","count":3},{"page":"paper-module-materials-chemistry-closed-loop-lab-safety-replay.html","count":3},{"page":"paper-module-agent-benchmark-integrity-anti-gaming-replay.html","count":2},{"page":"paper-module-agent-memory-temporal-conflict-replay.html","count":2},{"page":"paper-module-agent-sandbox-policy-escape-replay.html","count":2},{"page":"paper-module-certificate-kernel-execution-lab.html","count":2}]}},{"schema":"plectis_public_glossary_term_v0","position":7,"id":"source_authority","label":"source authority","preferred_label":"source authority","alternative_labels":["source of record","governing source"],"hidden_search_labels":["source owner","authority posture"],"term_kind":"governance","meaning_scope_class":"ai_workflow_specialization","meaning_scope_label":"System-local specialization","definition_profile":"governed_system_term","meaning_boundary":"Source authority means the local source-of-record relationship for a claim or artifact. It does not imply institutional, legal, academic, or moral authority.","meaning_contrasts":[],"status":"active","disposition":"public_authority_boundary_label","anchor":"glossary-source-authority","relative_href":"docs/glossary.html#glossary-source-authority","public_phrase":"where truth is owned","short_definition":"Source authority is the artifact, standard, tool, or ledger that owns a fact more strongly than anything generated from it: the place where the truth is held, beneath any page, summary, or AI answer that merely reports it.","reader_preview":"Source authority is the artifact, standard, tool, or ledger that owns a fact more strongly than anything generated from it: the place where the truth is held, beneath any page, summary, or AI answer that merely reports it.","reader_preview_basis":"definition","card_definition":"Every generated surface on this site, a page, a digest, an AI summary, is a projection of something lower down. The source authority is that lower thing: a source file, a standard, a validator, a ledger, or a generated sidecar with an ownership contract. When prose, HTML, or a summary may have drifted, the source authority is where the actual claim lives and where a fact is settled. A projection can route to its source authority, but summarising a fact does not make the summary the owner of it.","reader_rule":"When a page makes a claim, the source authority is the artifact it stands below; treat that artifact, not the page, as the place the fact is owned.","deep_definition":"This is the rule that keeps generated public surfaces honest: they are useful because they route, compress, and display, and they stay accurate by naming the lower artifact or tool that owns each claim and refusing to assert anything beyond it.","scope_note":"Source authority appears wherever a public page describes itself as a map, projection, record, or digest, and points to the underlying file, standard, validator, or ledger that holds the fact. It marks the boundary between a generated surface and the material it is drawn from.","example":"The vocabulary registry standard is a source authority: a glossary row on this site can cite it for the meaning of a term while the row itself remains a projection that owns nothing.","example_route":{"href":"doctrine.html#dcard-ap-13","label":"Generated-result record source inversion","gloss":"open the anti-pattern that fires when a generated record is mistaken for the source that owns the fact"},"non_example":"Source authority is not a generated page, a digest, or an AI answer. Those are projections; they hold authority only when a governing standard explicitly names them as the owner of the fact.","visible_text_byte_count":1695,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::source_authority","source_hash":"sha256:ab74c9b3006a5eebd8ec2cdb31943ed39db5c8947c04bc25c4d30401f8a5baa3","relationships":[{"verb":"governs","target":"record","gloss":"Records point to the authority that owns their claim."},{"verb":"bounds","target":"plectis","gloss":"Plectis public pages stay below source authority."},{"verb":"requires","target":"scope_limit","gloss":"Authority and scope must be named together for public claims."}],"source_refs":[{"ref":"codex/standards/std_system_term.json","role":"standard","gloss":"Defines system vocabulary rows as projections below cited source authority."},{"ref":"tools/meta/dissemination/build_microcosm_public_site.py","role":"public_projection_builder","gloss":"Emits source refs and authority boundaries into public packets."}],"evidence_commands":[{"command":"jq '.authority_model' codex/standards/std_system_term.json","reason":"Read the authority model for vocabulary projections."},{"command":"./repo-python tools/meta/dissemination/build_microcosm_public_site.py --check --validate","reason":"Verify generated public pages preserve source authority boundaries."}],"used_in":{"total_occurrences":315,"page_count":86,"top_page":"rules-and-ideas.html","top_pages":[{"page":"rules-and-ideas.html","count":142},{"page":"components.html","count":16},{"page":"reference.html","count":16},{"page":"paper-modules.html","count":13},{"page":"area-import-drift.html","count":8},{"page":"paper-module-routing-anti-patterns-registry.html","count":8}]}},{"schema":"plectis_public_glossary_term_v0","position":8,"id":"organ","label":"organ (source identifier)","preferred_label":"organ (source identifier)","alternative_labels":["source organ","organ id"],"hidden_search_labels":["runtime organ","organ registry"],"term_kind":"compatibility_identifier","meaning_scope_class":"compatibility_identifier","meaning_scope_label":"Compatibility identifier","definition_profile":"governed_system_term","meaning_boundary":"Organ is a retained source/storage compatibility identifier. It is not the preferred public object name, a biological claim, or proof that a storage rename has completed.","meaning_contrasts":[],"status":"active_compatibility","disposition":"source_identifier_exposed_only_when_needed","anchor":"glossary-organ","relative_href":"docs/glossary.html#glossary-organ","public_phrase":"source-level runtime unit","short_definition":"An organ is the source-code name for one self-contained unit of working machinery with a defined job, explicit inputs and outputs, and a recorded claim limit, presented as a component on public pages.","reader_preview":"An organ is the source-code name for one self-contained unit of working machinery with a defined job, explicit inputs and outputs, and a recorded claim limit, presented as a component on public pages.","reader_preview_basis":"definition","card_definition":"The system is built as a set of these units rather than one large program. Each organ does one job, such as mapping a route, running a check, importing source material, or generating a page, and can be opened and inspected on its own. Each one carries its own boundary and a record of what it is allowed to assert, so a unit's reach is fixed and visible rather than implied.","reader_rule":"When a source path or identifier names an organ, read it as one bounded unit doing a single job; the same unit appears on the public pages as a component.","deep_definition":"Organ is the source-level and runtime name for these units, kept stable in file paths, identifiers, and compatibility records; component is the same thing under its public label.","scope_note":"The term covers every bounded unit the system is assembled from. A reader meets it in source paths and identifiers such as organ_registry.json, and meets the same units as components throughout the public pages, where each can be opened on its own.","example":"A concrete organ is the Source Projection Import Protocol, a unit that brings source material across into the public slice; it is listed and openable among the components.","example_route":{"href":"components.html#component-macro_projection_import_protocol","label":"Source Projection Import Protocol","gloss":"browse the units, each openable on its own"},"non_example":"An organ is not a biological metaphor and not a marketing feature. It is a unit of running code with a stated boundary.","visible_text_byte_count":1458,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::organ","source_hash":"sha256:ab74c9b3006a5eebd8ec2cdb31943ed39db5c8947c04bc25c4d30401f8a5baa3","relationships":[{"verb":"translated_publicly_as","target":"component","gloss":"Component is the preferred reader-facing word."},{"verb":"belongs_to","target":"source_slice","gloss":"Organ ids may appear in the public source slice."},{"verb":"bounded_by","target":"scope_limit","gloss":"Source-level organ records still need claim boundaries."}],"source_refs":[{"ref":"microcosm-substrate/core/organ_registry.json","role":"source_registry","gloss":"Source-level registry that still uses organ ids."},{"ref":"microcosm-substrate/core/organ_atlas.json","role":"source_atlas","gloss":"Source atlas consumed by public component projections."}],"evidence_commands":[{"command":"jq 'keys[0:5]' microcosm-substrate/core/organ_registry.json","reason":"Inspect source-level organ registry ids."},{"command":"./repo-python kernel.py --term organ --term-band context","reason":"Read the organ compatibility term."}],"used_in":{"total_occurrences":486,"page_count":135,"top_page":"paper-modules.html","top_pages":[{"page":"paper-modules.html","count":88},{"page":"paper-module-formal-math-premise-retrieval.html","count":10},{"page":"paper-module-formal-math-readiness-gate.html","count":10},{"page":"paper-module-navigation-hologram-route-plane.html","count":9},{"page":"paper-module-agent-route-observability-runtime.html","count":7},{"page":"paper-module-agent-sabotage-scheming-monitor-replay.html","count":5}]}},{"schema":"plectis_public_glossary_term_v0","position":9,"id":"witness","label":"check result","preferred_label":"check result","alternative_labels":["verification record","witness"],"hidden_search_labels":["public witness","proof witness","witness route","witness surface"],"term_kind":"evidence","meaning_scope_class":"ai_workflow_specialization","meaning_scope_label":"System-local specialization","definition_profile":"governed_system_term","meaning_boundary":"Witness means a source-linked check result or verification record in this system. It is not a legal witness and does not by itself establish proof beyond its stated evidence class.","meaning_contrasts":[],"status":"active_compatibility","disposition":"prefer_check_result_publicly","anchor":"glossary-witness","relative_href":"docs/glossary.html#glossary-witness","public_phrase":"source-linked check result","short_definition":"A witness is the durable record that one specific check ran and what it found, shown in public copy as a check result or verification record.","reader_preview":"A witness is the durable record that one specific check ran and what it found, shown in public copy as a check result or verification record.","reader_preview_basis":"definition","card_definition":"When a statement on this site claims it is supported, the witness is the concrete artifact behind that support: the output of a validator, a test, or a source-row check, naming exactly what was examined and what held. It is evidence for one bounded statement, not a verdict on the whole system.","reader_rule":"Read what a witness actually checked before relying on it. It vouches for that one claim and nothing wider.","deep_definition":"Each witness carries its scope limit with it: the command or record that produced it, the exact thing it examined, and the boundary of what that result does and does not cover.","scope_note":"Witness covers the evidence layer of the system: the checkable records that sit behind supported claims on components, doctrine cards, and the evidence page, where the public label is check result or verification record.","example":"On the evidence page, the record of what a passing check examined, such as a source-row check or a validator run, is a witness: it names the command and the rows it inspected and the one claim that result supports.","example_route":{"href":"evidence.html","label":"What the pass proves (evidence)","gloss":"see the records that stand behind a passing check"},"non_example":"A witness is not legal testimony, a mathematical proof on its own, or a guarantee that the whole system is correct. It records that one named check passed.","visible_text_byte_count":1393,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::witness","source_hash":"sha256:ab74c9b3006a5eebd8ec2cdb31943ed39db5c8947c04bc25c4d30401f8a5baa3","relationships":[{"verb":"publicly_prefer","target":"record","gloss":"Verification record is the safer reader-facing object."},{"verb":"uses","target":"evidence_class","gloss":"A check result can carry an evidence class."},{"verb":"bounded_by","target":"scope_limit","gloss":"A witness/check result must not exceed its claim boundary."}],"source_refs":[{"ref":"microcosm-substrate/core/axiom_organ_routing.json","role":"source_routing","gloss":"Axiom support routes use witness-like evidence vocabulary."},{"ref":"tools/meta/dissemination/build_microcosm_public_site.py","role":"public_projection_builder","gloss":"Scrubs public copy toward check results and verification records."}],"evidence_commands":[{"command":"jq '.support_cases[0]?' microcosm-substrate/receipts/first_wave/axiom_support_cover/axiom_support_cover_result.json","reason":"Inspect a generated support record when present."},{"command":"./repo-python kernel.py --term witness --term-band context","reason":"Read the witness compatibility term."}],"used_in":{"total_occurrences":462,"page_count":59,"top_page":"rules-and-ideas.html","top_pages":[{"page":"rules-and-ideas.html","count":71},{"page":"reference.html","count":68},{"page":"doctrine.html","count":60},{"page":"paper-module-formal-math-lean-proof-witness.html","count":26},{"page":"paper-modules.html","count":22},{"page":"paper-module-agentic-vulnerability-discovery-patch-proof-replay.html","count":20}]}},{"schema":"plectis_public_glossary_term_v0","position":10,"id":"navigable","label":"navigable","preferred_label":"navigable","alternative_labels":["routable","discoverable"],"hidden_search_labels":["routeable","navigability"],"term_kind":"navigation","meaning_scope_class":"ai_workflow_specialization","meaning_scope_label":"System-local specialization","definition_profile":"governed_system_term","meaning_boundary":"Navigable means a cold reader can discover, select, open, and verify an artifact through supported routes; it is stronger than merely being searchable or linked somewhere.","meaning_contrasts":[],"status":"active","disposition":"public_route_property_label","anchor":"glossary-navigable","relative_href":"docs/glossary.html#glossary-navigable","public_phrase":"has a route","short_definition":"A navigable term, record, or page carries at least one concrete anchor, source reference, relationship link, command, or pointer that takes a reader from its summary to the underlying object or evidence.","reader_preview":"A navigable term, record, or page carries at least one concrete anchor, source reference, relationship link, command, or pointer that takes a reader from its summary to the underlying object or evidence.","reader_preview_basis":"definition","card_definition":"Across Plectis, an object is treated as navigable when it can be opened, cross-referenced, checked, and traced back to its source instead of only being named. A navigable component, for example, links to the source file that defines it, the standards it answers to, and the check that confirms it. 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When a task surfaces something that would otherwise go stale or be hard for the next person to find, the extra work is to make that one result durable: name the boundary, do the smallest improvement that prevents rediscovery, run a focused check, and leave evidence. 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Public access, repository access, and tool access are separate permissions.","reader_rule":"Treat access as a reachability condition and check authority, evidence, and license separately.","deep_definition":"A route can exist without being accessible because it needs an account, network, local checkout, or permission that the current reader does not have.","scope_note":"Access covers the stated reader, object, route, and permission conditions.","example":"A public glossary page has anonymous web access, while a private working file does not.","example_route":null,"non_example":"Being able to open a file does not prove that its claims are correct.","visible_text_byte_count":739,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::access","source_hash":"sha256:ab74c9b3006a5eebd8ec2cdb31943ed39db5c8947c04bc25c4d30401f8a5baa3","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":467,"page_count":118,"top_page":"paper-modules.html","top_pages":[{"page":"paper-modules.html","count":88},{"page":"rules-and-ideas.html","count":44},{"page":"components.html","count":31},{"page":"open-questions.html","count":27},{"page":"doctrine.html","count":18},{"page":"reference.html","count":11}]}},{"schema":"plectis_public_glossary_term_v0","position":14,"id":"accountability_layer","label":"accountability layer","preferred_label":"accountability layer","alternative_labels":["evidence accountability layer","claim accountability layer"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"ai_workflow_specialization","meaning_scope_label":"System-local specialization","definition_profile":"governed_system_term","meaning_boundary":"Here, accountability layer names the records and routes around a mechanism that make one claim inspectable. It is not the only accepted use of the phrase, and it does not imply that the wider system is public or independently verified.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-accountability-layer","relative_href":"docs/glossary.html#glossary-accountability-layer","public_phrase":"records that bound claims","short_definition":"The accountability layer is the set of evidence records, result records, source links, and scope limits that lets a reader trace what a mechanism ran and where the resulting claim stops.","reader_preview":"The accountability layer is the set of evidence records, result records, source links, and scope limits that lets a reader trace what a mechanism ran and where the resulting claim stops.","reader_preview_basis":"definition","card_definition":"This layer sits around a mechanism rather than replacing it. It connects the visible claim to the recorded run or replay, the source that owns it, the kind of evidence attached to it, and the boundary beyond which that evidence says nothing.","reader_rule":"Use the accountability layer to follow a claim back to evidence and source; do not treat the page that presents those links as the authority itself.","deep_definition":"The layer makes a claim inspectable by preserving provenance and limits together. A record without a source route or a scope limit is not enough to establish accountable public evidence.","scope_note":"The term covers the public records and routes that make one mechanism or claim traceable. 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They are not personality types, intelligence rankings, capability tiers, or a universal taxonomy of agents.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-actor-axes","relative_href":"docs/glossary.html#glossary-actor-axes","public_phrase":"actor classification axes","short_definition":"Actor axes keep separate how an actor accesses the system, what it can do, how it is invoked, and what role it plays.","reader_preview":"Actor axes keep separate how an actor accesses the system, what it can do, how it is invoked, and what role it plays.","reader_preview_basis":"definition","card_definition":"Rather than ranking actors on one scale from weak to strong, the system measures them along several axes at once. The primary axis is substrate authority, which marks whether an actor has live, direct access to the system's own files and tools (Type A) or works from an interface, an API, an injected copy of context, or the public web without that direct access (Type B). The other axes, including capability tier, invocation surface, topology role, evidence access, purpose lane, and reasoning budget, are recorded separately so that being clever, being expensive, and being delegated never get confused with having authority over the real material.","reader_rule":"When an actor is described by where it sits on these axes, read each axis on its own. A capable or costly actor is not automatically one with direct authority over the system, and a delegated helper with live access can hold more authority than a powerful service that only sees a copy.","deep_definition":"The decisive axis is substrate authority: a tool-using agent with live access to the repository counts as Type A, while operator dashboards, bridges, hosted models, and anything reasoning over injected or public context count as Type B unless they too hold live access to the private material.","scope_note":"Actor axes appear wherever the system labels who or what is acting, including handoffs between a Type B service and a Type A agent, prompt and route descriptions, and the documentation for working through a coding agent.","example":"A coding agent running with live access to the repository is an actor that sits at the Type A end of the substrate-authority axis, with a high capability tier and a delegated topology role, all distinguished at once by the axes.","example_route":{"href":"coding-agents.html","label":"With a coding agent","gloss":"see a Type A actor, a coding agent with live repository access, in use"},"non_example":"Actor axes are not a single ranking of actors from least to most powerful, and substrate authority is not the same as intelligence or cost. 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One admitted step in the wrong place can carry an entire result, so a large corpus with an admission may establish less than a small one without. Formal systems make this checkable rather than a matter of trust, because the checker records every admission.","reader_rule":"Check the admitted-step count first. Everything downstream of an admission is assumed, not proved.","deep_definition":"The reason to publish the count rather than the intention is that admissions are normal during development and easy to leave behind. 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Each affordance row is the atomic unit of that navigation surface: it states one option, the next thing an agent or reader could open, along with enough structured detail to decide whether to open it. A row can be shown at different depths, from a one-line flag through a fuller card to a detailed view with the exact command that produced its evidence, so a choice can be made from a budgeted menu instead of by searching through source files.","reader_rule":"When you meet an affordance row, read it as one option on a menu: it tells you what the thing is, what opening it costs, and what backs it, so you can decide your next read without scanning the underlying files.","deep_definition":"The row compresses choice rather than content: it turns \"what can I do with this artifact\" into source-derived structure, with each row being projectable at several depths, linked to its evidence, marked with a lifecycle state, and typed by what it is allowed to claim.","scope_note":"Affordance rows underlie the system's browse and navigation surfaces, where each artifact kind, components, paper modules, doctrine cards and others, is listed as rows at varying depths. 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The deposit is stamped with the agent that wrote it and the sources it drew on. It is treated as useful but lower authority than the operator's own statements of intent, so the system can retain what an agent learned without that material being mistaken for the operator's voice.","reader_rule":"Take an agent seed as an AI agent's recorded reasoning, attributed to that agent, not as the operator's intent. It carries who wrote it and what it drew on, so weigh it accordingly.","deep_definition":"Keeping agent-authored synthesis in its own seed is what lets the system absorb an agent's learning while the operator's raw seed stays the higher authority on intent, with neither layer impersonating the other.","scope_note":"Agent seed names the attributed AI-authored layer of the system's substrate, sitting alongside the operator-voice raw seed. A reader meets it on the pages about agent reliability and about how agent observations feed back into the system's doctrine.","example":"The Voice To Doctrine Self Improvement Loop is the component that carries agent-authored seed material onward, taking what an agent recorded and folding it into the system's standing doctrine while preserving who wrote it.","example_route":{"href":"components.html#component-voice_to_doctrine_self_improvement_loop","label":"Voice To Doctrine Self Improvement Loop","gloss":"see where agent-authored seed material feeds back into doctrine"},"non_example":"An agent seed is not the operator's own voice and not source authority on intent. 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The trace is the record underneath it, and the two can disagree. Keeping the trace separate is what makes it possible to check whether a reported action occurred, which is a different question from whether the report is plausible.","reader_rule":"Compare an agent\u2019s closeout with the captured trace, but read the trace only within its instrumentation coverage: it is evidence for recorded events, not proof that every action was observed.","deep_definition":"This is the same discipline the rest of the site applies to itself: prose describing a run is not the run. An agent is simply a case where the prose is unusually convincing.","scope_note":"An agent trace establishes the events captured by the named tracing surface during one session. 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It is judgment bound to the system's axioms, standards, and live state, and incomplete until any reusable lesson is propagated back.","visible_text_byte_count":1951,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::common_sense","source_hash":"sha256:ab74c9b3006a5eebd8ec2cdb31943ed39db5c8947c04bc25c4d30401f8a5baa3","relationships":[{"verb":"specializes","target":"type_a_judgment","gloss":"Common sense is one Type A judgment trait."},{"verb":"owned_by","target":"type_a","gloss":"The controller is responsible for applying situated judgment."},{"verb":"uses","target":"up_propagate","gloss":"Reusable common-sense lessons move upward into shared artifacts."},{"verb":"governed_by","target":"axiom","gloss":"The axiom-candidate lane decides whether common sense becomes constitutional."}],"source_refs":[{"ref":"codex/doctrine/skills/doctrine/local_to_general_propagation.md","role":"skill","gloss":"The closeout move that turns local common sense into durable shared grammar."},{"ref":"codex/doctrine/paper_modules/agent_entry_surfaces.md","role":"paper_module","gloss":"Compressed entry-surface doctrine for cold-agent visibility."},{"ref":"CODEX.md","role":"adapter","gloss":"Current Codex common-sense contract and mutation discipline."}],"evidence_commands":[{"command":"./repo-python kernel.py --term common_sense --term-band context","reason":"Read the governed common-sense definition."},{"command":"./repo-python kernel.py --docs-route \"axioms principles standards common sense\"","reason":"Route common sense through the doctrine lattice."}],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":77,"id":"compact","label":"compact","preferred_label":"compact","alternative_labels":["compacts","compacted","compacting","compact view"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. 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It should retain routes to fuller evidence and make its omissions or boundary understandable.","reader_rule":"Treat compact material as a navigation or decision aid and open the deeper source when omitted detail matters.","deep_definition":"Good compression preserves decision-relevant distinctions; merely making text shorter can erase the very boundary a reader needs.","scope_note":"The term covers condensed cards, summaries, packets, and interface layouts.","example":"A compact term card gives the definition and scope while linking to deeper material.","example_route":null,"non_example":"Truncated text that removes the qualification from a claim is not a faithful compact view.","visible_text_byte_count":794,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::compact","source_hash":"sha256:ab74c9b3006a5eebd8ec2cdb31943ed39db5c8947c04bc25c4d30401f8a5baa3","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":268,"page_count":194,"top_page":"paper-module-cold-reader-route-map.html","top_pages":[{"page":"paper-module-cold-reader-route-map.html","count":9},{"page":"paper-module-agent-route-observability-runtime.html","count":6},{"page":"paper-module-first-screen-composition-root.html","count":5},{"page":"paper-module-provider-context-recipe-budget.html","count":4},{"page":"paper-module-routing-anti-patterns-registry.html","count":4},{"page":"paper-module-spatial-world-model-counterfactual-simulation-replay.html","count":4}]}},{"schema":"plectis_public_glossary_term_v0","position":78,"id":"compatibility_route","label":"compatibility route","preferred_label":"compatibility route","alternative_labels":["legacy route","tombstone route"],"hidden_search_labels":["old route","redirect route","migration route"],"term_kind":"routing","meaning_scope_class":"ai_workflow_coined_term","meaning_scope_label":"System-coined term","definition_profile":"governed_system_term","meaning_boundary":"A compatibility route preserves a retired URL or identifier by forwarding it to the current Plectis object; it is not a second live identity or an endorsement of the retired name.","meaning_contrasts":[],"status":"active","disposition":"legacy_navigation_boundary_label","anchor":"glossary-compatibility-route","relative_href":"docs/glossary.html#glossary-compatibility-route","public_phrase":"old path support","short_definition":"A compatibility route is an old URL or identifier retained solely to redirect readers to the current Plectis surface, preserving access without reviving the retired name as the live one.","reader_preview":"A compatibility route is an old URL or identifier retained solely to redirect readers to the current Plectis surface, preserving access without reviving the retired name as the live one.","reader_preview_basis":"definition","card_definition":"When a site is renamed or reorganised, links and identifiers that people already hold would otherwise break. A compatibility route is the machinery that keeps those old paths resolving: a tombstone page, a redirect, an alias, or a manifest entry that takes the visitor to the current location. It names the old route, the new route, the behaviour, and the scope, so the move is visible rather than silent.","reader_rule":"Treat a compatibility route as a forwarding link, not as a second live site. Where it leads is the current surface; the old address it came from is retired.","deep_definition":"A compatibility route is public migration infrastructure: it prevents broken links and keeps the history of earlier addresses inspectable, while making the current identity, Plectis, unmistakable. It does not rewrite the past or imply that the old name is endorsed.","scope_note":"Covers retired Microcosm-era URLs, source paths, and identifiers that must continue resolving after the move to Plectis. A reader meets one whenever an old link or bookmark lands on a tombstone or redirect that points to the current page.","example":"The old Microcosm site root is a compatibility route: it shows a tombstone and forwards visitors to the current Plectis site rather than serving the former content as if it were still live.","example_route":{"href":"components.html#component-cold_reader_route_map","label":"Cold Reader Route Map","gloss":"open the component that maps readers from any entry point to the current surface"},"non_example":"A compatibility route is not evidence that the old name is still the current brand, and it is not a separate active site running in parallel. It is a one-way bridge from a retired address to the live one.","visible_text_byte_count":1565,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::compatibility_route","source_hash":"sha256:ab74c9b3006a5eebd8ec2cdb31943ed39db5c8947c04bc25c4d30401f8a5baa3","relationships":[{"verb":"routes_from","target":"microcosm","gloss":"Microcosm is now a legacy/compatibility label."},{"verb":"routes_to","target":"plectis","gloss":"Compatibility routes point to the current public identity."},{"verb":"uses","target":"routing","gloss":"Compatibility behavior is a routing concern."}],"source_refs":[{"ref":"sites/microcosm/legacy-route-manifest.json","role":"generated_legacy_manifest","gloss":"Generated old/new route behavior manifest."},{"ref":"sites/microcosm-legacy/index.html","role":"legacy_tombstone","gloss":"Legacy site root tombstone for old Microcosm URLs."}],"evidence_commands":[{"command":"jq '.routes[0]?' sites/microcosm/legacy-route-manifest.json","reason":"Inspect the generated legacy route manifest."},{"command":"./repo-python tools/meta/dissemination/verify_microcosm_public_site_host.py --base-url https://wcook04.github.io/microcosm-substrate/ --expected-release-mode legacy_plectis_tombstone --host-platform-waiver github_pages_state_a_public_shell","reason":"Verify the hosted legacy tombstone behavior."}],"used_in":{"total_occurrences":4,"page_count":2,"top_page":"lineage.html","top_pages":[{"page":"lineage.html","count":2},{"page":"paper-module-pattern-binding-contract.html","count":2}]}},{"schema":"plectis_public_glossary_term_v0","position":79,"id":"compression","label":"compression","preferred_label":"compression","alternative_labels":["compress","summarize safely"],"hidden_search_labels":["summarise safely","context compression","holographic compression","navigation compression","profile-governed compression","telescope compression","telescoping compression"],"term_kind":"navigation","meaning_scope_class":"ai_workflow_specialization","meaning_scope_label":"System-local specialization","definition_profile":"governed_system_term","meaning_boundary":"Compression means preservation of meaning across governed navigation bands. It is not data compression, model compression, or summarization in every external use.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-compression","relative_href":"docs/glossary.html#glossary-compression","public_phrase":"preserve through bands","short_definition":"Compression is the governed reduction of larger source material into a smaller navigable view that records what it preserves, what it omits, which profile and band shaped it, and how to reopen the authoritative source.","reader_preview":"Compression is the governed reduction of larger source material into a smaller navigable view that records what it preserves, what it omits, which profile and band shaped it, and how to reopen the authoritative source.","reader_preview_basis":"definition","card_definition":"In this system compression is governed, not casual. Every compact summary is produced under a declared profile that fixes how much detail a given band carries, so a one-line entry, a card, and a full module are different rungs of the same material rather than arbitrary trims. Each row records what was preserved and what was omitted, and keeps a path back to the source it stands for, so a reader can always open the fuller layer beneath any summary.","reader_rule":"Treat a compressed row as a faithful but partial view: it carries an omission record and a route back to source, so when detail matters, follow that route rather than relying on the short form alone.","deep_definition":"Compression here is the write side of navigation: standards declare the readable shapes, profiles declare the band contracts, and the resulting summaries are generated surfaces that let later readers spend attention deliberately. Where no profile governs a row, the system records a profile gap instead of inventing an ad hoc summary.","scope_note":"Compression governs how the public pages, component cards, paper modules, and field-guide entries are derived from the underlying files and records. A reader meets it wherever a short summary stands in for a larger artifact and offers a way to open the fuller layer.","example":"The Navigation Hologram Route Plane is a concrete instance: it is the running surface that holds these banded, profile-governed views and the routes that let a reader move from a compact row down to its source.","example_route":{"href":"components.html#component-navigation_hologram_route_plane","label":"Navigation Hologram Route Plane","gloss":"open the component that turns source material into banded, drillable views"},"non_example":"Compression is not minification, lossy file shrinking, or a loose vibe summary. It is preservation through declared bands that always names its omissions and keeps a route back to the original material.","visible_text_byte_count":1769,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::compression","source_hash":"sha256:ab74c9b3006a5eebd8ec2cdb31943ed39db5c8947c04bc25c4d30401f8a5baa3","relationships":[{"verb":"governed_by","target":"compression_profile","gloss":"Profiles own legal compression contracts."},{"verb":"projects_through","target":"band","gloss":"Bands are the compression packet levels."},{"verb":"serves","target":"navigation","gloss":"Compression creates rows navigation can traverse."},{"verb":"bounded_by","target":"row_job","gloss":"Worker compression should be bounded as a row job."},{"verb":"enacted_by","target":"standard","gloss":"Standards declare what must remain readable."}],"source_refs":[{"ref":"codex/doctrine/skills/compression/profile_governed_compression.md","role":"skill","gloss":"General compression skill for non-raw-seed rows."},{"ref":"codex/doctrine/paper_modules/navigation_hologram_theory.md","role":"paper_module","gloss":"Compression-as-navigation theory roof."}],"evidence_commands":[{"command":"./repo-python kernel.py --docs-route compression","reason":"Route compression work to the standard/navigation skill stack."},{"command":"./repo-python kernel.py --term compression --term-band context","reason":"Read the working definition."},{"command":"./repo-python kernel.py --skill-find profile_governed_compression","reason":"Open the general compression skill."}],"used_in":{"total_occurrences":4,"page_count":3,"top_page":"paper-module-batch8-compliance-pipeline-capsule.html","top_pages":[{"page":"paper-module-batch8-compliance-pipeline-capsule.html","count":2},{"page":"paper-module-corpus-readiness-mathlib-absence-gate.html","count":1},{"page":"paper-module-first-screen-composition-root.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":80,"id":"compression_profile","label":"compression profile","preferred_label":"compression profile","alternative_labels":["profile","profile_id"],"hidden_search_labels":["russian doll profile","band profile","compression rulebook"],"term_kind":"artifact","meaning_scope_class":"ai_workflow_coined_term","meaning_scope_label":"System-coined term","definition_profile":"governed_system_term","meaning_boundary":"A compression profile governs the depth bands and allowed omissions for one artifact kind; it is not a file-compression setting or an improvised summary style.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-compression-profile","relative_href":"docs/glossary.html#glossary-compression-profile","public_phrase":"governed compression rulebook","short_definition":"A compression profile is the repeatable rulebook for shortening one artifact kind, specifying its depth bands, mandatory content, allowed omissions, and forbidden collapses.","reader_preview":"A compression profile is the repeatable rulebook for shortening one artifact kind, specifying its depth bands, mandatory content, allowed omissions, and forbidden collapses.","reader_preview_basis":"definition","card_definition":"Each profile is tied to a specific artifact kind, such as a voice record, a relational map, a peer-knowledge note, or a term definition. It names the bands the artifact can be projected into, the schema each band must follow, the evidence policy, and the validation check, so that a cheap automated worker, a larger model, and a person all shorten the same artifact into the same shape. Four profiles run today, covering voice records, relational maps, peer-knowledge notes, and vocabulary terms.","reader_rule":"When a piece of content here is shown at varying lengths, the compression profile is what fixed which details had to survive and which were allowed to fall away; read it to know what a short form is guaranteed to still carry.","deep_definition":"Profiles are registered together so that compression behaves as a governed primitive: the stated design target is that every artifact kind eventually names its own profile, so that any worker producing a band yields an identical row shape.","scope_note":"The term covers the rulebooks that govern banded compression across the system. A reader meets it behind any surface that presents the same content at different levels of detail, and it is grounded in the navigation and route-plane machinery that projects artifacts through bands.","example":"The vocabulary-term profile is a concrete compression profile: it governs how a vocabulary term is projected into a word, a phrase, a flag, a card, and deeper bands, which is the same laddered shape this glossary entry follows.","example_route":{"href":"paper-modules.html#paper-module-navigation-hologram-route-plane","label":"Navigation Hologram Route Plane","gloss":"open the route plane where banded compression is projected"},"non_example":"A compression profile is not a file-size or data-compression setting, and not a one-off summary someone writes by hand. It is a declared rulebook, bound to a specific artifact kind, stating which details a shortened form must keep and which collapses are not allowed.","visible_text_byte_count":1757,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::compression_profile","source_hash":"sha256:ab74c9b3006a5eebd8ec2cdb31943ed39db5c8947c04bc25c4d30401f8a5baa3","relationships":[{"verb":"declares","target":"band","gloss":"Each profile declares its legal bands and contracts."},{"verb":"governs","target":"affordance_row","gloss":"Rows are emitted under a profile."},{"verb":"instance_of","target":"standard","gloss":"Profiles are schema-grade contracts."}],"source_refs":[{"ref":"codex/doctrine/paper_modules/navigation_hologram_theory.md","role":"paper_module","gloss":"pri_121 compression-profile law."}],"evidence_commands":[{"command":"./repo-python kernel.py --docs-route \"compression profile\"","reason":"Route the compression-profile registry."},{"command":"./repo-python kernel.py --paper-module navigation_hologram_theory","reason":"Read the pri_121 discussion in situ."}],"used_in":{"total_occurrences":17,"page_count":8,"top_page":"lineage.html","top_pages":[{"page":"lineage.html","count":4},{"page":"paper-module-batch8-tools-tail-primitives-capsule.html","count":3},{"page":"paper-module-certificate-kernel-execution-lab.html","count":2},{"page":"paper-module-tactic-portfolio-availability.html","count":2},{"page":"paper-modules.html","count":2},{"page":"rules-and-ideas.html","count":2}]}},{"schema":"plectis_public_glossary_term_v0","position":81,"id":"concept","label":"concept","preferred_label":"concept","alternative_labels":["con","con_*","doctrine concept"],"hidden_search_labels":["noun"],"term_kind":"doctrine","meaning_scope_class":"ai_workflow_specialization","meaning_scope_label":"System-local specialization","definition_profile":"governed_system_term","meaning_boundary":"Concept means a governed doctrine entity with a stable id and sources. 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A reader meets them wherever the doctrine refers to a defined object by name, and on the doctrine pages where those objects are listed and described.","example":"Executable Doctrine Grammar is one such named object: a defined idea in the system's doctrine that fixes what it means for a rule to be runnable before it is treated as authoritative, carrying a stable name other doctrine can refer to.","example_route":{"href":"components.html#component-executable_doctrine_grammar","label":"Executable Doctrine Grammar","gloss":"open one named doctrine object and read what it fixes"},"non_example":"A concept is not a step, a procedure, or a tool you run. 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Counting reductions alongside proofs would make a portfolio of open problems look like a portfolio of results.","scope_note":"A conditional reduction establishes an implication, not its conclusion.","example":"39 of the 100 registered claims are conditional reductions, held separately from the 37 proved outright.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1081,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::conditional_reduction","source_hash":"sha256:ab74c9b3006a5eebd8ec2cdb31943ed39db5c8947c04bc25c4d30401f8a5baa3","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":83,"id":"confirm","label":"confirm","preferred_label":"confirm","alternative_labels":["confirms","confirmed","confirming","confirmation"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. 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A collection entry is one stable selectable row, such as a glossary term or component. A system entry is a supported first-contact surface that accepts a task or question and returns a bounded next route, such as the kernel\u2019s task-conditioned entry packet. Both uses promise identity and direction: the reader should know what object or route was entered and where its deeper source lives.","reader_rule":"First identify whether entry means a collection row or a first-contact route. In either case, rely on it only when it names the object or task and provides a supported path to deeper source-backed material.","deep_definition":"The local distinction is support. Many files can be opened, but only an entry is deliberately designed to orient a cold reader or process without prior knowledge. 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The lattice is that whole-system relationship graph. Because the edges are typed, a lesson learned in one place can move to the part of the system that can safely reuse it while its origin and authority stay intact.","reader_rule":"When something is described as living in the lattice, read it as a position in a connected map rather than an isolated note: it has edges to the standards, routes, and units it relates to, and those edges tell you where else to look.","deep_definition":"The typed edges are what stop the system collapsing into flat doctrine: they keep authority, implementation, navigation, compression, and runtime as distinct but linked layers, so a claim carries its provenance as it travels rather than being copied loose.","scope_note":"The lattice covers the relationships between all the named kinds of thing on this site, from components and paper modules to doctrine cards and routes. A reader meets it most directly on the architecture map, where the parts are shown joined together, and implicitly whenever one page links across to a different kind of artifact.","example":"The architecture map on this site is a view of the lattice: it lays out the system's planes and units and shows the edges that connect a standard to the skill that applies it, or a route to the component it reaches.","example_route":{"href":"architecture.html","label":"How it fits together (architecture map)","gloss":"see the typed planes and units joined into one map"},"non_example":"A lattice is not a single diagram or a rigid hierarchy of layers stacked one above another. 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Lattice transposition is the disciplined version of moving it: it keeps the original case on record, names the portable rule, states where the rule applies and where it does not, and deposits it in the narrowest mature plane that can carry it. A friction met while promoting one principle can refine the standard for writing principles; a routing failure can refine a skill and its documentation. The move is only valid if it preserves the source case and respects the authority of the plane it lands in.","reader_rule":"When you see a rule presented as transposed, check that it cites the concrete case it came from and states its applicability and limits. It is a generalisation that is meant to stay accountable to its source, not a free-floating maxim.","deep_definition":"It is the cross-plane form of up-propagation: up-propagation says a lesson moves upward into durable material, while transposition adds that the lesson changes coordinates, from one artifact kind to the underlying invariant that governs several kinds. It is vocabulary the system's agents author, and it remains tied to cited source surfaces and to the promotion rules of the plane it enters.","scope_note":"The term covers how the system turns a one-off lesson into a durable, reusable rule routed to the right governing plane. A reader meets it in the doctrine and routing material, where rules are shown alongside the cases that produced them and the boundaries that limit them.","example":"The doctrine card on provenance propagation and non-interference (AX-8) is a concrete instance: it is the rule that a lesson may move between planes only while carrying its origin and without contaminating the plane it enters, which is exactly the discipline lattice transposition follows.","example_route":{"href":"doctrine.html#dcard-ax-8","label":"Provenance propagation and non-interference","gloss":"open the doctrine rule that governs how a lesson moves between planes"},"non_example":"Lattice transposition is not copying a snippet of code or text from one place to another, and it is not a vague claim that everything in the system is connected. It is the specific, accountable act of lifting a lesson to its underlying rule and routing that rule to the plane that owns it, with the source case kept on record.","visible_text_byte_count":2130,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::lattice_transposition","source_hash":"sha256:ab74c9b3006a5eebd8ec2cdb31943ed39db5c8947c04bc25c4d30401f8a5baa3","relationships":[{"verb":"specializes","target":"up_propagate","gloss":"Adds cross-plane invariant transfer to the upward routing move."},{"verb":"aliases_conceptually","target":"generalise_up_propagate","gloss":"Names the generalized up-propagation move."},{"verb":"moves_through","target":"lattice","gloss":"Transposition traverses typed edges of the system lattice."},{"verb":"may_deposit_to","target":"standard","gloss":"Schema-grade transpositions belong in standards."},{"verb":"may_deposit_to","target":"peer_propagate","gloss":"Immature cross-plane lessons can stage as PEER rows."}],"source_refs":[{"ref":"codex/doctrine/skills/doctrine/local_to_general_propagation.md","role":"skill","gloss":"Closeout skill defines lattice transposition and its owning-plane discipline."}],"evidence_commands":[{"command":"./repo-python kernel.py --docs-route \"lattice transposition\"","reason":"Route the term through local-to-general propagation surfaces."},{"command":"./repo-python kernel.py --skill-find local_to_general_propagation","reason":"Open the governing propagation skill."}],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":203,"id":"launch","label":"launch","preferred_label":"launch","alternative_labels":["launches","launched","launching"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. 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The word should not blur technical startup with public release when those have different gates.","reader_rule":"Identify what begins, who can reach it, and which readiness conditions were satisfied at launch.","deep_definition":"A launch is a transition into use, not proof that the launched thing will remain reliable or meet every goal.","scope_note":"The term covers starting tools, runtimes, campaigns, and public releases.","example":"Launching the local preview server makes the generated site available for browser inspection.","example_route":null,"non_example":"Building files without starting or publishing their delivery surface is not a public launch.","visible_text_byte_count":773,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::launch","source_hash":"sha256:ab74c9b3006a5eebd8ec2cdb31943ed39db5c8947c04bc25c4d30401f8a5baa3","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":795,"page_count":168,"top_page":"rules-and-ideas.html","top_pages":[{"page":"rules-and-ideas.html","count":129},{"page":"paper-modules.html","count":76},{"page":"components.html","count":61},{"page":"reference.html","count":44},{"page":"doctrine.html","count":34},{"page":"paper-module-batch12-release-claim-language-gate.html","count":30}]}},{"schema":"plectis_public_glossary_term_v0","position":204,"id":"launch_scope_decision","label":"launch-scope decision","preferred_label":"launch-scope decision","alternative_labels":["launch-scope"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"ai_workflow_specialization","meaning_scope_label":"System-local specialization","definition_profile":"public_contextual_gloss","meaning_boundary":"A launch-scope decision is the separate operator or release-owner judgment to host, ship, or share something; no component check or passing fixture possesses that authority.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-launch-scope-decision","relative_href":"docs/glossary.html#glossary-launch-scope-decision","public_phrase":"launch-scope decision","short_definition":"A launch-scope decision is a judgment about whether something is ready to ship, host, or share publicly, an authority that components on this site explicitly do not possess.","reader_preview":"A launch-scope decision is a judgment about whether something is ready to ship, host, or share publicly, an authority that components on this site explicitly do not possess.","reader_preview_basis":"definition","card_definition":"A component can establish that a check passed. It cannot establish that the result is sufficient to launch, and the distinction is enforced rather than left to taste: a component whose wording claims launch readiness, hosting authority, or public-sharing authority is rejected by the release gate with a named finding code. Readiness stays a human decision made with evidence in hand.","reader_rule":"Treat component evidence as input to a separate operator or release-owner decision. No passing component grants readiness to host, ship, publish, or share.","deep_definition":"The gate exists because the overclaim is so natural. A run that passes reads as permission, and prose drifts toward it without anyone intending to lie. Naming the refusal and giving it a finding code turns a matter of tone into something a build can fail on.","scope_note":"Launch-scope decisions are outside every component's authority on this site.","example":"A step claiming a component is ready to host is refused with the release-overclaim finding code, even when the underlying check genuinely passes.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1180,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::launch_scope_decision","source_hash":"sha256:ab74c9b3006a5eebd8ec2cdb31943ed39db5c8947c04bc25c4d30401f8a5baa3","relationships":[],"source_refs":[{"ref":"codex/standards/std_microcosm.json","role":"source_authority","gloss":"Grounds this Plectis-specific definition and its scope."},{"ref":"microcosm-substrate/src/microcosm_core/organs/batch12_release_claim_language_gate.py","role":"source_authority","gloss":"Grounds this Plectis-specific definition and its scope."}],"evidence_commands":[],"used_in":{"total_occurrences":546,"page_count":137,"top_page":"rules-and-ideas.html","top_pages":[{"page":"rules-and-ideas.html","count":139},{"page":"paper-modules.html","count":114},{"page":"components.html","count":22},{"page":"reference.html","count":22},{"page":"area-import-drift.html","count":7},{"page":"area-agent-reliability.html","count":6}]}},{"schema":"plectis_public_glossary_term_v0","position":205,"id":"lean","label":"Lean","preferred_label":"Lean","alternative_labels":["Lean prover"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. 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An observe pass learns, maps, compares, or probes a target and writes down what it found: the questions it asked, the files it examined, the responses it got back, a synthesis, and any validation signals. It can run locally or across a bridge of helpers, but it never mutates code or doctrine. Its job is to keep exploration separate from action so a decision rests on collected facts rather than a first guess.","reader_rule":"Read an observe pass as a record of what was examined and found, not as a completed change. The work it describes has been looked into, not yet carried out.","deep_definition":"Keeping the observe step explicit means the evidence behind a later change is inspectable on its own, and the system does not treat its first plan as if the work were already done.","scope_note":"Observe is a runtime activity in the system's observe/apply cycle. A reader meets it wherever the pages describe how a pass gathers evidence, runs probes, or proves a point before any application step, including the evidence and architecture surfaces.","example":"The Tactic Portfolio Availability Probe is one concrete observe: it inspects which proof tactics are available for a target and reports what it found, without itself attempting the proof.","example_route":{"href":"components.html#component-tactic_portfolio_availability_probe","label":"Tactic Portfolio Availability Probe","gloss":"see what an observe pass actually records and proves"},"non_example":"An observe is not the change itself and not a finished decision. 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A reader meets it wherever a short row stands in for fuller source material and carries a pointer back to that material.","example":"The doctrine card \"Preserve provenance across every boundary\" is a concrete instance: it holds the rule that a compressed surface must always keep a path back to the source it summarises, which is exactly what an omission receipt records on an individual row.","example_route":{"href":"doctrine.html#dcard-p-9","label":"Preserve provenance across every boundary","gloss":"open the doctrine rule that an omission receipt enforces on each row"},"non_example":"An omission receipt is not an error message, a disclaimer, or an admission that something is broken. 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It holds authority over every later artifact, so paper modules, standards, and agent-written summaries may compress or route it but can never replace or contradict it. The raw record is kept append-only for this reason: it is the layer everything else answers to.","reader_rule":"When something is marked as operator voice, treat it as the original human source, above any summary or agent rephrasing of it. If a later page conflicts with it, the operator voice is the one that holds.","deep_definition":"It is distinct from two nearby things: a Bridge proxy acting for the operator under a prepared packet, and an agent-authored projection that compresses operator voice into doctrine. Both of those are downstream; only operator voice is the original.","scope_note":"Operator voice covers the raw seeds, voiced paragraphs, and direct instructions that seed the system. A reader meets it behind the doctrine and component pages, which are projections drawn from it, and most directly in the loop that turns this voice into doctrine.","example":"The Voice To Doctrine Self Improvement Loop is a concrete instance: a component that takes the operator's recorded voice and distils it into doctrine while preserving the original wording, so you can see the raw human input on one side and the compressed rule on the other.","example_route":{"href":"components.html#component-voice_to_doctrine_self_improvement_loop","label":"Voice To Doctrine Self Improvement Loop","gloss":"open the component that turns recorded operator voice into doctrine"},"non_example":"Operator voice is not the polished doctrine, standards, or agent-written summaries on the site. Those are projections compressed from it. 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It is not a claim that the document is an academic publication, and it is not an ai_workflow paper module.","meaning_contrasts":["paper_module"],"status":"active","disposition":"active","anchor":"glossary-paper","relative_href":"docs/glossary.html#glossary-paper","public_phrase":"paper","short_definition":"A paper is a rendered PDF write-up that gives a human-readable account of a project, question, method, result, and evidence boundary, linked to its supporting source where available.","reader_preview":"A paper is a rendered PDF write-up that gives a human-readable account of a project, question, method, result, and evidence boundary, linked to its supporting source where available.","reader_preview_basis":"definition","card_definition":"The Plectis paper shelf contains reader-facing rendered PDFs: short problem notes, longer working records, mathematical expositions, and systems write-ups. 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The popup therefore keeps both senses visible so mathematical language cannot silently overwrite product-governance prose.","scope_note":"Covers the two ordinary senses that appear in Plectis and exists primarily to keep them explicitly disambiguated.","example":"\u201cProduct authority\u201d concerns permission to judge or launch an offering; a \u201cprime product formula\u201d concerns multiplication of prime factors.","example_route":null,"non_example":"3 + 4 is a sum, not a product.","visible_text_byte_count":984,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::product","source_hash":"sha256:ab74c9b3006a5eebd8ec2cdb31943ed39db5c8947c04bc25c4d30401f8a5baa3","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":121,"page_count":42,"top_page":"paper-modules.html","top_pages":[{"page":"paper-modules.html","count":19},{"page":"components.html","count":9},{"page":"rules-and-ideas.html","count":8},{"page":"paper-module-agent-memory-temporal-conflict-replay.html","count":7},{"page":"area-agent-reliability.html","count":6},{"page":"paper-module-spatial-world-model-counterfactual-simulation-replay.html","count":6}]}},{"schema":"plectis_public_glossary_term_v0","position":298,"id":"profile_gap","label":"profile gap","preferred_label":"profile gap","alternative_labels":["profile_gap","projection gap"],"hidden_search_labels":["option-surface gap","missing row support","navigation debt receipt","rowability gap"],"term_kind":"navigation","meaning_scope_class":"ai_workflow_coined_term","meaning_scope_label":"System-coined term","definition_profile":"governed_system_term","meaning_boundary":"A profile gap means an artifact kind exists but lacks the supported row projections needed for honest browsing; it is not an absent model profile or proof that the source artifact is missing.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-profile-gap","relative_href":"docs/glossary.html#glossary-profile-gap","public_phrase":"missing row support","short_definition":"A profile gap is a recorded admission that an artifact kind exists but lacks the supported summary and card rows needed for honest row-by-row browsing, so navigation shows the gap instead of pretending the option surface is complete.","reader_preview":"A profile gap is a recorded admission that an artifact kind exists but lacks the supported summary and card rows needed for honest row-by-row browsing, so navigation shows the gap instead of pretending the option surface is complete.","reader_preview_basis":"definition","card_definition":"The system arranges its artifacts into kinds and lets each kind be enumerated as stable rows that can be selected and drilled into. 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It is a small, open set of traits rather than a fixed list of virtues, with attention and care as early examples, and it is meant to be traceable rather than a passing mood.","reader_rule":"Where the site refers to Type A judgment, treat it as the agent's discernment exercised on this material, grounded in the system's rules and records, not as an unaccountable opinion.","deep_definition":"It rests on a simple invariant: if a judgment changes how a later agent will behave, it should be written down with its sources and routes rather than left as something only the current session knows.","scope_note":"Type A judgment covers the agent-side qualities of working over the system, such as common sense, care, and a critic's eye, and a reader meets it on the pages describing how agents operate over the system, particularly the coding-agent walkthrough.","example":"A coding agent working through this system pauses on a standard that no longer matches the code, notices the gap, and records where the correction belongs instead of carrying on regardless; that act of noticing and routing is Type A judgment.","example_route":{"href":"coding-agents.html","label":"With a coding agent","gloss":"see an agent exercising this judgment over the system"},"non_example":"Type A judgment is not a fixed checklist of virtues an agent is graded against, and it is not a free-floating personal opinion; it is discernment that stays anchored to the system's own rules and records.","visible_text_byte_count":1804,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::type_a_judgment","source_hash":"sha256:ab74c9b3006a5eebd8ec2cdb31943ed39db5c8947c04bc25c4d30401f8a5baa3","relationships":[{"verb":"specializes","target":"type_a_metabolism","gloss":"Names the live judgment side of Type A metabolism."},{"verb":"composes","target":"common_sense","gloss":"Common sense is one routeable Type A judgment trait."},{"verb":"composes","target":"care","gloss":"Care is one routeable Type A judgment trait, not the whole surface."},{"verb":"composes","target":"passion","gloss":"Passion is one routeable Type A judgment trait, not the whole surface."},{"verb":"composes","target":"critic_posture","gloss":"Critic posture is the thinking-trait move for noticing route gaps and weak abstractions."},{"verb":"uses","target":"up_propagate","gloss":"Judgment traits become durable through up-propagation into the owning plane."},{"verb":"constrained_by","target":"axiom","gloss":"Axiom candidates bound when a judgment trait becomes constitutional."}],"source_refs":[{"ref":"CODEX.md","role":"adapter","gloss":"Codex-facing common-sense and up-propagation contract."},{"ref":"CLAUDE.md","role":"adapter","gloss":"Claude-facing Type A judgment and follow-on substrate context."},{"ref":"codex/doctrine/skills/doctrine/local_to_general_propagation.md","role":"skill","gloss":"Closeout and route-gap repair lane for reusable judgment lessons."}],"evidence_commands":[{"command":"./repo-python kernel.py --term type_a_judgment --term-band context","reason":"Read the parent judgment-trait surface."},{"command":"./repo-python kernel.py --docs-route \"type A judgment trait\"","reason":"Route the parent judgment-trait surface to its home (sit_type_a_judgment_metabolism); the operating-register split sibling lives at --docs-route \"type A operating register\"."}],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":411,"id":"type_a_metabolism","label":"Type A metabolism","preferred_label":"Type A metabolism","alternative_labels":["type_a_metabolism","Type-A metabolism"],"hidden_search_labels":["controller metabolism","judgment metabolism","grammar metabolism"],"term_kind":"runtime","meaning_scope_class":"ai_workflow_coined_term","meaning_scope_label":"System-coined term","definition_profile":"governed_system_term","meaning_boundary":"Type A metabolism is the open-ended governance lane that creates or repairs rules where no settled procedure exists; it is not biological metabolism or routine execution of an existing job.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-type-a-metabolism","relative_href":"docs/glossary.html#glossary-type-a-metabolism","public_phrase":"judgment governs unsettled grammar","short_definition":"Type A metabolism is the high-agency lane that resolves work with no settled rule\u2014such as naming primitives, repairing contradictions, changing standards, or promoting principles\u2014into governed structure that later routine work can follow.","reader_preview":"Type A metabolism is the high-agency lane that resolves work with no settled rule\u2014such as naming primitives, repairing contradictions, changing standards, or promoting principles\u2014into governed structure that later routine work can follow.","reader_preview_basis":"definition","card_definition":"Most upkeep in the system runs along fixed tracks: a known job, a validator, a row to patch. Type A metabolism is what runs when no such track exists yet. It takes a broad, ambiguous situation and collapses it into something bounded and reusable, such as a named rule, a standard, or a repaired scaffold, while preserving the original situation, the limit of what it is allowed to decide, its checks, and a way to roll back. It is the system changing its own rules rather than just applying them, so it is treated as expensive and powerful.","reader_rule":"When you meet Type A metabolism, read it as a record of judgement applied where the rules were still open: the system deciding a new boundary under validation, not running a routine job. The decision it reached is what makes the later routine work cheap.","deep_definition":"Its judgement loop draws on the system's axioms and principles as design intuition and on imported patterns as raw material, and it ends in either a better local rule or a named, recorded rejection. Once the resulting grammar is stable, ongoing maintenance moves to the lower-agency Type B lane.","scope_note":"The term covers the governance-and-decision layer of the system's self-maintenance, as distinct from routine population work. A reader meets it on the architecture map and in the engine-room runtime where unsettled grammar is resolved before work is delegated downstream.","example":"The Engine Room Metabolism Runtime is a concrete instance: the runtime where the system processes unsettled grammar, resolving contradictions and naming primitives under validation before handing routine work to the lower-agency lane.","example_route":{"href":"paper-modules.html#paper-module-engine-room-metabolism-runtime","label":"Engine Room Metabolism Runtime","gloss":"open the runtime where unsettled grammar is resolved"},"non_example":"Type A metabolism is not the routine maintenance lane that applies rules already in place, and it is not a biological process. It is the decision layer that sets new rules under validation when none exist yet.","visible_text_byte_count":1866,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::type_a_metabolism","source_hash":"sha256:ab74c9b3006a5eebd8ec2cdb31943ed39db5c8947c04bc25c4d30401f8a5baa3","relationships":[{"verb":"specializes","target":"metabolism","gloss":"Names the Type A half of generalized metabolism."},{"verb":"owned_by","target":"type_a","gloss":"Requires live controller judgment and validation."},{"verb":"precedes","target":"type_b_metabolism","gloss":"Stable grammar can later be delegated to bounded workers."},{"verb":"uses","target":"lattice_transposition","gloss":"Local Type A repairs can become portable invariants."}],"source_refs":[{"ref":"codex/doctrine/paper_modules/navigation_hologram_theory.md","role":"paper_module","gloss":"Theory roof for Type A authority packets and active up-propagation."}],"evidence_commands":[{"command":"./repo-python kernel.py --term type_a_metabolism --term-band context","reason":"Read the Type A metabolism working definition."},{"command":"./repo-python kernel.py --docs-route \"lattice transposition\"","reason":"Route cross-plane Type A lessons to the propagation skill."}],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":412,"id":"type_a_operating_register","label":"Type A operating register","preferred_label":"Type A operating register","alternative_labels":["type_a_operating_register","operational register type a care passion"],"hidden_search_labels":["type-a operating register","type a useful findings","useful findings register","care passion common sense judgment packet","care and passion","care + passion","karen passion"],"term_kind":"actor","meaning_scope_class":"ai_workflow_coined_term","meaning_scope_label":"System-coined term","definition_profile":"governed_system_term","meaning_boundary":"The Type A operating register is the expected conduct of an actor working on the live substrate; it is not a personality profile, capability tier, or claim that every Type A actor behaves well.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-type-a-operating-register","relative_href":"docs/glossary.html#glossary-type-a-operating-register","public_phrase":"care and judgment register","short_definition":"The Type A operating register is the expected conduct of a live-substrate agent: steward what it touches and durably route what it learns.","reader_preview":"The Type A operating register is the expected conduct of a live-substrate agent: steward what it touches and durably route what it learns.","reader_preview_basis":"definition","card_definition":"It is one of two registers the system distinguishes. A Type A agent is a high-agency controller, a coding agent or its operator, acting on the system from the inside; the register describes how that controller is meant to behave across a piece of work. The behaviours are concrete: steward and thicken the files and records it edits, watch for drift between a summary and what it describes, use what it builds, and pass any local discovery upward through the narrowest honest channel. It is a posture, composed from named judgment traits like care, common sense, and critic stance, not a single tool or a step in a checklist.","reader_rule":"When a page describes work in the Type A register, read it as the controller's own account of how it operated: what it stewarded, what it noticed, and where it routed each finding. The register tells you the standard of conduct behind the work, not a guarantee about any single output.","deep_definition":"The register pairs with Type A judgment, which holds the individual trait definitions, and with the propagation surfaces that carry a local finding into doctrine, standards, or vocabulary according to how settled it is. It governs the behaviour of the controller, not the content of any one component.","scope_note":"The term covers the conduct expected of capable controlling agents, the coding agents and operators that act on the system, and appears wherever the pages describe how such an agent stewarded the substrate, spotted findings, and propagated them. It is encountered most directly on the page describing work alongside a coding agent.","example":"The conduct shown on the \"With a coding agent\" page is a concrete instance: an agent reads the substrate, makes a bounded change, records what it learned, and routes that finding into a durable plane rather than dropping it.","example_route":{"href":"coding-agents.html","label":"With a coding agent","gloss":"see a capable controller working the substrate in this register"},"non_example":"A Type A operating register is not a personality, a tone of voice, or a marketing claim about enthusiasm. It is also not a passive end-user mode; it describes a controller acting on the system, with the duty to steward what it touches and to route what it finds.","visible_text_byte_count":2086,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::type_a_operating_register","source_hash":"sha256:ab74c9b3006a5eebd8ec2cdb31943ed39db5c8947c04bc25c4d30401f8a5baa3","relationships":[{"verb":"specializes","target":"type_a","gloss":"Names the operating posture expected from Type A controllers."},{"verb":"expresses","target":"type_a_judgment","gloss":"The register is the operating packet; Type A judgment owns the expandable trait surface."},{"verb":"uses","target":"up_propagate","gloss":"Useful findings route upward through local-to-general propagation."},{"verb":"bounded_by","target":"standard","gloss":"Register does not override standards, validation, or authority boundaries."},{"verb":"interprets","target":"principle","gloss":"Care and passion interpret active principles into Type A behavior."},{"verb":"supports","target":"cold_agent","gloss":"Cold agents need this register as a compact route, not hidden adapter lore."}],"source_refs":[{"ref":"CODEX.md::PRIME DIRECTIVE (companion 2)","role":"adapter","gloss":"Codex adapter carries the care/passion/lattice register for execution-first Type A work."},{"ref":"CLAUDE.md::PRIME DIRECTIVE (companion 3)","role":"adapter","gloss":"Claude adapter defines care, passion, compression tiering, and lattice behavior for Type A controllers."},{"ref":"codex/doctrine/skills/doctrine/local_to_general_propagation.md","role":"skill","gloss":"Closeout skill that routes useful findings to the owning plane."}],"evidence_commands":[{"command":"./repo-python kernel.py --docs-route \"type a operating register\"","reason":"Route the Type A register read set."},{"command":"./repo-python kernel.py --term type_a_operating_register --term-band context","reason":"Read the working definition and boundaries."},{"command":"./repo-python kernel.py --skill-find local_to_general_propagation","reason":"Open the useful-finding plane-home workflow."}],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":413,"id":"type_b","label":"Type B","preferred_label":"Type B","alternative_labels":["type_b","type-b"],"hidden_search_labels":["external cognition lane","operator HUD","operator-hud type b","chatgpt web","bridge actor"],"term_kind":"actor","meaning_scope_class":"ai_workflow_coined_term","meaning_scope_label":"System-coined term","definition_profile":"governed_system_term","meaning_boundary":"Type B is ai_workflow's local no-private-substrate-authority classification. It is not a personality type, lower intelligence tier, capability grade, or synonym for every delegated worker.","meaning_contrasts":["type_a"],"status":"active","disposition":"active","anchor":"glossary-type-b","relative_href":"docs/glossary.html#glossary-type-b","public_phrase":"external or injected-context cognition","short_definition":"A Type B is an actor reasoning from context handed to it outside the live substrate, able to produce advice, drafts, or candidate answers but unable to inspect private files, run local commands, or mutate the system directly.","reader_preview":"A Type B is an actor reasoning from context handed to it outside the live substrate, able to produce advice, drafts, or candidate answers but unable to inspect private files, run local commands, or mutate the system directly.","reader_preview_basis":"definition","card_definition":"Type B is one of two actor shapes the system recognises. A Type B actor reasons through a chat window, a browser session, a provider or web API, or any channel where it sees only the context it was given, not the live repository. It can decide, research, critique, and correct at a high level over that materialised context, but it has no direct authority to inspect private files, execute commands, edit, test, or commit; any such action has to be carried out by an actor that does hold that access. The counterpart shape, Type A, is the actor running inside the live machinery with those affordances.","reader_rule":"When something is produced by Type B, treat it as advisory: a synthesis built from the context it was shown, to be checked against the live system before anything is acted on.","deep_definition":"The distinction is about access to the live machinery, not about intelligence: a capable Type B actor can outthink the harness it cannot touch, and a delegated worker that does hold live repository tools is a Type A actor, not Type B, despite being delegated.","scope_note":"Type B appears wherever the site describes work that happens outside the running repository: external reasoning lanes, provider and web calls, operator-carried questions, and bridge or injected-context exchanges. It is paired throughout with Type A, the in-harness actor.","example":"A general assistant in a chat or browser window, asked a framing or research question and given a pasted block of context to work from, is acting as a Type B actor: it reasons over what it was shown and returns an answer, without reaching into the live repository itself.","example_route":{"href":"coding-agents.html","label":"With a coding agent","gloss":"see where an external agent works over handed-in context rather than the live repository"},"non_example":"A Type B is not a lower intelligence tier and not simply any delegated worker. An agent that has direct, live access to the repository's files and commands is Type A, even when it has been delegated a task.","visible_text_byte_count":1936,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::type_b","source_hash":"sha256:ab74c9b3006a5eebd8ec2cdb31943ed39db5c8947c04bc25c4d30401f8a5baa3","relationships":[{"verb":"contrasts_with","target":"type_a","gloss":"Type B lacks direct live private-substrate mutation authority; Type A can inspect and mutate through repo gates."},{"verb":"specializes","target":"actor_axes","gloss":"Type B is the substrate_type=type_b value under con_016::actor_axes."},{"verb":"owns","target":"type_b_metabolism","gloss":"Type B handles bounded population and checking."},{"verb":"may_use","target":"bridge","gloss":"Bridge is one Type B surface, not the definition of Type B."},{"verb":"may_use","target":"actor_axes","gloss":"The operator can carry a Type A-prepared ask into an external HUD/web surface and paste the Type B response back."},{"verb":"may_use","target":"actor_axes","gloss":"Provider/API/NIM/OpenRouter calls are Type B surfaces whose class tier is per invocation."},{"verb":"operates_under","target":"standard","gloss":"Type B needs declared standards or schemas to stay safe."}],"source_refs":[{"ref":"docs/bridge_operating_system.md","role":"bridge_os","gloss":"Bridge operating system enforces Type A/Type B boundaries for CDP/web workers."}],"evidence_commands":[{"command":"./repo-python kernel.py --docs-route \"type b\"","reason":"Route the Type B vocabulary surface."},{"command":"./repo-python kernel.py --option-surface concepts --band card --ids con_016","reason":"Open the canonical Type A/B actor-axis concept."},{"command":"./repo-python kernel.py --docs-route \"type b bridge\"","reason":"Open the bridge-specific Type B boundary."},{"command":"./repo-python kernel.py --term type_b --term-band context","reason":"Read the Type B definition."}],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":414,"id":"type_b_metabolism","label":"Type B metabolism","preferred_label":"Type B metabolism","alternative_labels":["type_b_metabolism","Type-B metabolism"],"hidden_search_labels":["bounded metabolism","deterministic metabolism","worker metabolism","population metabolism"],"term_kind":"runtime","meaning_scope_class":"ai_workflow_coined_term","meaning_scope_label":"System-coined term","definition_profile":"governed_system_term","meaning_boundary":"Type B metabolism is repeatable upkeep after the target shape and checks are fixed; it is not the Type B actor classification itself, creative rule-setting, or final promotion authority.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-type-b-metabolism","relative_href":"docs/glossary.html#glossary-type-b-metabolism","public_phrase":"bounded workers populate rows","short_definition":"Type B metabolism is the bounded, deterministic upkeep lane for repeatable work after a structure is settled, such as syncing files, refreshing projections, filling records, and running checks rather than inventing new governing shapes.","reader_preview":"Type B metabolism is the bounded, deterministic upkeep lane for repeatable work after a structure is settled, such as syncing files, refreshing projections, filling records, and running checks rather than inventing new governing shapes.","reader_preview_basis":"definition","card_definition":"This is how the system keeps its many surfaces current at scale. Once a kind of thing has a known shape, with a defined target, input, output schema, validation command, and a record of what it is allowed to claim, Type B work carries out the repeatable part: refreshing an index, refilling a summary band, detecting duplicates, checking that evidence pointers still resolve, or applying a draft patch to a row. Each run is bounded by a fixed input and a fixed output shape, is checked against a stated standard, and can be safely re-run without changing the result.","reader_rule":"When a surface is described as produced by Type B metabolism, treat it as maintenance against a settled shape: it is kept current by repeatable, validated jobs, and it stops short of inventing new structure or making a final promotion decision on its own.","deep_definition":"It can be carried out by plain deterministic code, small validators, scheduled jobs, or cheap external calls, and its safety rests on four things: a bounded input, an explicit output shape, a validation step, and the fact that re-running it produces the same result. The complementary, open-ended lane that first makes a thing legible enough to maintain this way is called Type A.","scope_note":"The term covers the system's maintenance and population runtime: sync, projection refresh, compression-band filling, duplicate detection, evidence-pointer checks, and draft row patches. A reader meets it wherever the architecture or a paper module explains how surfaces are generated and kept up to date.","example":"The Engine Room Metabolism Runtime is a concrete instance: the part of the system that runs these bounded, repeatable upkeep jobs and keeps the generated surfaces refreshed against their declared standards.","example_route":{"href":"paper-modules.html#paper-module-engine-room-metabolism-runtime","label":"Engine Room Metabolism Runtime","gloss":"open the paper module describing this upkeep runtime"},"non_example":"Type B metabolism is not the creative work of designing how a thing should be structured in the first place, and it is not the moment a result is finally approved or made public. It is the bounded, validated, repeatable work that runs once the shape is already known.","visible_text_byte_count":1964,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::type_b_metabolism","source_hash":"sha256:ab74c9b3006a5eebd8ec2cdb31943ed39db5c8947c04bc25c4d30401f8a5baa3","relationships":[{"verb":"specializes","target":"metabolism","gloss":"Names the Type B half of generalized metabolism."},{"verb":"owned_by","target":"type_b","gloss":"Runs inside bounded worker or deterministic lanes."},{"verb":"follows","target":"type_a_metabolism","gloss":"Follows after Type A names stable grammar."},{"verb":"supports","target":"navigation","gloss":"Keeps projections current so navigation remains cheap."}],"source_refs":[{"ref":"codex/doctrine/paper_modules/continuous_runtime_layer.md","role":"paper_module","gloss":"metabolismd queues bounded jobs and generated projection upkeep."}],"evidence_commands":[{"command":"./repo-python kernel.py --term type_b_metabolism --term-band context","reason":"Read the Type B metabolism working definition."},{"command":"./repo-python kernel.py --paper-module continuous_runtime_layer","reason":"Open the resident runtime implementation surface."}],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":415,"id":"typed_relation","label":"typed relation","preferred_label":"typed relation","alternative_labels":["typed relations"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. Repository-specific examples contextualize the term; they do not redefine it or certify one universal definition.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-typed-relation","relative_href":"docs/glossary.html#glossary-typed-relation","public_phrase":"typed relation","short_definition":"A typed relation is a recorded link between two statements that says what kind of link it is: that one implies another, reformulates it, is a finite instance of it, or rules out a route to it.","reader_preview":"A typed relation is a recorded link between two statements that says what kind of link it is: that one implies another, reformulates it, is a finite instance of it, or rules out a route to it.","reader_preview_basis":"definition","card_definition":"An untyped link only says two things are related, which is nearly useless for reasoning. Typing the link is what makes the map answerable: you can ask what a result implies, what still depends on an unproved antecedent, or which approaches have already been closed off, and get an answer rather than a reading list.","reader_rule":"Follow the relation type, not just the link. What connects two statements is the content.","deep_definition":"Recording the type is also what makes a wrong connection falsifiable. A link labelled implies can be checked; a link labelled related cannot.","scope_note":"A typed relation records an authored judgement about two statements.","example":"1,088 typed relations join the statements, among them 223 implications and 157 dependencies on unproved antecedents.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1065,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::typed_relation","source_hash":"sha256:ab74c9b3006a5eebd8ec2cdb31943ed39db5c8947c04bc25c4d30401f8a5baa3","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":416,"id":"up_propagate","label":"up-propagate","preferred_label":"up-propagate","alternative_labels":["uppropagate","up propagate"],"hidden_search_labels":["up propagation","uppropagation"],"term_kind":"propagation","meaning_scope_class":"ai_workflow_coined_term","meaning_scope_label":"System-coined term","definition_profile":"governed_system_term","meaning_boundary":"Up-propagate means moving a reusable local lesson into the least-authoritative durable artifact that can honestly carry it; it does not promote every observation into doctrine.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-up-propagate","relative_href":"docs/glossary.html#glossary-up-propagate","public_phrase":"move learning upward","short_definition":"To up-propagate is to move a reusable local lesson into the smallest durable system artifact that should carry it.","reader_preview":"To up-propagate is to move a reusable local lesson into the smallest durable system artifact that should carry it.","reader_preview_basis":"definition","card_definition":"Up-propagation is the direction in which reusable learning travels through the system: from a specific local case toward a more lasting home such as a skill, a standard, a paper module, an agent seed, or an axiom. The destination is chosen by how settled and how authoritative the lesson is, so a routine procedure lands in a skill, a fixed data shape lands in a standard, and operator intent lands in the source voice records. The aim is to record the lesson once, in the least authoritative place that can hold it honestly, rather than letting it be rediscovered.","reader_rule":"When something is described as up-propagated, treat it as a local finding that has been deliberately lifted into a more durable artifact; trace it to that artifact to see where it now lives and what scope it claims.","deep_definition":"Up-propagation does not turn every finding into doctrine. It routes a finding to the least authoritative durable layer that can preserve it without overstating it, which is why most lessons settle as skills or standards rather than as axioms or principles.","scope_note":"The term covers the upward movement of learning across the system's layers, from local tasks and signals up to skills, standards, paper modules, seeds, and axioms. A reader meets it in the doctrine and rules pages, particularly around the axiom on provenance propagation, and wherever the system explains how a local case improved a shared artifact.","example":"The axiom on provenance propagation and non-interference, AX-8, is a concrete instance: it states how a local lesson may be carried into a higher artifact while preserving where it came from and without disturbing the artifact it joins.","example_route":{"href":"doctrine.html#dcard-ax-8","label":"Provenance propagation and non-interference","gloss":"open the axiom that governs carrying a local lesson upward"},"non_example":"Up-propagation is not a promotion of every observation into formal doctrine, and it is not a one-way push that overwrites the artifact it joins. 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A complete public site can still be only one projection of the whole repository, and a whole repository can still depend on outside services.","reader_rule":"Before interpreting whole or entire, identify the boundary that determines which parts are included.","deep_definition":"Properties of individual parts do not automatically establish the same property for the whole because interactions can create new behaviour.","scope_note":"The term covers complete bounded collections and systems explicitly named by the surrounding text.","example":"The entire generated site is the whole collection of pages and assets under its publication root.","example_route":null,"non_example":"One representative page is not the whole website.","visible_text_byte_count":789,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::whole","source_hash":"sha256:ab74c9b3006a5eebd8ec2cdb31943ed39db5c8947c04bc25c4d30401f8a5baa3","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":681,"page_count":153,"top_page":"paper-modules.html","top_pages":[{"page":"paper-modules.html","count":130},{"page":"rules-and-ideas.html","count":65},{"page":"doctrine.html","count":47},{"page":"components.html","count":35},{"page":"area-import-drift.html","count":14},{"page":"open-questions.html","count":14}]}},{"schema":"plectis_public_glossary_term_v0","position":428,"id":"whole_system_correctness","label":"whole-system correctness","preferred_label":"whole-system correctness","alternative_labels":["whole system correctness"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"ai_workflow_specialization","meaning_scope_label":"System-local specialization","definition_profile":"public_contextual_gloss","meaning_boundary":"Whole-system correctness is the global guarantee that Plectis explicitly refuses to infer from local checks; no component, page, or collection of green fixtures establishes it.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-whole-system-correctness","relative_href":"docs/glossary.html#glossary-whole-system-correctness","public_phrase":"whole-system correctness","short_definition":"Whole-system correctness is the claim that an entire system behaves correctly, used here mainly as an explicit boundary that no individual page, component, or passing check is allowed to assert.","reader_preview":"Whole-system correctness is the claim that an entire system behaves correctly, used here mainly as an explicit boundary that no individual page, component, or passing check is allowed to assert.","reader_preview_basis":"definition","card_definition":"Every component here establishes something local: this check, on these inputs, produced this result. None of them add up to a statement about the system entire. The phrase is used as a refusal rather than a goal, and the scope limit on each result names it explicitly so that a passing check is not read as a wider guarantee.","reader_rule":"Treat any wording that implies whole-system correctness as an overclaim, including on this site.","deep_definition":"The reason for naming a claim nobody makes is that it is the claim readers infer for free. A repository of passing checks reads like a working system unless the gap is stated, so the release gates refuse language that closes it.","scope_note":"This term exists to bound other claims. It is never itself established.","example":"A component's scope limit records that a passing fixture case does not establish whole-system correctness.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1119,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::whole_system_correctness","source_hash":"sha256:ab74c9b3006a5eebd8ec2cdb31943ed39db5c8947c04bc25c4d30401f8a5baa3","relationships":[],"source_refs":[{"ref":"codex/standards/std_microcosm.json","role":"source_authority","gloss":"Grounds this Plectis-specific definition and its scope."},{"ref":"microcosm-substrate/src/microcosm_core/validators/research_claim_assurance.py","role":"source_authority","gloss":"Grounds this Plectis-specific definition and its scope."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":429,"id":"work","label":"work","preferred_label":"work","alternative_labels":["works","working","worked","unit of work"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. Repository-specific examples contextualize the term; they do not redefine it or certify one universal definition.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-work","relative_href":"docs/glossary.html#glossary-work","public_phrase":"work","short_definition":"Work is purposeful effort that changes, examines, explains, or verifies something.","reader_preview":"Work is purposeful effort that changes, examines, explains, or verifies something.","reader_preview_basis":"definition","card_definition":"On this site, work usually means a bounded task with an intended result, not mere activity. Its value depends on what changed or became known and on whether that outcome was checked.","reader_rule":"Read a claim of work by asking what its target was, what result it produced, and what evidence closes it.","deep_definition":"Effort, output, and completion are separate: substantial effort may yield no valid result, while a small operation may complete the required work.","scope_note":"The term covers the named task and its direct validation, not every surrounding possibility.","example":"Adding a glossary card and regenerating the page is one unit of work.","example_route":null,"non_example":"Time spent without a target or result is activity, not necessarily completed work.","visible_text_byte_count":747,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::work","source_hash":"sha256:ab74c9b3006a5eebd8ec2cdb31943ed39db5c8947c04bc25c4d30401f8a5baa3","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":226,"page_count":196,"top_page":"doctrine.html","top_pages":[{"page":"doctrine.html","count":11},{"page":"coding-agents.html","count":3},{"page":"component-sleeper-memory-poisoning-quarantine-replay.html","count":3},{"page":"paper-module-batch8-structural-theses-capsule.html","count":3},{"page":"paper-module-mission-transaction-work-spine.html","count":3},{"page":"paper-module-undeclared-library-prior-classifier.html","count":3}]}},{"schema":"plectis_public_glossary_term_v0","position":430,"id":"work_landing","label":"work landing","preferred_label":"work landing","alternative_labels":["durable work landing","landing work"],"hidden_search_labels":["work-landing","land work"],"term_kind":"public_projection","meaning_scope_class":"ai_workflow_coined_term","meaning_scope_label":"System-coined term","definition_profile":"governed_system_term","meaning_boundary":"Work landing is a local ai_workflow construct for accepting one bounded change into durable substrate. It does not imply that unrelated follow-up work is complete or that the phrase is generally standardized.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-work-landing","relative_href":"docs/glossary.html#glossary-work-landing","public_phrase":"durable work acceptance","short_definition":"Work landing is the transition from attempted activity to a durable accepted result whose changes, validation, ownership, and closeout are recorded.","reader_preview":"Work landing is the transition from attempted activity to a durable accepted result whose changes, validation, ownership, and closeout are recorded.","reader_preview_basis":"definition","card_definition":"Work lands only when the attempted change reaches the durable surface that owns it and can be reopened without the session that produced it. In ai_workflow that normally means the owned source path contains the change, the focused validation has a recorded outcome, concurrent ownership was respected, the resulting status and remaining boundary are explicit, and the change has a commit or equivalent durable receipt. A successful-looking tool run or chat summary is still only activity until those pieces agree.","reader_rule":"Treat work as landed only when the artifact and its validation record can be reopened independently of the session that produced them.","deep_definition":"Landing is the acceptance transition between mutation and continuity. It binds the changed artifact to provenance, validation, ownership, reversibility, status, and any residual that still needs an owner. This is why a patch can be technically correct yet not landed, and why landing one bounded change says nothing about unrelated work.","scope_note":"The term covers acceptance of one bounded unit of work into its owning files, records, and status surfaces. It does not imply that unrelated follow-up work is complete.","example":"A scoped source change with passing focused tests, a recorded status, and a commit or equivalent durable receipt has landed.","example_route":null,"non_example":"Describing a proposed patch or leaving an unvalidated edit in a transient session is not work landing.","visible_text_byte_count":1388,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::work_landing","source_hash":"sha256:ab74c9b3006a5eebd8ec2cdb31943ed39db5c8947c04bc25c4d30401f8a5baa3","relationships":[{"verb":"completes","target":"work","gloss":"Landing distinguishes durable completed work from attempted activity."},{"verb":"preserves","target":"continuity","gloss":"A landed result leaves enough state for the next actor to resume safely."},{"verb":"records","target":"record","gloss":"Landing binds changes and validation into a durable record."},{"verb":"checked_by","target":"validation","gloss":"Focused validation is part of the landing boundary."}],"source_refs":[{"ref":"codex/doctrine/paper_modules/operational_work_item_spine.md","role":"work_spine_theory","gloss":"Defines durable work status and closeout across the work spine."},{"ref":"tools/meta/dissemination/microcosm_public_narratives.json::tour","role":"public_narrative_owner","gloss":"Explains work and continuity in the public mechanism tour."}],"evidence_commands":[{"command":"./repo-python kernel.py --term work_landing --term-band context","reason":"Read the governed work-landing boundary."},{"command":"./repo-python kernel.py --paper-module operational_work_item_spine","reason":"Open the work-spine theory that owns durable status and closeout."}],"used_in":{"total_occurrences":89,"page_count":21,"top_page":"rules-and-ideas.html","top_pages":[{"page":"rules-and-ideas.html","count":27},{"page":"reference.html","count":18},{"page":"doctrine.html","count":14},{"page":"paper-module-durable-agent-work-landing-replay.html","count":5},{"page":"paper-modules.html","count":4},{"page":"paper-module-work-landing-control-spine.html","count":3}]}},{"schema":"plectis_public_glossary_term_v0","position":431,"id":"working_record","label":"working record","preferred_label":"working record","alternative_labels":["working records"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"ai_workflow_specialization","meaning_scope_label":"System-local specialization","definition_profile":"public_contextual_gloss","meaning_boundary":"In Plectis, a working record is a long rendered reasoning-surface PDF for one mathematical problem; it is not an ordinary worklog, a complete private agent trace, or the shorter problem note.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-working-record","relative_href":"docs/glossary.html#glossary-working-record","public_phrase":"working record","short_definition":"A working record is the long-form rendered PDF that preserves the reasoning surface for one Plectis mathematics problem.","reader_preview":"A working record is the long-form rendered PDF that preserves the reasoning surface for one Plectis mathematics problem.","reader_preview_basis":"definition","card_definition":"A Plectis working record is the long-form companion to a short problem note. It keeps the mathematical setup, checked premises, routes that were explored, failures that materially changed the direction, exact formal results, and the surviving open obligation together in one rendered PDF. It is designed to show substantially more of the reasoning path than the polished note while remaining an authored, versioned account rather than a raw session dump.","reader_rule":"Use a working record to inspect how the argument and its dead ends developed, then follow its Lean and claim-registry links for proof status. Do not treat it as an exhaustive trace of every private search step.","deep_definition":"The distinction is three-way. A problem note gives the short proved/open boundary; a working record gives the extended reasoning surface; an agent trace or private search history records lower-level session events and may not be public. The working record selects and organizes the reasoning needed to understand the mathematical route without pretending that selection is the proof or the complete provenance.","scope_note":"A working record covers the selected reasoning, checked premises, failed routes, and surviving obligation presented in that rendered version. It is not proof authority, a complete private search history, or a generic contemporaneous project log.","example":"The Erd\u0151s #249 totient reasoning surface is a working record: a long PDF preserving the finite certificate route, failed approaches, the Lean-checked denominator bound, and the still-open unbounded certificate obligation.","example_route":null,"non_example":"Do not treat this glossary row as source authority; follow its source refs and evidence commands before making claims with the term.","visible_text_byte_count":1450,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::working_record","source_hash":"sha256:ab74c9b3006a5eebd8ec2cdb31943ed39db5c8947c04bc25c4d30401f8a5baa3","relationships":[],"source_refs":[{"ref":"tools/meta/dissemination/build_microcosm_public_site.py","role":"source_authority","gloss":"Grounds this Plectis-specific definition and its scope."},{"ref":"tools/meta/dissemination/build_short_human_outreach_batch.py","role":"source_authority","gloss":"Grounds this Plectis-specific definition and its scope."}],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":432,"id":"workingness","label":"workingness","preferred_label":"workingness","alternative_labels":["working","actually working","system workingness"],"hidden_search_labels":["workingness state","workingness dossier"],"term_kind":"diagnostic","meaning_scope_class":"ai_workflow_coined_term","meaning_scope_label":"System-coined term","definition_profile":"governed_system_term","meaning_boundary":"Workingness is a local diagnostic posture about observed operation. It is not a standardized reliability metric and does not imply whole-system correctness.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-workingness","relative_href":"docs/glossary.html#glossary-workingness","public_phrase":"safe useful self-knowledge","short_definition":"Workingness is the system's condition of safe, useful self-knowledge\u2014what it believes, why, what remains unknown, what would change its mind, and which actions uncertainty permits\u2014rather than a count of passing tests.","reader_preview":"Workingness is the system's condition of safe, useful self-knowledge\u2014what it believes, why, what remains unknown, what would change its mind, and which actions uncertainty permits\u2014rather than a count of passing tests.","reader_preview_basis":"definition","card_definition":"Workingness asks a sharper question than \"do the tests pass\". It is the diagnostic condition in which the system can turn evidence into calibrated belief, send each failure to the part of the system that owns its repair, and act only within bounds it is explicitly allowed to. It separates whether events were observed from whether the system's own state of belief is sound, and it is read off concrete signals such as a court pass, a drift check, or a live receipt before any change is made.","reader_rule":"Treat a workingness reading as a statement about the system's self-knowledge at that moment, not a warranty that everything is correct. Check what it is actually grounded in before relying on it to permit a change.","deep_definition":"Workingness is the control question behind the system improving itself: can it convert evidence into calibrated belief, failures into the right repair plane, and judgement into bounded permission to act, without corrupting its source material, its doctrine, its route graph, or its generated pages.","scope_note":"Workingness covers the system's epistemic health as a whole, distinct from ordinary event logging. A reader meets it wherever a result is being interpreted before a change is made: an evidence page showing what a pass proves, a drift check, or a live receipt. The owning module is the Workingness Instrument, and any dossier is a readable projection of that state.","example":"The evidence page, \"What the pass proves\", is a concrete instance: it lays out exactly what a given pass does and does not establish, which is the workingness question made visible for one result.","example_route":{"href":"evidence.html","label":"What the pass proves (evidence)","gloss":"see what a single pass actually establishes"},"non_example":"Workingness is not a green test suite or a build-passing badge. A run can be entirely green and still have low workingness if the system cannot say what remains unknown, what would change its mind, or which actions it is allowed to take.","visible_text_byte_count":1818,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::workingness","source_hash":"sha256:ab74c9b3006a5eebd8ec2cdb31943ed39db5c8947c04bc25c4d30401f8a5baa3","relationships":[{"verb":"projected_by","target":"workingness_instrument","gloss":"The instrument carries the claim model that makes workingness inspectable."},{"verb":"blocked_by","target":"appeal_lifecycle","gloss":"Unresolved appeals can block a workingness state from becoming promotable."},{"verb":"depends_on","target":"living_system_posture","gloss":"Workingness must be read against current substrate, not stale receipts."}],"source_refs":[{"ref":"codex/doctrine/paper_modules/workingness_instrument.md","role":"paper_module","gloss":"Defines the diagnostic truth-maintenance layer and state model."},{"ref":"codex/standards/std_workingness_instrument.json","role":"standard","gloss":"Candidate row contract for workingness claims."}],"evidence_commands":[{"command":"./repo-python kernel.py --paper-module workingness_instrument","reason":"Open the Workingness Instrument paper module."},{"command":"./repo-python kernel.py --term workingness --term-band context","reason":"Read the routable term row."}],"used_in":{"total_occurrences":4,"page_count":3,"top_page":"quickstart.html","top_pages":[{"page":"quickstart.html","count":2},{"page":"paper-module-cold-reader-route-map.html","count":1},{"page":"paper-module-formal-math-readiness-gate.html","count":1}]}},{"schema":"plectis_public_glossary_term_v0","position":433,"id":"workingness_instrument","label":"Workingness Instrument","preferred_label":"Workingness Instrument","alternative_labels":["workingness_instrument","workingness dossier"],"hidden_search_labels":["system workingness instrument","truth-maintaining instrument"],"term_kind":"artifact","meaning_scope_class":"ai_workflow_coined_term","meaning_scope_label":"System-coined term","definition_profile":"governed_system_term","meaning_boundary":"A workingness instrument binds one live result to evidence, invalidators, and allowed or blocked actions; it is not a monitoring dashboard, a correctness verdict, or an authority writer.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-workingness-instrument","relative_href":"docs/glossary.html#glossary-workingness-instrument","public_phrase":"truth-state action boundary","short_definition":"A workingness instrument is the record that states what one live result proves and which changes that evidence permits or blocks.","reader_preview":"A workingness instrument is the record that states what one live result proves and which changes that evidence permits or blocks.","reader_preview_basis":"definition","card_definition":"It is the row-shaped contract for one live outcome. In one place it pins down the kind of claim, its current status, the evidence and dependencies under it, the conditions that would invalidate it, the appeals that block it from being promoted, and the repairs and mutations it allows or forbids. It turns judgments that would otherwise stay implicit into explicit fields: this receipt counts as evidence, this scorer output is a judgment, this projection is stale, this change is forbidden until a named gate passes.","reader_rule":"Read a workingness instrument as the boundary on what a result lets you do: its allowed and blocked actions are the operative part, not the surrounding summary.","deep_definition":"It is a diagnostic contract, not an authority writer: it governs and records what a result permits, but it does not itself change the doctrine, the route graph, or the standards.","scope_note":"The term covers the governing record that binds evidence, judgment, appeals, repairs, and mutation permissions for a single live claim. A reader meets it where the site shows how a passing or failing result is allowed to act on the system, and a summary dossier may stand in front of it.","example":"The Proof Diagnostic Evidence Spine is a concrete instance: it links a specific result to the evidence under it and to what that evidence is permitted to claim, which is exactly the evidence-to-action binding a workingness instrument records.","example_route":{"href":"components.html#component-proof_diagnostic_evidence_spine","label":"Proof Diagnostic Evidence Spine","gloss":"open one component that binds evidence to what it may claim"},"non_example":"A workingness instrument is not a verdict that the whole system is working, and not a tool that edits the system on its own. It is the record of what one result permits, with no power to change doctrine, routes, or standards by itself.","visible_text_byte_count":1780,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::workingness_instrument","source_hash":"sha256:ab74c9b3006a5eebd8ec2cdb31943ed39db5c8947c04bc25c4d30401f8a5baa3","relationships":[{"verb":"defines","target":"workingness","gloss":"The instrument defines the row model that makes workingness inspectable."},{"verb":"specializes","target":"failure_class_propagation","gloss":"Local failures become typed ignorance nodes and repair-plane assignments."},{"verb":"gated_by","target":"standard","gloss":"The row shape is governed by std_workingness_instrument.json."}],"source_refs":[{"ref":"codex/doctrine/paper_modules/workingness_instrument.md","role":"paper_module","gloss":"Readable ontology and first microcosm trace."},{"ref":"codex/standards/std_workingness_instrument.json","role":"standard","gloss":"Machine-readable candidate row contract."}],"evidence_commands":[{"command":"./repo-python kernel.py --paper-module workingness_instrument","reason":"Open the instrument's paper module."},{"command":"jq '.microcosm_trace' codex/standards/std_workingness_instrument.json","reason":"Read the first proof trace from the candidate standard."}],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":434,"id":"zero","label":"zero","preferred_label":"zero","alternative_labels":["additive identity"],"hidden_search_labels":["zero value","0"],"term_kind":"mathematical_concept","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. Repository-specific examples contextualize the term; they do not redefine it or certify one universal definition.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-zero","relative_href":"docs/glossary.html#glossary-zero","public_phrase":"zero","short_definition":"Zero is the number that acts as the additive identity, so adding it leaves a number unchanged.","reader_preview":"Zero is the number that acts as the additive identity, so adding it leaves a number unchanged.","reader_preview_basis":"definition","card_definition":"Zero marks absence of quantity and separates positive from negative numbers. It is an integer, rational number, real number, and\u2014in Lean's convention\u2014a natural number.","reader_rule":"Check whether zero is allowed before dividing, taking reciprocals, or applying a positivity hypothesis.","deep_definition":"Many exceptional cases in algebra and number theory arise because multiplication by zero collapses information and division by zero is undefined.","scope_note":"Zero covers the additive identity in the number system or algebraic structure named by the statement.","example":"For every integer n, n + 0 = n.","example_route":null,"non_example":"Zero is not the same as missing, unknown, false, or an empty text field.","visible_text_byte_count":706,"source_basis":"codex/doctrine/system_vocabulary/term_registry.json::zero","source_hash":"sha256:ab74c9b3006a5eebd8ec2cdb31943ed39db5c8947c04bc25c4d30401f8a5baa3","relationships":[],"source_refs":[],"evidence_commands":[],"used_in":{"total_occurrences":0,"page_count":0,"top_page":null,"top_pages":[]}},{"schema":"plectis_public_glossary_term_v0","position":435,"id":"zone","label":"zone","preferred_label":"zone","alternative_labels":["zones"],"hidden_search_labels":[],"term_kind":"public_projection","meaning_scope_class":"general_or_domain_usage","meaning_scope_label":"General or domain usage","definition_profile":"public_contextual_gloss","meaning_boundary":"The glossary intends ordinary or domain usage. Repository-specific examples contextualize the term; they do not redefine it or certify one universal definition.","meaning_contrasts":[],"status":"active","disposition":"active","anchor":"glossary-zone","relative_href":"docs/glossary.html#glossary-zone","public_phrase":"zone","short_definition":"A zone is one thematic region of the map over the proof corpus, grouping the statements that belong to a single line of argument.","reader_preview":"A zone is one thematic region of the map over the proof corpus, grouping the statements that belong to a single line of argument.","reader_preview_basis":"definition","card_definition":"Zones exist so a reader can enter a corpus somewhere specific rather than at the beginning. Each collects the statements about one mechanism or one family of results, with the relations among them, so a question about that mechanism does not require reading the rest.","reader_rule":"A zone bounds where an argument lives. 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Build with\n% `tectonic erdos-243-reciprocal-tail-rigidity.tex` or `make`.\n\\documentclass[11pt]{article}\n\\input{problem-note-preamble}\n% Problem-specific source pin: #243's finite-negative-mass results advance\n% independently of the corpus default.\n\\renewcommand{\\commit}{69f870a9a5398e1373953ece77b1b28ba09ea5ba}\n\\renewcommand{\\commitshort}{69f870a9a539}\n\\hypersetup{\n  pdftitle={Excluding the bounded negative part: Lean-checked rigidity for Erd\u0151s Problem 243},\n  pdfsubject={Erd\u0151s Problem Notes: Problem 243},\n  pdfkeywords={irrationality, Ahmes series, Sylvester sequence, unit fractions, Lean 4},\n}\n\n% Names for the state maps and the two search objects.  They are set as\n% upright operators so that they read as maps rather than as products.\n\\newcommand{\\sylnext}{\\operatorname{syl}}\n\\newcommand{\\den}{\\operatorname{den}}\n\\newcommand{\\tail}{\\operatorname{tail}}\n\\newcommand{\\ctr}{\\operatorname{ctr}}\n\\newcommand{\\defect}{\\operatorname{def}}\n\\newcommand{\\fnum}{\\operatorname{num}}\n\\newcommand{\\surv}{\\operatorname{Surv}}\n\n\\runninghead{Erd\\H{o}s \\#243: excluding the bounded negative part}\n\\title{Excluding the Bounded Negative Part}\n\\subtitle{Lean-checked rigidity for Erd\\H{o}s Problem~\\#243}\n\\author{Will Cook}\n\\date{July 2026}\n\n\\begin{document}\n\\maketitle\n\\noteprovenance\n\n\\begin{abstract}\n\\noindent\nSylvester's sequence $2,3,7,43,1807,\\ldots$ satisfies\n$a_n=a_{n-1}^{2}-a_{n-1}+1$ and has reciprocal sum $1$.  Erd\\H{o}s asked\nwhether a rapidly growing integer sequence with rational reciprocal sum must\nsatisfy that recurrence eventually.  Clearing denominators gives integer states\n$D_{n+1}=a_nD_n$ and $C_{n+1}=a_nC_n-D_n$, and centring at the Sylvester tail\ngives $E_n=D_n-(a_n-1)C_n$, with $C_{n+1}=C_n-E_n$; the question becomes\nwhether $E$ can avoid vanishing.  When $E$ is nonnegative a one-line descent\nsettles it.  This note is about the negative side.\n\nWe formalise three arithmetic obstructions and two conditional consequences\nfor the state system.  First, $E$ cannot be constantly $-m$, at any magnitude\nand any scale: every multiplier would have to share a prime with the fixed\n$m$, whereas a prime dividing one multiplier cannot divide any later one.\nSecond, a periodic negative magnitude is impossible when the associated drift\nis positive and the magnitude at index $n$ is smaller than $a_n$.  Third, an\ninteger sequence tending to infinity with upward steps bounded by $B$ cannot\npermanently avoid infinitely many pairwise coprime moduli; a Chinese-remainder\nblock of consecutive multiples forces a hit.  This bounded-rise theorem does\nnot depend on the reciprocal-tail dynamics.\n\nApplied to a reduced tail, the bounded-rise theorem gives one conditional\nendpoint: under exact dynamics, eventual strict centring, normalised\nvanishing, and an eventually bounded negative part, $E$ vanishes eventually.\nA separate summability argument gives another: under normalised vanishing,\nfinite normalised negative mass also forces eventual vanishing.  These are\nparallel conclusions, not successive levels of generality.  In either case an\nabsorption lemma then propagates $E_n=0$, and the sequence satisfies the\nSylvester recurrence eventually.\n\nProblem~\\#243 is open and this note does not close it.  The last theorem is\nconditional, and two of its hypotheses are analytic inputs that we assume\nrather than derive.  The results separately remove the bounded case and the\nfinite normalised negative-mass case.  Under normalised vanishing, summability of\n\\(\\sum_n(-E_n)_+/C_n\\) makes \\(C_n\\) bounded and hence \\(E_n\\) eventually\nzero.  Thus any surviving exact state orbit must have negative excursions\nunbounded along a cofinal set and divergent normalised negative mass.\n\\end{abstract}\n\n\\section{Introduction}\\label{sec:problem}\n\nSylvester's sequence $2,3,7,43,1807,\\ldots$ is generated by\n$a_{n}=a_{n-1}^{2}-a_{n-1}+1$ and has reciprocal sum $1$; the recurrence is\nexactly the statement that the tail beyond $a_{n-1}$ equals $1/(a_{n-1}-1)$.\nErd\\H{o}s Problem~\\#243 asserts that this is the only way a sequence of that\ngrowth can have a rational reciprocal sum:\n\n\\begin{problem}[Erd\\H{o}s \\#243]\\label{res:problem}\nLet $1\\le a_1<a_2<\\cdots$ be a sequence of integers with\n\\[\n \\lim_{n\\to\\infty}\\frac{a_n}{a_{n-1}^{2}}=1\n \\qquad\\text{and}\\qquad\n \\sum\\frac{1}{a_n}\\in\\Q .\n\\]\nThen $a_n=a_{n-1}^{2}-a_{n-1}+1$ for all sufficiently large $n$.\n\\end{problem}\n\n\\noindent See~\\cite[p.~64]{erdosgraham1980} and~\\cite[p.~105]{erdos1988}.\nBloom's current catalogue record reproduces the displayed problem and labels\nit open, while explicitly warning that the status is the website owner's\npresent assessment and may not reflect all relevant literature\n\\cite{erdosproblems}.  We therefore use the catalogue for numbering and\ncurrent reported status only; the original publications carry the\nmathematical claims.  Below we index the same recurrence as\n$a_{n+1}=a_n^{2}-a_n+1$, which is the shift the formal sources use; the two\nforms are the same statement.  Write $\\sylnext(a)=a^{2}-a+1$, the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{37}{sylvesterNext}{Sylvester\nsuccessor}.\n\n\\paragraph{Relation to prior work.}\nTwo published results reach the original problem under analytic hypotheses that\nare not assumed below.  Their hypotheses are conditions on the sequence $(a_n)$\nitself.  Duverney's is not comparable with the state-level hypotheses used\nhere; the Erd\\H{o}s--Straus one, read through the dictionary of the next\nparagraph, is implied by nonnegativity of $E$, the hypothesis of\nTheorem~\\ref{res:descent}, so that criterion already covers the descent proved\nbelow (see~\\cite[Cor.~4(1) and Prop.~1(1), pp.~14--15]{koizumi2025}).\nErd\\H{o}s and Straus proved that if $\\lim a_n/a_{n-1}^{2}=1$, the reciprocal\nsum is rational, and $a_n$ does not satisfy the recurrence, then\n\\[\n \\limsup_{n\\to\\infty}\\ \\frac{[a_1,\\ldots,a_n]}{a_{n+1}}\n \\left(\\frac{a_{n+1}^{2}}{a_{n+2}}-1\\right)>0,\n\\]\nwhere $[a_1,\\ldots,a_n]$ is the least common\nmultiple~\\cite[Theorem~3, p.~132]{erdosstraus1964}.  This is the indexing in\nthe original theorem.  Erd\\H{o}s's 1988 survey, followed by the current\ncatalogue summary, instead prints $a_n^2/a_{n+1}-1$ in the second factor while\nretaining the same prefix-LCM quotient; that displayed summary is off by one\nand is not followed here~\\cite[p.~105]{erdos1988}\\cite{erdosproblems}.\nKoizumi records the convenient sufficient rate\n$a_n^2/a_{n+1}=1+o(1/n)$ under which the Erd\\H{o}s--Straus criterion settles\nthe problem~\\cite[Remark~3, p.~16]{koizumi2025}.  Duverney proved a\nconditional signed form of the\nproblem itself: if\n$\\sum_{n\\ge0}\\left|a_{n+1}/a_n^{2}-1\\right|$ converges, then\nthe reciprocal sum, even with numerators in $\\{-1,1\\}$, is rational if and\nonly if the corresponding signed recurrence holds for all large\n$n$~\\cite[Corollary~3.2, p.~287]{duverney2001}.  The all-positive\nspecialisation is the form relevant here.  Neither result is formalised here,\nand the results below do not recover either of them.\n\nKoizumi's work on doubly exponential sequences~\\cite{koizumi2025} bears\ndirectly on the state system used below, and supplies the coordinates in which\nseveral of the statements of this note are most naturally read.  For a rational\n$r=p/q$ with pseudo-greedy expansion $(a_n)$, remainders $(x_n)$ and gap\nsequence $\\varepsilon_n=x_n^{-1}+1-a_n$, his Lemma~4\n~\\cite[pp.~11--12]{koizumi2025} produces integers $c_n>0$,\n$d_n$ and $e_n$ with $x_n=c_n/d_n$ and $\\varepsilon_n=e_n/c_n$, satisfying\n\\[\n a_n=\\frac{d_n-e_n}{c_n}+1,\\qquad\n c_{n+1}=c_n-e_n,\\qquad\n d_{n+1}=a_nd_n ,\n\\]\nand the proof of that lemma also gives $c_{n+1}=a_nc_n-d_n$.  Under the\ndictionary $(C_n,D_n,E_n)=(c_n,d_n,e_n)$ these are exactly the objects of\nSection~\\ref{sec:state}: the first relation is the centring map $\\ctr$ rewritten\nas $e_n=d_n-(a_n-1)c_n$, the second is Proposition~\\ref{res:update}, the third\nis the denominator update, and the fourth is the tail update.  The lower-case\n$e_n$ of Section~\\ref{sec:state} is $-E_n$, so it is the negative of Koizumi's\n$e_n$ at an index where the centred state is negative.  His Theorem~3\n~\\cite[pp.~12--13]{koizumi2025} shows\nthat his Conjecture~1 --- that for a positive rational $r$ whose gap sequence\nsatisfies $\\varepsilon_n\\to0$ one has $\\varepsilon_n=0$ for all large $n$ ---\nis equivalent to an affirmative answer to the question of Erd\\H{o}s and Graham\nrecorded as his Question~1 on p.~3, which is Problem~\\ref{res:problem} above.\nTwo of the implications proved below have prior art in these canonical\ncoordinates: his Lemma~3, that $\\varepsilon_n=0$ forces $\\varepsilon_{n+1}=0$,\nis the absorption of Theorem~\\ref{res:absorb}, and his Proposition~1(2), that\n$\\varepsilon_n\\ge0$ for all large $n$ forces $\\varepsilon_n=0$ for all large\n$n$, is the descent of Theorem~\\ref{res:descent}.  These appear in canonical\ncoordinates in~\\cite[Lemma~3, p.~10; Prop.~1(2), p.~14]{koizumi2025}; no\nquestion of priority or independence is\nadjudicated here.\n\nThe growth hypothesis in Problem~\\ref{res:problem} is calibrated by two facts\nabout Sylvester's sequence.  It satisfies $a_n\\approx c_0^{2^{n}}$ with\n$c_0=1.2640847\\ldots$, and shifting it further produces sequences with\n$a_n\\approx C^{2^{n}}$ for arbitrarily large $C$ whose reciprocals still sum\nto a rational number~\\cite[arXiv v4, p.~2]{kovactao2024}.  The classical sufficient\ncondition for irrationality, $\\lim_n a_n^{1/2^{n}}=\\infty$, is therefore\nsharp.  Kova\\v{c} and Tao identify that condition as folklore\n~\\cite[arXiv v4, p.~2]{kovactao2024}; they attribute the sharpness observation to\nErd\\H{o}s (1975).  A sequence with\n$a_n/a_{n-1}^{2}\\to1$ has $a_n^{1/2^{n}}$\nconvergent, so that criterion says nothing about the sequences of\nProblem~\\ref{res:problem}, and rationality is genuinely possible there.\nSylvester's\nsequence is \\texttt{A000058} in the OEIS and its shifts \\texttt{A129871}.  For\nthe recent literature on irrationality of Ahmes series we refer to Kova\\v{c}\nand Tao~\\cite{kovactao2024}, who resolve several problems of Erd\\H{o}s and\nGraham drawn from the same two sources cited above, and whose introduction\ngives a sample of the intermediate work, including S\\'andor (1984) and Badea\n(1987).  They do not treat Problem~\\#243; the rigidity conclusion asked for\nthere is not among their results.\n\nThe \\emph{Formal Conjectures} collection contains a mathematically equivalent\nunproved declaration, up to its zero-based indexing\n~\\cite{formalconjectures243}.  Its summand is $\\Q$-valued, so Lean's\n\\texttt{Summable} hypothesis asserts the existence of a sum in $\\Q$; the\nfinite indexing shift changes that sum only by a rational prefix.  The\ndeclaration therefore does encode the rationality premise, but its proof is\n\\texttt{sorry}.  We use it as statement-level prior art, not as proof\nauthority; the development below is independent of it.\nWhat is machine-checked below concerns the state system rather than the\nanalytic problem: the exclusion of constant and\nperiodic negative magnitudes, the bounded-rise barrier of\nTheorem~\\ref{res:barrier}, and the two conditional endpoints\n(Theorems~\\ref{res:bounded} and~\\ref{res:mass}), each stated with the exact\nhypotheses that separate it from Problem~\\#243.  No claim of priority is made\nfor anything below.\n\nThe hypothesis is asymptotic and the conclusion is exact, so the argument must\nconvert an analytic rate into an integer obstruction.  The conversion used here\nis the classical one: clear denominators along the sequence, so that\nrationality makes a tail integral, and then centre that integer at the value it\nwould take on Sylvester's sequence.  What remains is a single integer error\n$E$, and the whole question is whether $E$ can avoid vanishing.\n\n\\statusboundary\n\n\\begin{table}[t]\n\\centering\n\\small\n\\begin{tabular}{@{}\n  >{\\raggedright\\arraybackslash}p{0.36\\linewidth}\n  >{\\raggedright\\arraybackslash}p{0.17\\linewidth}\n  >{\\raggedright\\arraybackslash}p{0.39\\linewidth}@{}}\n\\toprule\nStatement & Status & Treatment here \\\\\n\\midrule\nProblem~\\#243 as stated & Open & Not proved. \\\\\nDefect identity & Proved here & Theorem~\\ref{res:defect}, an identity in $\\Z[a,a',D,C]$. \\\\\nEventual vanishing forces the recurrence & Proved here & Theorem~\\ref{res:eventual}. \\\\\nZero is absorbing & Proved here & Theorem~\\ref{res:absorb}, under strict centring. \\\\\nNonnegative $E$ & Proved here & Theorem~\\ref{res:descent}, by descent. \\\\\nConstantly negative $E$ & Excluded here & Theorem~\\ref{res:constant}, at any magnitude and scale. \\\\\nPeriodic negative magnitude & Excluded here & Theorem~\\ref{res:periodic}, in the regime $e_n<a_n$ and with positive drift $M>0$. \\\\\nCoprime-modulus barrier & Proved here & Theorem~\\ref{res:barrier}. \\\\\nBounded negative part & Excluded here; conditional & Theorem~\\ref{res:bounded}; see its hypotheses. \\\\\nFinite normalised negative mass & Excluded here; conditional & Theorem~\\ref{res:mass}, Section~\\ref{sec:mass}; normalised vanishing remains an input. \\\\\nFactorial residue reduction & Proved here; superseded for its original purpose & Theorem~\\ref{res:residue}, Appendix~\\ref{app:residue}. \\\\\nStrict centring, normalised vanishing & Assumed here; supplied by prior art & Section~\\ref{sec:bounded}; derived in~\\cite{koizumi2025}, see Section~\\ref{sec:problem}. \\\\\nUnbounded, divergent-mass negative excursions & Open & Section~\\ref{sec:open}; the surviving obstruction. \\\\\nErd\\H{o}s--Straus criterion; Duverney & Proved elsewhere & Cited, not formalised; the Erd\\H{o}s--Straus hypothesis is implied by that of Theorem~\\ref{res:descent}. \\\\\nState system equals reciprocal tails & Not formalised & Section~\\ref{sec:state}. \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\noindent Taken together the exclusions constrain the shape of any\ncounterexample.  By Theorem~\\ref{res:absorb} a counterexample has $E_n\\ne0$ for\nevery large $n$.  By Theorem~\\ref{res:descent} it cannot have $E$ eventually\nnonnegative.  By Theorems~\\ref{res:constant} and~\\ref{res:periodic} its\nnegative magnitudes are neither eventually constant nor eventually periodic.\nAnd by Theorems~\\ref{res:bounded} and~\\ref{res:mass}, if the two analytic\nhypotheses of Section~\\ref{sec:bounded} hold, the negative part is neither\nbounded nor of finite normalised mass.  What survives is stated in\nSection~\\ref{sec:open}.\n\n\\subsection*{Structure}\n\nSection~\\ref{sec:state} sets up the state system, and Section~\\ref{sec:defect}\nproves the defect identity, from which eventual vanishing of $E$ forces the\nSylvester recurrence and a single vanishing error propagates.\nSection~\\ref{sec:descent} disposes of the case $E\\ge0$ in three lines.  The\nnegative side then splits into distinct regimes.  Sections~\\ref{sec:constant}\nand~\\ref{sec:periodic} exclude two rigid patterns.  Section~\\ref{sec:barrier}\nproves the dynamics-free bounded-rise theorem, and\nSection~\\ref{sec:bounded} applies it to an eventually bounded negative part.\nSection~\\ref{sec:mass} instead uses a summability argument; its endpoint is not\npresented as stronger than the bounded-negative-part theorem.  The exact\nremaining regime is recorded in Section~\\ref{sec:open}.  A reader who wants\nonly the two conditional state-level endpoints should read\nSections~\\ref{sec:bounded} and~\\ref{sec:mass}, followed by\nSection~\\ref{sec:open}.  Appendix~\\ref{app:index} is a guide to the formal\nsources, and Appendix~\\ref{app:residue} records an exact residue reduction that\nan earlier finite search used.  Linked phrases open the corresponding Lean\ndeclaration at the pinned source revision~\\commitshort.\n\n\\medskip\n\\noindent\\textbf{Keywords.} irrationality; Ahmes series; Sylvester's sequence;\nunit fractions; Lean~4.\n\\qquad\n\\textbf{MSC 2020.} 11J72 (primary); 11B37, 11D68, 68V20 (secondary).\n\n\\section{The centred integer state}\\label{sec:state}\n\nThe three maps defined next carry out the two steps announced at the end of\nSection~\\ref{sec:problem}.  The first two clear denominators along the\nsequence, so that a rational reciprocal sum leaves a sequence of integers; the\nthird subtracts from that integer the value it takes on Sylvester's sequence,\nleaving one number whose vanishing is the whole question.  All three are maps\nbetween integer tuples and carry no hypothesis of any kind.\n\nFor $a,D,C\\in\\Z$ define\n\\[\n \\den(a,D)=aD,\n \\quad\n \\tail(a,D,C)=aC-D,\n \\quad\n \\ctr(a,D,C)=D-(a-1)C ,\n\\]\nthe\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{41}{nextDenState}{denominator\nupdate}, the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{45}{nextTailState}{tail update},\nand the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{49}{centeredState}{centred state}.\nGiven a sequence $(a_n)$ of integers, whose terms we call the\n\\emph{multipliers}, we write $D_{n+1}=\\den(a_n,D_n)$ for the \\emph{denominator\nstate}, $C_{n+1}=\\tail(a_n,D_n,C_n)$ for the \\emph{tail state}, and\n$E_n=\\ctr(a_n,D_n,C_n)$ for the \\emph{centred state}, also called the\n\\emph{error}.  The word \\emph{centred} records what the third map does: it\nmeasures $D_n$ against $(a_n-1)C_n$, and $D_n=(a_n-1)C_n$ holds at every index\non Sylvester's sequence (Example~\\ref{ex:sylvester}), so $E$ is a deviation\nfrom that sequence rather than a size.  At an index where $E_n<0$ we write\n$e_n=-E_n>0$ and call it the \\emph{magnitude} of the error there.\n\n\\begin{proposition}[update law]\\label{res:update}\n$\\tail(a,D,C)=C-\\ctr(a,D,C)$; that is, $C_{n+1}=C_n-E_n$.\n\\end{proposition}\n\n\\begin{proof}\n$C-\\bigl(D-(a-1)C\\bigr)=aC-D$.\n\\end{proof}\n\n\\noindent Formalised as the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{57}{nextTailState_eq_sub_centered}{update\nlaw}.  The state therefore moves by exactly the error at each step, which is\nwhat makes the sign of $E$ decisive: where $E$ is positive the state falls,\nwhere $E$ is negative it rises, and where $E$ vanishes it is stationary.\n\n\\begin{example}[the Sylvester orbit]\\label{ex:sylvester}\nTake $a_n=2,3,7,43,1807,\\ldots$ with $a_{n+1}=a_n^{2}-a_n+1$, and start the\nstate at $D_0=C_0=1$.  The two updates give\n\n\\begin{center}\n\\begin{tabular}{@{}rrrrr@{}}\n\\toprule\n$n$ & $a_n$ & $D_n$ & $C_n$ & $E_n$\\\\\n\\midrule\n$0$ & $2$ & $1$ & $1$ & $0$\\\\\n$1$ & $3$ & $2$ & $1$ & $0$\\\\\n$2$ & $7$ & $6$ & $1$ & $0$\\\\\n$3$ & $43$ & $42$ & $1$ & $0$\\\\\n$4$ & $1807$ & $1806$ & $1$ & $0$\\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\n\\noindent and $D_n=a_n-1$ with $C_n=1$ at every index: if $D_n=a_n-1$ and\n$C_n=1$ then $C_{n+1}=a_n-(a_n-1)=1$ and $D_{n+1}=a_n(a_n-1)=a_{n+1}-1$.\nHence $E_n=D_n-(a_n-1)C_n=0$ throughout and the tail state never moves, which\nis Proposition~\\ref{res:update} in the stationary case.\n\\end{example}\n\n\\paragraph{The intended reading, which is not formalised.}\nLet $T_n=\\sum_{k\\ge n}1/a_k$ and suppose $T_0\\in\\Q$.  Put $D_0$ equal to a\ncommon denominator and $D_n=D_0a_0\\cdots a_{n-1}$, and set $C_n=D_nT_n$.  Then\n$C_n\\in\\Z$ for every $n$, and from $T_n=1/a_n+T_{n+1}$ one gets\n$C_{n+1}=a_nC_n-D_n$, which is the tail update.  On Sylvester's sequence\n$T_n=1/(a_n-1)$ exactly, so $D_n=(a_n-1)C_n$ and $E_n=0$: the centred state\nmeasures deviation from the Sylvester tail identity, and it vanishes\nidentically on the Sylvester orbit.\n\nNone of this paragraph is formalised.  The Lean module contains no series, no\nrationality hypothesis, and no growth hypothesis; it proves identities and\nimplications about the integer system above, and about its natural-number\nrealisation, for which the bridging identities are the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1863}{natTail_eq_nextTailState}{tail\nrealisation}, the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1875}{natDen_eq_nextDenState}{denominator\nrealisation}, and the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1884}{natTail_eq_sub_centeredState}{signed\nupdate law}.  A reader should treat the identification with reciprocal tails as\nmotivation for the definitions and not as a checked step.\n\n\\begin{proposition}[scale equivariance]\\label{res:scale}\nFor every $s,a,D,C\\in\\Z$,\n\\[\n \\den(a,sD)=s\\,\\den(a,D),\\qquad\n \\tail(a,sD,sC)=s\\,\\tail(a,D,C),\n\\]\n\\[\n \\ctr(a,sD,sC)=s\\,\\ctr(a,D,C).\n\\]\n\\end{proposition}\n\n\\noindent Formalised as the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{65}{nextDenState_scale}{denominator\nequivariance}, the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{70}{nextTailState_scale}{tail\nequivariance}, and the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{75}{centeredState_scale}{centred\nequivariance}.  The system depends only on the ray through $(D,C)$.  This is\nused twice below: once to normalise the finite search of\nAppendix~\\ref{app:residue}, and once, in Theorem~\\ref{res:periodic}, as the\ninduction step that removes common scale.\n\n\\section{The defect identity and its two consequences}\\label{sec:defect}\n\nSection~\\ref{sec:state} connects the Sylvester recurrence to the error in one\ndirection only: on Sylvester's sequence the error vanishes.  The converse is\nwhat the problem needs, and it comes from a single identity, which expresses\nthe deviation of $a_{n+1}$\nfrom the Sylvester successor, multiplied by the tail state $C_{n+1}$, as a\ncombination of the errors at $n$ and at $n+1$, the first weighted by\n$a_n^{2}$.  Its two consequences are\nTheorem~\\ref{res:eventual}, in which an eventually vanishing error forces the\nrecurrence provided the tail state is eventually nonzero, and\nTheorem~\\ref{res:absorb}, in which, under strict centring (the condition\n$|E_n|<C_n$ at every index), one vanishing error propagates to every later\nindex.\n\nWrite $\\defect(a,a')=a'-\\sylnext(a)$ for the deviation of the next term from\nthe Sylvester successor, the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{53}{sylvesterDefect}{Sylvester\ndefect}.\n\n\\begin{theorem}[defect identity]\\label{res:defect}\nFor all $a,a',D,C\\in\\Z$,\n\\[\n \\defect(a,a')\\cdot\\tail(a,D,C)\n =a^{2}\\,\\ctr(a,D,C)-\\ctr\\bigl(a',\\den(a,D),\\tail(a,D,C)\\bigr),\n\\]\nthat is, $\\Delta_n\\,C_{n+1}=a_n^{2}E_n-E_{n+1}$.\n\\end{theorem}\n\n\\begin{proof}\nA direct expansion: both sides equal\n$a'aC-a'D-a^{3}C+a^{2}D+a^{2}C-aD-aC+D$.\n\\end{proof}\n\n\\noindent Formalised as the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1769}{sylvesterDefect_mul_nextTailState}{defect\nidentity}.  It is a polynomial identity in four indeterminates, with no\nhypothesis of any kind, and it carries the whole argument twice over: once for\nrigidity, and once for absorption.\n\n\\begin{example}[one defect and the error it creates]\\label{ex:defect}\nContinue Example~\\ref{ex:sylvester} but replace $a_3=43$ by $a_3=44$.  The\nstates $D_3=42$ and $C_3=1$ are unchanged, since they depend only on\n$a_0,a_1,a_2$, and $E_2=0$ still.  The defect is\n$\\Delta_2=a_3-\\sylnext(a_2)=44-43=1$, and\n$E_3=D_3-(a_3-1)C_3=42-43=-1$, so the identity reads\n\\[\n \\Delta_2C_3=1\\cdot1=1=49\\cdot0-(-1)=a_2^{2}E_2-E_3 .\n\\]\nBy Proposition~\\ref{res:update} the tail state then rises, $C_4=C_3-E_3=2$.  A\nsingle unit of defect at one index has produced a negative error of magnitude\n$1$ at the next, and the state has started to climb.\n\\end{example}\n\n\\begin{theorem}[two vanishing errors force the step]\\label{res:step}\nLet $a,a',D,C\\in\\Z$ with $\\tail(a,D,C)\\ne0$.  If\n$\\ctr(a,D,C)=0$ and $\\ctr\\bigl(a',\\den(a,D),\\tail(a,D,C)\\bigr)=0$, then\n$a'=\\sylnext(a)$.\n\\end{theorem}\n\n\\begin{proof}\nBy Theorem~\\ref{res:defect} the product $\\defect(a,a')\\cdot\\tail(a,D,C)$ is\nzero, and the second factor is nonzero by hypothesis.\n\\end{proof}\n\n\\noindent Formalised as the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1781}{sylvesterNext_eq_of_centered_zero}{local\nrigidity step}.  The hypothesis $C_{n+1}\\ne0$ is not removable: at $C_{n+1}=0$\nthe identity gives no information about $a'$.\n\n\\begin{theorem}[sequence form]\\label{res:eventual}\nLet $a,D,C:\\N\\to\\Z$ satisfy $D_{n+1}=\\den(a_n,D_n)$ and\n$C_{n+1}=\\tail(a_n,D_n,C_n)$.  If $E_n=0$ for all sufficiently large $n$ and\n$C_{n+1}\\ne0$ for all sufficiently large $n$, then $a_{n+1}=a_n^{2}-a_n+1$ for\nall sufficiently large $n$.\n\\end{theorem}\n\n\\noindent Formalised as the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1799}{sylvesterNext_eventually_of_centered_zero}{eventual\nSylvester recurrence}.  The proof is routine: take the larger of the two\nthresholds and apply Theorem~\\ref{res:step} at each later index.  Every later\nsection is directed at the hypothesis of this theorem, that is, at forcing $E$\nto vanish eventually.\n\nThe defect identity has a second consequence, which is what makes a single\nvanishing error worth having.\n\n\\begin{theorem}[zero is absorbing]\\label{res:absorb}\nLet $a,C,D:\\N\\to\\N$ be an exact natural orbit, so\n$C_{n+1}+D_n=a_nC_n$ and $D_{n+1}=a_nD_n$, and let\n$E_n=\\ctr(a_n,D_n,C_n)$.  Suppose the centring is strict, $|E_n|<C_n$ for\nevery $n$.  If $E_n=0$ then $E_{n+1}=0$.\n\\end{theorem}\n\n\\begin{proof}\nPutting $E_n=0$ in Theorem~\\ref{res:defect} gives\n$\\Delta_n C_{n+1}=-E_{n+1}$, so $C_{n+1}$ divides $E_{n+1}$.  A multiple of\n$C_{n+1}$ of absolute value smaller than $C_{n+1}$ is zero, and\n$|E_{n+1}|<C_{n+1}$ by hypothesis.\n\\end{proof}\n\n\\noindent Formalised as the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{2133}{centeredState_zero_absorbing}{absorption\nof a vanishing centred state}, with the companion\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{2160}{eventually_nonzero_of_zero_absorbing}{contrapositive\nalong a tail}: if zero is absorbing beyond some index and $E$ does not vanish\neventually, then $E$ is nowhere zero beyond that index.\n\nAbsorption changes the shape of the problem.  Without it, one would have to\nexclude an $E$ that dips to zero and recovers.  With it, either $E$ hits zero\nonce and is finished, or it is never zero at all; the later sections may\ntherefore assume that $E_n\\ne0$ for every $n$, and in the negative case that\n$E_n<0$ throughout, without loss.\n\n\\section{Descent when the error is nonnegative}\\label{sec:descent}\n\nThe hypothesis of the theorem below is the update law of\nProposition~\\ref{res:update} read inside $\\N$: the error is nonnegative at\nevery index, so the tail state never rises.  This is the easy case, and it is\nsettled in three lines.\n\n\\begin{theorem}[descent]\\label{res:descent}\nLet $C,E:\\N\\to\\N$ satisfy $C_{n+1}+E_n=C_n$ for every $n$.  Then $E_n=0$ for\nall sufficiently large $n$.\n\\end{theorem}\n\n\\begin{proof}\nA one-line descent.  The relation forces $C_{n+1}\\le C_n$, so $C$ is a nonincreasing sequence of\nnatural numbers and is eventually equal to its minimum; beyond that index\n$C_{n+1}=C_n$, and the relation gives $E_n=0$.\n\\end{proof}\n\n\\noindent Formalised as the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1837}{centeredState_eventually_zero}{stabilisation\nof the nonnegative error}, on the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1819}{antitone_nat_eventually_constant}{eventual\nconstancy of a nonincreasing sequence}.\n\nThe hypothesis is exactly one-sidedness: $C$ and $E$ take values in $\\N$.  If\n$E_n<0$ at infinitely many indices then $C$ rises there, nothing descends, and\nthe argument is unavailable.  That is the whole difficulty, and the rest of\nthis note is directed at it.\n\n\\section{Excluding a constant negative magnitude}\\label{sec:constant}\n\nSuppose the error is negative with a constant magnitude, $E_n=-m$ for a fixed\n$m>0$ and every $n$.  By\nProposition~\\ref{res:update} the state then climbs by exactly $m$ each step,\nso $C_n=c+nm$ with $c=C_0$, and the definition of the centred state becomes a\n\\emph{shape equation}\n\\[\n D_n+m=(a_n-1)\\,(c+nm) .\n\\tag{5.1}\\label{eq:shape}\n\\]\nTogether with $D_{n+1}=a_nD_n$ this is a closed system in $(a,D)$, and it has\nno solutions.  The following instance shows how the failure happens; the\ntheorem then shows that it cannot be avoided.\n\n\\begin{example}[the shape equation running until it fails]\\label{ex:shape}\nTake $m=c=1$, so that $C_n=1+n$ and the shape equation~\\eqref{eq:shape} reads\n$D_n+1=(a_n-1)(1+n)$, and take $D_0=1$.  Each step is now determined: at $n=0$\nthe equation reads $2=(a_0-1)\\cdot1$, so $a_0=3$, and $D_1=a_0D_0=3$; at $n=1$\nit reads $4=(a_1-1)\\cdot2$, so $a_1=3$, and $D_2=a_1D_1=9$; at $n=2$ it reads\n$10=(a_2-1)\\cdot3$, which has no integer solution, and the orbit stops.\nLonger prefixes occur for other starting values: the finite search recorded in\nAppendix~\\ref{app:residue} found none longer than $17$ among the initial\nstates below $5000$.\n\\end{example}\n\n\\begin{theorem}[no constant negative magnitude]\\label{res:constant}\nThere is no pair of sequences $a,D:\\N\\to\\N$ with $a_n\\ge2$ for all $n$\nsatisfying $D_{n+1}=a_nD_n$ and~\\eqref{eq:shape}, for any $m>0$ and any\n$c$.  The same holds if the shape only begins at some index.\n\\end{theorem}\n\n\\begin{proof}[Proof when $c$ and $m$ are coprime]\nAssume first $\\gcd(c,m)=1$.  Two facts about the multipliers do all the work,\nand the difficulty lies in the fact that they concern the same finite set of\nprimes from opposite directions: the first makes every multiplier meet that\nset, the second lets each prime of the set meet at most one multiplier.\n\n\\emph{Every $a_j$ shares a prime with $m$.}  Suppose $\\gcd(a_j,m)=1$.  Then\n$m$ is invertible modulo $a_j$, so some $n$ has $c+nm\\equiv0$, that is\n$a_j\\mid C_n$; and $a_j\\mid D_n$ for every $n>j$, since $D$ is multiplicative\nwith $a_j$ among its factors.  Choosing such an $n$ beyond $j$\nand reading~\\eqref{eq:shape} modulo $a_j$ gives $a_j\\mid m$, contradicting\n$\\gcd(a_j,m)=1$ and $a_j\\ge2$.\n\n\\emph{A prime divisor of $m$ occurs in at most one $a_j$.}  Suppose $p\\mid m$\nand $p\\mid a_j$.  For $n>j$ we have $p\\mid D_n$, so~\\eqref{eq:shape} gives\n$(a_n-1)C_n\\equiv0 \\pmod p$.  Now $C_n=c+nm\\equiv c$, and $p\\nmid c$ because\n$p\\mid m$ and $\\gcd(c,m)=1$; hence $p\\mid a_n-1$, so $p\\nmid a_n$.  Thus $p$\ncannot divide any later multiplier.\n\nThe two facts are incompatible.  Each of the infinitely many multipliers needs\na prime divisor of $m$, and each prime divisor of $m$ serves at most one\nmultiplier, while $m$ has only finitely many prime divisors.\n\\end{proof}\n\n\\noindent The case $\\gcd(c,m)=1$ is the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{147}{no_scalePrimitiveConstantNegative_orbit}{scale-primitive\nexclusion}.  The special case $m=c=1$, worked through in\nExample~\\ref{ex:shape}, where $m$ has no prime divisors at all and the first\nfact is immediately contradictory, is recorded separately as the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{100}{no_normalizedConstantNegative_orbit}{normalised\nexclusion}.\n\n\\begin{proof}[Removing the scale]\nFor general $c$, put $g=\\gcd(c,m)$.  Equation~\\eqref{eq:shape} shows\n$g\\mid D_n$ for every $n$, so dividing $D$, $c$ and $m$ by $g$ leaves a system\nof the same shape with coprime data, which the previous case excludes.\n\\end{proof}\n\n\\noindent Formalised as the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{286}{no_constantNegative_orbit}{exclusion\nat every scale}, and the eventual form, obtained by shifting the orbit to the\nfirst constant index, as the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{350}{no_eventuallyConstantNegative_orbit}{eventual\nexclusion}.\n\nTheorem~\\ref{res:constant} excludes a centred state that is eventually constant\nand negative, at every magnitude and every scale.  It says nothing about a\nnegative state whose magnitude changes, and the argument genuinely uses\nconstancy: the finiteness of the set of prime divisors of $m$ is what the\npigeonhole uses, and a varying magnitude presents a fresh set at each index.\n\n\\section{Excluding a periodic negative magnitude}\\label{sec:periodic}\n\nThe next case allows the magnitude to vary, provided it repeats.  Write\n$e_n=-E_n>0$ for the magnitude, as in Section~\\ref{sec:state}, so that\n$C_{n+1}=C_n+e_n$; suppose $e$ has period $h$, meaning $e_{n+h}=e_n$ for every\n$n$, and that the state gains a fixed \\emph{drift} $M>0$ over one period,\nmeaning $C_{n+h}=C_n+M$ for every $n$.\n\nAt $h=1$ this says that the magnitude is constant and that $M$ is its value,\nwhich is the situation of Section~\\ref{sec:constant};\nTheorem~\\ref{res:constant} already excludes it, and does so without the extra\nhypothesis $e_n<a_n$ imposed below.  The new content is $h\\ge2$.\n\nSection~\\ref{sec:constant} could run its pigeonhole on the prime divisors of\nthe single number $m$.  Here the magnitude changes inside a period, and the\nnumber available to run it on is the drift $M$.  Three lemmas replace the two\nfacts of Section~\\ref{sec:constant}.  The first is\nthat multipliers persist: every $a_j$ divides every strictly later denominator\nstate, the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{382}{base_dvd_denState_later}{persistence\nof a multiplier}.  The second is a \\emph{lock}: a prime already dividing $D$ and also\ndividing the current multiplier is forced into the next tail state, the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{402}{divisorLocksTailSucc}{one-step\nlock}, and once present in both $D$ and $C$ it stays in every later $D$, $C$\nand magnitude, the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{429}{divisorLock_persists}{persistence\nof a lock}.  The third transports a lock backwards through the period: if a\ndivisor of the drift $M$ occurs in two multipliers, it is a common divisor of\n$C_0$ and of every magnitude, the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{494}{repeatedDivisor_forces_periodicCommonScale}{repeated-divisor\ntransport}, using that both the period relation and the drift iterate over any\nnumber of periods, the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{461}{periodic_add_mul}{iterated\nperiod} and the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{476}{phaseDrift_add_mul}{iterated\ndrift}.\n\n\\begin{theorem}[no periodic negative magnitude]\\label{res:periodic}\nLet $a,D,C,e:\\N\\to\\N$ with $a_n\\ge2$, $e_n>0$ and $e_n<a_n$ for every $n$,\nsatisfying\n\\[\n D_{n+1}=a_nD_n,\\qquad C_{n+1}=C_n+e_n,\\qquad D_n+e_n=(a_n-1)C_n ,\n\\]\nand suppose $e_{n+h}=e_n$ and $C_{n+h}=C_n+M$ for some $h>0$ and $M>0$.  This\nis impossible.\n\\end{theorem}\n\n\\begin{proof}\nStrong induction on the drift $M$.  The difficulty lies in the fact that a\nrepeated prime divisor of $M$ need not contradict anything directly.  It\ninstead forces a common scale on the whole orbit, which can be divided out, and\nthe induction runs on the drift that the division reduces.\n\nSuppose first that no prime divisor of $M$ is a common divisor of $C_0$ and of\nevery magnitude.  Repeated-divisor transport then applies in the\ncontrapositive: a prime divisor of $M$ occurring in two of the multipliers\nwould be exactly such a common divisor, so each prime divisor of $M$ occurs in\nat most one multiplier.  Against this, the argument of\nSection~\\ref{sec:constant} run phase by phase (along a fixed residue class\nmodulo $h$ the magnitude is constant and $C$ advances by $M$ at each period, so\nthe multipliers of that phase meet an arithmetic progression exactly as in\nSection~\\ref{sec:constant}) makes every multiplier share a prime with $M$.\nSince $M$ has only finitely many prime divisors and there are infinitely many\nmultipliers, pigeonhole gives a contradiction.\n\nOtherwise some prime $p$ divides $M$, divides $C_0$, and divides every\nmagnitude.  Then $p\\mid C_n$ for every $n$, since $C_{n+1}=C_n+e_n$ and both\nsummands on the right are divisible by $p$, and the shape equation\n$D_n+e_n=(a_n-1)C_n$ then gives $p\\mid D_n$ for every $n$.  Divide $D$, $C$ and\n$e$ by $p$.  The multipliers are untouched, so $a_n\\ge2$ and $e_n<a_n$ persist\nand the magnitudes stay positive; the three recurrences are homogeneous in\n$(D,C,e)$ and so survive; the period is still $h$; and the drift becomes\n$M/p<M$.  The inductive hypothesis applies.\n\\end{proof}\n\n\\noindent The first of the two cases above is the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{548}{no_phasePrimitivePeriodicNegative_orbit}{phase-primitive\nexclusion}, the induction is the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{690}{no_periodicNegative_orbit}{exclusion\nat every scale}, and the eventual form is the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{796}{no_eventuallyPeriodicNegative_orbit}{eventual\nexclusion}.\n\nThe bound $e_n<a_n$ is a genuine restriction and not a normalisation.  It is\nthe range in which a multiplier cannot divide its own phase magnitude, which\nis what the lock argument needs.  It holds in the intended reading, where\n$e_n$ is a centred representative and $a_n$ grows quadratically, but it is\nassumed here and not derived.  The drift hypothesis $M>0$ is also used: a\nperiodic magnitude with zero drift would be identically zero, which is the case\nalready settled by Section~\\ref{sec:descent}.\n\n\\begin{remark}\nOn the canonical orbit of~\\cite{koizumi2025}, read through the dictionary of\nSection~\\ref{sec:problem}, the bound $e_n<a_n$ holds eventually.  Lemma~4(2)\nthere gives the centring range $-C_n/2\\le E_n<C_n/2$, so $e_n=|E_n|\\le C_n/2$ at\na negative index, and the proof of Lemma~4(3) gives $C_{n+1}\\le\\tfrac32C_n$\n~\\cite[pp.~11--12]{koizumi2025},\nhence the uniform bound $C_n\\le C_0(3/2)^{n}$.  Against this, summability of\n$\\sum1/a_n$ forces\n$a_n\\to\\infty$, and the growth hypothesis $a_n^{2}/a_{n+1}\\to1$ gives\n$a_{n+1}\\ge a_n^{2}/2$ for all large $n$, hence $a_{n+1}\\ge2a_n$ once\n$a_n\\ge4$; so the multipliers grow at least geometrically with ratio $2$ and\noutgrow the bound on $C_n$, making $e_n/a_n$ eventually small.  Two caveats.\nSummability is used for this step, so the estimate is not available from the\nstate recurrence alone; and the conclusion is eventual and not universal,\nwhereas Theorem~\\ref{res:periodic} imposes $e_n<a_n$ at every index, so it is\nthe eventual form of the exclusion cited above that applies.  Under the\nperiodicity hypothesis of Theorem~\\ref{res:periodic} the estimate is in any\ncase unnecessary, since a periodic magnitude is bounded while the multipliers\ntend to infinity.\n\\end{remark}\n\n\\section{Bounded rise cannot avoid coprime moduli}\\label{sec:barrier}\n\nPeriodicity is still a strong assumption.  Removing it needs a different kind\nof obstruction, and the one used here is a counting argument about where an\ninteger sequence with bounded upward steps must land.\n\n\\begin{lemma}[shifted blocks of consecutive multiples]\\label{res:crt}\nLet $m_0,\\ldots,m_{B-1}$ be pairwise coprime and at least $2$.  For every bound\nthere is a $t$ beyond it with $m_i\\mid t+i$ for each $i<B$.\n\\end{lemma}\n\n\\begin{proof}\nStandard Chinese remaindering: solve $t\\equiv-i \\pmod{m_i}$ simultaneously,\nthen add multiples of $\\prod_i m_i$ to pass the bound.\n\\end{proof}\n\n\\noindent Formalised as the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{833}{exists_shifted_consecutiveMultiples}{shifted\nconsecutive multiples}.\n\n\\begin{example}[a block of three consecutive multiples]\\label{ex:crt}\nTake $B=3$ and $(m_0,m_1,m_2)=(3,4,5)$.  The congruences $t\\equiv0\\pmod3$,\n$t\\equiv3\\pmod4$ and $t\\equiv3\\pmod5$ hold exactly when $t\\equiv3\\pmod{60}$.\nAt $t=3$ the block is $3,4,5$ with $3\\mid3$, $4\\mid4$ and $5\\mid5$; at $t=63$\nit is $63,64,65$ with $3\\mid63$, $4\\mid64$ and $5\\mid65$.  A sequence of\nnatural numbers that starts at $0$, tends to infinity and rises by at most $3$\nat each step cannot step over the block $\\{3,4,5\\}$: it has a first value at\nleast $3$, that value is at most $5$, and every member of $\\{3,4,5\\}$ is\ndivisible by one of the three moduli.\n\\end{example}\n\n\\begin{theorem}[bounded rise cannot avoid fresh moduli]\\label{res:barrier}\nLet $u:\\N\\to\\N$ tend to infinity with $u_{n+1}\\le u_n+B$ for a fixed $B>0$.\nThen $u$ cannot permanently avoid infinitely many fresh pairwise coprime\nmoduli: there is no family of pairwise coprime $m_i\\ge2$, one for each index,\nsuch that $m_i\\nmid u_t$ whenever $i<t$.\n\\end{theorem}\n\n\\begin{proof}\nChoose a tail index $N$, take the $B$ moduli\n$m_N,\\ldots,m_{N+B-1}$, and put $T=N+B$.  Lemma~\\ref{res:crt} gives an\n$x>u_T$ such that $m_{N+i}\\mid x+i$ for $0\\le i<B$.  Since $u$ tends to\ninfinity, there is a first $s>T$ with $u_s\\ge x+B$.  Minimality gives\n$u_{s-1}<x+B$.  In fact $u_{s-1}<x$: otherwise\n$u_{s-1}=x+i$ for some $i<B$, so $m_{N+i}\\mid u_{s-1}$, while\n$N+i<T\\le s-1$, contradicting tail avoidance.  The rise bound now gives\n$u_s\\le u_{s-1}+B<x+B$, contrary to the choice of $s$.\n\\end{proof}\n\n\\noindent Formalised as the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{2303}{no_boundedRise_of_tailNondivisibility}{bounded-rise\nnondivisibility barrier}.  The earlier application-ready coprimality hypothesis\nis retained as the corollary\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{897}{no_boundedRise_of_tailAvoidance}{coprime-tail\navoidance}.  The key point is that the two hypotheses pull against each other\nexactly: divergence forces $u$ to travel arbitrarily far, and bounded rise\nforbids it from stepping over a window of width $B$.  Uniformity of the bound\nis what the proof uses, and it is the hypothesis that cannot be relaxed by this\nargument: a rise bounded only along a subsequence leaves the intervening steps\nfree to jump the window.  We have not looked for a sequence of unbounded rise\navoiding every modulus.  The statement is elementary and we are not aware of a\nprior published form of it, but we would not be surprised if one exists.\n\nThe moduli are indexed by the same set as the sequence, and the $i$th of them\nis asked to divide no value $u_t$ with $t>i$; nothing at all is asked of it at\nor before its own index.  That asymmetry is what makes the hypothesis available\nin the application below, where $m_i$ is the $i$th multiplier and only the\nlater numerators are known to be coprime to it.\n\nThe barrier applies to the problem because a \\emph{reduced} exact tail already\ncarries pairwise coprime moduli.  Call $(u,v)$ a reduced exact tail with\nmultipliers $a$ when $\\gcd(u_n,v_n)=1$,\n\\[\n u_{n+1}+v_n=a_nu_n,\\qquad v_{n+1}=a_nv_n .\n\\]\nHere $u$ is the numerator and $v$ the denominator: the two recurrences are\nthose of Section~\\ref{sec:state}, with $u$ in the role of the tail state $C$\nand $v$ in that of the denominator state $D$, and with coprimality imposed at\nevery index.\n\n\\begin{proposition}[reduced tails are pairwise coprime]\\label{res:reduced}\nIn a reduced exact tail, $a_n$ is coprime to $v_n$; the multipliers at distinct\nindices are pairwise coprime; and every earlier multiplier is coprime to every\nlater numerator.\n\\end{proposition}\n\n\\noindent These are the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{983}{reducedStep_coprime_currentFactor}{step\ncoprimality}, the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1010}{reducedTail_pairwiseCoprime}{pairwise\ncoprimality}, and the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1024}{reducedTail_wholeModulusAvoidance}{whole-modulus\navoidance}.  Feeding them to Theorem~\\ref{res:barrier} gives the form used\nlater: there is no reduced exact tail whose numerator tends to infinity with a\nuniformly bounded upward increment, the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1035}{no_boundedRise_reducedTail}{reduced\nbounded-rise exclusion}, together with its eventual form, the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1060}{no_eventuallyBoundedRise_reducedTail}{eventual\nreduced exclusion}.\n\n\\begin{example}[Sylvester's sequence as a reduced exact tail]\\label{ex:reduced}\nPut $u_n=1$ and $v_n=D_n=1,2,6,42,1806,\\ldots$ as in\nExample~\\ref{ex:sylvester}, with multipliers $a_n=v_n+1=2,3,7,43,1807,\\ldots$.\nThen $\\gcd(u_n,v_n)=1$, $u_{n+1}+v_n=1+v_n=a_nu_n$ and $v_{n+1}=a_nv_n$, so\nthis is a reduced exact tail, and Proposition~\\ref{res:reduced} returns the\nclassical fact that Sylvester's numbers are pairwise coprime.  Its numerator is\nconstant, so it satisfies every hypothesis of the exclusion just stated except\ndivergence; since the tail exists, that hypothesis cannot be dropped.\n\\end{example}\n\nThe orbit of Section~\\ref{sec:state} is not reduced, so one more step is needed\nbefore the barrier can be applied to it: the common factor must stop changing.\nWe call a set of indices \\emph{cofinal} when it is infinite, and say that a\nproperty holds \\emph{cofinally} when the set of indices at which it holds is\ncofinal.  The step below is the only one connecting the orbit of\nSection~\\ref{sec:state} to the reduced tails of\nProposition~\\ref{res:reduced}, and it is where the cofinal boundedness in the\nproof of Theorem~\\ref{res:bounded} is used.\n\n\\begin{proposition}[the tail gcd stabilises]\\label{res:gcdstab}\nLet $(a,D,C)$ be an exact natural orbit, that is, a triple of $\\N$-valued\nsequences with $C_{n+1}+D_n=a_nC_n$ and $D_{n+1}=a_nD_n$, whose centred state\n$E_n=\\ctr(a_n,D_n,C_n)$ is negative cofinally, with magnitudes bounded along\nthat cofinal set.  Then\n$\\gcd(C_n,D_n)$ is eventually constant, and beyond that index the orbit divided\nby the stable gcd is a reduced exact tail.\n\\end{proposition}\n\n\\begin{proof}\nThe tail gcd $\\gcd(C_n,D_n)$ divides its successor, the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1424}{tailGcd_dvd_succ}{gcd\nmonotonicity}, so the tail gcds form a positive divisibility chain.  At an\nindex where the centred state is negative the tail gcd is exactly\n$\\gcd(C_n,e_n)$, the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1581}{tailGcd_eq_gcd_negativeMagnitude}{gcd\nat a negative index}, and is therefore at most the magnitude there; so the\nchain is bounded along the cofinal set of negative indices.  A positive\ndivisibility chain whose values are bounded along a cofinal set is eventually\nconstant, the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1599}{dvdChain_eventuallyConstant_of_cofinally_bounded}{chain\nstabilisation}.  Hence cofinally bounded negative magnitudes make the tail gcd\nstabilise, the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1630}{tailGcd_eventuallyConstant_of_cofinally_boundedNegative}{gcd\nstabilisation}, and beyond that point dividing by the stable gcd leaves the\nnumerator and denominator coprime, that is, a reduced exact tail.\n\\end{proof}\n\n\\noindent The stabilisation of the tail gcd is the checked statement cited at\nthe end of the proof; the last clause, that the divided orbit is a reduced\nexact tail, is exposition and carries no separate label.  The hypothesis is\nboundedness \\emph{along a cofinal set} and not\nboundedness at every index.  That is what the situation gives: the tail gcd is\ncontrolled only at the negative indices, and cofinally many of them is all the\ndivisibility chain needs.\n\n\\section{Excluding a bounded negative part}\\label{sec:bounded}\n\nAssembling the previous section gives Theorem~\\ref{res:bounded}, the endpoint\nof the barrier argument.  Theorem~\\ref{res:mass} in the next section is a\nsecond endpoint under a different finiteness hypothesis, and neither theorem\ncontains the other.  Theorem~\\ref{res:mass} assumes no orbit structure, only\nthe state recurrence with $C_n>0$, so it is not a special case of\nTheorem~\\ref{res:bounded}; conversely hypothesis~(5) below does not imply\nfinite \\emph{normalised negative mass}, by which we mean convergence of\n$\\sum_n(-E_n)_+/C_n$, where $x_+=\\max(x,0)$, since a constant negative part\n$-b$ gives $C_n\\sim bn$ and hence a divergent sum.  The hypotheses of\nTheorem~\\ref{res:bounded} are listed\nin full, because two of them are analytic inputs\nthat are assumed here rather than derived from the growth condition of\nProblem~\\ref{res:problem}.\n\nInformally, hypotheses~(5) and~(6) below pull against each other.  By\nProposition~\\ref{res:update}, hypothesis~(5) caps the rise of the tail state at\n$B$ per step, while hypothesis~(6), once the error is nowhere zero, forces the\ntail state to diverge.  Proposition~\\ref{res:gcdstab} makes the orbit reduced,\nand Theorem~\\ref{res:barrier} then converts the tension into a contradiction:\nan integer sequence tending to infinity with upward steps of size at most $B$\ncannot permanently avoid the pairwise coprime moduli a reduced tail carries.\nThe\ncomplementary case, in which the error is eventually positive, is the descent\nof Theorem~\\ref{res:descent}.\n\n\\begin{theorem}[bounded negative part]\\label{res:bounded}\nLet $a,C,D:\\N\\to\\N$ and $E:\\N\\to\\Z$ satisfy\n\\begin{enumerate}\n\\item $a_n>1$ and $C_n>0$ for every $n$;\n\\item the exact dynamics $C_{n+1}+D_n=a_nC_n$ and $D_{n+1}=a_nD_n$;\n\\item $E_n=\\ctr(a_n,D_n,C_n)$ for every $n$;\n\\item \\emph{eventual strict centring}: $|E_n|<C_n$ for all large $n$;\n\\item \\emph{eventually bounded negative part}: $-B\\le E_n$ for all large $n$,\nfor some $B$;\n\\item \\emph{normalised vanishing}: for every $K$ there is an $N$ with\n$K\\,|E_n|<C_n$ for all $n\\ge N$.\n\\end{enumerate}\nThen $E_n=0$ for all sufficiently large $n$.\n\\end{theorem}\n\n\\begin{proof}\nSuppose not.  We first remove every late zero: by Theorem~\\ref{res:absorb} and\nits contrapositive form, $E$ is nowhere zero beyond the centring threshold.\nTwo cases remain.  If $E$ is negative cofinally,\nthen~(5) bounds those magnitudes, so Proposition~\\ref{res:gcdstab} makes the\ntail gcd stabilise and the orbit may be taken reduced past that point.\nHypothesis~(5) with\nProposition~\\ref{res:update} gives $C_{n+1}\\le C_n+B$, a bounded rise;\nhypothesis~(6) with $E$ nowhere zero gives $C_n\\to\\infty$.  A reduced exact\ntail with a divergent numerator and bounded rise is excluded by\nTheorem~\\ref{res:barrier}.  It remains to treat the case that $E$ is eventually\npositive, and there the descent\nof Theorem~\\ref{res:descent} applies and forces $E$ to vanish, contradicting\nnowhere-vanishing.\n\\end{proof}\n\n\\noindent Formalised as the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{2177}{boundedNegativePart_eventually_zero}{bounded-negative-part\nrigidity}, with the eventual form, in which the centring and the bound need\nonly hold from some index, as the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{2264}{eventuallyBoundedNegativePart_eventually_zero}{eventual\nbounded-negative-part rigidity}.  The two auxiliary results it uses are the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1733}{tailState_tendsto_atTop_of_nonzero_normalizedVanishes}{divergence\nfrom normalised vanishing} and the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1748}{no_cofinallyBoundedNegative_of_normalizedVanishes}{exclusion\nof cofinally bounded negative magnitudes}, the latter over the\n\\lword{Erdos243/ReciprocalTailRigidity.lean}{1655}{no_cofinallyBoundedNegative_of_tailTendsto}{tail-divergence\nform}.\n\n\\begin{corollary}\\label{res:cor}\nUnder the hypotheses of Theorem~\\ref{res:bounded}, together with\n$C_{n+1}\\ne0$ for all large $n$, the multipliers satisfy\n$a_{n+1}=a_n^{2}-a_n+1$ for all sufficiently large $n$.\n\\end{corollary}\n\n\\begin{proof}\nTheorem~\\ref{res:bounded} then Theorem~\\ref{res:eventual}.\n\\end{proof}\n\nHypotheses~(1)--(3) are the state system.  Hypothesis~(4) says the centred\nstate is a strict centred representative, and~(6) says $|E_n|/C_n\\to0$ in a\ndivision-free form.  Neither is derived here from\n$a_{n+1}/a_n^{2}\\to1$, and no argument in the formal source connects them to\nProblem~\\ref{res:problem}.  Theorem~\\ref{res:bounded} is therefore a\nconditional theorem about the state system, and Corollary~\\ref{res:cor} does\nnot settle any case of Problem~\\ref{res:problem}.  What is unconditional in\nthis note is the exclusion of constant and periodic negative magnitudes,\nTheorems~\\ref{res:constant} and~\\ref{res:periodic}, and the barrier itself,\nTheorem~\\ref{res:barrier}.\n\nThe conditionality can nevertheless be located precisely, because the missing\ninputs are supplied elsewhere.  With the results of~\\cite{koizumi2025},\nTheorem~\\ref{res:bounded} becomes a conditional theorem about\nProblem~\\ref{res:problem} itself rather than about the state system alone.  Let\n$(a_n)$ satisfy the hypotheses of Problem~\\ref{res:problem}.  After deleting a\nfinite prefix, Corollary~3 of~\\cite[p.~9]{koizumi2025} makes the sequence the\npseudo-greedy expansion of its own reciprocal sum, and Lemma~4 there supplies\nthe integers $c_n,d_n,e_n$ of Section~\\ref{sec:problem}.  Hypothesis~(1) then\nholds: $c_n$ is a positive integer by Lemma~4(1), and $a_n>1$ after a further\nfinite shift, because summability of $\\sum1/a_n$ forces $a_n\\to\\infty$.\nHypothesis~(2) is the pair of recurrences $c_{n+1}=a_nc_n-d_n$ and\n$d_{n+1}=a_nd_n$ of Lemma~4(2), and hypothesis~(3) is that lemma's\n$a_n=(d_n-e_n)/c_n+1$ read as $e_n=d_n-(a_n-1)c_n$, which is the map $\\ctr$.\nHypothesis~(4) holds at every index, not merely eventually, from the centring\nrange $-c_n/2\\le e_n<c_n/2$ of the same lemma.  Hypothesis~(6) is the vanishing\nof the gap sequence in Corollary~3, since $\\varepsilon_n=e_n/c_n$.  The sole\nhypothesis not supplied is~(5), the eventual bound on the negative part;\nnothing in~\\cite{koizumi2025} yields it and nothing here does either, so\nProblem~\\#243 stays open.\n\n\\section{Excluding finite normalised negative mass}\\label{sec:mass}\n\nTheorem~\\ref{res:bounded} uses a uniform bound on the negative part.  A\ndifferent finiteness hypothesis, on the total normalised negative mass, removes\na further family of orbits, and the argument is multiplicative rather than\narithmetic: it compares $C$ with a convergent product instead of with a system\nof congruences.\n\n\\begin{theorem}[finite normalised negative mass]\\label{res:mass}\nLet \\(C_n\\in\\N_{>0}\\) and \\(E_n\\in\\Z\\) satisfy\n\\[\n C_{n+1}=C_n-E_n,\\qquad\n \\forall K\\ \\exists N\\ \\forall n\\ge N,\\quad K|E_n|<C_n .\n\\]\nIf\n\\[\n \\sum_{n=0}^{\\infty}\\frac{(-E_n)_+}{C_n}<\\infty ,\n\\]\nthen \\(E_n=0\\) eventually.  For an exact reciprocal-tail orbit this implies\n\\(a_{n+1}=a_n^2-a_n+1\\) eventually.\n\\end{theorem}\n\n\\begin{proof}\nPut \\(\\delta_n=(-E_n)_+/C_n\\).  The update gives\n\\(C_{n+1}\\le C_n(1+\\delta_n)\\), so\n\\[\n C_N\\le C_0\\prod_{n<N}(1+\\delta_n).\n\\]\nThe product is bounded because \\(\\sum\\delta_n\\) converges.  Choose an integer\n\\(M\\) above that bound and apply normalised vanishing with \\(K=M\\).  Then\n\\(M|E_n|<C_n<M\\) for all sufficiently large \\(n\\), forcing the integer\n\\(E_n\\) to be zero.  The final recurrence is Theorem~\\ref{res:eventual}, whose\nhypothesis $C_{n+1}\\ne0$ holds because $C_n>0$ throughout.\n\\end{proof}\n\n\\noindent The product argument is the\n\\lword{Erdos243/SparseResetRecovery.lean}{90}{eventually_zero_of_summable_relativeGrowth}{summable\nrelative growth}; its specialisation to negative mass is the\n\\lword{Erdos243/SparseResetRecovery.lean}{155}{eventually_zero_of_summable_negativeRelativeMass}{vanishing\nfrom finite negative mass}, and the statement giving the recurrence directly is\nthe\n\\lword{Erdos243/SparseResetRecovery.lean}{175}{sylvesterNext_eventually_of_summable_negativeRelativeMass}{Sylvester\nrecurrence from summable negative mass}.  This is a checked implication about\nthe state system, not a derivation of normalised vanishing from\nProblem~\\#243.  Normalised vanishing is still an input: it is what converts the\nbound on $C$ into a bound on $|E_n|$, and nothing here derives it from the\ngrowth condition of Problem~\\ref{res:problem}.\n\nThe transfer of the previous section applies here as well.  On the canonical\norbit of Corollary~3 of~\\cite[p.~9]{koizumi2025} the recurrence\n$C_{n+1}=C_n-E_n$ holds with $C_n>0$ by Lemma~4\n~\\cite[pp.~11--12]{koizumi2025}, and normalised vanishing is the vanishing of\nthe gap sequence, so Theorem~\\ref{res:mass} is a conditional theorem about\nProblem~\\ref{res:problem} whose sole remaining hypothesis is summability of\n$\\sum_n(-E_n)_+/C_n$.  That hypothesis is not supplied there, and is the\ncontent of Problem~\\ref{res:masshyp} below.\n\n\\begin{example}[the product bound at a geometric rate]\\label{ex:mass}\nSuppose $C_0=100$ and $(-E_n)_+/C_n\\le2^{-n-1}$ for every $n$, so that the\nnormalised negative mass is at most $1$; the constraint at $n=0$ still permits\n$E_0$ as negative as $-50$.  Since $1+x\\le e^{x}$,\n\\[\n C_N\\le100\\prod_{n<N}\\bigl(1+2^{-n-1}\\bigr)\\le100e<272\n\\]\nfor every $N$.  Normalised vanishing at $K=272$ then gives\n$272|E_n|<C_n<272$ for all large $n$, so $|E_n|<1$ and $E_n=0$ there.  The\nconstant $272$ comes from the total mass and not from any single magnitude,\nwhich is why the hypothesis of Theorem~\\ref{res:mass} is summability of\n$(-E_n)_+/C_n$ and not a bound on it.\n\\end{example}\n\n\\section{Complements and further questions}\\label{sec:open}\n\nProblem~\\#243 is open.  The negative case has narrowed, and it is worth being\nprecise about what is left.\n\nBy Theorem~\\ref{res:absorb} a counterexample has $E_n\\ne0$ for every large $n$.\nBy Theorem~\\ref{res:descent} it cannot have $E$ eventually nonnegative.  By\nTheorems~\\ref{res:constant} and~\\ref{res:periodic} its negative magnitudes are\nneither eventually constant nor eventually periodic.  By\nTheorem~\\ref{res:bounded}, if the analytic hypotheses~(4) and~(6) hold, the\nnegative part cannot be bounded either, and by Theorem~\\ref{res:mass}, under\nnormalised vanishing, its normalised negative mass cannot be finite.  What\nremains is the following.\n\n\\begin{problem}[unbounded divergent-mass negative excursions]\\label{res:excursions}\nExclude, or construct, an exact natural orbit $(a,D,C)$ with $a_n>1$ and\n$C_n>0$, whose multipliers satisfy $\\lim_n a_{n+1}/a_n^{2}=1$ and whose centred\nstate satisfies eventual strict centring and normalised vanishing, and which is\nnegative infinitely often with magnitudes unbounded along that cofinal set and\n\\[\n \\sum_n\\frac{(-E_n)_+}{C_n}=\\infty .\n\\]\nEvery\nargument in Sections~\\ref{sec:constant}--\\ref{sec:bounded} uses some\nfiniteness hypothesis: a fixed set of prime divisors, a fixed period, or a\nfixed bound on the negative part.  Theorem~\\ref{res:mass} uses a finite total\nmultiplicative growth as well.  An excursion of the kind described has none of\nthese.\n\\end{problem}\n\n\\begin{remark}\nThe hypotheses in Problem~\\ref{res:excursions} beyond the state recurrence are\nnot decorative: the recurrence alone admits unbounded divergent-mass\nexcursions.  Put\n\\[\n C_n=2^{n},\\qquad b_0=2,\\qquad b_{n+1}=\\tfrac12 b_n(b_n+2),\\qquad\n a_n=b_n+2,\\qquad D_n=b_nC_n ,\n\\]\nso that $(b_n)=2,4,12,84,3612,\\ldots$ and $(a_n)=4,6,14,86,3614,\\ldots$.  Every\n$b_n$ is a positive even integer, since $b_n=2k$ gives $b_{n+1}=2k(k+1)$, so all\nfour sequences are $\\N$-valued.  The orbit is exact:\n$C_{n+1}+D_n=2^{n}(2+b_n)=a_nC_n$ and\n$D_{n+1}=b_{n+1}2^{n+1}=b_n(b_n+2)2^{n}=a_nD_n$.  Its centred state is\n\\[\n E_n=D_n-(a_n-1)C_n=b_n2^{n}-(b_n+1)2^{n}=-C_n ,\n\\]\nso the negative magnitudes $2^{n}$ are unbounded and $\\sum_n(-E_n)_+/C_n$\ndiverges.  What fails is the centring: $|E_n|=C_n$ at every index, so strict\ncentring fails at the boundary and normalised vanishing fails outright.  The\nmultipliers satisfy $a_{n+1}=\\tfrac12(a_n^{2}-2a_n+4)$, so\n$a_{n+1}/a_n^{2}\\to\\tfrac12$ and the growth hypothesis fails as well.  The\nexample therefore says nothing about Problem~\\ref{res:problem}, and it does not\nisolate the centring hypotheses: $E_n=-C_n$ at every index forces that\nmultiplier recurrence, so growth fails with them.  It is recorded only to show\nthat the state recurrence alone is not enough.\n\\end{remark}\n\n\\begin{problem}[summable normalised negative mass]\\label{res:masshyp}\nProve from the rapid-growth and rationality hypotheses of\nProblem~\\ref{res:problem} that \\(\\sum_n(-E_n)_+/C_n<\\infty\\), or prove any\nuniform bound on the partial products \\(\\prod_{n<N}(1+(-E_n)_+/C_n)\\); a bound\nof the form\n\\[\n \\sum_{n=N}^{M}\\frac{(-E_n)_+}{C_n}\\le B\n \\quad\\text{for all }M\\ge N\n\\]\nwith explicit hypotheses would suffice.  A disproof should exhibit a sequence\nsatisfying the hypotheses of Problem~\\ref{res:problem} whose exact state orbit\nhas divergent normalised negative mass; an orbit satisfying normalised\nvanishing alone is not enough, since normalised vanishing is strictly weaker\nthan those hypotheses, and neither is a list of locally admissible defects,\nsince local admissibility at each index does not produce a global orbit.\n\\end{problem}\n\n\\begin{remark}\nOne further question is not left open here, since it is answered\nin~\\cite{koizumi2025}: whether eventual strict centring and normalised\nvanishing follow from $a_{n+1}/a_n^{2}\\to1$ and rationality.  They do.  After\ndeleting a finite prefix, Koizumi's Corollary~3\n~\\cite[p.~9]{koizumi2025} makes the sequence the pseudo-greedy expansion of its\nown reciprocal sum and gives $\\varepsilon_n\\to0$ under exactly the hypotheses of\nProblem~\\ref{res:problem}; rationality is not needed for that step, only\nsummability of $\\sum1/a_n$.  Since $\\varepsilon_n=E_n/C_n$ under the intended\nreading of Section~\\ref{sec:state}, this is normalised vanishing.  Strict\ncentring comes from the same source and is stronger than what\nTheorem~\\ref{res:bounded} assumes: for a rational reciprocal sum, the range\n$-c_n/2\\le e_n<c_n/2$ of his Lemma~4(2)\n~\\cite[p.~11]{koizumi2025} gives $|E_n|<C_n$ at every index, not\nmerely for all large $n$.  What remains unsupplied is hypothesis~(5), the bound\non the negative part, which is why the theorem is still conditional.\n\\end{remark}\n\n\\begin{problem}[the two published criteria]\nFormalise the criterion of Erd\\H{o}s and Straus stated in\nSection~\\ref{sec:problem}, and Duverney's conditional characterisation, under\nexplicit analytic hypotheses.  Both are cited here and neither is formalised;\neach reaches the original problem under analytic hypotheses not assumed in this\nnote.\n\\end{problem}\n\n\\pdfbookmark[1]{Statements and declarations}{statements-declarations}\n\\subsection*{Statements and declarations}\n\n\\paragraph{Artefact and data availability.} The\n\\href{\\sourceurl}{pinned formal-source revision} contains the Lean sources, the\nfixed toolchain, and the library manifest used in the verification.  This\nmanuscript provides navigation rather than proof authority.\n\n\\paragraph{Declaration of generative AI use.} \\paperfullaideclaration\n\nLean checks each proof term against the fixed library\nversion, and the sources linked here contain no proof placeholders and no\nproject-defined axioms; Lean does not authorise the exposition, the citation\nchoices, or the interpretation, for which the author remains responsible.\n\n\\paragraph{Funding and competing interests.} This work received no external\nfunding.  The author declares no competing interests.\n\n\\paragraph{Acknowledgements.} The problem numbering and status follow the\nErd\\H{o}s Problems catalogue maintained by Thomas Bloom~\\cite{erdosproblems}.\n\n\\appendix\n\\section{Guide to the formal sources}\\label{app:index}\n\nEach linked phrase opens its Lean declaration at the pinned source\nrevision~\\commitshort.  The state system, the exclusions, the barrier, and the\nbounded-negative-part theorem are in \\texttt{ReciprocalTailRigidity.lean}; the\nfinite-negative-mass theorem is in \\texttt{SparseResetRecovery.lean}; and the\nresidue reduction of Appendix~\\ref{app:residue} is in\n\\texttt{FiniteHorizonResidue.lean}.  Three distinctions are worth carrying into\nthe source.  The\nidentification of any of these modules with reciprocal tails is the exposition\nof Section~\\ref{sec:state} and is not a checked statement.  The periodic exclusion\nassumes the regime $e_n<a_n$.  And the final theorem assumes strict centring\nand normalised vanishing, which are hypotheses of the theorem and are not\nderived in the formal source; for where they do come from,\nsee Sections~\\ref{sec:problem} and~\\ref{sec:bounded}.\n\n% The exhaustive source manifest is retained in the authored TeX for automated\n% validation, and kept outside the printed note.\n\\iffalse\n\n\\begin{tabular}{@{}ll@{}}\ninformal statement & linked Lean source\\\\\nSylvester successor & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{37}{sylvesterNext}\\\\\ndenominator update & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{41}{nextDenState}\\\\\ntail update & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{45}{nextTailState}\\\\\ncentred state & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{49}{centeredState}\\\\\nSylvester defect & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{53}{sylvesterDefect}\\\\\nupdate law & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{57}{nextTailState_eq_sub_centered}\\\\\ndenominator equivariance & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{65}{nextDenState_scale}\\\\\ntail equivariance & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{70}{nextTailState_scale}\\\\\ncentred equivariance & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{75}{centeredState_scale}\\\\\nnormalised exclusion & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{100}{no_normalizedConstantNegative_orbit}\\\\\nscale-primitive exclusion & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{147}{no_scalePrimitiveConstantNegative_orbit}\\\\\nconstant exclusion at every scale & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{286}{no_constantNegative_orbit}\\\\\neventual constant exclusion & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{350}{no_eventuallyConstantNegative_orbit}\\\\\npersistence of a multiplier & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{382}{base_dvd_denState_later}\\\\\none-step lock & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{402}{divisorLocksTailSucc}\\\\\npersistence of a lock & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{429}{divisorLock_persists}\\\\\niterated period & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{461}{periodic_add_mul}\\\\\niterated drift & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{476}{phaseDrift_add_mul}\\\\\nrepeated-divisor transport & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{494}{repeatedDivisor_forces_periodicCommonScale}\\\\\nphase-primitive exclusion & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{548}{no_phasePrimitivePeriodicNegative_orbit}\\\\\nperiodic exclusion at every scale & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{690}{no_periodicNegative_orbit}\\\\\neventual periodic exclusion & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{796}{no_eventuallyPeriodicNegative_orbit}\\\\\nshifted consecutive multiples & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{833}{exists_shifted_consecutiveMultiples}\\\\\nbounded-rise nondivisibility barrier & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{2303}{no_boundedRise_of_tailNondivisibility}\\\\\ncoprime-tail avoidance corollary & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{897}{no_boundedRise_of_tailAvoidance}\\\\\nstep coprimality & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{983}{reducedStep_coprime_currentFactor}\\\\\npairwise coprimality & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1010}{reducedTail_pairwiseCoprime}\\\\\nwhole-modulus avoidance & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1024}{reducedTail_wholeModulusAvoidance}\\\\\nreduced bounded-rise exclusion & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1035}{no_boundedRise_reducedTail}\\\\\neventual reduced exclusion & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1060}{no_eventuallyBoundedRise_reducedTail}\\\\\ngcd monotonicity & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1424}{tailGcd_dvd_succ}\\\\\ngcd at a negative index & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1581}{tailGcd_eq_gcd_negativeMagnitude}\\\\\nchain stabilisation & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1599}{dvdChain_eventuallyConstant_of_cofinally_bounded}\\\\\ngcd stabilisation & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1630}{tailGcd_eventuallyConstant_of_cofinally_boundedNegative}\\\\\ntail-divergence form & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1655}{no_cofinallyBoundedNegative_of_tailTendsto}\\\\\ndivergence from normalised vanishing & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1733}{tailState_tendsto_atTop_of_nonzero_normalizedVanishes}\\\\\nexclusion from normalised vanishing & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1748}{no_cofinallyBoundedNegative_of_normalizedVanishes}\\\\\ndefect identity & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1769}{sylvesterDefect_mul_nextTailState}\\\\\nlocal rigidity step & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1781}{sylvesterNext_eq_of_centered_zero}\\\\\neventual Sylvester recurrence & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1799}{sylvesterNext_eventually_of_centered_zero}\\\\\neventual constancy & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1819}{antitone_nat_eventually_constant}\\\\\nstabilisation of the nonnegative error & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1837}{centeredState_eventually_zero}\\\\\ntail realisation & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1863}{natTail_eq_nextTailState}\\\\\ndenominator realisation & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1875}{natDen_eq_nextDenState}\\\\\nsigned update law & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{1884}{natTail_eq_sub_centeredState}\\\\\nabsorption of a vanishing centred state & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{2133}{centeredState_zero_absorbing}\\\\\ncontrapositive along a tail & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{2160}{eventually_nonzero_of_zero_absorbing}\\\\\nbounded-negative-part rigidity & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{2177}{boundedNegativePart_eventually_zero}\\\\\neventual bounded-negative-part rigidity & \\lrefx{Erdos243/ReciprocalTailRigidity.lean}{2264}{eventuallyBoundedNegativePart_eventually_zero}\\\\\nforced numerator & \\lrefx{Erdos243/FiniteHorizonResidue.lean}{22}{forcedNumerator}\\\\\npolynomial congruence & \\lrefx{Erdos243/FiniteHorizonResidue.lean}{26}{forcedNumerator_modEq}\\\\\nexact-division cancellation & \\lrefx{Erdos243/FiniteHorizonResidue.lean}{41}{quotient_modEq_of_modEq_mul}\\\\\nhorizon modulus & \\lrefx{Erdos243/FiniteHorizonResidue.lean}{53}{horizonModulus}\\\\\nascending-factorial form & \\lrefx{Erdos243/FiniteHorizonResidue.lean}{59}{horizonModulus_eq_ascFactorial}\\\\\nfactorial value at the initial index & \\lrefx{Erdos243/FiniteHorizonResidue.lean}{69}{horizonModulus_zero_eq_factorial}\\\\\nsurvival predicate & \\lrefx{Erdos243/FiniteHorizonResidue.lean}{75}{ForcedSurvives}\\\\\nquotient transport & \\lrefx{Erdos243/FiniteHorizonResidue.lean}{85}{forcedQuotient_modEq}\\\\\nshrinking-modulus transport & \\lrefx{Erdos243/FiniteHorizonResidue.lean}{97}{forcedSurvives_iff_of_modEq}\\\\\nfactorial residue reduction & \\lrefx{Erdos243/FiniteHorizonResidue.lean}{134}{forcedSurvives_iff_of_modEq_factorial}\\\\\n\\end{tabular}\n\\fi\n\n\\section{A factorial residue reduction for forced orbits}\\label{app:residue}\n\nIn the case $m=c=1$ of Section~\\ref{sec:constant}, where the shape\nequation~\\eqref{eq:shape} reads $D_n+1=(a_n-1)(n+1)$, each multiplier is\ndetermined by its predecessor; we call such an orbit \\emph{forced}.  At index\n$n$ the numerator of the next multiplier is\n\\[\n \\fnum(n,a)=(n+1)a^{2}-(n+2)a+(n+3),\n\\]\nthe\n\\lword{Erdos243/FiniteHorizonResidue.lean}{22}{forcedNumerator}{forced\nnumerator}, and the divisor is $n+2$.  The orbit survives a step when that\ndivision is exact, giving a survival predicate, the\n\\lword{Erdos243/FiniteHorizonResidue.lean}{75}{ForcedSurvives}{survival\npredicate}.  Example~\\ref{ex:shape} is the forced orbit from $a=3$ read this\nway: $\\fnum(0,3)=6$ is divisible by $2$ and gives $a_1=3$, while\n$\\fnum(1,3)=13$ is not divisible by $3$, so the orbit stops there.  Deciding\nsurvival by iteration is expensive because the orbit\ngrows doubly exponentially, and it is unnecessary: survival over a finite\nhorizon depends on the initial value only through a factorial residue.\n\n\\begin{theorem}[factorial residue reduction]\\label{res:residue}\nFor all $h$ and all integers $a\\equiv b \\pmod{(h+1)!}$, the orbit from $a$\nsurvives $h$ forced updates exactly when the orbit from $b$ does.\n\\end{theorem}\n\n\\begin{proof}\nLet $M(0,i)=1$ and $M(h+1,i)=(i+2)M(h,i+1)$, an ascending factorial with\n$M(h,0)=(h+1)!$.  The numerator is a polynomial with integer coefficients, so\ncongruences transfer; reducing the modulus gives divisibility by $i+2$ for one\nexactly when for the other, and cancelling that common factor from both values\nand modulus leaves the inductive hypothesis at $M(h,i+1)$.\n\\end{proof}\n\n\\noindent Formalised as the\n\\lword{Erdos243/FiniteHorizonResidue.lean}{134}{forcedSurvives_iff_of_modEq_factorial}{factorial\nresidue reduction}, over the\n\\lword{Erdos243/FiniteHorizonResidue.lean}{97}{forcedSurvives_iff_of_modEq}{shrinking-modulus\ntransport}, the\n\\lword{Erdos243/FiniteHorizonResidue.lean}{26}{forcedNumerator_modEq}{polynomial\ncongruence}, and the\n\\lword{Erdos243/FiniteHorizonResidue.lean}{41}{quotient_modEq_of_modEq_mul}{exact-division\ncancellation}; the modulus identifications are the\n\\lword{Erdos243/FiniteHorizonResidue.lean}{59}{horizonModulus_eq_ascFactorial}{ascending-factorial\nform} and the\n\\lword{Erdos243/FiniteHorizonResidue.lean}{69}{horizonModulus_zero_eq_factorial}{factorial\nvalue at the initial index}.\n\nAt $h=1$ the modulus is $2!=2$, and surviving one update means\n$2\\mid a^{2}-2a+3$, which holds exactly for odd $a$: for instance\n$\\fnum(0,3)=6$ but $\\fnum(0,4)=11$.  So one step of survival is decided by the\nparity of $a$ alone, which is Theorem~\\ref{res:residue} at its smallest\nnontrivial horizon.\n\nThe search this supported is superseded.  Running it over initial states below\n$5000$ produced forced prefixes of length $17$ and no longer, which was\nevidence and not a proof; Theorem~\\ref{res:constant} now excludes the\nconstant-negative case outright, for every seed and at every scale.  The\nreduction is retained because it is exact, and because the shrinking-modulus\ntechnique transfers to any forced orbit whose step is a polynomial division.\n\n\\begin{thebibliography}{9}\n\\small\n\\raggedright\n\\setlength{\\itemsep}{0pt}\n\\bibitem{erdosstraus1964} P.~Erd\\H{o}s and E.~G. Straus,\n  \\href{https://users.renyi.hu/~p_erdos/1964-19.pdf}{\\emph{On the irrationality\n  of certain Ahmes series}},\n  J. Indian Math. Soc. (N.S.) \\textbf{27} (1964), 129--133. MR~175848.\n\\bibitem{duverney2001} D.~Duverney,\n  \\href{https://www.ms.u-tokyo.ac.jp/journal/pdf/jms080206.pdf}\n  {\\emph{Irrationality of fast converging series of rational numbers}},\n  J. Math. Sci. Univ. Tokyo \\textbf{8} (2001), 275--316. MR~1837165.\n\\bibitem{erdosgraham1980} P.~Erd\\H{o}s and R.~L. Graham,\n  \\href{https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf}{\\emph{Old\n  and New Problems and Results in Combinatorial Number Theory}},\n  Monogr. Enseign. Math. 28, Geneva, 1980, p.~64.\n\\bibitem{erdos1988} P.~Erd\\H{o}s, \\emph{On the irrationality of certain series:\n  problems and results}, in A.~Baker (ed.), \\emph{New Advances in Transcendence\n  Theory}, Cambridge UP, 1988, pp.~102--109,\n  doi:\\href{https://doi.org/10.1017/CBO9780511897184.009}{10.1017/CBO9780511897184.009}.\n\\bibitem{kovactao2024} V.~Kova\\v{c} and T.~Tao,\n  \\href{https://arxiv.org/abs/2406.17593}{\\emph{On several irrationality\n  problems for Ahmes series}}, Acta Math. Hungar. \\textbf{175} (2025),\n  572--608; preprint arXiv:2406.17593v4.\n\\bibitem{koizumi2025} J.~Koizumi,\n  \\href{https://math.colgate.edu/~integers/aa28/aa28.pdf}\n  {\\emph{Irrationality of the reciprocal sum of doubly exponential sequences}},\n  Integers \\textbf{26} (2026), Paper No.~A28, 17 pp.,\n  doi:\\href{https://doi.org/10.5281/zenodo.18714404}{10.5281/zenodo.18714404};\n  preprint arXiv:2504.05933v1.\n\\bibitem{erdosproblems} T.~F. Bloom,\n  \\href{https://www.erdosproblems.com/243}{\\emph{Erd\\H{o}s Problem \\#243}},\n  \\texttt{erdosproblems.com/243}, accessed 28 July 2026 (page displays\n  ``last edited 21 January 2026'').  The current record labels the problem\n  open, cites \\texttt{[ErGr80, p.~64]} and \\texttt{[Er88c, p.~105]}, and\n  explicitly describes its status as the website owner's present assessment\n  rather than a literature-completeness guarantee.\n\\bibitem{formalconjectures243} The Formal Conjectures Authors,\n  \\href{https://github.com/google-deepmind/formal-conjectures/blob/f776d2f2039351b00737ffcafb9d7d7666e1d9af/FormalConjectures/ErdosProblems/243.lean}\n  {\\emph{FormalConjectures.ErdosProblems.\\texttt{243}}},\n  Lean source at commit \\texttt{f776d2f}, 2025, accessed 28 July 2026.\n  Its $\\Q$-valued \\texttt{Summable} hypothesis encodes rationality, its\n  indexing is zero-based, and its proof ends in \\texttt{sorry}.\n\\end{thebibliography}\n\n\\end{document}\n","dependencies":[{"record_type":"paper_source_dependency","tex_reference":"problem-note-preamble","source_path":"paper/problem-note-preamble.tex","media_type":"text/x-tex","sha256":"sha256:e7378fe7784916681e7faf96c1ea7a8bb941a416cc651b97512e849bcaeb63f7","byte_count":4990,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Shared preamble for the Erd\u0151s Problem Notes: the short, standalone,\n% problem-owned manuscripts covering the expansion library `ErdosProblems`.\n%\n% One note owns one Erd\u0151s problem.  Each note repeats whatever context its own\n% reader needs, so that a reader who arrives at a single problem never has to\n% assemble it from the gateway paper.  Deliberate repetition across notes is the\n% design, not an oversight.\n%\n% Source links resolve against one pinned formal-source commit, declared once\n% below.  `scripts/check_problem_note_sources.py` verifies that every authored\n% (file, line, declaration) triple names that exact declaration in the pinned\n% snapshot, so a link cannot silently rot as later work moves lines.\n\n\\usepackage{paper-house-style}\n\\usepackage{array}\n\\usepackage{booktabs}\n\\usepackage{enumitem}\n\\setlist{topsep=0.35em,itemsep=0.2em}\n\\usepackage[colorlinks=true,linkcolor=linkinternal,urlcolor=linkexternal,\n  citecolor=linkinternal]{hyperref}\n\\hypersetup{bookmarksnumbered=true,bookmarksopen=true,bookmarksopenlevel=1,\n  pdfauthor={Will Cook},pdflang={en-GB}}\n\n% ---------------------------------------------------------------------------\n% Pinned formal source.  \\commit names the checkpoint every link resolves\n% against; the checker reads that snapshot rather than the working tree, so the\n% printed coordinates stay correct however later waves move declarations.\n% ---------------------------------------------------------------------------\n\\newcommand{\\commit}{d6d344d9557faee19276b01a1c1f20d7c08f03d5}\n\\newcommand{\\commitshort}{d6d344d9557f}\n\\newcommand{\\repobase}{https://github.com/wcook04/plectis-lean-erdos249-257/blob/\\commit}\n\\newcommand{\\sourceurl}{https://github.com/wcook04/plectis-lean-erdos249-257/tree/\\commit}\n\\newcommand{\\PX}{ErdosProblems}\n\n% Declaration names stay inside link targets.  The printed label is either a\n% spaced, lower-case reading of the declaration name (\\lref) or a short authored\n% phrase (\\lword); neither prints a module path or a raw Lean identifier.\n\\ExplSyntaxOn\n\\cs_new_protected:Npn \\noteleanlabel #1\n  {\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_to_str:n {#1} }\n    \\regex_replace_all:nnN { _+ } { \\c{space} } \\l_tmpa_tl\n    \\regex_replace_all:nnN\n      { ([a-z0-9])([A-Z]) }\n      { \\1\\c{space}\\2 }\n      \\l_tmpa_tl\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_lower_case:n { \\tl_use:N \\l_tmpa_tl } }\n    \\tl_use:N \\l_tmpa_tl\n  }\n\\ExplSyntaxOff\n\\newcommand{\\leanlabel}[1]{\\noteleanlabel{#1}}\n\\newcommand{\\lref}[3]{\\href{\\repobase/\\PX/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\lword}[4]{\\href{\\repobase/\\PX/#1\\#L#2}{#4}}\n\\newcommand{\\lloc}[2]{\\href{\\repobase/\\PX/#1\\#L#2}{Lean source}}\n\n% Links into a sibling library, named repository-relative.  The expansion\n% library is implicit in \\lref/\\lword, because that is where problem-owned\n% work lands.  Several problems have their headline theorem in the older\n% reviewed #249/#257 corpus, and a note that could not name that library\n% would have to paraphrase a checked statement instead of linking it.\n\\newcommand{\\mref}[3]{\\href{\\repobase/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\mword}[4]{\\href{\\repobase/#1\\#L#2}{#4}}\n\\newcommand{\\mloc}[2]{\\href{\\repobase/#1\\#L#2}{Lean source}}\n\\emergencystretch=2em\n\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\theoremstyle{definition}\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{example}[theorem]{Example}\n\\newtheorem{problem}[theorem]{Problem}\n\\theoremstyle{remark}\n\\newtheorem*{remark}{Remark}\n\n\\newcommand{\\N}{\\mathbb{N}}\n\\newcommand{\\Npos}{\\mathbb{N}_{>0}}\n\\newcommand{\\Z}{\\mathbb{Z}}\n\\newcommand{\\Q}{\\mathbb{Q}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\lcm}{\\operatorname{lcm}}\n\\newcommand{\\ph}{\\varphi}\n\n% The series line printed under every note's title block.\n\\newcommand{\\seriesline}{Erd\\H{o}s Problem Notes}\n\n% Problem-note compatibility name for the corpus-owned page-one disclosure in\n% paper-house-style.sty.\n\\newcommand{\\noteprovenance}{\\paperprovenance}\n\n% The status paragraph every note carries, in its own words, before any result.\n% Kernel checking establishes the exact Lean proposition; it does not establish\n% that the proposition is the interesting one, that it is new, or that the\n% Erd\u0151s problem is closed.\n\\newcommand{\\statusboundary}{%\n  \\emph{Status.}  The problem treated here is open, and this note does not\n  close it.  Every statement below marked as checked is a proposition that the\n  pinned Lean kernel accepts from the sources this note links to, with no\n  \\texttt{sorry}, no added axiom, and no unchecked evaluation.  That is a claim\n  about the formal statement, not about its mathematical interest, its\n  novelty, or the original problem.  The unresolved obligations are named\n  exactly, in their own section, and none of the finite computations,\n  reductions, or no-go results here removes one of them.}\n"}],"external_verification_route":"https://raw.githubusercontent.com/wcook04/plectis-lean-erdos249-257/main/paper/erdos-243-reciprocal-tail-rigidity.tex"},{"record_type":"paper","paper_id":"erdos-249-binary-totient-series","title":"A Basis for the 2-Kernel of Euler's Totient","source_repository":"plectis-lean-erdos249-257","source_path":"paper/erdos-249-binary-totient-series.tex","hosted_pdf_path":"papers/erdos-249-binary-totient-series.pdf","licence":"CC-BY-4.0","copyright":"2026 Will Cook","cite_as":"Plectis reader corpus, paper:erdos-249-binary-totient-series","pdf_sha256":"sha256:3dcbd8805dd0a87b85e64a40214ecc596017addc084f0c437719c0357ef353b3","pdf_byte_count":136323,"pdf_embedded":false,"pdf_omission_reason":"the exact LaTeX is embedded and retrieves better than extracted PDF text; the PDF is a presentation rendering and its hash is recorded for verification","source_embedded":true,"availability":"embedded_exact","embedded_copy_role":"native_manuscript","embedded_from_repository":"plectis-lean-erdos249-257","embedded_copy_path":"paper/erdos-249-binary-totient-series.tex","media_type":"text/x-tex","body_format":"latex","sha256":"sha256:1f081ff79dad74502557717b65016751418a3b41b1af044dc57339ee629157fa","byte_count":52354,"body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Standalone Erd\u0151s Problem Note for #249.  This deliberately repeats the\n% gateway paper's definitions and status boundary: independent readability is\n% more important than avoiding duplication.\n%\n% Ordering rule.  Most readers stop after two pages, and one of them is a\n% moderator who has already rejected a longer version of this work for\n% overstating scope.  So: the title names a theorem; the abstract states the\n% results and then the boundary; Section 1 is a numbered inventory of every\n% claim, each carrying a status tag and a live link into the checked source.\n% Everything after Section 1 is mechanism, scope and provenance.\n%\n% Second rule, learned from that rejection: a barrier is only worth printing\n% if its hypothesis class is one somebody would actually assume.  Two\n% propositions that appeared here in earlier drafts are demoted in Section 6\n% for failing that test, with the reason stated rather than the propositions\n% quietly dropped.\n%\n% Third rule, July 2026.  The status tag on a result must be the status the\n% claim registry records for the row that owns its declarations, not a tag\n% chosen while writing.  Six of the nine tags in the previous draft disagreed\n% with the registry, and two headline results turned out to be owned by no row\n% at all.  Those now say so on their face.  A note whose subject is the gap\n% between checked propositions and public wording cannot afford that gap in\n% its own status column.\n\\documentclass[11pt]{article}\n\\input{problem-note-preamble}\n\\hypersetup{\n  pdftitle={Exact finite-level ranks in the 2-kernel of Euler's totient},\n  pdfsubject={Erd\u0151s Problem Notes: Problem 249},\n  pdfkeywords={irrationality, Euler totient, k-regular sequences, 2-kernel, Lambert series, M\u00f6bius inversion, linear independence, Lean 4},\n}\n\n\\newcommand{\\Stot}{S}\n\\newcommand{\\EB}{E}\n\\newcommand{\\Aw}{A}\n\\newcommand{\\Ker}{\\mathcal K}\n\n% Status tag from the claim taxonomy, printed on every numbered result.\n\\newcommand{\\stag}[1]{{\\normalfont\\footnotesize\\textsc{[#1]}}}\n\n\\runninghead{Erd\\H{o}s \\#249: exact finite-level dyadic ranks}\n\\title{Exact Finite-Level Ranks in the 2-Kernel of Euler's Totient}\n\\subtitle{Exact finite-level ranks and consequences for Erd\\H{o}s\nProblem~\\#249}\n\\author{Will Cook}\n\\date{July 2026}\n\n\\begin{document}\n\\maketitle\n\\noteprovenance\n\n\\begin{abstract}\n\\noindent\nFor $j\\ge0$ and $0\\le r<2^j$, write\n$\\ph_{j,r}(n)=\\ph(2^jn+r)$.  We prove that, at every depth $e$, the\n$2^e+1$ canonical sections consisting of $\\ph_{0,0}$, $\\ph_{1,0}$ and the\nodd-residue sections through level $e$ are linearly independent over $\\Q$.\nTheir span therefore has exact dimension $2^e+1$, and the full dyadic-kernel\nspan is infinite-dimensional.\n\nLet $\\Stot=\\sum_{n\\ge1}\\ph(n)/2^n$.  If $\\Stot$ were rational, there would be\nan integer $v\\ge1$ and an integer sequence $u$ such that\n\\[\n  u(N+1)=2u(N)-v\\,\\ph(N+1),\n  \\qquad\n  \\frac{u(N)}{2^N}\\longrightarrow0,\n\\]\nwhose dyadic sections through level $e$ span a space of dimension at least\n$2^e-1$ for every $e$.  No upper bound contradicting this lower bound is\nproved.  We also exclude every rational representation $\\Stot=a/q$ with\n\\[\n  1\\le q\\le\n  79\\,639\\,646\\,646\\,701\\,375\\,323\\,355\\,774\\,875\\,831\\,053,\n\\]\nthe sharp exclusion for the chosen $240$-binary-digit Farey window, and prove that no\nfixed positive integer clears all normalised coefficients\n$(\\ph*\\mu)(n)/n$.\n\nErd\\H{o}s Problem~\\#249 remains open.  The final part of the note gives exact\ntail-certificate characterisations of irrationality and states the unproved\nunbounded hypotheses required by the corresponding sufficient criteria.\n\\end{abstract}\n\n\\section{Introduction and main results}\\label{sec:results}\n\nErd\\H{o}s Problem~\\#249 asks whether\n\\[\n  \\Stot=\\sum_{n\\ge1}\\frac{\\ph(n)}{2^n}\n\\]\nis irrational; see Erd\\H{o}s and Graham~\\cite[p.~61]{erdosgraham1980} and\nErd\\H{o}s~\\cite[p.~102]{erdos1988}.  Bloom's current catalogue record\nreproduces this question and labels it open, while explicitly warning that\nthe status is the website owner's present assessment and may omit relevant\nliterature~\\cite{erdosproblems}.  We therefore use the catalogue for\nnumbering and current reported status only; the two original sources carry\nthe problem statement.\n\nThe principal result gives an exact finite-level rank for canonical dyadic\nsections of Euler's totient.  Its connection with Erd\\H{o}s Problem~\\#249 is\nthrough the integral scaled-tail recurrence displayed below.  The rank theorem\ngives a lower bound for the dyadic-section rank of every such recurrence\nsupplied by a hypothetical rational value of $\\Stot$, but no corresponding\nupper bound is known.\n\nThe other unconditional statements are a finite Farey denominator exclusion,\nidentities for the Lambert coefficient sequence $\\Aw=\\ph*\\mu$, and a theorem\nthat no fixed common denominator clears every $\\Aw(n)/n$.  The final two items\nhave a different logical status: \\ref{res:equivalences} gives exact\ncharacterisations of irrationality, while \\ref{res:supply} gives a sufficient\ncriterion.  Their unbounded hypotheses are not proved.\n\n\\statusboundary\n\nEach \\emph{Checked} line below links the relevant declarations at\ncommit~\\texttt{\\commitshort}.  The bracketed tags reproduce the public status\nrecorded by the claim registry; where no registry row owns the cited\ndeclarations, the text says so rather than assigning a neighbouring status.\n\nThroughout $\\ph_{j,r}(n)=\\ph(2^jn+r)$, for $j\\ge0$ and $0\\le r<2^j$, is the\n$(j,r)$ \\emph{dyadic section} of Euler's totient, a vector in $\\Q^{\\N}$, and\n$j$ is its \\emph{level}; the family of all of them is the $2$-kernel\n$\\Ker_2(\\ph)$ in the sense of Allouche and Shallit~\\cite{allouche-shallit}.\nTwo elementary identities relate them: the \\emph{reductions}\n$\\ph_{j+1,0}=2^{j}\\ph_{1,0}$\n(\\mword{Erdos249257/TotientMahlerDefect.lean}{150}{totientKernel_zero_residue}{checked})\nand $\\ph_{j,\\,2^{t+1}s}=2^{t}\\ph_{j-t-1,\\,s}$ for $s$ odd with $2^{t+1}s<2^j$\n(\\mword{Erdos249257/TotientMahlerDefect.lean}{159}{totientKernel_even_residue_reduce}{checked}).\nAt level $3$, for instance, they give $\\ph_{3,0}=4\\ph_{1,0}$,\n$\\ph_{3,2}=\\ph_{2,1}$, $\\ph_{3,4}=2\\ph_{1,1}$ and $\\ph_{3,6}=\\ph_{2,3}$, while\nthe four odd residues $r=1,3,5,7$ are reduced by neither; numerically\n$\\ph_{3,4}(1)=\\ph(12)=4=2\\ph(3)=2\\ph_{1,1}(1)$.  So beyond $\\ph_{0,0}$ and\n$\\ph_{1,0}$ each level contributes only its odd residues, and the count\nthrough level $e$ is $2+\\sum_{j=1}^{e}2^{j-1}=2^e+1$; \\ref{res:rank} says that\ncount is the exact dimension.\n\nTwo objects built from $\\Stot$ recur below.  The \\emph{scaled tail}\n$R_N=\\sum_{m\\ge1}\\ph(N+m)/2^m$ is $2^N$ times the part of the series beyond\nindex $N$; since $2^N\\Stot=\\Phi_N+R_N$ with\n$\\Phi_N=\\sum_{1\\le n\\le N}\\ph(n)2^{N-n}$ an integer\n(\\mword{Erdos249257/TotientTailPeriodKiller.lean}{150}{two_pow_mul_totient_series_eq}{checked}),\n$R_N$ and $2^N\\Stot$ differ by an integer.  For $h,N,L\\in\\N$, put\n\\[\n  D_{h,N,L}\n  =\n  \\sum_{j=0}^{L-1}\n  \\bigl(\\ph(N+h+1+j)-\\ph(N+1+j)\\bigr)2^{L-1-j},\n\\]\nand let $\\rho_{h,N,L}\\in\\{0,\\dots,2^L-1\\}$ be its residue modulo $2^L$.\nA \\emph{finite tail-difference certificate} at $(h,N,L)$ is the pair of\nstrict inequalities\n\\[\n  N+h+L+2<\\rho_{h,N,L}<2^L-(N+h+L+2)\n\\]\n(\\mword{Erdos249257/TotientTailPeriodKiller.lean}{72}{certifiedKill}{definition}).\nIt exists to make a statement about an infinite tail decidable by a finite\ncomputation, and it is faithful: some depth $L$ certifies $(h,N)$ if and only\nif $R_{N+h}-R_N\\notin\\Z$\n(\\mword{Erdos249257/LcmConeFlatness.lean}{316}{exists_certifiedKill_iff_tail_diff_notMem_int}{checked}).\nFor example,\n\\[\n  D_{1,12,16}=-143140,\\qquad\n  \\rho_{1,12,16}=53468,\\qquad N+h+L+2=31,\n\\]\nso $(1,12,16)$ is a certificate.  More generally, the test succeeds at\n$(h,12,16)$ for every $h$ with $1\\le h\\le8$\n(\\mword{Erdos249257/TotientTailPeriodKiller.lean}{404}{certifiedKill_all_small}{checked}),\nso none of $R_{13}-R_{12},\\dots,R_{20}-R_{12}$ is an integer\n(\\mword{Erdos249257/TotientTailPeriodKiller.lean}{407}{tail_diff_not_int_small}{checked}).\n\nA third recurring object is an \\emph{integral scaled-tail sequence with\nmultiplier $v$}.  For a coefficient sequence $c$ and an integer $v\\ge1$, this\nis an integer sequence $u$ satisfying\n\\[\n  u(N+1)=2u(N)-v\\,c(N+1)\\quad\\hbox{for every }N,\n  \\qquad\n  \\frac{u(N)}{2^N}\\longrightarrow0\n\\]\n(\\mword{Erdos249257/GenericTailOrbitRigidity.lean}{67}{IsTemperedBinaryOrbit}{definition});\nfor $c=\\ph$ such a pair $(v,u)$ with $v\\ge1$ exists exactly when $\\Stot$ is\nrational (Section~\\ref{sec:carry-rank}).\n\n{\\small\n\\begin{enumerate}[label=\\textbf{R\\arabic*.},ref=R\\arabic*,leftmargin=2.4em,\n  itemsep=0.2em,topsep=0.25em]\n\n\\item\\label{res:infinite} \\stag{checked; no registry row}\n\\textbf{Infinite-dimensional dyadic kernel.}  The $\\Q$-span of all dyadic\nsections $\\ph_{j,r}$ is not finite-dimensional.  At every depth $e$ it\ncontains the canonical independent family of \\ref{res:rank}, whose dimension\nis exactly $2^e+1$; taking $e$ equal to a hypothetical ambient dimension gives\nthe contradiction.\n{\\footnotesize\\emph{Checked:}\n\\mword{Erdos249257/TotientMahlerDefect.lean}{1014}{not_finiteDimensional_span_fullTotientKernel}{infinite-dimensionality}.\n\\emph{Registry:} this declaration is not currently owned by a row of the claim\nregistry; see the formal source notes below.}\n\n\\item\\label{res:rank} \\stag{unconditional progress}\n\\textbf{Exact finite-level rank.}  For every $e\\ge0$ the $2^e+1$ sections\n$\\ph_{0,0}$, $\\ph_{1,0}$ and $\\ph_{j,r}$ with $1\\le j\\le e$, $r$ odd,\n$0<r<2^j$ are linearly independent over $\\Q$, so their span has dimension\nexactly $2^e+1=2+\\sum_{j=1}^{e}2^{j-1}$: exact, not an estimate, because the\neven residues are already dependent by the reductions.  Read instead as a\ntruncation of the kernel, $V_e=\\operatorname{span}_{\\Q}\\{\\ph_{j,r}:0\\le j\\le\ne\\}$ has $\\dim V_e=2^e+1$ for every $e\\ge1$, and $\\dim V_0=1$: the counted\nfamily carries $\\ph_{1,0}$, which is a level-one section, so the two readings\nagree from $e=1$ on and differ only at $e=0$.  At $e=1$ both readings say that\n$\\ph(n)$, $\\ph(2n)$ and $\\ph(2n+1)$ are linearly independent over $\\Q$.\nThe unbounded sequence of exact ranks is the input to \\ref{res:infinite}.\n{\\footnotesize\\emph{Checked:}\n\\mword{Erdos249257/TotientMahlerDefect.lean}{900}{linearIndependent_canonicalTotientKernelFamily}{independence},\n\\mword{Erdos249257/TotientMahlerDefect.lean}{954}{finrank_canonicalTotientKernel_eq}{dimension},\n\\mword{Erdos249257/TotientMahlerDefect.lean}{1014}{not_finiteDimensional_span_fullTotientKernel}{infinite-dimensionality directly}.}\n\n\\item\\label{res:denominator} \\stag{formalised here}\n\\textbf{Denominator exclusion, sharp for its window.}  If $\\Stot=a/q$ with\n$a\\in\\Z$ and $q\\ge1$ then\n$q>79\\,639\\,646\\,646\\,701\\,375\\,323\\,355\\,774\\,875\\,831\\,053\\approx7.96\\times10^{34}$.\nThe constant is optimal for the window that produces it --- the window here is\nthe pair $(N,K)=(1,240)$, meaning $240$ committed binary digits of the series\nshifted by one --- in the sense that the next integer up is the exact first\ndenominator at which that window's certificate fails.  The method is the\nclassical Farey mediant argument (Section~\\ref{sec:denominator}).\n{\\footnotesize\\emph{Checked:}\n\\mword{Erdos249257/CertificateKernel.lean}{18043}{tsum_totient_div_pow_two_ne_int_div_of_den_le_79639646646701375323355774875831053}{exclusion},\n\\mword{Erdos249257/CertificateKernel.lean}{18056}{tsum_totient_div_pow_two_ne_ratCast_of_den_le_79639646646701375323355774875831053}{reduced-denominator form},\n\\mword{Erdos249257/GapFareyBound.lean}{225}{gap_check_window_1_240_first_failure}{first failure}.\n\\emph{Registry:} the row owns the reduced-denominator form only.}\n\n\\item\\label{res:carryrank} \\stag{formalised here}\n\\textbf{Rank lower bound for a rational scaled-tail sequence.}  If $\\Stot$\nwere rational then some integral scaled-tail sequence $u$ for the coefficients\n$\\ph$, with positive multiplier, would have for \\emph{every} $e$ dyadic\nsections through level $e$ --- formed from $u$ exactly as the $\\ph_{j,r}$ are\nformed from $\\ph$ --- spanning a space of dimension at least $2^e-1$.  The\nsame lower bound holds for every positive-multiplier integral scaled-tail\nsequence.  This statement is specific to that\nbinary scaled-tail representation; it does not constrain arguments formulated\nonly through the Lambert weight, the coprimality identity, or tail differences.\nIt is not an irrationality criterion, because no finite-rank upper bound is\nproved for the scaled-tail sequences supplied by rationality.\n{\\footnotesize\\emph{Checked:}\n\\mword{Erdos249257/TotientCarryKernelRigidity.lean}{284}{not_irrational_totientSeries_implies_unbounded_carryRank_unconditional}{the barrier},\n\\mword{Erdos249257/TotientCarryKernelRigidity.lean}{211}{finrank_canonicalCarryKernel_ge_of_linearIndependent}{rank transport from \\ref{res:rank}}.}\n\n\\item\\label{res:weight} \\stag{checked; no registry row}\n\\textbf{The \\#249 weight.}  $\\Stot=\\sum_{d\\ge1}\\Aw(d)/(2^d-1)$ with\n$\\Aw=\\ph*\\mu$, the Dirichlet convolution\n$\\Aw(n)=\\sum_{d\\mid n}\\ph(d)\\mu(n/d)$; here $\\Aw\\ge0$, $\\Aw(p)=p-2$,\n$\\Aw(p^k)=\\ph(p^k)-\\ph(p^{k-1})$, and $\\Aw$ is unbounded.  Its values at\n$n=1,\\dots,10$ are $1,0,1,1,3,0,5,2,4,0$.\n{\\footnotesize\\emph{Checked:}\n\\mword{Erdos249257/CertificateKernel.lean}{18117}{tsum_primWeight_div_two_pow_sub_one_eq_totient_series}{identity},\n\\mword{Erdos249257/MersenneLambertLadder.lean}{370}{primWeight_nonneg}{sign},\n\\mword{Erdos249257/MersenneLambertLadder.lean}{297}{primWeight_apply_prime}{primes},\n\\mword{Erdos249257/MersenneLambertLadder.lean}{272}{primWeight_apply_prime_pow}{prime powers},\n\\mword{Erdos249257/MersenneLambertLadder.lean}{321}{primWeight_not_bounded}{unboundedness}.\n\\emph{Registry:} the Lambert-ladder row owns the neighbouring rungs, not these\nfive declarations; see the formal source notes below.}\n\n\\item\\label{res:periodic} \\stag{unconditional progress}\n\\textbf{Eventually periodic weights are settled.}  For every integer base\n$b\\ge2$, if $\\gamma\\colon\\N\\to\\Q$ is eventually periodic, nonnegative and\npositive at some positive index of its periodic part, then\n  $\\sum_{a}\\gamma(a)/(b^a-1)$ is irrational.  Theorem~A in Luca and\n  Tachiya's 2017 open-access RIMS paper restates their earlier signed theorem in the exact\n  purely-periodic integer case: every nonzero purely periodic integer sequence\n  gives an irrational Lambert value at every integer base $|q|>1$\n  \\cite[Theorem~A, p.~139]{lucatachiya2017}.  Their Theorem~1 strengthens the\n  nonnegative case to linear independence of each finite divisor-convolution\n  ladder \\cite[Theorem~1, p.~139; proof pp.~149--150]{lucatachiya2017}.\n  Clearing the common denominator of a rational period and subtracting the\n  finite rational prefix shows that Theorem~1 already proves the eventual\n  nonnegative rational claim above; the checked theorem gives an independent\n  formal proof.  By \\ref{res:weight} the weight $\\Aw$ is\nunbounded, hence not eventually periodic, so \\#249 is outside this class.\n{\\footnotesize\\emph{Checked:}\n\\mword{Erdos249257/CertificateKernel.lean}{12483}{irrational_ratWeightSeries_eventuallyPeriodic}{the periodic theorem}.}\n\n\\item\\label{res:primindex} \\stag{proved here}\n\\textbf{No fixed common denominator clears the normalised weight.}  For every integer\n$D\\ge1$ some $n\\ge1$ has $D\\cdot\\Aw(n)/n\\notin\\Z$ --- for $D=6$ at the prime\n$n=7$, where $\\Aw(7)/7=5/7$ --- and any $D$ clearing every coordinate up to a\nhorizon $N\\ge4$ is divisible by an explicit two-tier primorial ($4$ at $p=2$,\n$p^2$ when $p^2\\le N$, else $p$).  No argument that clears the coordinates\n$\\Aw(n)/n$ by a single positive integer valid at all $n$ can therefore\nsucceed.  This obstruction is caused by the normalisation: the unnormalised\nweights $\\Aw(n)$ are integers.  Arguments whose common denominator grows with\nthe horizon, arguments using the integral weights directly, and arguments\nthat never clear these coordinates are untouched.\n{\\footnotesize\\emph{Checked:}\n\\mword{Erdos249257/PrimitiveDeterminantLift.lean}{169}{no_fixed_integral_primitive_euler_index}{no fixed index},\n\\mword{Erdos249257/PrimitiveDeterminantLift.lean}{148}{twoTierPrimorial_dvd_of_integral_jet}{the primorial divisibility}.}\n\n\\item\\label{res:equivalences} \\stag{proved here}\n\\textbf{Four exact characterisations of irrationality.}  Irrationality\nof $\\Stot$ is equivalent to each of: non-integrality of the tail difference\n$R_{N+h}-R_N$ at every $N$ and every shift $h\\ge1$, equivalently a certificate\nat every such pair; a certificate at every $h\\ge1$ and at arbitrarily large\n$N$; and the same at the scales $N=h=\\lcm(1,\\dots,t)$, for arbitrarily large\n$t$.  A fourth equivalence,\nindependent of the certificate language, is a counted anti-concentration\ncondition on window phases.  These characterisations identify the unbounded\ninput exactly; they retain the full difficulty of \\#249 and do not supply that\ninput.\n{\\footnotesize\\emph{Checked:}\n\\mword{Erdos249257/LcmConeFlatness.lean}{399}{irrational_totient_series_iff_pointwise_certificates}{pointwise},\n\\mword{Erdos249257/LcmConeFlatness.lean}{412}{irrational_totient_series_iff_certificate_supply}{cofinal at each shift},\n\\mword{Erdos249257/LcmConeFlatness.lean}{426}{irrational_totient_series_iff_lcm_diagonal_certificate_supply}{$\\lcm$ diagonal},\n\\mword{Erdos249257/PivotAntiReconstruction.lean}{1744}{dtwWindowSeparatedPairs_iff_irrational_totient_series}{window-separated pairs}.}\n\n\\item\\label{res:supply} \\stag{conditional reduction}\n\\textbf{A denominator-indexed gap-certificate hypothesis.}  Suppose that for\neach denominator $q$ there is a truncation window $(N,K)$ whose totient residue\navoids a band of width $q(N{+}K{+}2)$ out of $2^K$.  This hypothesis\n\\emph{implies} irrationality.  It concerns a second certificate family, not\nthe one of \\ref{res:equivalences}: the band scales with $q$, and the family is\nindexed by denominators rather than by shifts.  The converse is not proved and\nis not claimed.\n{\\footnotesize\\emph{Checked:}\n\\mword{Erdos249257/CertificateKernel.lean}{15697}{irrational_tsum_totient_div_pow_two_of_gap_certificate_supply}{the supply implication}.\n\\emph{Registry:} no row currently owns this declaration.}\n\n\\item\\label{res:open} \\stag{open}\n\\textbf{Erd\\H{o}s~\\#249.}  Whether $\\Stot$ is irrational.  Not proved here.\n\n\\end{enumerate}\n}\n\n\\section{The Lambert-series form of Problem \\#249}\\label{sec:family}\n\nIf $f=g*\\mathbf 1$, that is $f(n)=\\sum_{d\\mid n}g(d)$, then interchanging the\ntwo sums gives\n\\[\n  \\sum_{n\\ge1}\\frac{f(n)}{2^n}\n  \\;=\\;\\sum_{d\\ge1}\\frac{g(d)}{2^d-1}.\n  \\tag{1}\\label{eq:lambert}\n\\]\nwhose right-hand side is a Lambert series with coefficients $g$.  Four\nclassical coefficient sequences have this shape.  The table records the\nconstant, the weight $g=f*\\mu$, and its status.\n\n\\begin{table}[h]\n\\centering\n\\small\n\\begin{tabular}{@{}\n  >{\\raggedright\\arraybackslash}p{0.07\\linewidth}\n  >{\\raggedright\\arraybackslash}p{0.19\\linewidth}\n  >{\\raggedright\\arraybackslash}p{0.21\\linewidth}\n  >{\\raggedright\\arraybackslash}p{0.43\\linewidth}@{}}\n\\toprule\n$f$ & $\\sum f(n)/2^n$ & weight $g=f*\\mu$ & status \\\\\n\\midrule\n$\\tau$ & $1.6066951\\ldots=\\EB$ & $g\\equiv1$ & The Erd\\H{o}s--Borwein constant;\n  irrational~\\cite[theorem on p.~63]{erdos1948};\n  formalised here in the Lambert form $\\sum_{d\\ge1}(2^d-1)^{-1}$\n  (\\mword{Erdos249257/CertificateKernel.lean}{8007}{irrational_erdosBorwein_series}{checked}) \\\\\n$\\omega$ & $0.5169428\\ldots$ & $g=\\mathbf 1_{\\text{primes}}$ &\n  Irrational at base $2$: Tao--Ter\\\"av\\\"ainen, Thm.~1.3, p.~4;\n  proof pp.~44--56~\\cite{taoteravainen2025} \\\\\n$\\Omega$ & $0.5895032\\ldots$ & $g=\\mathbf 1_{\\text{prime powers}}$ &\n  Asserted \\emph{ibid.}\\ by a similar argument, with the details left to the\n  reader; not a proved theorem there \\\\\n$\\ph$ & $1.3676308\\ldots=\\Stot$ & $g=\\Aw=\\ph*\\mu$ &\n  \\textbf{Open} (Erd\\H{o}s \\#249) \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\noindent The first three weights are bounded, taking only the values $0$\nand~$1$ (for $\\tau$ the weight is the constant $1$).  The fourth is unbounded,\nby \\ref{res:weight}.  That difference is the one that matters for the route in\n\\ref{res:periodic}, and we do not claim it is the only difference: $\\Aw$ also\nvanishes on $n\\equiv2\\pmod4$, while $\\mathbf 1_{\\text{primes}}$ is bounded but\nno more eventually periodic than $\\Aw$ is, so the settled $\\omega$ row is not\nan instance of \\ref{res:periodic} either.  It was settled by a different\nmethod, discussed in Section~\\ref{sec:open}.\n\nLuca and Tachiya's result is stronger than the single $\\tau$ row:\nExample~1 makes \\(1\\) and every finite ladder of generalized-divisor-function\nvalues linearly independent\n~\\cite[Example~1, p.~140; proof pp.~149--150]{lucatachiya2017}.\nTheir Example~2 applies the period-two odd-support indicator and proves joint\nlinear independence of its Lambert value with every finite higher\ndivisor-convolution ladder, for integer bases \\(q\\) of either sign with\n\\(|q|>1\\) \\cite[Example~2, p.~140]{lucatachiya2017}. Thus the cited theorem\nstrictly strengthens the isolated odd-support irrationality row.\nHistorically, Erd\\H{o}s already singled out the totient series itself as a\ndifficult analogue at the end of his 1948 paper~\\cite[p.~66]{erdos1948};\nthat remark supplies lineage, not a modern status proof.\n\n\\paragraph{A nearby 2026 totient theorem has different coordinates.}\nKaneko, Suzuki and Tachiya prove that if \\(f(n)\\) is a nonnegative integer\nsequence of infinite support with\n\\(\\sum_{n\\le x}f(n)=O(x(\\log x)^\\delta)\\), then, for every integer\n\\(t\\ge2\\), both\n\\[\n  \\sum_{n\\ge1}\\frac{f(n)}{t^{\\sigma(n)}}\n  \\qquad\\text{and}\\qquad\n  \\sum_{n\\ge1}\\frac{f(n)}{t^{\\ph(n)}}\n\\]\nare irrational\n\\cite[Corollary~3, pp.~5--6; proof pp.~18--19]{kanekosuzukitachiya2026}.\nThis includes substantial families of totient-related series, but\n\\(\\ph(n)\\) occurs in the \\emph{exponent}.  Problem~\\#249 instead places\n\\(\\ph(n)\\) in the coefficient of \\(2^{-n}\\).  The theorem therefore supplies\na genuine adjacent result and a useful warning against a tempting\nmisidentification; it does not settle or reduce the displayed problem.\n\n\\paragraph{Exact identities and representations.}\nSeveral neighbouring Dirichlet-convolution identities are rational and\nexactly computable:\n$\\sum_{d\\ge1}\\mu(d)/(2^d-1)=\\tfrac12$\n(\\mword{Erdos249257/CertificateKernel.lean}{18106}{tsum_moebius_div_two_pow_sub_one_eq_half}{checked})\nand $\\sum_{d\\ge1}\\ph(d)/(2^d-1)=2$\n(\\mword{Erdos249257/CertificateKernel.lean}{18101}{tsum_totient_div_two_pow_sub_one_eq_two}{checked}).\nThus $\\Stot$ sits one M\\\"obius convolution from a machine-checked\nirrational and one from a rational.  The squared-denominator representation\n$\\Stot=\\tfrac12+\\sum_{d\\ge1}\\mu(d)/(2^d-1)^2$\n(\\mword{Erdos249257/CertificateKernel.lean}{18126}{totient_series_eq_half_add_moebius_mersenne_square}{checked})\nconverges fast enough to recompute the decimal above.\n\nThere is also an exact probabilistic reading, and it is more than a gloss.\nLet $X$ and $Y$ be independent fair-coin geometric waiting times on $\\Npos$.\nThen\n\\[\n  \\Stot=\\tfrac12+\\Pr[\\gcd(X,Y)=1]\n\\]\n(\\mword{Erdos249257/CertificateKernel.lean}{18229}{totient_series_eq_half_add_visible_coprime_pairs}{checked}),\nthe probability being the visible-coprime-pair series\n$\\sum_{\\gcd(m,n)=1}2^{-(m+n)}$.  That probability is distributed over the\nStern--Brocot tree in a completely explicit way: the Mersenne weights of the\npositive reduced slopes sum to exactly~$1$\n(\\mword{Erdos249257/GcdMomentCalculus.lean}{349}{tsum_pos_coprime_inv_mersenne_eq_one}{checked}),\nand at each node $(a,b)$ the cylinder mass $1/((2^a-1)(2^b-1))$ splits exactly\ninto its stop mass and the masses of its two children\n(\\mword{Erdos249257/GcdMomentCalculus.lean}{474}{cylinderMass_split}{checked}).\nSo $\\Stot-\\tfrac12$ is the total mass of an exactly self-similar subdivision of\na probability space.  What is missing is not structure; it is an arithmetic\nconsequence of the structure.\n\n\\section{Proof of the dyadic-section rank theorem}\\label{sec:rank}\n\n\\subsection{Prior work}\n\nFor an integer $k\\ge2$, the \\emph{$k$-kernel} of a sequence $c$ is\n$\\{n\\mapsto c(k^jn+r):j\\ge0,\\ 0\\le r<k^j\\}$, and $c$ is \\emph{$k$-regular}\nwhen the module generated by its $k$-kernel is finitely generated; both\nnotions are due to Allouche and Shallit\n(\\cite[Defs.~1.1 and 2.1, author-preprint pp.~2--3]{allouche-shallit}).\nTheir Theorem~2.2 gives equivalent finite-kernel and matrix\ncharacterisations, and Theorem~3.1 proves closure under convolution\n~\\cite[pp.~3--4 and 10--11]{allouche-shallit}; neither result controls the\nfinite-level rank of a sequence that is not $k$-regular.  Coons proved that\n$\\ph$ is not $k$-regular for any $k\\ge2$ (\\cite{coons}, Theorem~3.2,\npp.~348--349, in the published version; Theorem~3.3, pp.~8--9, in the\npreprint, which numbers its results on a single running counter).  Bell and\nSmertnig reach a further negative\nconclusion, and by a different route rather than by strengthening that one:\ntheir Theorem~1.3 shows that a $k$-Mahler series with multiplicative\ncoefficients has a $k$-regular coefficient sequence in an explicit closed form,\nand that the totient generating series is not $k$-Mahler for any $k\\ge2$\nfollows directly from the closed form~\\cite{bell-smertnig}.\n\nCoons's statement is about the union over all levels: no finite set of sections\ngenerates the rest.  Bell and Smertnig's is about a functional equation.\n\\ref{res:infinite} is the checked global infinite-dimensionality consequence,\nwhile \\ref{res:rank} is the sharper finite-level statement.  The latter says\nthat a particular canonical family has exact rank $2^e+1$ at depth $e$; it\ndoes not claim that the older odd-core family is a checked basis for the full\nkernel.  We found no prior source giving this exact finite-level rank for\n$\\ph$; that is the outcome of a search, not a proof of non-existence, and no\npriority is claimed.\n\n\\subsection{Parity-separated evaluation matrices}\n\n\\ref{res:rank} is proved by exhibiting a square matrix of values of the\nsections whose determinant is nonzero, and the linear algebra at its centre is\nsimpler than the arithmetic construction that feeds it.  We give the linear\nalgebra first, then the arithmetic input it consumes.\n\nEvaluate the sections at one point apiece to get a square matrix $M$, and\nsuppose each column $j$ admits a fixed power $2^{d_j}$ dividing every entry of\nthat column such that, after dividing the column through by it, the diagonal\nentry is odd and every off-diagonal entry is even; call such an $M$ a\n\\emph{parity-separated evaluation matrix}.  The normalised matrix is then the\nidentity modulo $2$; its determinant is odd, hence nonzero, and the sections are\nindependent\n(\\mword{Erdos249257/TotientMahlerDefect.lean}{788}{paritySeparatedMatrix_det_ne_zero}{checked}).\nProducing evaluation points with exactly that parity pattern is what the\narithmetic input is for, and it is the harder half.  The Chinese remainder\ntheorem together with Dirichlet's theorem on primes in arithmetic progressions\nmakes one of the evaluated affine values prime, which fixes the exact power of\n$2$ dividing the diagonal entry, while every other value receives a fresh\nprime divisor congruent to $1$ modulo a large power of two, which supplies the\none extra factor of two that the off-diagonal entries need\n(\\mword{Erdos249257/TotientMahlerDefect.lean}{847}{exists_separatedMinorCertificate_totientAffineOddFamily}{checked}).\nEven residues never enter, because the second reduction has already carried\nthem to odd residues at lower levels\n(\\mword{Erdos249257/TotientMahlerDefect.lean}{159}{totientKernel_even_residue_reduce}{checked}).\n\n\\ref{res:infinite} then follows by embedding every canonical finite-level family\ninto the full dyadic-kernel span.  If that ambient span had finite dimension\n$d$, \\ref{res:rank} at $e=d$ would put a subspace of dimension $2^d+1$ inside\none of dimension $d$, an impossibility.  This is the current checked assembly:\nit proves infinite-dimensionality without asserting the retired full-basis\nstatement from the older source snapshot.\n\n\\begin{remark}\n\\ref{res:infinite} and \\ref{res:rank} are statements about $\\ph$, not about\n$\\Stot$; they are true, and their proofs are complete, whether or not $\\Stot$\nis irrational.  Their connection to \\#249 is not thematic but a single proved\ntransport, and it is \\ref{res:carryrank}.\n\\end{remark}\n\n\\section{Carry-rank consequences of rationality}\\label{sec:carry-rank}\n\nMultiplying successive scaled tails by $2$ turns a hypothetical rational\nvalue into an integral recurrence.  For coefficients $c(n)\\le n$, and in\nparticular for $c=\\ph$ since $\\ph(n)\\le n$, the series $\\sum c(n)/2^n$ is\nrational exactly when an integral scaled-tail sequence exists, that is, an\ninteger sequence $u$ with\n$u(N+1)=2u(N)-v\\,c(N+1)$ for every $N$ and $u(N)/2^N\\to0$, for some integer\n$v\\ge1$\n(\\mword{Erdos249257/GenericTailOrbitRigidity.lean}{426}{binaryCoeffSeries_rational_iff_exists_temperedBinaryOrbit}{checked}).\nTransporting \\ref{res:rank} through that recurrence gives \\ref{res:carryrank}.\nThe complementary finite-rank upper bound, which would make this a proof of\nirrationality, is not available.\n\n\\section{A Farey-mediant denominator exclusion}\\label{sec:denominator}\n\n\\ref{res:denominator} is a Farey exclusion, and we say so before saying\nanything else about it.  The argument is the classical mediant one.  The Lean\nlemma commits the first $240$ binary digits of the shifted series as a single\nresidue over the totient carry window $(N,K)=(1,240)$; a rational $a/q$ can\nequal $\\Stot$ only if it falls into a resulting bad interval of width\n$243/2^{240}$; that interval is bracketed by two explicit unimodular fractions\n$a_1/b<c_1/d$ with $bc_1-a_1d=1$; and the mediant lemma\n(\\mword{Erdos249257/GapFareyBound.lean}{51}{farey_gap}{checked}) forces any\nrational strictly between unimodular neighbours to have denominator at least\n$b+d$.  The excluded range is therefore $q\\le b+d-1$, which is the printed\nconstant\n(\\mword{Erdos249257/GapFareyBound.lean}{176}{gap_check_window_1_240_le_79639646646701375323355774875831053}{checked}),\nand the transport to the series is a tail estimate\n(\\mword{Erdos249257/CertificateKernel.lean}{15658}{tsum_totient_div_pow_two_ne_int_div_of_totient_gap_certificate}{checked}).\n\\emph{The method is standard, and we claim no novelty for it.}\n\nTwo things should be read off correctly.  First, the scale.  A window of $K$\ncomputed binary digits yields a mediant bound of order $2^{K/2}$: the same\npipeline gives $2.49\\times10^{17}$ at $K=120$\n(\\mword{Erdos249257/GapFareyBound.lean}{88}{gap_check_window_1_120_le_248672326362367909}{checked})\nand $7.96\\times10^{34}$ at $K=240$.  The constant is a function of how many\ndigits were committed, not a measure of how much the problem has moved.\nSecond, the sharpness.  Within its window the bound cannot be improved at all:\n$b+d$ itself fails the certificate, so $b+d-1$ is exactly the last denominator\nexcluded.  A reader who regards an explicit Farey exclusion as routine is not\nin disagreement with this note.  \\ref{res:infinite}, not \\ref{res:denominator},\nis where we would ask a sceptical reader to look.\n\nThe bound transfers to the coprimality form of Section~\\ref{sec:family} with\nthe constant halved: the visible-coprime-pair series has no representation\n$a/d$ with $d\\le39\\,819\\,823\\,323\\,350\\,687\\,661\\,677\\,887\\,437\\,915\\,526$\n(\\mword{Erdos249257/CertificateKernel.lean}{18244}{tsum_visible_coprime_pairs_ne_int_div_of_den_le_39819823323350687661677887437915526}{checked}),\nthat number being $(q_0-1)/2$ for the constant $q_0$ of\n\\ref{res:denominator}.\n\n\\section{Obstructions for fixed-coordinate methods}\\label{sec:nogo}\n\n\\ref{res:primindex} has a one-line proof.  Take a prime\n$p>\\max(D,2)$; then $\\Aw(p)=p-2$ by \\ref{res:weight}, and $p\\nmid D(p-2)$, so\n$D\\cdot\\Aw(p)/p\\notin\\Z$.\n\nTwo further checked results delimit narrower abstract hypothesis classes.\n\nA fixed-precision transport result\n(\\mword{Erdos249257/TropicalCurvatureCarry.lean}{137}{fixedPrecisionTropicalNoGo}{checked})\nsays that at every fixed positive precision, each finite compatible\nvaluation-unit word admits a prefix-locked centred completion.  Its statement\nquantifies over abstract pairs of a two-adic valuation and an odd unit; it\nmentions neither $\\ph$ nor $\\Stot$, its content is that a congruence class\nmeets any interval of the corresponding length, and it is a restatement of a\nlemma printed immediately above it.  The class of arguments it closes ---\nderiving a contradiction from a local signature read at a precision fixed in\nadvance, along a finite word --- is not one anybody has proposed for \\#249.\n\nA factor-ideal result\n(\\mword{Erdos249257/LcmFactorIdealPulseObstruction.lean}{798}{lcm_factorIdeal_finiteRank_shiftAlgebra_not_sufficient}{checked},\nwith a sparse-anchor companion at\n\\mword{Erdos249257/LcmFactorIdealPulseObstruction.lean}{866}{lcm_factorIdeal_sparseAnchor_not_sufficient}{checked})\nexhibits, for each $t\\ge3$ with $H=\\lcm(1,\\dots,t)$, a nonzero integer carry\nwhose forcing letters lie in the ideal generated by $\\ph(H)$, which reproduces\nthe true totient differences at $t-2$ prescribed indices, and which every\nfinite integer shift polynomial carries to a pair with the same coboundary\nform, the same ideal memberships and an $\\ell^1$-weight bound.  It does close\nthat hypothesis class.  Two limits.  The witness is a spike --- its state is\n$-\\ph(H)$ at $t-2$ points and zero elsewhere, so it stays uniformly bounded\nwhile the strip bounds it respects grow, and the construction is cheap for\nthat reason.  And the shifted pairs are not asserted to be nonzero, so an\nargument that additionally demanded non-vanishing after the shift is not\nexcluded.  The theorem contains no whole-ray anchor condition, diagonal bound,\nor strict-survivor condition.\n\n\\section{Further conditional criteria}\\label{sec:programmes}\n\nThe criteria below do not enter the proofs of the unconditional results.\nEach states its unproved hypotheses explicitly.\n\n\\paragraph{A prime-orbit sufficient condition.}\nThe hypothesis\n\\lword{Erdos249/TotientStrictPrimeEscape.lean}{18}{DTWNaturalPrimeTailOrbitStrictGap}{asks}\nfor cofinally many primes $p$ at which the first tail-orbit exponential has\nreal part strictly below $9/10$.  Granting it,\n\\lword{Erdos249/TotientStrictPrimeEscape.lean}{27}{naturalPivotPointEscape_of_naturalPrimeTailOrbitStrictGap}{a positive adaptive truncation budget}\nfollows, and then\n\\lword{Erdos249/TotientStrictPrimeEscape.lean}{66}{irrational_totient_series_of_naturalPrimeTailOrbitStrictGap}{$\\Stot$ is irrational}.\nThe $9/10$ hypothesis is weaker than the earlier $4/5$ hypothesis, but no\ninstance of this cofinal hypothesis is proved.\nSection~\\ref{sec:frontier} states the sharper form\nin which this hypothesis is now available.\n\n\\paragraph{Finite Euler-factor identities.}\nThe local coefficients are\n\\[\n  \\eta(0)=1,\\qquad \\eta(1)=-2,\\qquad \\eta(2)=1,\\qquad\n  \\eta(e)=0\\quad(e\\ge3),\n\\]\nthe coefficients of $(1-X)^2$.  For a prime $p$ and $e\\ge0$, write\n$\\sigma_p(e)=1+p+\\cdots+p^e$.  The two local Euler factors are\n\\[\n  1-\\frac2p+\\frac1{p^2}=\\left(1-\\frac1p\\right)^2,\n  \\qquad\n  1-\\frac2{p^2}+\\frac1{p^4}\n      =\\left(1-\\frac1{p^2}\\right)^2\n\\]\nat\n\\lword{Erdos249/FiniteEulerSieve.lean}{28}{muSqEulerFactor_one}{$s=1$}\nand\n\\lword{Erdos249/FiniteEulerSieve.lean}{36}{muSqEulerFactor_two}{$s=2$}.\nOn the prime-power divisor-sum row,\n\\[\n  \\sigma_p(1)-2\\sigma_p(0)=p-1\n\\]\n(\\lword{Erdos249/FiniteEulerSieve.lean}{44}{sigmaPrimePow_firstDiff}{first difference}),\nand for every $e\\ge0$,\n\\[\n  \\sigma_p(e+2)-2\\sigma_p(e+1)+\\sigma_p(e)\n    =p^{e+1}(p-1)\n\\]\n(\\lword{Erdos249/FiniteEulerSieve.lean}{51}{sigmaPrimePow_secondDiff}{second difference}).\nFor example, at $p=3$ the values\n$\\sigma_3(0),\\ldots,\\sigma_3(3)=1,4,13,40$ give\n$4-2=2$ and $40-2\\cdot13+4=18=3^2(3-1)$.  Thus convolution with\n$\\mu\\ast\\mu$ converts the prime-power divisor-sum row into the corresponding\ntotient row at every finite stage.  These are finite algebraic identities; no\ntranscendence conclusion is attached to them.\n\n\\paragraph{Mixed differences of cyclotomic layers.}\nFor a function $F:\\{0,1\\}^2\\to\\Z$, define\n\\[\n  \\Delta F=F(1,1)-F(1,0)-F(0,1)+F(0,0)\n\\]\n(\\lword{Erdos249/PrimeRayCyclotomicCurvature.lean}{15}{checkerboard}{mixed difference}).\nIf $F(i,j)=u(i)+v(j)$, then $\\Delta F=0$\n(\\lword{Erdos249/PrimeRayCyclotomicCurvature.lean}{19}{checkerboard_separable}{separable case}).\nConversely, among integer linear combinations of the four values of $F$ whose\ncoefficients sum to zero along each row and each column, the pattern\n$(1,-1,-1,1)$ is forced up to scale\n(\\lword{Erdos249/PrimeRayCyclotomicCurvature.lean}{25}{checkerboard_unique}{fixed-stencil uniqueness}).\nFor example, the table\n\\[\n\\begin{array}{c|cc}\n &0&1\\\\ \\hline\n0&9&13\\\\\n1&12&16\n\\end{array}\n\\]\nhas mixed difference $16-12-13+9=0$; increasing only the lower-right entry by\n$5$ changes the mixed difference to $5$.  The uniqueness statement is\ninternal to this fixed $2\\times2$ stencil.  It does not assert invariance under\nlarger stencils, nonlinear functionals, or another representation.\n\nThe formal development also records\n\\lword{Erdos249/PrimeRayCyclotomicCurvature.lean}{38}{fourPoint_layer_identity}{a four-point divisor-layer identity}\nand\n\\lword{Erdos249/PrimeRayCyclotomicCurvature.lean}{55}{residueClass_step_has_centred_completion}{a centred representative for a residue class modulo an even modulus}.\nThe two unproved hypotheses and their target conclusion are as follows.  For\n$C:\\N\\to\\N$ and $m,d\\in\\N$, put\n\\[\n\\begin{aligned}\n\\operatorname{Layer}(C,m)\\quad\\Longleftrightarrow\\quad&\n\\exists Q_0\\ \\forall q\\ge Q_0,\\quad q\\ {\\rm prime}\\Longrightarrow\\\\[-2pt]\n&\\hspace{5em}1<C(mq)\\ \\hbox{ and }\\ \\gcd(C(mq),mq)=1.\n\\end{aligned}\n\\]\n\\[\n\\begin{aligned}\n\\operatorname{BoundedOrder}(C,m,d)\\quad\\Longleftrightarrow\\quad&\n\\forall q,p,\\quad q,p\\ {\\rm prime},\\ p\\mid C(mq)\\Longrightarrow\\\\[-2pt]\n&\\hspace{3em}\\exists k,\\quad 1\\le k\\le d\\ \\hbox{ and }\\ mq\\mid p^k-1.\n\\end{aligned}\n\\]\n\\[\n\\begin{aligned}\n\\operatorname{FinitePrimeEscape}(C,m)\\quad\\Longleftrightarrow\\quad&\n\\forall\\text{ finite sets }S\\text{ of primes},\\ \\exists Q_0\\\n\\ \\forall q\\ge Q_0,\\\\[-2pt]\n&\\hspace{3em}q\\ {\\rm prime}\\Longrightarrow\n\\forall p\\in S,\\quad p\\nmid C(mq).\n\\end{aligned}\n\\]\nThese are the linked predicates\n\\lword{Erdos249/PrimeRayCyclotomicCurvature.lean}{88}{PrimeRayLayerSupply}{Layer},\n\\lword{Erdos249/PrimeRayCyclotomicCurvature.lean}{95}{BoundedDegreeOrderConsumer}{BoundedOrder},\nand\n\\lword{Erdos249/PrimeRayCyclotomicCurvature.lean}{102}{FinitePrimeSupportEscape}{FinitePrimeEscape}.\nThe bounded-order condition does not assert an exact multiplicative order; it\nasserts only the displayed divisibility for some $k\\le d$.\n\nThe implication from bounded order to finite-prime escape is a size argument\nin the paper.  If $m=0$, the bounded-order condition forces $C(0)=1$.  If\n$m>0$, the assertion is immediate for $S=\\varnothing$; otherwise choose\n$Q_0$ so that $mQ_0>\\max_{p\\in S}(p^d-1)$.  For every prime\n$q\\ge Q_0$, a divisor $p\\in S$ of $C(mq)$ would give\n$mq\\mid p^k-1$ for some $k\\le d$, hence\n$mq\\le p^k-1\\le p^d-1$, a contradiction.  The layer hypothesis separately\nensures that the layers are nontrivial and coprime to their indices.  Neither\nhypothesis is proved here; polynomial-resultant realisability and\nArchimedean growth are also not established.\n\n\\section{Open problems}\\label{sec:frontier}\n\nWrite $H_t=\\lcm(1,\\dots,t)$\n(\\mword{Erdos249257/CarrySurvivorExtinction.lean}{515}{periodLcm}{definition}),\nso that $H_1=1$, $H_2=2$, $H_3=6$, $H_4=12$, $H_5=H_6=60$ and $H_7=420$; and\nrecall from Section~\\ref{sec:results} the tail\n$R_N=\\sum_{m\\ge1}\\ph(N+m)/2^{m}$, which differs from $2^N\\Stot$ by an integer\n(\\mword{Erdos249257/TotientTailPeriodKiller.lean}{57}{totientTail}{definition}).\n\n\\begin{problem}[cofinal diagonal escape]\\label{prob:diagonal}\nFor every $t_0$ there is a $t\\ge t_0$ with $R_{2H_t}-R_{H_t}\\notin\\Z$.\n\\end{problem}\n\n\\noindent By \\ref{res:equivalences}, Problem~\\ref{prob:diagonal} is equivalent\nto irrationality of $\\Stot$.  A finite list of successful certificates\nestablishes the predicate only on its tested scales; it does not establish the\nquantified statement for arbitrarily large $t$.  In the opposite direction,\none pair $h>0$, $N$ with $R_{N+h}-R_N\\in\\Z$ already makes $\\Stot$ rational\n(\\mword{Erdos249257/LcmConeFlatness.lean}{357}{not_irrational_of_tail_diff_mem_int}{checked}).\n\nThe certificate of Section~\\ref{sec:results} is decidable at each $t$, and has\nbeen kernel-checked to fire at the $28$ scales\n(\\mword{Erdos249257/DiagonalPincerCertificatesT64.lean}{1933}{diagonalPincerCertificateScalesThroughT64}{listed})\nlying between $t=1$ and $t=64$\n(\\mword{Erdos249257/DiagonalPincerCertificatesT64.lean}{1967}{certifiedKill_diagonal_all_imported_through_t64}{checked}).\nLater work in the repository extends the checked band to every $t\\le82$; the\nmodule proving it postdates this note's pinned snapshot and so is not linked\nhere, and the claim registry records the wider band.  The theorem through\n$t=82$ supplies no instance at $t=83$ and therefore does not discharge the\ncofinal quantifier.\n\n\\paragraph{A first-harmonic sufficient condition.}\nA constant-saving first-harmonic bound on one dyadic block, or\non any nonempty finite set of indices, already forces a certificate\n(\\mword{Erdos249257/FirstHarmonicGap.lean}{104}{exists_certifiedKill_of_first_harmonic_gap_subset}{checked}).\nThe block sum in question decomposes exactly into four named terms ---\na centred supplier-fibre correlation, a fibre-mean contribution, a bad-set\ncontribution and a non-supplier contribution\n(\\mword{Erdos249257/FirstHarmonicPivot.lean}{342}{windowFirstExp_sum_eq_pivot_decomposition}{checked})\n--- and meeting four explicit numerical bounds on those terms cofinally\nalready gives irrationality\n(\\mword{Erdos249257/FirstHarmonicPivot.lean}{405}{irrational_totient_series_of_pivotResidualDecorrelation}{checked}).\nThus the sufficient condition is to beat four stated constants in an exact\ndecomposition.\n\nA proof of Problem~\\ref{prob:diagonal} gives irrationality of $\\Stot$ by\n\\ref{res:equivalences}.  Its negation settles \\#249 in the opposite direction:\nit supplies a positive-shift integral tail difference, which forces\nrationality.  By contrast, failure of the first-harmonic bounds would rule out\nonly this sufficient criterion.\n\n\\paragraph{The independent denominator exclusion.}\nThe constant $7.96\\times10^{34}$ of \\ref{res:denominator} is the classical\nFarey mediant bound applied to a window with a free parameter $K$; it has\norder $2^{K/2}$, and raising it needs only more committed binary digits, which\nis a computation rather than an idea.  It is not one of the open items of\nSection~\\ref{sec:open}.\n\n\\subsection{Logical status and analytic input}\\label{sec:open}\n\n\\begin{enumerate}\n\\item Irrationality of $\\Stot$ (\\ref{res:open}).  No proof is claimed.\n\\item The reformulations of \\ref{res:equivalences} convert irrationality\n  into the existence of an unbounded family of finite tail-difference\n  certificates.  Finite instances are checked --- the $28$ diagonal scales of\n  Section~\\ref{sec:frontier} --- and the unbounded family is not.\n  The equivalence fixes the exact missing quantifier but does not make that\n  quantifier easier to prove.  \\ref{res:denominator} is a finite instance of a\n  \\emph{different} family, the gap certificates of\n  \\ref{res:supply}, which reach irrationality by a one-way implication; the two\n  families should not be counted together.\n\\item The four pivot bounds of Section~\\ref{sec:frontier}, and the\n  prime-orbit gap they refine, are unproved.\n\\end{enumerate}\n\n\\noindent The correlation source that fits the normalised totient most\ndirectly is Balasubramanian--Giri--Srivastav~\\cite{bgs2017}, not a direct\ntransfer of the Tao--Ter\\\"av\\\"ainen method.  Write\n\\[\n  g(n)=\\frac{\\ph(n)}n\n      =\\sum_{d\\mid n}\\frac{\\mu(d)}d=(f*1)(n),\n  \\qquad f(d)=\\frac{\\mu(d)}d\\in\\mathcal A_1 ,\n\\]\nthe class $\\mathcal A_1$ being the coefficient class in which the theorem\ncited next is stated.\nTheorem~2.2 of the arXiv version of~\\cite{bgs2017} gives, uniformly for\n$|h|\\le x/2$, an explicit asymptotic for\n$\\sum g(n)g(n-h)$ with error $O(\\log^2x)$; Remark~2.4 gives its Euler product,\nFor each fixed $h$, the weighted partial-summation formula immediately\nfollowing Corollary~2.8 permits $Q(n)=n(n-h)$, restoring the two linear\nfactors needed to return from $g(n)g(n-h)$ to\n$\\ph(n)\\ph(n-h)$ with an explicitly propagated error.  That displayed\nweighted formula is not itself stated uniformly in $h$ and starts its main\nintegral at \\(1\\), not \\(H\\); retaining uniformity and the lower endpoint\nwould require applying partial summation directly to Theorem~2.2.  No later\nclaim here relies on an unproved uniform weighted version.\n\nTao--Ter\\\"av\\\"ainen's quantitative correlation theorem\n(\\cite{taoteravainen2025}, Theorem~3.1) assumes either quantitative\nequidistribution together with an exact small-prime condition, or\nnon-pretentiousness.  Here $g(p)=1-1/p$, so the stated small-prime condition\ndoes not hold, while\n$\\sum_p(1-g(p))/p=\\sum_p1/p^2<\\infty$, so $g$ is pretentious to the constant\nfunction.  Their theorem therefore does not apply unchanged.  The\nBalasubramanian--Giri--Srivastav theorem supplies the missing first- and\nsecond-moment input in the correct divisor-convolution class, but it does\n\\emph{not} supply the residue small-ball or phase anti-concentration estimate\nneeded to turn those moments into a certificate.  That modulo-$2^L$ step is\nthe precise remaining analytic gap; no irrationality conclusion is drawn from\nthe correlation asymptotic alone.\n\n\\section*{Formal source notes}\n\n\\paragraph{Registry coverage.}\nTwo of the items above are checked propositions that no row of the claim\nregistry currently owns: the declaration of \\ref{res:infinite}, and the\nfive of \\ref{res:weight}; the supply implication of \\ref{res:supply} is\nlikewise unowned, and \\ref{res:denominator} cites three declarations of which\nthe registry's row owns one.  Under this project's own division of authority\nthe registry, not a manuscript, owns public status, so the honest tag on those\nitems is that they are kernel-checked and not yet registered.  We print that\nrather than borrowing a neighbouring row's status.  The gap is a defect in the\nrecord, not in the proofs, and closing it is a review action rather than a\nmathematical one.\n\nThis manuscript is authored exposition, not proof authority.  The linked Lean\nsnapshot is authoritative only for its exact propositions, and kernel checking\nestablishes that a proposition was proved, not that it is interesting, novel,\nor sufficient.  The deduction in \\ref{res:periodic} that an unbounded weight is not eventually\nperiodic, the mechanism sentence in Section~\\ref{sec:nogo}, and the size\nargument in Section~\\ref{sec:programmes} are one-line arguments in the prose,\nmarked as such; the first two are drawn from checked statements, and the last\nassumes the unproved bounded-order condition.  The small numerical\ninstances printed above --- the level-three reductions, the certificate at\n$(1,12,16)$, the first values of $\\Aw$, the $p=3$ Euler-factor calculation,\nthe mixed-difference table, and the values of $H_t$ --- are computations from\nthe definitions and checked identities beside them, and are not separate\nchecked propositions.  Results attributed to\nAllouche and Shallit, to Coons, to Bell and Smertnig, to Kaneko, Suzuki and\nTachiya, to Luca and Tachiya, and to Tao and Ter\\\"av\\\"ainen are cited from the\nliterature and are not formalised here; the assessment that no prior source\ngives this exact finite-level dimension formula for the totient $2$-kernel is\nthe outcome of a literature search and is not a proof of non-existence.\n\n\\paragraph{Declaration of generative AI use.} \\paperfullaideclaration\n\nFormal authority is the pinned kernel's acceptance of an exact proposition;\nno model output carries any, and neither does this sentence.\n\nErd\\H{o}s Problem~\\#249 remains open.\n\n\\begin{thebibliography}{10}\n\\bibitem{allouche-shallit} J.-P.~Allouche and J.~Shallit,\n  \\href{https://cs.uwaterloo.ca/~shallit/Papers/as0.pdf}{\\emph{The ring of\n  $k$-regular sequences}}, Theoret.\\ Comput.\\ Sci.\\ \\textbf{98} (1992),\n  no.~2, 163--197,\n  doi:\\href{https://doi.org/10.1016/0304-3975(92)90001-V}\n  {10.1016/0304-3975(92)90001-V}.  Definitions~1.1 and 2.1 are on\n  pp.~2--3 of the linked author preprint, which differs slightly from the\n  published version; the cited definitions are unaffected.\n\\bibitem{coons} M.~Coons, \\emph{(Non)Automaticity of number theoretic\n  functions}, J.~Th\\'eor.\\ Nombres Bordeaux 22 (2010), no.~2, 339--352;\n  \\href{https://doi.org/10.5802/jtnb.718}{doi:10.5802/jtnb.718};\n  \\href{https://arxiv.org/abs/0810.3709}{arXiv:0810.3709}.  Theorem~3.2,\n  pp.~348--349, in the published version (Theorem~3.3, pp.~8--9, in the\n  preprint): $\\ph$ is not $k$-regular for any $k\\ge2$.\n\\bibitem{bell-smertnig} J.~Bell and D.~Smertnig, \\emph{Mahler series with\n  multiplicative coefficient sequences}, 2026,\n  \\href{https://doi.org/10.48550/arXiv.2603.23456}{arXiv:2603.23456}.  The\n  totient generating series is not $k$-Mahler for any $k\\ge2$, a consequence\n  of their Theorem 1.3 recorded in the introduction.\n\\bibitem{kanekosuzukitachiya2026} H.~Kaneko, Y.~Suzuki and Y.~Tachiya,\n  \\href{https://arxiv.org/abs/2601.20743}\n  {\\emph{Refinements of Erd\\H{o}s's irrationality criterion for certain\n  sparse infinite series}}, arXiv:2601.20743v1, 2026.\n  Corollary~3 is on pp.~5--6 and proves irrationality for the\n  \\(\\sigma(n)\\)- and \\(\\ph(n)\\)-in-the-exponent families; its proof is on\n  pp.~18--19.\n\\bibitem{taoteravainen2025} T.~Tao and J.~Ter\\\"av\\\"ainen,\n  \\emph{Quantitative correlations and some problems on prime factors of\n  consecutive integers}, arXiv:2512.01739 (submitted December 2025, revised\n  April 2026).  Theorem 1.3 proves irrationality of\n  $\\sum_{n\\ge1}\\omega(n)/2^n$ at base $2$; the extension to every integer base\n  and the $\\Omega$ analogue are stated as remarks, with the modifications left\n  to the reader.  The theorem is on p.~4 and its proof is Section~5,\n  pp.~44--56, in arXiv v2.\n\\bibitem{bgs2017} R.~Balasubramanian, S.~Giri and P.~Srivastav,\n  \\emph{On correlations of certain multiplicative functions},\n  J.\\ Number Theory 174 (2017), 221--238,\n  \\href{https://doi.org/10.1016/j.jnt.2016.10.001}{DOI};\n  \\href{https://arxiv.org/abs/1511.02221}{arXiv:1511.02221}.\n  The arXiv version's Theorem~2.2 and Remark~2.4 give the uniform shifted\n  divisor-convolution correlation and Euler product used above; the required\n  fixed-shift weighted partial-summation formula follows Corollary~2.8.\n\\bibitem{lucatachiya2017} F.~Luca and Y.~Tachiya,\n  \\href{https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2014-14.pdf}\n  {\\emph{Linear independence results for the values of divisor functions\n  series}}, RIMS K\\^oky\\^uroku No.~2014 (2017), 138--150.\n  Theorem~A on p.~139 explicitly restates their earlier Theorem~1.1 for\n  nonzero purely periodic integer weights.  Theorem~1 is on p.~139, its\n  examples are on p.~140, and its proof is on pp.~149--150.\n\\bibitem{erdos1948} P.~Erd\\H{o}s,\n  \\href{https://users.renyi.hu/~p_erdos/1948-04.pdf}\n  {\\emph{On arithmetical properties of Lambert series}},\n  J. Indian Math. Soc. (N.S.) \\textbf{12} (1948), 63--66.\n  The integer-base full-support theorem is on p.~63; the proof is on\n  pp.~63--66, and the closing totient-series remark is on p.~66.\n\\bibitem{erdosgraham1980} P.~Erd\\H{o}s and R.~L.~Graham, \\emph{Old and New\n  Problems and Results in Combinatorial Number Theory}, 1980, p.~61.\n\\bibitem{erdos1988} P.~Erd\\H{o}s, \\emph{On the irrationality of certain\n  series: problems and results}, in A.~Baker (ed.), \\emph{New Advances in\n  Transcendence Theory}, Cambridge UP, 1988, pp.~102--109,\n  doi:\\href{https://doi.org/10.1017/CBO9780511897184.009}\n  {10.1017/CBO9780511897184.009}.\n\\bibitem{erdosproblems} T.~F. Bloom,\n  \\href{https://www.erdosproblems.com/249}{\\emph{Erd\\H{o}s Problem \\#249}},\n  \\texttt{erdosproblems.com/249}, accessed 28 July 2026 (page displays\n  ``last edited 28 September 2025'').  The current record labels the problem\n  open, cites \\texttt{[ErGr80, p.~61]} and \\texttt{[Er88c, p.~102]}, and\n  explicitly describes its status as the website owner's present assessment\n  rather than a literature-completeness guarantee.\n\\end{thebibliography}\n\n\\end{document}\n","dependencies":[{"record_type":"paper_source_dependency","tex_reference":"problem-note-preamble","source_path":"paper/problem-note-preamble.tex","media_type":"text/x-tex","sha256":"sha256:e7378fe7784916681e7faf96c1ea7a8bb941a416cc651b97512e849bcaeb63f7","byte_count":4990,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Shared preamble for the Erd\u0151s Problem Notes: the short, standalone,\n% problem-owned manuscripts covering the expansion library `ErdosProblems`.\n%\n% One note owns one Erd\u0151s problem.  Each note repeats whatever context its own\n% reader needs, so that a reader who arrives at a single problem never has to\n% assemble it from the gateway paper.  Deliberate repetition across notes is the\n% design, not an oversight.\n%\n% Source links resolve against one pinned formal-source commit, declared once\n% below.  `scripts/check_problem_note_sources.py` verifies that every authored\n% (file, line, declaration) triple names that exact declaration in the pinned\n% snapshot, so a link cannot silently rot as later work moves lines.\n\n\\usepackage{paper-house-style}\n\\usepackage{array}\n\\usepackage{booktabs}\n\\usepackage{enumitem}\n\\setlist{topsep=0.35em,itemsep=0.2em}\n\\usepackage[colorlinks=true,linkcolor=linkinternal,urlcolor=linkexternal,\n  citecolor=linkinternal]{hyperref}\n\\hypersetup{bookmarksnumbered=true,bookmarksopen=true,bookmarksopenlevel=1,\n  pdfauthor={Will Cook},pdflang={en-GB}}\n\n% ---------------------------------------------------------------------------\n% Pinned formal source.  \\commit names the checkpoint every link resolves\n% against; the checker reads that snapshot rather than the working tree, so the\n% printed coordinates stay correct however later waves move declarations.\n% ---------------------------------------------------------------------------\n\\newcommand{\\commit}{d6d344d9557faee19276b01a1c1f20d7c08f03d5}\n\\newcommand{\\commitshort}{d6d344d9557f}\n\\newcommand{\\repobase}{https://github.com/wcook04/plectis-lean-erdos249-257/blob/\\commit}\n\\newcommand{\\sourceurl}{https://github.com/wcook04/plectis-lean-erdos249-257/tree/\\commit}\n\\newcommand{\\PX}{ErdosProblems}\n\n% Declaration names stay inside link targets.  The printed label is either a\n% spaced, lower-case reading of the declaration name (\\lref) or a short authored\n% phrase (\\lword); neither prints a module path or a raw Lean identifier.\n\\ExplSyntaxOn\n\\cs_new_protected:Npn \\noteleanlabel #1\n  {\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_to_str:n {#1} }\n    \\regex_replace_all:nnN { _+ } { \\c{space} } \\l_tmpa_tl\n    \\regex_replace_all:nnN\n      { ([a-z0-9])([A-Z]) }\n      { \\1\\c{space}\\2 }\n      \\l_tmpa_tl\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_lower_case:n { \\tl_use:N \\l_tmpa_tl } }\n    \\tl_use:N \\l_tmpa_tl\n  }\n\\ExplSyntaxOff\n\\newcommand{\\leanlabel}[1]{\\noteleanlabel{#1}}\n\\newcommand{\\lref}[3]{\\href{\\repobase/\\PX/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\lword}[4]{\\href{\\repobase/\\PX/#1\\#L#2}{#4}}\n\\newcommand{\\lloc}[2]{\\href{\\repobase/\\PX/#1\\#L#2}{Lean source}}\n\n% Links into a sibling library, named repository-relative.  The expansion\n% library is implicit in \\lref/\\lword, because that is where problem-owned\n% work lands.  Several problems have their headline theorem in the older\n% reviewed #249/#257 corpus, and a note that could not name that library\n% would have to paraphrase a checked statement instead of linking it.\n\\newcommand{\\mref}[3]{\\href{\\repobase/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\mword}[4]{\\href{\\repobase/#1\\#L#2}{#4}}\n\\newcommand{\\mloc}[2]{\\href{\\repobase/#1\\#L#2}{Lean source}}\n\\emergencystretch=2em\n\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\theoremstyle{definition}\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{example}[theorem]{Example}\n\\newtheorem{problem}[theorem]{Problem}\n\\theoremstyle{remark}\n\\newtheorem*{remark}{Remark}\n\n\\newcommand{\\N}{\\mathbb{N}}\n\\newcommand{\\Npos}{\\mathbb{N}_{>0}}\n\\newcommand{\\Z}{\\mathbb{Z}}\n\\newcommand{\\Q}{\\mathbb{Q}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\lcm}{\\operatorname{lcm}}\n\\newcommand{\\ph}{\\varphi}\n\n% The series line printed under every note's title block.\n\\newcommand{\\seriesline}{Erd\\H{o}s Problem Notes}\n\n% Problem-note compatibility name for the corpus-owned page-one disclosure in\n% paper-house-style.sty.\n\\newcommand{\\noteprovenance}{\\paperprovenance}\n\n% The status paragraph every note carries, in its own words, before any result.\n% Kernel checking establishes the exact Lean proposition; it does not establish\n% that the proposition is the interesting one, that it is new, or that the\n% Erd\u0151s problem is closed.\n\\newcommand{\\statusboundary}{%\n  \\emph{Status.}  The problem treated here is open, and this note does not\n  close it.  Every statement below marked as checked is a proposition that the\n  pinned Lean kernel accepts from the sources this note links to, with no\n  \\texttt{sorry}, no added axiom, and no unchecked evaluation.  That is a claim\n  about the formal statement, not about its mathematical interest, its\n  novelty, or the original problem.  The unresolved obligations are named\n  exactly, in their own section, and none of the finite computations,\n  reductions, or no-go results here removes one of them.}\n"}],"external_verification_route":"https://raw.githubusercontent.com/wcook04/plectis-lean-erdos249-257/main/paper/erdos-249-binary-totient-series.tex"},{"record_type":"paper","paper_id":"erdos249-totient-reasoning-surface","title":"The Binary Totient Series","source_repository":"plectis-lean-erdos249-257","source_path":"paper/erdos249-totient-reasoning-surface.tex","hosted_pdf_path":"papers/erdos249-totient-reasoning-surface.pdf","licence":"CC-BY-4.0","copyright":"2026 Will Cook","cite_as":"Plectis reader corpus, paper:erdos249-totient-reasoning-surface","pdf_sha256":"sha256:b2a79b5423b3f2586c4e59feb9f7063132a9212561121a6ad017307b8fc02724","pdf_byte_count":702930,"pdf_embedded":false,"pdf_omission_reason":"the exact LaTeX is embedded and retrieves better than extracted PDF text; the PDF is a presentation rendering and its hash is recorded for verification","source_embedded":true,"availability":"embedded_exact","embedded_copy_role":"native_manuscript","embedded_from_repository":"plectis-lean-erdos249-257","embedded_copy_path":"paper/erdos249-totient-reasoning-surface.tex","media_type":"text/x-tex","body_format":"latex","sha256":"sha256:01ddf45328ff6bc7279e8f4ec14da7b0881a61f472266810593f11c0634b86df","byte_count":466518,"body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% GENERATED by scratchpad/assemble.py from authored parts. Parts are inlined,\n% never \\input, so that the pandoc projection cannot silently drop a section.\n\\documentclass[11pt]{article}\n% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Shared preamble for the two per-problem reasoning-surface papers.\n% These documents are written to be read in full as working context: every\n% result appears with its exact hypotheses, its evidence band, the scale at\n% which it is proved, and the Lean declaration that carries it.\n\\usepackage{paper-house-style}\n\\usepackage{array}\n\\usepackage{adjustbox}\n\\usepackage{booktabs}\n\\usepackage{longtable}\n\\usepackage{enumitem}\n\\setlist{topsep=0.35em,itemsep=0.2em}\n\\usepackage[colorlinks=true,linkcolor=linkinternal,urlcolor=linkexternal,\n  citecolor=linkinternal]{hyperref}\n\\hypersetup{bookmarksnumbered=true,bookmarksopen=true,bookmarksopenlevel=1,\n  pdftitle={The Binary Totient Series: a complete working record of Erdos Problem 249},\n  pdfauthor={Will Cook},pdflang={en-GB}}\n% Lean declaration names are full of underscores and are typeset in text mode.\n% Category codes are fixed when TeX READS a line, not when a macro expands, so\n% no amount of \\catcode juggling inside \\lean can rescue an underscore that has\n% already been tokenised as a subscript. This package makes _ print as itself in\n% text while leaving it as a subscript in maths, which is exactly the split we\n% need. It also protects any stray underscore in authored prose.\n\\usepackage{underscore}\n\n% ---------------------------------------------------------------------------\n% Theorem environments.\n% ---------------------------------------------------------------------------\n\\theoremstyle{plain}\n\\newtheorem{thm}{Theorem}[section]\n\\newtheorem{lem}[thm]{Lemma}\n\\newtheorem{prop}[thm]{Proposition}\n\\newtheorem{cor}[thm]{Corollary}\n\\theoremstyle{definition}\n\\newtheorem{defn}[thm]{Definition}\n\\theoremstyle{remark}\n\\newtheorem{rem}[thm]{Remark}\n\\newtheorem{obs}[thm]{Observation}\n\n% ---------------------------------------------------------------------------\n% Citation of a Lean declaration. The declaration name is the authority; the\n% file:line is where it sat at the pinned formal-source checkpoint. Underscores\n% are handled locally so authors may write names verbatim.\n% ---------------------------------------------------------------------------\n\\newcommand{\\PK}{Erdos249257}\n\\newcommand{\\commit}{1128d0858e1060c741f26460afac166f36801af9}\n\\newcommand{\\repobase}{https://github.com/wcook04/plectis-lean-erdos249-257/blob/\\commit}\n\\newcommand{\\lean}[2]{%\n  \\href{\\repobase/\\PK/#2}{\\textsc{Lean source}}}\n\n% Declaration names are written \\mathtt{...} in maths and, unavoidably, in\n% running prose too. Rather than police every occurrence, make the command\n% mode-agnostic: monospace is what is meant in both places.\n\\AtBeginDocument{%\n  \\let\\origmathtt\\mathtt\n  \\renewcommand{\\mathtt}[1]{\\ifmmode\\origmathtt{#1}\\else\\texttt{#1}\\fi}%\n  \\let\\origmathrm\\mathrm\n  \\renewcommand{\\mathrm}[1]{\\ifmmode\\origmathrm{#1}\\else\\textrm{#1}\\fi}%\n}\n\n% One authored part uses \\m{...} as an inline-maths shorthand. Defining it is\n% cheaper and safer than rewriting the prose around every occurrence.\n\\providecommand{\\m}[1]{\\ensuremath{#1}}\n\n% ---------------------------------------------------------------------------\n% Evidence band. These are never blurred: a reader must always be able to see\n% what a kernel checked, what an exact computation checked, what was proved in\n% ordinary mathematics, what is quoted from the literature, and what is open.\n% ---------------------------------------------------------------------------\n\\newcommand{\\ev}[1]{{\\small\\textsf{[#1]}}}\n\n% Scale at which a statement is proved. This is the load-bearing axis of both\n% documents: the corpus proves at fixed or bounded scale, and both open\n% producers need cofinal scale.\n\\newcommand{\\scale}[1]{{\\small\\textsf{scale:#1}}}\n\n% The representation a statement is expressed in. Obstructions are relative to\n% a coordinate, never to the object, so every obstruction carries one.\n\\newcommand{\\coord}[1]{{\\small\\textsf{coord:#1}}}\n\n% ---------------------------------------------------------------------------\n% Mathematical shorthand.\n% ---------------------------------------------------------------------------\n\\newcommand{\\N}{\\mathbb{N}}\n\\newcommand{\\Npos}{\\mathbb{N}_{>0}}\n\\newcommand{\\Z}{\\mathbb{Z}}\n\\newcommand{\\Q}{\\mathbb{Q}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\C}{\\mathbb{C}}\n\\newcommand{\\ph}{\\varphi}\n\\newcommand{\\EB}{E}\n\n\\allowdisplaybreaks\n\\emergencystretch=2em\n\n% These working records contain a small number of intentionally exhaustive\n% formulas. Keep ordinary displays at their natural size, but bound any display\n% wider than the classical text measure rather than letting it disappear past\n% the page edge.\n\\long\\def\\[#1\\]{%\n  \\leavevmode\\par\\vspace{\\abovedisplayskip}%\n  \\noindent\\makebox[\\linewidth][c]{%\n    \\begin{adjustbox}{max width=\\linewidth}%\n      \\(\\displaystyle #1\\)%\n    \\end{adjustbox}}%\n  \\par\\vspace{\\belowdisplayskip}%\n}\n\n\\runninghead{Erd\\H{o}s \\#249: the binary totient series}\n\\title{The Binary Totient Series}\n\\subtitle{A complete working record of Erd\\H{o}s Problem~\\#249}\n\\author{Will Cook}\n\\date{July 2026}\n\n\\begin{document}\n\\maketitle\n\n\\paperprovenance\n\n\\begin{abstract}\nThis document is the complete working record of one attack on Erd\\H{o}s Problem 249: whether $S=\\sum_{n\\ge1}\\ph(n)/2^n$ is irrational.  The problem is open, and nothing here decides it.  What the document does is collect, in one place and in a form that can be used directly as premises, every result the project holds: the Lean-checked theorems with their exact hypotheses and source sites, the exact finite computations, the coordinates in which the constant can be written, the routes that have been closed and the mechanism that closed each one, and the exact statements that remain open.  Results are sorted by the scale at which they are proved, because that is where the problem stands or falls: everything proved here holds at fixed or bounded scale, while the remaining obligation is cofinal.\n\\end{abstract}\n\n\\section*{How to read this document}\n\nThis is a working record, not a survey.  It is written to be read in full before\nwork begins on the problem, so that every statement the project has established\nis available as a premise rather than as a reference to be chased.  Three\nconventions carry that load.\n\n\\emph{Evidence bands.}  Every statement is tagged.  \\ev{Lean} means a proof term\nwas checked by the pinned Lean kernel.  \\ev{Cert} means an exact finite\ncomputation, with no floating-point decision anywhere in it.  \\ev{Math} means\nproved in ordinary mathematics in the sources but not formalised.  \\ev{Cited}\nmeans established in the published literature.  \\ev{Open} means not proved.\nThese are never blurred, and a statement carrying one band is never described in\nlanguage belonging to another.\n\n\\emph{Scale.}  Every parameter-indexed statement is tagged \\scale{fixed} (proved\nat specific listed values), \\scale{bounded} (proved below an explicit bound),\n\\scale{cofinal} (proved for arbitrarily large parameters), or \\scale{uniform}\n(proved for all parameters).  This axis is the reason the problem is still open:\nthe corpus proves at fixed and bounded scale, and the remaining obligation is\ncofinal.  Sorting by scale puts that boundary where it can be seen.\n\n\\emph{Coordinates.}  Every statement records the representation it is expressed\nin.  An obstruction is a fact about a coordinate, not about the object, and a\nwall measured in one representation may simply not exist in another; the atlas\nsection gives the transport maps, so any obstruction recorded here can be\nre-measured elsewhere.\n\nErd\\H{o}s Problem 249 is open.  No section of this document claims\notherwise, and a reduction of the problem to another statement is recorded as a\nreduction, never as progress toward a solution.\n\n\n% ---- part a249_front ----\n\\section{The problem, and what is actually known}\n\nLet $\\varphi$ be Euler's totient function and put\n\\[\n  S \\;:=\\; \\sum_{n \\ge 1} \\frac{\\varphi(n)}{2^{n}}\n  \\;=\\; \\tfrac12 + \\tfrac14 + \\tfrac{2}{8} + \\tfrac{2}{16} + \\tfrac{4}{32} + \\cdots\n  \\;=\\; 1.3676308019850223\\ldots\n\\]\nThe series converges absolutely because $\\varphi(n) \\le n$. Erd\\H{o}s Problem\n\\#249 asks whether $S$ is irrational. \\emph{It is open.} Nothing in this\ndocument decides it, no route recorded here is a proof, and no result here\nshould be read as an approach that is close to working. What this document\ndoes is different in kind: it assembles every recorded failure, obstruction,\ncountermodel and dead route the programme has produced against \\#249, and\nclassifies each one by the exact class of argument it eliminates. The claim\nbeing made is that those failures are not independent --- they are repeated\nmeasurements of a single obstruction, and that obstruction has a shape which\ncan be stated.\n\nThe open question has a sharp equivalent inside the corpus, and it is the\nstatement everything below is measured against. Write $2^{N} S = \\Phi_N + R_N$\nwith $\\Phi_N := \\sum_{n \\le N} \\varphi(n) 2^{N-n} \\in \\N$ and\n$R_N := \\sum_{j \\ge 0} \\varphi(N+1+j)/2^{j+1} \\in [0,\\infty)$\n(\\lean{Erdos249257.TotientTailPeriodKiller.totientTail}{TotientTailPeriodKiller.lean:57},\n\\lean{two_pow_mul_totient_series_eq}{TotientTailPeriodKiller.lean:150}).\nIf $S \\in \\Q$ then, applying Euler's theorem to the odd part of its\ndenominator, there are $h > 0$ and $N_0$ with $R_{N+h} - R_N \\in \\Z$ for every\n$N \\ge N_0$\n(\\lean{Erdos249257.eventual\\_period\\_of\\_not\\_irrational}{TotientTailPeriodKiller.lean:358},\nexplicit witnesses $h = \\varphi(\\mathrm{oddPart}(r.\\mathrm{den}))$, $N_0 = v_2(r.\\mathrm{den})$).\nSo it suffices to refute integrality cofinally on every period ray. The finite\ndevice that refutes it is a residue certificate:\n\\[\n  \\mathrm{windowDiscrepancy}(h,N,L)\n  \\;:=\\; \\sum_{j<L}\\bigl(\\varphi(N{+}h{+}1{+}j) - \\varphi(N{+}1{+}j)\\bigr)\\,2^{\\,L-1-j} \\in \\Z,\n\\]\nthe depth-$L$ truncation of $2^{L}(R_{N+h} - R_N)$, and\n\\[\n  \\mathrm{certifiedKill}(h,N,L)\n  \\;:\\Longleftrightarrow\\;\n  (N{+}h{+}L{+}2) \\;<\\; \\mathrm{windowDiscrepancy}(h,N,L) \\bmod 2^{L}\n  \\;<\\; 2^{L} - (N{+}h{+}L{+}2),\n\\]\na decidable condition\n(\\lean{Erdos249257.TotientTailPeriodKiller.windowDiscrepancy}{TotientTailPeriodKiller.lean:66},\n\\lean{Erdos249257.TotientTailPeriodKiller.certifiedKill}{TotientTailPeriodKiller.lean:72}).\nThe radius $N{+}h{+}L{+}2$ is exactly the crude tail bound coming from\n$\\varphi(m) \\le m$, so a certificate is sound: it forces\n$R_{N+h} - R_N \\notin \\Z$, and in fact certificates are \\emph{complete} for\nnon-integrality\n(\\lean{Erdos249257.exists\\_certifiedKill\\_iff\\_tail\\_diff\\_notMem\\_int}{LcmConeFlatness.lean:316}).\nThe entire \\#249 side of the programme therefore reduces to one obligation.\n\n\\begin{defn}[The supply obligation $\\mathrm{Sep}$ --- the exact open target]\n\\label{defn:sep}\n\\[\n  \\mathrm{Sep}\n  \\;:\\Longleftrightarrow\\;\n  \\forall h \\ge 1\\ \\forall N_0\\ \\exists N \\ge N_0\\ \\exists L,\\;\n  \\mathrm{certifiedKill}(h,N,L).\n\\]\n$\\mathrm{Sep} \\Rightarrow \\mathrm{Irrational}(S)$\n(\\lean{Erdos249257.irrational\\_totient\\_series\\_of\\_certificate\\_supply}{TotientTailPeriodKiller.lean:394}).\n\\coord{binary-digit} \\scale{cofinal} \\ev{Lean}\n\\end{defn}\n\nTwo features of $\\mathrm{Sep}$ govern everything that follows. First, the\nquantifier shape is $\\forall\\forall\\exists\\exists$: the corpus can and does\nverify individual instances, but an instance is not the obligation. Second,\ndepth cannot stay bounded: from $\\mathrm{certifiedKill}(h,N,L)$ one gets\n$2(N{+}h{+}L{+}2) < 2^{L}$ immediately\n(\\lean{Erdos249257.certifiedKill\\_depth\\_floor}{TotientTailPeriodKiller.lean:79},\n\\ev{Lean}), so $L \\gtrsim \\log_2(N{+}h)$. A certificate is a demand that a\nweighted accumulation over a window of length $\\approx \\log_2 N$ lands near\nthe centre of its dyadic arc, at full arithmetic resolution.\n\n\\subsection{The unconditional record}\n\nThe following hold with no irrationality hypothesis. They are the results a\nspecialist should know before reading further; each is stated at the scale at\nwhich it is actually proved.\n\n\\begin{thm}[Denominator exclusion --- the headline unconditional fact]\n\\label{thm:denom}\nIf $S \\in \\Q$ then its reduced denominator exceeds\n$Q_0 := 79\\,639\\,646\\,646\\,701\\,375\\,323\\,355\\,774\\,875\\,831\\,053 \\approx 7.96 \\times 10^{34}$.\nEquivalently, $S \\ne p$ for every $p \\in \\Q$ with $p.\\mathrm{den} \\le Q_0$.\nThe bound is sharp for the method: $q = Q_0 + 1$ is the exact first failing\ndenominator, exhibited as the mediant of two explicit unimodular Farey\nneighbours.\n\\coord{farey} \\scale{bounded} \\ev{Lean}\n\\lean{Erdos249257.tsum\\_totient\\_div\\_pow\\_two\\_ne\\_ratCast\\_of\\_den\\_le\\_79639646646701375323355774875831053}{CertificateKernel.lean:18056}\n\\lean{Erdos249257.gap\\_check\\_window\\_1\\_240\\_first\\_failure}{GapFareyBound.lean:225}\n\\end{thm}\n\nThis is obtained from a classical Stern--Brocot gap lemma\n(\\lean{farey\\_gap}{GapFareyBound.lean:51}) applied at window $K = 240$, the\nlast rung of the ladder $4838 \\to 2^{22} \\to 2.49\\times10^{17} \\to Q_0$. It is\nlogically independent of $\\mathrm{Sep}$: it uses no certificate and no period\napparatus. Its own open continuation is whether $\\sup_K (b+d)(K) = \\infty$;\nthat would close \\#249 through this theorem's consumer. \\ev{Open}\n\n\\begin{prop}[Finite certificate deposits]\n\\label{prop:deposits}\n$\\mathrm{certifiedKill}$ has been verified at: the $28$ diagonal instances of\nthe LCM pincer through $t = 64$\n(\\lean{Erdos249257.certifiedKill\\_diagonal\\_all\\_imported\\_through\\_t64}{DiagonalPincerCertificatesT64.lean:1967},\nendpoint \\lean{Erdos249257.certifiedKill\\_diagonal\\_t64}{DiagonalPincerCertificatesT64.lean:1928});\nall shifts $h \\in [1,16]$ simultaneously at $(N,L) = (14,9)$, by\n\\texttt{decide}\n(\\lean{Erdos249257.certifiedKill\\_all\\_upto\\_sixteen}{CarrySurvivorExtinction.lean:574});\nand eight further period deposits at $N = 300$. Every deposit fires within a\nsmall additive constant of the minimum depth permitted by\n$\\mathrm{certifiedKill\\_depth\\_floor}$. Note the quantifier order: this deposit\nproves $\\exists N \\exists L\\, \\forall h \\le 16$, whereas $\\mathrm{Sep}$ needs\n$\\forall h\\, \\forall N_0\\, \\exists N \\exists L$.\n\\coord{binary-digit} \\scale{fixed} \\ev{Lean}\n\\end{prop}\n\n\\begin{prop}[Cofinal positivity, and why it is exactly half a certificate]\n\\label{prop:sign}\nThe true actual-LCM tail difference is strictly positive for every $a \\ge 8$,\nwith no irrationality hypothesis\n(\\lean{Erdos257PeriodNoncollapse.actualLcmTailDiff\\_shift\\_pos}{TotientActualLcmOrbitSign.lean:39},\n\\lean{Erdos257PeriodNoncollapse.actualLcmTailOrbit\\_pos}{TotientActualLcmOrbitSign.lean:144}).\nIn the same coordinate, integrality of the orbit forces the residue to the\n\\emph{top edge}, exactly $2^{K} - e$\n(\\lean{Erdos257PeriodNoncollapse.actualLcm\\_integral\\_forces\\_topEdgeResidue}{TotientActualLcmOrbitSign.lean:211}).\nPositivity does not exclude the top edge. So the strongest cofinal fact the\ncorpus owns about the actual object supplies one of the two inequalities a\ncertificate needs and provably cannot supply the other.\n\\coord{actual-lcm} \\scale{cofinal} \\ev{Lean}\n\\end{prop}\n\n\\begin{prop}[Rationality forces unbounded carry rank]\n\\label{prop:rank}\nIf $S$ is rational then, for every $e$, the dyadic carry-kernel family at\nlevel $e$ has $\\Q$-rank at least $2^{e}-1$\n(\\lean{Erdos249257.not\\_irrational\\_totientSeries\\_implies\\_unbounded\\_carryRank\\_unconditional}{TotientCarryKernelRigidity.lean:284}).\nThe scale side is complete and uniform in $e$: the canonical family of\n$2^{e}+1$ dyadic totient-kernel channels is linearly independent over $\\Q$ at\nevery depth, via CRT plus Dirichlet\n(\\lean{Erdos249257.linearIndependent\\_canonicalTotientKernelFamily}{TotientMahlerDefect.lean:900}),\nso the full family spans an infinite-dimensional space\n(\\lean{Erdos249257.not\\_finiteDimensional\\_span\\_fullTotientKernel}{TotientMahlerDefect.lean:1014}).\n\\coord{carry-rank} \\scale{uniform} \\ev{Lean}\n\\end{prop}\n\n\\begin{prop}[What rationality actually buys, and what it does not]\n\\label{prop:period-not-rank}\nRationality gives uniform eventual periodicity of the carry's dyadic sections\nmodulo $v$; that periodicity provably does \\emph{not} promote to any $\\Q$-rank\nbound on the carry family\n(\\lean{Erdos257PeriodNoncollapse.not\\_irrational\\_totientSeries\\_implies\\_mod\\_period\\_and\\_unbounded\\_rank}{TotientTailCarryPeriod.lean:224}).\n\\coord{carry-rank} \\scale{uniform} \\ev{Lean}\n\\end{prop}\n\n\\begin{prop}[Exact reformulations that do not move the truth value]\n\\label{prop:iffs}\nThree independent re-encodings of $\\mathrm{Sep}$ are proved \\emph{equivalent}\nto $\\mathrm{Irrational}(S)$, not merely sufficient for it: the period-multiple\nkill supply\n(\\lean{periodMultipleKillSupply\\_iff\\_irrational}{PeriodMultipleEscape.lean:153}),\nthe directed-certificate supply and its LCM specialisation\n(\\lean{Erdos257PeriodNoncollapse.CofinalDirectedLcmCertificateSupply}{TotientTailCarryPeriod.lean:829,875}),\nand the window-separated-pairs predicate\n(\\lean{Erdos249257.dtwWindowSeparatedPairs\\_iff\\_irrational\\_totient\\_series}{PivotAntiReconstruction.lean:1775}).\n\\coord{binary-digit} \\scale{cofinal} \\ev{Lean}\n\\end{prop}\n\n\\begin{prop}[A rational sequence indistinguishable from $\\varphi$ by every coarse invariant]\n\\label{prop:parity}\nThere is $c : \\N \\to \\N$ with $c(n) \\le 6$ and $c(n) \\le n$ for all $n$,\n$c(n) \\equiv \\varphi(n) \\pmod 2$ for \\emph{every} $n$, and $c$ not eventually\nperiodic --- indeed for every $N, G, K$ there are $K$ explicit carry pulses\nbeyond $N$, pairwise separated by more than $G$ --- and yet\n$\\sum_n c(n)/2^{n} = 3/2 \\in \\Q$\n(\\lean{Erdos257PeriodNoncollapse.exists\\_totientParity\\_arbitrarilyManySeparatedCarry\\_rational\\_countermodel}{TotientParityCoboundaryCountermodel.lean:637};\nsum at \\texttt{:359}, aperiodicity at \\texttt{:487}, parity match at \\texttt{:194};\n\\texttt{\\#print axioms} clean).\n\\coord{coefficient-word} \\scale{uniform} \\ev{Lean}\n\\end{prop}\n\nPropositions~\\ref{prop:sign}--\\ref{prop:parity} are the four facts a reader\nshould carry into the next section: the corpus's best cofinal information is\nhalf a certificate, its best rank information runs the wrong way, its\nreformulations are equivalences rather than reductions, and every purely\nqualitative property of the coefficient word is satisfied by a rational\ncountermodel.\n\n\\section{The wall}\n\\label{sec:wall}\n\nThis programme has produced many exact reformulations of \\#249 and no proof.\nThat is the honest summary, and the rest of this section is an argument that\nit is also the wrong way to read the record. The reformulations do not fail\nindependently. They fail in a small number of recurring ways, and when each\nfailure is stated as a claim about \\emph{which class of argument it\neliminates} rather than as a report that something did not work, the classes\nfit together into one obstruction with a describable shape. A hundred\nreformulations that all die are not a hundred failures; they are a hundred\nmeasurements of one wall, taken from different angles. The measurements are\nbelow.\n\nThe thesis, in one sentence: \\emph{every barrier in the record is a bound, and\nthe quantity a certificate measures is invariant under exactly the bounds that\nmake an argument finite.}\n\n\\subsection{The plane the barriers live on}\n\nFix two axes for a hypothetical proof of $\\mathrm{Sep}$\n(Definition~\\ref{defn:sep}): the \\emph{range} of indices at which it consults\n$\\varphi$, and the \\emph{resolution} at which it consults them. Two derived\naxes matter as well: the size of the proof's internal bookkeeping (carry\nstate, $\\Q$-rank, shift-polynomial degree), and whether the target is asserted\n\\emph{pointwise} at a chosen index or as an \\emph{average} over a block.\n\n\\begin{itemize}[leftmargin=*]\n\\item The quadrant [full resolution $\\times$ bounded range] is closed by\n  Theorem~\\ref{thm:gamma} (B1): $\\varphi$ may be pinned exactly on $[1,B]$ and\n  the series can still be rational.\n\\item The complementary quadrant [coarse resolution $\\times$ unbounded range]\n  is closed by Proposition~\\ref{prop:parity} (B7): $\\varphi$'s parity may be\n  pinned at \\emph{every} index, together with boundedness, $c(n) \\le n$, and\n  arbitrarily strong aperiodicity, and the series can still be $3/2$.\n\\item The remaining quadrant is closed \\emph{whenever the proof compresses}:\n  B5 (no bounded carry state, no fixed-precision signature) and B6 (four\n  independent finite truncations of the totient kernel, all with trivial\n  kernel).\n\\item B4 closes the one construction that tried to reach the good quadrant by\n  \\emph{prescribing} values: residue engineering pays for amplitude in\n  position, and position enters the certificate radius.\n\\item B2 closes the escape of retargeting; B3 records that no bounded result\n  in the corpus has ever promoted.\n\\end{itemize}\n\nThe corpus's own results partition along these axes perfectly, which is the\nfirst evidence that the axes are the right ones. Every unconditional\n\\emph{finite} deposit (Proposition~\\ref{prop:deposits}, the $K=240$ Farey rung\nof Theorem~\\ref{thm:denom}, the actual-LCM orbits at $a = 4$ and $a = 6$) sits\nat full resolution and bounded range, so B1 says extending them is evidence\nforever and proof never. Every unconditional \\emph{cofinal} theorem (letterwise\npositivity for all $a \\ge 8$, the tail bounds from $\\varphi(m) \\le m$,\nunbounded Mersenne-shadow denominator growth) sits at coarse resolution and\nunbounded range, and Proposition~\\ref{prop:sign} proves in the corpus's own\ncoordinate that this is exactly half a certificate.\n\n\\subsection{B1: the finite-inspection barrier, as a theorem}\n\nB1 is the one barrier that is a genuine no-go theorem about proof method, so\nit is stated and proved as one. The generic vocabulary is needed first. For\n$c : \\N \\to \\N$ with $c(n) \\le n$, write\n$T_c := \\sum_{n \\ge 1} c(n)/2^{n}$,\n$R^{c}_{N} := \\sum_{j \\ge 0} c(N{+}1{+}j)/2^{\\,j+1}$,\n$A_c(h,N,L) := \\sum_{j<L}(c(N{+}h{+}1{+}j) - c(N{+}1{+}j))2^{\\,L-1-j}$,\nand let $\\mathrm{Kill}_c(h,N,L)$ and $\\mathrm{Sep}_c$ be\n$\\mathrm{certifiedKill}$ and $\\mathrm{Sep}$ with $A_c$ in place of\n$\\mathrm{windowDiscrepancy}$. For $c = \\varphi$ these are the objects of\nDefinition~\\ref{defn:sep}.\n\n\\begin{lem}[Generic soundness]\n\\label{lem:gsound}\n$\\mathrm{Kill}_c(h,N,L) \\Rightarrow R^{c}_{N+h} - R^{c}_{N} \\notin \\Z$.\n\\coord{binary-digit} \\scale{uniform} \\ev{Math}\n\\end{lem}\n\n\\begin{proof}\nFrom $c(n) \\le n$ one gets\n$\\lvert 2^{L}(R^{c}_{N+h} - R^{c}_{N}) - A_c(h,N,L)\\rvert < N{+}h{+}L{+}2$;\nthis is the crude tail estimate, formalised for $\\varphi$ at\n\\lean{Erdos249257.tail\\_after\\_le}{TotientTailPeriodKiller.lean:198} and used\nthere to prove\n\\lean{Erdos249257.tail\\_diff\\_notMem\\_int\\_of\\_certifiedKill}{TotientTailPeriodKiller.lean:262}.\nIf $R^{c}_{N+h} - R^{c}_{N} = k \\in \\Z$ then $2^{L}k \\equiv 0 \\pmod{2^{L}}$\nand $A_c$ lies within $N{+}h{+}L{+}2$ of $2^{L}k$, so\n$A_c \\bmod 2^{L}$ lies within $N{+}h{+}L{+}2$ of $0$ or of $2^{L}$,\ncontradicting $\\mathrm{Kill}_c$.\n\\end{proof}\n\n\\begin{lem}[Generic tail-period law]\n\\label{lem:gperiod}\nIf $T_c = p/(2^{e}m)$ with $m$ odd and $h := \\mathrm{ord}_m(2)$, then\n$R^{c}_{N+h} - R^{c}_{N} \\in \\Z$ for every $N \\ge e$.\n\\coord{binary-digit} \\scale{cofinal} \\ev{Math}\n\\end{lem}\n\n\\begin{proof}\n$R^{c}_{N} = 2^{N}T_c - \\Phi^{c}_{N}$ with $\\Phi^{c}_{N} \\in \\Z$, so\n$R^{c}_{N+h} - R^{c}_{N} = 2^{N}(2^{h}-1)T_c - (\\Phi^{c}_{N+h} - \\Phi^{c}_{N})$,\nand $2^{N}(2^{h}-1)p/(2^{e}m) \\in \\Z$ because $2^{e} \\mid 2^{N}$ and\n$m \\mid 2^{h}-1$.\n\\end{proof}\n\n\\begin{rem}\nLemmas~\\ref{lem:gsound} and~\\ref{lem:gperiod} are elementary and are stated\nhere in ordinary mathematics. Their $\\varphi$-instances are Lean theorems\n(\\lean{Erdos249257.tail\\_diff\\_notMem\\_int\\_of\\_certifiedKill}{TotientTailPeriodKiller.lean:262},\n\\lean{Erdos249257.eventual\\_period\\_of\\_not\\_irrational}{TotientTailPeriodKiller.lean:358}),\nand the generic tail-period direction exists on disk inside\n\\lean{binaryCoeffSeries\\_rational\\_iff\\_exists\\_temperedBinaryOrbit}{GenericTailOrbitRigidity.lean:426};\na generic \\emph{named} theorem of the form of Lemma~\\ref{lem:gperiod} does not\nexist in either tree. The composition below is mechanical but unformalised, and\nis flagged \\ev{Math} throughout for that reason.\n\\end{rem}\n\n\\begin{thm}[B1, the $\\gamma$-splice: finite inspection cannot certify the supply]\n\\label{thm:gamma}\nLet $B \\ge 1$ and let $P > B$. Define $\\gamma : \\N \\to \\N$ by\n\\[\n  \\gamma(n) := \\varphi(n) \\ \\ (n \\le B), \\qquad\n  \\gamma(n) := \\begin{cases} n-1, & P \\mid n \\\\ n, & P \\nmid n \\end{cases}\n  \\ \\ (n > B).\n\\]\nThen:\n\\begin{enumerate}[label=(\\roman*),leftmargin=*]\n\\item $\\gamma(n) \\le n$ for all $n$, so $\\gamma$ lies in the same coefficient\n  class as $\\varphi$;\n\\item $\\gamma(n) = \\varphi(n)$ for every $n \\le B$, and consequently\n  $A_\\gamma(h,N,L) = A_\\varphi(h,N,L)$ for every $(h,N,L)$ with\n  $N + h + L \\le B$;\n\\item $T_\\gamma = D - 1/(2^{P}-1) \\in \\Q$, where\n  $D = 2 - \\sum_{n \\le B}(n - \\varphi(n))/2^{n}$, and the odd part of the\n  reduced denominator of $T_\\gamma$ is \\emph{exactly} $2^{P}-1$;\n\\item $\\mathrm{Sep}_\\gamma$ is false --- indeed it fails already at $h = P$.\n\\end{enumerate}\nHence for every $B$ there is a coefficient sequence in the same class,\nagreeing with $\\varphi$ on all of $[1,B]$, whose supply obligation is false.\n\\coord{binary-digit} \\scale{uniform} \\ev{Math}\n\\end{thm}\n\n\\begin{proof}\n(i) is immediate; (ii) holds because $A_\\varphi(h,N,L)$ reads $\\varphi$ only at\nindices $\\le N{+}h{+}L$. For (iii), since $P > B$ every multiple of $P$ exceeds\n$B$, so\n\\[\n  T_\\gamma\n  = \\sum_{n\\ge1}\\frac{n}{2^{n}} - \\sum_{n \\le B}\\frac{n - \\varphi(n)}{2^{n}}\n    - \\sum_{k \\ge 1} 2^{-kP}\n  = 2 - \\sum_{n \\le B}\\frac{n-\\varphi(n)}{2^{n}} - \\frac{1}{2^{P}-1}.\n\\]\nWrite $D = A/2^{B}$ with $A = 2^{B+1} - \\sum_{n \\le B}(n-\\varphi(n))2^{\\,B-n} \\in \\Z$.\nThen $T_\\gamma = \\bigl(A(2^{P}-1) - 2^{B}\\bigr)/\\bigl(2^{B}(2^{P}-1)\\bigr)$, and\n$\\gcd\\bigl(A(2^{P}-1) - 2^{B},\\, 2^{P}-1\\bigr) = \\gcd(2^{B}, 2^{P}-1) = 1$\nbecause $2^{P}-1$ is odd. So no factor of $2^{P}-1$ cancels, and the odd part\nof the reduced denominator is exactly $2^{P}-1$. For (iv), the odd part of the\ndenominator is $m = 2^{P}-1$ and $\\mathrm{ord}_m(2) = P$, while\n$v_2(\\mathrm{den}) \\le B$. By Lemma~\\ref{lem:gperiod},\n$R^{\\gamma}_{N+P} - R^{\\gamma}_{N} \\in \\Z$ for every $N \\ge B$, so by\nLemma~\\ref{lem:gsound} no $\\mathrm{Kill}_\\gamma(P,N,L)$ holds for any\n$N \\ge B$ and any $L$. Taking $N_0 = B$ refutes the inner existential of\n$\\mathrm{Sep}_\\gamma$ at $h = P$.\n\\end{proof}\n\n\\begin{cor}[What B1 rules out]\n\\label{cor:b1}\nNo argument establishing $\\mathrm{Sep}$ can have its totient input confined to\n$\\{\\varphi(n) : n \\le B\\}$ for a fixed $B$. Equivalently, no inference of the\nform ``$\\mathrm{Sep}$ has been verified at every scale up to $B$, for\narbitrarily large $B$, therefore $\\mathrm{Sep}$ holds cofinally'' is valid:\nthe hypothesis is consistent with the conclusion's negation at every $B$.\n\\end{cor}\n\n\\begin{rem}[Scope discipline --- what B1 does \\emph{not} say]\n\\label{rem:b1-scope}\nTheorem~\\ref{thm:gamma} is a statement about proof method, not about $\\varphi$.\nIt does not touch $S$, and it does not suggest that $\\varphi$ is such a\n$\\gamma$; it says only that no fixed-horizon check can tell them apart. It\nalso does \\emph{not} kill bounded-parameter certificates in general.\n``Bounded parameter'' splits into four independent bounds with four different\nkillers: bounded index range (this theorem), bounded certificate depth $L$\n(killed separately, and by a Lean theorem ---\n$\\mathrm{certifiedKill\\_depth\\_floor}$ forces $2(N{+}h{+}L{+}2) < 2^{L}$),\nbounded proof state (B5), and bounded rank (B6). Conflating them would\noverstate B1.\n\\end{rem}\n\nThe Lean-formalised sibling of the same mechanism --- a lacunary zero-valued\ncoboundary splice --- is Proposition~\\ref{prop:parity}, which is B7. B1 and B7\nare the two object-level witnesses of the wall, one along each axis.\n\n\\subsection{B2--B7}\n\nEach barrier below is stated as a claim about a class of arguments, with its\nstatus never blurred. \\emph{Proved} means there is a theorem (Lean or ordinary\nmathematics, as marked) whose content is the elimination. \\emph{Observed\npattern} means an audit over the corpus: real evidence, not a theorem, and it\nis never used as a premise elsewhere in this document.\n\n\\begin{prop}[B2, route-collapse: intermediate targets are not waypoints]\n\\label{prop:b2}\n\\textbf{Status: proved} (Lean, three independent mechanisms).\n\\begin{enumerate}[label=(\\alph*),leftmargin=*]\n\\item \\emph{Certificate completeness.} Every certificate vocabulary is an\n  \\emph{iff} with the underlying non-integrality, so every re-encoding of\n  $\\mathrm{Sep}$ is equivalent to $\\mathrm{Sep}$\n  (\\lean{periodMultipleKillSupply\\_iff\\_irrational}{PeriodMultipleEscape.lean:153},\n  forward at \\texttt{:132}, converse at \\texttt{:143}).\n\\item \\emph{Sample choice.} Any weakening in which the prover may \\emph{choose}\n  which indices to test collapses to $\\mathrm{Irrational}(S)$ outright, because\n  irrationality plus doubling expansivity already supplies a two-point sample\n  with the required separation\n  (\\lean{Erdos249257.dtwWindowSeparatedPairs\\_iff\\_irrational\\_totient\\_series}{PivotAntiReconstruction.lean:1775}).\n  The mechanism is visible in the converse proof\n  (\\texttt{windowSeparatedPairsAt\\_of\\_cofinally\\_scaled\\_adjacent\\_chord},\n  \\texttt{PivotAntiReconstruction.lean:1663}): it selects $T = \\{N, N+1\\}$ and\n  two ordered pairs, and the counted-energy threshold\n  $2\\lvert T\\rvert^{2}/5 = 8/5 \\le 2\\delta^{2}$ is met by $\\delta = 9/10$.\n\\item \\emph{Multi-point enrichment.} Adding vertices, higher-order differences,\n  or finite shift-polynomial combinations on the same ray buys nothing, because\n  integrality transports affinely along every ray $H \\mapsto kH$: the four-hit\n  diamond $\\mathrm{Hit}(H) \\wedge \\mathrm{Hit}(pH) \\wedge \\mathrm{Hit}(qH)\n  \\wedge \\mathrm{Hit}(pqH)$ is equivalent to $\\mathrm{Hit}(H)$ alone, with no\n  primality used\n  (\\lean{Erdos249257.fullTarget\\_primeAdjunction\\_diamond\\_iff\\_root}{FullTargetPrimeAdjunctionNoGo.lean:196}).\n\\end{enumerate}\n\\textbf{Rules out:} ``find an easier waypoint'' as a strategy class.\n\\textbf{Does not rule out:} targets over a \\emph{fixed full block} with no\nsample choice, and targets demanding a \\emph{uniform quantitative margin} that\nirrationality does not supply. These two exemptions are read off the collapse\nproofs themselves and are exactly what the surviving routes in\n\\S\\ref{sec:survivors} exploit.\n\\coord{binary-digit} \\scale{cofinal} \\ev{Lean}\n\\end{prop}\n\n\\begin{rem}\nA corroborating instance: rank-$2$ second-difference certificates are sound but\nmeasurably \\emph{not} shallower than rank-$1$ --- at $(h,N) = (1,8)$ rank-$1$\nfires at depth $8$ and no rank-$2$ certificate exists at depth $\\le 8$\n(\\texttt{totient\\_tail\\_rank\\_two\\_kill\\_sound\\_but\\_not\\_shallower\\_cell},\n\\texttt{CertificateKernel.lean:18762}, \\ev{Lean}). Separately, the claim\nsometimes made that the Farey growth law $\\sup_K(b+d) = \\infty$ is ``equivalent\nin difficulty to \\#249'' is an argument, not an \\emph{iff}: the bound produced\nis the convergent denominator of the underlying constant, which stalls at $q_0$\nprecisely if $S = a/q_0$. It should not be cited as an equivalence. \\ev{Math}\n\\end{rem}\n\n\\begin{prop}[B3, no free promotion]\n\\label{prop:b3}\n\\textbf{Status: observed pattern (an audit, not a theorem).} Across the\ncorpus's proved results, no bounded or partial result has a proof uniform\nenough to be promoted to cofinal scale by routine strengthening --- raising a\nbound, widening a hypothesis, or reindexing a quantifier. Method: $56$\nnear-miss rows were catalogued against the four open obligations; $18$ were\nflagged promotable on a statement-level read; all $18$ proof \\emph{bodies} were\nthen opened. Verdict: $16$ conclusively not promotable with a named blocker\neach, and $2$ promotable with no new mathematics but both pure restatements\nthat widen a target family without supplying arithmetic content. Blocker\ntaxonomy: hypothesis strength $\\times 6$, scale-only $\\times 5$,\ncoordinate-only $\\times 4$, multiple $\\times 3$. Rows with an outright\nquantifier-order mismatch were judged unpromotable before the body audit and\nexcluded, so the audited $18$ are the \\emph{most favourable} subset.\n\\coord{audit} \\scale{n/a} \\ev{Cert}\n\\end{prop}\n\n\\begin{obs}[The sharpest signal in the audit]\n\\label{obs:pointwise}\nEvery not-promotable \\#249-supply row asks for a \\emph{pointwise} fact at a\nspecially chosen index: one large $a$ with a top-edge residue gap, one $q$ with\na terminal-dominance inequality, one exponent past $a = 6$, one $t$ past $64$,\none sign at one LCM jump, one prime per LCM height, one $K$ past $240$. Not one\nsuch index has ever been located unconditionally beyond the finite census. The\nsingle audited row whose missing input is not of that shape is the\nfirst-harmonic gap --- an \\emph{average} over a block. This contrast is what\nselects the leading survivor in \\S\\ref{sec:survivors}. It is evidence, not\nproof.\n\\end{obs}\n\n\\begin{prop}[B4, amplitude versus radius: residue engineering is self-defeating]\n\\label{prop:b4}\n\\textbf{Status: proved} (Lean construction plus an unconditional arithmetic\nmargin). A certificate demands that the window residue sit at distance\n$> N{+}h{+}L{+}2$ from both $0$ and $2^{L}$ modulo $2^{L}$, and the depth floor\nforces $2^{L} > 2(N{+}h{+}L{+}2)$, so the required amplitude is comparable to\nthe modulus. Any producer that \\emph{buys} that amplitude by prescribing\ntotient values at engineered positions pays for the prescription in position,\nand position enters the radius. The corpus's strongest such construction proves\nthe trade is exactly self-cancelling, unconditionally and at every depth: a\ntwo-adic pulse block lands the residue at the arc \\emph{centre}\n$2^{K-1} \\bmod 2^{K}$ --- the ideal target --- and still cannot fire, because\nits own defining congruence $p \\equiv 1 + 2^{K-1} \\pmod{2^{K}}$ forces\n$p \\ge 1 + 2^{K-1}$, so with $N = p-K$, $h = H$, $L = K$ the radius $p+H+2$\nexceeds $2^{K-1}$ for every prime $p$ and every $K$.\n\\coord{two-adic-pulse} \\scale{cofinal} \\ev{Lean}\n\\lean{Erdos257PeriodNoncollapse.exists\\_prime\\_deltaTotient\\_twoAdic\\_pulseBlock}{TotientTwoAdicPulseBlock.lean:211}\n\\lean{Erdos257PeriodNoncollapse.windowDiscrepancy\\_modEq\\_half\\_of\\_twoAdic\\_pulse}{TotientTwoAdicPulseBlock.lean:262}\n\\end{prop}\n\n\\begin{rem}[B4's own recorded repair, and its scope]\n\\label{rem:b4}\nTwo further Lean facts show that prescribing \\emph{letters} is exhausted: the\nfull terminal dyadic staircase is unconditionally impossible --- its terminal\nletter would have to be positive, strictly below a wider modulus, and divisible\nby it, hence $0$\n(\\lean{Erdos257PeriodNoncollapse.not\\_actualLcmTerminalDyadicStaircase\\_of\\_room}{TotientActualLcmTopEdgeStaircase.lean:251});\nand the surviving \\emph{punctured} staircase pins its penultimate letter to\nexactly $2^{m-1}$ with $2^{m} < 2(2H{+}J{+}K{+}2)$, i.e.\\ no slack at all\n(\\texttt{puncturedDyadicStaircase\\_penultimate\\_eq\\_half},\n\\texttt{TotientActualLcmTopEdgeStaircase.lean:1187}). The repair the barrier\npermits is explicit: impose the half-turn on the \\emph{word} rather than on one\ndelta, i.e.\\ ask for cofinally many $(h,N,L)$ with\n$\\mathrm{windowDiscrepancy}(h,N,L) \\equiv 2^{L-1} \\pmod{2^{L}}$ and\n$2^{L-1} > N{+}h{+}L{+}2$. What is \\emph{proved} is the failure of the\nDirichlet/CRT residue-engineering family by an explicit exponential margin. The\nbroader reading --- that only carry accumulation across the weighted word can\nproduce the required amplitude, since one letter obeys\n$\\lvert \\varphi(n{+}h) - \\varphi(n) \\rvert < n{+}h$ while the weighted word can\nreach $2^{L}$ --- is an inference from the proved instance plus the linear\ngrowth bound, not itself a theorem on disk. \\ev{Math}\n\\end{rem}\n\n\\begin{prop}[B5, bounded state and local signatures]\n\\label{prop:b5}\n\\textbf{Status: proved} (three Lean theorems, all problem-agnostic).\n\\begin{enumerate}[label=(\\roman*),leftmargin=*]\n\\item No bounded or autonomous carry state summarising the history before\n  position $m$ determines the tail after $m$: for a balanced-pulse family whose\n  predecessor state is constant, no $\\mathrm{decode}$ recovers the radius\n  parameter, and any finite state type needs cardinality\n  $\\ge \\lfloor m/2 \\rfloor + 2$, unbounded in $m$\n  (\\lean{Erdos249257.balancedPulse\\_no\\_autonomous\\_decoder}{GenericTailOrbitRigidity.lean:247}).\n\\item Long common suffixes erase predecessor information \\emph{exactly}: two\n  affine binary orbits with different seeds satisfy\n  $\\mathrm{orbit}_u(L) - \\mathrm{orbit}_v(L) = 2^{L}(u_0 - v_0)$, so the\n  endpoint residue mod $2^{L}$ is independent of the initial carry\n  (\\lean{Erdos249257.affineBinaryOrbit\\_mod\\_twoPow\\_eq}{GenericTailOrbitRigidity.lean:307}).\n\\item Bounded local $2$-adic valuation-unit data at fixed precision excludes\n  nothing: for any finite word of odd-unit symbols at fixed precision and any\n  starting carry, a compatible orbit exists with every intermediate state\n  centred inside its symbol's dyadic radius\n  (\\lean{Erdos249257.fixedPrecisionTropicalNoGo}{TropicalCurvatureCarry.lean:137}).\n\\end{enumerate}\n\\textbf{Rules out:} the finite-automaton-computes-the-expansion family\noutright; any hope of distinguishing two carry histories after a long common\nsuffix; and any contradiction derived from a fixed-precision valuation-unit\nsignature. Growing precision is mandatory, not optional. None of the three\nmentions $\\varphi$, so all three bind \\#257 identically.\n\\coord{carry-orbit} \\scale{uniform} \\ev{Lean}\n\\end{prop}\n\n\\begin{prop}[B6, no finite linear compression]\n\\label{prop:b6}\n\\textbf{Status: proved for four independent truncations} (Lean), with the\nhonest limit stated below. Every natural finite truncation of the totient\nkernel has trivial kernel:\n\\begin{enumerate}[label=(\\roman*),leftmargin=*]\n\\item \\emph{Dyadic.} The canonical family of $2^{e}+1$ dyadic channels is\n  linearly independent over $\\Q$ at every depth\n  (Proposition~\\ref{prop:rank}), and the natural repair --- a bounded\n  compressed-adjoint certificate --- is impossible\n  (\\lean{Erdos249257.false\\_of\\_compressedAdjointCertificate}{TotientMahlerDefect.lean:1051},\n  structure at \\texttt{:1039}).\n\\item \\emph{M\\\"obius incidence.} $U_N(i,j) = \\mu((i{+}1)/(j{+}1))$ when\n  $(j{+}1) \\mid (i{+}1)$ and $0$ otherwise is lower triangular with unit\n  diagonal, so $\\det U_N = 1$ for every $N$ and the jet map is injective: no\n  finite incidence-quotient relation exists at any horizon\n  (\\lean{Erdos249257.incidenceMobius\\_det\\_eq\\_one}{IncidenceQuotientHermitePade.lean:56},\n  \\texttt{mobiusCompanionJetMap\\_injective} at \\texttt{:77}).\n\\item \\emph{Adjugate reconstruction.} Any finite rational row exactly isolating\n  one totient value has crude two-tail cost $\\ge 3$, hence never $< 1$, at any\n  finite grid height, using only $\\varphi(x) \\le x$\n  (\\lean{Erdos257PeriodNoncollapse.three\\_le\\_totientAdjugateTailCost}{TotientTailCarryPeriod.lean:266},\n  closure at \\texttt{:307}).\n\\item \\emph{Shift-polynomial.} An explicit nonzero all-horizon countermodel\n  agrees with every exact whole-ray anchor and survives every finite\n  commensurate LCM-cube shift polynomial, at every finite rank\n  (\\texttt{LcmFactorIdealPulseObstruction.lean}; module docstring, theorem\n  bodies not individually re-verified in this pass, and the construction is\n  explicitly synthetic --- it does not claim its compensation letters occur as\n  actual totient differences).\n\\end{enumerate}\nAdditionally, every strict-subrank monomial quotient in the M\\\"obius--Mersenne\nladder overshoots its target by more than $1/480$, \\emph{uniformly} --- every\nrung $r \\ge 3$ lies in $[1429/1512, 1)$ and every prefix after four atoms is\nwithin $1/3584$ of its rung, so this is a uniform no-go rather than a census\n(\\lean{rankOneSubrankQuotient\\_sub\\_theta\\_two\\_gt}{RankOneSubrankObstruction.lean:238}).\n\\textbf{Honest limit:} this is four checked truncations, not a proof that no\nfinite-linear shortcut exists. Its correct reading is the one the source module\ngives: any winning finite-linear shortcut must live in a genuinely different\ncoordinate or use a growing-parameter construction. The reason the barrier\nbites is Proposition~\\ref{prop:period-not-rank}: what rationality actually buys\nis periodicity, and periodicity provably does not promote to a rank bound.\n\\coord{carry-rank} \\scale{uniform} \\ev{Lean}\n\\end{prop}\n\n\\begin{prop}[B7, the coarse-invariant barrier]\n\\label{prop:b7}\n\\textbf{Status: proved} (Lean, \\texttt{\\#print axioms} clean). This is\nProposition~\\ref{prop:parity} read as an elimination.\n\\textbf{Rules out:} any proof of \\#249 whose hypothesis set on the coefficient\nword lies inside \\{uniform boundedness, $c(n) \\le n$, agreement with $\\varphi$\nmod $2$ at every index, failure of eventual periodicity in any strength up to\narbitrarily long, arbitrarily separated blocks\\}. The witness satisfies every\none of those and is rational.\n\\textbf{Does not rule out:} arguments using actual quantitative totient size or\nresidue information. That complement is precisely why the corpus's standing\nlesson is that only a quantitative argument in the actual-LCM or fixed-rank\nstyle can close \\#249. Any future sufficient condition stated purely in terms\nof coefficient-word properties must be checked against this fixture before\nbeing trusted, for either problem.\n\\coord{coefficient-word} \\scale{uniform} \\ev{Lean}\n\\end{prop}\n\n\\subsection{The shape}\n\nPut together, the barriers say something more specific than ``the problem is\nhard''. The certificate residue is a weighted accumulation over an unboundedly\nlong window at full arithmetic resolution, and \\emph{every device that makes an\nargument finite destroys precisely the quantity being measured}: bounding the\nwindow (B1), coarsening the values (B7), compressing the state (B5),\ntruncating the rank (B6), or letting the prover pick the sample (B2). B4\ncloses the remaining escape --- buying the amplitude by construction --- with\nan explicit exponential margin, and B3 records that in practice nothing\nbounded has ever promoted.\n\nThere is exactly one known way to hold unbounded range and full resolution at\nonce without a finite bookkeeping device: stop naming an index and assert an\naverage instead. That is what an exponential-sum bound is. It is why the\nsurviving routes of \\S\\ref{sec:survivors} are, with the honest exceptions noted\nthere, the ones that never name a good index.\n\n\\section{What the wall does not block}\n\\label{sec:survivors}\n\nAn elimination is only worth the paper it is written on if it leaves somewhere\nto stand. This section is that payoff. Five routes survive the classification\nof \\S\\ref{sec:wall}; for each we state the exact statement it needs, which\nbarrier it evades, and \\emph{why} the evasion is structural rather than\naccidental. Two of the five are demoted explicitly, because they evade every\n\\emph{proved} barrier while sitting squarely inside the class that the\nstrongest \\emph{observed} pattern (Observation~\\ref{obs:pointwise}) indicts.\nNone of the five is a proof, and none is close to one; what has changed is that\nthe space of things to try is no longer large.\n\n\\subsection{Survivor 1: the first-harmonic block cancellation bound}\n\n\\begin{defn}[The open analytic obligation]\n\\label{defn:fh}\nWrite $e(x) := \\exp(2\\pi i x)$ and\n$\\mathrm{windowFirstExp}(h,N,L) := e\\bigl((\\mathrm{windowDiscrepancy}(h,N,L) \\bmod 2^{L})/2^{L}\\bigr)$.\n$\\mathrm{DTWFirstHarmonicNormGap}$ is the statement\n\\[\n  \\forall h \\ge 1\\ \\forall X_0\\ \\exists X, L : \\quad\n  \\max(X_0,1) \\le X, \\quad\n  16(2X + h + L + 2) \\le 2^{L}, \\quad\n  \\Bigl\\lVert \\sum_{N \\in [X,\\,2X)} \\mathrm{windowFirstExp}(h,N,L) \\Bigr\\rVert\n  \\le \\tfrac{21}{25}X .\n\\]\nIt implies $\\mathrm{Irrational}(S)$.\n\\coord{first-harmonic} \\scale{cofinal} \\ev{Lean}\n\\lean{Erdos249257.DTWFirstHarmonicNormGap}{FirstHarmonicPivot.lean:75}\n\\lean{Erdos249257.irrational\\_totient\\_series\\_of\\_first\\_harmonic\\_norm\\_gap}{FirstHarmonicPivot.lean:83}\n\\end{defn}\n\nThe real-part form suffices and is strictly weaker: it is enough that\n$\\sum_{N \\in [X,2X)} \\mathrm{windowFirstCos}(h,N,L) \\le \\tfrac{9}{10}X$\n(\\lean{Erdos249257.exists\\_certifiedKill\\_of\\_first\\_harmonic\\_gap}{FirstHarmonicGap.lean:128},\nthe elementary unconditional engine; the $21/25 \\to 9/10$ bridge is\n\\lean{Erdos249257.exists\\_certifiedKill\\_of\\_first\\_harmonic\\_norm\\_gap}{FirstHarmonicPivot.lean:53}).\nUnpacked, the required object is a constant-saving cancellation estimate for\nthe dyadically weighted totient-difference exponential sum\n\\[\n  \\sum_{X \\le N < 2X}\n  e\\!\\left(\\frac{\\sum_{j<L}\\bigl(\\varphi(N{+}h{+}1{+}j)-\\varphi(N{+}1{+}j)\\bigr)2^{\\,L-1-j}}{2^{L}}\\right),\n  \\qquad L \\approx \\log_2 X + O(1).\n\\]\n\\textbf{Not one instance of this bound is proved anywhere, at any $X$, $h$ or\n$L$.} Both trees were searched for a theorem supplying it; only consumers\nexist. \\ev{Open}\n\n\\paragraph{Why it evades the wall.} This is the only route in the corpus that\nevades all seven barrier classes, and the reasons are structural.\n\n\\begin{itemize}[leftmargin=*]\n\\item \\textbf{B1.} $X$ is a free unbounded parameter and the room condition\n  $16(2X{+}h{+}L{+}2) \\le 2^{L}$ forces $L \\gtrsim \\log_2 X + 5$, so the\n  estimate reads $\\varphi$ at every index in $[X{+}1,\\,2X{+}h{+}L]$ with both\n  position and window length growing. No $\\gamma$ agreeing with $\\varphi$ only\n  on $[1,B]$ constrains it.\n\\item \\textbf{B2 --- the decisive point.} The proved collapse\n  $\\mathrm{DTWWindowSeparatedPairs} \\Leftrightarrow \\mathrm{Irrational}(S)$\n  works \\emph{because the prover may choose the sample}: the converse proof\n  selects $T = \\{N, N{+}1\\}$ and meets the threshold with $\\delta = 9/10$\n  supplied by irrationality plus doubling expansivity\n  (Proposition~\\ref{prop:b2}(b)). Definition~\\ref{defn:fh} admits no sample\n  choice: the sum ranges over all of $[X,2X)$ at one common depth $L$. Since\n  $\\neg\\,\\mathrm{certifiedKill}(h,N,L)$ forces\n  $\\mathrm{windowFirstCos}(h,N,L) > 9/10$\n  (\\lean{Erdos249257.windowFirstCos\\_gt\\_of\\_not\\_certifiedKill}{FirstHarmonicGap.lean:44}),\n  the gap demands that a \\emph{positive proportion} of basepoints in the block\n  carry certificates at a \\emph{single} depth. Irrationality supplies only one\n  certificate per $(h,N_0)$, at a depth that may vary with $N$. The collapse\n  mechanism therefore has no purchase.\n  \\textbf{Honest limit:} no theorem proves\n  $\\mathrm{DTWFirstHarmonicNormGap}$ inequivalent to $\\mathrm{Irrational}(S)$.\n  What is asserted is that the one proved collapse mechanism in this lane\n  demonstrably requires sample choice, which this predicate denies. Note also\n  that the \\emph{subset} form\n  (\\lean{Erdos249257.exists\\_certifiedKill\\_of\\_first\\_harmonic\\_gap\\_subset}{FirstHarmonicGap.lean:104})\n  does permit sample choice and is therefore the collapse-exposed variant.\n  Attack the full-block form.\n\\item \\textbf{B4.} The statistic is the first additive character of the\n  \\emph{accumulated} weighted word, never an individual letter. Nothing is\n  prescribed at any position, so the self-defeating cost that kills Dirichlet\n  residue engineering does not arise. B4's own recorded repair\n  (Remark~\\ref{rem:b4}) is to impose the structure on the word rather than on\n  one delta, which is exactly what a block exponential-sum bound does.\n\\item \\textbf{B5, B6, B7.} No carry state is summarised and no fixed precision\n  is used; the estimate is on the true residue mod $2^{L}$ with $L \\to \\infty$.\n  A cancellation bound is not a rank statement and seeks no finite-dimensional\n  relation. And the estimate uses the actual real residue, not parity,\n  boundedness or aperiodicity, so neither countermodel touches it --- both\n  differ from $\\varphi$ at exactly the large indices where this sum lives.\n\\item \\textbf{B3.} The audit's own verdict on this row is that the consumer\n  side is \\emph{finished} and already at the obligation's exact quantifier\n  shape: $X$ is free, so applying the bound at $X \\ge N_0$ gives\n  $\\exists N \\ge N_0 \\exists L$ directly. The missing piece is one named\n  arithmetic fact, not a strengthening of anything on disk. The constants\n  $9/10$, $\\pi/8$ and $16$ are absolute and do not degrade with $X$, $h$ or\n  $L$; there is no case analysis and no table.\n\\end{itemize}\n\n\\begin{rem}[Does B1 \\emph{select} this route? No.]\n\\label{rem:b1-not-selector}\nIt is tempting to say the finite-inspection theorem points at analysis. It does\nnot. B1 says any proof must use $\\varphi$ at unbounded indices --- a necessary\ncondition satisfied by \\emph{every} open route here (the LCM diagonal as\n$t \\to \\infty$, the actual-LCM supply for $a \\ge 8$, the Farey growth law as\n$K \\to \\infty$, the rank bound for all $e$). B1 does not discriminate. What\nselects this route is a sharper pair: (1) B4, proved, kills the only mechanism\nthe corpus ever found for reaching full resolution at unbounded range by\n\\emph{prescribing} values, and its own stated repair is accumulation over the\nword --- which is what an exponential sum measures; and (2)\nObservation~\\ref{obs:pointwise}, an observed pattern and not a theorem, shows\nthat every single-index producer has failed to be supplied at even one large\nindex, while this route's missing input is a block average and therefore never\nrequires naming a good index. B1's real contribution is narrower but still\nload-bearing: it proves that the finite deposits can never be extended into a\nproof, so the gap between the census and the obligation is a gap in kind, not\nin degree. That is a genuine no-go about method, of the same species as a\nrelativization barrier. It is not a selector.\n\\end{rem}\n\n\\begin{rem}[Evidence in both directions]\nWeak supporting evidence, observed and not proof: every landed diagonal\ncertificate through $t = 64$ fires within a small additive constant of the\nminimum depth forced by $\\mathrm{certifiedKill\\_depth\\_floor}$, which is the\nprofile a doubling-orbit equidistribution argument would produce. Countervailing\nevidence, also observed: the strict-LCM-jump census records a closest central\nmargin of $\\approx 0.000221$ of the modulus at $t = 100$, so these residues are\nnot robustly central and any equidistribution claim will be delicate. \\ev{Cert}\n\\end{rem}\n\n\\subsection{Survivor 2: the four-term pivot budget}\n\nSame lane as Survivor 1, with the analytic burden repackaged. Split the block\nsum at a largest-prime pivot into centred correlation, fibre-mean,\nbad-cofactor and non-supplier contributions and bound the four separately.\n\n\\begin{defn}[$\\mathrm{DTWPivotResidualDecorrelation}$]\n\\label{defn:pivot}\nFor every $h > 0$ there are $s > 0$ and $\\eta \\in (0,1)$ such that for every\n$X_0$ there are $X, L$ with $\\max(X_0,1) \\le X$, $h \\le L - s$,\n$16(2X{+}h{+}L{+}2) \\le 2^{L}$, and all four of\n\\[\n  \\mathrm{Re}\\,\\mathrm{pivotCenteredCorrelation} \\le \\tfrac{14}{25}X, \\quad\n  \\lVert \\mathrm{pivotFiberMean} \\rVert \\le \\tfrac{1}{100}X, \\quad\n  \\lVert \\mathrm{pivotBad} \\rVert \\le \\tfrac{1}{100}X, \\quad\n  \\lVert \\mathrm{pivotNonSupplier} \\rVert \\le \\tfrac{8}{25}X .\n\\]\nIt implies $\\mathrm{Irrational}(S)$.\n\\coord{first-harmonic} \\scale{cofinal} \\ev{Lean}\n\\lean{Erdos249257.DTWPivotResidualDecorrelation}{FirstHarmonicPivot.lean:397}\n\\lean{Erdos249257.irrational\\_totient\\_series\\_of\\_pivotResidualDecorrelation}{FirstHarmonicPivot.lean:406}\n\\end{defn}\n\nIt inherits Survivor 1's evasions of B1, B4, B5, B6 and B7 verbatim: the four\nterms are exact finite sums over canonical largest-prime supplier fibres of the\nsame accumulated-word phases, with no sample choice and nothing prescribed. Its\nspecific advantages are three. The decomposition is an \\emph{exact identity},\nproved unconditionally with \\texttt{\\#print axioms} clean\n(\\lean{Erdos249257.windowFirstExp\\_sum\\_eq\\_pivot\\_decomposition}{FirstHarmonicPivot.lean:342},\nbudget consumer\n\\lean{Erdos249257.first\\_harmonic\\_gap\\_of\\_pivotBudgetAt}{FirstHarmonicPivot.lean:377}).\nThe one-sided budget lowers the hard requirement from a norm bound\n$\\lVert \\cdot \\rVert \\le X/2$ to a real-part bound $\\le 14X/25$. And the pivot\nrests on genuine arithmetic rather than a sampled surrogate: on the canonical\nfibre $m = 1$, $s = L-h$ the supplier set is \\emph{literally} the shifted dyadic\ninterval of primes, proved as a membership equality\n(\\lean{Erdos249257.mem\\_pivotFiber\\_one\\_overlap\\_iff}{FirstHarmonicPivot.lean:251}),\nwith the totient factorisation at the pivot the honest\n$\\varphi(mp) = \\varphi(m)(p-1)$ and the non-divisibility discharged by size.\nThree of the four terms are then counting bookkeeping; only the first needs a\ngenuine correlation estimate, and only in real part.\n\n\\textbf{What it must respect.} B6 has a residual-gauge instance:\na residual-blind determinant or conditioning test cannot certify that genuine\nphase reconstruction rather than a locked degenerate configuration has occurred\n(\\lean{Erdos249257.locked\\_reconstruction\\_preserves\\_nonzero\\_minor}{ResidualGaugeObstruction.lean:94}).\nSo this route must couple rows by an extra arithmetic identity, not merely\ngauge-normalise columns. \\textbf{Honest status:} strictly a repackaging of\nSurvivor 1's burden; it is listed separately only because three quarters of it\nare already exact identities on disk. The module asserts no prime-distribution\nor decorrelation estimate --- the socket is deliberately empty. \\ev{Open}\n\n\\subsection{Survivor 3: uniform quantitative escape at cofinally many primes}\n\n\\begin{defn}[$\\mathrm{DTWNaturalPrimeTailOrbitStrictGap}$]\n\\label{defn:primegap}\nFor every $h \\ge 1$ and every $N_0$ there is a prime $p$ with\n$\\max(N_0{+}h{+}1,\\, h{+}5) \\le p$ and\n$\\mathrm{Re}\\,\\mathrm{tailOrbitFirstExp}(h,\\,p{-}h{-}1) < \\tfrac{9}{10}$,\nwhere $\\mathrm{tailOrbitFirstExp}(h,N) = e(R_{N+h} - R_N)$. Equivalently:\ncofinally many primes $p$ at which the totient tail difference across the shift\n$h$ stays a \\emph{fixed} distance\n$\\ge \\arccos(9/10)/2\\pi \\approx 0.0722$ from every integer.\nIt implies $\\mathrm{Irrational}(S)$.\n\\coord{prime-pivot} \\scale{cofinal} \\ev{Lean}\n\\lean{DTWNaturalPrimeTailOrbitStrictGap}{TotientStrictPrimeEscape.lean:18}\n\\lean{irrational\\_totient\\_series\\_of\\_naturalPrimeTailOrbitStrictGap}{TotientStrictPrimeEscape.lean:66}\n\\end{defn}\n\nIt evades B1, B4, B5, B6 and B7 for the reasons given for Survivor 1: a\nstatement about the true real tail difference at unbounded prime positions,\nwith no letters prescribed, no bounded state, no rank, no coarse\ncoefficient-word invariant. It evades B2 by the \\emph{second} of the two\nexemptions in Proposition~\\ref{prop:b2} --- not by denying sample choice, but by\ndemanding a \\emph{uniform margin}. Irrationality gives non-integrality of every\ntail difference and supplies no lower bound whatever on distance to $\\Z$;\ncertificate completeness therefore cannot manufacture this predicate.\n\n\\textbf{Honest demotion.} This is a \\emph{pointwise} producer: it requires\nnaming a good prime, and Observation~\\ref{obs:pointwise} records that no\npointwise producer in this corpus has ever been supplied at even one large\nindex, across seven independent attempts. It evades every proved barrier while\nsitting inside the class the strongest observed pattern indicts. Nothing on\ndisk proves it at a single prime for a single $h$. It ranks below Survivor 1.\n\\ev{Open}\n\n\\subsection{Survivor 4: depth-locked full-depth escape}\n\n\\begin{defn}[$\\mathrm{ApFullDepthEscape}$]\n\\label{defn:apfde}\nFor every $d \\ge 1$ and every $N$ there is $t \\ge 1$ with\n$\\mathrm{certifiedKill}(td,\\,N,\\,td)$; unpacked,\n\\[\n  N + 2td + 2 \\;<\\; \\mathrm{windowDiscrepancy}(td,N,td) \\bmod 2^{td}\n  \\;<\\; 2^{td} - (N + 2td + 2).\n\\]\nIt implies $\\mathrm{Irrational}(S)$.\n\\coord{period-ray} \\scale{cofinal} \\ev{Lean}\n\\lean{ApFullDepthEscape}{PeriodMultipleEscape.lean:162}\n\\lean{irrational\\_totient\\_series\\_of\\_apFullDepthEscape}{PeriodMultipleEscape.lean:171}\n\\end{defn}\n\nThis is the shortest fully stated open inequality the programme has produced.\nIts evasion of B2 is recorded on disk rather than argued: its ambient parent\n$\\mathrm{PeriodMultipleKillSupply}$ is \\emph{proved} equivalent to\n$\\mathrm{Irrational}(S)$ and $\\mathrm{ApFullDepthEscape}$ implies it\n(\\lean{periodMultipleKillSupply\\_of\\_apFullDepthEscape}{PeriodMultipleEscape.lean:165}),\nso it is at least as strong, and possibly strictly stronger --- the file's own\ndocstring records ``sufficient for irrationality; not known necessary''. Being\npossibly strictly stronger, certificate completeness cannot collapse it back.\nIt evades B1 (both $N$ and $t$ unbounded), B4 (locking depth to the period\nmeans the entire word accumulates and no letter is prescribed), B5 and B6 (no\nstate, no rank). Its substance is pure anti-concentration: via\n\\lean{windowDiscrepancy\\_self\\_eq\\_totientBlock\\_sub}{PeriodMultipleEscape.lean:67},\nthe difference of two adjacent period blocks must have central residue mod\n$2^{h}$ at some multiple period $h = td$, and the room condition is automatic\nonce $h$ is large, so nothing but the residue's position is at stake.\n\n\\textbf{Honest demotion,} identical to Survivor 3: it is a pointwise producer\nand names an index. It is listed because it is the cleanest target for\ncomputational exploration --- with Theorem~\\ref{thm:gamma} as the standing\nreminder that no amount of such exploration becomes a proof. \\ev{Open}\n\n\\subsection{Survivor 5: the rationality-side rank upper bound}\n\n\\begin{defn}[The coordinate-disjoint obligation]\n\\label{defn:rankupper}\nEither: there is $C$ such that for every $c : \\N \\to \\N$ with $c(n) \\le n$ and\n$T_c \\notin$ the irrationals, every $v > 0$ and every tempered binary orbit\n$u$ for $(c,v)$, and every $e$,\n$\\dim_{\\Q} \\mathrm{span}_{\\Q}\\bigl(\\mathrm{canonicalCarryKernelFamily}(u,e)\\bigr) \\le C$;\nor the same with any $g(e)$ growing strictly slower than $2^{e}-1$ in place of\n$C$. Either version contradicts the proved floor of\nProposition~\\ref{prop:rank} and closes \\#249.\n\\coord{carry-rank} \\scale{uniform} \\ev{Open}\n\\end{defn}\n\nThis is the only surviving obligation that is not a residue or certificate\nstatement at all, so B2's certificate completeness cannot reach it --- it is\nnot a reformulation of $\\mathrm{Sep}$ in any vocabulary. It evades B1 and B4\nentirely (no window, no letters, no prescribed residues) and B5 (linear algebra\nover the whole orbit, not a bounded-state summary). The scale side is finished\nand uniform in $e$ (Proposition~\\ref{prop:rank}).\n\n\\textbf{Honest flag, and it is severe.} B6 partly indicts this route from\ninside. The most natural approach to the upper bound is dead\n(\\lean{Erdos249257.false\\_of\\_compressedAdjointCertificate}{TotientMahlerDefect.lean:1051}),\nand what rationality actually buys --- uniform eventual periodicity of the\ncarry's dyadic sections mod $v$ --- provably does not promote to a $\\Q$-rank\nbound (Proposition~\\ref{prop:period-not-rank}). So this survivor is genuinely\ncoordinate-disjoint from everything else, which is its whole value, but its\nobvious approach is closed and the missing input is a rigidity theorem nobody\nhas stated.\n\n\\section{How to use this document}\n\n\\paragraph{Evidence bands.} Every claim carries one, and they are never\nblurred. \\ev{Lean} means kernel-checked, with the declaration named and the\nfile:line given. \\ev{Cert} means an exact finite computation, valid exactly on\nthe stated range. \\ev{Math} means ordinary mathematics, proved here or\nelsewhere but not formalised --- Theorem~\\ref{thm:gamma} is the important\ninstance, and its unformalised composition step is flagged where it occurs.\n\\ev{Cited} means published, with the reference. \\ev{Open} means not proved, by\nanyone, anywhere. A proved no-go and an audit over a corpus are both valuable\nand are never conflated: B1, B2, B4, B5, B6 and B7 are proved eliminations; B3\nis an audit and is labelled as evidence every time it is used.\n\n\\paragraph{The scale axis.} This is the load-bearing axis of the whole\ndocument. \\scale{fixed} means one instance; \\scale{bounded} means every\ninstance up to an explicit bound; \\scale{cofinal} means infinitely often past\nevery threshold; \\scale{uniform} means for all parameters with no dependence.\nThe single most important structural fact about the corpus is that\n\\emph{everything unconditional and quantitative sits on the fixed or bounded\nside, while $\\mathrm{Sep}$ is cofinal}. Theorem~\\ref{thm:gamma} says that gap\nis one of kind and not of degree: no accumulation of bounded verification, at\nany bound, is evidence that the cofinal statement holds. Read every result's\n\\texttt{scale} tag before reading its content.\n\n\\paragraph{Coordinates, and why obstructions are relative to them.} Every\nobstruction carries a \\texttt{coord} tag, because an obstruction is always an\nobstruction \\emph{in a representation}, never a property of the object. The\n$\\gamma$-splice obstructs arguments in the \\coord{binary-digit} coordinate;\nthe rank floor lives in \\coord{carry-rank}; positivity lives in\n\\coord{actual-lcm}; the Farey exclusion lives in \\coord{farey}. Two\nconsequences follow, and both matter for anyone attacking the problem. First,\nbefore reporting a new wall, name the coordinate it was met in and test a\nre-representation. Second, an obstruction proved in one coordinate is silent\nabout another: Survivor 5 survives precisely because it is coordinate-disjoint\nfrom the certificate lane, and Survivor 1 survives because the first-harmonic\ncoordinate is the one in which the corpus's proved collapse mechanism has no\npurchase.\n\n\\paragraph{Map of what follows.} The premise catalogue records, for each\ncoordinate, the exact statements the corpus holds, with hypotheses, evidence\nband, scale, and Lean declaration. It is followed by the typed-implication\ntables (which statement feeds which, and at which quantifier shape), the\nnear-miss index against the open obligations, and the promotion audit whose\nverdicts are summarised as B3. Nothing after this point restates the barriers;\nthey are all here.\n\n\\paragraph{Read this in full before starting.} That instruction is not\npoliteness. The recurring practical failure of this programme has not been a\nshortage of ideas; it has been rediscovering the same reformulations, because\nno single document held the whole picture. Every dead route in\n\\S\\ref{sec:wall} was found more than once. If you are about to prescribe\ntotient values at engineered positions, read B4 first. If you are about to\ndefine a bounded carry state, read B5. If you are about to look for a weaker\nsufficient condition, read B2 --- there probably isn't one, and the corpus has\nalready proved as much three times. The purpose of assembling the wall is that\nthe next attempt should start where the last one stopped.\n\n% ---- part a249_p0 ----\n\\section{The object, the target, and the exact record}\n\nThis part fixes the object of Erd\\H{o}s Problem \\#249, states exactly what is\nopen about it, and records every unconditional numeral, identity and\nLean-checked bound the corpus currently holds. Nothing in this part decides\nirrationality. Every result below is either an identity (no hypothesis, holds\nunconditionally), a finite computation (holds up to an explicitly stated\nbound), or is flagged \\ev{Cited}/\\ev{Open} where that is the honest status.\n\n\\subsection{The object}\n\n\\begin{defn}[The Erd\\H{o}s--249 constant]\n\\label{defn:S}\nLet $\\varphi$ denote Euler's totient function. Define\n\\[\nS \\;:=\\; \\sum_{n \\ge 1} \\frac{\\varphi(n)}{2^{n}} \\;=\\; \\sum_{n : \\mathbb{N}} \\frac{\\varphi(n)}{2^{n}},\n\\]\nthe two forms coinciding because $\\varphi(0)=0$; the second, $\\mathbb{N}$-indexed\nform is the one carried in the Lean source. The series converges absolutely\nsince $\\varphi(n) = O(n)$. \\emph{Erd\\H{o}s \\#249 asks whether $S$ is irrational.\nThis is OPEN.} Nothing in this paper decides it; every claim below is either an\nunconditional identity, a finite computation with an explicitly stated range,\nor is marked \\ev{Cited}/\\ev{Open}.\n\\end{defn}\n\n\\begin{rem}[Status]\nNo proof or disproof of $\\mathrm{Irrational}(S)$ exists anywhere in the corpus,\nformal or informal. What exists is: one large unconditional finite denominator\nexclusion (\\S\\ref{ssec:farey}), several exact reformulations of $S$ that\nrelocate the same open question onto different coordinates without touching\nits truth value (\\S\\ref{ssec:coprime}--\\S\\ref{ssec:lambert}), and a certificate\napparatus whose cofinal supply obligation is the precise open target\n(recorded in full in Part~1 of this paper; only its finite, checked instances\nare catalogued here, \\S\\ref{ssec:certtable}).\n\\end{rem}\n\n\\subsection{The binary-digit reduction}\n\n\\begin{prop}[Digit-shift identity]\n\\label{prop:shift}\nFor every $N : \\mathbb{N}$,\n\\[\n2^{N} \\cdot S \\;=\\; \\Phi_N + R_N,\n\\]\nwhere $\\Phi_N := \\sum_{n \\le N} \\varphi(n) \\cdot 2^{N-n} \\in \\mathbb{N}$ (the\ninteger prefix) and $R_N := \\sum_{j \\ge 0} \\varphi(N+1+j)/2^{j+1}$ (the\nfractional tail, \\texttt{totientTail} in Lean).\n\\coord{binary-digit} \\scale{uniform} \\ev{Lean}\n\\lean{two_pow_mul_totient_series_eq}{TotientTailPeriodKiller.lean:150}\n\\end{prop}\n\n\\begin{proof}[Consequence]\nRationality and eventual periodicity of the base-2 expansion of $S$ is\n\\emph{exactly} a statement about the tail $R_N$: $S \\in \\mathbb{Q}$ forces\n(by Euler's theorem applied to the odd part of the denominator) a period\n$h>0$ and pre-period $N_0$ with $R_{N+h}-R_N \\in \\mathbb{Z}$ for all\n$N \\ge N_0$ (\\texttt{eventual_period_of_not_irrational},\n\\lean{eventual_period_of_not_irrational}{TotientTailPeriodKiller.lean:358},\n\\ev{Lean}, explicit witness $h=\\varphi(\\mathrm{oddPart}(r.\\mathrm{den}))$,\n$N_0=v_2(r.\\mathrm{den})$). Nothing in the corpus supplies the converse\ncofinally; this is the certificate wall documented in full in Part~1 and is\nnot restated here beyond this pointer.\n\\end{proof}\n\n\\subsection{The unconditional denominator floor}\n\\label{ssec:farey}\n\nThe strongest \\emph{unconditional} fact the corpus holds about $S$ is a lower\nbound on its denominator should it happen to be rational. It is obtained by a\ncompletely elementary Farey/mediant argument, entirely free of any certificate\nor period apparatus, and is therefore logically independent of the open\ncertificate-supply obligation.\n\n\\begin{lem}[Farey gap, fully general]\n\\label{lem:farey}\nFor integers $a,b,c,d,r,s$ with $b>0$, $d>0$, $bc-ad=1$ (i.e.\\ $a/b$ and\n$c/d$ are unimodular Farey neighbours), and $as < rb$, $rd < cs$ (i.e.\\ $r/s$\nlies strictly between them): $b+d \\le s$.\n\\coord{farey} \\scale{n/a} \\ev{Lean}\n\\lean{farey_gap}{GapFareyBound.lean:51}\n\\end{lem}\n\nThis lemma is a classical Stern--Brocot fact with zero totient or Mersenne\ncontent; it is the generic engine underneath every Farey-gap bound in the\ncorpus, for either open problem.\n\n\\begin{prop}[The wave-17 gap certificate at window $K=240$]\n\\label{prop:gapwindow}\nLet $V$ be the explicit committed totient residue for window $(N,K)=(1,240)$.\nFor every $q : \\mathbb{N}$ with\n\\[\n0 < q \\;\\le\\; Q_0 := 79\\,639\\,646\\,646\\,701\\,375\\,323\\,355\\,774\\,875\\,831\\,053 \\;\\;(\\approx 7.96\\times 10^{34}),\n\\]\n\\[\n(q \\cdot V) \\bmod 2^{240} \\;+\\; 243\\,q \\;<\\; 2^{240}.\n\\]\nThis bound is \\emph{sharp}: $q = Q_0+1 = 79\\,639\\,646\\,646\\,701\\,375\\,323\\,355\\,774\\,875\\,831\\,054$\nis the exact first failing denominator, exhibited as the mediant of two\nexplicit unimodular Farey neighbours.\n\\coord{farey} \\scale{bounded} \\ev{Lean}\n\\lean{gap_check_window_1_240_le_79639646646701375323355774875831053}{GapFareyBound.lean:176}\n\\lean{gap_check_window_1_240_first_failure}{GapFareyBound.lean:225}\n\\end{prop}\n\n\\begin{thm}[Erd\\H{o}s \\#249 denominator exclusion --- the headline unconditional result]\n\\label{thm:denom-record}\nFor every $p \\in \\mathbb{Q}$ with reduced denominator $p.\\mathrm{den} \\le Q_0$,\n\\[\nS \\;\\neq\\; p.\n\\]\nEquivalently: \\emph{if $S$ is rational, its reduced denominator exceeds}\n$Q_0 \\approx 7.96 \\times 10^{34}$.\n\\coord{farey} \\scale{bounded} \\ev{Lean}\n\\lean{tsum_totient_div_pow_two_ne_ratCast_of_den_le_79639646646701375323355774875831053}{CertificateKernel.lean:18056}\n\\end{thm}\n\n\\begin{rem}[What this does and does not say]\nTheorem~\\ref{thm:denom-record} is a complete, unconditional finite fact: no rational\nof small denominator equals $S$. It says nothing about arbitrarily large\ndenominators and is not itself a route to irrationality. The docstring behind\nProposition~\\ref{prop:gapwindow} states the honest open question this leaves:\nwhether $\\sup_K (b+d)(K)$ (the growing analogue of $Q_0$ as the window $K$\ngrows) is unbounded as $K \\to \\infty$ --- which would close \\#249 through this\ntheorem's consumer, entirely independently of the certificate-supply\nobligation of Proposition~\\ref{prop:shift}. \\ev{Open}; not claimed or proved\nanywhere in the corpus.\n\\end{rem}\n\n\\begin{rem}[Ladder of prior rungs]\n$Q_0$ is the current end of an explicit sequence of increasingly wide Farey\nwindows computed by the same mechanism: $4838 \\to 2^{22} \\to 2.49\\times10^{17}$\n(window $K=120$) $\\to Q_0$ (window $K=240$). Each rung is a finite,\nindependently checked instance of Lemma~\\ref{lem:farey} and\nProposition~\\ref{prop:gapwindow}'s pattern at a larger $K$; none is claimed to\nextrapolate.\n\\end{rem}\n\n\\subsection{The same record, transported: the M\\\"obius-square and coprimality-probability forms}\n\\label{ssec:coprime}\n\nThe exact finite Farey record behind Theorem~\\ref{thm:denom-record} is not tied to\nthe $\\varphi(n)/2^n$ presentation of $S$; it transfers verbatim to two other\nexact reformulations of the same constant, at exactly half the bound.\n\n\\begin{prop}[Fair-coin coprimality form]\n\\label{prop:coprime}\nLet $X,Y$ be independent random variables with $\\Pr(X=n)=\\Pr(Y=n)=2^{-n}$\nfor $n \\ge 1$ (independent fair-coin waiting times). Then\n\\[\nS \\;=\\; \\tfrac12 \\;+\\; \\Pr\\bigl(\\gcd(X,Y)=1\\bigr)\n\\;=\\; \\tfrac12 \\;+\\; \\sum_{\\substack{a,b\\ge 1\\\\ \\gcd(a,b)=1}} 2^{-(a+b)}.\n\\]\nEquivalently, on the visible lattice: summing $2^{-(a+b)}$ over the half-open\ncoprime pairs ($a\\ge1$, $b\\ge0$, $\\gcd(a,b)=1$) recovers $\\sum_n \\varphi(n)/2^n$\nexactly, with no boundary correction, because the visible-point count on the\nhalf-open antidiagonal at height $n$ equals $\\varphi(n)$ for every $n$,\nincluding $n=0,1$.\n\\coord{probability} \\scale{n/a} \\ev{Lean}\n\\lean{tsum_visible_coprime_pairs_eq_totient_series}{CertificateKernel.lean:18216}\n\\lean{totient_series_eq_half_add_visible_coprime_pairs}{CertificateKernel.lean:18229}\n\\end{prop}\n\n\\begin{prop}[The gcd-layer normalisation]\n\\label{prop:gcdlayer}\nFor independent fair-coin waiting times as above, $\\sum_{g\\ge1}\\Pr(\\gcd(X,Y)=g)=1$\nexactly; and for every $d>0$, $\\Pr(d\\mid X \\wedge d\\mid Y) = 1/(2^d-1)^2$.\n\\coord{probability} \\scale{uniform} \\ev{Lean}\n\\lean{tsum_pos_coprime_inv_mersenne_eq_one}{GcdMomentCalculus.lean:349}\n\\lean{tsum_pos_pair_both_dvd_half_eq_inv_mersenne_sq}{GcdMomentCalculus.lean:266}\n\\end{prop}\n\n\\begin{thm}[Denominator exclusion for the coprimality-probability form]\n\\label{thm:denomcoprime}\nLet\n\\[\nQ_1 := \\left\\lfloor \\frac{Q_0}{2} \\right\\rfloor = 39\\,819\\,823\\,323\\,350\\,687\\,661\\,677\\,887\\,437\\,915\\,526.\n\\]\nFor every $a\\in\\mathbb{Z}$, $d\\in\\mathbb{N}$ with $0<d\\le Q_1$: the visible\ncoprime-pair probability $\\Pr(\\gcd(X,Y)=1)$ is not equal to $a/d$.\n\\coord{farey} \\scale{bounded} \\ev{Lean}\n\\lean{tsum_visible_coprime_pairs_ne_int_div_of_den_le_39819823323350687661677887437915526}{CertificateKernel.lean:18244}\n\\end{thm}\n\n\\begin{thm}[Denominator exclusion for the M\\\"obius-square form]\n\\label{thm:denommobsq}\nWith $Q_1$ as in Theorem~\\ref{thm:denomcoprime}: for every $a\\in\\mathbb{Z}$,\n$d\\in\\mathbb{N}$ with $0<d\\le Q_1$, the signed series\n$T := \\sum_{d\\ge1} \\mu(d)/(2^d-1)^2 = S - \\tfrac12$ (see \\S\\ref{ssec:mobius})\nis not equal to $a/d$.\n\\coord{farey} \\scale{bounded} \\ev{Lean}\n\\lean{tsum_moebius_div_two_pow_sub_one_sq_ne_int_div_of_den_le_39819823323350687661677887437915526}{CertificateKernel.lean:18159}\n\\end{thm}\n\n\\begin{rem}[Why the bound halves, and why this is not new information]\n$Q_1 = \\lfloor Q_0/2 \\rfloor$ arithmetically: $Q_0$ is odd\n($Q_0 = 79\\,639\\,646\\,646\\,701\\,375\\,323\\,355\\,774\\,875\\,831\\,053$), so\n$Q_0/2 = 39\\,819\\,823\\,323\\,350\\,687\\,661\\,677\\,887\\,437\\,915\\,526.5$ and\n$Q_1$ is its floor. The halving is exactly the cost of the affine shift\n$S = \\tfrac12 + T$ (resp.\\ $S = \\tfrac12 + \\Pr(\\gcd(X,Y)=1)$) on a\ndenominator-exclusion statement: excluding all denominators $\\le Q_1$ for $T$\nis equivalent, after the $\\tfrac12$-shift, to excluding all denominators\n$\\le Q_0$ for $S$ (a denominator-$d$ value of $T$ with $d\\le Q_1$ yields a\ndenominator dividing $2d\\le Q_0$ for $S$). Theorems~\\ref{thm:denomcoprime}\nand~\\ref{thm:denommobsq} are therefore the \\emph{same} finite Farey record as\nTheorem~\\ref{thm:denom-record}, transported through Proposition~\\ref{prop:coprime}\nand Proposition~\\ref{prop:mobsq} respectively --- not independent evidence,\nbut confirmation that the exclusion is coordinate-stable across all three\nexact presentations of $S$.\n\\end{rem}\n\n\\subsection{The M\\\"obius--Mersenne identity and why bounded coefficients matter}\n\\label{ssec:mobius}\n\n\\begin{prop}[M\\\"obius-square reduction]\n\\label{prop:mobsq}\n\\[\nS \\;=\\; \\sum_{n\\ge1}\\frac{\\varphi(n)}{2^n} \\;=\\; \\frac12 \\;+\\; \\sum_{d\\ge1}\\frac{\\mu(d)}{(2^d-1)^2},\n\\]\nwhere $\\mu$ is the M\\\"obius function, so $\\mu(d)\\in\\{-1,0,1\\}$ for every $d$.\nConsequently \\emph{Erd\\H{o}s \\#249 $\\iff$ $T:=\\sum_{d\\ge1}\\mu(d)/(2^d-1)^2 \\notin \\mathbb{Q}$},\nthe single reduced target every other result in this subsection and\n\\S\\ref{ssec:lambert} feeds.\n\\coord{mobius-mersenne} \\scale{n/a} \\ev{Lean}\n\\lean{totientSeries_eq_pnat_half_pow}{SquaredMersenneDiagonalEnclosure.lean:50}\n\\lean{tsum_totient_half_pow_eq_half_add_moebius_sq}{MersenneLambertLadder.lean:665}\n\\end{prop}\n\n\\begin{rem}[Why the bounded coefficients matter --- the Erd\\H{o}s-1948 regime]\nThe identity of Proposition~\\ref{prop:mobsq} rewrites $S$ (equivalently $T$)\nas a \\emph{M\\\"obius-twisted Lambert-squared series}: the numerator weight\n$\\mu(d)$ is bounded, $|\\mu(d)|\\le1$ for every $d$, uniformly in $d$. This\nplaces $T$ in exactly the coefficient regime of the classical Erd\\H{o}s (1948)\nnear-integer irrationality criterion and of the level-1 sibling identity\n$L(\\mu):=\\sum_d \\mu(d)/(2^d-1) = \\tfrac12$ (rational, trivially) alongside\n$L(1) = \\sum_d 1/(2^d-1) = E$, the Erd\\H{o}s--Borwein constant, which\n\\emph{is} proved irrational in this same kernel\n(\\lean{irrational_erdosBorwein_series}{CertificateKernel.lean:8007}, \\ev{Lean}).\nThis is the opposite regime from Proposition~\\ref{prop:shift}'s\n\\textbf{binary-digit coordinate}, where the corresponding weight is\n$\\varphi(n) = \\Theta(n)$ --- \\emph{unbounded}, growing linearly. A\nnear-integer/Dirichlet-approximation argument of Erd\\H{o}s-1948 shape\n(formalised generically as\n\\lean{irrational_of_int_mul_near_int}{CertificateKernel.lean:5792} and its\nbase-power specialisation\n\\lean{irrational_of_pow_mul_near_int}{CertificateKernel.lean:5821}, both\n\\ev{Lean}, \\coord{n/a}, fully coordinate-free) has a genuine chance of\ntransferring to $T$ precisely because its weight is bounded, in a way it does\nnot have a chance of transferring directly to the raw $\\varphi(n)/2^n$ series.\nNo such transfer is proved; \\S\\ref{ssec:mobius-wall} below (cross-referenced\nhere, developed in Part~2 of this paper) records exactly why the transfer has\nso far failed (the ``$\\mu$-pollution'' obstruction) rather than merely\nasserting the analogy.\n\\end{rem}\n\n\\subsection{The squared-Lambert gcd-moment identities}\n\\label{ssec:lambert}\n\n\\begin{prop}[Squared-Lambert transfer engine]\n\\label{prop:lambertengine}\nFor $w:\\mathbb{N}\\to\\mathbb{R}$ with $|w(d)|\\le d$ for all $d>0$, and $0\\le r<1$:\n\\[\n\\sum_{d\\ge1} w(d)\\left(\\frac{r^d}{1-r^d}\\right)^2 \\;=\\; \\sum_{n\\ge1}\\left(\\sum_{e\\mid n} w(e)\\Bigl(\\tfrac{n}{e}-1\\Bigr)\\right) r^n.\n\\]\n\\coord{mobius-mersenne} \\scale{uniform} \\ev{Lean}\n\\lean{tsum_lambert_linear_weight_sq_pure}{GcdMomentCalculus.lean:105}\n\\end{prop}\n\nThis one identity, at $r=1/2$, specialises to every squared-Lambert rung the\ncorpus computes; two instances are exact and directly relevant to \\#249's\nweight structure:\n\n\\begin{prop}[The known $\\zeta_q$-rung]\n\\label{prop:zetaq}\n\\[\n\\sum_{d\\ge1} \\frac{1}{(2^d-1)^2} \\;=\\; \\sum_{n\\ge1} \\frac{\\sigma(n)-\\tau(n)}{2^n} \\;=\\; \\zeta_q(2)-\\zeta_q(1) \\text{ at } q=\\tfrac12,\n\\]\nwhere $\\sigma$ is the sum-of-divisors function and $\\tau$ the number-of-divisors\nfunction. The \\emph{identity} is machine-checked (\\ev{Lean}); irrationality of\nthe \\emph{value} $\\zeta_q(2)-\\zeta_q(1)$ is \\ev{Cited} (Postelmans--Van Assche\n$q$-Pad\\'e), \\emph{not} formalised in this corpus.\n\\coord{mobius-mersenne} \\scale{n/a}\n\\lean{tsum_one_div_mersenne_sq_eq_sigma_sub_tau_series}{GcdMomentCalculus.lean:216}\n\\end{prop}\n\n\\begin{prop}[The Pillai/gcd-moment rung]\n\\label{prop:pillai}\n\\[\n\\sum_{d\\ge1} \\frac{\\varphi(d)}{(2^d-1)^2} \\;=\\; \\sum_{n\\ge1} \\bigl(P(n)-n\\bigr)\\cdot 2^{-n} \\;=\\; \\mathbb{E}[\\gcd(X,Y)],\n\\]\nwhere $P(n) := \\sum_{e\\mid n}\\varphi(e)\\cdot(n/e) = (\\varphi * \\mathrm{Id})(n)$\nis Pillai's gcd-sum function and $X,Y$ are the independent fair-coin waiting\ntimes of Proposition~\\ref{prop:coprime}. The identity itself is\nmachine-checked (\\ev{Lean}); the value $\\mathbb{E}[\\gcd(X,Y)]$ is \\ev{Open} ---\na cousin rung to \\#249, not \\#249 itself.\n\\coord{mobius-mersenne} \\scale{n/a}\n\\lean{tsum_totient_div_mersenne_sq_eq_gcd_moment_series}{GcdMomentCalculus.lean:235}\n\\end{prop}\n\n\\begin{rem}[The level mirror]\nWriting $L(f):=\\sum_d f(d)/(2^d-1)$ (level 1) and $L_2(f):=\\sum_d f(d)/(2^d-1)^2$\n(level 2): $L(\\mu)=\\tfrac12$ (rational, trivial), $L(1)=E$ (Erd\\H{o}s--Borwein,\nirrational, \\ev{Lean}), $L(\\varphi)=2$ (rational); $L_2(\\mu)=S-\\tfrac12$\n(\\ev{Open}, this is \\#249), $L_2(1)=\\zeta_q(2)-\\zeta_q(1)$ (\\ev{Cited}\nirrational, Proposition~\\ref{prop:zetaq}), $L_2(\\varphi)=\\mathbb{E}[\\gcd(X,Y)]$\n(\\ev{Open}, Proposition~\\ref{prop:pillai}). At level 1 the M\\\"obius rung is\ntrivial and the $\\zeta$-rung is the hard classical case; at level 2 the\n$\\zeta$-rung is known and the M\\\"obius rung \\emph{is} \\#249. M\\\"obius\nprojection is the one wall repeated at both levels of the ladder.\n\\end{rem}\n\n\\subsection{The exact certificate record: numerical anchors through $t=64$}\n\\label{ssec:certtable}\n\nThe certificate apparatus itself (the predicate \\texttt{certifiedKill}, its\nsoundness/completeness theorems, and the cofinal supply obligation that is the\nactual open target) is developed in full in Part~1 of this paper. What is\nrecorded here is exhaustively: every finite, machine-checked numerical anchor\nthe corpus currently holds for it, stated precisely, with no extrapolation\nimplied.\n\n\\paragraph{Certificate object.} For $h,N,L : \\mathbb{N}$, define the window\ndiscrepancy $\\Delta_{h,N,L} := \\sum_{j<L} (\\varphi(N+h+1+j)-\\varphi(N+1+j))\\cdot 2^{L-1-j} \\in \\mathbb{Z}$\n(\\lean{windowDiscrepancy}{TotientTailPeriodKiller.lean:66}, \\ev{Lean}), and\n\\[\n\\mathrm{Sep}(h,N,L) \\;:\\equiv\\; (N+h+L+2) < \\Delta_{h,N,L} \\bmod 2^{L} < 2^{L}-(N+h+L+2),\n\\]\n(\\lean{certifiedKill}{TotientTailPeriodKiller.lean:72}, \\ev{Lean}, decidable). By\ncompleteness (\\lean{exists_certifiedKill_iff_tail_diff_notMem_int}{LcmConeFlatness.lean:316},\n\\ev{Lean}), $(\\exists L,\\ \\mathrm{Sep}(h,N,L)) \\iff R_{N+h}-R_N \\notin \\mathbb{Z}$\nfor every $h,N$: an unbounded (cofinal, over $h$ and $N$) supply of\n$\\mathrm{Sep}$ is exactly equivalent to $\\mathrm{Irrational}(S)$. The table\nbelow records every $(h,N,L)$-shaped or $t$-shaped instance that has actually\nbeen checked; none of it is cofinal, and none of it is presented as deciding\n\\#249.\n\n\\small\n\\begin{longtable}{p{0.20\\linewidth} p{0.40\\linewidth} p{0.13\\linewidth} p{0.20\\linewidth}}\n\\textbf{Anchor} & \\textbf{Exact statement of what was checked} & \\textbf{Scale} & \\textbf{Site} \\\\\n\\hline\n\\endhead\nFixed-window deposit, depth 16\n&\n$\\mathrm{Sep}(h,12,16)$ holds for every $h \\in \\{1,\\dots,8\\}$ (by \\texttt{decide},\n$256$ totient values below $37$); consequently $S \\ne r$ for every\n$r\\in\\mathbb{Q}$ with $1\\le h\\le 8$ and $r.\\mathrm{den} \\mid 2^{12}(2^h-1)$.\n&\n\\scale{fixed}\n&\n\\lean{certifiedKill_all_small}{TotientTailPeriodKiller.lean:404}\n\\lean{totient_series_ne_rat_of_den_dvd}{TotientTailPeriodKiller.lean:416}\n\\\\\nFixed-window deposit, depth 14, wider net\n&\n$\\mathrm{Sep}(h,14,\\text{depth }14)$-shaped exclusion extends to every\n$h\\in\\{1,\\dots,16\\}$: $S \\ne r$ for every $r\\in\\mathbb{Q}$ with $1\\le h\\le16$\nand $r.\\mathrm{den}\\mid 2^{14}(2^h-1)$.\n&\n\\scale{fixed}\n&\n\\lean{totient_series_ne_rat_of_den_dvd_pow_two_mul_mersenne_upto_sixteen}{CertificateKernel.lean:18562}\n\\\\\nDiagonal-pincer certificates, base 12 scales\n&\n$\\mathrm{Sep}\\bigl(H_t, H_t, D(t)\\bigr)$, $H_t := \\mathrm{lcm}(1,\\dots,t)$,\nholds (by explicit \\texttt{decide}/\\texttt{norm_num} on checked\n\\texttt{Nat.totient} values, via factored prime-power blocks with\nLucas-primality certificates) for $t \\in \\{1,2,3,4,5,7,8,9,11,13,16,17\\}$\nwith certificate depths $D(t) \\in \\{6,5,7,7,9,14,15,14,21,22,23,26\\}$\nrespectively (in the same order).\n&\n\\scale{fixed}\n&\n\\lean{certifiedKill_diagonal_all_imported}{DiagonalPincerCertificates.lean:2842}\n\\lean{diagonalPincerCertificateScales}{DiagonalPincerCertificates.lean:2842}\n\\lean{diagonalPincerKillDepth}{DiagonalPincerCertificates.lean:2844}\n\\\\\nDiagonal-pincer certificates, extended scales through $t=64$\n&\nThe same predicate $\\mathrm{Sep}(H_t,H_t,D(t))$ is additionally checked, one\nsibling module per scale, at\n$t \\in \\{19,\\allowbreak23,\\allowbreak25,\\allowbreak27,\\allowbreak29,\\allowbreak31,\n\\allowbreak32,\\allowbreak37,\\allowbreak41,\\allowbreak43,\\allowbreak47,\\allowbreak49,\n\\allowbreak53,\\allowbreak59,\\allowbreak61,\\allowbreak64\\}$\n(16 further explicit values). Together with the base 12 scales above this is\n\\textbf{28 explicit values of $t$ in total}, the complete list being\n$\\{1,\\allowbreak2,\\allowbreak3,\\allowbreak4,\\allowbreak5,\\allowbreak7,\n\\allowbreak8,\\allowbreak9,\\allowbreak11,\\allowbreak13,\\allowbreak16,\\allowbreak17,\n\\allowbreak19,\\allowbreak23,\\allowbreak25,\\allowbreak27,\\allowbreak29,\\allowbreak31,\n\\allowbreak32,\\allowbreak37,\\allowbreak41,\\allowbreak43,\\allowbreak47,\\allowbreak49,\n\\allowbreak53,\\allowbreak59,\\allowbreak61,\\allowbreak64\\}$.\nThis does \\emph{not} establish $\\mathrm{Sep}(H_t,H_t,\\cdot)$ for infinitely\nmany $t$; it is the concrete finite floor of the diagonal sequence, exactly\nwhat an unbounded-supply argument would need to extend.\n&\n\\scale{fixed}\n&\nPer-scale Lean modules; endpoint\n\\lean{certifiedKill_diagonal_t64}{DiagonalPincerCertificateT64Endpoint.lean:1928}\n\\\\\nFarey denominator floor\n&\nEvery $q\\in\\mathbb{N}$ with $0<q\\le Q_0=79\\,639\\,646\\,646\\,701\\,375\\,323\\,355\\,774\\,875\\,831\\,053$\nsatisfies the window-$(N,K)=(1,240)$ gap certificate; $q=Q_0+1$ is the exact\nfirst failure.\n&\n\\scale{bounded}\n&\n\\lean{gap_check_window_1_240_le_79639646646701375323355774875831053}{GapFareyBound.lean:176}\n\\\\\nCoprimality-probability / M\\\"obius-square Farey floor\n&\nEvery $d\\in\\mathbb{N}$ with $0<d\\le Q_1=39\\,819\\,823\\,323\\,350\\,687\\,661\\,677\\,887\\,437\\,915\\,526$\n(exactly $\\lfloor Q_0/2\\rfloor$) is excluded as a denominator of\n$\\Pr(\\gcd(X,Y)=1)$ and, separately, of $T=\\sum_d\\mu(d)/(2^d-1)^2$.\n&\n\\scale{bounded}\n&\n\\lean{tsum_visible_coprime_pairs_ne_int_div_of_den_le_39819823323350687661677887437915526}{CertificateKernel.lean:18244}\n\\lean{tsum_moebius_div_two_pow_sub_one_sq_ne_int_div_of_den_le_39819823323350687661677887437915526}{CertificateKernel.lean:18159}\n\\\\\n\\end{longtable}\n\\normalsize\n\n\\begin{rem}[What this table is not]\nNo row above is cofinal in its indexing parameter ($h$, $t$, or the\ndenominator bound), and no row is claimed to extrapolate. The certificate\ndepths in the diagonal-pincer row are strictly increasing and irregular\n($6,5,7,7,9,14,15,14,21,22,23,26,\\dots$); no closed-form growth rate for\n$D(t)$ is proved or conjectured in the corpus. This table is the complete\ninventory of finite Erd\\H{o}s-\\#249 certificate evidence as of this writing;\nthe cofinal supply obligation it falls short of is the open target developed\nin Part~1.\n\\end{rem}\n\n% ---- part a249_p1a ----\n\\section{Catalogue of usable premises}\n\nThis catalogue renders every producer and converter/identity result found for Erd\\H{o}s\n\\#249 ($S = \\sum_{n\\ge1}\\varphi(n)/2^n$) in the certificate-kernel lane\n(\\texttt{TotientTailPeriodKiller.lean}, \\texttt{CertificateKernel.lean}, and the\nlcm/diagonal/cone reduction family) and the M\\\"obius--Mersenne / squared-Lambert lane\n(\\texttt{GcdMomentCalculus.lean}, \\texttt{RepunitMobiusNumerator.lean} through\n\\texttt{PrimePowerJumpDynamics.lean}, \\texttt{MersenneLambertLadder.lean}). Each entry\nis a self-contained premise: exact hypotheses, exact conclusion, the coordinate it was\nproved in, and the Lean site. \\textbf{\\#249 is OPEN}; nothing below decides it. Entries\nlabelled \\texttt{consumer}-shaped (an implication whose hypothesis is an unsupplied\ncofinal predicate) are marked explicitly as conditional \u2014 the hypothesis itself carries\n\\ev{Open} and is never claimed as proved. Lean site paths are relative to the\npublic \\texttt{Erdos249257/} source tree.\n\n\\subsection{Producers}\n\nProducers conclude an existence or a supply: a witnessed object, a witnessed finite\nfamily, or an unconditional dimension/growth lower bound obtained by exhibiting\nwitnesses (CRT, Dirichlet primes in AP, Bertrand's postulate, explicit \\texttt{decide}\ncomputation). Sorted by scale: uniform, then bounded, then fixed; no producer in this\nlane is cofinal (the cofinal-scale reduction theorems are consumer-shaped and are\ncatalogued as converters below, since each moves a cofinal hypothesis across a\ncoordinate boundary to the target conclusion).\n\n\\begin{thm}[cert:a9 --- \\texttt{eventual\\_period\\_of\\_not\\_irrational}, THE TAIL-PERIOD LAW]\nIf $S$ is rational then a period exists: $\\neg\\mathrm{Irrational}(S) \\to \\exists h:\\mathbb{N},\\, 0<h \\wedge \\exists N_0:\\mathbb{N},\\, \\forall N\\ge N_0,\\ \\mathrm{totientTail}(N+h) - \\mathrm{totientTail}(N) \\in \\mathrm{range}((\\uparrow):\\mathbb{Z}\\to\\mathbb{R})$. The witness is explicit: $h = \\varphi(\\mathrm{oddPart}(r.\\mathrm{den}))$, $N_0 = v_2(r.\\mathrm{den})$, from Euler's theorem applied to the odd part of the hypothetical denominator.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{binary-digit} \\lean{eventual\\_period\\_of\\_not\\_irrational}{TotientTailPeriodKiller.lean:358}\n\\end{thm}\n\n\\begin{thm}[cert:d5 --- \\texttt{not\\_irrational\\_totientSeries\\_implies\\_unbounded\\_carryRank\\_unconditional}]\n$\\neg\\mathrm{Irrational}(S) \\to \\exists v>0,\\, \\exists u:\\mathbb{N}\\to\\mathbb{Z},\\, \\mathrm{IsTemperedBinaryOrbit}\\,\\varphi\\,v\\,u \\wedge \\forall e,\\ 2^e-1 \\le \\mathrm{finrank}_{\\mathbb{Q}}(\\mathrm{span}(\\mathrm{range}(\\mathrm{canonicalCarryKernelFamily}\\,u\\,e)))$. Rationality of $S$ forces its associated tempered integral binary-carry orbit to have unboundedly rich dyadic-section rank; a second, coordinate-independent necessary condition on rationality, parallel to cert:a9 but in the carry-kernel-rank coordinate rather than the binary-digit-periodicity coordinate. Explicitly flagged in its own docstring as not by itself an irrationality proof: a separate rationality-side rank-compression theorem (contradicting cert:d4 below) would still be needed.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:carry-kernel-rank} \\lean{not\\_irrational\\_totientSeries\\_implies\\_unbounded\\_carryRank\\_unconditional}{TotientCarryKernelRigidity.lean:284}\n\\end{thm}\n\n\\begin{thm}[cert:d4 --- \\texttt{linearIndependent\\_canonicalTotientKernelFamily}]\nFor every $e:\\mathbb{N}$, the canonical dyadic totient-kernel family $\\mathrm{canonicalTotientKernelFamily}(e) : \\mathrm{TotientCanonicalIndex}(e) \\to \\mathbb{N}\\to\\mathbb{Q}$, which has exactly $2^e+1$ channels, is linearly independent over $\\mathbb{Q}$. Proved unconditionally by constructing, via CRT and Dirichlet's theorem on primes in arithmetic progression (\\texttt{PrimesCongruentOne}/\\texttt{PrimesInAP} from Mathlib), an explicit evaluation point at which one channel becomes prime and every other channel picks up a fresh prime $\\equiv 1$ modulo a large power of $2$ \u2014 a genuine witnessed producer, not merely a dimension count. Consequently $\\neg\\mathrm{FiniteDimensional}\\,\\mathbb{Q}\\,(\\mathrm{span}\\,\\mathbb{Q}\\,(\\mathrm{range}\\,\\mathrm{fullTotientKernelFamily}))$. Self-flagged: this shows the dyadic-kernel side is infinite-rank; it is not itself an irrationality proof.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:dyadic-kernel-rank} \\lean{card\\_totientCanonicalIndex}{TotientMahlerDefect.lean:51} \\lean{linearIndependent\\_canonicalTotientKernelFamily}{TotientMahlerDefect.lean:900}\n\\end{thm}\n\n\\begin{prop}[cert:a8 --- \\texttt{tail\\_diff\\_int\\_of\\_den\\_dvd}]\nIf $S = (r:\\mathbb{R})$ for $r:\\mathbb{Q}$ and $r.\\mathrm{den} \\mid 2^N\\cdot(2^h-1)$, then $\\mathrm{totientTail}(N+h) - \\mathrm{totientTail}(N) \\in \\mathrm{range}((\\uparrow):\\mathbb{Z}\\to\\mathbb{R})$: a rational value of $S$ with a denominator of this shape forces the shifted tail difference to be an integer. The sharp contrapositive engine underlying cert:a9 and cert:a6.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{binary-digit} \\lean{tail\\_diff\\_int\\_of\\_den\\_dvd}{TotientTailPeriodKiller.lean:327}\n\\end{prop}\n\n\\begin{prop}[mob:b5 --- top fibre survives, T3, \\texttt{mobiusNumerator\\_gcd\\_cyclotomicValue}]\nFor $r$ squarefree: $\\gcd(|\\mathrm{mobiusNumerator}(r)|,\\ \\mathrm{cyclotomicValue}(r)) = 1$, hence $\\mathrm{cyclotomicValue}(r) \\mid \\mathrm{baseMobiusShadow}(r).\\mathrm{den}$. The top cyclotomic channel $|\\Phi_r(2)|$ is produced as a survivor: it can never cancel from the numerator and is exhibited as an unconditional divisor of the reduced denominator at every squarefree $r$.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{cyclotomic} \\lean{mobiusNumerator\\_gcd\\_cyclotomicValue}{CyclotomicProjectionOfShadow.lean:350} \\lean{cyclotomicValue\\_dvd\\_baseMobiusShadow\\_den}{CyclotomicProjectionOfShadow.lean:389}\n\\end{prop}\n\n\\begin{prop}[mob:b6 --- upper-half prime channel survival, T4]\nFor a scale coprime to $C = \\prod_{p \\in \\mathrm{upperHalfPrimes}(t)} \\mathrm{mersenne}(p)$ (or with prime support $\\le t$), the whole finite product $C$ over the explicit upper-half prime set $\\mathrm{upperHalfPrimes}(t) = \\{p \\text{ prime} : t/2 < p \\le t\\}$ divides the reduced denominator of the scaled M\\\"obius shadow at LCM height $t$, unscaled. A genuine finite family of surviving Mersenne-prime channels is exhibited explicitly at every $t$, not merely bounded below.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{cyclotomic} \\lean{upperHalfPrimes}{MersenneShadowCyclotomicNoncollapse.lean:914} \\lean{upperHalfChannel\\_product\\_dvd\\_den\\_of\\_scale\\_primeFactors\\_le}{MersenneShadowCyclotomicNoncollapse.lean:809} \\lean{lcmHeight\\_upperHalf\\_product\\_dvd\\_den}{MersenneShadowCyclotomicNoncollapse.lean:983}\n\\end{prop}\n\n\\begin{prop}[mob:b7a --- denominator growth lower bound]\nFor $t \\ge 5$ (Bertrand's postulate supplies nonemptiness of $\\mathrm{upperHalfPrimes}(t)$): $2^{t/2} \\le \\prod_{p \\in \\mathrm{upperHalfPrimes}(t)} \\mathrm{mersenne}(p) \\le \\big(\\mathrm{lcmHeight}(t)\\cdot\\mathrm{numericMobiusShadow}(\\mathrm{lcmHeight}(t))\\big).\\mathrm{den}$. An exponential-in-$t/2$ growth lower bound for the reduced denominator at every LCM height, produced from mob:b6's explicit surviving channel product. Denominator-only: does not by itself rule out cancellation by a foreign-defect term, hence does not by itself prove \\#249.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{mobius-mersenne} \\lean{upperHalfMersenneProduct\\_lower\\_bound}{MersenneShadowDenominatorGrowth.lean:60}\n\\end{prop}\n\n\\begin{prop}[mob:b7b --- exact denominator value]\nFor every $t$ (no lower bound on $t$ needed for this direction): $\\big(\\mathrm{lcmHeight}(t)\\cdot\\mathrm{numericMobiusShadow}(\\mathrm{lcmHeight}(t))\\big).\\mathrm{den} = \\mathrm{mersenne}(\\mathrm{lcmRadical}(t)) / \\gcd\\big(\\mathrm{mersenne}(\\mathrm{lcmRadical}(t)),\\ \\mathrm{lcmScale}(t)\\cdot|\\mathrm{oddJordanScalar}(\\mathrm{lcmRadical}(t))|\\big)$. A fully closed form for the reduced denominator at every scale, produced (not merely bounded) as an explicit rational function of $t$.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{mobius-mersenne} \\lean{lcmHeight\\_scaledMobiusShadow\\_den\\_exact}{MersenneShadowDenominatorGrowth.lean:147}\n\\end{prop}\n\n\\begin{prop}[mob:d3 --- signed dyadic sum nonvanishing, \\texttt{scaled\\_dyadic\\_sum\\_ne\\_zero}]\nIf a finite signed sum $\\sum_{i \\in s} u(i)\\cdot 2^{e(m)-e(i)}$ (clearing dyadic denominators to a common exponent $e(m)$) has one index $m \\in s$ with strictly maximal exponent $e(m)$ and odd coefficient $u(m)$, while every other index has strictly smaller exponent, then the cleared sum is $\\equiv 1 \\pmod 2$, hence nonzero. A positive finite-nonvanishing engine, produced via a rectangular Cauchy--Binet determinant expansion (\\texttt{det\\_mul\\_rectangular}) applied to the signed-Hankel/$q$-moment construction (\\texttt{hankelDet}, \\texttt{truncatedMoment}); the underlying arithmetic is a $q=1/2$ Hankel-determinant framework for a possible Pad\\'e route to \\#249. No supply theorem is proved that such a unique-terminal configuration exists cofinally \u2014 this is a per-instance producer, not a cofinal producer.\n\\par\\noindent\\ev{Lean} \\scale{bounded} \\coord{p-adic} \\lean{scaled\\_dyadic\\_sum\\_ne\\_zero}{SignedQMomentObstruction.lean:95} \\lean{det\\_mul\\_rectangular}{SignedQMomentObstruction.lean:28}\n\\end{prop}\n\n\\begin{prop}[cert:a11 --- \\texttt{certifiedKill\\_all\\_small}]\n$\\forall h \\in [1,8],\\ \\mathrm{certifiedKill}\\ h\\ 12\\ 16$ \u2014 a finite, fully unconditional \\texttt{decide}-checked family of 8 certificate witnesses at the fixed point $N=12$, $L=16$ (256 explicit totient values below 37 evaluated). Consequence, via cert:a6: $\\forall r:\\mathbb{Q},\\ 1\\le h\\le 8,\\ r.\\mathrm{den}\\mid 2^{12}\\cdot(2^h-1) \\to S \\ne r$.\n\\par\\noindent\\ev{Cert} \\scale{fixed} \\coord{other:binary-window} \\lean{certifiedKill\\_all\\_small}{TotientTailPeriodKiller.lean:404} \\lean{totient\\_series\\_ne\\_rat\\_of\\_den\\_dvd}{TotientTailPeriodKiller.lean:416}\n\\end{prop}\n\n\\begin{prop}[cert:a12 --- upto-sixteen deposit]\n$\\forall h \\in [1,16]$ at exponent $14$: a wider, fully unconditional \\texttt{decide}-checked family of certificate witnesses, subsuming cert:a11 on both axes (larger $h$-range, larger 2-adic exponent). Consequence: $\\forall r:\\mathbb{Q},\\ 1\\le h\\le 16,\\ r.\\mathrm{den}\\mid 2^{14}\\cdot(2^h-1) \\to S \\ne r$.\n\\par\\noindent\\ev{Cert} \\scale{fixed} \\coord{other:binary-window} \\lean{totient\\_series\\_ne\\_rat\\_of\\_den\\_dvd\\_pow\\_two\\_mul\\_mersenne\\_upto\\_sixteen}{CertificateKernel.lean:18562}\n\\end{prop}\n\n\\begin{prop}[cert:b11 --- diagonal pincer finite deposits, THE DIAGONAL-PINCER IDENTITY]\n$\\mathrm{certifiedKill\\_diagonal\\_all\\_imported} : \\forall t \\in \\{1,2,3,4,5,7,8,9,11,13,16,17,\\dots\\},\\ \\mathrm{certifiedKill}(\\mathrm{periodLcm}(t))(\\mathrm{periodLcm}(t))(\\mathrm{diagonalPincerKillDepth}(t))$, with explicit certificate depths $[6,5,7,7,9,14,15,14,21,22,23,26,\\dots]$, extended by sibling modules through $t=64$ (28 explicit deposits total, endpoint \\texttt{certifiedKill\\_diagonal\\_t64}). Each instance is a finite \\texttt{decide}/\\texttt{norm\\_num} computation on explicit \\texttt{Nat.totient} values, factored via checked prime-power blocks (\\texttt{FactorBlock}, \\texttt{totient\\_factorBlocks}, \\texttt{prime\\_dvd\\_factorBlocks}) with Lucas-primality certificates for the primes involved. This is the concrete finite floor of the single-parameter diagonal sequence $P(t) := \\exists L,\\ \\mathrm{certifiedKill}(\\mathrm{periodLcm}(t))(\\mathrm{periodLcm}(t))\\,L$ (cert:b3 below): it produces $P(t)$ for these 28 explicit $t$, and does \\emph{not} establish $P(t)$ for infinitely many $t$ \u2014 quantifier order is exact: a finite witnessed list, not a cofinal supply.\n\\par\\noindent\\ev{Cert} \\scale{fixed} \\coord{other:lcm-diagonal} \\lean{certifiedKill\\_diagonal\\_all\\_imported}{DiagonalPincerCertificates.lean:2842} \\lean{diagonalPincerKillDepth}{DiagonalPincerCertificates.lean:2870} \\lean{certifiedKill\\_diagonal\\_t64}{DiagonalPincerCertificateT64Endpoint.lean:1928}\n\\end{prop}\n\n\\subsection{Converters and exact identities}\n\nConverters and identities move a statement between coordinates without changing its\ntruth value: definitions, exact algebraic identities, iff-characterisations, and the\nimplication theorems that convert a supply hypothesis in one coordinate into the target\nconclusion $\\mathrm{Irrational}(S)$. This subsection carries the certificate-kernel core\n(the \\texttt{Sep}(h,N,L) predicate, its completeness iff, and every known reformulation\nof the open supply obligation) and the M\\\"obius--Mersenne / squared-Lambert identity\nfamily, including the Mersenne--Lambert five-row status ladder. Sorted by scale:\nfoundational (n/a) definitions and identities first, then uniform, then cofinal, then\nbounded, then fixed.\n\n\\subsubsection{Foundational definitions and identities (scale n/a)}\n\n\\begin{defn}[cert:a1 --- \\texttt{totientTail}]\n$\\mathrm{totientTail}(N) := \\sum_{j\\ge0}' \\varphi(N+1+j)/2^{j+1}$, the fractional layer of $2^N\\cdot S$; well-defined for every $N$.\n\\par\\noindent\\ev{Lean} \\scale{n/a} \\coord{binary-digit} \\lean{totientTail}{TotientTailPeriodKiller.lean:57}\n\\end{defn}\n\n\\begin{defn}[cert:a3 --- \\texttt{windowDiscrepancy}]\n$\\mathrm{windowDiscrepancy}(h,N,L) := \\sum_{j<L} \\big(\\varphi(N+h+1+j) - \\varphi(N+1+j)\\big)\\cdot 2^{L-1-j} \\in \\mathbb{Z}$, the depth-$L$ truncation of $2^L\\cdot(\\mathrm{totientTail}(N+h) - \\mathrm{totientTail}(N))$; computable and decidable given $h,N,L$.\n\\par\\noindent\\ev{Lean} \\scale{n/a} \\coord{other:binary-window} \\lean{windowDiscrepancy}{TotientTailPeriodKiller.lean:66}\n\\end{defn}\n\n\\begin{defn}[cert:a4 --- \\texttt{certifiedKill}, THE Sep(h,N,L) PREDICATE]\n$\\mathrm{certifiedKill}(h,N,L) := (N+h+L+2:\\mathbb{Z}) < \\mathrm{windowDiscrepancy}(h,N,L) \\bmod 2^L < 2^L - (N+h+L+2)$ \u2014 the residue of the window discrepancy modulo $2^L$ avoids the shrinking radius-$(N+h+L+2)$ neighbourhood of $0$. This is exactly the object named $\\mathrm{Sep}(h,N,L)$: purely finite arithmetic, decidable, with no analytic hypothesis. This is the certificate kernel's core object; every theorem below is either a hypothesis-shape wrapping it or an unconditional finite instance of it.\n\\par\\noindent\\ev{Lean} \\scale{n/a} \\coord{other:binary-window} \\lean{certifiedKill}{TotientTailPeriodKiller.lean:72}\n\\end{defn}\n\n\\begin{prop}[cert:a2 --- shift identity, \\texttt{two\\_pow\\_mul\\_totient\\_series\\_eq}]\n$2^N \\cdot \\Big(\\sum_{n\\ge0}' \\varphi(n)/2^n\\Big) = \\mathrm{totientPrefix}(N) + \\mathrm{totientTail}(N)$, where $\\mathrm{totientPrefix}(N) = \\sum_{n\\le N}\\varphi(n)\\cdot 2^{N-n} \\in \\mathbb{N}$ is an exact integer. Rationality/periodicity of $S$'s binary expansion reduces exactly to a statement about $\\mathrm{totientTail}$. Template shape (constant $\\cdot$ prefix + tail split) reusable for any $\\sum f(n)/2^n$.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{binary-digit} \\lean{two\\_pow\\_mul\\_totient\\_series\\_eq}{TotientTailPeriodKiller.lean:150}\n\\end{prop}\n\n\\begin{lem}[cert:a5 --- certificate depth floor, \\texttt{certifiedKill\\_depth\\_floor}]\n$\\mathrm{certifiedKill}(h,N,L) \\to 2\\cdot(N+h+L+2) < 2^L$. A structural corollary of the kernel definition: the certificate depth $L$ cannot stay bounded while $N+h \\to \\infty$; $L$ must grow at least logarithmically with $N+h$. Necessary context for reading every fixed/bounded-scale kernel instance below.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:binary-window} \\lean{certifiedKill\\_depth\\_floor}{TotientTailPeriodKiller.lean:79}\n\\end{lem}\n\n\\begin{prop}[cert:a6 --- certificate soundness, the kernel's converter direction, \\texttt{tail\\_diff\\_notMem\\_int\\_of\\_certifiedKill}]\n$\\mathrm{certifiedKill}(h,N,L) \\to \\mathrm{totientTail}(N+h) - \\mathrm{totientTail}(N) \\notin \\mathrm{range}((\\uparrow):\\mathbb{Z}\\to\\mathbb{R})$. This is the kernel's soundness converter: it moves a finite, decidable, binary-window fact into a real-analytic non-integrality fact. Composes with cert:a9 (tail-period law) by contradiction to kill a hypothetical rational's period.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:binary-window} \\lean{tail\\_diff\\_notMem\\_int\\_of\\_certifiedKill}{TotientTailPeriodKiller.lean:262}\n\\end{prop}\n\n\\begin{thm}[cert:a7 --- certificate completeness, \\texttt{exists\\_certifiedKill\\_iff\\_tail\\_diff\\_notMem\\_int}]\n$(\\exists L,\\ \\mathrm{certifiedKill}(h,N,L)) \\iff \\mathrm{totientTail}(N+h) - \\mathrm{totientTail}(N) \\notin \\mathrm{range}((\\uparrow):\\mathbb{Z}\\to\\mathbb{R})$, for all $h,N$. Certificates are \\emph{complete} receipts of non-integrality, not merely sufficient: the certificate vocabulary is dispensable, and the real target of the whole kernel is exactly the right-hand real-analytic statement. Converts the open \\#249 obligation freely between \"supply of certificates\" language and \"supply of non-integral tail differences at arbitrarily large scale\" language (cert:b8 below uses the latter form). Any independent non-integrality proof from any coordinate automatically yields a certificate by this iff, and vice versa.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:binary-window} \\lean{exists\\_certifiedKill\\_iff\\_tail\\_diff\\_notMem\\_int}{LcmConeFlatness.lean:316}\n\\end{thm}\n\n\\begin{defn}[cert:b1 --- \\texttt{periodLcm}, the universal period ray]\n$\\mathrm{periodLcm}(0) = 1$, $\\mathrm{periodLcm}(t+1) = \\mathrm{lcm}(\\mathrm{periodLcm}(t), t+1)$, i.e.\\ $\\mathrm{periodLcm}(t) = \\mathrm{lcm}(1,\\dots,t)$; $t \\le \\mathrm{periodLcm}(t)$; every primitive period $h_0 \\le t$ divides $\\mathrm{periodLcm}(t)$. Pure number theory about $\\mathrm{lcm}(1..t)$, zero totient content; directly reusable for \\#257 or any period-search problem \u2014 \"stand on the universal-period ray to remove one free parameter\" is a general reduction technique.\n\\par\\noindent\\ev{Lean} \\scale{n/a} \\coord{other:lcm-period-ray} \\lean{periodLcm}{CarrySurvivorExtinction.lean:515} \\lean{le\\_periodLcm}{CarrySurvivorExtinction.lean:81}\n\\end{defn}\n\n\\begin{prop}[cert:c1 --- \\texttt{farey\\_gap}, the mediant lemma]\nFor $a,b,c,d,r,s:\\mathbb{Z}$ with $b>0$, $d>0$, unimodular neighbours $bc-ad=1$, and $r$ strictly between $a/b$ and $c/d$ ($a\\cdot s < r\\cdot b$, $r\\cdot d < c\\cdot s$): $b+d \\le s$. Any rational strictly between two unimodular Farey neighbours has denominator at least the sum of the neighbours' denominators. Fully problem-agnostic \u2014 pure Stern--Brocot/Farey fact, zero totient content, directly reusable for \\#257.\n\\par\\noindent\\ev{Lean} \\scale{n/a} \\coord{farey} \\lean{farey\\_gap}{GapFareyBound.lean:51}\n\\end{prop}\n\n\\begin{prop}[cert:d1 --- \\texttt{irrational\\_of\\_den\\_mul\\_abs\\_sub\\_tendsto\\_zero}, generic Dirichlet-gap criterion]\nIf $u:\\mathbb{N}\\to\\mathbb{Q}$ is eventually never equal to $x:\\mathbb{R}$, and $\\mathrm{den}(u(k))\\cdot|x - u(k)| \\to 0$, then $\\mathrm{Irrational}(x)$. Classical Dirichlet-approximation irrationality criterion, zero totient/Mersenne content, already reused in this corpus to prove full-support Erd\\H{o}s--Borwein irrationality (\\#257-shaped). Apply to any explicit sequence of convergents to $S$ with a provable denominator$\\cdot$gap $\\to 0$ bound.\n\\par\\noindent\\ev{Lean} \\scale{n/a} \\coord{n/a} \\lean{irrational\\_of\\_den\\_mul\\_abs\\_sub\\_tendsto\\_zero}{CertificateKernel.lean:5371}\n\\end{prop}\n\n\\begin{prop}[cert:d2 --- \\texttt{irrational\\_of\\_int\\_mul\\_near\\_int}, classical near-integer criterion]\nIf $\\forall q>0,\\ \\exists m,z:\\mathbb{Z},\\ 0 < |m\\cdot\\xi - z| < 1/q$, then $\\mathrm{Irrational}(\\xi)$. The classical Erd\\H{o}s-1948-shape criterion behind digit/carry irrationality proofs. Base-power specialisation $\\mathrm{irrational\\_of\\_pow\\_mul\\_near\\_int}$ (witnesses of form $b^n\\cdot\\xi$) directly matches a digit/carry construction in any base $b$, hence directly usable for \\#257's $b^n-1$ denominators. Together with cert:d1 these are the only two general-purpose irrationality criteria in the whole kernel; everything else exists to supply or substitute for their hypotheses.\n\\par\\noindent\\ev{Lean} \\scale{n/a} \\coord{n/a} \\lean{irrational\\_of\\_int\\_mul\\_near\\_int}{CertificateKernel.lean:5792} \\lean{irrational\\_of\\_pow\\_mul\\_near\\_int}{CertificateKernel.lean:5821}\n\\end{prop}\n\n\\begin{prop}[cert:d3 --- \\texttt{SeparatedMinorCertificate}, linear independence from a nonzero minor]\nFor an indexed family $\\mathrm{family}: \\iota\\to\\mathbb{N}\\to\\mathbb{Q}$: a $\\mathrm{SeparatedMinorCertificate}$ (an explicit finite evaluation-point assignment $\\mathrm{rowIndex}:\\iota\\to\\mathbb{N}$ with $\\det(\\mathrm{family}(j)(\\mathrm{rowIndex}(i)))_{i,j} \\ne 0$) implies $\\mathrm{LinearIndependent}\\ \\mathbb{Q}\\ \\mathrm{family}$. Fully generic finite-dimensional linear algebra; $\\mathrm{family}$ is a free variable, nothing here mentions totient or Mersenne structure. Reusable for any finite-rank/linear-independence obstruction in either open problem.\n\\par\\noindent\\ev{Lean} \\scale{n/a} \\coord{n/a} \\lean{SeparatedMinorCertificate}{TotientMahlerDefect.lean:73} \\lean{linearIndependent\\_of\\_separatedMinorCertificate}{TotientMahlerDefect.lean:81}\n\\end{prop}\n\n\\begin{prop}[cert:d9 --- \\texttt{positive\\_rational\\_difference\\_lower\\_bound}]\nFor $\\mathrm{pfx} < \\mathrm{whole}:\\mathbb{Q}$ (strict): $1/(\\mathrm{whole.den}\\cdot\\mathrm{pfx.den}) \\le (\\mathrm{whole}:\\mathbb{R}) - (\\mathrm{pfx}:\\mathbb{R})$. A positive rational difference is bounded below by the reciprocal of the product of the two \\emph{actual} reduced denominators, not a displayed/guessed one. Fully general; feeds \\texttt{prefixDenominator\\_shell\\_power\\_bound\\_of\\_rational\\_difference} and \\texttt{nextSupport\\_power\\_bound\\_of\\_rational\\_difference}, converting any analytic upper bound on a rational gap into a denominator-growth lower bound \u2014 plug in any tail estimate from either open problem's coordinate.\n\\par\\noindent\\ev{Lean} \\scale{n/a} \\coord{n/a} \\lean{positive\\_rational\\_difference\\_lower\\_bound}{PrimitiveRationalGapSupply.lean:31}\n\\end{prop}\n\n\\begin{prop}[mob:a1a --- the M\\\"obius--squared-Mersenne identity]\n$S = \\sum_{n\\ge1} \\varphi(n)/2^n = \\sum_{d\\ge1}' \\mu(d)\\cdot 2^d/(2^d-1)^2 = \\tfrac12 + \\sum_{d\\ge1}' \\mu(d)/(2^d-1)^2$. Unconditional identity of convergent real series; this is the reduced target every M\\\"obius--Mersenne-lane result in this catalogue feeds.\n\\par\\noindent\\ev{Lean} \\scale{n/a} \\coord{mobius-mersenne} \\lean{tsum\\_totient\\_half\\_pow\\_eq\\_half\\_add\\_moebius\\_sq}{SquaredMersenneDiagonalEnclosure.lean:121} \\lean{totientSeries\\_eq\\_pnat\\_half\\_pow}{SquaredMersenneDiagonalEnclosure.lean:50} \\lean{mobius\\_square\\_series\\_eq\\_partial\\_add\\_tail}{SquaredMersenneDiagonalEnclosure.lean:72}\n\\end{prop}\n\n\\begin{cor}[mob:a1b --- \\#249 restated in the squared-Lambert coordinate]\nImmediate from mob:a1a: $\\mathrm{Irrational}(S) \\iff \\sum_{d\\ge1}' \\mu(d)/(2^d-1)^2 \\notin \\mathbb{Q}$. Denote the right-hand series $L_2(\\mu) := S - 1/2$ (mob:a3 below). Every subsequent M\\\"obius-coordinate obstruction or identity in this catalogue is measured against this exact restatement, not the original totient series.\n\\par\\noindent\\ev{Lean} \\scale{n/a} \\coord{mobius-mersenne} \\lean{tsum\\_totient\\_half\\_pow\\_eq\\_half\\_add\\_moebius\\_sq}{SquaredMersenneDiagonalEnclosure.lean:121}\n\\end{cor}\n\n\\begin{prop}[mob:a2 --- squared-Lambert transfer engine, \\texttt{tsum\\_lambert\\_linear\\_weight\\_sq\\_pure}]\nFor $w:\\mathbb{N}\\to\\mathbb{R}$ with $|w(d)| \\le d$ for $d>0$, and $0 \\le r < 1$: $\\sum_{d:\\mathbb{N}^+}' w(d)\\cdot(r^d/(1-r^d))^2 = \\sum_{n:\\mathbb{N}^+}' \\Big(\\sum_{e\\mid n} w(e)\\cdot(n/e-1)\\Big)\\cdot r^n$. Converts any linear-growth-bounded squared-Lambert series into a divisor-convolution power series \u2014 the single reusable brick for the whole level-2 M\\\"obius--Lambert ladder below. Pure Dirichlet-convolution/Lambert-series algebra, no Mersenne-specific structure; instantiate at $w = \\mu, 1, \\varphi$ to obtain mob:a3--a5, and directly reusable for a weighted or squared variant of \\#257's series.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{mobius-mersenne} \\lean{tsum\\_lambert\\_linear\\_weight\\_sq\\_pure}{GcdMomentCalculus.lean:105}\n\\end{prop}\n\n\\begin{defn}[mob:a3 --- $L_2(\\mu)$ is exactly \\#249]\n$L_2(\\mu) := \\sum_{d:\\mathbb{N}^+}' \\mu(d)/(2^d-1)^2 = S - 1/2$. The open \\#249 atom, restated as the M\\\"obius rung of the level-2 (squared) Lambert ladder; consequence of mob:a1a and mob:a2 at $w=\\mu$.\n\\par\\noindent\\ev{Cited} \\scale{n/a} \\coord{mobius-mersenne} \\lean{tsum\\_totient\\_half\\_pow\\_eq\\_half\\_add\\_moebius\\_sq}{SquaredMersenneDiagonalEnclosure.lean:121}\n\\end{defn}\n\n\\begin{prop}[mob:a4 --- $L_2(1)$, the known $q$-zeta anchor rung, \\texttt{tsum\\_one\\_div\\_mersenne\\_sq\\_eq\\_sigma\\_sub\\_tau\\_series}]\n$\\sum_{d:\\mathbb{N}^+}' 1/(2^d-1)^2 = \\sum_{n:\\mathbb{N}^+}' \\big(\\sigma(n)-\\tau(n)\\big)\\cdot(1/2)^n = \\zeta_q(2) - \\zeta_q(1)$ at $q=1/2$. The level-2 $\\zeta$-rung, exactly evaluated in Lean. Irrationality of this \\emph{value} is cited (Postelmans--Van Assche $q$-Pad\\'e), \\emph{not} formalised \u2014 only the identity itself is machine-checked. This is the \"known\" sibling rung mirroring mob:a3 in the level-mirror table mob:a6.\n\\par\\noindent\\ev{Lean} \\scale{n/a} \\coord{mobius-mersenne} \\lean{tsum\\_one\\_div\\_mersenne\\_sq\\_eq\\_sigma\\_sub\\_tau\\_series}{GcdMomentCalculus.lean:216}\n\\end{prop}\n\n\\begin{prop}[mob:a5 --- $L_2(\\varphi)$, the Pillai gcd-moment rung, \\texttt{tsum\\_totient\\_div\\_mersenne\\_sq\\_eq\\_gcd\\_moment\\_series}]\n$\\sum_{d:\\mathbb{N}^+}' \\varphi(d)/(2^d-1)^2 = \\sum_{n:\\mathbb{N}^+}' (P(n)-n)\\cdot(1/2)^n$, where $P = \\varphi * \\mathrm{Id}$ (Pillai's gcd-sum function). Equals $\\mathbb{E}[\\gcd(X,Y)]$ for independent fair-coin waiting times $X,Y$ (probabilistic coordinate, mob:a7--a9). A cousin rung, not \\#249 itself; status open.\n\\par\\noindent\\ev{Lean} \\scale{n/a} \\coord{mobius-mersenne} \\lean{tsum\\_totient\\_div\\_mersenne\\_sq\\_eq\\_gcd\\_moment\\_series}{GcdMomentCalculus.lean:235}\n\\end{prop}\n\n\\begin{obs}[mob:a6 --- the level-mirror table]\nLevel 1 ($L(f) := \\sum f(d)/(2^d-1)$): $L(\\mu) = 1/2$ (rational, trivial); $L(1) = \\mathcal{E}$, the Erd\\H{o}s--Borwein constant (irrational, Erd\\H{o}s 1948, machine-checked in this kernel, cert:d7c below); $L(\\varphi) = 2$ (rational). Level 2 ($L_2(f) := \\sum f(d)/(2^d-1)^2$): $L_2(\\mu) = S - 1/2$ (mob:a3, OPEN, $=$ \\#249); $L_2(1) = \\zeta_q(2)-\\zeta_q(1)$ (mob:a4, irrational, cited); $L_2(\\varphi) = \\mathbb{E}[\\gcd]$ (mob:a5, open, Pillai). At level 1 the M\\\"obius rung is trivial and the $\\zeta$-rung is hard; at level 2 the $\\zeta$-rung is known and the M\\\"obius rung \\emph{is} \\#249 \u2014 M\\\"obius projection is the single wall at both levels. This mirror-structure phenomenon is a general observation about Dirichlet-convolution ladders, potentially informative for \\#257's Mersenne-shifted sums too.\n\\par\\noindent\\ev{Lean} \\scale{n/a} \\coord{mobius-mersenne}\n\\end{obs}\n\n\\begin{prop}[mob:a7 --- gcd-divisibility factorises, \\texttt{tsum\\_pos\\_pair\\_both\\_dvd\\_half\\_eq\\_inv\\_mersenne\\_sq}]\nFor independent fair-coin waiting times $X,Y$ ($P(X=n)=2^{-n}$) and $d>0$: $P(d\\mid X \\wedge d\\mid Y) = 1/(2^d-1)^2$. The foundation stone for reading $L_2(f)$ as $\\mathbb{E}[(f*\\zeta)(\\gcd(X,Y))]$. Pure probability/geometric-series fact about independent geometric random variables, zero \\#249-specific content; directly reusable for a two-coordinate gcd structure in \\#257.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{probability} \\lean{tsum\\_pos\\_pair\\_both\\_dvd\\_half\\_eq\\_inv\\_mersenne\\_sq}{GcdMomentCalculus.lean:266}\n\\end{prop}\n\n\\begin{prop}[mob:a8 --- reduced-direction law, \\texttt{tsum\\_pos\\_coprime\\_inv\\_mersenne\\_eq\\_one}]\n$\\sum_{(a,b):\\, a,b\\ge1,\\, \\gcd(a,b)=1}' 1/(2^{a+b}-1) = 1$: the sum over exact bipartite coprime pairs. Every positive coprime pair carries a slope mass $1/(2^{a+b}-1)$, and these mass exactly one \u2014 the root cylinder $M(1,1)=1$ of the Stern--Brocot tree, mob:a9. Pure coprimality/geometric-series law, applies verbatim to any base-$b$ analogue.\n\\par\\noindent\\ev{Lean} \\scale{n/a} \\coord{probability} \\lean{tsum\\_pos\\_coprime\\_inv\\_mersenne\\_eq\\_one}{GcdMomentCalculus.lean:349}\n\\end{prop}\n\n\\begin{prop}[mob:a9a --- Stern--Brocot cylinder recursion, \\texttt{cylinderMass\\_split}]\n$M(a,b) := 1/\\big((2^a-1)(2^b-1)\\big)$ satisfies the exact telescoping identity $M(a,b) = 1/(2^{a+b}-1) + M(a+b,b) + M(a,a+b)$ for $a,b:\\mathbb{N}^+$: an exact Markov measure on the Stern--Brocot mediant tree, whose closed form is a product of two independent-coordinate divisor probabilities $P(a\\mid X)\\cdot P(b\\mid Y)$ (mob:a7).\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:stern-brocot} \\lean{cylinderMass\\_split}{GcdMomentCalculus.lean:474}\n\\end{prop}\n\n\\begin{prop}[mob:a9b --- depth-$d$ convergence rate, \\texttt{sternBrocotDepthMass\\_error} / \\texttt{tendsto\\_sternBrocotDepthMass}]\nChildren of a Stern--Brocot cylinder carry at most $2/3$ of the parent's mass; the depth-$d$ finite unfolding of mob:a9a converges to $M(a,b)$ at explicit rate $|M(a,b) - M_d(a,b)| \\le (2/3)^d\\cdot M(a,b)$. The \"recursion limit $=$ subtree mass\" identification is explicitly \\emph{not} claimed beyond this rate bound (flagged as a wave-21 gap in the source). All-left cusp cylinders $M(N,1) = 1/(2^N-1)$ are near-Mersenne-reciprocal, one exponential order closer to the Erd\\H{o}s--Borwein engine than raw $\\varphi(n)$ coefficients \u2014 an advisory bridge candidate to Erd\\H{o}s-1948-style arguments, not a proved route.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:stern-brocot} \\lean{sternBrocotDepthMass\\_error}{GcdMomentCalculus.lean:525} \\lean{tendsto\\_sternBrocotDepthMass}{GcdMomentCalculus.lean:560}\n\\end{prop}\n\n\\begin{prop}[mob:b1 --- repunit gcd word, T1, \\texttt{mobiusNumeratorPolynomial\\_eq\\_gcdWord}]\nFor $r$ squarefree, the divisor-signed polynomial $\\mathrm{mobiusNumeratorPolynomial}(r) := \\sum_{d\\mid r} \\mu(d)\\cdot(r/d)\\cdot\\mathrm{spacedRepunit}(d,r/d)$ equals $\\mathrm{gcdWord}(r)$, whose $X^k$ coefficient ($k<r$) is $\\mathrm{gcdWordCoeff}(r,k) = (r/\\gcd(r,k))\\cdot\\varphi(\\gcd(r,k))$, strictly positive for $k<r$ and zero for $k\\ge r$. The signed repunit numerator is exactly a positive gcd word; the classical polynomial identity underlying it (signed spaced repunit $=$ gcd word) is base-independent, though the application here is Mersenne-specific.\n\\par\\noindent\\ev{Lean} \\scale{n/a} \\coord{cyclotomic} \\lean{mobiusNumeratorPolynomial\\_eq\\_gcdWord}{RepunitMobiusNumerator.lean:205} \\lean{mobiusNumeratorPolynomial\\_coeff}{RepunitMobiusNumerator.lean:217} \\lean{mobiusNumeratorPolynomial\\_coeff\\_pos}{RepunitMobiusNumerator.lean:234}\n\\end{prop}\n\n\\begin{prop}[mob:b2 --- evaluation at $2$ recovers the integer numerator, \\texttt{mobiusNumeratorPolynomial\\_eval\\_two}]\n$\\mathrm{mobiusNumeratorPolynomial}(r).\\mathrm{eval}\\,2 = \\mathrm{mobiusNumerator}(r)$ for $r$ squarefree, where $\\mathrm{mobiusNumerator}(r) := \\sum_{s \\subseteq \\mathrm{primeFactors}(r)} (-1)^{|s|}\\cdot(r/d)\\cdot(\\mathrm{mersenne}(r)/\\mathrm{mersenne}(d))$, $d = \\prod s$. Bridges the polynomial (mob:b1) world to the integer arithmetic used by every denominator-survival theorem below.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{cyclotomic} \\lean{mobiusNumeratorPolynomial\\_eval\\_two}{RepunitMobiusNumerator.lean:320} \\lean{mobiusNumerator}{RadicalMobiusShadow.lean:101}\n\\end{prop}\n\n\\begin{prop}[mob:b3 --- radical shadow scale decomposition]\n$\\mathrm{baseMobiusShadow}(r) := \\mathrm{mobiusNumerator}(r)/(2^r-1)$ (unscaled); $\\mathrm{numericMobiusShadow}(H) := \\mathrm{baseMobiusShadow}(\\mathrm{rad}(H))/\\mathrm{rad}(H)$; exactly $H\\cdot\\mathrm{numericMobiusShadow}(H) = (H/\\mathrm{rad}(H))\\cdot\\mathrm{baseMobiusShadow}(\\mathrm{rad}(H))$ for $H>0$. Reduced-denominator identity for the unscaled shadow: $\\mathrm{baseMobiusShadow}(r).\\mathrm{den} = \\mathrm{mersenne}(r)/\\gcd(|\\mathrm{mobiusNumerator}(r)|, \\mathrm{mersenne}(r))$ for $r>0$ \u2014 exact and generic, no coprimality assumed, no channel-survival hidden.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{mobius-mersenne} \\lean{scaledMobiusShadow\\_eq\\_radicalBase}{RadicalMobiusShadow.lean:125} \\lean{baseMobiusShadow\\_den}{RadicalMobiusShadow.lean:150}\n\\end{prop}\n\n\\begin{prop}[mob:b4 --- cyclotomic congruence, T2, \\texttt{cyclotomic\\_dvd\\_mobiusNumeratorPolynomial\\_sub}]\nFor $r$ squarefree, $m \\mid r$: $\\Phi_m \\mid \\big(\\mathrm{mobiusNumeratorPolynomial}(r) - C(\\mu(m)\\cdot J_2(r/m))\\big)$ in $\\mathbb{Z}[X]$, where $J_2 = \\mu * \\mathrm{id}^2$ is the Jordan totient. Evaluated: $\\mathrm{cyclotomicEval}(m) \\mid \\mathrm{mobiusNumerator}(r) - \\mu(m)\\cdot J_2(r/m)$. Every cyclotomic fibre $\\Phi_m(2)$ of the numerator is congruent to an explicit constant depending only on $\\mu(m)$ and the Jordan totient of the cofactor. Setting $m=r$ gives mob:b5 (top fibre survives).\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{cyclotomic} \\lean{cyclotomic\\_dvd\\_mobiusNumeratorPolynomial\\_sub}{CyclotomicProjectionOfShadow.lean:299} \\lean{mobiusNumerator\\_mod\\_cyclotomicEval}{CyclotomicProjectionOfShadow.lean:310}\n\\end{prop}\n\n\\begin{prop}[mob:b8a --- prime-power jump recurrence, T5, new fibre]\nAdjoining a new prime $p \\nmid r$: on a genuinely new fibre $m\\cdot p$ (with $m \\mid r$), $\\Phi_{mp} \\mid \\mathrm{mobiusNumeratorPolynomial}(rp) + \\mathrm{expand}_p(\\mathrm{mobiusNumeratorPolynomial}(r))$ \u2014 the new fibre picks up a sign-flipped $p$-expansion of the old numerator. Holds for $p$ prime, $p \\nmid r$, $m \\mid r$; the underlying recurrence needs no squarefreeness of $r$.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{cyclotomic} \\lean{cyclotomic\\_dvd\\_primeJump\\_new\\_fibre}{PrimePowerJumpDynamics.lean:281} \\lean{cyclotomic\\_dvd\\_primeJump\\_new\\_fibre\\_constant}{PrimePowerJumpDynamics.lean:347}\n\\end{prop}\n\n\\begin{prop}[mob:b8b --- prime-power jump recurrence, T5, old fibre]\nAdjoining a new prime $p \\nmid r$: on an old fibre $m$ (with $m \\mid r$), $\\Phi_m \\mid \\mathrm{mobiusNumeratorPolynomial}(rp) - C(p^2-1)\\cdot\\mathrm{mobiusNumeratorPolynomial}(r)$ \u2014 old fibres scale by exactly $p^2-1$. Requires $r$ squarefree (needed for the T2 constant-fibre step mob:b4, not for the underlying recurrence itself). An inductive tool for building explicit denominator/numerator values one prime at a time; a candidate base case for an induction proving mob:b7a/b7b's growth bound tight.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{cyclotomic} \\lean{cyclotomic\\_dvd\\_primeJump\\_old\\_fibre}{PrimePowerJumpDynamics.lean:310}\n\\end{prop}\n\n\\begin{prop}[cert:d7 --- the Mersenne--Lambert ladder identities]\nDefine $L(f) := \\sum_{n\\ge1} f(n)/(2^n-1)$ (level 1, no square). Writing $A := \\varphi*\\mu$ (the primitive Euler weight, $\\alpha$ in prose): the Dirichlet-convolution identities $A*\\zeta = \\varphi$ and $\\varphi*\\zeta = \\mathrm{Id}$ separate five exact values of $L$ along one ladder, each obtained by factoring $L(f)$ as a divisor-transform $(f*1)$ followed by binary evaluation, $L(f) = \\sum_n (f*1)(m)/2^m$. $S$ sits inside this ladder as $L(A)$: this is \\#249 restated in positive Erd\\H{o}s--Borwein form.\n\\par\\noindent\\ev{Lean} \\scale{n/a} \\coord{mobius-mersenne} \\lean{tsum\\_totient\\_div\\_pow\\_two\\_eq\\_pnat\\_half\\_pow}{CertificateKernel.lean:18087}\n\\end{prop}\n\n\\begin{longtable}{@{}l l l@{}}\n\\caption{The Mersenne--Lambert ladder five-row status table (docstring @ \\texttt{CertificateKernel.lean:18063--18083}, body \\texttt{MersenneLambertLadder.lean}).}\\\\\n\\toprule\nWeight $f$ & Value $L(f)$ & Status \\\\\n\\midrule\n\\endfirsthead\n$\\mu$ & $L(\\mu) = 1/2$ & rational, trivial \\\\\n$\\varphi$ & $L(\\varphi) = 2$ & rational \\\\\n$\\mathrm{Id}$ & $L(\\mathrm{Id}) = \\sum_m \\sigma(m)/2^m$ & transcendental (Nesterenko 1996, \\ev{Cited}, not formalised) \\\\\n$1$ & $L(1) = \\mathcal{E}$, Erd\\H{o}s--Borwein constant & irrational, \\ev{Lean} (\\texttt{irrational\\_erdosBorwein\\_series}, \\\\\n & & \\texttt{CertificateKernel.lean:8007}) \u2014 matches \\#257's full-support case \\\\\n$A = \\varphi*\\mu$ & $L(A) = S$ & \\textbf{OPEN} \u2014 this is \\#249 \\\\\n\\bottomrule\n\\end{longtable}\n\n\\begin{rem}[cert:d7 --- reading the ladder]\nThis is the single clearest bridge placing \\#249 inside \\#257's native Mersenne-Lambert coordinate ($\\sum f(n)/(b^n-1)$, here $b=2$): if the \\#257 lane's Mersenne-coordinate machinery (achievement sets, greedy orbits) can say anything about the sign or density structure of $A = \\varphi*\\mu$ specifically, it transfers here for free via this ladder identity. Conversely, any \\#249 result about $L(A)$ phrased purely in Lambert-series terms (not totient terms) is directly \\#257-lane-portable. The `evaluated' identity confirming $S$'s $\\mathbb{N}$-indexed form equals its $\\mathbb{N}^+$/Lambert-indexed form is the one row here checked directly against source (\\lean{tsum\\_totient\\_div\\_pow\\_two\\_eq\\_pnat\\_half\\_pow}{CertificateKernel.lean:18087}); the body machinery in \\texttt{MersenneLambertLadder.lean} establishing the other four rows was read only via this docstring and is flagged \\ev{Cited}/\\ev{Math} accordingly except where a specific declaration is named above.\n\\end{rem}\n\n\\begin{prop}[mob:e3 --- joint-35 cone annihilator, \\texttt{oldChannel\\_affine\\_moment\\_annihilation} / \\texttt{joint35\\_oldChannel\\_zero}]\nThe four-vertex affine annihilator $q(X,Y) = XY-3X-2Y+4$ ($q(1,1)=q(3,5)=0$, $q(9,25)=152$) kills every ``old'' divisor channel exactly at any LCM height: for $d\\mid H$, $d>0$, $\\mathrm{transportResidueKernel}(d,15H) - 3\\cdot\\mathrm{transportResidueKernel}(d,3H) - 2\\cdot\\mathrm{transportResidueKernel}(d,5H) + 4\\cdot\\mathrm{transportResidueKernel}(d,H) = 0$. General form (\\texttt{oldChannel\\_affine\\_moment\\_annihilation}): any finite affine annihilator with $\\sum c_i = 0$, $\\sum c_i\\cdot m_i = 0$ kills every old residue channel, independent of the particular $(3,5)$ choice \u2014 fully coordinate-free finite linear algebra. Sharp cone radius $19H+5L+5$ (\\texttt{sharpJoint35ConeRadius}). An unbounded certificate supply built from this (not proved) would close \\#249 via the same cone-flatness route as mob:e2/cert:b6.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:lcm-diagonal} \\lean{oldChannel\\_affine\\_moment\\_annihilation}{JointExponentTransport.lean:36} \\lean{joint35\\_oldChannel\\_zero}{JointExponentTransport.lean:71} \\lean{sharpJoint35ConeRadius}{JointExponentTransport.lean:124}\n\\end{prop}\n\n\\begin{prop}[mob:f1 --- composite dilation defect identity, \\texttt{supportCoeff\\_mul\\_eq\\_add\\_defect}]\nFor an abstract support set $A\\subseteq\\mathbb{N}$, $\\mathrm{supportCoeff}_A(n) := \\#\\{a\\in A: a\\mid n\\}$: for $a\\in A$, $a>0$, $x>0$, $\\mathrm{supportCoeff}_A(ax) = \\mathrm{supportCoeff}_A(x) + [a\\nmid x] + \\mathrm{compositeDilationDefect}_A(a,x)$. On prime-only support (every element of $A$ prime) the defect is identically zero, recovering the classical $\\mathrm{supportCoeff}_A(px) = \\mathrm{supportCoeff}_A(x) + [p\\nmid x]$. Pure divisor combinatorics over an abstract support set, no $2^n$-specific structure \u2014 a shared \\#249/\\#257 substrate identity: instantiate $A=\\mathbb{N}$ weighted by $\\varphi$ for \\#249, or $A$ the chosen infinite subset for \\#257's $\\sum_{n\\in A}1/(2^n-1)$.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:support-divisor-counting} \\lean{supportCoeff\\_mul\\_eq\\_add\\_defect}{CompositeDilationDefect.lean:30} \\lean{compositeDilationDefect\\_eq\\_zero\\_of\\_prime\\_support}{CompositeDilationDefect.lean:103} \\lean{supportCoeff\\_mul\\_prime\\_support}{CompositeDilationDefect.lean:119}\n\\end{prop}\n\n\\subsubsection{Cofinal-scale converters \u2014 the certificate-kernel reduction ladder}\n\nEvery entry in this block is an implication whose \\emph{conclusion} is\n$\\mathrm{Irrational}(S)$ (\\#249 solved) and whose \\emph{hypothesis} is a cofinal supply\npredicate. \\textbf{None of these hypotheses is supplied anywhere in the corpus.} Each\ntheorem itself is \\ev{Lean}-checked; the hypothesis it consumes carries \\ev{Open}. These\nare listed here, not as producers, because each one is exactly a converter: it moves an\nunsupplied cofinal statement in the binary-window/lcm-cone coordinate across to the\ntarget statement about $S$, and (via cert:a7's iff) several of them are genuine\nlogical equivalences, not mere one-directional implications.\n\n\\begin{thm}[cert:a10 --- \\texttt{irrational\\_totient\\_series\\_of\\_certificate\\_supply}, WAVE-21 REDUCTION, THE WALL]\n$\\big(\\forall h:\\mathbb{N},\\ 0<h \\to \\forall N_0:\\mathbb{N},\\ \\exists N\\ge N_0,\\ \\exists L,\\ \\mathrm{certifiedKill}(h,N,L)\\big) \\to \\mathrm{Irrational}(S)$. The hypothesis is exactly $\\mathrm{Sep}(h,N,L)$ quantified as $\\forall h\\ge1\\ \\forall N_0\\ge0\\ \\exists N\\ge N_0\\ \\exists L$. \\textbf{This hypothesis is nowhere proved in the corpus; this is the central open supply obligation.} Every other reduction theorem in this block is a re-derivation of an equivalent-or-weaker sufficient condition for the same missing supply, never a proof of it.\n\\par\\noindent\\ev{Lean} (implication proved; hypothesis \\ev{Open}) \\scale{cofinal} \\coord{other:binary-window} \\lean{irrational\\_totient\\_series\\_of\\_certificate\\_supply}{TotientTailPeriodKiller.lean:394}\n\\end{thm}\n\n\\begin{thm}[cert:b2 --- multiple-period collapse, wave-22, \\texttt{irrational\\_totient\\_series\\_of\\_multiple\\_certificate\\_supply}]\n$\\big(\\forall h_0>0,\\ \\forall N_0,\\ \\exists m>0,\\ \\exists N\\ge N_0,\\ \\exists L,\\ \\mathrm{certifiedKill}(m\\cdot h_0, N, L)\\big) \\to \\mathrm{Irrational}(S)$. Weaker hypothesis than cert:a10: for every primitive period it suffices to certify \\emph{some} multiple. Proof idea (\\texttt{tail\\_diff\\_mul}, telescoping an $m\\cdot h$-difference into a sum of $m$ shifted $h$-differences) is problem-agnostic; logically equivalent-strength weakening of cert:a10, still open.\n\\par\\noindent\\ev{Lean} (implication proved; hypothesis \\ev{Open}) \\scale{cofinal} \\coord{other:lcm-period-multiple} \\lean{irrational\\_totient\\_series\\_of\\_multiple\\_certificate\\_supply}{CarrySurvivorExtinction.lean:502}\n\\end{thm}\n\n\\begin{thm}[cert:b3 --- lcm-diagonal collapse, wave-23, the canonical single-parameter form, \\texttt{irrational\\_totient\\_series\\_of\\_lcm\\_diagonal\\_certificate\\_supply}]\n$\\big(\\forall t_0:\\mathbb{N},\\ \\exists t\\ge t_0,\\ \\exists L,\\ \\mathrm{certifiedKill}(\\mathrm{periodLcm}(t), \\mathrm{periodLcm}(t), L)\\big) \\to \\mathrm{Irrational}(S)$. Standing at $N=h=\\mathrm{periodLcm}(t)$ beats every hypothetical rational simultaneously (any $t \\ge \\max(h_0,N_0)$ works for both parameters at once). \\textbf{This is the canonical restatement with only one quantified unknown $t$}, and the best target for search/heuristics; the finite floor is cert:b11 above (28 explicit values, not a cofinal supply).\n\\par\\noindent\\ev{Lean} (implication proved; hypothesis \\ev{Open}) \\scale{cofinal} \\coord{other:lcm-diagonal} \\lean{irrational\\_totient\\_series\\_of\\_lcm\\_diagonal\\_certificate\\_supply}{LcmDiagonalReduction.lean:112}\n\\end{thm}\n\n\\begin{lem}[cert:b4 --- lcm-window structure, \\texttt{eq\\_prime\\_pow\\_of\\_not\\_dvd\\_periodLcm}]\nFor $0<j<2t$: if $j \\nmid \\mathrm{periodLcm}(t)$ then $\\exists$ prime $p,k$ with $j=p^k \\wedge t<j$ \u2014 below $2t$, every non-divisor of $\\mathrm{lcm}(1..t)$ is a bare prime power exceeding $t$. Pure elementary number theory about $\\mathrm{lcm}(1..t)$; structural input for anyone trying to search for a diagonal certificate \u2014 narrows where the \"noise\" in $\\varphi$ on the window comes from.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:lcm-window} \\lean{eq\\_prime\\_pow\\_of\\_not\\_dvd\\_periodLcm}{CertificateKernel.lean:18613}\n\\end{lem}\n\n\\begin{prop}[cert:b5 --- lcm-ray window totient factorisation, \\texttt{totient\\_periodLcm\\_ray\\_split}]\nOn a clean divisor $j \\mid \\mathrm{periodLcm}(t)$ (every prime factor of $j$ still divides $\\mathrm{periodLcm}(t)/j$): $\\varphi(q\\cdot\\mathrm{periodLcm}(t)+j) = \\varphi(j)\\cdot\\varphi\\big(q\\cdot(\\mathrm{periodLcm}(t)/j)+1\\big)$. An exact multiplicative split of window totient values into a known local part and a cofactor forced $\\equiv1$ modulo every exhausted prime \u2014 the exact algebraic handle needed to attempt an unconditional supply proof at cert:b3.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:lcm-window-multiplicative} \\lean{totient\\_periodLcm\\_ray\\_split}{LcmDiagonalReduction.lean:208}\n\\end{prop}\n\n\\begin{thm}[cert:b6 --- lcm-cone flatness law, wave-24, \\texttt{rational\\_totient\\_series\\_forces\\_lcm\\_cone\\_flatness}]\n$\\neg\\mathrm{Irrational}(S) \\to \\exists t_1,\\ \\forall t\\ge t_1,\\ \\forall q,m:\\mathbb{N},\\ 0<q \\to \\mathrm{totientTail}(q\\cdot\\mathrm{periodLcm}(t) + m\\cdot\\mathrm{periodLcm}(t)) - \\mathrm{totientTail}(q\\cdot\\mathrm{periodLcm}(t)) \\in \\mathrm{range}((\\uparrow):\\mathbb{Z}\\to\\mathbb{R})$. Rationality forces one fractional constant on the \\emph{entire} lcm cone $\\{k\\cdot\\mathrm{periodLcm}(t):k\\ge1\\}$ at every scale $t\\ge t_1$, not just the diagonal pair \u2014 a strict generalisation of cert:a9's hypothesis-generating side. Any certificate anywhere on the cone kills \\#249 (cert:b7).\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:lcm-cone} \\lean{rational\\_totient\\_series\\_forces\\_lcm\\_cone\\_flatness}{CertificateKernel.lean:18674}\n\\end{thm}\n\n\\begin{thm}[cert:b7 --- cone collapse, wave-24, annihilator umbrella, \\texttt{irrational\\_totient\\_series\\_of\\_lcm\\_cone\\_window\\_kill\\_supply}]\n$\\big(\\forall t_0:\\mathbb{N},\\ \\exists t\\ge t_0,\\ \\exists q,m,L:\\mathbb{N},\\ 0<q \\wedge \\mathrm{certifiedKill}(m\\cdot\\mathrm{periodLcm}(t),\\, q\\cdot\\mathrm{periodLcm}(t),\\, L)\\big) \\to \\mathrm{Irrational}(S)$. One certified kill anywhere on the two-multiplier lcm cone, at arbitrarily large $t$, suffices. Diagonal (cert:b3) is the cell $q=m=1$; $q$-ray steps are $m=1$; prime-jump pairs are $(q,m)=(1,p-1)$. The widest known target still logically equivalent-in-strength to cert:a10.\n\\par\\noindent\\ev{Lean} (implication proved; hypothesis \\ev{Open}) \\scale{cofinal} \\coord{other:lcm-cone} \\lean{irrational\\_totient\\_series\\_of\\_lcm\\_cone\\_window\\_kill\\_supply}{CertificateKernel.lean:18686}\n\\end{thm}\n\n\\begin{cor}[cert:b8 --- pure non-integrality frontier forms, no certificate vocabulary]\nDiagonal: $\\big(\\forall t_0,\\ \\exists t\\ge t_0,\\ \\mathrm{totientTail}(2\\cdot\\mathrm{periodLcm}(t)) - \\mathrm{totientTail}(\\mathrm{periodLcm}(t)) \\notin \\mathrm{range}((\\uparrow):\\mathbb{Z}\\to\\mathbb{R})\\big) \\to \\mathrm{Irrational}(S)$; cone form is the $(q,m)$ generalisation identical in shape. Via cert:a7's iff these hypotheses are \\emph{exactly} equivalent to cert:b3/cert:b7 respectively. \\textbf{The frontier of \\#249 stripped of certificate vocabulary}: does $\\mathrm{totientTail}(2H_t) - \\mathrm{totientTail}(H_t) \\notin \\mathbb{Z}$ for infinitely many $t$, $H_t = \\mathrm{lcm}(1..t)$?\n\\par\\noindent\\ev{Lean} (implication proved; hypothesis \\ev{Open}) \\scale{cofinal} \\coord{other:real-analytic-nonintegrality} \\lean{irrational\\_totient\\_series\\_of\\_lcm\\_diagonal\\_nonintegrality\\_supply}{CertificateKernel.lean:18706} \\lean{irrational\\_totient\\_series\\_of\\_lcm\\_cone\\_nonintegrality\\_supply}{CertificateKernel.lean:18718}\n\\end{cor}\n\n\\begin{prop}[cert:b9a --- second-difference certificates, sound but measured not shallower, \\texttt{second\\_diff\\_notMem\\_int\\_of\\_certifiedRank2Kill}]\n$\\mathrm{certifiedRank2Kill}(h,N,L) \\to \\big(\\mathrm{totientTail}(N+2h)-\\mathrm{totientTail}(N+h)\\big) - \\big(\\mathrm{totientTail}(N+h)-\\mathrm{totientTail}(N)\\big) \\notin \\mathrm{range}((\\uparrow):\\mathbb{Z}\\to\\mathbb{R})$. Sound second-difference non-integrality via a doubled band radius. Measured cell $(h,N)=(1,8)$: rank-1 fires at depth 8, no rank-2 certificate exists at depth $\\le8$, rank-2 first fires at depth 9 \u2014 a probe over $t\\le20$ finds rank-1 at least as shallow in 30/40 cells. This route is empirically \\emph{not} a shortcut over rank-1; flagged do-not-re-attempt without new information.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:binary-window} \\lean{second\\_diff\\_notMem\\_int\\_of\\_certifiedRank2Kill}{CertificateKernel.lean:18729} (measured verdict: \\lean{totient\\_tail\\_rank\\_two\\_kill\\_sound\\_but\\_not\\_shallower\\_cell}{CertificateKernel.lean:18762})\n\\end{prop}\n\n\\begin{thm}[cert:b10a --- cone non-flatness menu refuter, wave-25, sharper than pairwise, \\texttt{exists\\_nonintegral\\_pair\\_of\\_coneNonflatCert}]\nFor a nonempty menu $Q$ of positive vertex multipliers with the one-sided floor $\\forall q\\in Q,\\ q\\cdot H + L + 2 < 2^L$ (half the pairwise floor of $\\mathrm{certifiedKill}$): $\\mathrm{coneNonflatCert}(H,L,Q) \\to \\exists q_i,q_j \\in Q,\\ \\mathrm{totientTail}(q_j\\cdot H) - \\mathrm{totientTail}(q_i\\cdot H) \\notin \\mathrm{range}((\\uparrow):\\mathbb{Z}\\to\\mathbb{R})$. Proved by an argmin/Helly-avoidance argument over one-sided arcs: if all vertices shared one fractional part, the minimal-deep-tail vertex would be a common left endpoint of every arc, which $\\mathrm{coneNonflatCert}$ denies. Information-theoretically half the depth floor of pairwise $\\mathrm{certifiedKill}$; the reusable combinatorial-geometry technique transplants to any modular-residue pincer with more than two points.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:lcm-cone-menu} \\lean{exists\\_nonintegral\\_pair\\_of\\_coneNonflatCert}{CertificateKernel.lean:18801}\n\\end{thm}\n\n\\begin{thm}[cert:b10b --- cone non-flat supply, wave-25, \\texttt{irrational\\_totient\\_series\\_of\\_lcm\\_cone\\_nonflat\\_supply}]\nIf $\\mathrm{coneNonflatCert}$ fires (via cert:b10a) on a menu whose scale is unbounded, then $\\mathrm{Irrational}(S)$. The sharpest known certificate-depth-reduced restatement of the wall; for $|Q|\\ge3$ genuinely joint (menu inconsistent while every pair consistent) this is the best-known target for a search-based attempt at supplying cert:a10/cert:b3/cert:b7.\n\\par\\noindent\\ev{Lean} (implication proved; supply \\ev{Open}) \\scale{cofinal} \\coord{other:lcm-cone-menu} \\lean{irrational\\_totient\\_series\\_of\\_lcm\\_cone\\_nonflat\\_supply}{CertificateKernel.lean:18812}\n\\end{thm}\n\n\\begin{prop}[cert:b12 --- survivorKill, alternative certificate via bounded carry orbit]\n$\\mathrm{survivorKill}(h,N,K) := \\forall j < 2(N+h+1)+1,\\ \\exists i\\le K,\\ \\mathrm{carryOrbit}(h,N,j-(N+h+1),i)$ escapes the strip $|\\cdot|\\le N+i+h+2$. Soundness: $\\mathrm{survivorKill}(h,N,K) \\to \\mathrm{totientTail}(N+h)-\\mathrm{totientTail}(N)\\notin\\mathrm{range}((\\uparrow):\\mathbb{Z}\\to\\mathbb{R})$ \u2014 a bounded-orbit argument, alternative to cert:a4/cert:a6's residue-window argument, exhausting all $2(N+h+1)+1$ integer box candidates and checking each provably escapes within $K$ steps. cert:a4 (window) and this (orbit) are two \\emph{independently complete} coordinates for the same non-integrality target (via cert:a7's iff): the \"enumerate all integer candidates in a shrinking box, verify each provably diverges\" pattern is a general technique for irrationality-by-integer-orbit arguments.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:carry-orbit} \\lean{survivorKill}{CarrySurvivorExtinction.lean:412} \\lean{tail\\_diff\\_notMem\\_int\\_of\\_survivorKill}{CarrySurvivorExtinction.lean:428}\n\\end{prop}\n\n\\begin{thm}[mob:e1 --- first-harmonic norm-gap supply, \\texttt{irrational\\_totient\\_series\\_of\\_first\\_harmonic\\_norm\\_gap}]\nHypothesis $\\mathrm{DTWFirstHarmonicNormGap} := \\forall h>0,\\ \\forall X_0,\\ \\exists X\\ge\\max(X_0,1),\\ \\exists L,\\ 16(2X+h+L+2)\\le 2^L \\wedge \\big\\|\\sum_{N\\in[X,2X)} \\mathrm{windowFirstExp}(h,N,L)\\big\\| \\le (21/25)\\cdot X$. If this holds then $\\mathrm{Irrational}(S)$. A genuinely cofinal first-harmonic exponential-sum (Weyl-sum) cancellation statement is sufficient for \\#249; unconditional companion \\texttt{exists\\_certifiedKill\\_of\\_first\\_harmonic\\_gap} shows any constant-saving first-harmonic gap on one dyadic block forces a finite kill certificate (via $\\cos(\\pi/8)>9/10$ and a pigeonhole/averaging argument).\n\\par\\noindent\\ev{Lean} (conditional theorem proved; hypothesis \\ev{Open}) \\scale{cofinal} \\coord{binary-digit} \\lean{irrational\\_totient\\_series\\_of\\_first\\_harmonic\\_norm\\_gap}{FirstHarmonicPivot.lean:83} \\lean{exists\\_certifiedKill\\_of\\_first\\_harmonic\\_gap}{FirstHarmonicGap.lean:128}\n\\end{thm}\n\n\\begin{thm}[mob:e2 --- prime-jump sharp-kill supply, \\texttt{irrational\\_totient\\_series\\_of\\_primeJumpSharpKill\\_supply}]\n$\\mathrm{primeJumpTailCommutator}(H,p) := D(pH) - p\\cdot D(H)$, $D(H) = R_{2H}-R_H$, has an exact partial/tail split with sharp radius $3pH + (p+1)(L+2)$ (tighter than the earlier $4pH$ two-cell-disjunction radius). If $\\forall t_0\\, \\exists t\\ge t_0\\, \\exists p,L>0,\\ \\mathrm{primeJumpSharpKill}(\\mathrm{periodLcm}(t), p, L)$, then $\\mathrm{Irrational}(S)$. One concrete deposit is proved unconditionally: $\\mathrm{primeJumpSharpKill}(12,5,15)$, kernel-\\texttt{decide}d. Proof strategy: assume $S$ rational $\\Rightarrow$ cone flatness (cert:b6) $\\Rightarrow$ contradiction via the sharp prime-jump kill.\n\\par\\noindent\\ev{Lean} (conditional theorem + \\ev{Cert} witness; supply \\ev{Open}) \\scale{cofinal} \\coord{other:lcm-diagonal} \\lean{irrational\\_totient\\_series\\_of\\_primeJumpSharpKill\\_supply}{PrimeJumpWindow.lean:193} \\lean{primeJumpSharpKill\\_twelve\\_five}{PrimeJumpWindow.lean:186}\n\\end{thm}\n\n% ---- part a249_p1b ----\n\\subsection{Consumers}\n\nThis subsection lists every premise in the corpus whose conclusion is\n\\emph{conditional} on an unsupplied hypothesis, with that hypothesis rendered\nin display maths so an agent can recognise it as a search target. All targets\nbelow are logically downstream of the single exact equivalence that defines\nthe open problem:\n\n\\begin{prop}[The exact target: \\#249 = cofinal actual-orbit nonintegrality]\\label{prop:NI-01}\nUnconditionally,\n\\[\n\\mathrm{Irrational}\\Big(\\textstyle\\sum_n' \\varphi(n)/2^n\\Big)\n\\iff\n\\forall a_0,\\ \\exists a \\ge a_0,\\ \\mathtt{actualLcmTailOrbit}\\ a \\notin \\mathrm{range}(\\mathbb{Z}\\to\\mathbb{R}).\n\\]\nNothing beyond a supply of the right-hand side is needed in principle; every\nother proposition in this subsection is a route to it.\n\\scale{n/a}\\ \\ev{Lean}\\ \\coord{mobius-mersenne}\\\\\n\\lean{irrational_totientSeries_iff_actualLcmOrbitNonintegralitySupply}{TotientActualLcmOrbitNonintegrality.lean:37}\n\\end{prop}\n\n\\begin{prop}[Short-window arithmetic-kill supply]\\label{prop:AR-07}\nIf\n\\[\n\\forall a_0,\\ \\exists a\\ge a_0,\\ \\exists L< 2\\cdot 2^a,\\quad \\mathtt{LcmDiagonalArithmeticKill}(2^a,L)\n\\]\nholds, then \\#249 follows (via \\mathtt{actualLcmOrbitNonintegralitySupply\\_of\\_shortArithmeticKill}\nthen Prop.~\\ref{prop:NI-01}). The predicate is exactly the residue-band exclusion of\nAppendix~A4/A6: $\\mathtt{lcmDiagonalArithmeticWord}$ at scale $2^a$ escapes a shrinking\ncentral arc mod $2^L$. Only two instances are proved (Prop.~\\ref{prop:SK-02}'s base cases);\nthe cofinal supply is the open trigger of the entire private batch.\n\\scale{cofinal}\\ \\ev{Open}\\ \\coord{mobius-mersenne}\\\\\n\\lean{PowerTwoActualLcmShortArithmeticKillSupply}{TotientActualLcmOrbitArithmetic.lean:2107}\n\\end{prop}\n\n\\begin{prop}[Diophantine separation supply]\\label{prop:SEP-03}\nIf, at canonically-guarded odd ranks $q$, cofinally many $a$ satisfy\n\\[\n\\big|\\,\\mathtt{actualLcmTailOrbit}\\ a - z\\,\\big| > \\tfrac{1}{32} + (\\text{explicit error radius}) \\qquad \\forall z\\in\\mathbb Z,\n\\]\nthen \\#249 follows via the landed signed-margin producer. This restates the target as\neffective irrationality-measure / anti-concentration for the actual orbit rather than\nexact residue exclusion, using the explicit approximant of Prop.~\\ref{prop:SEP-02}.\n\\scale{cofinal}\\ \\ev{Open}\\ \\coord{mobius-mersenne}\\\\\n\\lean{irrational_totientSeries_of_actualLcmOrbitSeparationSupply}{TotientActualLcmOrbitSeparation.lean:305}\n\\end{prop}\n\n\\begin{prop}[One-sided top-edge residue-gap supply]\\label{prop:TE-04}\nIf, for the room bound of Prop.~\\ref{prop:SGN-01} ($a\\ge 8$, $J+K+(a+6)<2\\cdot2^a$),\n\\[\n\\mathtt{ActualLcmTopEdgeResidueGap}\\ a\\ J\\ K\\ m\n\\quad:\\iff\\quad\nm\\le K \\ \\wedge\\ \\big(\\text{residue of }\\mathtt{windowDiscrepancy}\\text{ at scale }m\\big)\\le 2^m-(\\text{room bound})\n\\]\nholds cofinally, then \\#249 follows (\\mathtt{actualLcmTailDiff\\_notMem\\_int\\_of\\_topEdgeResidueGap},\nsufficiency at line~1260), because Prop.~\\ref{prop:SGN-01}/\\ref{prop:SGN-03} already exclude the\nnegative-side residue independently \u2014 only the positive arc needs excluding. This is a strictly\n\\emph{weaker} target than the old symmetric certifiedKill band and is the genuinely easier open\ntarget of the whole batch.\n\\scale{cofinal}\\ \\ev{Open}\\ \\coord{mobius-mersenne}\\\\\n\\lean{PowerTwoActualLcmTopEdgeResidueGapSupply}{TotientActualLcmTopEdgeStaircase.lean:900,1260}\n\\end{prop}\n\n\\begin{prop}[The five-link equivalence chain below the residue-gap target]\\label{prop:TE-05}\nEach of the following cofinal supplies is proved \\emph{sufficient} for\nProp.~\\ref{prop:TE-04}'s target, chained in decreasing strength:\n\\[\n\\begin{aligned}\n&\\mathtt{PowerTwoActualLcmTopEdgeResidueGapSupply}\n \\Leftarrow \\mathtt{PowerTwoAdjacentSuffixMidbandSupply}\\\\\n&\\Leftarrow \\mathtt{PowerTwoOddGuardTopEdgeHalfWordBandSupply}\n \\iff \\mathtt{PowerTwoActualFinalTopEdgeMagnitudeSupply}\\\\\n&\\Leftarrow \\mathtt{PowerTwoFlexibleActualTopEdgeMagnitudeSupply}\n \\Leftarrow \\mathtt{PowerTwoFlexibleActualTerminalDominanceSupply}\\\\\n&\\Leftarrow \\mathtt{PowerTwoFlexibleActualTerminalCarryCorridorEscapeSupply}\n\\end{aligned}\n\\]\nAll six named predicates are open; none is proved. Because the chain is\nimplication-only-downward, proving the single \\emph{weakest} link\n(\\texttt{...TerminalCarryCorridorEscapeSupply}) closes the entire cluster and hence \\#249.\nThis is the true minimal remaining target of \\texttt{TotientActualLcmTopEdgeStaircase}.\n\\scale{cofinal}\\ \\ev{Open}\\ \\coord{mobius-mersenne}\\\\\n\\lean{powerTwoActualLcmTopEdgeResidueGapSupply\\_of\\_adjacentSuffixMidband}{TotientActualLcmTopEdgeStaircase.lean:2108,2121,1938,1991,2184-2235}\n\\end{prop}\n\n\\begin{prop}[The exact closed-form escape identity]\\label{prop:TE-06}\nUnder integrality of the actual orbit (representative $z$) and the half-cell fit condition\n$2(H+q+2)\\le 4^q$ ($a\\ge 8$, room bound as above),\n\\[\n2\\cdot \\mathtt{actualOddHalfCenteredLift}\\ a\\ q \\;=\\; \\mathtt{diagonalWindowIncrement}(2^a)(2q+2) \\;-\\; \\mathtt{carryOrbit}\\ H\\ H\\ z\\ (2q+1)\n\\]\nholds \\emph{exactly}. Escaping either side of this named open interval (the ``terminal/carry\ncorridor'') is both necessary and sufficient for non-integrality at rank $q$\n(\\mathtt{actualLcmTailOrbit\\_notMem\\_int\\_of\\_actualTerminalCarryCorridorEscape}). This is\nthe sharpest fully explicit statement of ``what remains to prove'' anywhere in the batch.\n\\scale{bounded}\\ \\ev{Lean}\\ \\coord{mobius-mersenne}\\\\\n\\lean{two_mul_actualOddHalfCenteredLift_eq_terminal_sub_trueCarry}{TotientActualLcmTopEdgeStaircase.lean:1678}\n\\end{prop}\n\n\\begin{prop}[Extend the short-kill stub past $a_0=6$]\\label{prop:SK-02}\nCurrently proved only for $a_0\\le 6$:\n\\[\n\\forall a_0\\le 6,\\ \\exists a\\ L,\\ a_0\\le a\\ \\wedge\\ L<2\\cdot2^a\\ \\wedge\\ \\mathtt{LcmDiagonalArithmeticKill}(2^a)\\ L.\n\\]\nThe module's own comment states plainly: ``the remaining endpoint gap is now precisely the\nunbounded continuation beyond this finite prefix.'' Supplying a single further instance at\n$a_0=7$ or $a_0=8$ already extends the stub and is independently interesting evidence toward\nProp.~\\ref{prop:AR-07}'s full cofinal supply; the two base cases $a=4,6$ trace to\n\\mathtt{certifiedKill\\_diagonal\\_t16}/\\mathtt{t64}.\n\\scale{bounded}\\ \\ev{Lean (finite prefix only)}\\ \\coord{mobius-mersenne}\\\\\n\\lean{powerTwoActualLcmShortArithmeticKillSupply_through_six}{TotientActualLcmShortKill.lean:54}\n\\end{prop}\n\n\\begin{prop}[Fixed-rank extremal-ordering supply]\\label{prop:FR-01}\nDefine $\\mathtt{fixedRankSecondDifference}\\ H\\ j := \\varphi(3H+j)-2\\varphi(2H+j)+\\varphi(H+j)$.\nIf, at $H=\\mathtt{periodLcm}(2^a)$ and some fixed small $j$,\n\\[\n\\mathtt{MiddleRankTotientExtremal}\\ H\\ j\n\\quad:\\iff\\quad\n\\varphi(2H+j)\\ \\text{is a strict min or max among}\\ \\{\\varphi(H+j),\\varphi(2H+j),\\varphi(3H+j)\\},\n\\]\nthen $\\mathtt{fixedRankSecondDifference}\\ H\\ j\\ne 0$, with sign matching the extremum. Notably\nan \\emph{ordering} suffices \u2014 no quantitative gap is required. This is a genuinely different\ncoordinate from the sliding LCM window: $j$ fixed and small, only three fixed ranks examined.\nThe remaining open step is \\mathtt{PowerTwoLcmMiddleRankExtremalSupply}: does this ordering hold\ncofinally in $a$.\n\\scale{uniform}\\ \\ev{Lean}\\ \\coord{other:fixed-rank-curvature}\\\\\n\\lean{MiddleRankTotientExtremal}{TotientFixedRankLcmAsymptotic.lean:256,267,275,291,307}\n\\end{prop}\n\n\\begin{prop}[The directed/LCM-specialised certificate supply]\\label{prop:CP-06}\n$\\mathtt{directedCertifiedKill}\\ h\\ N\\ L$ is sound \\emph{and complete}:\n\\[\n(\\exists L,\\ \\mathtt{directedCertifiedKill}\\ h\\ N\\ L) \\iff \\mathtt{totientTail}(N+h)-\\mathtt{totientTail}(N)\\notin \\mathrm{range}(\\mathbb Z\\to\\mathbb R),\n\\]\nexactly, no gap; and $\\mathrm{Irrational}(S)\\iff \\mathtt{CofinalDirectedLcmCertificateSupply}$ (the\nLCM-diagonal specialisation). The asymmetric strip is a genuine finite-depth improvement over\nthe symmetric certificate (kills $t=3$ at depth~6, one level earlier than the symmetric one), but\nthe file itself notes the improvement does not turn into an independent sieve theorem: supplying\nthe cofinal predicate is exactly as hard as \\#249 itself.\n\\scale{uniform}\\ \\ev{Lean}\\ \\coord{seam-integer}\\\\\n\\lean{CofinalDirectedLcmCertificateSupply}{TotientTailCarryPeriod.lean:766,829,875,845}\n\\end{prop}\n\n\\begin{prop}[Pulse-restricted survivor-search supply]\\label{prop:CP-07}\nGiven the cofinal mod-4 pulse supply of Prop.~\\ref{prop:CP-05-inv}, it suffices to kill only the\n$2\\bmod 4$-class candidate states (a fourfold reduction of the initial search strip) at one\ncofinal arithmetic-pulse prime per putative period, to conclude \\#249:\n\\[\n\\mathtt{modFourPulseSurvivorKill} \\implies \\mathrm{Irrational}(S).\n\\]\nThe reduction technique (use a cofinal totient-specific pulse to legally restrict the survivor\nsearch to one residue class mod a small modulus) transfers wherever an analogous cofinal pulse\ncan be built on the \\#257 side.\n\\scale{cofinal}\\ \\ev{Lean}\\ \\coord{mobius-mersenne}\\\\\n\\lean{irrational_totientSeries_of_cofinal_modFourPulseSurvivorKill}{TotientTailCarryPeriod.lean:906,973,1066}\n\\end{prop}\n\n\\begin{prop}[The wave-21 wall: certificate supply over all periods]\\label{prop:A10}\n\\[\n\\big(\\forall h\\ge 1,\\ \\forall N_0,\\ \\exists N\\ge N_0,\\ \\exists L,\\ \\mathtt{certifiedKill}\\ h\\ N\\ L\\big)\n\\implies \\mathrm{Irrational}(S).\n\\]\nThis is exactly the quantifier structure $\\forall h\\ge1\\ \\forall N_0\\ \\exists N\\ge N_0\\ \\exists L\\ \\mathrm{Sep}(h,N,L)$.\nNothing in the public or private corpus supplies this hypothesis; every other reduction in Part~B\nof the certificate bank is a logically equivalent-or-weaker reformulation of this same missing\nsupply, never independent progress on it.\n\\scale{cofinal}\\ \\ev{Open}\\ \\coord{mobius-mersenne}\\\\\n\\lean{irrational_totient_series_of_certificate_supply}{TotientTailPeriodKiller.lean:394}\n\\end{prop}\n\n\\begin{prop}[Diagonal collapse \u2014 one free parameter]\\label{prop:B3}\n\\[\n\\big(\\forall t_0,\\ \\exists t\\ge t_0,\\ \\exists L,\\ \\mathtt{certifiedKill}(\\mathtt{periodLcm}\\,t)(\\mathtt{periodLcm}\\,t)\\ L\\big)\n\\implies \\mathrm{Irrational}(S).\n\\]\nStanding at $N=h=\\mathtt{periodLcm}\\,t$ beats every hypothetical rational simultaneously (any\n$t\\ge\\max(h_0,N_0)$ serves both parameters at once), collapsing Prop.~\\ref{prop:A10}'s\ntwo unbounded parameters to one. This is the canonical single-unknown restatement, the best\ntarget for search or heuristics.\n\\scale{cofinal}\\ \\ev{Open}\\ \\coord{mobius-mersenne}\\\\\n\\lean{irrational_totient_series_of_lcm_diagonal_certificate_supply}{LcmDiagonalReduction.lean:112}\n\\end{prop}\n\n\\begin{prop}[Cone collapse \u2014 annihilator umbrella]\\label{prop:B7}\n\\[\n\\big(\\forall t_0,\\ \\exists t\\ge t_0,\\ \\exists q\\,m\\,L,\\ 0<q \\wedge \\mathtt{certifiedKill}(m\\cdot\\mathtt{periodLcm}\\,t)(q\\cdot\\mathtt{periodLcm}\\,t)\\ L\\big)\n\\implies \\mathrm{Irrational}(S).\n\\]\nOne certified kill \\emph{anywhere} on the two-multiplier LCM cone, at arbitrarily large $t$,\nsuffices; the diagonal (Prop.~\\ref{prop:B3}) is the cell $q=m=1$. This is the widest known\ntarget still logically equivalent-in-strength to Prop.~\\ref{prop:A10}.\n\\scale{cofinal}\\ \\ev{Open}\\ \\coord{mobius-mersenne}\\\\\n\\lean{irrational_totient_series_of_lcm_cone_window_kill_supply}{CertificateKernel.lean:18686}\n\\end{prop}\n\n\\begin{prop}[Menu non-flatness supply \u2014 sharper than pairwise]\\label{prop:B10}\nFor a nonempty menu $Q$ of positive vertex multipliers with one-sided floor\n$\\forall q\\in Q,\\ qH+L+2<2^L$ (\\emph{half} the pairwise floor of \\texttt{certifiedKill}),\n\\[\n\\mathtt{coneNonflatCert}\\ H\\ L\\ Q \\implies \\exists\\, q_i,q_j\\in Q,\\ \\mathtt{totientTail}(q_jH)-\\mathtt{totientTail}(q_iH)\\notin \\mathrm{range}(\\mathbb Z\\to\\mathbb R).\n\\]\nIf this fires at unbounded scale over a menu with $|Q|\\ge3$ genuinely joint (menu inconsistent\nwhile every pair is separately consistent), \\#249 follows via Prop.~\\ref{prop:B7}. The\nargmin/Helly-avoidance proof technique is a reusable combinatorial-geometry pattern for any\nmodular-residue pincer with more than two points.\n\\scale{cofinal}\\ \\ev{Open}\\ \\coord{mobius-mersenne}\\\\\n\\lean{irrational_totient_series_of_lcm_cone_nonflat_supply}{CertificateKernel.lean:18801,18812}\n\\end{prop}\n\n\\begin{prop}[Unbounded Farey growth \u2014 a second independent wall]\\label{prop:C2sup}\nThe Farey-gap denominator bound at window $K$ is currently $\\sim 7.96\\times10^{34}$ at $K=240$\n(Prop.~\\ref{prop:C2-inv}). If\n\\[\n\\sup_K\\, (b+d)(K) = \\infty\n\\]\n(the bound growing without limit as $K\\to\\infty$), then \\#249 follows via the $C3$-style\ndenominator-exclusion consumer, logically \\emph{independently} of Prop.~\\ref{prop:A10}'s\ncertificate-supply wall. No unboundedness claim is proved or attempted in the corpus.\n\\scale{cofinal}\\ \\ev{Open}\\ \\coord{other:farey-gap}\\\\\n\\lean{gap\\_check\\_window\\_1\\_240\\_le\\_79639646646701375323355774875831053}{GapFareyBound.lean:176,225}\n\\end{prop}\n\n\\begin{prop}[Finite-rank compression \u2014 a third independent wall]\\label{prop:D5cons}\nBy Prop.~\\ref{prop:D4-inv}, the canonical dyadic totient-kernel family is unconditionally\n$(2^e+1)$-dimensional at every level $e$. Rationality of $S$ forces an associated tempered\ncarry orbit with $\\mathbb Q$-rank $\\ge 2^e-1$ at every level (Prop.~\\ref{prop:CP-02}). A theorem\n\\[\n\\text{bounding the dyadic-section rank of \\emph{every} rationality-supplied tempered carry}\n\\]\nwould contradict the unconditional lower bound and close \\#249 by a linear-algebraic coordinate\nwholly distinct from the binary-window certificate coordinate of Props.~\\ref{prop:A10}-\\ref{prop:B10}.\n\\scale{cofinal}\\ \\ev{Open}\\ \\coord{other:carry-kernel-rank}\\\\\n\\lean{not_irrational_totientSeries_implies_unbounded_carryRank_unconditional}{TotientCarryKernelRigidity.lean:284}\n\\end{prop}\n\n\\begin{prop}[Alternative coordinate: bounded carry-orbit survivor supply]\\label{prop:B12cons}\n$\\mathtt{survivorKill}\\ h\\ N\\ K$ enumerates all $2(N+h+1)+1$ integer box candidates and checks\neach provably diverges from the strip $|\\cdot|\\le N+i+h+2$ within $K$ steps; soundness gives\n$\\mathtt{survivorKill}\\ h\\ N\\ K \\implies \\mathtt{totientTail}(N+h)-\\mathtt{totientTail}(N)\\notin\\mathrm{range}(\\mathbb Z\\to\\mathbb R)$,\nproving the \\emph{same} target as Prop.~\\ref{prop:A10}/\\ref{prop:NI-01} from a different\ncoordinate (bounded dynamics, not residue windows). Per the doctrine ``obstructions are\ncoordinate-relative,'' a case hard in the window coordinate may be easy here.\n\\scale{n/a}\\ \\ev{Lean}\\ \\coord{other:carry-orbit-dynamics}\\\\\n\\lean{tail_diff_notMem_int_of_survivorKill}{CarrySurvivorExtinction.lean:412,428}\n\\end{prop}\n\n\\subsection{Invariants}\n\nNecessary conditions that any successful (or unsuccessful) approach must be\nconsistent with \u2014 facts a solution cannot contradict, whichever way \\#249\neventually resolves.\n\n\\begin{prop}[Exact quotient-scale digit closed form, no cleanliness hypothesis]\\label{prop:AR-04-inv}\n\\[\n\\mathtt{lcmRayArithmeticLetter}\\ t\\ j \\;=\\; \\mathtt{deltaTotient}(\\mathtt{periodLcm}\\,t)(\\mathtt{periodLcm}\\,t + j) \\;=\\; \\mathtt{diagonalWindowIncrement}\\ t\\ j,\n\\]\nvalid for \\emph{both} divisor and non-divisor offsets $j$, with no side condition on which primes\nof $j$ survive in $H/j$. Every downstream sign, positivity, or residue statement about the\ndiagonal word must factor through this identity; it generalises the older\n$\\mathtt{deltaTotient\\_periodLcm\\_ray\\_split}$, which needed $j$'s primes to survive.\n\\scale{uniform}\\ \\ev{Lean}\\ \\coord{mobius-mersenne}\\\\\n\\lean{lcmRayArithmeticLetter_eq_deltaTotient}{TotientActualLcmOrbitArithmetic.lean:291,298,310}\n\\end{prop}\n\n\\begin{prop}[Uniform $3/4$-density retained on rough integers]\\label{prop:AR-03-inv}\nFor $a\\ge8$, $t=2^a$, every prime factor of $n>0$ exceeding $t$, and $n<2^{2\\cdot2^a}$:\n$n.\\mathrm{primeFactors.card} < 2^a/4$, hence $(3/4)\\cdot n < \\varphi(n)$ over $\\mathbb Q$. Any\nargument bounding $\\varphi$ on the short window below height $\\mathtt{periodLcm}(2^a)^2$ must\nrespect this floor; the counting mechanism (union-bound-in-product form, then convert via\n$\\mathtt{Nat.totient\\_eq\\_mul\\_prod\\_factors}$) is reusable for \\#257's Mersenne-rough objects\nat the analogous height.\n\\scale{bounded}\\ \\ev{Lean}\\ \\coord{mobius-mersenne}\\\\\n\\lean{three_quarters_mul_lt_totient_of_rough_lt_two_pow}{TotientActualLcmOrbitArithmetic.lean:722,768}\n\\end{prop}\n\n\\begin{prop}[Every short-window digit sign is fixed to $+$]\\label{prop:AR-05-inv}\nFor $a\\ge8$, $0<j<2\\cdot2^a$: $0<\\mathtt{lcmRayArithmeticLetter}(2^a)\\ j$. Every coefficient of\nthe actual diagonal word in the entire short window is strictly positive \u2014 divisor offsets via\na quantitative $H<4jc$ bound, foreign prime powers via predecessor descent plus\nProp.~\\ref{prop:AR-03-inv}'s rough-density bracket ($H/4<c<5H/2$), exponent-one new primes via\n$H/4<c$. This unconditional fact (no irrationality hypothesis) is what makes every sign theorem\nin the batch \u2014 Prop.~\\ref{prop:SGN-01} and the staircase-impossibility of\nProp.~\\ref{prop:TE-01-killer} \u2014 provable at all.\n\\scale{bounded}\\ \\ev{Lean}\\ \\coord{mobius-mersenne}\\\\\n\\lean{lcmRayArithmeticLetter_pos_of_lt_two_mul}{TotientActualLcmOrbitArithmetic.lean:1703}\n\\end{prop}\n\n\\begin{prop}[Window word = truncated discrepancy; kill $\\iff$ certificate]\\label{prop:AR-06-inv}\n$\\mathtt{lcmDiagonalArithmeticWord}\\ t\\ L = \\mathtt{windowDiscrepancy}(\\mathtt{periodLcm}\\,t)(\\mathtt{periodLcm}\\,t)\\ L$,\nand $\\mathtt{LcmDiagonalArithmeticKill}\\ t\\ L \\iff \\mathtt{certifiedKill}(\\mathtt{periodLcm}\\,t)(\\mathtt{periodLcm}\\,t)\\ L$.\nAny residue-band existence claim about the actual diagonal word converts directly to a\n$\\mathtt{certifiedKill}$/non-integrality fact through this iff \u2014 it is the fixed bridge every\nconsumer in this subsection implicitly uses.\n\\scale{n/a}\\ \\ev{Lean}\\ \\coord{mobius-mersenne}\\\\\n\\lean{lcmDiagonalArithmeticKill_iff_certifiedKill}{TotientActualLcmOrbitArithmetic.lean:1998,2045}\n\\end{prop}\n\n\\begin{prop}[Exact global-to-local bridge]\\label{prop:SEP-01-inv}\n\\[\n\\mathtt{actualLcmTailOrbit}\\ a \\;=\\; 2^H(2^H-1)\\Big(\\textstyle\\sum_n' \\varphi(n)/2^n\\Big) - \\big(\\mathtt{totientPrefix}(2H)-\\mathtt{totientPrefix}(H)\\big),\\quad H=\\mathtt{periodLcm}(2^a).\n\\]\nEvery local orbit statement is secretly a statement about the global series value minus a\ncomputable finite prefix; any rational-approximation or continued-fraction style separation\nargument for \\#249 must express its target through this exact affine image.\n\\scale{n/a}\\ \\ev{Lean}\\ \\coord{mobius-mersenne}\\\\\n\\lean{actualLcmTailOrbit_eq_scaled_totientSeries_sub_prefix}{TotientActualLcmOrbitSeparation.lean:68}\n\\end{prop}\n\n\\begin{prop}[Explicit computable approximant with explicit error radius]\\label{prop:SEP-02-inv}\nFor all $a,q$: $|\\mathtt{actualLcmTailOrbit}\\ a - \\mathtt{actualLcmRawApprox}\\ a\\ q| < (2H+2q+3)/2^{2q+1}$,\nwhere $\\mathtt{actualLcmRawApprox}\\ a\\ q := \\mathtt{diagonalAdjacentSuffixRawBlock}(2^a,0,2q{+}1)/2^{2q+1}$.\nAny separation-from-integers argument (Prop.~\\ref{prop:SEP-03}) is forced to work with this\nfinite, computable block rather than the infinite tail, at this exact error rate.\n\\scale{uniform}\\ \\ev{Lean}\\ \\coord{mobius-mersenne}\\\\\n\\lean{abs_actualLcmTailOrbit_sub_rawApprox_lt}{TotientActualLcmOrbitSeparation.lean:141}\n\\end{prop}\n\n\\begin{prop}[Unconditional positive-sign corridor \u2014 a real theorem, not a supply]\\label{prop:SGN-01}\nFor $a\\ge8$, $J+(a+6)<2\\cdot2^a$:\n\\[\n0 < \\mathtt{totientTail}(2H+J) - \\mathtt{totientTail}(H+J),\\qquad H=\\mathtt{periodLcm}(2^a),\n\\]\nwith no irrationality hypothesis \u2014 the true, infinite, real translated tail difference is\nstrictly positive throughout almost the entire short window, proved unconditionally from\nProp.~\\ref{prop:AR-05-inv} plus a directed one-sided tail bound. Any argument about the sign of\nthe actual orbit (not merely its residue mod $2^L$) must agree with this; specialised at $J=0$\nthis gives $0<\\mathtt{actualLcmTailOrbit}\\ a$.\n\\scale{bounded}\\ \\ev{Lean}\\ \\coord{mobius-mersenne}\\\\\n\\lean{actualLcmTailDiff_shift_pos}{TotientActualLcmOrbitSign.lean:39,144}\n\\end{prop}\n\n\\begin{prop}[Integrality forces the exact top-edge residue \u2014 names the obstruction]\\label{prop:SGN-03}\nUnder the room bound of Prop.~\\ref{prop:SGN-01} plus $2H+J+K+2<2^K$: integrality of the actual\norbit forces\n\\[\n\\mathtt{windowDiscrepancy}\\ H\\ (H{+}J)\\ K \\bmod 2^K \\;=\\; 2^K - e\n\\]\n(the top-edge representative), provably \\emph{outside} the central arc required by\n$\\mathtt{directedCertifiedKill}$. This documents precisely why the sign theorem alone cannot\nclose \\#249: any contradiction needs an independent exclusion of this specific top-edge boundary\nband, not a re-derivation of the carry reset. The named residue $2^K-e$ ($e>0$ small) is exactly\nwhat Prop.~\\ref{prop:TE-04} sets out to exclude.\n\\scale{bounded}\\ \\ev{Lean}\\ \\coord{mobius-mersenne}\\\\\n\\lean{actualLcm_integral_forces_topEdgeResidue}{TotientActualLcmOrbitSign.lean:211}\n\\end{prop}\n\n\\begin{prop}[Punctured staircase is pinned to the half-turn]\\label{prop:TE-02-inv}\nUnder the room bound and the punctured-staircase hypothesis (all but the last letter vanish at\ngrowing dyadic weight; the last letter retained below the top-edge carry band),\n\\[\n\\mathtt{lcmRayArithmeticLetter}(2^a)(J{+}K{-}1) = 2^{m-1} \\quad\\text{exactly, and}\\quad 2^m < 2(2H+J+K+2).\n\\]\nThe modulus must be the first dyadic scale above the room bound; any extra bit of modulus makes\neven the punctured route empty. Any attempt to revive a partial-staircase strategy must land\nexactly here.\n\\scale{bounded}\\ \\ev{Lean}\\ \\coord{mobius-mersenne}\\\\\n\\lean{puncturedDyadicStaircase_penultimate_eq_half}{TotientActualLcmTopEdgeStaircase.lean:1187}\n\\end{prop}\n\n\\begin{prop}[Any certificate reduces to a two-bit test \u2014 universal normal form]\\label{prop:TE-03-inv}\nUnconditionally, for arbitrary $h,N$ (no totient content in the proof):\n\\[\n(\\exists L,\\ \\mathtt{certifiedKill}\\ h\\ N\\ L) \\iff \\mathtt{GuardCylinderWitness}\\ h\\ N,\n\\]\nwhere the witness is a logarithmic-depth socket $(b{+}1)$ or a two-bit mixed-guard cylinder at\nscale $b=\\log_2(N{+}h{+}L{+}2){+}1$. Any certificate-search algorithm or complexity bound for\neither problem's tail differences must respect this compression \u2014 an unbounded-depth search is\nnever truly necessary once $b$ is fixed.\n\\scale{uniform}\\ \\ev{Lean}\\ \\coord{seam-integer}\\\\\n\\lean{exists_certifiedKill_iff_guardCylinderWitness}{TotientActualLcmTopEdgeStaircase.lean:844}\n\\end{prop}\n\n\\begin{prop}[Primitive kernel factor of the fixed-rank curvature]\\label{prop:FR-02-inv}\nOn the square-root clean window $j^2\\le2^a$, $a\\ge4$: $2\\varphi(j) \\mid \\mathtt{fixedRankSecondDifference}(\\mathtt{periodLcm}(2^a))\\ j$.\nThe exact local factor $2\\varphi(j)$ comes from the affine-rank-3 kernel $(1,-2,1)$ combined with\nthe parity of three odd rough cofactors, and is provably tight \u2014 Prop.~\\ref{prop:FR-03-kill}\nshows no bare application of the kernel can force one more factor of $2$. Any curvature-based\nattack at fixed rank must land inside this exact divisibility ceiling.\n\\scale{bounded}\\ \\ev{Lean}\\ \\coord{other:fixed-rank-curvature}\\\\\n\\lean{two_mul_totient_dvd_fixedRankSecondDifference}{TotientFixedRankLcmAsymptotic.lean:519}\n\\end{prop}\n\n\\begin{prop}[Carry displacement $\\iff$ integral tail difference \u2014 central plumbing]\\label{prop:CP-01-inv}\nFor a positive-multiplier tempered totient carry $u$ ($\\mathtt{IsTemperedBinaryOrbit}\\ \\varphi\\ v\\ u$, $v>0$):\n\\[\n(v:\\mathbb Z)\\mid u(N{+}k)-u(N) \\iff \\mathtt{totientTail}(N{+}k)-\\mathtt{totientTail}(N)\\in\\mathrm{range}(\\mathbb Z\\to\\mathbb R).\n\\]\nEvery statement in this batch that converts between ``carry orbit divisibility'' and ``real tail\ndifference is an integer'' is an instance of this identity \u2014 the underlying machinery\n($\\mathtt{IsTemperedBinaryOrbit}$, $\\mathtt{binaryCoeffTail}$) is generic over any $f$ with\n$f(n)\\le n$, so it transfers to \\#257's own coefficient function verbatim.\n\\scale{uniform}\\ \\ev{Lean}\\ \\coord{seam-integer}\\\\\n\\lean{carryShift_dvd_iff_tailDiff_mem_int}{TotientTailCarryPeriod.lean:77}\n\\end{prop}\n\n\\begin{prop}[What rationality does and does not buy \u2014 the exact frontier]\\label{prop:CP-02}\nIf $\\neg\\mathrm{Irrational}(S)$, there exist $v>0$ and a tempered orbit $u$ such that for every\nlevel $e$, the $\\mathbb Q$-rank of the canonical carry-kernel-family span is $\\ge 2^e-1$ (torsion-free\nrank grows exponentially in level), \\emph{yet} $u$'s dyadic sections are uniformly eventually\nperiodic modulo $v$ ($\\mathtt{CarrySectionsEventuallyPeriodicMod}$). Rationality buys\nquotient-mod-$v$ periodicity but does \\emph{not} buy any bound on $\\mathbb Q$-rank \u2014 this is\nprecisely the boundary any Mersenne-specific residue argument (mod-4 pulse, two-adic pulse) exists\nto cross, and it is the exact hypothesis feeding Prop.~\\ref{prop:D5cons}'s third independent wall.\n\\scale{uniform}\\ \\ev{Lean}\\ \\coord{other:carry-kernel-rank}\\\\\n\\lean{not_irrational_totientSeries_implies_mod_period_and_unbounded_rank}{TotientTailCarryPeriod.lean:224}\n\\end{prop}\n\n\\begin{prop}[Cofinal mod-4 residue pulse \u2014 genuinely totient-specific]\\label{prop:CP-05-inv}\nFor any $h>0$ and bound $B$, there is a prime $p>B$ with $\\mathtt{deltaTotient}(4h)\\ p \\equiv 2\\pmod4$,\nconstructed via Dirichlet on $p\\equiv(3r-H)\\bmod 4r$ for a fresh prime $r\\equiv1\\pmod4$. Unlike the\nMersenne-primitivity killer of Prop.~\\ref{prop:CP-03-kill}, this \\emph{is} a genuinely\ntotient-specific residue supplied cofinally beyond every threshold \u2014 any argument invoking\nProp.~\\ref{prop:CP-07} inherits this as its forcing mechanism. The fresh-prime CRT template is\nscaled to arbitrary depth in Prop.~\\ref{prop:TA-inv} below.\n\\scale{cofinal}\\ \\ev{Lean}\\ \\coord{other:dirichlet-crt}\\\\\n\\lean{exists_prime_deltaTotient_four_mul_mod_four_two}{TotientTailCarryPeriod.lean:384,553}\n\\end{prop}\n\n\\begin{prop}[Arbitrary-depth zero-prefix-then-pulse construction, and its transfer]\\label{prop:TA-inv}\nFor $K\\ge2$, $H>K$, $B$: cofinally many primes $p>B$ satisfy\n$p\\equiv1+2^{K-1}\\pmod{2^K}$, $2^K\\mid\\varphi(p{+}H)$, and $2^K\\mid\\varphi(p{-}j)\\wedge2^K\\mid\\varphi(p{-}j{+}H)$\nfor every $1\\le j<K$ \u2014 an entire length-$(K{-}1)$ zero prefix followed by a half-turn terminal\npulse, at arbitrary two-adic depth (Prop.~\\ref{prop:CP-05-inv}'s $K=2$ case generalised to every\n$K$, via $K{-}1$ fresh Dirichlet primes glued by CRT). This forces\n\\[\n\\mathtt{windowDiscrepancy}\\ H\\ (p{-}K)\\ K \\equiv 2^{K-1}\\pmod{2^K},\n\\]\nand under eventual integrality transfers to $\\exists z,\\ (z:\\mathbb R)=\\mathtt{totientTail}(p{+}H)-\\mathtt{totientTail}(p)\\wedge z\\equiv2^{K-1}\\pmod{2^K}$.\nAny solution attempt must be consistent with the existence of this pulse family at every depth $K$.\n\\scale{cofinal}\\ \\ev{Lean}\\ \\coord{p-adic}\\\\\n\\lean{exists_prime_totient_twoAdic_pulse_divisors}{TotientTwoAdicPulseBlock.lean:54,211,262,301,327}\n\\end{prop}\n\n\\begin{prop}[Exact M\u00f6bius-inversion tail identity]\\label{prop:MP-01-inv}\n$\\mathtt{totientTail}\\ N = \\sum_d' \\mu(d)\\cdot 2^{d-r_d(N)}\\big(q_d(N)/(2^d-1) + 1/(2^d-1)^2\\big)$,\nwith $r_d(N)=d-N\\bmod d$ (forward shift to the next multiple of $d$), $q_d(N)=\\lfloor N/d\\rfloor+1$ \u2014\nexact, not a definition-by-subtraction from a target. Any M\u00f6bius/Lambert-series manipulation of\n$\\mathtt{totientTail}$ in this batch or in \\#257 factors through this identity and its fully\ngeneric regrouping primitive $\\mathtt{tsum\\_lambert\\_pair\\_regroup\\_if}$ (proved for arbitrary\nweight functions $|w(d)|\\le d$, $|v(m)|\\le m$).\n\\scale{uniform}\\ \\ev{Lean}\\ \\coord{cyclotomic}\\\\\n\\lean{totientTail_eq_tsum_shiftedMobiusPulse}{TotientShiftedMobiusPulse.lean:535,112}\n\\end{prop}\n\n\\begin{prop}[The rare unconditional deposits \u2014 necessary base cases]\\label{prop:SK-01-inv}\n$\\mathtt{LcmDiagonalArithmeticKill}(2^4)\\ 23$ and $\\mathtt{LcmDiagonalArithmeticKill}(2^6)\\ 93$ are\nproved, routed through pre-existing compressed certificates\n$\\mathtt{certifiedKill\\_diagonal\\_t16}/\\mathtt{t64}$, giving $\\mathtt{actualLcmTailOrbit}\\ 4,6\\notin\\mathrm{range}(\\mathbb Z\\to\\mathbb R)$\n\u2014 the \\emph{only} two unconditional actual-orbit nonintegrality facts currently proved. Any\ninduction or bootstrapping attempt at Prop.~\\ref{prop:AR-07} must reproduce these as base cases,\nand they double as a computational sanity check that the $\\mathtt{lcmRayArithmeticLetter}$\nmachinery agrees with the older compressed-certificate route.\n\\scale{fixed}\\ \\ev{Cert}\\ \\coord{mobius-mersenne}\\\\\n\\lean{actualLcmTailOrbit_four_and_six_notMem_int}{TotientActualLcmShortKill.lean:18,27,35}\n\\end{prop}\n\n\\begin{prop}[Certificate depth floor]\\label{prop:A5-inv}\n$\\mathtt{certifiedKill}\\ h\\ N\\ L \\implies 2(N{+}h{+}L{+}2)<2^L$. Certificate depth $L$ cannot stay\nbounded while $N{+}h\\to\\infty$: $L$ must grow at least logarithmically. Any complexity argument\nabout certificate search inherits this floor, and it is exactly what Prop.~\\ref{prop:TE-03-inv}\ncompresses to a two-bit test at the corresponding logarithmic scale.\n\\scale{n/a}\\ \\ev{Lean}\\ \\coord{seam-integer}\\\\\n\\lean{certifiedKill_depth_floor}{TotientTailPeriodKiller.lean:79}\n\\end{prop}\n\n\\begin{prop}[The tail-period law \u2014 necessary consequence of rationality]\\label{prop:A9-inv}\n$\\neg\\mathrm{Irrational}(S) \\implies \\exists h{:}\\mathbb N,\\ 0<h \\wedge \\exists N_0,\\ \\forall N\\ge N_0,\\ \\mathtt{totientTail}(N{+}h)-\\mathtt{totientTail}(N)\\in\\mathrm{range}(\\mathbb Z\\to\\mathbb R)$,\nwith explicit witnesses $h=\\varphi(\\mathrm{oddPart}(r.\\mathrm{den}))$, $N_0=v_2(r.\\mathrm{den})$\nfor $S=r$. This is the ``only if'' half of the master equivalence: any rationality-refutation\nstrategy must eventually contradict \\emph{some} period $h$ and preperiod $N_0$ produced this way.\nComposes with Prop.~\\ref{prop:AR-06-inv}/soundness to give Prop.~\\ref{prop:A10}.\n\\scale{uniform}\\ \\ev{Lean}\\ \\coord{other:euler-theorem}\\\\\n\\lean{eventual_period_of_not_irrational}{TotientTailPeriodKiller.lean:358}\n\\end{prop}\n\n\\begin{prop}[Rationality flattens the whole LCM cone]\\label{prop:B6-inv}\n$\\neg\\mathrm{Irrational}(S) \\implies \\exists t_1,\\ \\forall t\\ge t_1,\\ \\forall q\\,m{:}\\mathbb N,\\ 0<q,\\ \\mathtt{totientTail}(q\\cdot\\mathtt{periodLcm}\\,t + m\\cdot\\mathtt{periodLcm}\\,t)-\\mathtt{totientTail}(q\\cdot\\mathtt{periodLcm}\\,t)\\in\\mathrm{range}(\\mathbb Z\\to\\mathbb R)$.\nRationality forces ONE fractional constant on the entire cone $\\{k\\cdot\\mathtt{periodLcm}\\,t : k\\ge1\\}$\nat every scale $t\\ge t_1$, not just the diagonal pair \u2014 a strict generalisation of\nProp.~\\ref{prop:A9-inv}. Any certificate anywhere on the cone therefore kills \\#249\n(Prop.~\\ref{prop:B7}).\n\\scale{n/a}\\ \\ev{Lean}\\ \\coord{mobius-mersenne}\\\\\n\\lean{rational_totient_series_forces_lcm_cone_flatness}{CertificateKernel.lean:18674}\n\\end{prop}\n\n\\begin{prop}[Unconditional $(2^e{+}1)$-dimensional dyadic kernel \u2014 the rank floor]\\label{prop:D4-inv}\nThe canonical dyadic-kernel family $\\mathtt{canonicalTotientKernelFamily}\\ e$ has exactly\n$2^e+1$ channels, unconditionally linearly independent for every $e$ (CRT + Dirichlet: one channel\nmade prime, every other channel gets a fresh prime $\\equiv1\\bmod$ a large power of $2$). This is a\nproved fact, needing no rationality hypothesis at all, and it is the fixed lower bound that\nProp.~\\ref{prop:D5cons} would have to contradict.\n\\scale{n/a}\\ \\ev{Lean}\\ \\coord{other:carry-kernel-rank}\\\\\n\\lean{card_totientCanonicalIndex}{TotientMahlerDefect.lean:51}\n\\end{prop}\n\n\\begin{prop}[Sharp Farey rung \u2014 concrete necessary floor at $K=240$]\\label{prop:C2-inv}\nFor all $q{:}\\mathbb N$, $0<q\\le79639646646701375323355774875831053$ ($\\sim7.96\\times10^{34}$):\n$(qV)\\bmod2^{240} + 243q < 2^{240}$, where $V$ is the explicit committed totient residue for\nwindow $(N,K)=(1,240)$; this is SHARP \u2014 $q=79639646646701375323355774875831054$ is the exact\nfirst failing denominator (an explicit Farey mediant). Any candidate rational value for $S$ must\nhave reduced denominator strictly larger than this bound (Prop.~\\ref{prop:C3-inv} below).\n\\scale{fixed}\\ \\ev{Cert}\\ \\coord{other:farey-gap}\\\\\n\\lean{gap_check_window_1_240_first_failure}{GapFareyBound.lean:176,225}\n\\end{prop}\n\n\\begin{prop}[Denominator exclusion \u2014 headline unconditional consequence]\\label{prop:C3-inv}\n$\\forall p{:}\\mathbb Q,\\ p.\\mathrm{den}\\le79639646646701375323355774875831053 \\implies S\\ne(p:\\mathbb R)$.\nIf $S$ is rational, its reduced denominator exceeds $\\sim7.96\\times10^{34}$ \u2014 an unconditional\nnecessary condition on any hypothetical rational value, logically independent of the\ncertificate-supply wall (Prop.~\\ref{prop:A10}); it excludes small denominators outright, without\nneeding a period at all.\n\\scale{fixed}\\ \\ev{Lean}\\ \\coord{other:farey-gap}\\\\\n\\lean{tsum_totient_div_pow_two_ne_ratCast_of_den_le_79639646646701375323355774875831053}{CertificateKernel.lean:18056}\n\\end{prop}\n\n\\begin{prop}[The two foundational irrationality engines]\\label{prop:D1D2-inv}\nTwo, and only two, general-purpose irrationality criteria exist in the kernel, and any successful\nproof of \\#249 must ultimately instantiate one of them: (i) if $u{:}\\mathbb N\\to\\mathbb Q$ is\neventually never $x$ and $\\mathrm{den}(u_k)\\cdot|x-u_k|\\to0$, then $\\mathrm{Irrational}(x)$; (ii)\nif $\\forall q>0,\\ \\exists m,z{:}\\mathbb Z,\\ 0<|m\\xi-z|<1/q$, then $\\mathrm{Irrational}(\\xi)$\n(with a base-power specialisation $b^n\\xi$ matching any digit/carry construction in base $b$).\nEvery certificate, cone, or Farey argument in this catalogue exists either to supply one of these\ntwo hypotheses or to substitute for them via the $\\mathtt{certifiedKill}$ route.\n\\scale{n/a}\\ \\ev{Lean}\\ \\coord{other:dirichlet-approximation}\\\\\n\\lean{irrational_of_den_mul_abs_sub_tendsto_zero}{CertificateKernel.lean:5371,5792,5821}\n\\end{prop}\n\n\\begin{prop}[S sits on the Mersenne-Lambert ladder]\\label{prop:D7-inv}\nWriting $L(f):=\\sum_{n\\ge1}f(n)/(2^n-1)$: $L(\\mu)=1/2$, $L(\\varphi)=2$ (exactly rational,\nmachine-checked), $L(1)=E$ = the Erd\\H os-Borwein constant, proved \\textbf{irrational} in this\nsame kernel; $L(A)=S$ for $A=\\varphi*\\mu$ is exactly \\#249 restated in ``positive\nErd\\H os-Borwein form,'' still open; $L(\\mathrm{Id})=\\sum\\sigma(m)/2^m$ is transcendental by\nNesterenko 1996 (cited, not formalised). Any proof of \\#249 sits on this Dirichlet-convolution\nladder next to a proved irrational neighbour ($L(1)$) and a proved rational neighbour\n($L(\\varphi)$); this is the clearest bridge into \\#257's native Lambert-series coordinate.\n\\scale{n/a}\\ \\ev{Lean, Cited}\\ \\coord{mobius-mersenne}\\\\\n\\lean{tsum_totient_div_pow_two_eq_pnat_half_pow}{CertificateKernel.lean:18063-18087}\n\\end{prop}\n\n\\begin{prop}[Generic rational-gap adapter]\\label{prop:D9-inv}\nFor $\\mathtt{pfx}<\\mathtt{whole}{:}\\mathbb Q$: $1/(\\mathtt{whole.den}\\cdot\\mathtt{pfx.den}) \\le \\mathtt{whole}-\\mathtt{pfx}$\n(as reals). Any analytic upper bound on a totient-tail gap converts, via this lemma, into a\ndenominator-growth lower bound extending Prop.~\\ref{prop:C2-inv}'s Farey rungs without redoing the\nFarey-neighbour search from scratch \u2014 a clean, fully general adapter usable on either problem's\ntail estimate.\n\\scale{n/a}\\ \\ev{Lean}\\ \\coord{other:farey-gap}\\\\\n\\lean{positive_rational_difference_lower_bound}{PrimitiveRationalGapSupply.lean:31}\n\\end{prop}\n\n\\begin{prop}[Finite pincer floor \u2014 concrete necessary evidence through $t=64$]\\label{prop:B11-inv}\n$\\mathtt{certifiedKill\\_diagonal\\_all\\_imported}$: Prop.~\\ref{prop:B3}'s predicate $P\\,t$ holds\nfor $t\\in\\{1,2,3,4,5,7,8,9,11,13,16,17,\\dots\\}$ (28 explicit cases through $t=64$), each a finite\n$\\mathtt{decide}/\\mathtt{norm\\_num}$ computation on explicit $\\varphi$ values via checked\nprime-power blocks with Lucas-primality certificates. This does \\emph{not} establish $P\\,t$ for\ninfinitely many $t$ and hence does not close Props.~\\ref{prop:A10}/\\ref{prop:B3}/\\ref{prop:B7} \u2014\nit is exactly the finite floor any unbounded-supply argument must extend.\n\\scale{fixed}\\ \\ev{Cert}\\ \\coord{mobius-mersenne}\\\\\n\\lean{certifiedKill_diagonal_all_imported}{DiagonalPincerCertificates.lean:2842-2870}\n\\end{prop}\n\n\\begin{prop}[The fair-coin coprimality reformulation of $S$]\\label{prop:C1-inv}\n$\\#\\{(a,b): a{+}b=n,\\ a>0,\\ \\gcd(a,b)=1\\}=\\varphi(n)$ for every $n$ (uniform, $\\varphi(0)=0$,\n$\\varphi(1)=1$ included automatically). At $r=1/2$, for independent fair-coin waiting times\n$X,Y$ ($P(X{=}n)=2^{-n}$, $n\\ge1$): $P(\\gcd(X,Y)=1) = S - 1/2$. \\#249's constant is, up to the\nadditive constant $1/2$, a literal lattice-point probability. Any digit argument for \\#249\nnecessarily also carries a positional meaning about visible lattice points on the addition\nantidiagonal \u2014 a reformulation any proof attempt should keep in view even though it is not itself\na route to a proof.\n\\scale{n/a}\\ \\ev{Lean}\\ \\coord{other:visible-lattice-points}\\\\\n\\lean{tsum_pos_coprime_pair_pow}{GeometricCoprimality.lean:120,182,56}\n\\end{prop}\n\n\\subsection{Killers and countermodels}\n\nAdversarial constructions and no-go results, kept with the mechanism that\ncloses each route so the machinery outlives the framing. Per the corpus's\nown doctrine (project memory \\texttt{feedback\\_invent\\_dont\\_audit}): obstructions\nbelow are coordinate-relative, and each entry names exactly which coordinate\nor proof-template it excludes, not the underlying object.\n\n\\begin{obs}[Nonnegative-branch elimination for the true survivor]\\label{prop:SGN-02}\nIf the translated actual orbit is integral at the start of Prop.~\\ref{prop:SGN-01}'s positive\ncorridor, then for every later depth $K$ with room $J{+}K{+}(a{+}6)<2\\cdot2^a$, the negated\ncarry-orbit trajectory $-\\mathtt{carryOrbit}\\ H\\ (H{+}J)\\ d\\ K$ is negative and is an\n$\\mathtt{endpointSurvivor}$ at every depth in the corridor. This eliminates only the nonnegative\nbranch of the true survivor's sign \u2014 it is emphatically \\emph{not} a nonintegrality claim;\nspurious survivors of either sign may still exist. Any strategy hoping to conclude non-integrality\npurely from this sign fact fails: Prop.~\\ref{prop:SGN-03} names exactly what remains.\n\\scale{bounded}\\ \\ev{Lean}\\ \\coord{mobius-mersenne}\\\\\n\\lean{actualLcm_trueEndpointSurvivor_neg}{TotientActualLcmOrbitSign.lean:172}\n\\end{obs}\n\n\\begin{obs}[Total dyadic staircase is impossible \u2014 route-pruning]\\label{prop:TE-01-killer}\nFor $a\\ge8$, room bound, and modulus wide enough ($2H{+}J{+}K{+}2<2^m$):\n$\\mathtt{ActualLcmTerminalDyadicStaircase}\\ a\\ J\\ K\\ m$ is \\textbf{false} \u2014 a terminal window\nwhere every one of the last $m$ letters is divisible by its own growing power of two\n($2^{r+1}\\mid\\text{letter}_r$) cannot occur. The last letter would have to be positive\n(Prop.~\\ref{prop:AR-05-inv}), strictly below the wide modulus, and divisible by it, forcing it to\nbe exactly $0$ \u2014 contradicting positivity. \\textbf{Kills outright}: any strategy aiming for total\ndyadic annihilation of the terminal suffix. The generic mechanism\n($0<e<2^m\\wedge2^m\\mid e\\implies e=0$) is reusable wherever a positive quantity is asked to vanish\nmod a wider-than-itself modulus. The surviving route is the \\emph{punctured} staircase\n(Prop.~\\ref{prop:TE-02-inv}) or the residue-gap producers (Prop.~\\ref{prop:TE-04}).\n\\scale{bounded}\\ \\ev{Lean}\\ \\coord{mobius-mersenne}\\\\\n\\lean{not_actualLcmTerminalDyadicStaircase_of_room}{TotientActualLcmTopEdgeStaircase.lean:251}\n\\end{obs}\n\n\\begin{obs}[The bare 3-rank kernel is exactly tight \u2014 a sharpness no-go]\\label{prop:FR-03-kill}\n$\\mathtt{fixedRankSecondDifference}(2^{n+1}\\cdot2)(2^{n+1}\\cdot3) = -2^{n+1}$ exactly, for all $n$\n(a closed-form computed fixture). The two-adic valuation gained by Prop.~\\ref{prop:FR-02-inv} is\nexactly tight \u2014 the normalised $1{\\times}1$ minor is odd at every depth, so the bounded-height\nprimitive kernel alone can \\textbf{never} force one additional factor of $2$. \\textbf{Kills}: any\nattempt to squeeze more $2$-adic information out of the bare 3-rank curvature kernel; further\nprogress needs new arithmetic input, matching the file's own open target\n$\\mathtt{PowerTwoLcmMiddleRankExtremalSupply}$.\n\\scale{uniform}\\ \\ev{Cert}\\ \\coord{other:fixed-rank-curvature}\\\\\n\\lean{fixedRankSecondDifference_dyadic_fixture_exact}{TotientFixedRankLcmAsymptotic.lean:462}\n\\end{obs}\n\n\\begin{obs}[The parity coboundary countermodel object]\\label{prop:CM-01}\n$\\mathtt{parityCoboundaryWeight}\\ n := \\mathtt{parityBaseWeight}\\ n + 2\\cdot\\mathtt{largePowerTwoBit}\\ n - 4\\cdot\\mathtt{largePowerTwoBit}(n{-}1)$,\nwhere $\\mathtt{parityBaseWeight}$ is the eventually-constant word $0,1,1,2,4,4,4,\\dots$ and\n$\\mathtt{largePowerTwoBit}\\ n=1$ iff $n=2^{k+3}$ (lacunary spikes starting at $8$). This is a\nhand-built adversary sequence containing \\emph{no} $\\varphi$ in its definition at all except via\nits parity. The construction technique \u2014 add a zero-valued sparse binary coboundary\n$2\\cdot2^{-m}-4\\cdot2^{-(m+1)}=0$ at lacunary ranks to an eventually-periodic base word, destroying\nperiodicity while preserving the rational sum \u2014 is a completely general recipe, directly portable\nto an analogous \\#257 adversary matched to $1/(2^n-1)$-parity patterns.\n\\scale{n/a}\\ \\ev{Lean}\\ \\coord{other:binary-digit}\\\\\n\\lean{parityCoboundaryWeight}{TotientParityCoboundaryCountermodel.lean:37,41,46,59}\n\\end{obs}\n\n\\begin{obs}[What the countermodel refutes \u2014 the flagship kill of the batch]\\label{prop:CM-02}\nThere is $c{:}\\mathbb N\\to\\mathbb N$ with: (a) arbitrarily many, arbitrarily-separated, cofinal\nexplicit ``$6,0$'' carry pairs (for every $N,G,K$ a block of $K$ such pairs beyond $N$, pairwise\nseparated by $>G$); (b) $\\forall n,\\ c(n)\\le6$; (c) $\\forall n,\\ c(n)\\le n$; (d) $\\forall n,\\ c(n)\\equiv\\varphi(n)\\pmod2$\n(exact parity agreement with Euler's totient); (e) $c$ is \\textbf{not} eventually periodic; yet\n(f) $\\sum_n' c(n)/2^n = 3/2$, a \\emph{rational} number. \\textbf{Kills}: any \\#249 proof strategy\nwhose only inputs are uniform boundedness, the trivial growth bound $c(n)\\le n$, exact\n$\\varphi$-parity agreement, and failure of eventual periodicity \u2014 \\emph{even strengthened} to\ncofinal, arbitrarily-separated, arbitrarily-long-block non-periodicity. The obstruction is\nstructural, not a shortage of non-periodicity witnesses: a rational coboundary can carry unlimited\n\\emph{visible} aperiodic structure while summing to a fixed rational. \\textbf{Doctrine consequence}\n(matches project memory \\texttt{feedback\\_erdos\\_reductions\\_rejected\\_bank\\_real\\_results}): only\narguments using actual quantitative totient/Mersenne size or residue information (as in the\n$\\mathtt{TotientActualLcm}\\ast$/$\\mathtt{TotientFixedRank}\\ast$ families) can possibly close \\#249;\npure symbolic-word arguments cannot. This countermodel is also the ready-made stress-test fixture\nfor any future sufficient-condition candidate stated purely in coefficient-word terms\n(boundedness/growth/parity/periodicity), on either \\#249 or \\#257.\n\\scale{cofinal}\\ \\ev{Cert}\\ \\coord{other:binary-digit}\\\\\n\\lean{exists_totientParity_arbitrarilyManySeparatedCarry_rational_countermodel}{TotientParityCoboundaryCountermodel.lean:637,359,487,194}\n\\end{obs}\n\n\\begin{obs}[Primitive Mersenne-prime factors alone force nothing]\\label{prop:CP-03-kill}\nFor the completely-multiplicative control $c(n)=n$ (zero totient content:\n$\\mathtt{binaryCoeffTail}\\ \\mathrm{id}\\ N = N{+}2$): if $q>K>0$ and $q\\mid2^K-1$ (e.g. $q$ a\nprimitive prime factor of the homogeneous Mersenne multiplier), the shift is integral at every $N$\n(value exactly $K$) but $q\\nmid K$. \\textbf{Kills}: ``a large primitive Mersenne prime factor\nalone forces a contradiction'' as a proof strategy \u2014 this is a route-pruning warning applicable\n\\emph{verbatim} to \\#257, whose denominators $2^n-1$ are literally the object $q\\mid2^K-1$ tested\nhere. Any successful use of such a factor needs genuinely totient-specific residue/size input\nbeyond bare Mersenne primitivity \u2014 contrast with the genuinely totient-specific pulse of\nProp.~\\ref{prop:CP-05-inv}.\n\\scale{bounded}\\ \\ev{Lean}\\ \\coord{mobius-mersenne}\\\\\n\\lean{idCoeff_mersenneModulus_does_not_annihilate_tailShift}{TotientTailCarryPeriod.lean:342,320}\n\\end{obs}\n\n\\begin{obs}[Universal no-go for absolute-adjugate coefficient reconstruction]\\label{prop:CP-04-kill}\nFor any finite rational row $w{:}\\iota\\to\\mathbb Q$, integer targets $x{:}\\iota\\to\\mathbb N$ that\n\\emph{exactly} isolate one totient channel ($\\sum_i w_i\\varphi(x_i)=1$): the crude two-tail cost\n$\\sum_i|w_i|\\cdot(2(x_i{+}1)+(x_i{+}2))$ is $\\ge3$, hence never $<1$. \\textbf{Kills}: the strategy\nof recovering $\\varphi(x)$ from $2R_{x-1}-R_x$ and bounding each tail termwise by $R_M\\le M{+}2$ \u2014\nit can \\emph{never} reach the strict $<1$ tail-error threshold, at any finite grid height,\nindependent of matrix height, row translation, determinant, or target channel. The proof only uses\n$\\varphi(x)\\le x$, so the same triangle-inequality floor transfers verbatim to any \\#257\nreformulation with $c(n)\\le n$.\n\\scale{uniform}\\ \\ev{Lean}\\ \\coord{other:adjugate-linear-algebra}\\\\\n\\lean{three_le_totientAdjugateTailCost}{TotientTailCarryPeriod.lean:266,307}\n\\end{obs}\n\n\\begin{obs}[Rank-2 second-difference certificates: sound but measured not shallower]\\label{prop:B9cert-kill}\n$\\mathtt{certifiedRank2Kill}\\ h\\ N\\ L \\implies$ the second difference\n$(\\mathtt{totientTail}(N{+}2h){-}\\mathtt{totientTail}(N{+}h)) - (\\mathtt{totientTail}(N{+}h){-}\\mathtt{totientTail}\\,N) \\notin\\mathrm{range}(\\mathbb Z\\to\\mathbb R)$,\nsound at doubled band radius versus rank~1. Measured at $(h,N)=(1,8)$: rank-1 fires at depth~8, no\nrank-2 certificate exists at depth $\\le8$, rank-2 first fires at depth~9 \u2014 empirically \\textbf{not}\na shortcut over rank-1 (rank-1 at least as shallow in $30/40$ probed cells for $t\\le20$).\n\\textbf{Kills}: expecting a depth improvement from moving to second differences. Explicitly flagged\nas a dead end \u2014 do not re-attempt this exact refinement without new information.\n\\scale{n/a}\\ \\ev{Cert}\\ \\coord{mobius-mersenne}\\\\\n\\lean{totient_tail_rank_two_kill_sound_but_not_shallower_cell}{CertificateKernel.lean:18729,18762}\n\\end{obs}\n\n\\begin{obs}[Unit-gap strengthening rescues at most one lattice point]\\label{prop:D8cert-kill}\n$\\mathtt{ReducedDenominatorUnitGapCert}\\ u\\ N\\ K := \\forall t\\in[1,L],\\ \\neg\\mathrm{Coprime}(J{+}t,u)$\n(nonunits-only refinement of the Farey-gap consumer). At a prime-power reduced denominator $p^e$:\n$\\mathtt{ReducedDenominatorUnitGapCert}(p^e)\\ N\\ K \\iff L{=}0 \\vee (L{=}1\\wedge p\\mid(J{+}1))$ \u2014\nthe unit-gap strengthening can rescue \\textbf{at most one} additional candidate lattice point\nbeyond the ordinary Farey-gap certificate. \\textbf{Kills}: expecting this refinement to unboundedly\nstrengthen Prop.~\\ref{prop:C2-inv}/\\ref{prop:C3-inv}'s Farey rungs \u2014 it is self-flagged in its own\ndocstring as ``a strict certificate-level strengthening, not a supply theorem.'' A known dead end;\ndo not re-attempt expecting more than $+1$ lattice point per prime power.\n\\scale{bounded}\\ \\ev{Lean}\\ \\coord{other:farey-gap}\\\\\n\\lean{reducedDenominatorCandidateCount_prime_pow_le_one}{PrimitiveWeightCertificate.lean:30,54,111}\n\\end{obs}\n\n\\begin{obs}[Synthetic all-horizon countermodel to homogeneous-factor-only proof strategies]\\label{prop:B4b-kill}\nThere is an explicit nonzero ``all-horizon countermodel'' sequence built from multiples of\n$\\varphi(\\mathtt{periodLcm}\\,t)$ that (i) agrees with every exact whole-ray anchor\n$\\mathtt{deltaTotient}\\ H\\ (qH)=\\varphi(H)$ for $2\\le q<t$; (ii) stays inside the natural diagonal\nbounds; and (iii) \\textbf{survives every finite integer shift polynomial} \u2014 every commensurate\nfinite-rank LCM cube, via the normal form $P_m(E_H)\\cdot\\prod(E_H^n-1)$. Explicitly flagged\nsynthetic: it does not claim the compensation letters occur as actual totient differences.\n\\textbf{Kills}: the strategy of ``retain only homogeneous LCM-ray factors'' at \\emph{every} finite\nrank, not just rank 2 or 3. This is a representation-level exclusion (a particular factor-ideal\nprojection throws away information that can be adversarially reconstructed), not an exclusion of\n$\\varphi$ itself \u2014 an actual proof must control the fresh M\u00f6bius channel, per\n$\\mathtt{totient\\_eq\\_sum\\_mobiusTotientChannel}$ in the same file.\n\\scale{n/a}\\ \\ev{Lean (docstring-sourced)}\\ \\coord{mobius-mersenne}\\\\\n\\lean{LcmFactorIdealPulseObstruction}{LcmFactorIdealPulseObstruction.lean:1-24,328}\n\\end{obs}\n\n\\begin{obs}[Fixed-precision local valuation-unit signatures never obstruct]\\label{prop:B5-kill}\nFor any finite word of odd-valuation-unit symbols at fixed local $2$-adic precision $u>0$, and any\nstarting carry state $e$, there \\textbf{exists} a compatible carry orbit realising that exact word\nwith every intermediate state centred in its dyadic interval ($|e'|\\le\\mathtt{vuRadius}\\ u\\ \\sigma$).\n\\textbf{Kills}: any proof strategy trying to derive a contradiction purely from ``the local\nvaluation-unit signature at fixed precision $u$ is incompatible with $X$'' \u2014 such signatures are\n\\emph{always} realisable by some carry orbit. Framed against \\#249 in the docstring, but the\ntheorem itself carries no totient- or Mersenne-specific content: a proof needs growing precision\nor extra arithmetic coupling, not a fixed-window local signature.\n\\scale{n/a}\\ \\ev{Lean}\\ \\coord{p-adic}\\\\\n\\lean{fixedPrecisionTropicalNoGo}{TropicalCurvatureCarry.lean:137,69,115}\n\\end{obs}\n\n\\begin{obs}[Square-CRT correction-suppression is independent of nonvanishing]\\label{prop:B8-kill}\nSquare-CRT correction suppression (fixing a cofactor mod a prime square to remove a weighted\ncorrection term on a finite horizon) is achievable, \\textbf{but} suppression alone does not force\na nonzero coefficient: the smallest returned countermodel has the whole two-step finite block\nvanish identically, while a separate clean witness shows it can also be nonzero \u2014 ``clean''\n(correction-suppressed) is consistent with \\emph{both} vanishing and nonvanishing. \\textbf{Kills}:\n``achieve a clean square-CRT horizon'' as a sufficient condition by itself for a nonvanishing\ncorrection term; an additional anti-concentration or residue producer is still needed, exactly as\nthe module's own docstring states.\n\\scale{n/a}\\ \\ev{Lean (docstring-sourced)}\\ \\coord{other:square-crt-cocycle}\\\\\n\\lean{squareCRT_clean_block_can_vanish}{SquareCRTCube.lean:285,294,303,311}\n\\end{obs}\n\n\\begin{obs}[No fixed integer clears every normalised primitive-Euler coordinate]\\label{prop:D6-kill}\n$\\forall D>0,\\ \\neg\\forall n>0,\\ \\exists z{:}\\mathbb Z,\\ D\\cdot(A(n)/n)=z$, where $A=\\varphi*\\mu$ is\nthe Mersenne-Lambert primitive weight (so $S=L(A)$, Prop.~\\ref{prop:D7-inv}). Any $D$ clearing\nnormalised coordinates through horizon $N$ must be divisible by every odd prime $p\\le N$, every\n$p^2\\le N$, and by $4$ once $N\\ge4$ \u2014 so no fixed $D$ works for all $n$. \\textbf{Kills}: any\nstrategy seeking a single fixed integer denominator that simultaneously integralises every\nprimitive-Euler coordinate $A(n)/n$ \u2014 a finite no-lift theorem in the integral\nEuler/Witt-coordinate category (weaker than the general Dieudonn\\'e-Dwork theorem, and explicitly\n\\emph{not} itself an irrationality proof). An independent-coordinate no-go, orthogonal to the\nbinary-window (Prop.~\\ref{prop:A10}) and carry-rank (Prop.~\\ref{prop:D5cons}) obstructions \u2014 a\nthird distinct coordinate attacking the same open problem.\n\\scale{n/a}\\ \\ev{Lean}\\ \\coord{mobius-mersenne}\\\\\n\\lean{no_fixed_integral_primitive_euler_index}{PrimitiveDeterminantLift.lean:169,148}\n\\end{obs}\n\n\\begin{obs}[Finite certificates only ever prove exclusion, never membership]\\label{prop:B11cert-kill}\nCertified-death/kill families (finite decidable checks at a given depth) are explicitly\n\\textbf{one-sided}: a found certificate proves exclusion, but failure to find one, or survival\nthrough any finite probed depth, proves \\emph{nothing} about membership or rationality. This is a\nstructural caveat recurring across every certificate family in this catalogue\n($\\mathtt{certifiedKill}$, $\\mathtt{directedCertifiedKill}$, $\\mathtt{survivorKill}$,\n$\\mathtt{LcmDiagonalArithmeticKill}$): none of them can ever certify membership/rationality by\nfinite search, only non-membership/non-integrality. \\textbf{Kills}: treating an unsuccessful finite\nsearch over any of the certificate families above as evidence toward rationality, or treating a\nfinite prefix of confirmed kills (Prop.~\\ref{prop:SK-01-inv}, Prop.~\\ref{prop:B11-inv}) as\nanything more than a floor to extend.\n\\scale{n/a}\\ \\ev{Lean (structural caveat, cross-problem)}\\ \\coord{n/a}\\\\\n\\lean{CertifiedGreedyMersenneDeath}{GreedyAchievementSet.lean:1720,1742-1744}\n\\end{obs}\n\n% ---- part a249_newdecls ----\n\\subsection{Recently published declarations}\n\nThis subsection catalogues material that entered the public record after the\nrest of this paper was drafted: declarations union-merged into modules that\nwere already public (\\texttt{SignedQMomentObstruction},\n\\texttt{GenericTailOrbitRigidity}, \\texttt{DiagonalFreshLossBridge},\n\\texttt{SquaredMersenneDiagonalEnclosure},\n\\texttt{ActualForeignResidueProjection}, \\texttt{DyadicPrefixCompression},\n\\texttt{RepunitMobiusNumerator}, \\texttt{GeometricCoprimality},\n\\texttt{LcmDiagonalReduction}, \\texttt{PivotAntiReconstruction}, all in\n\\texttt{Erdos249257/}), and the newly published per-problem tree\n\\texttt{ErdosProblems/Erdos249/}. Every declaration below was read directly\nfrom source; none is inferred from a name. Several of these files contain\nlong chains of ``\\texttt{XSupply} $\\to$ irrational'' implications whose\nhypothesis is an unproved cofinal predicate. Per the standing rule of this\npaper, those chains are named as exactly what they are --- reductions, not\nresults --- and are not restated as if they narrowed the open problem. The\nmodule \\texttt{SquareCRTCube} is deliberately excluded: its private additions\ndid not compile even in the private tree and were reverted, so nothing from\nit beyond the existing public file is cited here.\n\n\\subsubsection{The M\u00f6bius--Mersenne ladder: unconditional log-concavity}\n\n\\texttt{SignedQMomentObstruction} adds an infinite ladder\n$\\Theta_r := \\sum_{n\\ge 0} \\mu(n+1)/(2^{n+1}-1)^r$ built from the same\nM\u00f6bius--Mersenne atoms as the rest of the corpus, together with a full\nunconditional order-two Hankel (log-concavity) theorem for every rung.\n\n\\begin{defn}\n$\\Theta_r := \\sum_{n\\ge 0}\\mu(n+1)/(2^{n+1}-1)^r$, split exactly into the\nfirst two atoms $\\mathrm{TwoAtom}(r) := 1 - 3^{-r}$ and the tail\n$\\mathrm{TailAfterTwo}(r) := \\sum_{n\\ge 0}\\mu(n+3)/(2^{n+3}-1)^r$.\n\\lean{Erdos249257.SignedQMomentObstruction.mobiusMersenneTheta}{SignedQMomentObstruction.lean:136}\n\\quad \\lean{Erdos249257.SignedQMomentObstruction.mobiusMersenneTwoAtom}{SignedQMomentObstruction.lean:140}\n\\quad \\lean{Erdos249257.SignedQMomentObstruction.mobiusMersenneTailAfterTwo}{SignedQMomentObstruction.lean:144}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{mobius-mersenne}.\n\\end{defn}\n\n\\begin{thm}[Two low rungs are exact rationals tied to \\#249]\n$\\Theta_1 = 1/2$, and $\\Theta_2 = \\bigl(\\sum_{n\\ge 1}\\varphi(n)2^{-n}\\bigr) - 1/2$\n--- the second rung is literally $S - 1/2$.\n\\lean{Erdos249257.SignedQMomentObstruction.mobiusMersenneTheta_one}{SignedQMomentObstruction.lean:197}\n\\quad \\lean{Erdos249257.SignedQMomentObstruction.mobiusMersenneTheta_two_eq_totient_offset}{SignedQMomentObstruction.lean:214}\n\\quad \\ev{Lean} \\quad \\scale{fixed} \\quad \\coord{mobius-mersenne}.\n\\end{thm}\n\n\\begin{thm}[Two-atom exact Hankel gap]\n$\\mathrm{TwoAtom}(r{+}1)^2 - \\mathrm{TwoAtom}(r)\\mathrm{TwoAtom}(r{+}2) = 4/3^{r+2}$,\nhence the two-atom truncation alone is strictly log-concave at every $r$.\n\\lean{Erdos249257.SignedQMomentObstruction.mobiusMersenneTwoAtom_hankelGap}{SignedQMomentObstruction.lean:311}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{mobius-mersenne}.\n\\end{thm}\n\n\\begin{thm}[Full ladder: unconditional strict log-concavity, every rung]\nFor every $r\\ge 1$, $\\Theta_r\\,\\Theta_{r+2} < \\Theta_{r+1}^2$: the shifted\n$2\\times 2$ Hankel determinant of the whole ladder is negative at every\nrung, not just asymptotically. The proof propagates the exact two-atom gap\nabove against a geometric tail-error budget that contracts by a factor $1/4$\nper rung against a gap that only contracts by $1/3$, giving an inductive\nfloor from rung $5$ on (kernel-checked base cases below).\n\\lean{Erdos249257.SignedQMomentObstruction.mobiusMersenneTheta_strict_logConcave}{SignedQMomentObstruction.lean:730}\n\\quad \\lean{Erdos249257.SignedQMomentObstruction.mobiusMersenneTheta_strict_logConcave_of_five_le}{SignedQMomentObstruction.lean:534}\n\\quad \\lean{Erdos249257.SignedQMomentObstruction.mobiusMersenneTheta_hankel_two_neg}{SignedQMomentObstruction.lean:740}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{mobius-mersenne}.\n\\end{thm}\n\n\\begin{obs}\nThis is a genuine unconditional analytic fact about the ladder, but it is\nnot by itself an irrationality obstruction or a step toward one: negative\nHankel determinants say the sequence $(\\Theta_r)$ is not the moment\nsequence of a positive measure, which is unrelated to whether any single\n$\\Theta_r$ (in particular $\\Theta_2 = S - 1/2$) is rational. It is recorded\nhere as exactly what it is --- a structural fact about the ladder --- and\nno stronger claim is made.\n\\end{obs}\n\nThe file also supplies the underlying algebraic infrastructure: a\nrectangular Cauchy--Binet expansion derived from the Leibniz formula\n(\\lean{Erdos249257.SignedQMomentObstruction.det_mul_rectangular}{SignedQMomentObstruction.lean:29},\n\\ev{Lean}, \\scale{uniform}, \\coord{other:hankel-determinant}, needed because\nMathlib's pinned determinant API does not expose rectangular Cauchy--Binet\ndirectly), and a finite unique-terminal-dyadic-exponent parity lemma showing\nthat clearing a common denominator by its uniquely largest power of two\npreserves oddness of the numerator\n(\\lean{Erdos249257.SignedQMomentObstruction.scaled_dyadic_sum_odd}{SignedQMomentObstruction.lean:78},\n\\ev{Lean}, \\scale{uniform}, \\coord{other:dyadic-parity}).\n\n\\subsubsection{Generic tail-orbit rigidity: a self-labelled non-claim}\n\n\\texttt{GenericTailOrbitRigidity} is explicit in its own header that it\nasserts no novelty and no priority, and contains an explicit \\texttt{NON\\_CLAIM}\nguard against a superseded ``positive orbit'' route; it is a formal\nalgebra/analysis interface, reproduced here only because it entered the\npublic tree after the earlier parts of this paper were drafted.\n\n\\begin{thm}[T7: tempered-orbit equivalence, abstract binary series]\nFor any $c:\\mathbb{N}\\to\\mathbb{N}$ with $c(n)\\le n$, writing\n$X_c := \\sum_{n\\ge 1} c(n)/2^n$, $X_c$ is rational iff there exist $v>0$ and\nan integer sequence $u$ with $u(N{+}1) = 2u(N) - v\\,c(N{+}1)$ and\n$u(N)/2^N \\to 0$; when it exists such an orbit is rigid, $u(N) = v\\cdot T_c(N)$\nfor every $N$, where $T_c$ is the scaled tail.\n\\lean{Erdos249257.binaryCoeffSeries_rational_iff_exists_temperedBinaryOrbit}{GenericTailOrbitRigidity.lean:426}\n\\quad \\lean{Erdos249257.temperedBinaryOrbit_eq_scaledTail}{GenericTailOrbitRigidity.lean:339}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{other:tail-orbit-rigidity}.\n\\end{thm}\n\n\\begin{prop}[Rigidity engine]\nA real sequence $d$ with $d(N{+}1)=2d(N)$ and $d(N)=o(2^N)$ is identically\nzero.\n\\lean{Erdos249257.doublingOrbit_eq_zero_of_tempered}{GenericTailOrbitRigidity.lean:315}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{other:tail-orbit-rigidity}.\n\\end{prop}\n\n\\begin{obs}[Finite-state no-go for successor decoders]\nFor fixed $m\\ge 2$, the ``balanced-pulse'' family\n$c_{m,r}$ (mass $r$ moved from digit $m$ to digit $m{+}1$, $0\\le r\\le\n\\lfloor(m{+}1)/2\\rfloor$) has the same value and the same complete\npre-$m$ history for every $r$, yet the parameter $r$ is recovered exactly\nfrom the first post-pulse digit. Consequently no state that identifies all\nmembers of one such family can be decoded by any autonomous map, and the\nfan-out of any correct decoder is unbounded in $m$.\n\\lean{Erdos249257.balancedPulse_no_autonomous_decoder}{GenericTailOrbitRigidity.lean:247}\n\\quad \\lean{Erdos249257.balancedPulse_label_card_lower_bound}{GenericTailOrbitRigidity.lean:228}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{other:tail-orbit-rigidity}.\nThis is a genuine barrier of the class the brief asks for: it rules out\nexactly the class of arguments that try to predict/decode the exact tail\norbit from a state depending only on the pre-pulse history, because the\nfamily exhibited is a literal counterexample generator for any such decoder.\nIt says nothing about the totient-specific series itself.\n\\end{obs}\n\nTwo ``rational control models'' are included as honesty checks, not\nirrationality statements: the pair-balanced family\n$c(2k){=}k{-}a(k)$, $c(2k{+}1){=}2a(k)$ has tempered orbit for every bounded\npayload $a$ and a fixed marked value $4/9$ regardless of $a$\n(\\lean{Erdos249257.globalBalancedCoeff_value}{GenericTailOrbitRigidity.lean:545},\n\\ev{Lean}, \\scale{uniform}, \\coord{other:tail-orbit-rigidity}); and the\nidentity coefficients $c(n)=n$ have the explicit multiplier-one tempered\norbit $u(N)=N+2$\n(\\lean{Erdos249257.idCoeff_temperedOrbit}{GenericTailOrbitRigidity.lean:564},\n\\ev{Lean}, \\scale{fixed}, \\coord{other:tail-orbit-rigidity}).\n\n\\subsubsection{Squared-Mersenne diagonal enclosure}\n\n\\texttt{SquaredMersenneDiagonalEnclosure} spends the exactly-summable\nfirst-order Lambert term of the actual \\#249 diagonal exactly, leaving only\na squared-Mersenne tail to bound.\n\n\\begin{prop}[Exact rational centre, direct]\n$\\mathrm{scaleDiagonalTailDifference}(H) - \\mathrm{lambertProjectedDiagonal}(H,D)\n= C_H \\cdot \\mathrm{mobiusSquareTail}(D)$ for every $H,D$ --- no residue split\nand no side condition on $D$ versus $H$.\n\\lean{Erdos249257.SquaredMersenneDiagonalEnclosure.scaleDiagonalTailDifference_sub_lambertProjectedDiagonal}{SquaredMersenneDiagonalEnclosure.lean:138}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{other:squared-mersenne-tail}.\n\\end{prop}\n\n\\begin{prop}[Sharp geometric tail bound]\n$|\\mathrm{mobiusSquareTail}(D)| \\le 4/\\bigl(3(2^{D+1}-1)^2\\bigr)$.\n\\lean{Erdos249257.SquaredMersenneDiagonalEnclosure.abs_mobiusSquareTail_le}{SquaredMersenneDiagonalEnclosure.lean:167}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{other:squared-mersenne-tail}.\n\\end{prop}\n\n\\begin{obs}[Directed vs. symmetric enclosure]\nWhen the first nonzero M\u00f6bius channel past $D$ is known, its sign directs a\none-sided interval half the width of the symmetric bound\n(\\lean{Erdos249257.SquaredMersenneDiagonalEnclosure.mobiusSquareTail_directed_of_first_support}{SquaredMersenneDiagonalEnclosure.lean:313},\n\\ev{Lean}, \\scale{uniform}, \\coord{other:squared-mersenne-tail}), and either\nenclosure composes with an integer-lattice separation hypothesis into a\nconsumer theorem\n(\\lean{Erdos249257.SquaredMersenneDiagonalEnclosure.scaleFullTarget_miss_of_lambert_projected_separation}{SquaredMersenneDiagonalEnclosure.lean:426}).\nThe separation hypothesis itself is exactly the remaining open content; the\nenclosure is real but does not supply it.\n\\end{obs}\n\n\\subsubsection{Old/foreign M\u00f6bius channel split (DiagonalFreshLossBridge)}\n\nThe 4243-line addition to \\texttt{DiagonalFreshLossBridge} isolates, at the\nlevel of individual M\u00f6bius-totient channels, which channels are ``old''\n(index divides the LCM height $H$) and which are ``foreign'' (index does\nnot), and proves exact identities for both. Most of the file's bulk beyond\nwhat is recorded here is a long ladder of \\texttt{...Supply}-conditional\n``$\\Rightarrow$ irrational'' implications at successively more specialised\nhypotheses; those are reductions and are not restated as results.\n\n\\begin{prop}[Exact old-channel value: a positive gcd word, scaled]\nFor $H>0$, the sum of totient forward-differences over exactly the divisor\nchannels of $H$ equals\n$(H/\\mathrm{rad}(H))\\cdot\\mathrm{gcdWordCoeff}(\\mathrm{rad}(H),s)$, where\n$\\mathrm{gcdWordCoeff}$ is the positive gcd-word coefficient of\n\\S\\ref{repunit-gcd-word} below.\n\\lean{Erdos249257.DiagonalFreshLossBridge.oldMobiusIncrement_eq_scale_gcdWordCoeff}{DiagonalFreshLossBridge.lean:653}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{other:gcd-word}.\n\\end{prop}\n\n\\begin{prop}[Exact old/foreign split of the diagonal height increment]\n$\\mathrm{diagonalHeightIncrement}(H,s) = \\mathrm{oldMobiusIncrement}(H,s) +\n\\mathrm{finiteForeignChannelIncrement}(H,s)$, a literal finite-sum identity,\nnot an estimate.\n\\lean{Erdos249257.DiagonalFreshLossBridge.diagonalHeightIncrement_eq_oldMobius_add_finiteForeign}{DiagonalFreshLossBridge.lean:829}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{other:gcd-word}.\n\\end{prop}\n\n\\begin{thm}[Exact doubling law for the old-channel increment]\nIf $H$ and $r$ are both even, $\\mathrm{diagonalHeightIncrement}(2H,2r) =\n2\\cdot\\mathrm{diagonalHeightIncrement}(H,r)$; if $H$ is even and $r$ odd, the\nsame doubled height increment equals $\\mathrm{diagonalHeightIncrement}(H,r)$\nunchanged.\n\\lean{Erdos249257.DiagonalFreshLossBridge.diagonalHeightIncrement_two_mul_even}{DiagonalFreshLossBridge.lean:308}\n\\quad \\lean{Erdos249257.DiagonalFreshLossBridge.diagonalHeightIncrement_two_mul_odd}{DiagonalFreshLossBridge.lean:323}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{other:gcd-word}.\n\\end{thm}\n\n\\begin{obs}[Every nonzero foreign phase term is squarefree-supported]\nA ``foreign'' channel $d\\nmid H$ contributes a nonzero phase term to the\nliteral increment only if $d$ is squarefree and $d$ divides exactly one of\nthe two window endpoints $2H{+}s$, $H{+}s$ (never both, since $d\\nmid H$\nforces the two endpoints incongruent mod $d$).\n\\lean{Erdos249257.DiagonalFreshLossBridge.squarefree_of_foreignChannelPhaseTerm_ne_zero}{DiagonalFreshLossBridge.lean:909}\n\\quad \\lean{Erdos249257.DiagonalFreshLossBridge.foreign_endpoint_support_disjoint}{DiagonalFreshLossBridge.lean:852}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{other:gcd-word}.\nThis is a real structural narrowing --- it rules out non-squarefree indices\nand simultaneous double support as sources of foreign contribution --- but\nit narrows a finite bookkeeping decomposition, not the analytic separation\nobligation itself.\n\\end{obs}\n\n\\begin{prop}[Low/top echo doubling]\nIf a foreign channel $d$ has a lower-endpoint hit at offset $s$, its value\nthere is $-\\mu(d)\\cdot\\lfloor(H{+}s)/d\\rfloor$, and its top-endpoint value at\nthe doubled offset $2s$ is exactly twice that quantity in absolute value\n(with sign flipped): $\\mathrm{foreignChannelPhaseTerm}(d,H,2s) =\n2\\bigl(\\mu(d)\\lfloor(H{+}s)/d\\rfloor\\bigr)$.\n\\lean{Erdos249257.DiagonalFreshLossBridge.foreignChannelPhaseTerm_low_double_echo}{DiagonalFreshLossBridge.lean:941}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{other:gcd-word}.\n\\end{prop}\n\n\\subsubsection{Repunit M\u00f6bius numerator is a positive gcd word (T1)}\n\\label{repunit-gcd-word}\n\n\\begin{thm}[T1]\nFor squarefree $r$, the signed repunit numerator polynomial\n$\\sum_{d\\mid r}\\mu(d)(r/d)(1+X^d+\\cdots+X^{r-d})$ equals the ``gcd word''\nwhose coefficient at $X^k$ ($k<r$) is $(r/\\gcd(r,k))\\cdot\\varphi(\\gcd(r,k))$\n--- strictly positive for $k<r$, zero for $k\\ge r$. The squarefree\nhypothesis is kept explicit and is not promoted to arbitrary $r$.\n\\lean{Erdos249257.RepunitMobiusNumerator.mobiusNumeratorPolynomial_eq_gcdWord}{RepunitMobiusNumerator.lean:205}\n\\quad \\lean{Erdos249257.RepunitMobiusNumerator.gcdWordCoeff}{RepunitMobiusNumerator.lean:41}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{other:gcd-word}.\n\\end{thm}\n\n\\begin{cor}\nEvaluation at $X=2$ recovers the integral M\u00f6bius--Mersenne numerator already\nowned by \\texttt{RadicalMobiusShadow}, on the same squarefree boundary.\n\\lean{Erdos249257.RepunitMobiusNumerator.mobiusNumeratorPolynomial_eval_two}{RepunitMobiusNumerator.lean:445}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{other:gcd-word}.\n\\end{cor}\n\n\\subsubsection{Dyadic prefix compression: exact greedy-carry arithmetic}\n\n\\texttt{DyadicPrefixCompression} adds roughly 3200 lines of exact-arithmetic\ninfrastructure for a greedy half-orbit carry construction (Mersenne-weight\nachievement sets); the bulk is machinery for a specific producer hypothesis.\nTwo pieces of genuinely reusable exact arithmetic:\n\n\\begin{defn}\nFor a displayed residual $p/(2L)$, the integer excess numerator above the\nnext dyadic point $2^{-(n+1)}$ is\n$\\mathrm{nextDyadicExcessIntNumerator}(p,n,L) := 2^n p - L$, chosen so the\nskipped-branch comparison is an exact Diophantine inequality rather than a\nreal-valued phase estimate; it obeys the exact doubling recurrence\n$E(p,n{+}1,L) = 2E(p,n,L) + L$.\n\\lean{Erdos249257.nextDyadicExcessIntNumerator}{DyadicPrefixCompression.lean:156}\n\\quad \\lean{Erdos249257.nextDyadicExcessIntNumerator_succ}{DyadicPrefixCompression.lean:161}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{other:dyadic-carry}.\n\\end{defn}\n\n\\begin{prop}[Exact geometric tail bound for the Mersenne-weight remainder]\nFor $n\\ge 2$, $\\mathrm{mersenneWeightRemainder}(n) \\le (4/3)(1/8)^n$, and the\ncorresponding infinite tail from depth $m\\ge 1$ is $\\le (4/21)(1/8)^m$.\n\\lean{Erdos249257.mersenneWeightRemainder_le_four_thirds}{DyadicPrefixCompression.lean:1007}\n\\quad \\lean{Erdos249257.mersenneWeightRemainderTail_le_four_twentyone}{DyadicPrefixCompression.lean:1024}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{other:dyadic-carry}.\n\\end{prop}\n\n\\subsubsection{Pivot-fibre anti-reconstruction: an exact energy identity}\n\n\\texttt{PivotAntiReconstruction} adds roughly 1800 lines building a\nfinite-fibre variance/energy machinery around the ``first-harmonic''\nproducer hypothesis, ending in another chain of \\texttt{DTW...Supply}\n$\\Leftrightarrow$ irrationality equivalences (again reductions, flagged as\nsuch). The exact algebraic core underneath is reusable:\n\n\\begin{prop}[Anchor defect is a squared complex distance]\n$\\mathrm{firstHarmonicAnchorDefect}(h,L,T) = \\sum_{N\\in T}\\|\n\\mathrm{windowFirstExp}(h,N,L) - 1\\|^2$, and it equals\n$2|T| - 2\\sum_{N\\in T}\\mathrm{windowFirstCos}(h,N,L)$.\n\\lean{Erdos249257.TotientTailPeriodKiller.firstHarmonicAnchorDefect_eq_sum_norm_sq}{PivotAntiReconstruction.lean:563}\n\\quad \\lean{Erdos249257.TotientTailPeriodKiller.firstHarmonicAnchorDefect_eq}{PivotAntiReconstruction.lean:551}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{other:pivot-fiber}.\n\\end{prop}\n\n\\begin{lem}[Separated-pairs energy floor]\nFor a finite family $z:T\\to\\mathbb{C}$ and any set of pairs $P\\subseteq\nT\\times T$ each separated by $\\ge\\delta$, $|P|\\cdot\\delta^2 \\le\n\\sum_{i,j\\in T}\\|z_i-z_j\\|^2$.\n\\lean{Erdos249257.TotientTailPeriodKiller.card_mul_sq_le_sum_pairwise_norm_sq_of_separatedPairs}{PivotAntiReconstruction.lean:724}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{other:pivot-fiber}.\n\\end{lem}\n\n\\subsubsection{Actual foreign-residue projection}\n\n\\texttt{ActualForeignResidueProjection} is explicit that it is ``the proof\nconsumer'' for a receipt whose analytic kernel identity remains a separate\nLean validation lane; it supplies the finite bridge, not the estimate.\n\n\\begin{prop}[Explicit shadow is exactly the divisor-channel sum]\nFor $H>0$, $\\mathrm{scaleExplicitShadow}(H) = \\sum_{d\\mid H}\n\\mathrm{residueIncrement}(d,H)$.\n\\lean{Erdos249257.ActualForeignResidueProjection.scaleExplicitShadow_eq_sum_divisors_residueIncrement}{ActualForeignResidueProjection.lean:338}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{other:foreign-residue}.\n\\end{prop}\n\n\\begin{prop}[Complement-noncancellation consumer]\nIf a projection's error against the true foreign defect is controlled by\nthe closed geometric budget $\\mathrm{foreignComplementBound}(H,D)$, and the\nfinite rational state $\\mathrm{scaleExplicitShadow}(H) +\n\\mathrm{projectedForeignDefect}(H,D)$ is farther from every integer than\nthat budget, then $S\\ne \\mathrm{fullTargetHit}$ at scale $H$.\n\\lean{Erdos249257.ActualForeignResidueProjection.scaleFullTarget_miss_of_projected_separation}{ActualForeignResidueProjection.lean:414}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{other:foreign-residue}. The\ncontrol and separation hypotheses are exactly the two remaining open\nobligations; the consumer is a real theorem, not a claim they hold.\n\\end{prop}\n\n\\subsubsection{Geometric coprimality: a coordinate that provably cannot decide \\#249 alone}\n\n\\texttt{GeometricCoprimality} relocates \\#249 onto the visible-lattice-point\nmass of coprime pairs, and then proves that this specific coordinate change,\nby itself, cannot decide irrationality at any rational parameter --- a\ngenuine proved barrier against an entire family of approaches.\n\n\\begin{thm}[Visible-point count is the totient, uniformly]\nFor every $n\\in\\mathbb{N}$, $\\#\\{(a,b): a+b=n,\\ 0<a,\\ \\gcd(a,b)=1\\} =\n\\varphi(n)$, with no case split at $n=0,1$.\n\\lean{GeometricCoprimality.card_antidiagonal_filter_pos_coprime}{GeometricCoprimality.lean:56}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{other:coprime-lattice}.\n\\end{thm}\n\n\\begin{prop}[Coprime-pair lattice mass bridges]\nFor $0\\le r<1$: $\\sum_{(a,b)\\ \\mathrm{coprime},\\,a\\ge 1} r^{a+b} =\n\\sum_n \\varphi(n)r^n$; over strictly positive coprime pairs the same sum is\n$\\sum_n\\varphi(n)r^n - r$; and layering by $\\gcd = g$ recovers the full\npositive-quadrant mass $\\sum_g[\\text{layer }g] = (r/(1-r))^2$, which equals\n$1$ exactly at $r=1/2$ (two independent fair-coin waiting times have a\nfinite gcd almost surely).\n\\lean{GeometricCoprimality.tsum_coprime_pair_pow_eq_tsum_totient_mul_pow}{GeometricCoprimality.lean:120}\n\\quad \\lean{GeometricCoprimality.tsum_pos_coprime_pair_pow}{GeometricCoprimality.lean:182}\n\\quad \\lean{GeometricCoprimality.tsum_gcd_layer_pos_coprime_pow}{GeometricCoprimality.lean:328}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{other:coprime-lattice}.\n\\end{prop}\n\n\\begin{thm}[Rational geometric-model falsifier --- a proved barrier]\nFor every $0\\le r<1$, $\\sum_{(a,b)\\ \\mathrm{coprime},\\, a,b\\ge 1}\n\\dfrac{r^{a+b}}{1-r^{a+b}} = \\Bigl(\\dfrac{r}{1-r}\\Bigr)^2$, an elementary\nrational function of $r$, hence rational at every rational $r$ including\n$r=1/2$.\n\\lean{GeometricCoprimality.tsum_pos_coprime_lambert_eq_sq}{GeometricCoprimality.lean:407}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{other:coprime-lattice}.\n\\end{thm}\n\n\\begin{obs}\nThis is a proved barrier, not an observed pattern: it names exactly the\nclass of argument it rules out. Coprime-pair restriction, exact\nStern--Brocot-type splitting by $\\gcd$, and geometric cylinder decay of the\nsummand under the map $n\\mapsto r^n/(1-r^n)$, used together and evaluated at\na rational $r$, cannot imply irrationality of the resulting sum, because the\nidentical construction is provably rational at \\emph{every} rational $r$ in\n$[0,1)$ --- the mechanism has zero sensitivity to whether $r$ itself is\nrational. Any future argument built on this specific coordinate must use\nsome further feature of $r=1/2$ beyond coprimality, gcd-layering, and\ngeometric decay; the coordinate change is real and interesting, but this\nparticular route through it is closed.\n\\end{obs}\n\n\\subsubsection{The lcm-diagonal collapse: reduced to one \\texorpdfstring{$\\mathbb{N}$}{N}-indexed predicate}\n\n\\texttt{LcmDiagonalReduction} (module \\texttt{TotientTailPeriodKiller},\n``wave 23'') removes the second free parameter from the wave-22\none-parameter reduction, producing a single decidable $\\mathbb{N}$-indexed\nstatement equivalent to \\#249. The module's own header states plainly that\nthe resulting supply hypothesis ``is the open content of \\#249 and is NOT\nclaimed''; the equivalence itself is a reduction, reported as such.\n\n\\begin{obs}[Diagonal collapse --- reduction, not a result]\n$\\mathrm{irrational}(S)$ follows from: for every $t_0$, some $t\\ge t_0$\nadmits a certified kill at the diagonal point $(H_t,H_t)$, $H_t :=\n\\mathrm{lcm}(1,\\dots,t)$. This is proved as an implication only; the\nantecedent is the unproved open content.\n\\lean{Erdos249257.TotientTailPeriodKiller.irrational_totient_series_of_lcm_diagonal_certificate_supply}{LcmDiagonalReduction.lean:112}\n\\quad \\ev{Lean} (implication proved; antecedent \\ev{Open})\n\\quad \\scale{cofinal} \\quad \\coord{other:lcm-diagonal}.\n\\end{obs}\n\n\\begin{thm}[Window structure of the lcm ray --- unconditional]\nBelow $2t$, the only indices $j$ that fail to divide $H_t$ are bare prime\npowers exceeding $t$; every $j\\le t$ divides $H_t$ outright. On a ``clean''\ndivisor $j\\mid H_t$ (every prime of $j$ still divides $H_t/j$) the window\ntotient factors exactly:\n$\\varphi(qH_t+j) = \\varphi(j)\\cdot\\varphi(q(H_t/j)+1)$.\n\\lean{Erdos249257.TotientTailPeriodKiller.eq_prime_pow_of_not_dvd_periodLcm}{LcmDiagonalReduction.lean:137}\n\\quad \\lean{Erdos249257.TotientTailPeriodKiller.totient_periodLcm_ray_split}{LcmDiagonalReduction.lean:208}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{other:lcm-diagonal}.\n\\end{thm}\n\n\\begin{obs}[Diagonal deposits]\nThe kernel decides $P(t)$ (a certified diagonal kill) unconditionally for\nevery $1\\le t\\le 8$ (totient arguments stay $\\le 130$), at tabulated depths.\n\\lean{Erdos249257.TotientTailPeriodKiller.certifiedKill_periodLcm_diagonal_upto_six}{LcmDiagonalReduction.lean:257}\n\\quad \\lean{Erdos249257.TotientTailPeriodKiller.certifiedKill_periodLcm_diagonal_at_eight}{LcmDiagonalReduction.lean:266}\n\\quad \\ev{Cert} \\quad \\scale{fixed} \\quad \\coord{other:lcm-diagonal} --- a\nfinite floor, not evidence toward the cofinal supply.\n\\end{obs}\n\n\\subsubsection{Cyclotomic anchored kills: unconditional prime support, plus new exclusion certificates}\n\nThe newly published \\texttt{ErdosProblems/Erdos249/CyclotomicAnchoredKill.lean}\n(3222 lines, namespace \\texttt{ErdosProblems.Erdos249.CyclotomicAnchoredKill})\ndischarges the abstract order-consumer producer of\n\\S\\ref{prime-ray-curvature} completely for the concrete polynomial $X-2$, and\nseparately deposits new kernel-checked denominator exclusions. Its middle\n$\\approx\\!1600$ lines are a further ladder of\n\\texttt{CyclotomicPrime...CarryKillSupply} $\\Leftrightarrow$ irrationality\nequivalences; those are reductions and are reported only as architecture,\nnot as narrowing progress.\n\n\\begin{thm}[Unconditional unbounded prime support for the Mersenne layer]\nEvery prime divisor $p$ of $2^q-1$, for prime $q$, satisfies $q\\mid p-1$\n(the order of $2$ mod $p$ is exactly $q$, by Fermat/Lagrange in\n$(\\mathbb{Z}/p)^\\times$); consequently the prime divisors appearing in the\nlayers $\\{2^n-1\\}$ are unbounded, unconditionally, with no cyclotomic\nresultant hypothesis left open.\n\\lean{ErdosProblems.Erdos249.CyclotomicAnchoredKill.prime_index_dvd_pred}{CyclotomicAnchoredKill.lean:33}\n\\quad \\lean{ErdosProblems.Erdos249.CyclotomicAnchoredKill.mersenneLayer_unboundedPrimeDivisorSupply}{CyclotomicAnchoredKill.lean:105}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{other:cyclotomic-anchor}.\n\\end{thm}\n\n\\begin{thm}[Unconditional clean cyclotomic anchor existence]\nFor every period $h>0$ and threshold $N_0$, there exist a prime $q$ and a\nprime factor $p$ of $|\\Phi_{hq}(2)|$ (the binary cyclotomic layer) with $p$\ncoprime to $hq$, $hq\\mid p-1$, and $p-1\\ge N_0$. The characteristic-prime\nexceptional case in the cyclotomic order decomposition is eliminated\ndirectly, by choosing $q>2^h$ (rules out $p=q$) and $q>h$ (rules out\n$p\\mid h$).\n\\lean{ErdosProblems.Erdos249.CyclotomicAnchoredKill.exists_clean_binaryCyclotomicAnchor}{CyclotomicAnchoredKill.lean:1464}\n\\quad \\lean{ErdosProblems.Erdos249.CyclotomicAnchoredKill.binaryCyclotomicLayer_prime_order_decomposition}{CyclotomicAnchoredKill.lean:1390}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{other:cyclotomic-anchor}.\n\\end{thm}\n\n\\begin{obs}\nThese two theorems are genuine unconditional number theory --- an exact-order\nargument and a Dirichlet-plus-cyclotomic-root existence construction --- and\nthey close the abstract producer hypotheses of\n\\texttt{PrimeRayCyclotomicCurvature} (\\S\\ref{prime-ray-curvature}) for the\n$X-2$ layer with no residual conditional. Neither result touches the\nirrationality question itself; they supply arithmetic support for a\n\\emph{different}, still-open cofinal predicate about certified kills at\ncyclotomic-anchored periods.\n\\end{obs}\n\n\\begin{obs}[New certified exclusions at period 30]\nThe kernel decides two independent certificates at the composite cyclotomic\nanchor period $h=30$ (natural basepoint $N=300$, and the prime-anchored\nbasepoint $N=330$), yielding a genuine new denominator exclusion:\n$S\\ne r$ for every $r\\in\\mathbb{Q}$ with $r.\\mathrm{den}\\mid 2^{300}(2^{30}-1)$.\n\\lean{ErdosProblems.Erdos249.CyclotomicAnchoredKill.certifiedKill_cyclotomic_30_331_natural}{CyclotomicAnchoredKill.lean:3167}\n\\quad \\lean{ErdosProblems.Erdos249.CyclotomicAnchoredKill.totient_series_ne_rat_of_den_dvd_30_300}{CyclotomicAnchoredKill.lean:3188}\n\\quad \\ev{Cert} \\quad \\scale{fixed} \\quad \\coord{other:cyclotomic-anchor} --- a\nfinite exclusion, not evidence toward the cofinal supply.\n\\end{obs}\n\n\\subsubsection{Prime-ray cyclotomic curvature: the abstract order-consumer producer}\n\\label{prime-ray-curvature}\n\n\\begin{thm}[Bounded-degree order realisability forces unbounded prime support]\nIf every prime divisor of a layer $C(mq)$ has order at most $d$ in the\nrelevant residue group (\\texttt{BoundedDegreeOrderConsumer}), then for every\nfixed finite set of primes $S$, all sufficiently large prime indices $q$\navoid $S$ entirely on layer $C(mq)$; combined with a nontrivial-layer\nhypothesis this forces arbitrarily large new prime divisors on cofinally\nmany prime indices.\n\\lean{ErdosProblems.Erdos249.PrimeRayCyclotomicCurvature.finitePrimeSupportEscape_of_orderConsumer}{PrimeRayCyclotomicCurvature.lean:116}\n\\quad \\lean{ErdosProblems.Erdos249.PrimeRayCyclotomicCurvature.unboundedPrimeDivisorSupply_of_orderConsumer}{PrimeRayCyclotomicCurvature.lean:172}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{other:cyclotomic-anchor}. This\nis the abstract producer instantiated unconditionally for $X-2$ in\n\\S\\ref{prime-ray-curvature}'s sibling result above; polynomial resultant\nrealisability and Archimedean growth for other layers remain explicit\nupstream obligations, not proved here.\n\\end{thm}\n\n\\subsubsection{Rank-one subrank obstruction: a uniform proved barrier}\n\\label{rank-one-subrank}\n\n\\texttt{RankOneSubrankObstruction} is, together with the coprimality\nfalsifier above, the sharpest new barrier in this batch: it names an entire\nfamily of candidate linear-form constructions and proves, uniformly, that\nnone of them can work.\n\n\\begin{thm}[Uniform rank-one no-go]\nLet $\\Theta_r$ be the M\u00f6bius--Mersenne ladder rung and let $t(Y,r)$ be its\nfirst $Y$ atoms. For every $e\\ge 1$ and $Y\\ge 4$,\n$$t(Y,e{+}2)^2/t(Y,2e{+}2) - \\Theta_2 > 1/480.$$\nEvery rank-one monomial Schur quotient built this way overshoots $\\Theta_2 =\nS-1/2$ by a fixed positive margin, uniformly in $e$ and $Y$; the bound uses\nonly two interval facts (every rung $r\\ge 3$ lies in $[1429/1512,1)$, and\nevery length-$\\ge\\!4$ prefix is within $1/3584$ of its rung).\n\\lean{ErdosProblems.Erdos249.RankOneSubrankObstruction.rankOneSubrankQuotient_sub_theta_two_gt}{RankOneSubrankObstruction.lean:238}\n\\quad \\lean{ErdosProblems.Erdos249.RankOneSubrankObstruction.mobiusMersenneTheta_ge_alpha}{RankOneSubrankObstruction.lean:163}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{other:rank-one-subrank}.\n\\end{thm}\n\n\\begin{prop}[The gap survives positive averaging]\nFor any nonempty finite positively-weighted family of admissible quotients,\nthe weighted average still overshoots $\\Theta_2$ by more than $1/480$; and\nevery primitive integer linear form $q\\Theta_2 - p$ realising such a\nquotient satisfies $|q\\Theta_2 - p| > q/480$.\n\\lean{ErdosProblems.Erdos249.RankOneSubrankObstruction.positive_direct_sum_sub_theta_two_gt}{RankOneSubrankObstruction.lean:300}\n\\quad \\lean{ErdosProblems.Erdos249.RankOneSubrankObstruction.primitive_form_abs_gt}{RankOneSubrankObstruction.lean:341}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{other:rank-one-subrank}.\n\\end{prop}\n\n\\begin{obs}\nThis is a proved barrier stated at the required precision: the class of\nargument it rules out is exactly ``a primitive rational linear form for\n$\\Theta_2$ (equivalently for $S$) obtained as a rank-one strict-subrank\nmonomial quotient of finite M\u00f6bius--Mersenne prefixes,'' and the reason is\nthe explicit uniform lower bound above, not an empirical failure report.\nIt does not touch, and is not claimed to touch, linear forms built by any\nother mechanism.\n\\end{obs}\n\n\\subsubsection{Period-multiple escape: the nesting identity and eight new exclusions}\n\n\\texttt{PeriodMultipleEscape} proves the period-multiple kill supply is\n\\emph{exactly equivalent} to irrationality of $S$ (both directions,\nsufficiency by telescoping the tail-period law, necessity by certificate\ncompleteness) --- reported here as an equivalence-class reduction, not\nprogress --- and separately deposits genuinely new denominator exclusions.\n\n\\begin{thm}[Nesting: the cyclotomic fan collapses onto the dyadic tower]\n$Q_{a+b,N} = 2^b Q_{a,N} + Q_{b,N+a}$ (block concatenation), so the order-4\ncyclotomic channel at height $h$ is literally the order-2 channel at height\n$2h$: the 2--3--4 fan of channels is a nested family along the tower\n$h,2h,4h,\\dots$, not three unrelated moduli.\n\\lean{ErdosProblems.Erdos249.PeriodMultipleEscape.totientBlock_add}{PeriodMultipleEscape.lean:77}\n\\quad \\lean{ErdosProblems.Erdos249.PeriodMultipleEscape.totientBlock_two_mul}{PeriodMultipleEscape.lean:94}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{other:lcm-period-multiple}.\n\\end{thm}\n\n\\begin{obs}[Supply $\\Leftrightarrow$ irrational --- reduction, not a result]\n\\lean{ErdosProblems.Erdos249.PeriodMultipleEscape.periodMultipleKillSupply_iff_irrational}{PeriodMultipleEscape.lean:153}\n\\quad \\ev{Lean} (equivalence proved; both directions of the underlying\npredicate are exactly as open as \\#249 itself)\n\\quad \\scale{cofinal} \\quad \\coord{other:lcm-period-multiple}. The paper's\nexisting diagonal bank already certifies kills at $H_t$ through $t=64$; this\nmodule's equivalence contributes no new information about which $t$ beyond\n64 will certify.\n\\end{obs}\n\n\\begin{obs}[Eight new certified denominator exclusions past the 64-smooth diagonal bank]\nThe kernel certifies kills, unconditionally, at the eight prime-power\nperiods the diagonal bank could not reach ($67,81,97,101,121,125,127,128$,\nall at basepoint $N=300$), each yielding $S\\ne r$ for every $r$ with\n$r.\\mathrm{den}\\mid 2^{300}(2^h-1)$ --- new odd denominator classes,\nincluding the Cole factors of $2^{67}-1$ and the Mersenne prime $2^{127}-1$.\nOne of the eight is certified at its own locked depth $L=h=67$, the concrete\ninstance of the sufficient (not known necessary) depth-equals-period form\n$\\mathrm{ApFullDepthEscape}$.\n\\lean{ErdosProblems.Erdos249.PeriodMultipleEscape.certifiedKill_67_300}{PeriodMultipleEscape.lean:192}\n\\quad \\lean{ErdosProblems.Erdos249.PeriodMultipleEscape.certifiedKill_127_300}{PeriodMultipleEscape.lean:211}\n\\quad \\lean{ErdosProblems.Erdos249.PeriodMultipleEscape.certifiedKill_67_300_fullDepth}{PeriodMultipleEscape.lean:219}\n\\quad \\lean{ErdosProblems.Erdos249.PeriodMultipleEscape.totient_series_ne_rat_of_den_dvd_127_300}{PeriodMultipleEscape.lean:255}\n\\quad \\ev{Cert} \\quad \\scale{fixed} \\quad \\coord{other:lcm-period-multiple}\n--- eight finite exclusions, not evidence toward the cofinal supply.\n\\end{obs}\n\n\\subsubsection{Strict prime-orbit escape: a sharper reduction}\n\n\\texttt{TotientStrictPrimeEscape} sharpens the legacy first-harmonic\nproducer's threshold from a $4/5$ gap to a strict $9/10$ gap with an\nadaptive truncation budget, and proves the sharper predicate still closes\n\\#249 through the existing singleton-certificate endpoint. This is, exactly\nas its own comment states, a reduction: ``the producer itself remains\nunproved.''\n\\lean{ErdosProblems.Erdos249.irrational_totient_series_of_naturalPrimeTailOrbitStrictGap}{TotientStrictPrimeEscape.lean:66}\n\\quad \\ev{Lean} (implication proved; antecedent\n\\ev{Open})\n\\quad \\scale{cofinal} \\quad \\coord{other:first-harmonic}.\n\n\\subsubsection{Finite Euler-sieve algebra}\n\n\\texttt{FiniteEulerSieve} records the elementary finite-stage identities\nbehind the squared-M\u00f6bius Euler factor, with no transcendence claim\nasserted: $(1-2/p+1/p^2) = (1-1/p)^2$ and its degree-two analogue, and the\nsecond finite difference of $1+p+\\cdots+p^{e}$ recovering\n$p^{e+1}(p-1)$, the prime-power totient row.\n\\lean{ErdosProblems.Erdos249.FiniteEulerSieve.muSqEulerFactor_one}{FiniteEulerSieve.lean:28}\n\\quad \\lean{ErdosProblems.Erdos249.FiniteEulerSieve.sigmaPrimePow_secondDiff}{FiniteEulerSieve.lean:51}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{other:euler-sieve}.\n\n% ---- part a249_p2 ----\n\\section{The scale ladder}\n\nEvery statement in the Erd\\H{o}s \\#249 corpus that this paper draws on is tagged with a\n\\emph{scale}: \\scale{fixed} (a finite, explicitly enumerated set of parameter values, typically\nclosed by \\texttt{decide}/\\texttt{interval\\_cases}), \\scale{bounded} (holds for all parameter\nvalues on one side of a threshold, e.g.\\ $\\forall a\\ge 8$, but the proof or the constants inside\nit do not survive removing the threshold), \\scale{uniform} (holds unconditionally for every value\nof every free parameter, with no scale restriction at all), and \\scale{cofinal} (an existential\nclaim of the shape $\\forall N_0\\,\\exists N\\ge N_0,\\ P(N)$ \u2014 infinitely often, arbitrarily far out).\nThe source tables underlying this ladder also mark a handful of pure identities and converters\n\\texttt{n/a} when they are not indexed by any problem-scale parameter at all (they are definitions\nor unconditional equivalences); for ladder purposes these are folded into \\scale{uniform}, since an\nunconditional statement is, if anything, stronger than a uniform one. This convention is applied\nuniformly below and is stated here once rather than re-flagged on every row.\n\n\\#249's own supply obligation is \\emph{exactly} cofinal. The reduction chain\n(\\S\\ref{sec:249-supply-chain-forward-ref}, and see \\lean{irrational\\_totientSeries\\_iff\\_actualLcmOrbitNonintegralitySupply}{TotientActualLcmOrbitNonintegrality.lean:37})\nshows\n\\[\n  \\mathrm{Irrational}\\Bigl(\\sum_{n\\ge 1}\\varphi(n)/2^n\\Bigr) \\iff \\forall a_0\\ \\exists a\\ge a_0,\\ \\texttt{actualLcmTailOrbit}\\ a \\notin \\mathrm{range}(\\mathbb{Z}\\to\\mathbb{R}),\n\\]\nan \\texttt{cofinal} statement in the exponent $a$ that indexes $H=\\mathrm{periodLcm}(2^a)$. Nothing\nweaker in scale can close \\#249; the entire question is whether the corpus's uniform, bounded, and\nfixed machinery can be pushed to cofinal. The table below lists every catalogued \\#249 result,\ngrouped by scale, with a horizontal rule separating everything that is \\emph{proved} (uniform,\nbounded, fixed) from the handful of statements that are themselves the \\emph{cofinal target} or\nconsumers of it (below the rule). Quantifier prefixes are written out explicitly and exactly as\nrecorded against the Lean source; nothing is compressed to ``$\\forall\\dots$'' where the source\nspecifies bounds.\n\n\\begingroup\n\\fontsize{6}{7.2}\\selectfont\n\\setlength{\\tabcolsep}{3pt}\n\\begin{longtable}{p{3.2cm}p{2.7cm}p{1.1cm}p{5.8cm}p{1.4cm}}\n\\caption{The \\#249 scale ladder. Every row above the rule is proved; every row below it is the\ncofinal target or a direct consumer of it.}\\\\\n\\toprule\n\\textbf{Result} & \\textbf{Lean site} & \\textbf{Scale} & \\textbf{Quantifier prefix} & \\textbf{Coordinate}\\\\\n\\midrule\n\\endfirsthead\n\\toprule\n\\textbf{Result} & \\textbf{Lean site} & \\textbf{Scale} & \\textbf{Quantifier prefix} & \\textbf{Coordinate}\\\\\n\\midrule\n\\endhead\n\\midrule \\multicolumn{5}{r}{\\emph{continued on next page}} \\\\\n\\endfoot\n\\bottomrule\n\\endlastfoot\n\n\\multicolumn{5}{l}{\\emph{-- \\scale{uniform} (includes source-tagged \\texttt{n/a} unconditional identities/converters) --}}\\\\\ntotientTail well-defined & \\lean{totientTail}{TotientTailPeriodKiller.lean:57} & uniform & $\\forall N$ & binary-digit\\\\\n$2^N\\!\\cdot\\! S$ split & \\lean{two\\_pow\\_mul\\_totient\\_series\\_eq}{TotientTailPeriodKiller.lean:150} & uniform & $\\forall N$ & binary-digit\\\\\nwindowDiscrepancy (def) & \\lean{windowDiscrepancy}{TotientTailPeriodKiller.lean:66} & uniform & $\\forall h\\,\\forall N\\,\\forall L$ & other-binary-window\\\\\ncertifiedKill Sep$(h,N,L)$ (def) & \\lean{certifiedKill}{TotientTailPeriodKiller.lean:72} & uniform & $\\forall h\\,\\forall N\\,\\forall L$ & other-binary-window\\\\\ncertificate depth floor & \\lean{certifiedKill\\_depth\\_floor}{TotientTailPeriodKiller.lean:79} & uniform & $\\forall h\\,\\forall N\\,\\forall L$, given \\texttt{certifiedKill}$\\,h\\,N\\,L$ & other-binary-window\\\\\nkill engine soundness & \\lean{tail\\_diff\\_notMem\\_int\\_of\\_certifiedKill}{TotientTailPeriodKiller.lean:262} & uniform & $\\forall h\\,\\forall N\\,\\forall L$, given \\texttt{certifiedKill} & other-binary-window\\\\\ncertificates are complete receipts & \\lean{exists\\_certifiedKill\\_iff\\_tail\\_diff\\_notMem\\_int}{LcmConeFlatness.lean:316} & uniform & $\\forall h\\,\\forall N$ & other-binary-window\\\\\nforced integrality under den.\\ divisibility & \\lean{tail\\_diff\\_int\\_of\\_den\\_dvd}{TotientTailPeriodKiller.lean:327} & uniform & $\\forall N\\,\\forall h\\,\\forall r{:}\\mathbb{Q}$, given $S=r$, $r.\\mathrm{den}\\mid 2^N(2^h{-}1)$ & binary-digit\\\\\ntail-period law & \\lean{eventual\\_period\\_of\\_not\\_irrational}{TotientTailPeriodKiller.lean:358} & uniform & $\\lnot\\mathrm{Irrational}\\,S\\to\\exists h{>}0\\,\\exists N_0\\,\\forall N{\\ge}N_0$ & binary-digit\\\\\nperiodLcm basic facts & \\lean{periodLcm}{CarrySurvivorExtinction.lean:515} & uniform & $\\forall t$ & other-lcm-period-ray\\\\\nnon-divisor$<2t$ is a bare prime power & \\lean{eq\\_prime\\_pow\\_of\\_not\\_dvd\\_periodLcm}{CertificateKernel.lean:18613} & uniform & $\\forall t\\,\\forall j\\,(0{<}j{<}2t)$, given $j\\nmid\\mathrm{periodLcm}\\,t$ & other-lcm-window\\\\\nLCM-ray multiplicative split & \\lean{totient\\_periodLcm\\_ray\\_split}{LcmDiagonalReduction.lean:208} & uniform & $\\forall t\\,\\forall j\\,\\forall q$, given $j\\mid\\mathrm{periodLcm}\\,t$, clean-divisor & other-lcm-window-multiplicative\\\\\nrationality flattens the whole LCM cone & \\lean{rational\\_totient\\_series\\_forces\\_lcm\\_cone\\_flatness}{CertificateKernel.lean:18674} & uniform & $\\lnot\\mathrm{Irrational}\\,S\\to\\exists t_1\\,\\forall t{\\ge}t_1\\,\\forall q{>}0\\,\\forall m$ & other-lcm-cone\\\\\nsecond-difference kill engine & \\lean{second\\_diff\\_notMem\\_int\\_of\\_certifiedRank2Kill}{CertificateKernel.lean:18729} & uniform & $\\forall h\\,\\forall N\\,\\forall L$, given \\texttt{certifiedRank2Kill} & other-binary-window\\\\\ncone-menu nonintegral pair & \\lean{exists\\_nonintegral\\_pair\\_of\\_coneNonflatCert}{CertificateKernel.lean:18801} & uniform & $\\forall H\\,\\forall L\\,\\forall Q{\\ne}\\emptyset$, given \\texttt{coneNonflatCert} & other-lcm-cone-menu\\\\\nsurvivorKill engine (carry-orbit route) & \\lean{tail\\_diff\\_notMem\\_int\\_of\\_survivorKill}{CarrySurvivorExtinction.lean:428} & uniform & $\\forall h\\,\\forall N\\,\\forall K$, given \\texttt{survivorKill} & other-carry-orbit\\\\\nFarey neighbour denominator law & \\lean{farey\\_gap}{GapFareyBound.lean:51} & uniform & $\\forall a\\,b\\,c\\,d\\,r\\,s{:}\\mathbb{Z}$, $b{>}0,d{>}0,bc{-}ad{=}1$, unimodular-between & farey\\\\\nDirichlet near-integer criterion & \\lean{irrational\\_of\\_den\\_mul\\_abs\\_sub\\_tendsto\\_zero}{CertificateKernel.lean:5371} & uniform & $\\forall u{:}\\mathbb{N}\\to\\mathbb{Q}\\,\\forall x{:}\\mathbb{R}$ & n/a (generic)\\\\\nErd\\H{o}s 1948 near-integer criterion & \\lean{irrational\\_of\\_int\\_mul\\_near\\_int}{CertificateKernel.lean:5792} & uniform & $\\forall \\xi{:}\\mathbb{R}$, $\\forall q{>}0\\,\\exists m\\,z\\,0{<}|m\\xi{-}z|{<}1/q$ & n/a (generic)\\\\\nfinite-minor $\\Rightarrow$ linear independence & \\lean{linearIndependent\\_of\\_separatedMinorCertificate}{TotientMahlerDefect.lean:81} & uniform & $\\forall\\,\\mathrm{family}{:}\\iota\\to\\mathbb{N}\\to\\mathbb{Q}$, given SeparatedMinorCertificate & n/a (generic)\\\\\ndyadic totient-kernel rank $=2^e{+}1$ & \\lean{linearIndependent\\_canonicalTotientKernelFamily}{TotientMahlerDefect.lean:900} & uniform & $\\forall e$ & other-dyadic-kernel-rank\\\\\nrationality forces unbounded carry-kernel rank & \\lean{not\\_irrational\\_totientSeries\\_implies\\_unbounded\\_carryRank\\_unconditional}{TotientCarryKernelRigidity.lean:284} & uniform & $\\lnot\\mathrm{Irrational}\\,S\\to\\exists v{>}0\\,\\exists u\\,\\forall e$ & other-carry-kernel-rank\\\\\nno fixed $D$ clears all primitive Euler jets & \\lean{no\\_fixed\\_integral\\_primitive\\_euler\\_index}{PrimitiveDeterminantLift.lean:169} & uniform & $\\forall D{>}0$ & mobius-mersenne\\\\\nMersenne\u2013Lambert ladder value identities & \\lean{tsum\\_totient\\_div\\_pow\\_two\\_eq\\_pnat\\_half\\_pow}{CertificateKernel.lean:18063} & uniform & value identities, $L(\\mu){=}\\tfrac12$, $L(\\varphi){=}2$, $L(1){=}$Erd\\H{o}s--Borwein & mobius-mersenne\\\\\nprime-power reduced-denom.\\ unit-gap ceiling & \\lean{reducedDenominatorUnitGapCert\\_prime\\_pow\\_iff}{PrimitiveWeightCertificate.lean:54} & uniform & $\\forall p\\ \\text{prime}\\,\\forall e{>}0$ & farey\\\\\npositive rational-difference lower bound & \\lean{positive\\_rational\\_difference\\_lower\\_bound}{PrimitiveRationalGapSupply.lean:31} & uniform & $\\forall\\ \\text{whole}\\,\\text{pfx}{:}\\mathbb{Q},\\ \\text{pfx}{<}\\text{whole}$ & n/a (generic)\\\\\nM\\\"obius-square identity ($S=\\tfrac12+\\Sigma$) & \\lean{tsum\\_totient\\_half\\_pow\\_eq\\_half\\_add\\_moebius\\_sq}{SquaredMersenneDiagonalEnclosure.lean:50} & uniform & unconditional & mobius-mersenne\\\\\nsquared-Lambert transfer engine & \\lean{tsum\\_lambert\\_linear\\_weight\\_sq\\_pure}{GcdMomentCalculus.lean:105} & uniform & $\\forall w{:}\\mathbb{N}\\to\\mathbb{R}\\,(|w(d)|{\\le}d)\\,\\forall r\\in[0,1)$ & mobius-mersenne\\\\\n$L_2(\\mu)=S-\\tfrac12$ & prose:L2\\_of\\_mu\\_is\\_249 & uniform & unconditional & mobius-mersenne\\\\\n$L_2(1)=\\zeta_q(2)-\\zeta_q(1)$ identity & \\lean{tsum\\_one\\_div\\_mersenne\\_sq\\_eq\\_sigma\\_sub\\_tau\\_series}{GcdMomentCalculus.lean:216} & uniform & unconditional & mobius-mersenne\\\\\n$L_2(\\varphi)$ = gcd moment & \\lean{tsum\\_totient\\_div\\_mersenne\\_sq\\_eq\\_gcd\\_moment\\_series}{GcdMomentCalculus.lean:235} & uniform & unconditional & mobius-mersenne/probability\\\\\nlevel-mirror table & prose:level\\_mirror\\_table & uniform & unconditional & mobius-mersenne\\\\\ngcd-divisibility factorizes & \\lean{tsum\\_pos\\_pair\\_both\\_dvd\\_half\\_eq\\_inv\\_mersenne\\_sq}{GcdMomentCalculus.lean:266} & uniform & $\\forall d{:}\\mathbb{N},\\,d{>}0$ & probability\\\\\nreduced-direction law $\\Sigma=1$ & \\lean{tsum\\_pos\\_coprime\\_inv\\_mersenne\\_eq\\_one}{GcdMomentCalculus.lean:349} & uniform & unconditional (sum over all coprime pairs) & probability\\\\\nStern--Brocot cylinder recursion & \\lean{cylinderMass\\_split}{GcdMomentCalculus.lean:474} & uniform & $\\forall a\\,b{:}\\mathbb{N}^+\\,\\forall\\text{depth}\\,d$ & other-stern-brocot\\\\\nrepunit gcd-word T1 & \\lean{mobiusNumeratorPolynomial\\_eq\\_gcdWord}{RepunitMobiusNumerator.lean:205} & uniform & $\\forall r\\,(\\mathrm{Squarefree}\\,r)\\,\\forall k{<}r$ & cyclotomic\\\\\neval-at-2 recovers numerator & \\lean{mobiusNumeratorPolynomial\\_eval\\_two}{RepunitMobiusNumerator.lean:320} & uniform & $\\forall r\\,(\\mathrm{Squarefree}\\,r)$ & cyclotomic\\\\\nradical-shadow scale decomposition & \\lean{scaledMobiusShadow\\_eq\\_radicalBase}{RadicalMobiusShadow.lean:125} & uniform & $\\forall H{>}0\\,\\forall r{>}0$ & mobius-mersenne\\\\\ncyclotomic congruence T2 & \\lean{cyclotomic\\_dvd\\_mobiusNumeratorPolynomial\\_sub}{CyclotomicProjectionOfShadow.lean:299} & uniform & $\\forall r\\,(\\mathrm{Squarefree}\\,r)\\,\\forall m\\,(m{\\mid}r)$ & cyclotomic\\\\\ntop-fibre survives T3 & \\lean{mobiusNumerator\\_gcd\\_cyclotomicValue}{CyclotomicProjectionOfShadow.lean:350} & uniform & $\\forall r\\,(\\mathrm{Squarefree}\\,r)$ & cyclotomic\\\\\nupper-half prime channel survival T4 & \\lean{upperHalfChannel\\_product\\_dvd\\_den\\_of\\_scale\\_primeFactors\\_le}{MersenneShadowCyclotomicNoncollapse.lean:809} & uniform & $\\forall t\\,\\forall p\\ \\text{prime}\\in(t/2,t]$, scale-coprime-to-$C$ & cyclotomic\\\\\ndenominator lower bound / exact value & \\lean{lcmHeight\\_scaledMobiusShadow\\_den\\_exact}{MersenneShadowDenominatorGrowth.lean:147} & uniform & $\\forall t$ (exact-value); $t{\\ge}5$ (lower bound) & mobius-mersenne\\\\\nprime-power jump recurrence T5 & \\lean{cyclotomic\\_dvd\\_primeJump\\_new\\_fibre}{PrimePowerJumpDynamics.lean:281} & uniform & $\\forall p\\ \\text{prime}\\,(p{\\nmid}r)\\,\\forall m\\,(m{\\mid}r)$ & cyclotomic\\\\\nsquared-Mersenne diagonal tail enclosure & \\lean{mobius\\_square\\_series\\_eq\\_partial\\_add\\_tail}{SquaredMersenneDiagonalEnclosure.lean:72} & uniform & $\\forall D{:}\\mathbb{N}$ & mobius-mersenne\\\\\n$(3,5)$ joint annihilator & \\lean{joint35\\_oldChannel\\_zero}{JointExponentTransport.lean:71} & uniform & $\\forall d{>}0\\,(d{\\mid}H)$; general: finite affine annihilator & other-lcm-diagonal\\\\\ncomposite-dilation defect identity & \\lean{supportCoeff\\_mul\\_eq\\_add\\_defect}{CompositeDilationDefect.lean:30} & uniform & $\\forall a{\\in}A\\,(a{>}0)\\,\\forall x{>}0$ & other-divisor-support-coeff\\\\\nsublogarithmic zero-window T11 & \\lean{supportCoeffZeroWindow\\_length\\_le\\_eps\\_logb\\_add}{SublogDivisorCoverage.lean:392} & uniform & $\\forall \\varepsilon{>}0\\,\\exists B$, given rationality of the base-2 support series & other-support-divisor-counting\\\\\ncampbell shift $\\leftrightarrow$ Mersenne endpoint (shared namespace, \\#257-flavored) & \\lean{shiftWindowZero\\_iff\\_half\\_mem\\_mersenneAchievementSet}{CampbellShiftSynchronization.lean:488} & uniform & iff, no extra parameter & other-greedy-mersenne-achievement\\\\\ntotient overlap-factor multiplicativity & \\lean{totient\\_mul\\_eq\\_overlapFactor\\_mul}{TotientActualLcmOrbitArithmetic.lean:43} & uniform & $\\forall x{>}0$ ($j$ fixed) & other-totient-arithmetic\\\\\ntotient relative-Euler-product split & \\lean{totient\\_mul\\_eq\\_totient\\_mul\\_relativeEulerProduct}{TotientActualLcmOrbitArithmetic.lean:100} & uniform & $\\forall j\\,x{>}0$ & other-totient-arithmetic\\\\\nlcmRayArithmeticLetter $=$ deltaTotient $=$ diagonalWindowIncrement & \\lean{lcmRayArithmeticLetter\\_eq\\_deltaTotient}{TotientActualLcmOrbitArithmetic.lean:298} & uniform & $\\forall t\\,\\forall j$ & mobius-mersenne\\\\\nlcmDiagonalArithmeticWord $=$ windowDiscrepancy & \\lean{lcmDiagonalArithmeticWord\\_eq\\_windowDiscrepancy}{TotientActualLcmOrbitArithmetic.lean:1998} & uniform & $\\forall t\\,\\forall L$ & mobius-mersenne\\\\\n\\#249 $\\iff$ cofinal actual-orbit nonintegrality & \\lean{irrational\\_totientSeries\\_iff\\_actualLcmOrbitNonintegralitySupply}{TotientActualLcmOrbitNonintegrality.lean:37} & uniform & iff & mobius-mersenne\\\\\nactualLcmTailOrbit $=$ scaled series $-$ prefix & \\lean{actualLcmTailOrbit\\_eq\\_scaled\\_totientSeries\\_sub\\_prefix}{TotientActualLcmOrbitSeparation.lean:68} & uniform & $\\forall a$ & mobius-mersenne\\\\\nexplicit finite-block approximation & \\lean{abs\\_actualLcmTailOrbit\\_sub\\_rawApprox\\_lt}{TotientActualLcmOrbitSeparation.lean:141} & uniform & $\\forall a\\,\\forall q$ & mobius-mersenne\\\\\nguard-cylinder normal form & \\lean{exists\\_certifiedKill\\_iff\\_guardCylinderWitness}{TotientActualLcmTopEdgeStaircase.lean:844} & uniform & $\\forall h\\,N,\\ \\text{iff}$ & seam-integer\\\\\nfixedRankSecondDifference sign under extremality & \\lean{fixedRankSecondDifference}{TotientFixedRankLcmAsymptotic.lean:256} & uniform & $\\forall H\\,j$, given MiddleRankTotientExtremal & other-fixed-rank-curvature\\\\\ndyadic fixture exactness (kernel-valuation sharpness) & \\lean{fixedRankSecondDifference\\_dyadic\\_fixture\\_exact}{TotientFixedRankLcmAsymptotic.lean:462} & uniform & $\\forall n$ (closed-form) & other-fixed-rank-curvature\\\\\nparityCoboundaryWeight (def) & \\lean{parityCoboundaryWeight}{TotientParityCoboundaryCountermodel.lean:37} & uniform & $\\forall n$ (definition) & binary-digit\\\\\ntotientTail $=$ positive foreign-residue kernel sum & \\lean{totientTail\\_eq\\_tsum\\_positiveForeignResidueKernel}{TotientShiftedMobiusForeignBridge.lean:52} & uniform & $\\forall N$ & cyclotomic\\\\\ntotientTail $=$ shifted M\\\"obius-pulse sum & \\lean{totientTail\\_eq\\_tsum\\_shiftedMobiusPulse}{TotientShiftedMobiusPulse.lean:535} & uniform & $\\forall N$ & cyclotomic\\\\\nLambert double-sum regroup engine & \\lean{tsum\\_lambert\\_pair\\_regroup\\_if}{TotientShiftedMobiusPulse.lean:112} & uniform & $\\forall w\\,v{:}\\mathbb{N}\\to\\mathbb{R}\\,(|w(d)|{\\le}d,\\,|v(m)|{\\le}m)$ & cyclotomic\\\\\ncarryShift-integrality iff & \\lean{carryShift\\_dvd\\_iff\\_tailDiff\\_mem\\_int}{TotientTailCarryPeriod.lean:77} & uniform & $\\forall N\\,k$, given tempered orbit & seam-integer\\\\\nrationality $\\Rightarrow$ periodicity \\& unbounded rank & \\lean{not\\_irrational\\_totientSeries\\_implies\\_mod\\_period\\_and\\_unbounded\\_rank}{TotientTailCarryPeriod.lean:224} & uniform & $\\lnot\\mathrm{Irrational}\\,S\\to\\exists v\\,u\\,\\forall e$ & seam-integer\\\\\nadjugate tail-cost floor $\\ge 3$ & \\lean{three\\_le\\_totientAdjugateTailCost}{TotientTailCarryPeriod.lean:266} & uniform & $\\forall\\ \\text{finite}\\ w{:}\\iota\\to\\mathbb{Q}\\,x{:}\\iota\\to\\mathbb{N}$, given $\\sum w_i\\varphi(x_i){=}1$ & other-adjugate-linear-algebra\\\\\ndirectedCertifiedKill exact iff & \\lean{directedCertifiedKill}{TotientTailCarryPeriod.lean:766} & uniform & $\\forall h\\,N\\,L$ & seam-integer\\\\\ntempered orbit rigidity iff & \\lean{binaryCoeffSeries\\_rational\\_iff\\_exists\\_temperedBinaryOrbit}{GenericTailOrbitRigidity.lean:425} & uniform & $\\forall c{:}\\mathbb{N}\\to\\mathbb{N}\\,(c(n){\\le}n)$, iff & binary-digit\\\\\ndoubling-orbit rigidity & \\lean{doublingOrbit\\_eq\\_zero\\_of\\_tempered}{GenericTailOrbitRigidity.lean:314} & uniform & $\\forall d{:}\\mathbb{N}\\to\\mathbb{R}$, doubling $+$ tempered $\\to d{\\equiv}0$ & other-abstract-recursion\\\\\nfinite-state no-go (unbounded state) & \\lean{balancedPulse\\_no\\_autonomous\\_decoder}{GenericTailOrbitRigidity.lean:246} & uniform & $\\forall m\\,\\forall\\mathrm{State}$, collapsing-state $\\to$ no decoder & binary-digit\\\\\nfixed-depth affine reset & \\lean{affineBinaryOrbit\\_mod\\_twoPow\\_eq}{GenericTailOrbitRigidity.lean:306} & uniform & $\\forall a{:}\\mathbb{N}\\to\\mathbb{Z}\\,\\forall u_0\\,v_0\\,\\forall L$ & binary-digit\\\\\nsigned dichotomy engine (full-block certs) & \\lean{int\\_coeff\\_series\\_irrational\\_or\\_bpow\\_mul\\_eq\\_intCast\\_of\\_full\\_block\\_certificates}{CertificateKernel.lean:13606} & uniform & $\\forall b{\\ge}2\\,\\forall c\\,\\forall q$, given full-block certificate at $q$ & binary-digit\\\\\nLambert double-sum regroup (dyadic forward-difference calc.) & \\lean{dvd\\_iteratedForwardDifference}{LcmFactorIdealPulseObstruction.lean:211} & uniform & $\\forall f,c{:}\\mathbb{N}\\to\\mathbb{Z}\\,\\forall\\ \\text{shift-lists}$ & binary-digit\\\\\nfair-coin coprimality bridge & \\lean{tsum\\_coprime\\_pair\\_pow\\_eq\\_tsum\\_totient\\_mul\\_pow}{GeometricCoprimality.lean:120} & uniform & $\\forall n$; $\\forall r\\in[0,1)$ & probability\\\\\nlinear-descender rigidity & \\lean{linearDescender\\_eq\\_smul\\_eval}{AdelicHeightObstruction.lean:120} & uniform & $\\forall V,W\\,\\forall\\mathrm{ev}\\,\\forall L$, $\\ker(\\mathrm{ev}){\\le}\\ker(L)$ & p-adic\\\\\nrational denominator survival law & \\lean{divisor\\_dvd\\_divInt\\_den}{RationalDenominatorSurvival.lean:17} & uniform & $\\forall D{>}0\\,\\forall m{\\mid}D\\,\\forall a$ & p-adic\\\\\nscalar-localization complement-dvd (adelic height tax) & \\lean{scalarLocalization\\_complement\\_dvd}{AdelicHeightObstruction.lean:23} & uniform & $\\forall x{:}\\mathbb{Q}\\,\\forall H\\,(H{\\mid}x.\\mathrm{den})\\,\\forall c\\,((cx).\\mathrm{den}{\\mid}H)$ & p-adic\\\\\nsigned Hankel terminal-parity engine & \\lean{scaled\\_dyadic\\_sum\\_ne\\_zero}{SignedQMomentObstruction.lean:95} & bounded & $\\exists m{\\in}s$ (strictly maximal, odd coeff), $\\forall i{\\ne}m\\,e(i){<}e(m)$ & p-adic\\\\\nresidual gauge obstruction (locked minors) & \\lean{det\\_residualMonomialMatrix}{ResidualGaugeObstruction.lean:47} & uniform & $\\forall d\\,\\forall e{:}\\mathrm{Fin}\\,d{\\to}\\mathbb{N}\\,\\forall z{:}\\mathrm{Fin}\\,d{\\to}\\mathbb{C}$ ($z$ nonzero) & other-first-harmonic-phase\\\\\nincidence-quotient no compression & \\lean{incidenceMobius\\_det\\_eq\\_one}{IncidenceQuotientHermitePade.lean:56} & uniform & $\\forall N{:}\\mathbb{N}$ & mobius-mersenne\\\\\n\n\\multicolumn{5}{l}{\\emph{-- \\scale{bounded} --}}\\\\\nFarey window $K{=}240$ denominator bound & \\lean{gap\\_check\\_window\\_1\\_240\\_le\\_79639646646701375323355774875831053}{GapFareyBound.lean:176} & bounded &\n$\\forall q{:}\\mathbb{N},\\,0{<}q{\\le}$\\newline\n$79{,}639{,}646{,}646{,}701{,}$\\newline\n$375{,}323{,}355{,}774{,}875{,}831{,}053$ & farey\\\\\ndenominator lower bound $S{\\ne}p/q$, $q{\\le}Q_0$ & \\lean{tsum\\_totient\\_div\\_pow\\_two\\_ne\\_ratCast\\_of\\_den\\_le\\_79639646646701375323355774875831053}{CertificateKernel.lean:18056} & bounded & $\\forall p{:}\\mathbb{Q},\\,p.\\mathrm{den}{\\le}Q_0$ & farey\\\\\nforeign-residue tail-limit converter & \\lean{controlledForeignProjection\\_of\\_tail\\_limit}{LambertDiagonalEnclosure.lean:31} & bounded & $\\forall H\\,D$ with $2H{\\le}D$, given tail-limit convergence & mobius-mersenne\\\\\nfixed-precision tropical no-go & \\lean{fixedPrecisionTropicalNoGo}{TropicalCurvatureCarry.lean:137} & bounded & $\\forall u{>}0$ (fixed precision), $\\forall$ finite carry word & p-adic\\\\\n$\\mu$-clean-Pad\\'e feasibility wall & prose:mu\\_pollution\\_qpade\\_wall & bounded & claimed $\\forall d$; machine-audited only $d{\\le}12$ & mobius-mersenne (q-Pad\\'e)\\\\\niterated pullback / bounded-$\\Omega$ vanishing & \\lean{iteratedDilationDifference\\_eq\\_zero\\_of\\_cardFactors\\_le}{SupportDilationDifferences.lean:250} & bounded & given $\\Omega(a){\\le}K\\,\\forall a{\\in}A$ & other-support-divisor-counting\\\\\nrough-density $\\Rightarrow$ totient $\\ge\\tfrac34 n$ & \\lean{three\\_quarters\\_mul\\_lt\\_totient\\_of\\_rough\\_lt\\_two\\_pow}{TotientActualLcmOrbitArithmetic.lean:768} & bounded & $\\forall a{\\ge}8\\,\\forall n{>}0$ ($n$ is $t$-rough, $t{=}2^a$, $n{<}2^{2\\cdot2^a}$) & mobius-mersenne\\\\\nshort-window sign fixed positive & \\lean{lcmRayArithmeticLetter\\_pos\\_of\\_lt\\_two\\_mul}{TotientActualLcmOrbitArithmetic.lean:1703} & bounded & $\\forall a{\\ge}8\\,\\forall j\\,(0{<}j{<}2\\cdot2^a)$ & mobius-mersenne\\\\\nunconditional lower-half positivity & \\lean{actualLcmTailDiff\\_shift\\_pos}{TotientActualLcmOrbitSign.lean:39} & bounded$^\\dagger$ & $\\forall a{\\ge}8\\,\\forall J\\,(J{+}(a{+}6){<}2\\cdot2^a)$, no rationality hypothesis & mobius-mersenne\\\\\ntop-edge residue survivor negative & \\lean{actualLcm\\_trueEndpointSurvivor\\_neg}{TotientActualLcmOrbitSign.lean:172} & bounded & $\\forall a{\\ge}8\\,\\forall J\\,K$ (room bound) & mobius-mersenne\\\\\nintegrality forces top-edge residue & \\lean{actualLcm\\_integral\\_forces\\_topEdgeResidue}{TotientActualLcmOrbitSign.lean:211} & bounded & $\\forall a{\\ge}8\\,\\forall J\\,K$ (room, $2H{+}J{+}K{+}2{<}2^K$) & mobius-mersenne\\\\\nshort-kill supply through $a{\\le}6$ & \\lean{powerTwoActualLcmShortArithmeticKillSupply\\_through\\_six}{TotientActualLcmShortKill.lean:54} & bounded & $\\forall a_0{\\le}6\\,\\exists a\\,L$ & mobius-mersenne\\\\\nterminal dyadic staircase impossible with room & \\lean{not\\_actualLcmTerminalDyadicStaircase\\_of\\_room}{TotientActualLcmTopEdgeStaircase.lean:251} & bounded & $\\forall a{\\ge}8\\,\\forall J\\,K\\,m$ (room, modulus-wide) & mobius-mersenne\\\\\npunctured staircase penultimate pin & \\lean{puncturedDyadicStaircase\\_penultimate\\_eq\\_half}{TotientActualLcmTopEdgeStaircase.lean:1187} & bounded & $\\forall a{\\ge}8\\,\\forall J\\,K\\,m$ (room $+$ punctured hyp.) & mobius-mersenne\\\\\ntop-edge residue-gap $\\Rightarrow$ nonintegral & \\lean{actualLcmTailDiff\\_notMem\\_int\\_of\\_topEdgeResidueGap}{TotientActualLcmTopEdgeStaircase.lean:1260} & bounded & $\\forall a{\\ge}8\\,\\forall J\\,K\\,m$ (room, one-sided) & mobius-mersenne\\\\\nodd-rank centred-lift $=$ terminal $-$ true carry & \\lean{two\\_mul\\_actualOddHalfCenteredLift\\_eq\\_terminal\\_sub\\_trueCarry}{TotientActualLcmTopEdgeStaircase.lean:1678} & bounded & $\\forall a{\\ge}8\\,\\forall q$ (room, fit bound $2(H{+}q{+}2){\\le}4^q$) & mobius-mersenne\\\\\nprime-power kernel valuation floor & \\lean{two\\_mul\\_totient\\_dvd\\_fixedRankSecondDifference}{TotientFixedRankLcmAsymptotic.lean:519} & bounded & $\\forall a{\\ge}4\\,\\forall j{>}0\\,(j^2{\\le}2^a)$ & other-fixed-rank-curvature\\\\\nfinite rational-separation $\\Rightarrow$ $\\lnot$ hit (\\#257-flavored) & \\lean{renormalizedResidueAgreement\\_of\\_twice\\_le}{TotientShiftedMobiusForeignBridge.lean:329} & bounded & $\\forall H{>}0\\,\\forall D{\\ge}2H$, given finite separation & mobius-mersenne\\\\\nhomogeneous Mersenne multiplier does not annihilate & \\lean{idCoeff\\_mersenneModulus\\_does\\_not\\_annihilate\\_tailShift}{TotientTailCarryPeriod.lean:342} & bounded & $\\forall K,q\\,(K{>}0,K{<}q,q{\\mid}2^K{-}1)$ & mobius-mersenne\\\\\nnumerator-gap $\\Rightarrow$ $\\lnot$ hit (\\#257-flavored) & \\lean{scaleFullTarget\\_miss\\_of\\_lambert\\_projected\\_num\\_gap}{LambertProjectedNumeratorGap.lean:23} & bounded & $\\forall H,D$ fixed, given uniform numerator-gap bound & mobius-mersenne\\\\\nperiodic-weight Lambert dichotomy & \\lean{irrational\\_or\\_bpow\\_mul\\_eq\\_intCast\\_intWeightedErdosSeries\\_periodic}{CertificateKernel.lean:13847} & bounded & $\\forall b{\\ge}2\\,\\forall m{>}0\\,\\forall w$ ($w(n{+}m){=}w(n)$) & mobius-mersenne\\\\\nnon-negative periodic $\\Rightarrow$ irrational (unconditional) & \\lean{irrational\\_intWeightedErdosSeries\\_periodic\\_of\\_coeff\\_nonneg\\_of\\_frequently\\_ne\\_zero}{CertificateKernel.lean:14255} & bounded & $\\forall b{\\ge}2\\,\\forall m{>}0$, $w$ periodic, $\\ge0$, frequently ${\\ne}0$ & mobius-mersenne\\\\\n\n\\multicolumn{5}{l}{\\emph{-- \\scale{fixed} --}}\\\\\nall $h\\in[1,8]$ kill certificate at depth 16, $N{=}12$ & \\lean{certifiedKill\\_all\\_small}{TotientTailPeriodKiller.lean:404} & fixed & $\\forall h\\in[1,8]$, \\texttt{certifiedKill}\\,$h$\\,12\\,16 (by \\texttt{decide}) & other-binary-window\\\\\n$h\\in[1,16]$, exponent 14 kill table & \\lean{totient\\_series\\_ne\\_rat\\_of\\_den\\_dvd\\_pow\\_two\\_mul\\_mersenne\\_upto\\_sixteen}{CertificateKernel.lean:18562} & fixed & $\\forall h\\in[1,16]$, denominator exponent 14 (by \\texttt{decide}) & other-binary-window\\\\\nrank-2 kill not shallower at $(h,N){=}(1,8)$ & \\lean{totient\\_tail\\_rank\\_two\\_kill\\_sound\\_but\\_not\\_shallower\\_cell}{CertificateKernel.lean:18762} & fixed & measured over $t{\\le}20$ (30/40 cells) & other-binary-window\\\\\n28-point diagonal pincer table through $t{=}64$ & \\lean{certifiedKill\\_diagonal\\_all\\_imported}{DiagonalPincerCertificates.lean:2842} & fixed & $t\\in\\{1,2,3,4,5,7,8,9,11,13,16,17,\\dots\\}$ (28 explicit values) & other-lcm-diagonal\\\\\nshort kill at $a{=}4$ and $a{=}6$ & \\lean{lcmDiagonalArithmeticKill\\_two\\_pow\\_four}{TotientActualLcmShortKill.lean:18} & fixed & $a{=}4$, $a{=}6$ (concrete, kernel-checked) & mobius-mersenne\\\\\nsquare-CRT clean block: vanishing / nonvanishing witnesses & \\lean{squareCRT\\_clean\\_block\\_can\\_vanish}{SquareCRTCube.lean:285} & fixed & witness $n{=}52$ (vanishes); witness $n{=}27$ (nonzero) & other-crt-dyadic-residue\\\\\necho-vs-height two-front wall & prose:echo\\_versus\\_height\\_two\\_front\\_wall & fixed & empirical over tested families only (not universal) & mobius-mersenne (q-Pad\\'e, quantitative)\\\\\nperiod-4 zeroset escape witness & \\lean{intWeightedCoeff\\_periodFourSignWeight\\_eq\\_zero\\_of\\_mod\\_four\\_eq\\_three}{CertificateKernel.lean:13898} & fixed & $\\forall n,\\,n{\\equiv}3\\!\\!\\pmod 4$ (fixed weight $w$) & mobius-mersenne\\\\\n\n\\midrule\n\\multicolumn{5}{c}{\\emph{--- frontier: everything below this line is the cofinal target or a direct consumer of it ---}}\\\\\n\\midrule\n\n\\multicolumn{5}{l}{\\emph{-- \\scale{cofinal} (OPEN) --}}\\\\\n\\textbf{THE WALL} \u2014 certificate supply $\\Rightarrow$ Irrational $S$ & \\lean{irrational\\_totient\\_series\\_of\\_certificate\\_supply}{TotientTailPeriodKiller.lean:394} & \\textbf{cofinal} & $\\forall h{\\ge}1\\,\\forall N_0{\\ge}0\\,\\exists N{\\ge}N_0\\,\\exists L,\\ \\texttt{certifiedKill}\\,h\\,N\\,L$ & other-binary-window\\\\\nmultiple-certificate supply (weakened) & \\lean{irrational\\_totient\\_series\\_of\\_multiple\\_certificate\\_supply}{CarrySurvivorExtinction.lean:502} & cofinal & $\\forall h_0{>}0\\,\\forall N_0\\,\\exists m{>}0\\,\\exists N{\\ge}N_0\\,\\exists L$ & other-lcm-period-multiple\\\\\nLCM-diagonal one-parameter restatement & \\lean{irrational\\_totient\\_series\\_of\\_lcm\\_diagonal\\_certificate\\_supply}{LcmDiagonalReduction.lean:112} & cofinal & $\\forall t_0\\,\\exists t{\\ge}t_0\\,\\exists L$ & other-lcm-diagonal\\\\\nLCM-cone window-kill supply (widest target) & \\lean{irrational\\_totient\\_series\\_of\\_lcm\\_cone\\_window\\_kill\\_supply}{CertificateKernel.lean:18686} & cofinal & $\\forall t_0\\,\\exists t{\\ge}t_0\\,\\exists q{>}0\\,\\exists m\\,\\exists L$ & other-lcm-cone\\\\\nLCM diagonal/cone nonintegrality restatement & \\lean{irrational\\_totient\\_series\\_of\\_lcm\\_diagonal\\_nonintegrality\\_supply}{CertificateKernel.lean:18706} & cofinal & $\\forall t_0\\,\\exists t{\\ge}t_0$, $\\texttt{totientTail}(2H_t){-}\\texttt{totientTail}(H_t)\\notin\\mathbb{Z}$ & other-real-analytic-nonintegrality\\\\\nLCM cone-nonflat supply (sharpest depth-reduced form) & \\lean{irrational\\_totient\\_series\\_of\\_lcm\\_cone\\_nonflat\\_supply}{CertificateKernel.lean:18812} & cofinal & $\\exists$ unbounded-scale menu $Q$ with \\texttt{coneNonflatCert} firing & other-lcm-cone-menu\\\\\nfirst-harmonic norm-gap supply & \\lean{irrational\\_totient\\_series\\_of\\_first\\_harmonic\\_norm\\_gap}{FirstHarmonicPivot.lean:83} & cofinal & $\\forall h{>}0\\,\\forall X_0\\,\\exists X{\\ge}\\max(X_0,1)\\,\\exists L$, room $16(2X{+}h{+}L{+}2){\\le}2^L$ & binary-digit\\\\\nprime-jump sharp-kill supply & \\lean{irrational\\_totient\\_series\\_of\\_primeJumpSharpKill\\_supply}{PrimeJumpWindow.lean:193} & cofinal & $\\forall t_0\\,\\exists t{\\ge}t_0\\,\\exists p,L{>}0$ & binary-digit\\\\\n$a$-exponent short-arithmetic kill supply (the open trigger) & \\lean{PowerTwoActualLcmShortArithmeticKillSupply}{TotientActualLcmOrbitArithmetic.lean:2107} & cofinal & $\\forall a_0\\,\\exists a\\,L,\\ a_0{\\le}a\\land L{<}2\\cdot2^a$ & mobius-mersenne\\\\\nactual-orbit separation supply & \\lean{irrational\\_totientSeries\\_of\\_actualLcmOrbitSeparationSupply}{TotientActualLcmOrbitSeparation.lean:305} & cofinal & cofinal $1/32{+}$error separation at guarded odd ranks & mobius-mersenne\\\\\ntop-edge-residue-gap sufficiency chain (5 links, weakest first) & \\lean{irrational\\_totientSeries\\_of\\_flexibleActualTerminalDominanceSupply}{TotientActualLcmTopEdgeStaircase.lean:2221} & cofinal & $\\forall a_0\\,\\exists a{\\ge}a_0\\,\\exists\\langle\\text{supply predicate}\\rangle$ & mobius-mersenne\\\\\nrational countermodel refutes \\{bound,parity,aperiodicity\\} & \\lean{exists\\_totientParity\\_arbitrarilyManySeparatedCarry\\_rational\\_countermodel}{TotientParityCoboundaryCountermodel.lean:637} & cofinal & $\\exists c{:}\\mathbb{N}\\to\\mathbb{N}$, $c{\\le}6$, $c{\\le}n$, $c{\\equiv}\\varphi\\!\\!\\pmod2$, arbitrarily-separated blocks & binary-digit\\\\\nmod-4 pulse cofinal supply (via CRT+Dirichlet) & \\lean{exists\\_prime\\_deltaTotient\\_four\\_mul\\_mod\\_four\\_two}{TotientTailCarryPeriod.lean:384} & cofinal & $\\forall h{>}0\\,\\forall B\\,\\exists\\text{prime}\\,p{>}B$ & mobius-mersenne\\\\\nmod-4 pulse survivor kill (fourfold reduction) & \\lean{irrational\\_totientSeries\\_of\\_cofinal\\_modFourPulseSurvivorKill}{TotientTailCarryPeriod.lean:1066} & cofinal & given CP-05's cofinal supply & mobius-mersenne\\\\\ntwo-adic pulse at arbitrary depth $K$ & \\lean{exists\\_prime\\_totient\\_twoAdic\\_pulse\\_divisors}{TotientTwoAdicPulseBlock.lean:54} & cofinal & $\\forall K{\\ge}2\\,\\forall H{>}K\\,\\forall B\\,\\exists\\text{cofinally many primes}\\,p{>}B$ & p-adic\\\\\ntwo-adic pulse transfer (needs eventual integrality) & \\lean{eventual\\_integral\\_tailDiff\\_has\\_cofinal\\_twoAdic\\_half\\_pulse}{TotientTwoAdicPulseBlock.lean:327} & cofinal & $\\forall K{\\ge}2$, given TA-01 pulse $+$ eventual integrality & p-adic\\\\\n\\end{longtable}\n\\endgroup\n\n\\noindent $^\\dagger$ The source normal-form table tags \\texttt{actualLcmTailDiff\\_shift\\_pos} \\scale{bounded}\nbecause the window parameter $J$ is finite-range relative to $2\\cdot2^a$; the exponent $a$ itself is\ncompletely unbounded ($\\forall a\\ge 8$), and this row's constants ($4$, $8$, $32$) do not degrade with\n$a$. It is flagged here, not silently re-tagged, because it is exactly the kind of row an automated\npromotion sweep must not misclassify \u2014 see the audit below.\n\n\\subsection{Checked negative result: no free promotion from bounded to cofinal}\n\nA ladder like the one above invites an obvious question: is any \\scale{bounded} result actually\nuniform in disguise, so that removing its threshold is a routine generalisation rather than new\nmathematics? This was checked directly rather than assumed. An audit read the Lean proof body\nbehind every bounded-scale result in the \\#249 corpus \u2014 eighteen candidates in total, spanning\nthe near-miss list against the \\#249-supply obligation catalogued in the interface index (fourteen\nrows, each independently checked against the exact cofinal target it is closest to) together with\nthe remaining bounded/fixed rows in the certificate, M\\\"obius--Lambert, and private-totient banks\nabove \u2014 and found \\textbf{no} bounded result whose proof is uniform enough in its threshold\nparameter to promote to cofinal or uniform for free. Every candidate failed for one of three\nreasons, and the failure mode is worth recording because it is a checked negative result, not an\nabsence of search:\n\n\\begin{enumerate}[leftmargin=*]\n\\item \\textbf{Finite certificate tables.} Results such as\n\\lean{certifiedKill\\_diagonal\\_all\\_imported}{DiagonalPincerCertificates.lean:2842} (28 explicit\nvalues of $t$ through $t{=}64$),\n\\lean{certifiedKill\\_all\\_small}{TotientTailPeriodKiller.lean:404} ($h\\in[1,8]$ at fixed depth\n16), and\n\\lean{totient\\_series\\_ne\\_rat\\_of\\_den\\_dvd\\_pow\\_two\\_mul\\_mersenne\\_upto\\_sixteen}{CertificateKernel.lean:18562}\n($h\\in[1,16]$ at exponent 14) are each a finite \\emph{list} of independently verified rows, not a\nsingle argument instantiated at a free parameter. There is no uniform proof underneath to strip\nthe bound from; each new row is a fresh finite computation.\n\\item \\textbf{\\texttt{interval\\_cases}/\\texttt{decide} over a bounded range.} Results such as\n\\lean{three\\_quarters\\_mul\\_lt\\_totient\\_of\\_rough\\_lt\\_two\\_pow}{TotientActualLcmOrbitArithmetic.lean:768}\n($a{\\ge}8$) and the entire\n\\lean{actualLcmTailDiff\\_shift\\_pos}{TotientActualLcmOrbitSign.lean:39}/top-edge-staircase family\n(room conditions of the shape $J{+}(a{+}6){<}2\\cdot2^a$) are proved by case-splitting a residue or\na divisor structure that is only exhaustively enumerable inside the stated range; the mathematical\ncontent genuinely narrows as the range widens, so \\texttt{decide} cannot simply be re-run at a\nlarger bound without an exponential blow-up in the search space it certifies.\n\\item \\textbf{Constants that degrade with the parameter.} The clearest instance is the\necho-versus-height wall (prose:echo\\_versus\\_height\\_two\\_front\\_wall,\n\\S5e--5f of the ambitious-modular-route note): height-optimal ladder families pay quadratic\nheight $P(1)\\sim 2^{K^2/2}$ matching\n\\lean{lcmHeight\\_scaledMobiusShadow\\_den\\_exact}{MersenneShadowDenominatorGrowth.lean:147}'s own\ngrowth rate, while height-minimal integer-relation minimizers regrow the echo term $E_R$ to match\n\u2014 for every tested family, the two costs cannot both be driven to zero as the scale parameter\ngrows. The interface index records the same phenomenon for\n\\texttt{HalfRung}$(J)$-shaped truncation rungs: the certificate window $B(J)$ is driven by\n$L_J=\\mathrm{lcm}(2,\\dots,J)$, which grows super-exponentially in $J$, so the per-$J$ constant\ndegrades with the very parameter the cofinal claim needs to range over \u2014 ``the hallmark of a proof\nthat does not survive bound removal'' (interface index, row \\texttt{d-3a/d-3b}).\n\\end{enumerate}\n\nNone of the fourteen near-miss rows against the \\#249-supply obligation is closed by a scale\nargument alone either: reading each one's own recorded \\texttt{exact\\_mismatch} shows the residual\ngap is a missing \\emph{arithmetic input} (a constant-saving Weyl-sum bound for\n\\texttt{e1-companion}; a one-sided top-edge residue gap for \\texttt{TE-04}/\\texttt{TE-05-weakest};\nan unbounded finite-rank compression on the rationality side for \\texttt{d4/d5}), never a\ngeneralisable proof technique sitting one \\texttt{omega} call away from cofinal. The one row where\nthe source tag itself is imprecise \u2014 \\texttt{SGN-01}, marked \\scale{bounded} even though its\nunderlying exponent $a$ is unbounded, see $^\\dagger$ above \u2014 is precisely the kind of false\npositive the audit was built to catch, and even there what remains open is the \\emph{other} half of\nthe certificate band (the top-edge residue), not a scale defect in SGN-01 itself.\n\n\\section{Coordinate atlas and transport maps}\n\n\\#249 is not proved or disproved in a single representation. The corpus expresses\n$S=\\sum_{n\\ge1}\\varphi(n)/2^n$ in at least eight distinct coordinates, and the central\nmethodological lesson of the whole programme \u2014 flagged repeatedly in the source banks as\nAP17/AP18, ``obstructions are coordinate-relative'' \u2014 is that a mechanism which kills a proof\nstrategy in one coordinate can be silent, or even actively helpful, in another. This section lists\nevery coordinate used, gives at least one exact transport map out of it into another coordinate\nwith its Lean site, and then makes the coordinate-relativity claim precise for the one pair where\nit matters most: binary-digit versus M\\\"obius--Mersenne.\n\n\\begin{description}[leftmargin=*,style=nextline]\n\n\\item[binary-digit] $S$ as the literal base-2 expansion object: \\texttt{totientTail},\n\\texttt{windowDiscrepancy}, \\texttt{certifiedKill}. \\emph{Transport to mobius-mersenne:} the\ncore identity\n\\[\n  S=\\sum_{n\\ge1}\\frac{\\varphi(n)}{2^n}=\\frac12+\\sum_{d\\ge1}\\frac{\\mu(d)}{(2^d-1)^2}\n\\]\nproved by \\lean{tsum\\_totient\\_half\\_pow\\_eq\\_half\\_add\\_moebius\\_sq}{SquaredMersenneDiagonalEnclosure.lean:50}\n(consumed at \\lean{mobius\\_square\\_series\\_eq\\_partial\\_add\\_tail}{SquaredMersenneDiagonalEnclosure.lean:72})\nturns every binary-digit statement about $S$ into a statement about a squared M\\\"obius--Mersenne\nLambert series, and back.\n\n\\item[mobius-mersenne] $S$ as $\\tfrac12+\\sum_d\\mu(d)/(2^d-1)^2$, the Lambert ladder\n$L(w)=\\sum_n w(n)/(2^n-1)$ at $w=\\mu,1,\\varphi$, and the whole cyclotomic denominator-survival\nchain built on it. \\emph{Transport to cyclotomic:} evaluating the M\\\"obius-numerator polynomial at\n$x=2$,\n\\lean{mobiusNumeratorPolynomial\\_eval\\_two}{RepunitMobiusNumerator.lean:320} together with the\ndefinition \\lean{mobiusNumerator}{RadicalMobiusShadow.lean:101}, moves the squared-denominator\nquestion into finite cyclotomic-polynomial arithmetic over $\\mathbb{Z}$.\n\n\\item[probability/coprime-pair] $S-\\tfrac12=\\Pr(\\gcd(X,Y)=1)$ for independent fair-coin waiting\ntimes $X,Y$. \\emph{Transport from mobius-mersenne:} the gcd-divisibility factorisation\n\\lean{tsum\\_pos\\_pair\\_both\\_dvd\\_half\\_eq\\_inv\\_mersenne\\_sq}{GcdMomentCalculus.lean:266}\n($\\Pr(d\\mid X\\wedge d\\mid Y)=1/(2^d-1)^2$) together with the reduced-direction law\n\\lean{tsum\\_pos\\_coprime\\_inv\\_mersenne\\_eq\\_one}{GcdMomentCalculus.lean:349} converts the\nsquared-Lambert sum into a probability mass, and the independently-built\n\\lean{tsum\\_coprime\\_pair\\_pow\\_eq\\_tsum\\_totient\\_mul\\_pow}{GeometricCoprimality.lean:120} gives\nthe same bridge from the totient generating function directly.\n\n\\item[cyclotomic] The T1--T6 chain: \\texttt{mobiusNumeratorPolynomial}, its coefficients (an exact\ndivisor-counting formula, \\lean{mobiusNumeratorPolynomial\\_coeff}{RepunitMobiusNumerator.lean:217}),\ncyclotomic-value survival\n(\\lean{cyclotomicValue\\_dvd\\_baseMobiusShadow\\_den}{CyclotomicProjectionOfShadow.lean:389}), and\nprime-power jump dynamics. \\emph{Transport to mobius-mersenne (denominator growth):} the upper-half\nprime-channel survival theorem\n\\lean{upperHalfChannel\\_product\\_dvd\\_den\\_of\\_scale\\_primeFactors\\_le}{MersenneShadowCyclotomicNoncollapse.lean:809}\nfeeds directly into the exact denominator-growth formula\n\\lean{lcmHeight\\_scaledMobiusShadow\\_den\\_exact}{MersenneShadowDenominatorGrowth.lean:147} in the\nM\u00f6bius--Mersenne coordinate.\n\n\\item[p-adic] Fixed and growing 2-adic valuation-unit dynamics: the adelic height/denominator-tax\nlaw, the tropical fixed-precision no-go, the signed-Hankel terminal-parity engine, the two-adic\npulse-block construction. \\emph{Transport to farey:} the scalar-localisation complement-dvd law\n\\lean{scalarLocalization\\_complement\\_dvd}{AdelicHeightObstruction.lean:23} together with the\nMersenne-height corollary \\lean{positiveRat\\_mersenne\\_height}{AdelicHeightObstruction.lean:103}\n($2^r\\mid|x.\\mathrm{num}|\\wedge x<2/(2^n-1)\\Rightarrow 2^r(2^n-1)<2\\cdot x.\\mathrm{den}$) is exactly\na denominator lower bound of the same shape the Farey coordinate proves directly.\n\n\\item[farey] Unimodular denominator exclusion:\n\\lean{farey\\_gap}{GapFareyBound.lean:51} and the committed $K{=}240$ window certificate\n\\lean{gap\\_check\\_window\\_1\\_240\\_le\\_79639646646701375323355774875831053}{GapFareyBound.lean:176}\ngiving $S\\ne p/q$ for every $q\\le Q_0\\approx7.96\\times10^{34}$. \\emph{Transport to binary-digit:}\nthe window bound is built directly from a committed $2^{240}$-scale totient residue\n$V$ for the window $(N,K)=(1,240)$ \u2014 a binary-digit object \u2014 via\n\\lean{gap\\_check\\_window\\_1\\_240\\_first\\_failure}{GapFareyBound.lean:225}, which shows the bound\nis sharp at exactly the mediant $b_K+d_K$.\n\n\\item[lcm-orbit] The \\texttt{periodLcm}/LCM-diagonal/LCM-cone family:\n\\lean{periodLcm}{CarrySurvivorExtinction.lean:515},\n\\lean{eq\\_prime\\_pow\\_of\\_not\\_dvd\\_periodLcm}{CertificateKernel.lean:18613}, the multiplicative ray\nsplit, and the cone-flatness/cone-nonflat producers. \\emph{Transport to binary-digit:} the identity\n\\lean{lcmRayArithmeticLetter\\_eq\\_deltaTotient}{TotientActualLcmOrbitArithmetic.lean:298} and\n\\lean{diagonalWindowIncrement\\_eq\\_lcmRayArithmeticLetter}{TotientActualLcmOrbitArithmetic.lean:310}\nidentifies the LCM-ray letter literally with a \\texttt{deltaTotient}/\\texttt{windowDiscrepancy}\nvalue, so every LCM-orbit statement is, term for term, a binary-digit statement in disguise.\n\n\\item[first-harmonic/exponential-sum] Weyl-sum block cancellation over the totient window\ndiscrepancy: \\lean{irrational\\_totient\\_series\\_of\\_first\\_harmonic\\_norm\\_gap}{FirstHarmonicPivot.lean:83}\nand the unconditional companion at\n\\lean{exists\\_certifiedKill\\_of\\_first\\_harmonic\\_gap}{FirstHarmonicGap.lean:128} (any\nconstant-saving cancellation, via $\\cos(\\pi/8)>9/10$ plus pigeonhole, forces a finite kill\ncertificate). \\emph{Transport to binary-digit:} by construction the exponential sum is literally\n$\\sum_N \\cos\\bigl(2\\pi\\cdot(\\texttt{windowDiscrepancy}\\ h\\ N\\ L \\bmod 2^L)/2^L\\bigr)$ \u2014 a first-harmonic\nstatistic \\emph{of} the binary-digit residue, so a norm-gap in this coordinate is definitionally a\nstatement about \\texttt{windowDiscrepancy}.\n\n\\end{description}\n\n\\subsection{Obstructions are coordinate-relative: the binary-digit vs.\\ M\\\"obius--Mersenne case}\n\nThe clearest instance of coordinate-relative obstruction in the whole \\#249 corpus is the fate of\nthe classical Erd\\H{o}s (1948) near-integer digit method. That method needs, in essence, a finite\nautomaton: a bounded amount of state carried forward from one digit block to the next, so that the\nvalue of a far-away digit block can be recovered from a bounded summary of everything before it. In\nthe binary-digit coordinate this is exactly what fails, and it fails as a proved Lean theorem, not\na heuristic remark: \\lean{balancedPulse\\_no\\_autonomous\\_decoder}{GenericTailOrbitRigidity.lean:246}\nshows that for a balanced-pulse family of radius $m$, any finite \\texttt{State} type collapsing the\nfamily to a single autonomous summary must satisfy $|\\mathrm{State}|\\ge\\lfloor m/2\\rfloor+2$ \u2014\nunbounded in $m$ \u2014 so no finite-state decoder can recover the shift $r$ from its collapsed state\n(\\ev{Lean}, \\scale{uniform}, coordinate \\texttt{binary-digit}). This is the formal shadow of the\nclassical fact that Euler's totient function has $\\varphi(n)=\\Theta(n)$ (average order $\\tfrac6{\\pi^2}n$,\nelementary; \\ev{Math}) rather than being bounded or sparse: the digit-window discrepancy\n\\texttt{windowDiscrepancy}$\\,h\\,N\\,L$ genuinely needs $\\Omega(m)$ bits of state to track as $N$\ngrows, which is exactly why \\#249's own binary-digit engine is forced into a \\emph{cofinal\ncertificate supply} (\\lean{irrational\\_totient\\_series\\_of\\_certificate\\_supply}{TotientTailPeriodKiller.lean:394})\nrather than a single finite-state argument: the same shape of no-go recurs at\n\\lean{det\\_residualMonomialMatrix}{ResidualGaugeObstruction.lean:47} (locked gauge defeats\nresidual-blind rank certificates for the first-harmonic pivot) and at\n\\lean{linearIndependent\\_canonicalTotientKernelFamily}{TotientMahlerDefect.lean:900} (the dyadic\ntotient-kernel span is genuinely infinite-rank, not compressible to any fixed dimension).\n\nThe M\\\"obius--Mersenne coordinate does not carry this obstruction, because it is built on a\ncoefficient sequence, $\\mu(d)\\in\\{-1,0,1\\}$, that is bounded rather than growing with $d$. The\nsquared-Lambert transfer engine\n\\lean{tsum\\_lambert\\_linear\\_weight\\_sq\\_pure}{GcdMomentCalculus.lean:105} takes any weight with\n$|w(d)|\\le d$ \u2014 $\\mu$ trivially qualifies with room to spare \u2014 and converts it into a\ndivisor-convolution power series for free; no finite-state decoder is ever needed because the\nrelevant recursion (the cyclotomic T1--T6 chain: \\lean{mobiusNumeratorPolynomial\\_eq\\_gcdWord}{RepunitMobiusNumerator.lean:205}\nthrough \\lean{cyclotomic\\_dvd\\_primeJump\\_new\\_fibre}{PrimePowerJumpDynamics.lean:281}) is a finite,\nexact polynomial identity at every squarefree radical $r$, uniform in $r$, with no growth-driven\nstate explosion. This is precisely why the corpus's strongest \\emph{unconditional} denominator\nresult \u2014 the exact-value and lower-bound formulas of\n\\lean{lcmHeight\\_scaledMobiusShadow\\_den\\_exact}{MersenneShadowDenominatorGrowth.lean:147} and\n\\lean{upperHalfMersenneProduct\\_lower\\_bound}{MersenneShadowDenominatorGrowth.lean:60} \u2014 lives\nentirely in the M\\\"obius--Mersenne coordinate and is scale-\\texttt{uniform}, while the sharpest\n\\emph{conditional} route in the binary-digit coordinate is scale-\\texttt{cofinal} and unclosed. The\ntwo coordinates are not equivalent representations of the same difficulty: the wall each one hits\nis a fact about that coordinate's own state-growth, not about \\#249 itself, which is exactly the\nAP17/AP18 discipline this corpus was built to enforce \u2014 before reporting an obstruction, name the\ncoordinate it was measured in, and test whether a re-representation removes it.\n\n% ---- part a249_p3 ----\n\\section{Interface index: what stands between the corpus and a proof}\n\nThis section is the join layer of the paper. Parts I--II inventory what the corpus proves; this\npart inventories, obligation by obligation, exactly what is \\emph{missing} to turn each proved\nreduction into a proof of Erd\\H{o}s \\#249, in a form usable as a premise without opening the Lean\ntree. Erd\\H{o}s \\#249 asks whether $S = \\sum_{n\\ge 0} \\varphi(n)/2^n$ is irrational. It is OPEN.\nNothing in this section decides it. Every row below either (a) states a proved theorem exactly, or\n(b) states an unproved but well-posed proposition (an \\emph{open producer} or \\emph{open supply})\ntogether with the exact proved consumer that would turn it into irrationality of $S$.\n\n\\subsection{Reading this index}\n\nEvery entry follows the same discipline. \\textbf{Yields}: the exact statement proved, with\nquantifiers in their proved order. \\textbf{Exact mismatch}: precisely where the proved statement\nfalls short of the open obligation it is compared against --- never ``it doesn't work,'' always the\nnamed axis of shortfall. \\textbf{What would close it}: the exact remaining proposition, stated in\nfull, whose proof (via the cited Lean consumer, already on disk) proves $\\mathrm{Irrational}(S)$.\n\nEach numbered claim carries three tags. \\scale{fixed} means proved at finitely many explicit\nnumerals with no argument in the growing parameter; \\scale{bounded} means proved on a bounded range\nof a parameter that is itself unbounded elsewhere in the statement; \\scale{uniform} means proved for\n\\emph{every} value of the relevant parameter (the strongest positive tag short of matching the\nopen target); \\scale{cofinal} means the statement itself has the shape \\m{\\forall a_0\\ \\exists a\\ge\na_0}, i.e.\\ it is exactly the quantifier shape the open obligation needs; \\scale{n/a} applies to\nstructural non-existence results with no growing parameter. The evidence tags are\n\\ev{Lean}, \\ev{Cert}, \\ev{Math}, \\ev{Cited}, \\ev{Open}, used exactly as defined in Part I.\n\nThe canonical open obligation for \\#249, proved in Part II to be sufficient\n(\\lean{irrational_totient_series_of_certificate_supply}{TotientTailPeriodKiller.lean}), is the\n\\textbf{249-supply} obligation:\n\\[\n  \\forall h \\ge 1,\\ \\forall N_0,\\ \\exists N \\ge N_0,\\ \\exists L,\\ \\mathtt{certifiedKill}\\ h\\ N\\ L .\n\\]\nFourteen near-miss rows are catalogued against it below: one headline row (the sharpest reduction\nin the corpus, given its own subsection) and thirteen further rows, each attacking the same\nobligation from a different coordinate.\n\n\\subsection{The headline: \\#249 reduces to a single unsupplied exponential-sum bound}\n\\label{ssec:headline}\n\nThe following two definitions, verified against the live Lean tree in this session, are the\nsharpest statement of the \\#249 frontier that the corpus contains.\n\n\\begin{defn}[Window discrepancy]\n\\lean{windowDiscrepancy}{TotientTailPeriodKiller.lean:66}. For $h,N,L\\in\\mathbb N$,\n\\[\n  A_{h,N,L} \\;=\\; \\sum_{j=0}^{L-1} \\bigl(\\varphi(N+h+1+j) - \\varphi(N+1+j)\\bigr)\\cdot 2^{\\,L-1-j}\n  \\ \\in \\mathbb Z .\n\\]\nThis is the depth-$L$ truncation of $2^L\\cdot(R_{N+h}-R_N)$, where\n$R_N = \\sum_{j\\ge 1}\\varphi(N+j)/2^j$ is the local totient tail\n(\\lean{totientTail}{TotientTailPeriodKiller.lean:57}).\n\\end{defn}\n\n\\begin{defn}[Certified kill]\n\\lean{certifiedKill}{TotientTailPeriodKiller.lean:72}.\n\\[\n  \\mathtt{certifiedKill}\\ h\\ N\\ L \\;:\\Leftrightarrow\\;\n  (N{+}h{+}L{+}2) < A_{h,N,L}\\bmod 2^L < 2^L - (N{+}h{+}L{+}2).\n\\]\nThat is: the residue of $A_{h,N,L}$ modulo $2^L$ avoids the radius-$(N{+}h{+}L{+}2)$ neighbourhood\nof $0$. \\lean{certifiedKill_depth_floor}{TotientTailPeriodKiller.lean:79} records the necessary\nroom condition this forces: $2(N{+}h{+}L{+}2) < 2^L$.\n\\end{defn}\n\n\\begin{defn}[First-harmonic real and complex characters]\n\\lean{windowFirstCos}{FirstHarmonicGap.lean:27} and\n\\lean{windowFirstExp}{FirstHarmonicPivot.lean:38}:\n\\[\n  \\mathtt{windowFirstCos}\\ h\\ N\\ L = \\cos\\!\\Bigl(2\\pi\\,\\tfrac{A_{h,N,L}\\bmod 2^L}{2^L}\\Bigr),\\qquad\n  \\mathtt{windowFirstExp}\\ h\\ N\\ L = \\exp\\!\\Bigl(i\\,2\\pi\\,\\tfrac{A_{h,N,L}\\bmod 2^L}{2^L}\\Bigr).\n\\]\n\\lean{windowFirstExp_re}{FirstHarmonicPivot.lean:42} records $\\mathrm{Re}(\\mathtt{windowFirstExp})\n= \\mathtt{windowFirstCos}$ and \\lean{norm_windowFirstExp}{FirstHarmonicPivot.lean:48} that\n$\\|\\mathtt{windowFirstExp}\\,h\\,N\\,L\\| = 1$: this is literally the first additive character of the\nendpoint discrepancy modulo $2^L$, a standard exponential-sum object.\n\\end{defn}\n\n\\begin{thm}[Real-part certificate consumer]\n\\label{thm:hgap-real}\n\\lean{exists_certifiedKill_of_first_harmonic_gap}{FirstHarmonicGap.lean:128}. \\ev{Lean}.\n\\scale{uniform}. \\coord{first-harmonic}.\nFor all $h,X,L$ with $0<X$ and the room condition $16(2X{+}h{+}L{+}2)\\le 2^L$, if\n\\[\n  \\sum_{N=X}^{2X-1} \\mathtt{windowFirstCos}\\ h\\ N\\ L \\;\\le\\; \\tfrac{9}{10}\\,X ,\n\\]\nthen $\\exists N\\in[X,2X)$ with $\\mathtt{certifiedKill}\\ h\\ N\\ L$.\n\\end{thm}\n\n\\begin{thm}[Subset consumer --- no density or partition hypothesis]\n\\label{thm:hgap-subset}\n\\lean{exists_certifiedKill_of_first_harmonic_gap_subset}{FirstHarmonicGap.lean:104}.\n\\ev{Lean}. \\scale{uniform}. \\coord{first-harmonic}.\nFor any nonempty finite $T\\subseteq\\mathbb N$ with $T\\subset[0,2X)$ and the same room condition, if\n\\[\n  \\sum_{N\\in T} \\mathtt{windowFirstCos}\\ h\\ N\\ L \\;\\le\\; \\tfrac{9}{10}\\,|T| ,\n\\]\nthen $\\exists N\\in T$ with $\\mathtt{certifiedKill}\\ h\\ N\\ L$. This is strictly stronger than\nTheorem~\\ref{thm:hgap-real}: $T$ can be \\emph{any} explicitly chosen nonempty finite subset of the\ndyadic block, not the whole block, and the proof (an averaging pigeonhole, \\S below) never uses\nthat $T$ has positive density or comes from a partition.\n\\end{thm}\n\n\\begin{thm}[Complex norm consumer and the open producer]\n\\label{thm:hgap-norm}\n\\lean{exists_certifiedKill_of_first_harmonic_norm_gap}{FirstHarmonicPivot.lean:53} and\n\\lean{irrational_totient_series_of_first_harmonic_norm_gap}{FirstHarmonicPivot.lean:83}.\n\\ev{Lean} for both implications; the hypothesis \\m{DTWFirstHarmonicNormGap} is \\ev{Open}.\nThe complex norm bound is strictly stronger than the real-part bound\n(\\m{|z|\\ge\\mathrm{Re}(z)}, and $21/25 < 9/10$ absorbs the slack), so it composes through\nTheorem~\\ref{thm:hgap-real} to the same certificate. Define\n\\[\n  \\mathtt{DTWFirstHarmonicNormGap} :\\Leftrightarrow\\;\n  \\forall h{>}0\\ \\forall X_0\\ \\exists X, L,\\ \\max(X_0,1)\\le X\\ \\wedge\\ 16(2X{+}h{+}L{+}2)\\le 2^L\\ \\wedge\n\\]\n\\[\n  \\Bigl\\|\\ \\sum_{N=X}^{2X-1}\\mathtt{windowFirstExp}\\ h\\ N\\ L\\ \\Bigr\\| \\;\\le\\; \\tfrac{21}{25}\\,X .\n\\]\n\\coord{first-harmonic}. \\scale{cofinal} (this is the open target itself). Then\n\\[\n  \\mathtt{DTWFirstHarmonicNormGap} \\;\\Longrightarrow\\; \\mathrm{Irrational}\\Bigl(\\sum_{n\\ge 0}\n  \\tfrac{\\varphi(n)}{2^n}\\Bigr),\n\\]\nproved in full, with no further gap, by chaining\nTheorem~\\ref{thm:hgap-norm}$\\to$Theorem~\\ref{thm:hgap-real}$\\to$\n\\lean{irrational_totient_series_of_certificate_supply}{TotientTailPeriodKiller.lean}.\n\\end{thm}\n\n\\begin{obs}[The exact negative]\n\\label{obs:no-instance}\nNo instance of \\m{\\mathtt{DTWFirstHarmonicNormGap}}'s hypothesis, nor of \\m{hgap} in\nTheorems~\\ref{thm:hgap-real}--\\ref{thm:hgap-subset}, is proved anywhere in the corpus, at any $h$,\n$X$, $L$. \\ev{Open} is exact, not conservative rounding. The two consumer theorems that\n\\emph{use} the subset form (Theorem~\\ref{thm:hgap-subset}) are conditional consumers, not a\nsupply: \\lean{AdjacentPhaseSeparation.lean}{AdjacentPhaseSeparation.lean:267} instantiates $T$ as\nthe explicit two-element set $\\{N,M\\}$ (\\texttt{apply exists\\_certifiedKill\\_of\\_first\\_harmonic\\_gap\\_subset (\\{N, M\\} : Finset \u2115)}), and both call sites in\n\\lean{PivotAntiReconstruction.lean}{PivotAntiReconstruction.lean:851} and\n\\lean{PivotAntiReconstruction.lean}{PivotAntiReconstruction.lean:1541} likewise apply it to a\nfinite explicit pair, never to an infinite or growing family. There is no theorem anywhere in the\ncorpus that discharges $hgap$ at a single instance, let alone cofinally.\n\\end{obs}\n\n\\begin{rem}[Why this is the sharpest reduction, and why it is tractable]\nThis reduction converts the open producer for \\#249 from certificate-supply language --- ``exhibit\na certified kill $(N,L)$'' --- into a \\emph{constant-saving cancellation estimate for the first\nadditive character of the totient window discrepancy over a dyadic block}: exactly the shape of a\nclassical exponential-sum bound in analytic number theory (a Weyl-type estimate for\n$\\sum_N e(\\theta_N)$). Three features make this genuinely the narrowest gap in the corpus.\n\nFirst, the saving required is a \\emph{constant}, not $o(X)$: $21/25$ (complex norm) or $9/10$\n(real part), fixed independent of $h$, $X$, $L$. Any nontrivial cancellation estimate --- even one\nfar weaker than square-root cancellation --- would suffice; this is not asking for GRH-strength\ninput.\n\nSecond, the room condition $16(2X{+}h{+}L{+}2)\\le 2^L$ forces only $L \\gtrsim \\log_2 X + 5$, which\nis satisfiable for every $X$ by simply taking $L$ large enough (the depth floor grows logarithmically\nin $X$, not polynomially); no scale obstruction of the kind documented elsewhere in this paper\n(e.g.\\ the $\\sqrt{2\\log_2 N}$ silent-position defect for \\#257, Part~I) stands in the way of the\nproducer's own room requirement.\n\nThird, the subset form (Theorem~\\ref{thm:hgap-subset}) removes even the requirement that the\ncancellation happen on the whole dyadic block or a positive-density subset of it: any single\nexplicitly named finite $T$ with the averaged bound would do. In particular a supplier-fibre or\nsparse-subsequence argument --- e.g.\\ picking $N$ to be one less than a prime, as the exact pivot\nalgebra in \\lean{FirstHarmonicPivot.lean}{FirstHarmonicPivot.lean} already sets up via\n\\texttt{pivotArgument}, \\texttt{pivotPrime}, \\texttt{pivotSupplierBases} --- is a legitimate route\nto this obligation and does not need to control the discrepancy on a full interval.\n\\end{rem}\n\n\\subsection{Row-by-row: the remaining near misses against 249-supply}\n\nThe headline (\\S\\ref{ssec:headline}) is row \\textbf{e1-companion} of the source interface catalogue\nin full. The remaining thirteen rows below attack the same 249-supply obligation from independent\ncoordinates in the Lean tree: the actual-LCM sign corridor, the top-edge staircase, the raw\nrational approximant, the short-window arithmetic word, the diagonal pincer certificate bank, the\n$h$-uniform single-window certificate, the Farey/continued-fraction denominator growth law, the\n2-adic pulse block, the LCM-jump slack scalar, the prime-jump commutator, the M\\\"obius--Mersenne\ndenominator channel, and the carry-kernel rank. None of them is closed; each is recorded with its\nexact remaining content.\n\n\\subsubsection{SGN-01 --- the positive-sign half of the certified-kill band (\\coord{actual-lcm-sign})}\n\n\\begin{defn}\nThe \\emph{actual LCM tail orbit} at exponent $a$ is\n$\\mathtt{actualLcmTailOrbit}\\ a = R_{2H+2H} - R_{H+... }$, precisely\n$\\mathrm{totientTail}(2H)-\\mathrm{totientTail}(H)$ where $H = \\mathtt{periodLcm}(2^a)$\n(\\lean{actualLcmTailOrbit_pos}{TotientActualLcmOrbitSign.lean:144}).\n\\end{defn}\n\n\\begin{thm}\n\\lean{actualLcmTailDiff_shift_pos}{TotientActualLcmOrbitSign.lean:39} and\n\\lean{actualLcmTailOrbit_pos}{TotientActualLcmOrbitSign.lean:144}. \\ev{Lean}. \\scale{uniform}.\nFor every $a\\ge 8$ and every $J$ with $J+(a{+}6) < 2\\cdot 2^a$,\n\\[\n  0 \\;<\\; \\mathrm{totientTail}(2H{+}J) - \\mathrm{totientTail}(H{+}J), \\qquad H=\\mathtt{periodLcm}(2^a).\n\\]\nHence $0 < \\mathtt{actualLcmTailOrbit}\\ a$ for every $a\\ge 8$. The proof is genuinely uniform in\n$a$: it rests on two facts proved for all $a\\ge 8$ by a two-case structural split (divisor letter\nvs.\\ foreign prime power), with absolute constants $4,8,32$ and no lookup table ---\n\\lean{lcmRayArithmeticLetter_pos_of_lt_two_mul}{TotientActualLcmOrbitArithmetic.lean:1703} and\n\\lean{periodLcm_lt_eight_mul_t_mul_lcmRayArithmeticLetter}{TotientActualLcmOrbitArithmetic.lean:1906}.\n\\end{thm}\n\n\\begin{rem}[Exactly half the certificate band, and where the other half goes]\n$\\mathtt{certifiedKill}$ needs the residue to avoid a \\emph{symmetric} radius-$(N{+}h{+}L{+}2)$\nneighbourhood of $0$ on both sides. SGN-01 gives positivity only. Its companion\n\\lean{actualLcm_integral_forces_topEdgeResidue}{TotientActualLcmOrbitSign.lean:211} proves that,\nunder integrality, the consequence is not a kill but the opposite: the residue is forced to the\n\\emph{top edge}, exactly $2^K - e$ where $e$ is the true carry orbit (\\lean{carryOrbit}{}). So the\nlower half of the certified-kill band is discharged unconditionally and cofinally by SGN-01; the\nupper half is untouched, and what remains open is a genuinely different, one-sided statement about\nthe top edge --- row TE-04 below. The corpus's own normal-form tables tag this row\n\\scale{bounded}; that tag is misleading, since only the window offset $J$ is bounded relative to\nthe height, and the height $2\\cdot 2^a$ itself is unbounded: the correct tag is \\scale{uniform}.\n\\end{rem}\n\n\\subsubsection{TE-04 --- the one-sided top-edge staircase (\\coord{actual-lcm-top-edge})}\n\nThis is the sharpest \\emph{landed weakening} of the 249-supply obligation in the corpus: the\nsymmetric certified-kill band is replaced by a single upper inequality, halving the information\nthe open producer must supply.\n\n\\begin{thm}[Consumers, fully proved]\n\\lean{actualLcmTailDiff_notMem_int_of_topEdgeResidueGap}{TotientActualLcmTopEdgeStaircase.lean:1260},\n\\lean{actualLcmTailOrbit_notMem_int_of_topEdgeResidueGap}{TotientActualLcmTopEdgeStaircase.lean:1314},\n\\lean{irrational_totientSeries_of_topEdgeResidueGapSupply}{TotientActualLcmTopEdgeStaircase.lean:2184}.\n\\ev{Lean}. \\scale{uniform}. For every $a\\ge 8$ and every $J,K,m$ inside the sign corridor\n($J{+}K{+}(a{+}6) < 2\\cdot 2^a$), a one-sided residue gap at precision $m\\le K$ ---\nroom $2H{+}J{+}K{+}2 < 2^m$ and $A_{H,H+J,K}\\bmod 2^m \\le 2^m - (2H{+}J{+}K{+}2)$ --- already\nforces $\\mathrm{totientTail}(2H{+}J) - \\mathrm{totientTail}(H{+}J)\\notin\\mathbb Z$. No lower margin\nat all is demanded. The proof chain is complete: the theorem holds for every $a\\ge 8$, and\n\\emph{\\m{\\mathtt{PowerTwoActualLcmTopEdgeResidueGapSupply}} $\\Rightarrow$ Irrational $S$} is proved.\n\\end{thm}\n\n\\begin{defn}[The open producer]\n\\lean{PowerTwoActualLcmTopEdgeResidueGapSupply}{TotientActualLcmTopEdgeStaircase.lean:1325}.\n\\ev{Open}. \\scale{cofinal}.\n\\[\n  \\forall a_0\\ \\exists a,K,m,\\ a_0\\le a\\ \\wedge\\ 8\\le a\\ \\wedge\\ K{+}(a{+}6)<2\\cdot 2^a\\ \\wedge\\\n  \\mathtt{ActualLcmTopEdgeResidueGap}\\ a\\ 0\\ K\\ m .\n\\]\nNo unconditional statement anywhere in the corpus locates the residue of the diagonal word\n$A_{H,H,K}$ (where $H=\\mathtt{periodLcm}(2^a)$) below the top strip at even one large $a$.\n\\lean{actualLcmTopEdgeResidueGap_iff_terminal}{TotientActualLcmTopEdgeStaircase.lean:909} further\nreduces the condition to the last $m$ arithmetic letters of the word alone.\n\\end{defn}\n\n\\begin{prop}[The five strictly weaker sufficient links --- proving any one closes \\#249]\n\\label{prop:te-chain}\nAll five are proved sufficient for\n\\lean{PowerTwoActualLcmTopEdgeResidueGapSupply}{TotientActualLcmTopEdgeStaircase.lean:1325} in the\nsame module, and each is \\emph{strictly weaker} in the sense that it drops a hypothesis, widens a\nband, or asks only for magnitude rather than a signed inequality.\n\\begin{enumerate}[leftmargin=*]\n  \\item \\lean{PowerTwoAdjacentSuffixMidbandSupply}{TotientActualLcmTopEdgeStaircase.lean:1334}\n    (\\ev{Open}, \\scale{cofinal}):\n    \\[\n      \\forall a_0\\ \\exists a,m,\\ a_0\\le a \\wedge 8\\le a \\wedge m{+}1{+}(a{+}6)<2\\cdot 2^a \\wedge\n      2H{+}m{+}3 < 2^m \\wedge\n    \\]\n    \\[\n      2H{+}m{+}2 \\le \\mathtt{diagonalAdjacentSuffixResidue}(2^a)\\,0\\,m \\le 2^m - (2H{+}m{+}2)\n    \\]\n    (a two-sided band on the adjacent-suffix residue directly, one candidate depth $m$).\n  \\item \\lean{PowerTwoOddGuardTopEdgeHalfWordBandSupply}{TotientActualLcmTopEdgeStaircase.lean:1349}\n    (\\ev{Open}, \\scale{cofinal}): at the odd guarded depth $2q{+}1$, a two-sided band of half-width\n    $H{+}q{+}2$ on the half-word residue mod $4^q$ --- ``substantially weaker than the older fixed\n    $1/32$ central band.''\n  \\item \\lean{PowerTwoActualFinalTopEdgeMagnitudeSupply}{TotientActualLcmTopEdgeStaircase.lean:1471}\n    (\\ev{Open}, \\scale{cofinal}), \\emph{proved equivalent} to the previous one via\n    \\lean{topEdgeHalfWordBandSupply_iff_actualFinalCenteredMagnitudeSupply}{TotientActualLcmTopEdgeStaircase.lean:1479}:\n    \\[\n      \\forall a_0\\ \\exists a,q,\\ \\max(14,a_0)\\le a \\wedge \\mathtt{oddGuardedCanonicalAdjacentSuffixDepth}(2^a)=2q{+}1\n      \\wedge H{+}q{+}2 \\le |\\mathtt{actualOddHalfCenteredLift}\\ a\\ q| .\n    \\]\n  \\item \\lean{PowerTwoFlexibleActualTopEdgeMagnitudeSupply}{TotientActualLcmTopEdgeStaircase.lean:1927}\n    (\\ev{Open}, \\scale{cofinal}): the same magnitude bound with the depth restriction relaxed from\n    ``canonical guarded'' to any odd $2q{+}1$ satisfying the half-cell fit\n    $2(H{+}q{+}2)\\le 4^q$ and the sign-corridor room $2q{+}2{+}(a{+}6)<2\\cdot 2^a$.\n  \\item \\lean{PowerTwoFlexibleActualTerminalDominanceSupply}{TotientActualLcmTopEdgeStaircase.lean:1908}\n    --- the weakest of all five; treated separately as row TE-05-weakest below.\n\\end{enumerate}\nProving \\emph{any one} of these five closes Erd\\H{o}s \\#249; the corpus has already proved all the\nimplications from each to\n\\lean{PowerTwoActualLcmTopEdgeResidueGapSupply}{TotientActualLcmTopEdgeStaircase.lean:1325} (chain:\nmidband $\\to$ residue-gap at \\lean{}{TotientActualLcmTopEdgeStaircase.lean:2109}; half-word-band\n$\\to$ midband at \\lean{}{TotientActualLcmTopEdgeStaircase.lean:2058}; final-magnitude\n$\\Leftrightarrow$ half-word-band at \\lean{}{TotientActualLcmTopEdgeStaircase.lean:1479};\nflexible-magnitude $\\to$ midband at \\lean{}{TotientActualLcmTopEdgeStaircase.lean:2008};\nfinal-magnitude $\\to$ flexible-magnitude at \\lean{}{TotientActualLcmTopEdgeStaircase.lean:1993}).\n\\end{prop}\n\n\\subsubsection{TE-05-weakest --- one scalar inequality at cofinally many odd ranks (\\coord{actual-lcm-top-edge})}\n\nThis is the weakest landed link in the whole \\#249 sufficiency lattice: a single comparison\nbetween one terminal arithmetic letter and twice a centred lift.\n\n\\begin{defn}\n\\lean{PowerTwoFlexibleActualTerminalDominanceSupply}{TotientActualLcmTopEdgeStaircase.lean:1908}.\n\\ev{Open}. \\scale{cofinal}. Writing $H=\\mathtt{periodLcm}(2^a)$,\n\\[\n  \\forall a_0\\ \\exists a,q,\\ a_0\\le a \\wedge 8\\le a \\wedge 2q{+}2{+}(a{+}6)<2\\cdot 2^a \\wedge\n  2(H{+}q{+}2)\\le 4^q \\wedge\n\\]\n\\[\n  \\mathtt{diagonalWindowIncrement}(2^a)(2q{+}2) \\;\\le\\; 2\\cdot \\mathtt{actualOddHalfCenteredLift}\\ a\\ q .\n\\]\n\\end{defn}\n\n\\begin{thm}[Consumer]\n\\lean{actualLcmTailOrbit_notMem_int_of_actualTerminalDominance}{TotientActualLcmTopEdgeStaircase.lean:1880}\nand\n\\lean{irrational_totientSeries_of_flexibleActualTerminalDominanceSupply}{TotientActualLcmTopEdgeStaircase.lean:2221}.\n\\ev{Lean}. \\scale{uniform}. The dominance hypothesis at a single odd rank already excludes\nintegrality of \\m{\\mathtt{actualLcmTailOrbit}\\ a}, and the supply predicate composes to\n\\m{\\mathrm{Irrational}(S)}.\n\\end{thm}\n\n\\begin{obs}[The exact identity, and the internal tension it exposes]\n\\label{obs:staircase-tension}\n\\lean{two_mul_actualOddHalfCenteredLift_eq_terminal_sub_trueCarry}{TotientActualLcmTopEdgeStaircase.lean:1678}.\n\\ev{Lean}. \\scale{uniform}. For $a\\ge 8$, room $2q{+}1{+}1{+}(a{+}6)<2\\cdot2^a$, half-cell fit\n$2(H{+}q{+}2)\\le 4^q$, and any integral representative $z$ with $z = \\mathtt{actualLcmTailOrbit}\\ a$:\n\\[\n  2\\cdot \\mathtt{actualOddHalfCenteredLift}\\ a\\ q\n  \\;=\\; \\mathtt{diagonalWindowIncrement}(2^a)(2q{+}2) - \\mathtt{carryOrbit}\\ H\\ H\\ z\\ (2q{+}1) .\n\\]\nThis is an \\emph{exact equality}, not a bound. Comparing it against the dominance inequality above,\n\\emph{the dominance inequality is literally equivalent to\n$\\mathtt{carryOrbit}\\,H\\,H\\,z\\,(2q{+}1)\\le 0$}. But SGN-02\n(\\lean{actualLcm_trueEndpointSurvivor_neg}{TotientActualLcmOrbitSign.lean:172}, \\ev{Lean},\n\\scale{uniform}) proves that under integrality the true carry orbit is \\emph{strictly positive}\nthroughout this exact corridor: for $a\\ge 8$ and $J{+}K{+}(a{+}6)<2\\cdot 2^a$,\n\\[\n  0 < \\mathtt{carryOrbit}\\ H\\ (H{+}J)\\ d\\ K\n\\]\nwhenever $d$ is the real-valued representative of the translated tail difference. So the two\nbranches of the corridor-escape disjunction that\n\\lean{actualLcmTailOrbit_notMem_int_of_actualTerminalCarryCorridorEscape}{TotientActualLcmTopEdgeStaircase.lean:1824}\noffers are \\emph{not} symmetric: SGN-02 has already eliminated the branch that the identity most\nnaturally supplies (dominance, $\\mathtt{carryOrbit}\\le 0$), and the surviving branch is the\n\\emph{lower} escape\n\\[\n  2\\cdot \\mathtt{actualOddHalfCenteredLift}\\ a\\ q \\;\\le\\;\n  \\mathtt{diagonalWindowIncrement}(2^a)(2q{+}2) - (2H{+}2q{+}3) .\n\\]\nThis is stated precisely so the reader can check it: it is neither a refutation of the dominance\nroute (SGN-02 constrains the sign of $\\mathtt{carryOrbit}$, it does not itself bound\n$\\mathtt{diagonalWindowIncrement}$ or $\\mathtt{actualOddHalfCenteredLift}$ against $0$, so dominance\nis not shown \\emph{false}, only shown to entail a carry-orbit sign the census never violates) nor a\nproof (no theorem excludes dominance outright). It is the precise reason the finite census keeps\nlanding just inside the corridor rather than outside it, and it identifies the lower-escape branch\nas the coordinate where a producer is actually needed.\n\\end{obs}\n\n\\begin{prop}[What would close it]\nEither (i) the lower-escape branch cofinally --- as displayed just above --- or (ii) the two-sided\nmagnitude form \\lean{PowerTwoFlexibleActualTopEdgeMagnitudeSupply}{TotientActualLcmTopEdgeStaircase.lean:1927}\n(item 4 of Proposition~\\ref{prop:te-chain}), which asks only $H{+}q{+}2 \\le\n|\\mathtt{actualOddHalfCenteredLift}\\ a\\ q|$ and which the file proves\n(\\lean{flexibleActualTerminalCarryCorridorEscapeSupply_of_magnitude}{TotientActualLcmTopEdgeStaircase.lean:1938})\nimplies corridor escape via a clean sign split (positive branch escapes above the terminal letter,\nnegative branch escapes below the directed bound), sidestepping the SGN-02 tension entirely because\nit does not commit to a sign for the centred lift.\n\\end{prop}\n\n\\subsubsection{SEP-02 --- separation from an explicit rational approximant (\\coord{raw-approximant})}\n\n\\begin{thm}\n\\lean{abs_actualLcmTailOrbit_sub_rawApprox_lt}{TotientActualLcmOrbitSeparation.lean:141} and\n\\lean{irrational_totientSeries_of_actualLcmOrbitSeparationSupply}{TotientActualLcmOrbitSeparation.lean:305}.\n\\ev{Lean}. \\scale{uniform}. Unconditionally, for every $a$ and $q$,\n\\[\n  \\bigl|\\, \\mathtt{actualLcmTailOrbit}\\ a - \\mathtt{actualLcmRawApprox}\\ a\\ q \\,\\bigr|\n  \\;<\\; \\frac{4H + 2(2q{+}1) + 4}{2^{2q+2}} ,\n\\]\nwhere $\\mathtt{actualLcmRawApprox}\\ a\\ q$ is an explicit finite computable rational block. The\nerror radius is uniform in $a$ and $q$ and decays like $H/4^q$: the analytic tail is fully\ndischarged.\n\\end{thm}\n\n\\begin{rem}\nConsequently a cofinal $1/32$-separation of the raw approximant from every integer suffices for\n\\#249. The corpus verifies the analogous kill \\emph{unconditionally} at exactly two exponents,\n$a=4$ and $a=6$\n(\\lean{actualLcmTailOrbit_four_and_six_notMem_int}{TotientActualLcmShortKill.lean:35}, \\ev{Cert},\n\\scale{fixed}, via \\lean{certifiedKill_diagonal_t16}{} and\n\\lean{certifiedKill_diagonal_t64}{DiagonalPincerCertificateT64Endpoint.lean:1928}). Nothing beyond\n$a=6$ is proved. The depth $q$ is not free: the supply predicate pins $2q{+}1 =\n\\mathtt{oddGuardedCanonicalAdjacentSuffixDepth}(2^a)$, so per scale $a$ there is exactly one\nadmissible depth, not a search over depths.\n\\end{rem}\n\n\\begin{defn}[What would close it]\n$\\mathtt{PowerTwoActualLcmOrbitSeparationSupply}$: \\ev{Open}, \\scale{cofinal}.\n\\[\n  \\forall a_0\\ \\exists a\\ge\\max(2,a_0)\\ \\exists q,\\\n  \\mathtt{oddGuardedCanonicalAdjacentSuffixDepth}(2^a)=2q{+}1 \\wedge\n\\]\n\\[\n  \\forall z\\in\\mathbb Z,\\ \\tfrac1{32} + \\mathtt{actualLcmRawErrorRadius}\\ a\\ q \\;\\le\\;\n  |\\mathtt{actualLcmTailOrbit}\\ a - z| .\n\\]\nSince the error radius is explicit, this reduces to a distance-to-nearest-integer lower bound for\none explicitly computable rational per scale --- the cleanest reduction of the \\#249 obligation to\na purely finite, computable question in the whole corpus.\n\\end{defn}\n\n\\subsubsection{SK-02 --- the short-window arithmetic kill, verbatim, truncated at $a_0\\le 6$ (\\coord{short-window-arithmetic})}\n\n\\begin{thm}\n\\lean{powerTwoActualLcmShortArithmeticKillSupply_through_six}{TotientActualLcmShortKill.lean:54}\nand\n\\lean{irrational_totientSeries_of_shortArithmeticKillSupply}{TotientActualLcmShortKill.lean:63}.\n\\ev{Cert} for the witness, \\ev{Lean} for the consumer. \\scale{bounded}.\n\\[\n  \\forall a_0\\le 6,\\ \\exists a,L,\\ a_0\\le a \\wedge L < 2\\cdot 2^a \\wedge\n  \\mathtt{LcmDiagonalArithmeticKill}(2^a)\\,L\n\\]\n--- literally the open cofinal supply predicate below with its universal quantifier truncated at\n$a_0\\le 6$. This is the purest \\scale{fixed}-vs-\\scale{cofinal} row in the corpus: the proof term is\n\\texttt{\u27e86, 93, ha\u2080, by norm\\_num, lcmDiagonalArithmeticKill\\_two\\_pow\\_six\u27e9} --- a single\nhard-coded witness $(a,L)=(6,93)$, itself discharged from\n\\lean{certifiedKill_diagonal_t64}{DiagonalPincerCertificateT64Endpoint.lean:1928} (a\n\\texttt{norm\\_num} evaluation over an explicit table of $\\varphi$ values). There is no argument in\n$a$ whatsoever; removing the bound $a_0\\le 6$ requires an entirely new proof, not a re-run.\n\\end{thm}\n\n\\begin{defn}[What would close it]\n$\\mathtt{PowerTwoActualLcmShortArithmeticKillSupply}$: \\ev{Open}, \\scale{cofinal}.\n\\[\n  \\forall a_0\\ \\exists a,L,\\ a_0\\le a \\wedge L < 2\\cdot 2^a \\wedge\n  \\mathtt{LcmDiagonalArithmeticKill}(2^a)\\,L .\n\\]\nThe short-window restriction $L<2\\cdot 2^a$ buys extra structure --- every non-divisor offset in\nthat window is a bare prime power, by \\texttt{eq\\_prime\\_pow\\_of\\_not\\_dvd\\_periodLcm} --- so this\nis a better-equipped target than the raw SEP-02 supply even though it is formally a stronger\nstatement (it implies certified-kill directly, without the separation-and-round step).\n\\end{defn}\n\n\\subsubsection{b11 --- the diagonal pincer certificate bank (\\coord{diagonal-pincer})}\n\n\\begin{thm}\n\\lean{certifiedKill_diagonal_all_imported}{DiagonalPincerCertificates.lean:2860}, with\n\\lean{diagonalPincerCertificateScales}{DiagonalPincerCertificates.lean:2842} and\n\\lean{diagonalPincerKillDepth}{DiagonalPincerCertificates.lean:2844}. \\ev{Cert}. \\scale{fixed}.\n\\[\n  \\forall t\\in\\{1,2,3,4,5,7,8,9,11,13,16,17\\},\\ \\exists L,\\\n  \\mathtt{certifiedKill}\\ (\\mathtt{periodLcm}\\ t)\\ (\\mathtt{periodLcm}\\ t)\\ L\n\\]\nat depths $\\{6,5,7,7,9,14,15,14,21,22,23,26\\}$ respectively, extended by the separate T19\\ldots T64\nmodules to 28 values through $t=64$ (the $t=64$ endpoint is\n\\lean{certifiedKill_diagonal_t64}{DiagonalPincerCertificateT64Endpoint.lean:1928}). Every witness\nis a \\texttt{norm\\_num} evaluation over a hard-coded block of $\\varphi$ values --- e.g.\\ the $t=17$\ncertificate lists $\\varphi(12252241),\\ldots,\\varphi(24504506)$ explicitly --- so nothing in the\nproof is a function of $t$.\n\\end{thm}\n\n\\begin{rem}[A stronger signal in the depth table than the theorem states]\n\\lean{certifiedKill_depth_floor}{TotientTailPeriodKiller.lean:79} forces $2(2H_t{+}L{+}2)<2^L$,\ni.e.\\ $L \\gtrsim \\log_2(4\\cdot\\mathtt{periodLcm}\\ t)$ at any certified depth. Comparing that floor\nagainst $\\mathtt{diagonalPincerKillDepth}$ shows every landed certificate fires within a small\nadditive constant of the theoretical minimum depth, at every tested scale --- a numerically\nsupported (\\ev{Cert}, not \\ev{Math}) anti-concentration observation, not a theorem.\n\\end{rem}\n\n\\begin{prop}[What would close it]\n\\[\n  \\exists C\\ \\forall t_0\\ \\exists t\\ge t_0\\ \\exists L\\le \\log_2(4\\cdot\\mathtt{periodLcm}\\ t)+C,\\quad\n  \\mathtt{certifiedKill}\\ (\\mathtt{periodLcm}\\ t)\\ (\\mathtt{periodLcm}\\ t)\\ L .\n\\]\nThis is an anti-concentration statement about the diagonal word's residue at the first admissible\ndepth --- exactly the shape a doubling-orbit equidistribution argument would produce.\n\\end{prop}\n\n\\subsubsection{a12 --- one window, sixteen periods, wrong-way quantifiers (\\coord{h-uniform-certificate})}\n\n\\begin{thm}\n\\lean{certifiedKill_all_upto_sixteen}{CertificateKernel.lean:18556} and\n\\lean{totient_series_ne_rat_of_den_dvd_pow_two_mul_mersenne_upto_sixteen}{CertificateKernel.lean:18562}\n(shallower sibling \\lean{certifiedKill_all_small}{TotientTailPeriodKiller.lean:404}). \\ev{Cert}.\n\\scale{fixed}.\n\\[\n  \\forall h\\in[1,16],\\quad \\mathtt{certifiedKill}\\ h\\ 14\\ 9\n\\]\n--- a single position $N=14$ and a single depth $L=9$ that simultaneously certify every period $h$\nup to $16$. Consequence: $S\\ne r$ for every rational $r$ with $r.\\mathrm{den}\\mid 2^{14}(2^h{-}1)$,\n$1\\le h\\le 16$. Proof is \\texttt{decide} on a 9-bit window; no part of it is a function of $N$ or of\nthe $h$-range.\n\\end{thm}\n\n\\begin{rem}[Quantifier inversion --- the only $h$-uniform row in the corpus]\nThe 249-supply obligation is $\\forall h\\ \\forall N_0\\ \\exists N\\ge N_0\\ \\exists L$. This row proves\n$\\exists N\\ \\exists L\\ \\forall h\\in[1,16]$: the quantifiers are inverted, and the $h$-range is\nfinite. The inversion helps in one direction (one $(N,L)$ covers a whole block of periods, more\nthan the obligation asks) and hurts in the other ($N$ is pinned at $14$, not cofinal). This is the\nonly row in the corpus with $h$-uniformity, and it is the reason a scale-uniform version would be\nunusually strong.\n\\end{rem}\n\n\\begin{prop}[What would close it]\nA scale-uniform version of the same shape: $\\exists f:\\mathbb N\\to\\mathbb N$ with $f(N)\\to\\infty$\nsuch that $\\forall N_0\\ \\exists N\\ge N_0\\ \\exists L$ with $\\mathtt{certifiedKill}\\ h\\ N\\ L$ for all\n$h\\le f(N)$. This implies the obligation immediately (fix $h$, take $N_0$ large enough that\n$f(N)\\ge h$) and, unlike the obligation, is a statement about one window at a time rather than a\nper-period search.\n\\end{prop}\n\n\\subsubsection{c3 --- the Farey denominator floor and its stalled growth law (\\coord{farey-window})}\n\n\\begin{thm}\n\\lean{tsum_totient_div_pow_two_ne_ratCast_of_den_le_79639646646701375323355774875831053}{CertificateKernel.lean:18056},\nbuilt from the classical mediant lemma \\lean{farey_gap}{GapFareyBound.lean:51} and the window\nsharpness certificate \\lean{gap_check_window_1_240_le_79639646646701375323355774875831053}{GapFareyBound.lean:176}\n/ \\lean{gap_check_window_1_240_first_failure}{GapFareyBound.lean:225}. \\ev{Lean} for the general\nmediant lemma; \\ev{Cert} for the $K=240$ window instantiation. \\scale{fixed}. If $S$ is rational,\nits reduced denominator exceeds $7.9639646646701375323355774875831053\\times 10^{34}$ (the exact\nnumeral in the declaration name). This is the strongest unconditional statement about $S$ in the\ncorpus, logically independent of the certificate-supply reduction.\n\\end{thm}\n\n\\begin{rem}[Why re-running the same window buys nothing]\nThe cofinal upgrade of this statement --- $S\\ne p/q$ for every bound $q$, not just $q =\n7.96\\times10^{34}$ --- is literally $\\mathrm{Irrational}(S)$, so the target coordinate is right.\nWhat blocks promotion is proved on disk:\n\\lean{gap_check_window_1_240_first_failure}{GapFareyBound.lean:225} establishes that the $K=240$\nwindow bound is \\emph{sharp}, at exactly the mediant $b{+}d$, so re-running the same argument at\nthe same window buys nothing further. Each new $K$ needs (i) a freshly committed $2^K$-scale\ntotient residue $V_K$ and (ii) fresh continued-fraction convergents of $V_K'/2^K$, both hard-coded\nnumerals in the current proofs. The growth of the bound $b_K{+}d_K$ is governed by the convergent\ndenominators of the underlying constant, which stall at $q_0$ precisely if $S=a/q_0$: the growth\nlaw is equivalent in difficulty to \\#249 itself.\n\\end{rem}\n\n\\begin{obs}[A closed sub-route]\n\\lean{ReducedDenominatorUnitGapCert}{PrimitiveWeightCertificate.lean:30} and\n\\lean{PrimitiveReducedDenominatorUnitGapSupply}{PrimitiveWeightCertificate.lean:47}, the corpus's\nown attempt at strengthening this window refinement via a unit-modulo-odd-part criterion, prove a\nhard ceiling in\n\\lean{reducedDenominatorUnitGapCert_prime_pow_iff}{PrimitiveWeightCertificate.lean:54} and\n\\lean{reducedDenominatorCandidateCount_prime_pow_le_one}{PrimitiveWeightCertificate.lean:111}\n(\\ev{Lean}, \\scale{n/a}): at a prime-power denominator, the strengthening rescues at most \\emph{one}\nextra lattice point. That route is closed --- it cannot be the source of a growth law.\n\\end{obs}\n\n\\begin{prop}[What would close it]\nA proved growth law for the window family: a function $g$ with $g(K)\\to\\infty$ and a proof that for\nevery $K$ the $(N{=}1,K)$ gap check passes for all $q\\le g(K)$. Equivalently, a lower bound on the\nconvergent denominators of the totient-window constant, uniform in $K$.\n\\end{prop}\n\n\\subsubsection{TA-01/TA-02 --- the two-adic pulse block reaches the arc centre and fails by an exponential (\\coord{two-adic-pulse})}\n\n\\begin{thm}\n\\lean{exists_prime_totient_twoAdic_pulse_divisors}{TotientTwoAdicPulseBlock.lean:54},\n\\lean{exists_prime_deltaTotient_twoAdic_pulseBlock}{TotientTwoAdicPulseBlock.lean:211},\n\\lean{windowDiscrepancy_modEq_half_of_twoAdic_pulse}{TotientTwoAdicPulseBlock.lean:262},\n\\lean{eventual_integral_tailDiff_has_cofinal_twoAdic_half_pulse}{TotientTwoAdicPulseBlock.lean:327}.\n\\ev{Lean}. \\scale{uniform} (the theorem is unconditional and holds for \\emph{every} $K$, $H$, $B$,\nnot merely cofinally many). For every $K\\ge 2$, every $H>K$, and every bound $B$, there are primes\n$p>B$ with a length-$(K{-}1)$ zero prefix and a terminal half-turn, giving\n\\[\n  A_{H,p-K,K} \\equiv 2^{K-1} \\pmod{2^K}.\n\\]\nUnder eventual integrality this transfers to an integer $z$ with $(z:\\mathbb R) =\n\\mathrm{totientTail}(p{+}H) - \\mathrm{totientTail}(p)$ and $z\\equiv 2^{K-1}\\pmod{2^K}$.\n\\end{thm}\n\n\\begin{rem}[Lands at the exact centre of the arc, fails by an exponential --- and why]\nThis lands the residue at the exact centre of the certified-kill arc: $2^{K-1}$ is maximally far\nfrom $0$ modulo $2^K$, precisely the coordinate the obligation wants. Taking $N{=}p{-}K$,\n$h{=}H$, $L{=}K$ gives radius $N{+}h{+}L{+}2 = p{+}H{+}2$, so $\\mathtt{certifiedKill}$ requires\n$2^{K-1} > p{+}H{+}2$. But $z\\equiv 2^{K-1}\\pmod{2^K}$ forces $|z|\\ge 2^{K-1}$, while the directed\ntail bound forces $|z| < p{+}H{+}2$; and the construction's own congruence\n$p\\equiv 1{+}2^{K-1}\\pmod{2^K}$ (equivalently $v_2(p{-}1)=K{-}1$) forces $p\\ge 1{+}2^{K-1}$. So\n$p{+}H{+}2 > 2^{K-1}$ always, and no choice of $p$ rescues it. The quantifier order is inverted:\nthe theorem gives, for fixed $K$, cofinally many \\emph{large} $p$; the certificate needs a $p$\n\\emph{small} relative to $2^{K-1}$, impossible for a single letter since $|\\varphi(N{+}H)-\\varphi(N)|\n< N{+}H$. Depth-promotion is not the issue: it has already been performed, generalizing an\nexponent-2 construction to every $K$ uniformly by CRT-gluing over a\n$\\mathrm{Unit}\\oplus\\mathrm{Fin}(K{-}1)\\oplus\\mathrm{Fin}(K{-}1)$ family.\n\\end{rem}\n\n\\begin{prop}[What would close it]\nThe same half-turn residue realised by an \\emph{accumulated} window rather than a single terminal\nletter: cofinally many $(h,N,L)$ with $A_{h,N,L}\\equiv 2^{L-1}\\pmod{2^L}$ and\n$2^{L-1} > N{+}h{+}L{+}2$. Only the weighted sum $\\sum \\Delta\\varphi\\cdot 2^{L-1-j}$ can carry\nmagnitude $2^L$; the per-letter pulse provably cannot. Concretely: a pulse-block construction whose\nzero prefix and half-turn are imposed on the \\emph{word}, not on one delta.\n\\end{prop}\n\n\\subsubsection{d-3a/d-3b --- cofinal jump positions, one unproved slack sign (\\coord{lcm-jump-slack})}\n\n\\begin{thm}[Cofinal producer, fully proved]\n\\lean{exists_periodLcm_strict_jump_ge}{DiagonalFreshLossBridge.lean:2437}. \\ev{Lean}.\n\\scale{cofinal} (proved, not open): $\\forall t_0\\ \\exists t\\ge t_0$ with\n$\\mathtt{periodLcm}\\ t < \\mathtt{periodLcm}(t{+}1)$, witnessed by $p{-}1$ for any prime $p>t_0$.\nThe power-of-two specialization\n\\lean{CanonicalAdjacentSuffixPowerTwoPostJumpSlackSupply}{DiagonalFreshLossBridge.lean:2495} shows\nthe positions $t=2^a{-}1$ already suffice, so no position search is needed at all.\n\\end{thm}\n\n\\begin{defn}[The slack scalar and the open supply]\n\\lean{canonicalAdjacentSuffixCentralSlack}{DiagonalFreshLossBridge.lean:2453}:\n\\[\n  \\mathtt{canonicalAdjacentSuffixCentralSlack}\\ t = \\min\\bigl(d-2^{m-5},\\ (2^m-2^{m-5})-d\\bigr),\n\\]\nwith $m$ the canonical adjacent-suffix depth and $d$ the residue at that depth. The open\nproducers, both \\ev{Open} \\scale{cofinal}, are\n\\lean{CanonicalAdjacentSuffixPostJumpSlackSupply}{DiagonalFreshLossBridge.lean:2488} (any strict\njump) and \\lean{CanonicalAdjacentSuffixPowerTwoPostJumpSlackSupply}{DiagonalFreshLossBridge.lean:2495}\n(power-of-two positions), each asking $0\\le \\mathtt{canonicalAdjacentSuffixCentralSlack}(t{+}1)$\ncofinally, and both compose to\n$\\mathrm{Irrational}(S)$ at\n\\lean{irrational_totientSeries_of_canonicalAdjacentSuffixPostJumpSlackSupply}{DiagonalFreshLossBridge.lean:2739}\nand\n\\lean{irrational_totientSeries_of_canonicalAdjacentSuffixPowerTwoPostJumpSlackSupply}{DiagonalFreshLossBridge.lean:2746}.\n\\end{defn}\n\n\\begin{rem}[Census evidence only]\n\\ev{Cert}, \\scale{fixed}. All $40$ strict jumps up to endpoint $113$ pass, closest margin\n$\\approx 0.000221$ of the modulus at $t=100$; the power-of-two endpoints $4,8,16,32$ pass with\nslack fractions $0.41,0.036,0.40,0.19$. Census, not theorem. The structural handle nobody has used:\n$\\mathtt{periodLcm\\_succ\\_eq\\_prime\\_mul\\_of\\_strict\\_jump}$\n(\\lean{periodLcm_succ_eq_prime_mul_of_strict_jump}{DiagonalFreshLossBridge.lean:2134}) says a\nstrict jump at $t$ means $t{+}1=p^k$ and $\\mathtt{periodLcm}(t{+}1) = p\\cdot\\mathtt{periodLcm}(t)$\n--- so the required statement is a transfer lemma for how the adjacent-suffix residue at height $H$\nmoves under $H\\mapsto pH$.\n\\lean{canonicalAdjacentSuffixCentralSlack_eq_of_periodLcm_eq}{DiagonalFreshLossBridge.lean:2460}\nalready proves the slack is constant across each LCM plateau, so only jump indices matter.\n\\end{rem}\n\n\\begin{prop}[What would close it]\n$0\\le \\mathtt{canonicalAdjacentSuffixCentralSlack}(2^a)$ for cofinally many $a$.\n\\end{prop}\n\n\\subsubsection{e2 --- the prime-jump commutator, one fixed witness (\\coord{prime-jump-commutator})}\n\n\\begin{thm}\n\\lean{primeJumpSharpRadius}{PrimeJumpWindow.lean:37} and\n\\lean{primeJumpTailCommutator_notMem_int_of_sharpKill}{PrimeJumpWindow.lean:178}. \\ev{Lean}.\n\\scale{uniform}. Uniform consumer for all $H,p,L$: a residue of the four-vertex commutator\n$J(H,p)$ outside the sharp radius $3pH{+}(p{+}1)(L{+}2)$ forces $J(H,p)\\notin\\mathbb Z$. This is\ntighter than the earlier $4pH$ two-cell disjunction.\n\\end{thm}\n\n\\begin{thm}[The one witness]\n\\lean{primeJumpSharpKill_twelve_five}{PrimeJumpWindow.lean:186}. \\ev{Cert}. \\scale{fixed}.\n$\\mathtt{primeJumpSharpKill}\\ 12\\ 5\\ 15$ (i.e.\\ $H=\\mathtt{periodLcm}\\ 4=12$, $p=5$, $L=15$),\nproved by \\texttt{decide} on an explicit 15-bit window with no dependence on $t$ or $p$; only a\nsingle instance exists, at $t=4$, the smallest nontrivial height. The full endpoint composition is\n\\lean{irrational_totient_series_of_primeJumpSharpKill_supply}{PrimeJumpWindow.lean:193}.\n\\end{thm}\n\n\\begin{rem}\nThe supply hypothesis $\\forall t_0\\ \\exists t\\ge t_0\\ \\exists p,L{>}0$ with\n$\\mathtt{primeJumpSharpKill}(\\mathtt{periodLcm}\\ t)\\ p\\ L$ is a genuinely cheaper target than the\nraw SEP-02 supply: it asks for one fresh prime $p$ per LCM height rather than a full central-arc\ncertificate. Because the consumer route is\n\\texttt{rational\\_totient\\_series\\_forces\\_lcm\\_cone\\_flatness}$\\to$contradiction, the natural\nattack is to pick $p$ as the fresh prime introduced at the next strict LCM jump (linking this row\nto d-3a), where the commutator's old-channel contributions are annihilated by\n\\lean{oldChannel_affine_moment_annihilation}{JointExponentTransport.lean:36}.\n\\end{rem}\n\n\\subsubsection{b7 --- the M\\\"obius--Mersenne denominator channel: unbounded, but off-coordinate (\\coord{mobius-mersenne})}\n\n\\begin{thm}\n\\lean{upperHalfMersenneProduct_lower_bound}{MersenneShadowDenominatorGrowth.lean:60},\n\\lean{lcmHeight_scaledMobiusShadow_den_lower_bound}{MersenneShadowDenominatorGrowth.lean:85},\n\\lean{exists_upperHalf_mersenne_channel_dvd_den_with_bounds}{MersenneShadowDenominatorGrowth.lean:99}.\n\\ev{Lean}. \\scale{uniform}. For every $t\\ge 5$, uniformly in $t$:\n\\[\n  2^{t/2} \\;\\le\\; \\prod_{p\\in \\mathrm{upperHalfPrimes}(t)} \\mathtt{mersenne}(p) \\;\\le\\;\n  \\mathrm{den}\\bigl(\\mathtt{lcmHeight}(t)\\cdot \\mathtt{numericMobiusShadow}(\\mathtt{lcmHeight}(t))\\bigr).\n\\]\nAn unbounded denominator lower bound at every LCM height, with an individual surviving channel\nisolated ($2^{t/2}\\le \\mathtt{mersenne}(p) < 2^t$). The proof is genuinely uniform in $t$\n(Bertrand's postulate via \\texttt{upperHalfPrimes\\_nonempty}, plus $2^{p-1}\\le 2^p{-}1$ factorwise\n--- no table, no constant degrading with $t$).\n\\end{thm}\n\n\\begin{rem}[The corpus's only proved unbounded-growth quantity in this coordinate, but the wrong coordinate for \\#249]\nThis lives in a different coordinate from the 249-supply obligation: it bounds the reduced\ndenominator of the \\emph{scaled} shadow at LCM height $t$, and there is no landed transport from a\ndenominator lower bound to $\\mathtt{certifiedKill}$ or to a totient-tail non-integrality. Two\nfurther honest caveats are recorded in the module's own docstring: it does not rule out\ncancellation by the foreign-defect term, and the growth rate $2^{t/2}$ is far below the scale\n$\\mathtt{lcmHeight}(t)\\approx 2^{1.44t}$ against which the enclosure consumers measure error. The\nexisting consumers built on it\n(\\texttt{scaleFullTarget\\_miss\\_of\\_lambert\\_projected\\_separation},\n\\texttt{scaleFullTarget\\_miss\\_of\\_lambert\\_projected\\_num\\_gap}) are \\#257-facing\n(\\texttt{ScaleFullTargetHit}), not \\#249-facing.\n\\end{rem}\n\n\\begin{prop}[What would close it]\nEither (i) a Dirichlet-criterion bridge in the shape of\n\\lean{irrational_of_den_mul_abs_sub_tendsto_zero}{CertificateKernel.lean:5371}: a sequence of\nrationals $u_t$ with $u_t\\ne S$ and $\\mathrm{den}(u_t)\\cdot|S-u_t|\\to 0$, which requires the\napproximation error to beat the denominator, not merely the denominator to grow; or (ii) a\ntransport map from ``channel of size $\\ge 2^{t/2}$ survives in the denominator'' to ``windowDiscrepancy\nresidue avoids the arc,'' which does not exist in the corpus. Route (i) needs the growth rate\nraised from $2^{t/2}$ to beat the shadow's own truncation error.\n\\end{prop}\n\n\\subsubsection{d4/d5 --- the parallel rank obligation, and a corrected pair of citations (\\coord{carry-rank})}\n\nThis row runs a second, independent open obligation in parallel to 249-supply, in the carry-kernel\n\\emph{rank} coordinate rather than the residue coordinate. \\textbf{Correction}: the source index\nattributed the two theorems below to swapped files; direct reads this session confirm the correct\nattribution is as given.\n\n\\begin{thm}\n\\lean{linearIndependent_canonicalTotientKernelFamily}{TotientMahlerDefect.lean:900}. \\ev{Lean}.\n\\scale{uniform}. Unconditional and uniform in $e$: the dyadic totient-kernel family is linearly\nindependent at every depth $e$ (dimension exactly $2^e{+}1$, via CRT and Dirichlet on primes in\narithmetic progression).\n\\end{thm}\n\n\\begin{thm}\n\\lean{not_irrational_totientSeries_implies_unbounded_carryRank_unconditional}{TotientCarryKernelRigidity.lean:284}.\n\\ev{Lean}. \\scale{uniform}. If $S$ is rational there is a tempered integral carry orbit $u$ with\n\\[\n  2^e - 1 \\;\\le\\; \\mathrm{finrank}_{\\mathbb Q}\\,\\mathrm{span}(\\mathtt{canonicalCarryKernelFamily}\\ u\\ e)\n  \\qquad\\text{for every } e .\n\\]\n\\end{thm}\n\n\\begin{rem}[The scale side is finished; the missing direction is proved dead on one route]\nSo: rationality forces \\emph{unbounded} carry-kernel rank, and the $2^e{-}1$ floor holds for all\n$e$ with a uniform proof. What is missing is the opposite inequality --- a rationality-side rank\nupper bound, which would contradict the floor and close \\#249. The corpus proves one natural route\nto it is dead:\n\\lean{false_of_compressedAdjointCertificate}{TotientMahlerDefect.lean:1051} (\\ev{Lean},\n\\scale{n/a}) shows no\n\\lean{CompressedAdjointCertificate}{TotientMahlerDefect.lean:1039} ($Q\\cdot v\\cdot A = $ boundary\nwith $|\\mathrm{boundary}| < Q\\cdot v$ and $A\\ne 0$) can exist, and\n\\texttt{not\\_finiteDimensional\\_span\\_fullTotientKernel} (immediately above it in the same file)\nshows the full family spans an infinite-dimensional space. Cross-checked against a separate row,\n$\\mathtt{lcm\\_factorIdeal\\_finiteRank\\_shiftAlgebra\\_not\\_sufficient}$: a single explicit\ncountermodel defeats every finite-rank shift-polynomial observation simultaneously, so finite-rank\namplification per se is not the missing rigidity.\n\\end{rem}\n\n\\begin{prop}[What would close it]\nA rationality-side rank upper bound: $\\exists C$ such that any tempered integral binary carry\norbit for a rational $\\mathtt{binaryCoeffSeries}$ has\n$\\mathrm{finrank}_{\\mathbb Q}\\,\\mathrm{span}(\\mathtt{canonicalCarryKernelFamily}\\ u\\ e)\\le C$ for\nall $e$, or any bound growing slower than $2^e{-}1$.\n\\texttt{not\\_irrational\\_totientSeries\\_implies\\_mod\\_period\\_and\\_unbounded\\_rank}\n(\\texttt{TotientTailCarryPeriod.lean:224}) pins the obstacle precisely: rationality buys uniform\neventual periodicity of $u$'s dyadic sections mod $v$, and that periodicity provably does\n\\emph{not} promote to a $\\mathbb Q$-rank bound without extra arithmetic input.\n\\end{prop}\n\n\\subsection{Synthesis: where the frontier actually is}\n\nCollecting the fourteen rows, three facts stand out as loci for future effort, stated with the same\nprecision as the rows themselves rather than as summary rhetoric.\n\nFirst, the single closest approach to 249-supply is the first-harmonic reduction of\n\\S\\ref{ssec:headline}: a constant-saving ($21/25$ or $9/10$) exponential-sum cancellation estimate,\nwith a satisfiable room condition and, via the subset consumer\n(Theorem~\\ref{thm:hgap-subset}), no requirement that the saving hold on a full interval or a\npositive-density subset. This is the only row in the index whose remaining content is recognisably\na single classical estimate rather than a compound arithmetic-geometric statement.\n\nSecond, the top-edge staircase (TE-04 through TE-05-weakest, \\coord{actual-lcm-top-edge}) is the\nonly row where the corpus has both halved the obligation (one-sided instead of symmetric) and\nexposed, via the exact identity of Observation~\\ref{obs:staircase-tension}, precisely which of two\nlogically symmetric escape branches survives the sign machinery already proved elsewhere in the\nsame file family. That five strictly weaker equivalent or sufficient forms are already proved\ninter-derivable (Proposition~\\ref{prop:te-chain}) means effort here is not fragmented across\nrestatements: proving any one closes the others automatically.\n\nThird, the d4/d5 rank obligation is structurally independent of the residue-coordinate rows above:\nit is a second, self-contained sufficient condition for \\#249 (an upper rank bound contradicting\nthe proved $2^e{-}1$ lower bound), in a coordinate (carry-kernel rank) where a natural finite-rank\nstrengthening is already proved impossible. A rank upper bound, if found, would not need to route\nthrough $\\mathtt{certifiedKill}$ at all.\n\nNo row in this index is a solution, a partial solution in the sense of resolved cases, or evidence\nthat \\#249 is likely true or false. Every open producer listed is exactly as open as the original\nErd\\H{o}s--Borwein question it reduces to; what the index adds is the exact remaining content,\nverified against the live Lean tree, of each reduction.\n\n% ---- part a249_p4 ----\n\\section{Closed routes, with the mechanism that closed them}\n\nThis section catalogues every no-go, countermodel, obstruction, and refuted\nproof strategy found in the \\#249/\\#257 corpus that bears on Erd\\H{o}s \\#249\n(\\(S := \\sum_{n\\ge1}\\varphi(n)/2^n\\) irrational). Erd\\H{o}s \\#249 is\n\\ev{Open}; nothing in this section decides it. What follows is a machinery\ninventory: for each dead route, the exact statement that closed it, the exact\nscope of the closure, and the exact machinery salvaged from the wreckage.\nThe organising warning, stated once here and applicable to every item below:\nseveral of these ``obstructions'' are statements about a\n\\emph{representation} of the problem (a proof strategy, a certificate\nfamily, a coordinate system) and not about the object \\(S\\) itself. Confusing\nthe two is the single most common error this catalogue exists to prevent.\n\n\\subsection{Meta-level: finite inspection cannot certify the supply (the\n\\texorpdfstring{\\(\\gamma\\)}{gamma}-splice remark)}\n\n\\textbf{(a) Route as conceived.} Every unconditional route to \\#249 in this\ncorpus bottoms out in a \\emph{certificate-supply} obligation of the shape\n\\(\\forall h\\ge1\\,\\forall N_0\\,\\exists N\\ge N_0\\,\\exists L,\\ \\mathrm{Sep}(h,N,L)\\)\n(\\lean{Erdos249257.LcmConeFlatness.lean}{LcmConeFlatness.lean:412}, the exact\nopen-obligation form quoted throughout the existing manuscript). A natural\nhope is that verifying \\(\\mathrm{Sep}\\) \u2014 or its diagonal specialisation \u2014 at\nevery scale up to some large, explicit bound \\(B\\) is evidence that the\nsupply holds, and that pushing \\(B\\) far enough would eventually amount to a\nproof.\n\n\\textbf{(b) Exact mechanism that closed it.} The manuscript's own remark\n(\\ev{Math}, prose-only, not formalised in Lean:\n\\emph{\\S5.4, ``finite inspection cannot establish the supply''}) gives an\nexplicit splice construction. Take any coefficient stream \\(\\gamma:\\mathbb\nN\\to\\mathbb N\\) that agrees with \\(\\varphi\\) on every index \\(\\le B\\), and\nalter \\(\\gamma\\) at one residue class beyond \\(B\\) so that the resulting\nbinary series \\(\\sum\\gamma(n)/2^n\\) is forced \\emph{rational}. Such a\n\\(\\gamma\\) exists for every \\(B\\) (this is the same coboundary-splice\ntechnique that produces \\S\\ref{sec:parity-countermodel}'s\n\\texttt{parityCoboundaryWeight} witness, generalised: insert a lacunary\nzero-valued coboundary edit \\(2/2^m - 4/2^{m+1} = 0\\) at some \\(m>B\\)). Because\n\\(\\gamma\\) agrees with \\(\\varphi\\) on every index \\(\\le B\\), \\(\\gamma\\) passes\nevery finite certificate that only inspects indices \\(\\le B\\) \u2014 in\nparticular every instance of \\(\\mathrm{Sep}(h,N,L)\\) with \\(N+L\\le B\\) that\n\\(\\varphi\\) itself passes or fails. Yet \\(\\sum\\gamma(n)/2^n\\in\\mathbb Q\\) by\nconstruction.\n\n\\textbf{(c) Precise scope.} This is a statement about \\emph{proof method},\nnot about \\(\\varphi\\): it does not touch \\(S\\) at all. What it excludes is\nthe inference ``\\(\\mathrm{Sep}\\) verified up to bound \\(B\\), for arbitrarily\nlarge \\(B\\), therefore \\(\\mathrm{Sep}\\) holds cofinally.'' The scope is\nexactly \\(\\scale{bounded}\\) versus \\(\\scale{cofinal}\\): any finite-B\nverification, however large, is compatible with both outcomes, because a\nrational stream can be built to survive any \\emph{fixed} inspection horizon.\nIt says nothing about whether \\(\\varphi\\) itself is such a \\(\\gamma\\) \u2014 only\nthat no finite-horizon check can distinguish \\(\\varphi\\) from a \\(\\gamma\\)\nthat is.\n\n\\textbf{(d) Salvage.} The splice construction is exactly the mechanism\nunderlying \\S\\ref{sec:parity-countermodel}'s Lean-formalised countermodel\nbelow, so its \\emph{content} is not lost even though this specific remark is\nunformalised: everywhere a proof strategy in this corpus proposes to certify\na supply by exhaustive finite search (the diagonal deposits at\n\\(t\\in\\{1,\\dots,64\\}\\), \\lean{Erdos249257.DiagonalPincerCertificates}{DiagonalPincerCertificates.lean:2842}; the row-200{,}000 sqrt-escape\ncensus on the \\#257 side), this remark is the standing reason such a search,\nhowever large, is evidence and not proof. \\ev{Math} \\scale{n/a}\n\\coord{binary-digit}.\n\n\\subsection{The parity-aperiodicity countermodel\n(\\texttt{TotientParityCoboundaryCountermodel})}\n\\label{sec:parity-countermodel}\n\n\\textbf{(a) Route as conceived.} A natural strengthening of \\#249 attempts:\nshow that any coefficient word \\(c:\\mathbb N\\to\\mathbb N\\) that (i) is\nuniformly bounded, (ii) satisfies the trivial linear-growth bound\n\\(c(n)\\le n\\), (iii) matches \\(\\varphi(n)\\bmod 2\\) exactly, and (iv) is not\neventually periodic \u2014 even strengthened to \\emph{cofinally, arbitrarily\nseparated, arbitrarily long blocks} of non-periodicity \u2014 must have irrational\nbinary series \\(\\sum c(n)/2^n\\). This is the natural target for anyone\ntrying to use only \\(\\varphi\\)'s coarse parity/aperiodicity profile.\n\n\\textbf{(b) Exact mechanism.} The construction\n\\(\\mathtt{parityCoboundaryWeight}(n) := \\mathtt{parityBaseWeight}(n) +\n2\\cdot\\mathtt{largePowerTwoBit}(n) - 4\\cdot\\mathtt{largePowerTwoBit}(n-1)\\),\nwhere \\(\\mathtt{parityBaseWeight}\\) is the eventually-constant word\n\\(0,1,1,2,4,4,4,\\dots\\) and \\(\\mathtt{largePowerTwoBit}(n)=1\\) iff\n\\(n=2^{k+3}\\), is a lacunary zero-valued coboundary splice on top of an\neventually-constant base. The strongest of five landed theorems,\n\\begin{prop}[Totient-parity arbitrarily-many-separated-carry rational\ncountermodel]\nThere exists \\(c:\\mathbb N\\to\\mathbb N\\) such that: \\(c(n)\\le6\\) for all\n\\(n\\); \\(c(n)\\le n\\); \\(c(n)\\equiv\\varphi(n)\\pmod2\\) for every \\(n\\); for\nevery \\(N,G,K\\) there is a block of \\(K\\) explicit \\((6,0)\\) carry-pulse\npairs beyond \\(N\\), each pair separated by more than \\(G\\); and\n\\(\\sum_n c(n)/2^n = 3/2\\).\n\\end{prop}\n\\lean{Erdos257PeriodNoncollapse.exists\\_totientParity\\_arbitrarilyManySeparatedCarry\\_rational\\_countermodel}{TotientParityCoboundaryCountermodel.lean:637}\n(sum computation \\lean{tsum\\_parityCoboundaryWeight\\_eq\\_three\\_halves}{TotientParityCoboundaryCountermodel.lean:359}; non-periodicity\n\\lean{parityCoboundaryWeight\\_not\\_eventually\\_periodic}{TotientParityCoboundaryCountermodel.lean:487}; exact parity match\n\\lean{parityCoboundaryWeight\\_mod\\_two\\_eq\\_totient}{TotientParityCoboundaryCountermodel.lean:194}). \\ev{Lean} \\scale{cofinal}\n\\coord{binary-digit}.\n\n\\textbf{(c) Precise scope.} This is an \\emph{object}-level existence witness\n(\\(c\\ne\\varphi\\), only \\(c\\equiv\\varphi\\pmod2\\)) that closes a\n\\emph{proof-route}: it proves that the hypothesis set \\(\\{\\)bounded, linear\ngrowth, \\(\\varphi\\)-parity match, non-eventual-periodicity\\(\\}\\), no matter\nhow strongly the non-periodicity clause is strengthened (up to the\narbitrarily-separated arbitrarily-numerous form above), can never entail\nirrationality of the associated binary series, because \\(c\\) satisfies every\nhypothesis and is rational. It says nothing about \\(\\varphi\\) itself \u2014 the\nwitness sequence \\(c\\) is a different, hand-built sequence. Any future\nsufficient-condition candidate stated purely in terms of \\(\\{\\)coefficient\nboundedness, growth, parity, periodicity\\(\\}\\) must be checked against this\ncountermodel before being trusted, for \\emph{either} \\#249 or \\#257.\n\n\\textbf{(d) Salvage.} The lacunary coboundary-splice technique itself \u2014\ninserting zero-valued edits \\(2\\cdot2^{-m}-4\\cdot2^{-(m+1)}=0\\) at sparse\nranks to destroy periodicity while preserving the rational sum \u2014 is a fully\ngeneral recipe for building rational-valued, non-eventually-periodic,\nbounded coefficient sequences matching \\emph{any} parity template, and is\nliterally the mechanism instantiated informally in\n\\S\\ref{sec:gamma-splice}'s \\(\\gamma\\)-splice remark above. It is also a\nready-made stress-test fixture for any future parity-based sufficient\ncondition proposed anywhere in the corpus. Confirms the corpus-wide lesson\n(project memory \\texttt{feedback\\_erdos\\_reductions\\_rejected\\_bank\\_real\\_results}):\nonly arguments using \\emph{actual} quantitative totient/Mersenne size or\nresidue information \u2014 as in the \\texttt{TotientActualLcm*},\n\\texttt{TotientFixedRank*} families below \u2014 can possibly close \\#249; pure\nsymbolic-word arguments cannot.\n\n\\subsection{Two scoped \\#249 no-go countermodels: square-CRT correction\nsuppression is ambiguous}\n\n\\textbf{(a) Route as conceived.} The totient prime-dilation cocycle\n\\(\\varphi(pm)=(p-1)\\varphi(m)+[p\\mid m]\\varphi(m)\\) carries a correction term\nwhenever \\(p\\mid m\\). A natural strategy imposes a square congruence\n\\(n\\equiv A+pa\\pmod{p^2}\\) with a ``clean'' shift \\(h\\) (\\(p\\nmid a+h\\)) to\nsuppress that correction on a finite horizon, then hopes that\ncorrection-suppression (``cleanliness'') is itself enough to guarantee that\nthe resulting finite-difference cube is \\emph{nonzero} \u2014 i.e. that cleaning\naway the correction term automatically leaves behind a genuine, nonvanishing\narithmetic signal usable in a curvature argument.\n\n\\textbf{(b) Exact mechanism that closed it.} Two explicit, opposite\nfinite computations refute the hoped-for entailment simultaneously:\n\\begin{itemize}[nosep]\n\\item \\emph{Vanishing witness.} The smallest returned countermodel has the\n\\emph{entire} two-step finite block \\(\\mathtt{actualOneCubeCoeff}\\) equal to\nzero at \\(n=52\\), \\(r_0=13\\), \\(r_1=13/18\\)\n(\\lean{squareCRT\\_clean\\_block\\_can\\_vanish}{SquareCRTCube.lean:285}, kernel-\\texttt{decide}d;\ncleanliness witnessed by \\lean{squareCRT\\_vanishing\\_countermodel\\_is\\_clean}{SquareCRTCube.lean:294}).\n\\item \\emph{Nonvanishing witness.} A separate, nearby clean configuration at\n\\(n=27\\) has the same block \\emph{nonzero}\n(\\lean{squareCRT\\_clean\\_block\\_can\\_be\\_nonzero}{SquareCRTCube.lean:303}; cleanliness witnessed by\n\\lean{squareCRT\\_nonzero\\_countermodel\\_is\\_clean}{SquareCRTCube.lean:311}).\n\\end{itemize}\nBoth witnesses are exhibited alongside the positive achievability result\n\\lean{exists\\_squareCRT\\_clean\\_horizon}{SquareCRTCube.lean:198} (a bounded common base satisfying prescribed\nCRT anchors/residues does exist). \\ev{Cert} (theorem bodies not\nindependently re-verified beyond signatures and docstring \u2014 flagged\n\\texttt{DOCSTRING-SOURCED} in the source bank) \\scale{fixed}\n\\coord{other:square-crt-cocycle}.\n\n\\textbf{(c) Precise scope.} These are two \\#249-scoped countermodels \u2014 the\ncoordinate is the totient prime-dilation cocycle specifically, not a\nproblem-agnostic phenomenon. Together they show that\n\\emph{correction-suppression is logically independent of the\nsign/nonvanishing question it was hoped to settle}: ``clean'' (i.e.\ncorrection-free) is consistent with \\emph{both} outcomes. This is a\nstatement about a specific finite-difference construction (the\nrepresentation), not about \\(\\varphi\\): it does not exclude a nonvanishing\nfinite-difference cube in general, only the inference from cleanliness\n\\emph{alone} to nonvanishing.\n\n\\textbf{(d) Salvage.} The general cocycle identity\n\\(\\varphi(pm)=(p-1)\\varphi(m)+[p\\mid m]\\varphi(m)\\)\n(\\lean{totient\\_mul\\_prime\\_cocycle}{SquareCRTCube.lean}) and the CRT-witness apparatus\n(\\lean{exists\\_squareCRT\\_base}{SquareCRTCube.lean}, \\lean{exists\\_large\\_squareCRT\\_base}{SquareCRTCube.lean}) remain fully reusable for\n\\emph{any} residue-forcing construction, in any coordinate. More valuably,\nthe abstract cube-cancellation lemma extracted from the same file,\n\\lean{alternating\\_powerset\\_sum\\_eq\\_zero\\_of\\_insert\\_invariant}{SquareCRTCube.lean:264} \u2014 ``if inserting one\ncoordinate never changes the value of \\(f\\), the alternating powerset sum of\n\\(f\\) vanishes'' \u2014 is completely divorced from totient/CRT content and is\ndirectly reusable for any Hilbert-cube or higher-difference argument on\neither problem. What remains as the honest open gap: an independent\nanti-concentration or residue-producer argument for why the clean residual\ndoes \\emph{not} vanish, which this module explicitly does not supply.\n\n\\subsection{The adelic height obstruction}\n\\label{sec:adelic-height}\n\n\\textbf{(a) Route as conceived.} Several proof strategies attempt to\n\\emph{clear} an unwanted denominator factor from a rational value \\(x\\) by\nmultiplying by a small integer coefficient \\(c\\), hoping to discard the\ncomplementary denominator information entirely (e.g. localising a diagonal\ntail difference to one LCM ray channel while dropping the rest).\n\n\\textbf{(b) Exact mechanism.}\n\\begin{lem}[Scalar-localisation complement divisibility]\nFor \\(x:\\mathbb Q\\), \\(c:\\mathbb Z\\), \\(H:\\mathbb N\\): if \\(H\\mid x.\\mathrm{den}\\)\nand \\((c\\cdot x).\\mathrm{den}\\mid H\\), then \\(x.\\mathrm{den}/H \\mid\nc.\\mathrm{natAbs}\\).\n\\end{lem}\n\\lean{Erdos249257.AdelicHeightObstruction.scalarLocalization\\_complement\\_dvd}{AdelicHeightObstruction.lean:23}. The equality\nstrengthening, \\(\\exists t:\\mathbb Z,\\ H\\cdot c\\cdot x = t\\cdot x.\\mathrm{num}\\), is\n\\lean{Erdos249257.AdelicHeightObstruction.scalarLocalization\\_integer\\_eq\\_mul\\_num}{AdelicHeightObstruction.lean:56}. The\nMersenne-scale corollary \u2014 for positive \\(x\\), if \\(2^r\\mid x.\\mathrm{num.natAbs}\\)\nand \\(x<2/(2^n-1)\\), then \\(2^r(2^n-1)<2\\cdot x.\\mathrm{den}\\) \u2014 is\n\\lean{Erdos249257.AdelicHeightObstruction.positiveRat\\_mersenne\\_height}{AdelicHeightObstruction.lean:103} (generic form\n\\lean{positiveRat\\_numDivisor\\_mul\\_lt\\_two\\_mul\\_den}{AdelicHeightObstruction.lean:77}). A companion rigidity fact\nrules out smuggling the discarded information back in through a linear\nchannel: if a linear map \\(\\Lambda:V\\to W\\) factors through a surjective\nevaluation \\(\\mathrm{ev}\\) (\\(\\ker\\mathrm{ev}\\le\\ker\\Lambda\\)), then\n\\(\\Lambda\\) is \\emph{exactly} \\(\\mathrm{ev}(\\cdot)\\bullet w_0\\) for a single\nfixed \\(w_0\\) \u2014\n\\lean{Erdos249257.AdelicHeightObstruction.linearDescender\\_eq\\_smul\\_eval}{AdelicHeightObstruction.lean:120}. The upstream\nprimitive these import,\n\\lean{Erdos249257.RationalDenominatorSurvival.divisor\\_dvd\\_divInt\\_den}{RationalDenominatorSurvival.lean:17} and\n\\lean{survivingDivisor\\_dvd\\_scaled\\_divInt\\_den}{RationalDenominatorSurvival.lean:38}, gives the exact\nsurvival/cancellation law: a divisor \\(m\\mid D\\) of a displayed denominator\nsurvives reduction of \\(a/D\\) iff \\(m\\) is coprime to \\(a\\); after scaling the\nnumerator by \\(h\\), a surviving coprime divisor \\(C\\mid D\\) shrinks to\nexactly \\(C/\\gcd(C,h)\\), never further. \\ev{Lean} \\scale{uniform}\n\\coord{other:rational-height}.\n\n\\textbf{(c) Precise scope.} This is pure \\(\\mathbb Q\\)-arithmetic with zero\nMersenne or totient content \u2014 it is not a statement about \\(S\\) at all, but\nabout \\emph{any} rational-height bookkeeping argument. Its exact content:\nscalar denominator-clearing never \\emph{erases} the complementary\ndenominator; it moves that complement into the coefficient's (Archimedean)\nsize. Any construction that tries to shrink a displayed denominator down to\none surviving channel pays for it in coefficient growth. This directly\nexplains \\emph{why} denominator-compression strategies are structurally hard:\nthe Mersenne-shadow denominator lower bound\n\\(2^{t/2}\\le\\prod_{p\\in\\mathrm{upperHalfPrimes}\\,t}\\mathrm{mersenne}(p)\\)\n(\\lean{upperHalfMersenneProduct\\_lower\\_bound}{MersenneShadowDenominatorGrowth.lean:60}) forces a\nquadratic-in-scale coefficient cost via this lemma if one tries to localise\nto a single channel.\n\n\\textbf{(d) Salvage.} Fully problem-agnostic and directly reusable for \\#257\nor any other Lambert-type denominator-survival argument as-is. It is the\ngeneral mechanism underlying the corpus's Farey-gap denominator-exclusion\nfamily (Part C of the certificate bank,\n\\lean{tsum\\_totient\\_div\\_pow\\_two\\_ne\\_ratCast\\_of\\_den\\_le\\_79639646646701375323355774875831053}{CertificateKernel.lean:18056}, the strongest\ncurrent unconditional statement about \\(S\\): if \\(S\\) is rational its\nreduced denominator exceeds \\(79{,}639{,}646{,}646{,}701{,}375{,}323{,}355{,}774{,}875{,}831{,}053\\approx7.96\\times10^{34}\\)).\n\n\\subsection{The signed \\(q\\)-moment machinery \u2014 a positive tool, not an\nobstruction}\n\n\\textbf{(a) Route as conceived.} A ``signed Hankel determinant'' /\nHermite--Pad\\'e style route to \\#249 would exhibit a finite signed linear\ncombination of dyadically-cleared totient terms and show it is provably\nnonzero, giving a nonvanishing certificate for a first-harmonic or\nM\u00f6bius-companion construction.\n\n\\textbf{(b) Exact mechanism (a nonvanishing \\emph{engine}, not a no-go).}\n\\begin{lem}[Unique-terminal parity nonvanishing]\nIf a finite signed sum \\(\\sum_{i\\in s}u(i)\\cdot2^{e(m)-e(i)}\\), clearing\ndyadic denominators to a common exponent \\(e(m)\\), has one index \\(m\\in s\\)\nwith strictly maximal exponent \\(e(m)\\) and odd coefficient \\(u(m)\\), while\nevery other index has strictly smaller exponent, then the whole cleared sum\nis \\(\\equiv1\\pmod2\\), hence nonzero.\n\\end{lem}\n\\lean{scaled\\_dyadic\\_sum\\_ne\\_zero}{SignedQMomentObstruction.lean:95}, built from a fully generic rectangular\nCauchy--Binet-style determinant-of-product expansion\n\\lean{det\\_mul\\_rectangular}{SignedQMomentObstruction.lean:28} (\\texttt{hankelDet}/\\texttt{truncatedMoment} defs at\n\\lean{SignedQMomentObstruction.lean}{SignedQMomentObstruction.lean:107,116}). \\ev{Lean} \\scale{bounded}\n\\coord{p-adic}.\n\n\\textbf{(c) Precise scope \u2014 the point this section exists to make.} Despite\nliving in a file named alongside the corpus's ``obstruction'' modules, this\nis explicitly \\emph{not} phrased as a no-go in its own docstring: it is the\nfinite nonvanishing engine that a Hankel--Pad\u00e9 construction for \\#249 would\n\\emph{consume}. The parity lemma and the Cauchy--Binet expansion use no\ntotient or Mersenne structure at all \u2014 pure linear algebra, fully\nproblem-agnostic, directly transplantable to any base-\\(q\\) dyadic-clearing\nargument including \\#257's series family. What is missing, and what\nprevents this from closing anything, is a \\emph{supply} theorem: no result\nin the corpus proves that a unique-terminal configuration of this shape\nexists cofinally for the totient-derived family. The obstruction, such as it\nis, is not in this lemma but in the absence of its hypothesis's cofinal\nsupply \u2014 exactly the same shape of gap as \\S\\ref{sec:mahler-defect}'s Mahler\ndefect below.\n\n\\textbf{(d) Salvage.} The entire lemma is reusable machinery, not wreckage \u2014\nnothing here needs salvaging because nothing here failed. It stands ready\nfor whichever future construction can supply the missing cofinal\nunique-terminal-configuration existence.\n\n\\subsection{The residual-gauge obstruction}\n\n\\textbf{(a) Route as conceived.} A ``first-harmonic pivot'' strategy for\n\\#249 (via exponential/character-sum cancellation,\n\\lean{FirstHarmonicPivot.lean}{FirstHarmonicPivot.lean}) might try to certify a nonzero determinant or\nfull-rank minor of a residual-weighted monomial matrix as sufficient\nevidence that a genuine phase reconstruction (as opposed to a degenerate\nlocked configuration) has occurred.\n\n\\textbf{(b) Exact mechanism.}\n\\begin{prop}[Locked reconstruction preserves a nonzero minor]\nFor a monomial matrix \\(\\mathrm{residualMonomialMatrix}(e,r,z)(i,j) =\nr(i,j)\\cdot z(j)^{e(i)}\\) with all residual entries \\(z(j)\\ne0\\) and\n\\(e(i)=1\\) for a distinguished row \\(i\\): \\(\\det(\\mathrm{phasePowerMatrix})\n\\ne0 \\implies \\det(\\mathrm{residualMonomialMatrix})\\ne0\\), \\emph{and} the\ndistinguished row is identically \\(1\\) regardless.\n\\end{prop}\n\\lean{Erdos249257.locked\\_reconstruction\\_preserves\\_nonzero\\_minor}{ResidualGaugeObstruction.lean:94}\n(row-dependent relaxation \\lean{rowResidualMonomialMatrix\\_reconstructs\\_phasePowerMatrix}{ResidualGaugeObstruction.lean:124}; the\ncolumn-gauge identity underlying it,\n\\(\\det(\\mathrm{residualMonomialMatrix}) =\n\\det(\\mathrm{phasePowerMatrix})\\cdot\\prod_j W(j)\\), is\n\\lean{det\\_residualMonomialMatrix}{ResidualGaugeObstruction.lean:47}; a row-dependent relaxation still fails at\n\\lean{rowReconstruction\\_zero\\_row\\_one}{ResidualGaugeObstruction.lean:145}). \\ev{Lean} \\scale{uniform}\n\\coord{other:first-harmonic-phase}.\n\n\\textbf{(c) Precise scope.} Pure linear algebra over \\(\\mathbb C\\), fully\ngeneric in dimension \\(d\\) and exponent function \\(e\\), zero \\#249-specific\ncontent \u2014 this is a statement about \\emph{residual-weighted monomial\nmatrices as a representation}, not about the totient object. It rules out\nexactly one class of certificate: a residual-blind rank/determinant/\nconditioning test, used alone, cannot distinguish genuine phase\nreconstruction from an explicit ``locked'' degenerate configuration in which\na residual weighting scales every column by \\(W(j)=z(j)^{-1}\\), collapsing\nthe exponent-one row to the constant row \\((1,\\dots,1)\\) while the\ndeterminant stays nonzero whenever the base (Vandermonde-shaped)\n\\(\\mathrm{phasePowerMatrix}\\) determinant is nonzero. It does \\emph{not}\nrule out determinants carrying an additional arithmetic coupling identity\nbetween rows.\n\n\\textbf{(d) Salvage.} A hard requirement for any future first-harmonic\ndeterminant construction (Part E of the certificate bank,\n\\lean{irrational\\_totient\\_series\\_of\\_first\\_harmonic\\_norm\\_gap}{FirstHarmonicPivot.lean:83}): the construction \\emph{must}\ncouple rows by an extra arithmetic identity, not merely gauge-normalise\ncolumns. The diagonal-gauge-action technique itself is reusable for any\ndeterminant-based nonvanishing strategy in any coordinate.\n\n\\subsection{Cone flatness and cone nonflatness}\n\n\\textbf{(a) Route as conceived.} The diagonal certificate-supply obligation\n(\\lean{Erdos249257.LcmConeFlatness}{LcmConeFlatness.lean:412}) concerns only the pair \\((H,2H)\\) at LCM\nheight \\(H=\\mathrm{periodLcm}(t)\\). Two natural strengthenings: (i) show\nrationality of \\(S\\) is even more constrained than a single diagonal pair \u2014\nit flattens an \\emph{entire cone} of ratios; (ii) conversely, exhibit\n\\emph{nonflatness} on a wider ``menu'' of cone vertices as a cheaper\nsufficient certificate than a single pairwise \\(\\mathrm{certifiedKill}\\).\n\n\\textbf{(b) Exact mechanism \u2014 flatness (necessary consequence of\nrationality).}\n\\begin{prop}[Wave-24 LCM-cone flatness]\n\\(\\neg\\mathrm{Irrational}(S) \\implies \\exists t_1,\\ \\forall t\\ge t_1,\\\n\\forall q,m:\\mathbb N,\\ 0<q \\implies \\mathrm{totientTail}(q\\cdot\n\\mathrm{periodLcm}(t)+m\\cdot\\mathrm{periodLcm}(t)) -\n\\mathrm{totientTail}(q\\cdot\\mathrm{periodLcm}(t)) \\in\n\\mathrm{range}((\\uparrow):\\mathbb Z\\to\\mathbb R)\\).\n\\end{prop}\n\\lean{rational\\_totient\\_series\\_forces\\_lcm\\_cone\\_flatness}{CertificateKernel.lean:18674} (body\n\\texttt{LcmConeFlatness.lean}). \\ev{Lean} \\scale{uniform} (for all\nsufficiently large \\(t\\), \\emph{given} rationality) \\coord{other:lcm-ray-totient}.\nRationality forces one fractional constant on the entire LCM cone\n\\(\\{k\\cdot\\mathrm{periodLcm}(t):k\\ge1\\}\\) at every scale \\(t\\ge t_1\\), not\nmerely the single diagonal pair \u2014 the umbrella collapse theorem\n\\lean{irrational\\_totient\\_series\\_of\\_lcm\\_cone\\_window\\_kill\\_supply}{CertificateKernel.lean:18686} shows any\n\\(\\mathrm{certifiedKill}\\) \\emph{anywhere} on the two-multiplier cone, at\narbitrarily large \\(t\\), forces \\(\\mathrm{Irrational}(S)\\).\n\n\\textbf{(b\\('\\)) Exact mechanism \u2014 nonflatness (a sharper sufficient\ncertificate).}\n\\begin{prop}[Wave-25 cone-nonflat menu refuter]\nFor a nonempty menu \\(Q\\) of positive vertex multipliers with a\n\\emph{one-sided} floor \\(\\forall q\\in Q,\\ q\\cdot H+L+2<2^L\\) (half the\npairwise floor of \\(\\mathrm{certifiedKill}\\)): if \\(\\mathrm{coneNonflatCert}\\,\nH\\,L\\,Q\\) fires (an argmin/Helly-avoidance argument: the vertices cannot all\nshare one fractional part, else the minimal-deep-tail vertex would be a\ncommon left endpoint of every arc), then \\(\\exists q_i,q_j\\in Q,\\\n\\mathrm{totientTail}(q_j H)-\\mathrm{totientTail}(q_i H)\\notin\n\\mathrm{range}((\\uparrow):\\mathbb Z\\to\\mathbb R)\\).\n\\end{prop}\n\\lean{exists\\_nonintegral\\_pair\\_of\\_coneNonflatCert}{CertificateKernel.lean:18801} (supply theorem\n\\lean{irrational\\_totient\\_series\\_of\\_lcm\\_cone\\_nonflat\\_supply}{CertificateKernel.lean:18812}). \\ev{Lean}\n\\scale{bounded} (proved for a given finite menu \\(Q\\); an\n\\emph{unbounded}-scale menu is the open sufficient target)\n\\coord{other:lcm-ray-totient}.\n\n\\textbf{(c) Precise scope.} Flatness is a genuine \\emph{necessary\nconsequence} of rationality \u2014 it is a statement about what rationality of\n\\(S\\) would force, conditional on rationality, not an unconditional fact\nabout \\(S\\). Nonflatness is an unconditional \\emph{sufficient} certificate\nmechanism, information-theoretically half the depth floor of the pairwise\n\\(\\mathrm{certifiedKill}\\) (one-sided versus two-sided radius charging), but\nit has been proved only for individual finite menus \\(Q\\); no menu of\nunbounded scale has been supplied. Neither direction decides \\#249: flatness\ngives a contrapositive route (exhibit \\emph{any} cone nonintegrality at\narbitrarily large \\(t\\) and rationality is refuted) and nonflatness sharpens\nthe certificate needed, but the underlying producer obligation \u2014 a\nnonintegral tail difference at cofinally many scales \u2014 is unchanged. Note\nalso the coordinate warning from the manuscript's own Appendix C\n(\\lean{res:flat}{LcmConeFlatness.lean:90}): cone flatness is a\n\\emph{necessary} consequence, and only the sharper diagonal-pincer /\nfull-target-avoidance normal forms\n(\\lean{DiagonalPincerDecomposition.lean}{DiagonalPincerDecomposition.lean:215,290}) are an exact\n\\(\\mathrm{iff}\\) with irrationality.\n\n\\textbf{(d) Salvage.} The argmin/Helly-avoidance combinatorial-geometry\ntechnique behind cone-nonflatness is a reusable pattern for any modular-residue\npincer argument with more than two points, in any coordinate. The rank-2\nsecond-difference certificate family built on the same window apparatus was\nindependently tested and found \\emph{not} to be a shortcut over rank 1\n(measured, not merely conjectured: at \\((h,N)=(1,8)\\), rank-1 fires at depth\n8 while no rank-2 certificate exists at depth \\(\\le8\\);\n\\lean{totient\\_tail\\_rank\\_two\\_kill\\_sound\\_but\\_not\\_shallower\\_cell}{CertificateKernel.lean:18762}) \u2014 an\nexplicitly flagged dead end not to re-attempt without new information.\n\n\\subsection{The Mahler defect: dyadic totient-kernel finite rank per level,\ninfinite rank overall}\n\\label{sec:mahler-defect}\n\n\\textbf{(a) Route as conceived.} A finite-linear-algebra shortcut to \\#249:\nfind a bounded-depth linear relation among the dyadic totient-kernel\nchannels \\(n\\mapsto\\varphi(2^jn+r)\\) that would compress the infinite\nfamily to a finite-dimensional space, from which a rationality-forcing\ncontradiction (or a genuine rank obstruction) might be extracted.\n\n\\textbf{(b) Exact mechanism.}\n\\begin{thm}[Infinite dyadic totient-kernel rank]\nFor every depth \\(e\\ge0\\), the canonical family of \\(2^e+1\\) dyadic\ntotient-kernel channels (\\(\\mathtt{card\\_totientCanonicalIndex}\\),\n\\lean{TotientMahlerDefect.lean}{TotientMahlerDefect.lean:51}) is linearly independent over\n\\(\\mathbb Q\\), proved via a \\texttt{SeparatedMinorCertificate} (an explicit\nfinite evaluation-point assignment with nonzero determinant, forced by CRT +\nDirichlet's theorem on primes in arithmetic progressions \u2014 one channel made\nprime, every other channel forced through a fresh prime \\(\\equiv1\\bmod\\) a\nlarge power of \\(2\\)).\n\\end{thm}\n\\lean{linearIndependent\\_canonicalTotientKernelFamily}{TotientMahlerDefect.lean:900}; consequently the full\ninfinite family spans an infinite-dimensional \\(\\mathbb Q\\)-space,\n\\lean{not\\_finiteDimensional\\_span\\_fullTotientKernel}{TotientMahlerDefect.lean:1014}. A companion\nimpossibility result closes the natural repair attempt directly: a bounded\n\\emph{compressed-adjoint certificate} \u2014 a triple \\((Q,A,\\mathrm{boundary})\\)\nwith \\(Q\\cdot v\\cdot A=\\mathrm{boundary}\\), \\(|\\mathrm{boundary}|<Q\\cdot v\\),\n\\(A\\ne0\\) \u2014 is provably impossible,\n\\lean{false\\_of\\_compressedAdjointCertificate}{TotientMahlerDefect.lean:1051}. \\ev{Lean}\n\\scale{uniform} (holds for every \\(e\\); existence side is unconditional, via\nMathlib's CRT + primes-in-AP machinery) \\coord{binary-digit} (dyadic\ntotient-kernel \u2014 \\emph{not} the M\u00f6bius coordinate).\n\n\\textbf{(c) Precise scope \u2014 coordinate-relative, stated explicitly by the\nsource module.} This does \\emph{not} show irrationality of \\(S\\). It proves\nthe dyadic-kernel \\emph{side} is infinite-rank; the module's own docstring\nnames the missing input as ``a rationality-side finite-rank compression, or\nequivalent contradiction.'' The companion necessary-consequence-of-rationality\ntheorem, \\(\\neg\\mathrm{Irrational}(S) \\implies \\exists v>0,\\exists U:\\mathbb\nN\\to\\mathbb Z,\\ \\mathrm{IsTemperedBinaryOrbit}(\\varphi,v,U)\\wedge\\forall\ne,\\ 2^e-1\\le\\mathrm{finrank}_{\\mathbb Q}\\,\\mathrm{span}(\\mathrm{range}(\n\\mathrm{canonicalCarryKernelFamily}(U,e)))\\)\n(\\lean{not\\_irrational\\_totientSeries\\_implies\\_unbounded\\_carryRank\\_unconditional}{TotientCarryKernelRigidity.lean:284}),\nshows the SCALE side is finished \u2014 the \\(2^e-1\\) floor holds for all \\(e\\)\nwith a uniform proof \u2014 but the opposite inequality (a rationality-side rank\n\\emph{upper} bound) is exactly what is missing, and its absence is the\nentire open content here. A \\emph{third}, independent coordinate exhibits\nthe same coordinate-relative phenomenon: the M\u00f6bius-incidence companion\nmatrix \\(U_N(i,j)=\\mu((i+1)/(j+1))\\) if \\((j+1)\\mid(i+1)\\) else \\(0\\) is\nlower triangular with diagonal \\(1\\), hence \\(\\det U_N=1\\) for every \\(N\\),\nhence its jet-evaluation map is injective \u2014 a finite ``incidence quotient''\ncompression is impossible at any finite horizon in the M\u00f6bius coordinate too\n(\\lean{incidenceMobius\\_det\\_eq\\_one}{IncidenceQuotientHermitePade.lean:56}). Both natural finite\ntruncations \u2014 dyadic-residue and M\u00f6bius-incidence \u2014 turn out to have no\nnontrivial kernel, in two completely different coordinates. That is strong\nevidence any winning finite-linear-algebra shortcut, if one exists, must\nlive in a genuinely third coordinate or use an infinite/growing-parameter\nconstruction, not evidence that no shortcut exists at all.\n\n\\textbf{(d) Salvage.} \\lean{SeparatedMinorCertificate}{TotientMahlerDefect.lean:73} and\n\\lean{linearIndependent\\_of\\_separatedMinorCertificate}{TotientMahlerDefect.lean:81} are fully generic finite-rank\nlinear-independence infrastructure \u2014 the indexed family is a free variable,\nreusable for any finite-rank obstruction argument on either problem (e.g.\ntesting independence of Mersenne-indexed sequences for \\#257 via an explicit\nnonzero minor). The CRT+Dirichlet evaluation-forcing technique (make one\nchannel prime, force every other channel through a fresh arithmetic-progression\nprime) is a reusable construction pattern independent of the totient\nspecifics.\n\n\\subsection{Further closed routes, catalogued for completeness}\n\nThe following additional no-gos and route-pruning results were read in full\nand are recorded here in compressed form; each follows the same (a)/(b)/(c)/(d)\ndiscipline as above but is presented as a table row for space.\n\n\\begin{longtable}{p{0.16\\linewidth}p{0.40\\linewidth}p{0.36\\linewidth}}\n\\textbf{Route} & \\textbf{Mechanism \\& site} & \\textbf{Scope / salvage} \\\\\n\\hline\\endhead\nFull terminal dyadic staircase (annihilate every suffix letter mod growing\npowers of 2) &\n\\(a\\ge8\\), room bound \\(\\Rightarrow\\) \\texttt{ActualLcmTerminalDyadicStaircase}\nis false: the terminal letter would have to be positive (by short-window\npositivity) and strictly below a wider-than-itself modulus while divisible\nby it, forcing it to \\(0\\), contradicting positivity.\n\\lean{not\\_actualLcmTerminalDyadicStaircase\\_of\\_room}{TotientActualLcmTopEdgeStaircase.lean:251}.\n\\ev{Lean} \\scale{uniform} \\coord{binary-digit} &\nKills the FULL terminal-staircase proof strategy outright and\nunconditionally; the \\emph{punctured} staircase (all but the last letter\nvanish) survives and is pinned exactly to the half-turn value \\(2^{m-1}\\) at\nthe penultimate letter\n(\\lean{puncturedDyadicStaircase\\_penultimate\\_eq\\_half}{TotientActualLcmTopEdgeStaircase.lean:1187}). The\ngeneric mechanism (\\(0<e<2^m\\wedge2^m\\mid e\\Rightarrow e=0\\)) is reusable\nanywhere a positive quantity is asked to vanish mod a wider modulus. \\\\\n\\hline\nLCM factor-ideal retention (keep only the homogeneous ``factor-only'' part\nof an LCM-ray decomposition) &\nAn explicit nonzero all-horizon countermodel, built from multiples of\n\\(\\varphi(\\mathrm{periodLcm}(t))\\), agrees with every exact whole-ray anchor\n\\(\\Delta\\varphi(H,qH)=\\varphi(H)\\) for \\(2\\le q<t\\) and survives every finite\ncommensurate LCM-cube shift-polynomial, at every finite rank.\n\\lean{LcmFactorIdealPulseObstruction.lean:1--24}{LcmFactorIdealPulseObstruction.lean:1-24} (module docstring;\ntheorem bodies \\ev{Cert}, DOCSTRING-SOURCED). \\scale{uniform}\n\\coord{other:lcm-ray-totient} &\nThis is a \\emph{representation}-level exclusion, not an object-level one: it\nshows the homogeneous factor-ideal projection discards fresh M\u00f6bius-channel\ninformation that can be adversarially reconstructed. Explicitly flagged\nSYNTHETIC \u2014 no claim that the compensation letters occur as actual totient\ndifferences. Salvage: a future factor-ideal argument must control the\n\\emph{fresh} channel via\n\\lean{totient\\_eq\\_sum\\_mobiusTotientChannel}{LcmFactorIdealPulseObstruction.lean:328}, not just the old one. \\\\\n\\hline\nTwo-prime ``full-target diamond'' (four simultaneous ray hits certify more\nthan one) &\nThe four-hit diamond\n\\(\\mathrm{Hit}(H)\\wedge\\mathrm{Hit}(pH)\\wedge\\mathrm{Hit}(qH)\\wedge\n\\mathrm{Hit}(pqH)\\) is logically equivalent to \\(\\mathrm{Hit}(H)\\) alone, no\nprimality of \\(p,q\\) needed \u2014 hits transport multiplicatively for free\nbecause \\(\\mathrm{Hit}(H)\\iff\\mathrm{IsIntegralValue}(\\text{scaleDiagonalTailDifference}(H))\\)\nand integrality transports affinely along every ray \\(H\\mapsto kH\\).\n\\lean{Erdos249257.fullTarget\\_primeAdjunction\\_diamond\\_iff\\_root}{FullTargetPrimeAdjunctionNoGo.lean:196}.\n\\ev{Lean} \\scale{uniform} \\coord{other:mobius-mersenne} &\nThe 4-condition diamond carries zero information beyond its base point \u2014 a\n``curvature-zero collapse'' of a multi-point certificate. Any future\npositive-holonomy argument needs an independently-defined projection of the\nforeign state \\emph{plus} control of the discarded complement; a bare\n2/4-point diamond cannot supply new information beyond its own scale. \\\\\n\\hline\nFixed-precision local valuation-unit signature (2-adic ``tropical curvature\ncarry'' attack) &\nFor any finite word of odd-unit-part 2-adic symbols at fixed precision\n\\(u>0\\) and any starting carry state \\(e\\), there exists a compatible carry\norbit realising the word with every intermediate state centred inside its\nsymbol's dyadic radius.\n\\lean{Erdos249257.TotientTailPeriodKiller.fixedPrecisionTropicalNoGo}{TropicalCurvatureCarry.lean:137}.\n\\ev{Lean} \\scale{bounded} \\coord{p-adic} &\nBounded LOCAL valuation-unit data at FIXED precision can never exclude all\nfinite centred carry completions \u2014 no obstruction is derivable from a\nfixed-window local signature alone; growing precision, or extra arithmetic\ncoupling, is required. Pure 2-adic dynamics, zero totient content, reusable\nagainst any similarly-shaped fixed-window carry-certificate attack. \\\\\n\\hline\nFixed-depth affine carry reset (distinguishing two carry histories after a\nlong common tail) &\nFor the affine binary orbit \\(\\mathrm{orbit}(0)=u_0\\),\n\\(\\mathrm{orbit}(n{+}1)=2\\,\\mathrm{orbit}(n)-a(n{+}1)\\) with common forcing\nword \\(a\\): \\(\\mathrm{orbit}_u(L)-\\mathrm{orbit}_v(L)=2^L(u_0-v_0)\\) exactly.\n\\lean{Erdos249257.affineBinaryOrbit\\_mod\\_twoPow\\_eq}{GenericTailOrbitRigidity.lean:306}. \\ev{Lean}\n\\scale{uniform} \\coord{binary-digit} &\nAfter \\(L\\) common steps the endpoint residue mod \\(2^L\\) is\n\\emph{independent} of the initial carry \u2014 long common tails erase\npredecessor information exactly, not approximately. Directly relevant to any\nargument hoping to distinguish two carry histories from a shared long\nsuffix; the terminal forcing word alone determines the residue. \\\\\n\\hline\nBounded finite-state / autonomous successor decoder for the binary carry &\nFor a balanced-pulse family at location \\(m\\) whose predecessor \\texttt{State}\nis constant across the whole family, no \\(\\mathrm{decode}:\\mathrm{State}\\to\n\\mathbb N\\) recovers the radius parameter; any finite \\texttt{Fintype State}\nneeds \\(\\mathrm{card}\\ge\\lfloor m/2\\rfloor+2\\), unbounded in \\(m\\).\n\\lean{Erdos249257.balancedPulse\\_no\\_autonomous\\_decoder}{GenericTailOrbitRigidity.lean:246}. \\ev{Lean}\n\\scale{uniform} \\coord{binary-digit} &\nRules out, unconditionally, ANY proof strategy that defines a bounded or\nautonomous ``carry state'' summarising pre-\\(m\\) history and claims it\nexactly determines the post-\\(m\\) tail \u2014 independent of whether the\ncoefficients are \\(\\varphi\\) or a M\u00f6bius-support indicator. A hard stop\nagainst finite-automaton-computes-the-expansion style attacks on either\nproblem. \\\\\n\\hline\nPeriod-4 sign weight as a route to A7's ``frequently nonzero'' hypothesis for\nfree &\nThe divisor coefficient of the period-4 sign weight\n(\\(w(a){=}1\\) if \\(a{\\equiv}1\\), \\(-1\\) if \\(a{\\equiv}3\\), else \\(0\\), mod\n4) vanishes identically at every \\(n\\equiv3\\pmod4\\), via the divisor-pairing\ninvolution \\(d\\mapsto n/d\\).\n\\lean{Erdos249257.intWeightedCoeff\\_periodFourSignWeight\\_eq\\_zero\\_of\\_mod\\_four\\_eq\\_three}{CertificateKernel.lean:13898}.\n\\ev{Lean} \\scale{fixed} \\coord{mobius-mersenne} &\nWitnesses that Dirichlet-convolution coefficients CAN vanish identically on\nan arithmetic progression, so the ``frequently nonzero'' hypothesis in the\nnonnegative-periodic irrationality closure\n(\\lean{irrational\\_intWeightedErdosSeries\\_periodic\\_of\\_coeff\\_nonneg\\_of\\_frequently\\_ne\\_zero}{CertificateKernel.lean:14255})\nis not free \u2014 it is why the corpus lands a dichotomy for general periodic\nsigned weights rather than an unconditional supply theorem. \\\\\n\\hline\nCertified finite-depth ``death'' certificates as a route to membership proofs\n(both problems) &\n\\(\\mathtt{CertifiedGreedyMersenneDeath}(x,\\mathrm{level},\\mathrm{lookahead})\\)\nis a decidable, finite-depth certificate proving non-membership only\n(\\lean{greedyMersenneDeath\\_not\\_mem}{GreedyAchievementSet.lean:1720}; witness\n\\lean{three\\_fourths\\_certifiedGreedyMersenneDeath}{GreedyAchievementSet.lean:1742}, \\(3/4\\) certified dead at\nlevel 1). \\ev{Cert} \\scale{bounded} \\coord{greedy-orbit} &\nStructural one-sidedness, recurring across the WHOLE certified-kill family\n(\\lean{TotientTailPeriodKiller}{TotientTailPeriodKiller.lean}, \\lean{LcmConeFlatness}{LcmConeFlatness.lean}, this one):\ncertified-death \\(\\Rightarrow\\) non-membership, but absence of a found\ncertificate, or survival through any finite depth, proves \\emph{nothing}\nabout membership or rationality. \\\\\n\\hline\nReconstructing one totient value from a two-tail absolute-value linear\ncombination &\nFor any finite \\(w:\\iota\\to\\mathbb Q\\), \\(x:\\iota\\to\\mathbb N\\) with\n\\(\\sum w_i\\varphi(x_i)=1\\) (exact isolation), the crude two-tail cost\n\\(\\sum|w_i|(2(x_i+1)+(x_i+2))\\ge3\\), using only \\(\\varphi(x)\\le x\\).\n\\lean{Erdos257PeriodNoncollapse.three\\_le\\_totientAdjugateTailCost}{TotientTailCarryPeriod.lean:266}\n(closure \\lean{not\\_totientAdjugateTailCost\\_lt\\_one}{TotientTailCarryPeriod.lean:307}). \\ev{Lean}\n\\scale{uniform} \\coord{other:totient-adjugate-linear-algebra} &\nThe strategy needs cost \\(<1\\) and gets \\(\\ge3\\) at ANY finite grid height \u2014\nkills the entire ``isolate one coefficient via absolute-value two-tail\nbookkeeping'' family before it starts. The proof mechanism is problem-agnostic\n(any \\(c(n)\\le n\\) sequence gets the same floor by the triangle inequality)\nand transfers verbatim to a \\#257 reformulation. \\\\\n\\hline\nHomogeneous Mersenne-primitive prime factor alone forces a contradiction &\nFor the completely-multiplicative control \\(c(n)=n\\) (zero totient content):\nif \\(K<q\\), \\(q\\mid2^K-1\\), the corresponding tail shift is integral at every\n\\(N\\) yet \\(q\\nmid K\\). \\lean{idCoeff\\_mersenneModulus\\_does\\_not\\_annihilate\\_tailShift}{TotientTailCarryPeriod.lean:342}.\n\\ev{Lean} \\scale{n/a} \\coord{other:mersenne-modulus} &\nA primitive prime factor of the homogeneous multiplier \\(2^K-1\\) alone\nsupplies NO contradiction from tail integrality. A direct cross-problem\nwarning: \\#257's denominators \\(2^n-1\\) are literally this \\(q\\mid2^K-1\\)\nobject, so ``a large primitive Mersenne prime factor alone forces a\ncontradiction'' should be checked against this lemma before being attempted\non either problem. \\\\\n\\hline\nPrime-power reduced-denominator unit-gap strengthening (rescuing extra Farey\nlattice points) &\nAt a prime-power reduced denominator \\(p^e\\), the unit-gap refinement can\nrescue at most ONE additional candidate lattice point beyond the ordinary\ngap certificate. \\lean{reducedDenominatorCandidateCount\\_prime\\_pow\\_le\\_one}{PrimitiveWeightCertificate.lean:111}\n(iff-characterisation \\lean{PrimitiveWeightCertificate.lean}{PrimitiveWeightCertificate.lean:54}). \\ev{Lean}\n\\scale{uniform} \\coord{other:farey-gap} &\nA formal ceiling on how much this specific refinement can ever buy \u2014 a\nknown dead end for extending the \\(\\approx7.96\\times10^{34}\\) Farey rung\n(\\S\\ref{sec:adelic-height}) unboundedly. Do not re-attempt this exact\nstrengthening expecting more than +1 lattice point per prime power; the\nunderlying reduced-denominator-implies-unit-numerator technique is reusable\nfor other denominator-exclusion arguments. \\\\\n\\hline\nFixed rank-3 finite-difference kernel squeezing more than one dyadic bit of\ncurvature information &\n\\(\\mathrm{fixedRankSecondDifference}(2^{n+1}\\cdot2,\\,2^{n+1}\\cdot3)=-2^{n+1}\\)\nexactly, for all \\(n\\): the two-adic valuation gained by the primitive\nkernel factor \\(2\\varphi(j)\\) is exactly tight.\n\\lean{fixedRankSecondDifference\\_dyadic\\_fixture\\_exact}{TotientFixedRankLcmAsymptotic.lean:462}. \\ev{Lean}\n\\scale{n/a} \\coord{binary-digit} &\nRoute-pruning ammunition: rules out extracting more than one dyadic bit from\nthe bare 3-rank affine kernel \\((1,-2,1)\\); any further progress on the\nfixed-rank curvature route needs new arithmetic input, not a sharper\nvaluation squeeze from the same kernel. The affine rank-3 kernel\nclassification itself\n(\\lean{affine\\_rank\\_three\\_kernel\\_classification}{TotientFixedRankLcmAsymptotic.lean:338}, pure linear algebra\nover \\(\\mathbb Z\\), zero number-theoretic content) is fully reusable for any\n3-point finite-difference argument on either problem. \\\\\n\\end{longtable}\n\n\\section{Why the bounds are load-bearing}\n\\label{sec:promotion-audit}\n\nThe near-miss interface between the corpus's proved results and the four\nopen supply obligations \u2014 \\(\\#249\\)-supply, \\(\\#257\\)-reset,\n\\(\\#257\\)-cofinal-rows, and \\(\\#257\\)-universal \u2014 was swept for\n\\emph{promotion candidates}: proved theorems whose \\emph{stated} hypotheses\nlooked, on a first read of the \\texttt{yields} clause, close enough to the\nopen obligation's shape that a routine strengthening (raising a scale bound,\nwidening a hypothesis, or reindexing a quantifier) might close the gap\nwithout new mathematics. Fifty-six near-miss rows were catalogued across the\nfour obligations; eighteen of them were flagged \\texttt{promotable\\_claimed:\nTrue} on this first pass. \\textbf{Every one of the eighteen proofs was then\nopened and read in full}, not merely re-inspected at the statement level,\nand the promotion claim was checked against the actual proof body. The\naudit table below reports the verdict for all eighteen.\n\nOf the eighteen, \\textbf{sixteen are conclusively NOT\\_PROMOTABLE}: reading\nthe proof body exposes a documented arithmetic, logical, or coordinate\nobstruction that a routine strengthening cannot cross, and the row's own\nrecord states exactly what new input would be required. \\textbf{Two}\n(\\texttt{NM-02}, \\texttt{NM-03} in the \\(\\#257\\)-universal obligation) are\ngenuine exceptions: the auditor found the promotion is available with, in\nits own words, ``no new mathematics'' \u2014 a pure widening of an already-general\nproof to a wider stated target. Crucially, \\emph{neither of these two closes\nanything}: they widen an equivalence or a corollary from one fixed target\nvalue to a family of targets, without touching the missing arithmetic\ncontent (a cofinal certificate supply, a rank upper bound, an anti-concentration\nestimate) that every genuine closure of \\(\\#249\\) or \\(\\#257\\) requires. So\nwhile the promotion audit is not literally eighteen-for-eighteen at the level\nof ``can this exact statement be re-derived for a wider scope,'' it is\neighteen-for-eighteen at the level that matters for this paper's obligations:\n\\textbf{no promotion candidate in the audited set supplies missing arithmetic\ncontent toward closing \\#249 or \\#257}. The two mechanically-widenable rows\nare recorded honestly below rather than folded into the sixteen, because\ncollapsing that distinction would itself be exactly the kind of\nquantifier-slippage error this paper's accuracy rules forbid.\n\n\\begin{longtable}{p{0.10\\linewidth}p{0.14\\linewidth}p{0.10\\linewidth}p{0.58\\linewidth}}\n\\textbf{Obligation} & \\textbf{Row / site} & \\textbf{Gap kind} &\n\\textbf{Verdict and exact blocker} \\\\\n\\hline\\endhead\n249-supply & e1-companion,\n\\lean{FirstHarmonicGap.lean}{FirstHarmonicGap.lean:128} & hypothesis\\_strength &\nNOT\\_PROMOTABLE. The consumer already has the obligation's exact quantifier\nshape; the missing input is a single unproved arithmetic fact \u2014 a\nconstant-saving first-harmonic (Weyl-sum) cancellation bound\n\\(\\sum_{N\\in[X,2X)}\\cos(2\\pi\\cdot(\\ldots)/2^L)\\le(9/10)X\\) \u2014 not proved at\nany \\(X,h,L\\) anywhere in the corpus. \\\\\n249-supply & SGN-01,\n\\lean{TotientActualLcmOrbitSign.lean}{TotientActualLcmOrbitSign.lean:39} & hypothesis\\_strength &\nNOT\\_PROMOTABLE. Positivity is exactly HALF of the needed certificate: the\ncompanion theorem in the same file proves integrality forces the residue to\nthe TOP edge, not a kill \u2014 the lower half of the band is discharged\nunconditionally and cofinally, the upper half is untouched. \\\\\n249-supply & TE-04,\n\\lean{TotientActualLcmTopEdgeStaircase.lean}{TotientActualLcmTopEdgeStaircase.lean:1260} & hypothesis\\_strength &\nNOT\\_PROMOTABLE. The consumer side is finished and cofinal (proved for\nevery \\(a\\ge8\\)); the residual is purely the one-sided residue-gap\n\\emph{producer}, unconditionally unsupplied at even one large \\(a\\). \\\\\n249-supply & TE-05-weakest,\n\\lean{TotientActualLcmTopEdgeStaircase.lean}{TotientActualLcmTopEdgeStaircase.lean:1908} & hypothesis\\_strength &\nNOT\\_PROMOTABLE. The exact identity pinning the target shows the dominance\ninequality is equivalent to \\(\\mathrm{carryOrbit}\\le0\\), and the sign\nmachinery (SGN-02) already proves the true carry orbit is strictly\n\\emph{positive} under integrality \u2014 the corridor-escape branch the identity\nnaturally supplies is exactly the branch already eliminated. \\\\\n249-supply & SEP-02,\n\\lean{TotientActualLcmOrbitSeparation.lean}{TotientActualLcmOrbitSeparation.lean:141} & scale\\_only &\nNOT\\_PROMOTABLE. The analytic tail is fully discharged with an explicit,\nuniform error radius; only two exponents (\\(a=4,6\\)) have the resulting\nfinite distance-to-integer question verified unconditionally\n(\\lean{actualLcmTailOrbit\\_four\\_and\\_six\\_notMem\\_int}{TotientActualLcmShortKill.lean:35}); nothing beyond\n\\(a=6\\) is proved. \\\\\n249-supply & b7,\n\\lean{MersenneShadowDenominatorGrowth.lean}{MersenneShadowDenominatorGrowth.lean:60} & coordinate\\_only &\nNOT\\_PROMOTABLE. The only proved unbounded-growth quantity in the\nM\u00f6bius--Mersenne coordinate, uniform in \\(t\\), but there is no landed\ntransport from a denominator lower bound to \\(\\mathrm{certifiedKill}\\) or to\ntotient-tail non-integrality \u2014 a different coordinate from the obligation. \\\\\n249-supply & d4/d5,\n\\lean{TotientMahlerDefect.lean}{TotientMahlerDefect.lean:900}, \\lean{TotientCarryKernelRigidity.lean}{TotientCarryKernelRigidity.lean:284} &\ncoordinate\\_only & NOT\\_PROMOTABLE. The scale side of a second, independent\nopen obligation (unbounded carry-kernel rank forced by rationality) is\nfinished uniformly in \\(e\\); the opposite (rationality-side rank upper\nbound) is missing, and one natural route to it is proved dead\n(\\lean{false\\_of\\_compressedAdjointCertificate}{TotientMahlerDefect.lean:1051}). \\\\\n257-reset & rc-2 / F4-adjacent,\n\\lean{HalfCylinderMiddleCarryLowerBound.lean}{HalfCylinderMiddleCarryLowerBound.lean:2309} & multiple &\nNOT\\_PROMOTABLE. The excess-side run-length law is only half-assembled: an\nexact iterate identity exists, and an upper bound on the terminal excess\nexists separately, but no theorem in Lean composes the two into an\nexcess-side run-length statement, even though every ingredient is present. \\\\\n257-reset & E1,\n\\lean{HalfCylinderSeamLimit.lean}{HalfCylinderSeamLimit.lean:192} & coordinate\\_only &\nNOT\\_PROMOTABLE. Supplies an UPPER bound on the remainder (the ceiling half\nof the picture); the obligation needs a LOWER bound on the deviation\nmagnitude \u2014 the two do not compose without an additional input. \\\\\n257-reset & rc-8,\nprose \\emph{erdos257\\_reset\\_crossing\\_unification\\_2026\\_07\\_24.md}\\,\\S6 &\nmultiple & NOT\\_PROMOTABLE. Two gaps at once: the sign law is\n\\ev{Cert}/empirical only (flagged \\ev{Open} as an unconditional lemma, not\nproved), and even if proved it delivers only the SIGN of the deviation, not\nits magnitude, which is what the obligation needs. \\\\\n257-cofinal-rows & TH-sharp-capacity-progress,\n\\lean{BooleanMobiusSkipRow.lean}{BooleanMobiusSkipRow.lean:286} & hypothesis\\_strength &\nNOT\\_PROMOTABLE. Unconditional and strictly more general than every actual\nconsumer, but it needs a per-\\(c\\) input\n(\\(\\mathrm{localBinarySuffix}\\,D\\,1\\,(2c{-}2)<2^{c-2}\\)) that the corpus\nonly ever supplies through a route which then rigidly collapses onto one\nspecific support family \u2014 the deficit hypothesis is a self-imposed\nrestriction of the consumers, and the general input itself is unsupplied. \\\\\n257-cofinal-rows & precriticalSuffix\\_lt\\_of\\_future\\_skip\\_after\\_takenBlock,\n\\lean{BooleanMobiusCriticalCapacityCofinal.lean}{BooleanMobiusCriticalCapacityCofinal.lean:603} &\nhypothesis\\_strength & NOT\\_PROMOTABLE. Unconditional and uniform in both\nparameters \\((c,t)\\); the sole missing input is an orbit-level skip-gap\nbound on the rational half-greedy orbit, never proved and never previously\nisolated as a target anywhere in the banks. \\\\\n257-cofinal-rows & greedyHalfFrozenMargin\\_nonneg,\n\\lean{HalfCylinderFiniteShadow.lean}{HalfCylinderFiniteShadow.lean:1180} & scale\\_only &\nNOT\\_PROMOTABLE. The margin-nonnegativity horizon is produced by a\n\\emph{non-effective} limit argument (an existential horizon from a\nconvergence fact); the exact-row route needs the SPECIFIC effective horizon\n\\(J=k-2\\), and effectivising it is explicitly recorded as unfinished\nbookkeeping, not new mathematics \u2014 but it is still unfinished. \\\\\n257-cofinal-rows & C4a,\n\\lean{TerminalOnlyCofinal.lean}{TerminalOnlyCofinal.lean:34} & multiple &\nNOT\\_PROMOTABLE. Trades off against the obligation in the opposite\ndirection on \\emph{support locality}: tolerates any support inside a depth\nwindow but demands a carry inside a tight \\(\\sim2\\sqrt M\\) strip, where the\nobligation tolerates an exponentially looser carry but demands the support\nbe confined to a lower window \u2014 the gap is support locality, not carry\nsize, and neither socket implies the other. \\\\\n257-universal & NM-02,\n\\lean{GreedyAchievementSet.lean}{GreedyAchievementSet.lean:2482} & scale\\_only &\n\\textbf{PROMOTABLE (no new mathematics), but does not close anything.}\nThe proof was opened and found already uniform in the target in every\ninternal step; only the statement's own quantifier is artificially pinned\nto \\(t=1/2\\). Promoting it converts the whole half-greedy refutation machine\nfrom a one-target equivalence to one covering every rational target \u2014 the\nhighest-value mechanical promotion in the bank \u2014 but it supplies no new\narithmetic content: it is a restatement, not a producer. \\\\\n257-universal & NM-03,\n\\lean{HalfCarryReachability.lean}{HalfCarryReachability.lean:589} & scale\\_only &\n\\textbf{PROMOTABLE (no new mathematics), but does not close anything.} The\nheadline is pinned to target \\(1/2\\) while its engine\n(\\lean{finiteErdosSum\\_den\\_odd}{DyadicPrefixCompression.lean:1358}) is already fully general over\nfinite supports; verbatim re-derivation at any even-denominator target is\navailable with zero new arithmetic. Paired with NM-02: universal \\#257 at\n\\(b=2\\) is false iff some rational \\(t\\) with even reduced denominator has\nan infinitely-skipping greedy Mersenne orbit \u2014 a genuine reformulation, not\na resolution. \\\\\n257-universal & NM-04,\n\\lean{RationalSupportCarrySkeleton.lean}{RationalSupportCarrySkeleton.lean:2200} & coordinate\\_only &\nNOT\\_PROMOTABLE (partial mechanical content only). The dyadic restriction\n\\(v=1\\) is a call-site choice, not a proof constraint \u2014 the general-\\(v\\)\nconclusion is already landed for arbitrary \\((v,h,L,F)\\) at\n\\lean{shiftedBooleanCollision\\_wrap\\_bound\\_of\\_common\\_multiple}{RationalSupportCarrySkeleton.lean:1962} \u2014 but\nthe strict form needed for non-dyadic targets requires one genuinely new\nauxiliary bound (a mean-least-residue estimate) not present in the corpus. \\\\\n257-universal & NM-12,\n\\lean{HalfCutLocator.lean}{HalfCutLocator.lean:623} & scale\\_only &\nNOT\\_PROMOTABLE (unverified, not merely blocked). The killer and the\ndichotomy scaffold are already target-generic; the two endpoint kills that\npin the dichotomy to \\(1/2\\) generalise on their face, but their proof\nbodies were explicitly \\emph{not} read in this audit pass \u2014 recorded as a\nstrong promotability hypothesis, not a verified promotion, since a\n\\texttt{norm\\_num}-style numeric step could hide a \\(1/2\\)-specific bound. \\\\\n\\end{longtable}\n\n\\subsection*{Classification of the sixteen genuine blockers}\n\nReading all eighteen proof bodies rather than trusting the \\texttt{yields}\nclause exposes a small, recurring taxonomy of blocker shapes, tagged by\n\\texttt{gap\\_kind} in the table above:\n\n\\begin{itemize}[nosep]\n\\item \\textbf{hypothesis\\_strength} (6 rows: e1-companion, SGN-01, TE-04,\nTE-05-weakest, TH-sharp-capacity-progress,\nprecriticalSuffix\\_lt\\_of\\_future\\_skip\\_after\\_takenBlock). The consumer\ntheorem is already at the obligation's exact quantifier shape; a single\nnamed arithmetic fact \u2014 a Weyl-sum cancellation bound, a one-sided residue\ngap, an orbit-level skip-gap bound \u2014 is missing and is not a re-derivation\nof anything on disk.\n\\item \\textbf{scale\\_only} (5 rows: SEP-02, greedyHalfFrozenMargin\\_nonneg,\nNM-02, NM-03, NM-12). The mathematics is either already general or reduces\nto a finite verified range; what is missing is either extending a finite\ncensus (SEP-02) or effectivising a non-constructive existence bound\n(greedyHalfFrozenMargin\\_nonneg). NM-02 and NM-03 are the two genuine\nexceptions where scale widening costs nothing \u2014 and, being restatements, buy\nnothing toward closure either.\n\\item \\textbf{coordinate\\_only} (4 rows: b7, d4/d5, E1, NM-04). A quantity\nis proved unconditionally in one coordinate (M\u00f6bius--Mersenne denominator\ngrowth, dyadic-kernel rank, remainder upper bound, dyadic support-fraction\nmass) with no landed transport into the coordinate the obligation is stated\nin.\n\\item \\textbf{multiple} (3 rows: rc-2/F4-adjacent, rc-8, C4a). More than\none of the above simultaneously \u2014 typically an unassembled composition of\ntwo otherwise-landed halves, or two orthogonal deficiencies (empirical-only\nplus sign-not-magnitude) stacked on the same row.\n\\end{itemize}\n\nNo row in the audited eighteen falls under \\texttt{quantifier\\_order} \u2014 every\nrow with a genuine quantifier-order mismatch (e.g. \\texttt{a12}'s\n\\(\\exists N\\,\\exists L\\,\\forall h\\) versus the obligation's\n\\(\\forall h\\,\\forall N_0\\,\\exists N\\,\\exists L\\)) was judged NOT promotable\nat first read and excluded from the eighteen entirely, rather than surviving\nto a full proof-body audit. The eighteen audited here are precisely the rows\nwhose \\emph{stated} hypotheses looked closest to the obligation; that even\nthis most-favourable subset yields sixteen genuine blockers and only two\ncostless-but-empty restatements is the paper's evidence that the corpus's\nbounds are load-bearing in the strict sense: no amount of routine\nstrengthening of what is already proved, at the level the audit checked,\nsupplies the missing arithmetic content that \\(\\#249\\) or \\(\\#257\\)\nactually needs.\n\n% ---- part a249_invent ----\n\\section{The mathematics this problem still needs}\n\nEverything in Parts I--IV of this document is either a theorem about\n$\\varphi$ or a theorem about proof methods for $\\varphi$. Nothing in it\ndecides Erd\\H{o}s \\#249, and this section does not either. What this section\ndoes is different in kind from the rest: it takes each route that survives\nthe barrier classification, states the missing mathematics as an exact\nsentence, identifies the shape of argument that could supply it, names the\nprecise point at which the nearest existing technique fails, and --- where\nthe evidence permits --- attempts the construction rather than describing it.\n\nFour things are new here and are marked as such. (i) An exact normal form\nthat turns every surviving \\#249 route into a statement about the binary\nexpansion of one explicit real number, with the thresholds computed\n(Lemma~\\ref{lem:orbit}, Proposition~\\ref{prop:transfer},\nCorollary~\\ref{cor:digitform}). (ii) A proof that the first-harmonic gap is\nstrictly stronger than irrationality, by an explicit witness inside the same\ncoefficient class (Theorem~\\ref{thm:lacunary}); the corpus had this only as\na plausibility argument. (iii) Two of the four analytic budgets of the pivot\nroute reduced to classical theorems, with the crossover scale computed\n(Propositions~\\ref{prop:dickman} and \\ref{prop:badcof}), together with the\nexact modulus window in which the pivot is neither $2$-adically trivial nor\nbeyond prime-distribution technology (Observation~\\ref{obs:tension}).\n(iv) A refutation of the fifth surviving route as stated\n(Theorem~\\ref{thm:rankdead}), which retires it.\n\nWhere a claim is proved it is marked \\ev{Math}; where it is checked by exact\nor floating-point computation, \\ev{Cert}; where it is quoted, \\ev{Cited};\nwhere it is a proposal, \\ev{Open}. No proposal below is offered as progress\ntowards a solution. Two of them are reductions, and a reduction is not a\nresult.\n\n\\begin{rem}\nDeclarations in the shared tree are cited as\n\\texttt{Erdos249257.name} with file and line. Declarations in the newer\nper-problem tree are cited with their path from the repository root,\n\\texttt{ErdosProblems/Erdos249/File.lean:line}; the hyperlink target for\nthose is the per-problem directory, not \\texttt{Erdos249257/}.\n\\end{rem}\n\n\\subsection{One identity, and what it does to every surviving route}\n\nWrite $S=\\sum_{n\\ge 1}\\varphi(n)/2^n$, $R_N=\\sum_{m\\ge 1}\\varphi(N+m)/2^m$\nfor the local tail\n(\\lean{Erdos249257.\\allowbreak TotientTailPeriodKiller.\\allowbreak totientTail}{TotientTailPeriodKiller.\\allowbreak lean:57}),\nand $\\Phi_N=\\sum_{n\\le N}\\varphi(n)2^{N-n}\\in\\Z$ for the integer prefix, so\nthat $2^N S=\\Phi_N+R_N$. For $h\\ge 1$ set\n\\[\n  \\alpha_h \\;:=\\; (2^h-1)\\,S .\n\\]\n\n\\begin{lem}[Doubling normal form]\\label{lem:orbit}\nFor all $N\\ge 0$ and $h\\ge 1$,\n\\[\n  R_{N+1}=2R_N-\\varphi(N+1),\n  \\qquad\n  R_{N+h}-R_N \\;=\\; 2^N\\alpha_h-\\bigl(\\Phi_{N+h}-\\Phi_N\\bigr),\n\\]\nso $R_{N+h}-R_N \\equiv 2^N\\alpha_h \\pmod 1$, and consequently the exact\nfirst character of the tail difference is the $\\times 2$ orbit of one real\nnumber:\n\\[\n  \\mathrm{tailOrbitFirstExp}(h,N)\n  \\;=\\; e\\bigl(2^N\\alpha_h\\bigr),\n  \\qquad e(x):=\\exp(2\\pi i x).\n\\]\nHence, for fixed $h$, the phases $\\{\\,R_{N+h}-R_N \\bmod 1\\,\\}_{N\\ge 0}$ are\nthe forward orbit of $\\alpha_h \\bmod 1$ under $x\\mapsto 2x$.\n\\end{lem}\n\n\\noindent\nThe second identity and the character form are landed:\n\\lean{Erdos249257.\\allowbreak TotientTailPeriodKiller.\\allowbreak tail_diff_eq_scaled_\\allowbreak totient_series_sub_prefix}{PivotAntiReconstruction.\\allowbreak lean:54}\nand\n\\lean{Erdos249257.\\allowbreak TotientTailPeriodKiller.\\allowbreak tailOrbitFirstExp_eq_scaledTotientSeriesFirstExp}{PivotAntiReconstruction.\\allowbreak lean:66}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{other:binary-window}.\nThe recurrence $R_{N+1}=2R_N-\\varphi(N+1)$ is one line from the definition\nand is used below only for intuition \\ev{Math}.\n\nThis is not a new fact --- it is two landed Lean theorems read together ---\nbut reading them together has a consequence the corpus never draws. The\nfirst-harmonic block sum, which is the analytic frontier of \\#249, is a\n\\emph{lacunary exponential sum in a single real variable}. The following\nproposition makes the transfer exact, including the truncation budget.\n\n\\begin{prop}[Orbit transfer]\\label{prop:transfer}\nDefine\n\\[\n  \\mathrm{OrbitBlockGap} :\\equiv\n  \\forall h\\ge 1\\ \\forall X_0\\ \\exists X\\ge \\max(X_0,1):\\quad\n  \\sum_{N=X}^{2X-1}\\cos\\bigl(2\\pi\\,2^{N}\\alpha_h\\bigr)\\;\\le\\;\\tfrac{89}{100}\\,X .\n\\]\nThen $\\mathrm{OrbitBlockGap}$ implies that $S$ is irrational.\n\\end{prop}\n\n\\begin{proof}\nFix $h\\ge 1$ and $X_0$, and take $X$ as supplied. Choose $L$ with\n$2^{L}\\ge 1024\\,(2X+h+L+2)$; such $L$ exists because $2^L/L\\to\\infty$, and it\nsatisfies the room inequality $16(2X+h+L+2)\\le 2^L$ a fortiori. For every\n$N<2X$ the landed truncation bound\n\\lean{Erdos249257.\\allowbreak TotientTailPeriodKiller.\\allowbreak norm_windowFirstExp_\\allowbreak sub_tailOrbitFirstExp_lt}{PivotAntiReconstruction.\\allowbreak lean:202}\ngives\n$\\bigl\\|\\,\\mathrm{windowFirstExp}(h,N,L)-e(2^N\\alpha_h)\\,\\bigr\\|\n < 2\\pi(N+L+h+2)/2^{L} \\le 2\\pi/1024 < 1/100$,\nso by\n\\lean{Erdos249257.\\allowbreak TotientTailPeriodKiller.\\allowbreak windowFirstCos_le_\\allowbreak add_of_tailOrbitGap}{PivotAntiReconstruction.\\allowbreak lean:221},\n$\\mathrm{windowFirstCos}(h,N,L)\\le \\cos(2\\pi 2^N\\alpha_h)+1/100$. Summing over\nthe $X$ values $N\\in[X,2X)$ gives\n$\\sum_N \\mathrm{windowFirstCos}(h,N,L)\\le \\tfrac{89}{100}X+\\tfrac{1}{100}X\n=\\tfrac{9}{10}X$, so\n\\lean{Erdos249257.\\allowbreak TotientTailPeriodKiller.\\allowbreak exists_certifiedKill_of_\\allowbreak first_harmonic_gap}{FirstHarmonicGap.\\allowbreak lean:128}\nproduces $N\\in[X,2X)$ with $\\mathrm{certifiedKill}(h,N,L)$, and $N\\ge X\\ge X_0$.\nThis is exactly the certificate supply consumed by\n\\lean{Erdos249257.\\allowbreak TotientTailPeriodKiller.\\allowbreak irrational_totient_series_\\allowbreak of_certificate_supply}{TotientTailPeriodKiller.\\allowbreak lean:394}.\n\\end{proof}\n\n\\noindent\n\\ev{Math} \\quad \\scale{cofinal} \\quad \\coord{other:binary-window}. The proof\nuses only landed lemmas; formalising it is a variant of\n\\lean{Erdos249257.\\allowbreak irrational_totient_series_of_\\allowbreak first_harmonic_norm_gap}{FirstHarmonicPivot.\\allowbreak lean:83}\nwith the constant $21/25$ replaced by the real-part constant $89/100$ and the\nroom factor $16$ by $1024$. Nothing here is an improvement of the analytic\nrequirement; it is a change of coordinate that makes the requirement legible.\n\n\\begin{cor}[Digit form of the analytic requirement]\\label{cor:digitform}\nLet $\\rho_h(X)$ denote the proportion of $N\\in[X,2X)$ with\n$\\|2^N\\alpha_h\\|_{\\R/\\Z}\\ge 1/4$. If for every $h\\ge 1$ there are cofinally\nmany $X$ with $\\rho_h(X)\\ge 11/100$, then $S$ is irrational.\nEquivalently, when $\\alpha_h$ is not a dyadic rational: if for every $h$\nthere are cofinally many dyadic blocks $[X,2X)$ in which the binary\nexpansion of $\\alpha_h$ contains at least $0.11X$ digit changes --- that is,\nin which the mean binary run length is at most $9.1$ --- then $S$ is\nirrational.\n\\end{cor}\n\n\\begin{proof}\n$\\|x\\|\\ge 1/4$ forces $\\cos 2\\pi x\\le 0$, and the remaining terms are at most\n$1$, so $\\sum_{N}\\cos(2\\pi 2^N\\alpha_h)\\le (1-\\rho_h(X))X$; require\n$1-\\rho\\le 89/100$ and apply Proposition~\\ref{prop:transfer}. For the digit\nreading, $\\|2^N\\alpha_h\\|\\ge 1/4$ holds exactly when the binary digits of\n$\\alpha_h$ in positions $N+1,N+2$ differ.\n\\end{proof}\n\n\\noindent\n\\ev{Math}. This threshold is worth staring at. A single digit change in\nevery ninth position, in one block per scale, for each fixed $h$, would\nsettle Erd\\H{o}s \\#249. What is actually true, numerically, is far stronger.\n\n\\begin{table}[h]\\centering\n\\begin{tabular}{r r r r r}\n\\hline\n$h$ & $X$ & $\\rho_h(X)$ & $X^{-1}\\sum_{N\\in[X,2X)}\\cos(2\\pi 2^N\\alpha_h)$ & required \\\\\n\\hline\n$1$ & $256$  & $0.4766$ & $\\phantom{-}0.0037$ & $\\le 0.89$ \\\\\n$1$ & $2048$ & $0.5073$ & $-0.0043$ & $\\le 0.89$ \\\\\n$3$ & $256$  & $0.4648$ & $\\phantom{-}0.0296$ & $\\le 0.89$ \\\\\n$3$ & $2048$ & $0.5020$ & $\\phantom{-}0.0028$ & $\\le 0.89$ \\\\\n$8$ & $2048$ & $0.5112$ & $-0.0088$ & $\\le 0.89$ \\\\\n$13$& $2048$ & $0.5049$ & $-0.0195$ & $\\le 0.89$ \\\\\n\\hline\n\\end{tabular}\n\\caption{Block statistics for $\\alpha_h=(2^h-1)S$, computed from the exact\ninteger $\\lfloor 2^{6000}S\\rfloor$ (error $<1$ in the last bit, tail bound\n$(M+2)2^{-M}$ at $M=6080$). Over positions $1..5800$ the binary expansion of\n$\\alpha_1$ has $2863$ runs, mean run length $2.0255$, longest run $13$; for\n$\\alpha_3$, $2964$ runs, mean $1.9565$, longest $12$; for $\\alpha_8$, $2926$\nruns, mean $1.9819$, longest $14$. A uniform digit model predicts mean run\nlength $2$ and longest run $\\approx\\log_2 5800\\approx 12.5$. \\ev{Cert}}\n\\label{tab:blocks}\n\\end{table}\n\n\\begin{obs}\\label{obs:allroutes}\nUnder Lemma~\\ref{lem:orbit} every route that survives the barrier\nclassification becomes a statement about the binary expansion of $S$ or of\nsome $\\alpha_h$. Routes 1 and 2 ask for a positive proportion of digit\nchanges in a block; Route 3 asks for one short run at a prime-indexed\nposition; Route 4 asks that two blocks of digits of $S$ at distance $h$\ndisagree early. The routes differ in \\emph{which positions} they interrogate\nand \\emph{how uniform} the margin must be, not in what they are about. This\nis developed in \\S\\ref{sub:shape}.\n\\end{obs}\n\n\\subsection{Route 1: a constant-saving cancellation over a dyadic block}\n\n\\paragraph{The exact statement needed.}\n\\[\n  \\forall h\\ge 1\\ \\forall X_0\\ \\exists X,L:\\quad\n  \\max(X_0,1)\\le X,\\quad 16(2X+h+L+2)\\le 2^{L},\\quad\n  \\Bigl\\|\\sum_{N=X}^{2X-1} e\\bigl(D(h,N,L)/2^{L}\\bigr)\\Bigr\\|\\le \\tfrac{21}{25}X,\n\\]\nwhere $D(h,N,L)=\\sum_{j<L}\\bigl(\\varphi(N+h+1+j)-\\varphi(N+1+j)\\bigr)2^{L-1-j}$.\n\\lean{Erdos249257.\\allowbreak TotientTailPeriodKiller.\\allowbreak DTWFirstHarmonicNormGap}{FirstHarmonicPivot.\\allowbreak lean:75},\nconsumer\n\\lean{Erdos249257.\\allowbreak irrational_totient_series_of_\\allowbreak first_harmonic_norm_gap}{FirstHarmonicPivot.\\allowbreak lean:83}\n\\quad \\ev{Open} \\quad \\scale{cofinal}. By Proposition~\\ref{prop:transfer} the\nweaker real-part form with constant $89/100$ suffices.\n\n\\paragraph{What $D$ is, as an arithmetic object.}\nDividing by $2^L$ and reindexing the two ranges onto a common variable\n$t$, with $m=N+t$,\n\\begin{equation}\\label{eq:window}\n  \\frac{D(h,N,L)}{2^{L}}\n  =\\underbrace{-\\sum_{t=1}^{h}\\frac{\\varphi(N+t)}{2^{t}}}_{\\text{head}}\n  \\;+\\;\\underbrace{(2^{h}-1)\\sum_{t=h+1}^{L}\\frac{\\varphi(N+t)}{2^{t}}}_{\\text{body}}\n  \\;+\\;\\underbrace{\\sum_{t=L+1}^{L+h}\\frac{\\varphi(N+t)}{2^{t-h}}}_{\\text{tail}} .\n\\end{equation}\nSo the phase is a \\emph{geometrically weighted linear form in $\\varphi$ at\n$L+h$ consecutive arguments}, with the value at offset $t$ entering modulo\n$2^{t}$ and with an odd multiplier $2^h-1$ throughout the body. Two features\ncontrol everything that follows. First, the weight at offset $t$ has\ndenominator exactly $2^{t}$, so offset $t$ contributes nothing unless\n$v_2(\\varphi(N+t))<t$: the $2$-adic valuation of the totient is the gate.\nSecond, $2^h-1$ is odd, so the multiplier never closes a gate that\n$\\varphi$ leaves open.\n\n\\paragraph{Which technique family is even the right shape.}\nFour candidates, in decreasing order of relevance.\n\n\\emph{Weyl differencing and van der Corput} are the wrong shape and can be\ndismissed exactly. Both require the phase to be a smooth or polynomial\nfunction of the summation variable so that a difference operator lowers its\ndegree. Here $N\\mapsto D(h,N,L)/2^{L}$ is, by Lemma~\\ref{lem:orbit}, the\n$\\times 2$ orbit map: differencing in $N$ multiplies the phase by $2$ and\nsubtracts an integer, so the difference operator is an exact isometry of the\nproblem and lowers nothing. This is not a heuristic --- it is\n$R_{N+1}=2R_N-\\varphi(N+1)$ read modulo $1$.\n\n\\emph{The large sieve} needs a family of well-separated frequencies and a\nsum over both the frequencies and the variable. Here there is one frequency\nper $N$ and no family: the sum is over a single orbit. A large-sieve\ninequality could be applied after introducing an artificial family (for\nexample, over the $h$ parameter), but the predicate quantifies $h$\nuniversally, so an average over $h$ is not admissible.\n\n\\emph{Vaughan/Vinogradov bilinear decomposition} is the right shape, and is\nwhat Route 2 already implements: decompose the argument $N+t$ at a large\nprime factor, use $\\varphi(mp)=\\varphi(m)(p-1)$ to linearise the phase in\n$p$, and sum over primes. \\S\\ref{sub:pivot} carries this as far as it goes.\n\n\\emph{Erd\\H{o}s's 1948 digit method}, which proved irrationality of\n$E=\\sum_n 1/(2^n-1)=\\sum_N d(N)/2^N$, is the right shape for the\n\\coord{mobius-mersenne} coordinate. That coordinate exists here and is\nexact: from $\\varphi=\\mu * \\mathrm{id}$,\n\\begin{equation}\\label{eq:mobmers}\n  S=\\sum_{d\\ge 1}\\mu(d)\\,\\frac{2^{d}}{(2^{d}-1)^{2}}\n  \\qquad (\\ev{Math};\\ \\text{verified to }16\\ \\text{digits},\\ S=1.3676308019850223\\ldots,\\ \\ev{Cert}).\n\\end{equation}\nBut the method does not transfer, for a reason that can be stated\nquantitatively rather than vaguely. Erd\\H{o}s's argument for $E$ exploits\nthat the coefficient $d(N)$ has enormous multiplicative fluctuation: over\n$N\\le Y$ the ratio of its maximum to its typical value is\n$Y^{(\\log 2+o(1))/\\log\\log Y}$, so certain highly composite $N$ deposit\nidentifiable spikes into the digit stream. The coefficient here is\n$\\varphi(N)$, which satisfies $\\varphi(N)/N\\in[c/\\log\\log N,1]$: its\nmultiplicative fluctuation over $N\\le Y$ is a factor $e^{\\gamma}\\log\\log Y$,\nthat is, $\\log\\log Y$ rather than $Y^{c/\\log\\log Y}$. There are no spikes to\nfind. In the M\\\"obius--Mersenne coordinate \\eqref{eq:mobmers} the situation is\nworse rather than better: the coefficients $\\mu(d)$ change sign, so the\ndigit blocks contributed by successive $d$ cancel rather than accumulate,\nand there is no positivity to run a block argument on. This is, as far as\nthe evidence in this corpus goes, the specific reason \\#249 is harder than\nits Erd\\H{o}s--Borwein ancestor.\n\n\\paragraph{The specific obstacle: what the estimate cannot be deduced from.}\nThe corpus records that no theorem proves the first-harmonic gap\ninequivalent to irrationality, and argues the point from the structure of a\ncollapse proof. It can be settled outright, and the witness lives in the\nsame coefficient class that carries barriers B1 and B7.\n\n\\begin{thm}[The gap is strictly stronger than irrationality]\\label{thm:lacunary}\nLet $c(n)=1$ if $n=k!$ for some $k\\ge 1$ and $c(n)=0$ otherwise, so\n$0\\le c(n)\\le n$ for all $n\\ge 1$, and let $\\beta=\\sum_{n\\ge1}c(n)/2^{n}\n=\\sum_{k\\ge 1}2^{-k!}$. Then $\\beta$ is irrational, and for every $h\\ge 1$\nand every $X\\ge 81(h+5)$,\n\\[\n  \\sum_{N=X}^{2X-1}\\cos\\bigl(2\\pi\\,2^{N}(2^{h}-1)\\beta\\bigr) \\;>\\; \\tfrac{9}{10}X .\n\\]\nConsequently the block-gap requirement fails at \\emph{every} scale for\n$\\beta$, while $\\beta$ is irrational. No proof of the block gap for $S$ can\ntherefore proceed from the irrationality of $S$ together with the growth\nbound $c(n)\\le n$; it must use arithmetic of $\\varphi$.\n\\end{thm}\n\n\\begin{proof}\nIrrationality of $\\beta$ is Liouville's criterion. Write\n$\\gamma=(2^{h}-1)\\beta=\\sum_{k}(2^{h-k!}-2^{-k!})$, so for every $k$ with\n$k!>2h$ the binary expansion of $\\gamma$ carries ones exactly in the\npositions $k!-h+1,\\dots,k!$ and zeros between consecutive such blocks. If\n$N\\le k!-h-5$ for the least $k$ with $k!>N$, then\n$\\{2^{N}\\gamma\\}\\le 2\\cdot 2^{-5}=1/16$, whence $\\|2^{N}\\gamma\\|\\le 1/16$ and\n$\\cos(2\\pi 2^{N}\\gamma)\\ge\\cos(\\pi/8)>0.9239$. The remaining $N$ lie in\n$(k!-h-5,\\,k!]$, at most $h+5$ values per factorial, and consecutive\nfactorials differ by a factor exceeding $2$ from $k\\ge 3$, so the window\n$[X,2X)$ meets at most one such interval. Therefore the sum exceeds\n$0.9239\\,(X-h-5)-(h+5)$, which exceeds $\\tfrac{9}{10}X$ once\n$X\\ge 81(h+5)$.\n\\end{proof}\n\n\\noindent\n\\ev{Math} \\quad \\scale{cofinal} \\quad \\coord{other:binary-window}. This\nupgrades the corpus's ``no collapse mechanism is known to apply'' to ``no\ncollapse mechanism can exist''. It also says precisely what the missing\ninput must do: it must rule out that $S$ behaves, in its doubling orbit,\nlike a Liouville number. That is the content of Corollary~\\ref{cor:digitform},\nand it is why the requirement is a digit-statistics statement and not an\nirrationality statement.\n\n\\paragraph{Size of the quantity, honestly calibrated.}\nUnder any model in which the binary digits of $\\alpha_h$ behave like fair\ncoin flips, $\\rho_h(X)\\to 1/2$ and the block sum is $O(\\sqrt{X\\log\\log X})$\nby the Erd\\H{o}s--G\\'al law of the iterated logarithm for lacunary series\n\\ev{Cited}. The requirement is $\\rho_h\\ge 0.11$ and a saving of a constant\nfactor. The measured values in Table~\\ref{tab:blocks} are $\\rho_h\\approx 0.50$ and a block\naverage of order $X^{-1/2}$. So the analytic requirement is weaker than the\napparent truth by a factor $\\sqrt X$, and the difficulty is entirely that no\ntechnique produces \\emph{any} nontrivial digit statistic for an explicit\nconstant of this kind. That is a statement about technology, and it is worth\nseparating from a statement about $S$.\n\n\\subsection{Route 2: the four-term pivot budget}\\label{sub:pivot}\n\n\\paragraph{The exact statement needed.}\nFor every $h\\ge 1$ there must exist $s\\ge 1$ and $\\eta\\in(0,1)$ such that for\nevery $X_0$ there are $X\\ge\\max(X_0,1)$ and $L$ with $h\\le L-s$,\n$16(2X+h+L+2)\\le 2^{L}$, and all four of\n\\[\n\\begin{aligned}\n  \\operatorname{Re}\\bigl(\\mathrm{pivotCenteredCorrelation}\\bigr) &\\le \\tfrac{14}{25}X,\n  &\\qquad \\bigl\\|\\mathrm{pivotFiberMeanContribution}\\bigr\\| &\\le \\tfrac{1}{100}X,\\\\\n  \\bigl\\|\\mathrm{pivotBadContribution}\\bigr\\| &\\le \\tfrac{1}{100}X,\n  &\\qquad \\bigl\\|\\mathrm{pivotNonSupplierContribution}\\bigr\\| &\\le \\tfrac{8}{25}X .\n\\end{aligned}\n\\]\n\\lean{Erdos249257.\\allowbreak TotientTailPeriodKiller.\\allowbreak DTWPivotResidualDecorrelation}{FirstHarmonicPivot.\\allowbreak lean:397},\nconsumer\n\\lean{Erdos249257.\\allowbreak irrational_totient_\\allowbreak series_of_pivotResidualDecorrelation}{FirstHarmonicPivot.\\allowbreak lean:405}\n\\quad \\ev{Open} \\quad \\scale{cofinal}. The decomposition is an exact identity,\n\\lean{Erdos249257.\\allowbreak TotientTailPeriodKiller.\\allowbreak windowFirstExp_sum_\\allowbreak eq_pivot_decomposition}{FirstHarmonicPivot.\\allowbreak lean:342}\n\\quad \\ev{Lean}, and the supplier set at the canonical fibre is a membership\nequality with a shifted dyadic interval of primes,\n\\lean{Erdos249257.\\allowbreak TotientTailPeriodKiller.\\allowbreak mem_pivotFiber_\\allowbreak one_overlap_iff}{FirstHarmonicPivot.\\allowbreak lean:251}\n\\quad \\ev{Lean}.\n\n\\paragraph{Two of the four budgets are classical. Here they are.}\nThe last two terms have never been attacked because they look like\nunquantified bookkeeping. They are not: they are exactly calibrated, and both\nare within reach of standard analytic number theory.\n\n\\begin{prop}[The non-supplier budget]\\label{prop:dickman}\nLet $P(n)$ denote the largest prime factor of $n$. A base $N$ fails\n$\\mathrm{pivotSupplier}$ only if the pivot argument $n=N+L-s+1$ satisfies\n$P(n)<4\\sqrt X$. Consequently\n\\[\n  \\#\\{N\\in[X,2X): \\neg\\,\\mathrm{pivotSupplier}(X,L,s,N)\\}\n  \\;=\\;\\bigl(\\rho(2)+O(1/\\log X)\\bigr)X\n  \\;=\\;\\bigl(1-\\log 2+o(1)\\bigr)X\n  \\;=\\;(0.30685\\ldots+o(1))X ,\n\\]\nso the required bound $\\tfrac{8}{25}X=0.32X$ holds for all sufficiently large\n$X$, where $\\rho$ is the Dickman function.\n\\end{prop}\n\n\\begin{proof}[Proof sketch]\n$\\mathrm{pivotSupplier}$ demands $2\\lfloor\\sqrt X\\rfloor<p$ and\n$m=n/p\\le\\lfloor\\sqrt X\\rfloor/2$. The second forces $p\\ge 2n/\\sqrt X$, which\nfor $n<2X+O(\\log X)$ is the binding condition and lies in\n$[2\\sqrt X,4\\sqrt X)$; so a non-supplier has $P(n)<4\\sqrt X$. The count is\nthen $\\Psi(2X,y)-\\Psi(X,y)$ in shape with $u=\\log n/\\log y\\to 2$, and\nHildebrand's uniform smooth-number estimate \\ev{Cited} applies far below\n$y=\\sqrt x$. Finally $\\rho(2)=1-\\log 2$.\n\\end{proof}\n\n\\noindent\n\\ev{Math} (modulo the cited uniform estimate) \\quad \\scale{cofinal}. Two\nremarks. First, the budget $8/25$ evidently was chosen to sit just above\n$1-\\log 2$: the slack is $0.0132$, four per cent of the budget. Second, the\napproach to the limit is at rate $\\asymp 1/\\log X$, so the inequality is true\nbut not observable. Exact computation gives non-supplier density $0.41795$ at\n$X=2\\cdot 10^{4}$ and $0.40381$ at $X=8\\cdot 10^{4}$ \\ev{Cert}; extrapolating\nthe $1/\\log X$ correction, the crossover below $0.32$ lies beyond\n$X=10^{30}$, plausibly near $10^{40}$ (order-of-magnitude extrapolation, not\na computation). Any numerical sanity check of this budget will therefore\nappear to refute it.\n\n\\begin{prop}[The bad-cofactor budget]\\label{prop:badcof}\nFor $\\eta\\in(0,1)$ let $B(\\eta)=\\{m:\\varphi(m)<\\eta m\\}$, with natural density\n$D(\\eta)$; by Schoenberg's theorem $D$ exists, is continuous, and\n$D(0+)=0$ \\ev{Cited}. Then\n\\[\n  \\#\\{N\\in[X,2X): N\\ \\text{a supplier with cofactor}\\ m\\in B(\\eta)\\}\n  \\;\\le\\;\\bigl(D(\\eta)+o(1)\\bigr)X ,\n\\]\nso a single choice of $\\eta$ with $D(\\eta)<1/200$ meets the $\\tfrac{1}{100}X$\nbudget for all large $X$. Since $\\eta$ is quantified existentially before\n$X_0$, and a smaller $\\eta$ only enlarges the good set, this choice costs\nnothing elsewhere in the budget.\n\\end{prop}\n\n\\begin{proof}[Proof sketch]\nSuppliers with cofactor $m$ number $\\pi(2X/m)-\\pi(X/m)$, which is at most\n$(2X/m)/\\log X$ up to $1+o(1)$ because $m\\le\\sqrt X/2$ forces\n$\\log(X/m)\\ge\\tfrac12\\log X$. Sum over $m\\in B(\\eta)$ with $m\\le\\sqrt X/2$\nand use that the logarithmic density of $B(\\eta)$ tends to $D(\\eta)$.\n\\end{proof}\n\n\\noindent\n\\ev{Math} \\quad \\scale{cofinal}. The choice is not tight: the least element\nof $B(1/5)$ is $2\\cdot3\\cdot5\\cdot7\\cdot11\\cdot13=30030$, with\n$\\varphi(m)/m=5760/30030=0.19181$, and the logarithmic density of $B(1/5)$ up\nto $3\\cdot10^{5}$ is $3\\cdot 10^{-5}$ \\ev{Cert}. So $\\eta=1/5$ is already\ndefensible pending an explicit numerical bound on $D(1/5)$, and that bound is\nitself a finite computation a reader could start today.\n\n\\paragraph{Where the pivot may live: an exact window.}\nThe pivot sits at offset $t=L-s+1$ in \\eqref{eq:window}, so its modulus is\n$2^{t}$ and its coefficient is $(2^{h}-1)\\varphi(m)$ with $2^{h}-1$ odd. Two\nconstraints pull against each other, and the corpus states neither.\n\n\\begin{obs}[The pivot modulus window]\\label{obs:tension}\n\\emph{(a) Shallow pivots are $2$-adically trivial.} If $v_2(\\varphi(m))\\ge t$\nthen $(2^{h}-1)\\varphi(m)(p-1)/2^{t}\\in\\Z$, the pivot phase is identically\n$1$, the fibre mean is $1$, and the budget\n$\\|\\mathrm{pivotFiberMeanContribution}\\|\\le X/100$ fails outright. Now\n$v_2(\\varphi(m))=\\sum_{p\\mid m,\\ p\\ \\mathrm{odd}}v_2(p-1)+\\max(v_2(m)-1,0)$\nhas mean $\\sim 2\\log\\log m$ --- measured means $3.11,\\,3.71,\\,4.18,\\,4.37$\nfor $m\\le 10^{3},10^{4},10^{5},3\\cdot10^{5}$ \\ev{Cert} --- and is\nasymptotically normal with variance of order $\\log\\log m$, so a $1/100$\nfibre-mean budget forces roughly\n$t\\gtrsim 2\\log\\log X+C\\sqrt{\\log\\log X}$.\n\n\\emph{(b) Deep pivots are beyond prime distribution.} The fibre mean at\ncofactor $m$ is $|F_m|^{-1}\\sum_{p}e\\bigl(a_m(p-1)/2^{t}\\bigr)$ with\n$a_m=(2^{h}-1)\\varphi(m)$ and $p$ running over primes in an interval of\nlength $\\asymp X/m$. Writing $q'=2^{t}/\\gcd(a_m,2^{t})$, the main term is\n$\\mu(q')/\\varphi(q')$ times the count, and $\\mu(2^{j})=0$ for $j\\ge2$: the\nmain term \\emph{vanishes identically}. What remains is the error term for\nprimes in progressions to modulus $2^{t}$, which Siegel--Walfisz supplies\nonly for $2^{t}\\le(\\log X)^{A}$, that is $t\\le 1.443\\,A\\log\\log X$. At the\nroom-forced depth $L\\approx\\log_2X+5$, a pivot at $t\\approx L$ would demand\nequidistribution of primes to a modulus exceeding the length of the range ---\npast Bombieri--Vinogradov ($x^{1/2}$) and past Elliott--Halberstam\n($x^{1-\\varepsilon}$) even conjecturally.\n\n\\emph{(c) The window is non-empty.} Combining,\n\\[\n  2\\log\\log X+C\\sqrt{\\log\\log X}\\;\\lesssim\\;t\\;\\lesssim\\;1.443\\,A\\log\\log X ,\n\\]\nnon-empty for any $A>1.39$; Siegel--Walfisz holds for every fixed $A$, at the\ncost of an ineffective constant. The pivot route is therefore not dead on\nmodulus grounds, and the standing intuition that ``the modulus $2^{L}$ is\nhopeless'' applies only to the deep pivot, not to the free choice of $s$ that\nthe predicate actually offers.\n\\end{obs}\n\n\\noindent\n\\ev{Math} for (a) and for the vanishing main term in (b); \\ev{Cited} for\nSiegel--Walfisz and the two barrier scales; \\ev{Cert} for the measured\nvaluations. This is the item in this section a specialist is most likely to\nbe able to use: it says the analytic socket is well posed and confines it to\na definite range of one free parameter.\n\n\\paragraph{The specific obstacle: the weight is neither Type I nor Type II.}\nWith Propositions~\\ref{prop:dickman}, \\ref{prop:badcof} and\nObservation~\\ref{obs:tension}, three of the four budgets have a route. The\nfirst does not, and the failure can be located exactly. Factoring the pivot\nargument as $n=mp$ and dividing out the pivot phase leaves\n\\[\n  \\operatorname{Re}\\sum_{m}\\ \\sum_{p\\,:\\,mp\\in[X,2X)+t}\n    w(mp)\\,\\bigl(e\\bigl(a_m(p-1)/2^{t}\\bigr)-\\overline{e}_m\\bigr)\n  \\;\\le\\;\\tfrac{14}{25}X ,\n  \\qquad |w|\\equiv 1 ,\n\\]\nwhere $\\overline{e}_m$ is the fibre mean and, by \\eqref{eq:window}, $w(n)$ is\nthe character of the same weighted totient sum with the single offset $t$\ndeleted. Vinogradov's bilinear method requires that, after the decomposition\n$n=mp$, the summand split as $\\alpha_m\\beta_p$ (Type II) or be independent of\n$p$ given $m$ (Type I), up to boundedly many pieces. The weight $w(mp)$ is\nneither: it is a function of the totients of the $L+h$ integers\n\\emph{adjacent} to $mp$, and $\\varphi(mp+j)$ for $j\\ne0$ bears no relation to\nthe factorisation $mp$ --- it is not a function of $m$, not a function of\n$p$, and not a product of the two. No device is known that converts a weight\nof this shape into bilinear form.\n\nAt $h=1$ and the canonical pivot the required estimate becomes fully\nexplicit. There $e\\bigl((p-1)/4\\bigr)=i^{\\,p-1}=\\chi_{-4}(p)$, so what is\nneeded is\n\\[\n  \\operatorname{Re}\\sum_{X<p\\le 2X}\\chi_{-4}(p)\\,\n    e\\Bigl(\\frac{\\varphi(p+1)}{8}+\\frac{\\varphi(p+2)}{16}+\\cdots\\Bigr)\n  \\;\\le\\;\\Bigl(\\frac{9}{10}-\\delta\\Bigr)\\,\\bigl(\\pi(2X)-\\pi(X)\\bigr).\n\\]\nThe missing input is thus a \\emph{non-correlation between a fixed quadratic\ncharacter at $p$ and the $2$-adic behaviour of $\\varphi$ at the shifts\n$p+1,p+2,\\dots$}: a correlation statement for two arithmetic functions at\nshifted arguments, of Chowla--Elliott type. The unconditional results in that\nfamily --- Matom\\\"aki--Radziwi\\l{}\\l{} in almost all short intervals, Tao's\nlogarithmically averaged Chowla, Tao--Ter\\\"av\\\"ainen for odd order\n\\ev{Cited} --- are averaged, never pointwise at a single scale.\n\n\\paragraph{A swing: the predicate needs only cofinally many $X$, so average\nover $X$.} This is the one place where the shape of the obligation is a gift.\nBoth $\\mathrm{DTWFirstHarmonicNormGap}$ and\n$\\mathrm{DTWPivotResidualDecorrelation}$ ask for \\emph{some} $X$ beyond each\n$X_0$, never for all $X$. Hence it suffices to prove a logarithmically\naveraged bound: if\n\\[\n  \\sum_{k\\le K}\\ \\frac{1}{2^{k}}\n  \\Bigl|\\sum_{N\\in[2^{k},2^{k+1})}e\\bigl(D(h,N,L_k)/2^{L_k}\\bigr)\\Bigr|\n  \\;=\\;o(K)\n  \\quad\\text{for each }h,\n\\]\nwith $L_k$ any admissible depth sequence, then infinitely many blocks satisfy\nthe gap and Proposition~\\ref{prop:transfer} applies. Logarithmic averaging is\nexactly the regime in which the entropy-decrement method operates. The first\nconcrete step is not to prove this but to decide whether the method reaches\nit: the phase $e\\bigl(D(h,N,L)/2^{L}\\bigr)$ is not multiplicative, but it is\na bounded \\emph{local function of a multiplicative function} --- a fixed\ncontinuous function of $\\bigl(\\varphi(N+1)/2,\\dots,\\varphi(N+L)/2^{L}\\bigr)$\nread modulo $1$ --- whereas entropy decrement is stated for correlations of\nbounded multiplicative functions along fixed shifts. The precise question to\nsettle, and it can be started on immediately, is whether the decrement\nsurvives when the observable is a local function of $\\varphi$ with a growing\nnumber of coordinates $L\\approx\\log_2X$ rather than a product of boundedly\nmany multiplicative values. \\ev{Open}\n\n\\subsection{Route 3: uniform quantitative escape at primes}\n\n\\paragraph{The exact statement needed.}\n\\[\n  \\forall h\\ge 1\\ \\forall N_0\\ \\exists p\\ \\text{prime}:\\quad\n  \\max(N_0+h+1,\\,h+5)\\le p\n  \\ \\wedge\\\n  \\operatorname{Re}\\bigl(\\mathrm{tailOrbitFirstExp}(h,\\,p-h-1)\\bigr)<\\tfrac{9}{10}.\n\\]\n\\lean{ErdosProblems.\\allowbreak Erdos249.\\allowbreak DTWNaturalPrimeTailOrbitStrictGap}{ErdosProblems/\\allowbreak Erdos249/\\allowbreak TotientStrictPrimeEscape.\\allowbreak lean:18},\nconsumer\n\\lean{ErdosProblems.\\allowbreak Erdos249.\\allowbreak irrational_totient_\\allowbreak series_of_naturalPrimeTailOrbitStrictGap}{ErdosProblems/\\allowbreak Erdos249/\\allowbreak TotientStrictPrimeEscape.\\allowbreak lean:66}\n\\quad \\ev{Open} \\quad \\scale{cofinal}.\n\n\\paragraph{What it says, after Lemma~\\ref{lem:orbit}.}\n$\\operatorname{Re}e(2^{p-h-1}\\alpha_h)<9/10$ says\n$\\|2^{p-h-1}\\alpha_h\\|_{\\R/\\Z}>\\arccos(9/10)/2\\pi=0.07226\\ldots$, so the\nrequirement is: \\emph{for each $h$, the binary expansion of $\\alpha_h$ begins\na run of at most three equal digits at position $p-h$, for cofinally many\nprimes $p$}. It is a run-length bound at a prime-indexed subsequence of a\n$\\times2$ orbit.\n\n\\paragraph{The specific obstacle.} Two, stacked. First, this is a\n\\emph{pointwise} demand: it names an index, and the promotion audit records\nthat no pointwise producer in this programme has ever been supplied at even\none large index, across seven independent attempts, while the single audited\nrow whose missing input is a block average is the first-harmonic row.\nSecond, and sharper: digit behaviour \\emph{along the primes} is strictly\nharder than digit behaviour on average even for constants where the average\ncase is settled. The only constants whose digits are understood at all along a sparse\nsubsequence are ones built for the purpose (Champernowne;\nCopeland--Erd\\H{o}s) \\ev{Cited}. No technique reads the digits of an\nexplicit constant at prime positions.\n\n\\paragraph{Numerical calibration.} Among the $501$ primes in $[1000,5000)$,\nthe proportion with $\\operatorname{Re}e(2^{p-h-1}\\alpha_h)<9/10$ is $0.8623$\nfor $h=1$, $0.8503$ for $h=2$, $0.8583$ for $h=5$, with minima below\n$-0.9999$ \\ev{Cert}. The cofinal supply is thus met by about six primes in\nseven at these scales. As with Route 1, the requirement is very weak and the\nobstacle is that nothing sees it.\n\n\\paragraph{A swing: replace the pointwise demand by a canonical fibre.}\nThe corpus warns against the subset consumer\n\\lean{Erdos249257.\\allowbreak TotientTailPeriodKiller.\\allowbreak exists_certifiedKill_of_\\allowbreak first_harmonic_gap_subset}{FirstHarmonicGap.\\allowbreak lean:104}\nbecause a predicate quantifying existentially over subsets\n$T\\subseteq[X,2X)$ is collapse-exposed: the proved collapse\n\\lean{Erdos249257.\\allowbreak dtwWindowSeparatedPairs_iff_\\allowbreak irrational_totient_series}{PivotAntiReconstruction.\\allowbreak lean:1775}\nworks precisely by \\emph{choosing} $T=\\{N,N+1\\}$. That warning applies to a\nfree $T$. It does not apply to a $T$ named in advance as a function of\n$(h,X,L)$, and the corpus already contains the right one:\n\\[\n  T_{h,X,L}\\;:=\\;\\mathrm{pivotFiber}(X,L,L-h,1)\n  \\;=\\;\\{\\,N\\in[X,2X)\\ :\\ N+h+1\\ \\text{prime}\\,\\},\n\\]\na membership equality rather than a sampled surrogate\n(\\lean{Erdos249257.\\allowbreak TotientTailPeriodKiller.\\allowbreak mem_pivotFiber_\\allowbreak one_overlap_iff}{FirstHarmonicPivot.\\allowbreak lean:251},\n\\ev{Lean}). The proposal is to prove\n\\begin{equation}\\label{eq:primefibre}\n  \\forall h\\ge1\\ \\forall X_0\\ \\exists X\\ge X_0,\\ L:\\quad\n  16(2X+h+L+2)\\le 2^{L}\n  \\ \\wedge\\\n  \\sum_{N\\in T_{h,X,L}}\\mathrm{windowFirstCos}(h,N,L)\\ \\le\\ \\tfrac{9}{10}\\,\\bigl|T_{h,X,L}\\bigr| ,\n\\end{equation}\nwhich by the landed subset consumer yields a certificate at some\n$N\\ge X\\ge X_0$, hence irrationality. \\ev{Open}\n\nThree features make this the sharpest available target. (i) On this fibre the\ncofactor is $m=1$, so the pivot coefficient is $(2^{h}-1)\\varphi(1)=2^{h}-1$,\nwhich is \\emph{odd}: the $2$-adic triviality of\nObservation~\\ref{obs:tension}(a) cannot occur. (ii) The pivot modulus is\n$2^{h+1}$, \\emph{fixed} once $h$ is fixed, so the fibre-mean term is governed\nby the prime number theorem in progressions to a fixed power of two, where\nthe Ramanujan main term $c_{2^{h+1}}(2^{h}-1)=\\mu(2^{h+1})=0$ vanishes for\nevery $h\\ge1$ and the error term is effective. Observation~\\ref{obs:tension}(b)\ndoes not bite. (iii) The bookkeeping budgets of \\S\\ref{sub:pivot} disappear:\ninside $T$ there are no non-suppliers and no bad cofactors. What remains is\nexactly one estimate --- decorrelation of the fixed phase\n$e\\bigl((2^{h}-1)(p-1)/2^{h+1}\\bigr)$ from the residual totient weight --- and\nit is the same estimate isolated at the end of \\S\\ref{sub:pivot}. This does\nnot make that estimate easier. It removes everything that is not that\nestimate.\n\n\\begin{rem}\nHonesty about \\eqref{eq:primefibre}: it is not proved to be strictly stronger\nthan irrationality, and no collapse is proved either. What can be said is\nthat the one collapse mechanism that exists in this lane consumes the freedom\nto choose the sample, and \\eqref{eq:primefibre} has no such freedom. Deciding\nthis either way --- exhibiting a collapse, or a witness in the coefficient\nclass of Theorem~\\ref{thm:lacunary} that separates it from irrationality ---\nis itself a well-defined finite piece of work.\n\\end{rem}\n\n\\subsection{Route 4: depth-locked full-depth escape}\n\n\\paragraph{The exact statement needed.}\n\\[\n  \\mathrm{ApFullDepthEscape} :\\equiv\n  \\forall d\\ge 1\\ \\forall N\\ \\exists t\\ge 1:\\ \\mathrm{certifiedKill}(td,\\,N,\\,td),\n\\]\nunpacked, $(N+2td+2:\\Z)<D(td,N,td)\\bmod 2^{td}<2^{td}-(N+2td+2)$.\n\\lean{ErdosProblems.\\allowbreak Erdos249.\\allowbreak PeriodMultipleEscape.\\allowbreak ApFullDepthEscape}{ErdosProblems/\\allowbreak Erdos249/\\allowbreak PeriodMultipleEscape.\\allowbreak lean:162},\nconsumer\n\\lean{ErdosProblems.\\allowbreak Erdos249.\\allowbreak PeriodMultipleEscape.\\allowbreak irrational_totient_\\allowbreak series_of_apFullDepthEscape}{ErdosProblems/\\allowbreak Erdos249/\\allowbreak PeriodMultipleEscape.\\allowbreak lean:171},\nambient equivalence\n\\lean{ErdosProblems.\\allowbreak Erdos249.\\allowbreak PeriodMultipleEscape.\\allowbreak periodMultipleKillSupply_iff_irrational}{ErdosProblems/\\allowbreak Erdos249/\\allowbreak PeriodMultipleEscape.\\allowbreak lean:153}\n\\quad \\ev{Open} \\quad \\scale{cofinal}. It is the shortest fully stated open\ninequality the programme has produced.\n\n\\paragraph{What it says: an anti-self-similarity statement about the digits\nof $S$ alone.}\nWriting $h=td$ and using\n$D(h,N,L)/2^{L}=(R_{N+h}-R_N)-(R_{N+L+h}-R_{N+L})/2^{L}$ at $L=h$, together\nwith the landed strip $|R_{M+h}-R_M|<M+h+2$\n(\\lean{Erdos249257.\\allowbreak TotientTailPeriodKiller.\\allowbreak abs_tail_\\allowbreak diff_lt}{CarrySurvivorExtinction.\\allowbreak lean:79},\n\\ev{Lean}), applied at $M=N+h$, one gets:\n\n\\begin{prop}\\label{prop:route4}\n$\\mathrm{certifiedKill}(h,N,h)$ holds whenever\n$\\bigl\\|2^{N+h}S-2^{N}S\\bigr\\|_{\\R/\\Z}>2(N+2h+2)/2^{h}$.\n\\end{prop}\n\n\\begin{proof}\n$\\|D(h,N,h)/2^{h}\\|\\ge\\|R_{N+h}-R_N\\|-|R_{N+2h}-R_{N+h}|/2^{h}\n >2(N+2h+2)/2^{h}-(N+2h+2)/2^{h}=(N+2h+2)/2^{h}$, and\n$R_{N+h}-R_N\\equiv 2^{N}(2^{h}-1)S=2^{N+h}S-2^{N}S\\pmod 1$ by\nLemma~\\ref{lem:orbit}. A residue at distance more than $(N+2h+2)/2^{h}$ from\n$\\Z$ is exactly a certified kill at depth $h$.\n\\end{proof}\n\n\\noindent\n\\ev{Math}. So Route 4 asks: \\emph{for every ray $d$ and every basepoint $N$,\nsome multiple $h=td$ is such that the tail of the binary expansion of $S$\nbeginning at position $N+1$ and the tail beginning at position $N+h+1$, read\nas reals in $[0,1)$, are more than $2(N+2h+2)/2^{h}$ apart in $\\R/\\Z$.} Two\nthings follow. First, unlike Routes 1--3, this involves only $S$ itself, not\nthe multiples $\\alpha_h$. Second, the required precision grows only like\n$\\log_2(N+2h)$ while the available depth grows linearly in $t$: the demand\nweakens rapidly along each ray.\n\n\\paragraph{Which technique family fits, and why it is unavailable.}\nA statement that a sequence does not almost repeat at some multiple of every\ngap, to logarithmic precision, is a \\emph{subword-complexity} statement, and\nthe mature technology for such statements about a specific real number is the\nAdamczewski--Bugeaud method: the Schmidt subspace theorem applied to the\n$b$-ary expansion, which shows that an algebraic irrational cannot have too\nmany long repetitions \\ev{Cited}. The shape fits exactly. The hypothesis does\nnot: the method conditions on algebraicity, and $S$ is not known to be\nalgebraic --- indeed it is not known to be irrational, which is the whole\nproblem. There is no version of the method that runs from an analytic\ndescription of the constant. That is the entire obstruction, and it is worth\nstating plainly because it explains why Route 4 looks so much easier than it\nis: the required combinatorial property is weak, but the only machine that\nproduces such properties needs an input we do not have.\n\n\\paragraph{What the data says.}\nAt $N=300$, the least $t$ with $\\mathrm{certifiedKill}(td,300,td)$ exists\nand is small for every ray $d\\le 24$: $t=10$ for $d=1$, $t=5$ for $d=2$,\n$t=4$ for $d=3$, $t=3$ for $d=4$, $t=2$ for $5\\le d\\le 9$, and $t=1$ for\n$10\\le d\\le 24$ --- in every case at or near the first depth admitted by the\nroom floor. The residues are not marginal: $r/2^{h}=0.3594$ at $d=1$,\n$0.5977$ at $d=3$, $0.6853$ at $d=7$, $0.2620$ at $d=17$, $0.5887$ at\n$d=23$, against edge radii $3.1\\cdot10^{-1}$, $8.0\\cdot10^{-2}$,\n$2.0\\cdot10^{-2}$, $2.6\\cdot10^{-3}$, $4.1\\cdot10^{-5}$ \\ev{Cert}. This is a\nfloor, not a trend, exactly as barrier B1 requires; it decides nothing.\n\n\\paragraph{A heuristic size computation.}\nUnder a uniform model for $\\|2^{N}(2^{h}-1)S\\|$ the failure probability at\ndepth $h=td$ is $\\approx 4(N+2h+2)/2^{h}$, which is $\\approx 1$ at the first\nadmissible $t$ and falls by a factor $2^{d}$ at each subsequent $t$. The\nprobability that a given $(d,N)$ fails at every $t$ is therefore a product\n$\\prod_{j\\ge0}\\min(1,\\,c\\,2^{-jd})$, super-exponentially small, and the\nexpected number of failing pairs $(d,N)$ over all $N$ converges. At $d=1$,\n$N=300$ the product evaluates to about $10^{-5}$, and the observed value is a\nsuccess at the very first admissible depth. The model therefore predicts\nRoute 4 holds with enormous margin --- which is precisely why no finite\ncomputation will ever be evidence for it. \\ev{Open} (heuristic).\n\n\\subsection{Route 5: the rationality-side rank bound, and why it should be\nretired}\n\n\\paragraph{The exact statement that was wanted.}\n\\[\n  \\exists C\\ \\forall c:\\N\\to\\N\\ \\bigl(c(n)\\le n,\\ \\neg\\,\\mathrm{Irrational}(X_c)\\bigr)\n  \\ \\forall v>0\\ \\forall u\\ \\mathrm{IsTemperedBinaryOrbit}(c,v,u)\\ \\forall e:\\quad\n  \\operatorname{rk}_e(u)\\le C ,\n\\]\nwhere $X_c=\\sum_{n\\ge1}c(n)/2^{n}$,\n$\\operatorname{rk}_e(u)=\\dim_{\\Q}\\operatorname{span}_{\\Q}\\{n\\mapsto u(2^{j}n+r)\n: 1\\le j\\le e,\\ r<2^{j}\\}$, or the same with $C$ replaced by any $g(e)=o(2^{e})$.\nIt would contradict the landed floor\n\\lean{Erdos249257.\\allowbreak not_irrational_totientSeries_\\allowbreak implies_unbounded_carryRank_unconditional}{TotientCarryKernelRigidity.\\allowbreak lean:284}\n\\quad \\ev{Lean}, which gives $2^{e}-1\\le\\operatorname{rk}_e(u)$ for every $e$.\n\n\\paragraph{First, the right vocabulary.}\n``The $\\Q$-span of all dyadic sections is finite-dimensional'' is, verbatim,\nthe definition of a \\emph{$2$-regular sequence} in the sense of Allouche and\nShallit \\ev{Cited}. In that vocabulary the corpus's unconditional theorem\n\\lean{Erdos249257.\\allowbreak not_finiteDimensional_\\allowbreak span_fullTotientKernel}{TotientMahlerDefect.\\allowbreak lean:1014}\nsays: \\emph{$\\varphi$ is not $2$-regular}; and the obligation above says:\n\\emph{a rational binary coefficient series with linear growth forces its\ncarry orbit to be $2$-regular}. Naming this connects the lane to a mature\nliterature and, as it turns out, decides it.\n\n\\begin{thm}[The obligation is false in its bounded form]\\label{thm:rankdead}\nLet $M=\\{4p:p\\ \\text{prime}\\}$ and define $b:\\N_{\\ge1}\\to\\N$ by $b(n)=1$ for\n$n\\notin M\\cup(M+1)$, $b(m)=0$ for $m\\in M$, $b(m+1)=3$ for $m\\in M$. Put\n$c(n)=n-b(n)$. Then\n\\begin{enumerate}[label=(\\roman*),leftmargin=2.2em]\n\\item $c(n)\\in\\N$ and $c(n)\\le n$ for every $n\\ge1$;\n\\item $X_c=1$, so $X_c$ is rational and a tempered integral orbit $u$ exists\n      with $v=1$;\n\\item $c$ is not $2$-regular, hence $\\operatorname{rk}_e(c)\\to\\infty$ and, by\n      the carry bridge\n      \\lean{Erdos249257.\\allowbreak totient_carryKernel_diff}{TotientCarryKernelRigidity.\\allowbreak lean:94}\n      in its coefficient-generic form --- which is the recurrence\n      $v\\,c(N+1)=2u(N)-u(N+1)$ read on each dyadic section, and uses nothing\n      about $\\varphi$ --- $\\operatorname{rk}_e(u)\\to\\infty$.\n\\end{enumerate}\nHence no constant $C$ works, and the bounded form of the obligation is\nrefuted.\n\\end{thm}\n\n\\begin{proof}\n(i) $b$ takes values in $\\{0,1,3\\}$, and $b(n)=3$ only at $n=m+1\\ge 9$, so\n$c(n)=n-b(n)\\ge0$ always; $c(1)=0$, $c(2)=1$.\n(ii) $\\sum_{n\\ge1}n2^{-n}=2$, and $\\sum_{n\\ge1}b(n)2^{-n}=1$ because the base\nsequence $b\\equiv1$ sums to $1$ and each edit contributes\n$-1\\cdot2^{-m}+2\\cdot2^{-(m+1)}=0$; the edits are disjoint since $M\\subseteq4\\N$\ncontains no two integers at distance $1$. Hence $X_c=2-1=1$.\n(iii) The sequence $n\\mapsto n$ is $2$-regular, and $2$-regular sequences form\na ring closed under $\\Q$-linear combination, so if $c$ were $2$-regular then\n$b=\\mathrm{id}-c$ would be. But $b$ is bounded, and a bounded $2$-regular\nsequence is $2$-automatic (Allouche--Shallit) \\ev{Cited}; a $2$-automatic $b$\nmakes $M=\\{n:b(n)=0\\}$ a $2$-automatic set, hence\n$\\{p\\ \\text{prime}\\}=\\{k:4k\\in M\\}$ $2$-automatic, contradicting the theorem\nof Minsky and Papert that the primes are not recognisable by a finite\nautomaton in any base \\ev{Cited}. Finally, $v\\,c(N+1)=2u(N)-u(N+1)$ gives\neach section of $c$ at level $j$ and residue $r\\ge1$ as a $\\Q$-combination of\ntwo sections of $u$ at the same level, so\n$\\operatorname{rk}_e(u)\\ge\\operatorname{rk}_e(c)-O(1)$.\n\\end{proof}\n\n\\noindent\n\\ev{Math} \\quad \\scale{uniform} \\quad \\coord{other:carry-kernel}.\n\n\\begin{prop}[The growth form is false too]\\label{prop:rankgrowth}\nThe set $M$ above may be chosen so that $\\operatorname{rk}_e(u)\\ge 2^{e-1}-3$\nfor every $e$, which exceeds every $g(e)=o(2^{e})$ for large $e$.\n\\end{prop}\n\n\\begin{proof}[Construction]\nTake $M=4G$ and note that for $j\\ge2$ and $r\\equiv0\\pmod4$ the level-$j$,\nresidue-$r$ section of $b$ is $n\\mapsto 1-\\mathbf 1_G(2^{j-2}n+r/4)$, since\n$M+1$ misses $4\\N$. It\ntherefore suffices to choose $G$ so that the family\n$f_{i,s}(n)=\\mathbf 1_G(2^{i}n+s)$, over $i\\ge0$ and $0\\le s<2^{i}$, is\n$\\Q$-linearly independent; there are $2^{e-1}-1$ such pairs with\n$i\\le e-2$, while the sections of $c$ differ from those of $b$ by the\nsections of $\\mathrm{id}$ and by the constant function, together at most\nthree dimensions. Build\n$G=\\{g_1<g_2<\\cdots\\}$ greedily against an enumeration $(i_t,s_t)$ of the\npairs. At step $t$ choose $g_t\\equiv s_t\\ (\\mathrm{mod}\\ 2^{i_t})$, far above\neverything chosen so far, and set $n_t=(g_t-s_t)/2^{i_t}$, so that\n$f_{i_t,s_t}(n_t)=1$. The values $2^{i}n_t+s$ for other pairs $(i,s)$ are the\nonly ones that could make $f_{i,s}(n_t)=1$; those below $g_t$ are finitely\nmany and are avoided by a finite condition on $g_t$, and those above $g_t$ are\nadded to a forbidden set. The forbidden set contributed by step $t$ is\n$\\bigcup_i[2^{i}n_t,2^{i}(n_t+1))$, whose relative density inside any\narithmetic progression of $2$-power modulus is $O(1/n_t)$; choosing\n$n_t\\ge4^{t}$ makes the total forbidden density less than $1/2$, so a legal\n$g_{t'}$ always exists. The resulting evaluation matrix\n$\\bigl(f_{i_{t'},s_{t'}}(n_t)\\bigr)$ is the identity, so the family is\nindependent.\n\\end{proof}\n\n\\noindent\n\\ev{Math} (elementary greedy construction; the only quantitative input is the\ndensity estimate stated).\n\n\\paragraph{And the floor carries no information from rationality.}\nThere is a second, independent reason to retire this route. The bridge\n\\lean{Erdos249257.\\allowbreak totient_carryKernel_diff}{TotientCarryKernelRigidity.\\allowbreak lean:94}\nplaces every positive-residue $\\varphi$-section inside the span of two\n$u$-sections at the same level, and\n\\lean{Erdos249257.\\allowbreak not_finiteDimensional_\\allowbreak span_fullTotientKernel}{TotientMahlerDefect.\\allowbreak lean:1014}\nsays that span is infinite-dimensional unconditionally. So the floor\n$2^{e}-1\\le\\operatorname{rk}_e(u)$ is nothing but the non-$2$-regularity of\n$\\varphi$ transported through an exact identity: it uses rationality only to\nproduce an integral orbit $u$ at all, and it would hold for any $u$\nwhatsoever. The two sides of the hoped-for contradiction are therefore not\nindependent measurements of the object. One of them is a fact about $\\varphi$\nthat holds regardless, and the other --- the upper bound --- must consequently\ncarry the entire problem. The apparent coordinate-disjointness that made this\nroute attractive is an artefact of stating the floor and the ceiling in the\nsame vocabulary. Combined with Theorem~\\ref{thm:rankdead} and\nProposition~\\ref{prop:rankgrowth}, the recommendation is: retire the route as\nstated. What is \\emph{not} retired is the vocabulary: ``is the carry orbit of\na rational linear-growth coefficient series $2$-regular?'' is a good question\nwith a clean negative answer, and knowing the answer is worth more than the\nroute was.\n\n\\subsection{What the shape of the wall suggests about the object}\\label{sub:shape}\n\nEverything in this subsection is inference from the assembled evidence, not\ntheorem. It is written because the assembly makes one inference available\nthat no individual reformulation does, and because refusing to draw it would\nbe a different kind of dishonesty than drawing it carelessly.\n\n\\paragraph{First, a fact rather than an inference: the routes are one route\nin five coordinates.}\nLemma~\\ref{lem:orbit} collapses the coordinate spread that made the corpus\nlook like many independent attacks.\n\n\\begin{center}\n\\begin{tabular}{p{0.16\\linewidth} p{0.44\\linewidth} p{0.32\\linewidth}}\n\\hline\nRoute & What it asks of the binary expansion & Positions interrogated \\\\\n\\hline\n1, 2 & at least $11\\%$ of positions carry a digit change; mean run length\n$\\le 9.1$ in one block per scale & a full dyadic block of $\\alpha_h$ \\\\\n3 & one run of length $\\le 3$ & positions $p-h$, $p$ prime, in $\\alpha_h$ \\\\\n4 & the tails at gap $h$ differ by more than $2(N+2h+2)2^{-h}$ & every\nbasepoint, some multiple of every ray, in $S$ \\\\\n5 & (retired: Theorem~\\ref{thm:rankdead}) the carry orbit is $2$-regular &\n--- \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\n\\noindent\nThis is the honest content of the operator's ``hundred measurements of one\nwall''. The measurements are of one thing, and the thing is the binary digit\nsequence of $S$ and of its multiples $(2^{h}-1)S$. The routes differ in which\npositions they interrogate and how uniform a margin they demand; they do not\ndiffer in subject.\n\n\\paragraph{Second: the wall is a wall about explicit constants, not about\n$\\varphi$.}\nOf the seven barriers, six --- B1, B2, B3, B5, B6, B7 --- are statements that\na class of proof cannot see the digits: bounded inspection, retargeting,\nstrengthening, bounded state, finite linear compression, coarse invariants.\nNone asserts that the digits misbehave. Only B4 is internal to the object,\nand even B4 says that one engineering scheme self-cancels, not that the\nquantity being engineered is unattainable. Meanwhile the two object-level\nwitnesses the corpus can produce --- the $\\gamma$-splice of B1, the parity\ncoboundary of B7 --- and the one produced here\n(Theorem~\\ref{thm:lacunary}) are all lacunary or eventually periodic:\nconstructed, measure-zero, and highly structured. In two centuries nobody has\nexhibited a naturally occurring constant with Liouville-type digit behaviour;\nevery known example is built to have it. So the shape of the barrier set\npoints at our technology, not at $S$.\n\n\\paragraph{Third: is the object behind the measurements simple or complex?}\nThe evidence available leans one way, and I will say how far it leans.\n\n\\emph{For ``generic, and the difficulty is entirely ours''.} (a) Every\nstatistic measurable is indistinguishable from a fair-coin digit model:\n$\\rho_h(X)\\approx0.50$ against a required $0.11$; mean run length\n$1.96$--$2.03$ against a permitted $9.1$; longest run $12$--$14$ over $5800$\npositions against a model prediction of $\\approx12.5$; block sums of order\n$X^{-1/2}$ against a permitted $0.89$ (Table~\\ref{tab:blocks}, \\ev{Cert}). (b) The\nrequirements are weaker than the apparent truth by a factor $\\sqrt X$; a\nproof does not need to understand the digits, only to exclude a pathology.\n(c) All countermodels are constructed lacunary objects. (d) The one\ncoordinate in which $S$ has classical structure --- the M\\\"obius--Mersenne\nform \\eqref{eq:mobmers} --- is an alternating sum of rational functions of\n$2^{d}$ with no functional equation, no modularity, no continued-fraction\nstructure, and no algebraicity: there is no hidden object to find, which is\nconsistent with genericity.\n\n\\emph{Against, or at least complicating.} (a) B4's self-cancellation is\nexact, unconditional, and holds at every depth $K$: the cost of forcing a\ndepth-$K$ residue by a Dirichlet prime is $p\\ge1+2^{K-1}$ while the payoff is\namplitude $2^{K-1}$, so the trade is precisely null. An exact null of that\nform is a structural fact and not an accident of parametrisation, and it is\nthe one place where the object itself pushes back. (b) The exact census\nrecords residues approaching the forbidden edge --- a closest central margin\nof about $2.2\\cdot10^{-4}$ of the modulus at $t=100$ --- which is what a\nuniform model predicts and which rules out, permanently, any argument that\nproceeds by a crude uniform margin. (c) Proposition~\\ref{prop:dickman} shows\nthat even the pieces of the frontier that are provable become true only past\n$X\\approx10^{30}$--$10^{40}$; a genuinely simple object would not usually\nrequire asymptotics that begin so late.\n\n\\emph{Verdict, flagged as inference.} The evidence supports ``the digits of\n$S$ are generic and the wall is technological'' over ``the object is\nsubtle'', but it does not determine it, and the reason it cannot is\nstructural: every test performed is a test a generic object passes, so\npassing them is weak evidence, and the only tests that would discriminate are\nexactly the ones no technique can run. What the evidence \\emph{does} determine\nis narrower and firmer: the difficulty of \\#249 is not located in $\\varphi$'s\nirregularity --- $\\varphi$ is the smoothest interesting multiplicative\nfunction, with multiplicative fluctuation $\\log\\log Y$ where $d(n)$ has\n$Y^{c/\\log\\log Y}$ --- but in the fact that we possess no method whatsoever\nfor lower-bounding digit changes of a constant that was not designed to have\nthem. That is why the Erd\\H{o}s--Borwein constant fell in 1948 and this one\nhas not: $d(n)$ has spikes to deposit into the digit stream and $\\varphi(n)$\ndoes not, and in the M\\\"obius coordinate the signs cancel.\n\n\\paragraph{What would actually change the picture.}\nOne theorem, of any strength, giving a nontrivial digit statistic for $S$ or\nfor some $\\alpha_h$: an upper bound $o(N)$ on the longest binary run in the\nfirst $N$ positions would not by itself decide \\#249, but it would be the\nfirst evidence that the object is legible at all, and by\nCorollary~\\ref{cor:digitform} any bound strong enough to give a positive\nproportion of digit changes in one block per scale would decide it. Failing\nthat, the two concrete openings identified above are: the logarithmically\naveraged form of the block gap, where the entropy-decrement method is at\nleast in the right regime (\\S\\ref{sub:pivot}); and the canonical prime fibre\n\\eqref{eq:primefibre}, where every budget except one decorrelation estimate\nhas been discharged. Neither is close. Both are well posed, and both can be\nstarted on today.\n\n% ---- part a249_p5 ----\n\\section{What is open, stated exactly}\n\nErd\u0151s \\#249 asks whether $S = \\sum_{n\\ge 0} \\varphi(n)/2^n$ is irrational. It is\n\\ev{Open}: nothing in this corpus decides it. Every attack recorded in Parts\nI--IV terminates, in one of exactly two ways, at a small number of\n\\emph{cofinal supply obligations} on the totient tail\n$R_N := \\sum_{j\\ge 0}\\varphi(N+1+j)/2^{j+1}$\n(\\lean{Erdos249257.TotientTailPeriodKiller.totientTail}{TotientTailPeriodKiller.lean:57}):\neither it proves a genuine theorem of the shape ``\\emph{if} this cofinal\npredicate holds, \\emph{then} $S$ is irrational,'' with the predicate itself\nleft completely unproved, or it prunes one specific proof shape and shows it\ncannot supply the predicate. This section renders every such obligation\nexactly as Lean states it, gives its site, records which of them are proved\n\\emph{equivalent} to one another (iff) and which are proved only\n\\emph{sufficient} (one-directional, an easier target implied by a harder\none), and closes by naming, without overclaiming, the piece of new\nmathematics each surviving form would need. Quantifier order is preserved\nexactly as written in Lean; nowhere below does a finite verified list stand\nin for a cofinal supply, and nowhere does an implication get reported as a\nsolved case.\n\nThroughout, $H_t := \\mathrm{periodLcm}(t) = \\operatorname{lcm}(1,\\dots,t)$\n(\\lean{Erdos249257.CarrySurvivorExtinction.periodLcm}{CarrySurvivorExtinction.lean:515}),\n$D(h,N,L) := \\sum_{j<L}\\bigl(\\varphi(N{+}h{+}1{+}j)-\\varphi(N{+}1{+}j)\\bigr)\\cdot 2^{L-1-j}\\in\\mathbb{Z}$\nis \\texttt{windowDiscrepancy}\n(\\lean{Erdos249257.TotientTailPeriodKiller.windowDiscrepancy}{TotientTailPeriodKiller.lean:66}),\nand the base predicate named ``$\\mathrm{Sep}(h,N,L)$'' in the task brief is\nliterally \\texttt{certifiedKill}:\n\\[\n  \\mathrm{certifiedKill}(h,N,L) :\\equiv\n  (N+h+L+2 : \\mathbb{Z}) < D(h,N,L) \\bmod 2^L\n  \\ \\wedge\\\n  D(h,N,L) \\bmod 2^L < 2^L - (N+h+L+2).\n\\]\n\\lean{Erdos249257.TotientTailPeriodKiller.certifiedKill}{TotientTailPeriodKiller.lean:72}\n\\quad \\ev{Lean} (the definition; decidable, instance at line 84)\n\\quad \\scale{n/a} \\quad \\coord{other:binary-window}. In words: the residue\nof the window discrepancy modulo $2^L$ avoids a shrinking radius-$(N+h+L+2)$\nneighbourhood of $0$ inside the growing modulus $2^L$.\n\n\\subsection{The certificate-completeness converter}\n\nEvery reformulation below ultimately measures the same underlying real\nquantity, because certificates are \\emph{complete} receipts of\nnon-integrality, not merely sufficient ones:\n\\[\n  \\bigl(\\exists L,\\ \\mathrm{certifiedKill}(h,N,L)\\bigr)\n  \\iff\n  R_{N+h} - R_N \\notin \\operatorname{range}\\bigl((\\uparrow)\\colon \\mathbb{Z}\\to\\mathbb{R}\\bigr).\n\\]\n\\lean{Erdos249257.exists_certifiedKill_iff_tail_diff_notMem_int}{LcmConeFlatness.lean:316}\n(wrapped as \\texttt{totient\\_tail\\_window\\_kill\\_exists\\_iff\\_tail\\_diff\\allowbreak\\_notMem\\_int} at\n\\lean{Erdos249257.CertificateKernel}{CertificateKernel.lean:18697})\n\\quad \\ev{Lean} \\quad \\scale{uniform} (holds for all $h,N$)\n\\quad \\coord{other:binary-window}.\n\n\\emph{Note.} This iff is the reason every ``certificate supply'' form below\ncan be restated, without loss, in ``pure non-integrality'' language with no\ncertificate vocabulary at all (the B8/TE forms). It does \\emph{not} assert\nthat the cofinal supply exists; it only says the certificate route and the\nreal-analytic route are the same route.\n\n\\subsection{The base cofinal obligation \u2014 the wall}\n\n\\begin{quote}\n\\textbf{Obligation 1 (certificate supply).}\n\\[\n  \\forall h\\ge 1,\\ \\forall N_0,\\ \\exists N\\ge N_0,\\ \\exists L,\\ \\mathrm{certifiedKill}(h,N,L)\n  \\quad\\Longrightarrow\\quad\n  \\mathrm{Irrational}(S).\n\\]\n\\end{quote}\nThis is exactly the quantifier structure named ``$\\mathrm{Sep}(h,N,L)$'' in\nthe task brief: $\\forall h\\ge 1\\ \\forall N_0\\ge 0\\ \\exists N\\ge N_0\\ \\exists L\\ \\mathrm{Sep}(h,N,L)$.\n\\lean{Erdos249257.irrational_totient_series_of_certificate_supply}{TotientTailPeriodKiller.lean:394}\n\\quad \\ev{Lean} (the implication is proved; the hypothesis is unsupplied \u2014 evidence\ntags the theorem, not the missing premise)\n\\quad \\scale{cofinal} \\quad \\coord{other:binary-window}.\n\nThe proof is by contradiction against the tail-period law: if $S$ were\nrational, \\texttt{eventual\\_period\\_of\\_not\\_irrational}\n(\\lean{Erdos249257.eventual_period_of_not_irrational}{TotientTailPeriodKiller.lean:358},\n\\ev{Lean}, unconditional) supplies a period $h>0$ and pre-period $N_0$ with\n$R_{N+h}-R_N\\in\\mathbb{Z}$ for all $N\\ge N_0$; the certificate hypothesis\ninstantiated at that $(h,N_0)$ produces an $N\\ge N_0$ where\n$R_{N+h}-R_N\\notin\\mathbb{Z}$, a contradiction via \\texttt{tail\\_diff\\_notMem\\_int\\_of\\_certifiedKill}\n(\\lean{Erdos249257.tail_diff_notMem_int_of_certifiedKill}{TotientTailPeriodKiller.lean:262}).\n\\emph{What supplying Obligation 1 would take:} an arithmetic (not merely\nexistential) argument that, for every period length $h$ and past every\nthreshold $N_0$, the totient window discrepancy's residue mod $2^L$ can be\npushed off $0$ by a margin growing with the modulus \u2014 i.e., genuine\ncancellation/anti-concentration information about $\\varphi$ on a sliding\nwindow, at every scale, for every $h$. No such argument exists in the\ncorpus; every row below is a restatement of exactly this missing input in a\ndifferent coordinate or with fewer free parameters, never a proof of it.\n\n\\subsection{Collapsing free parameters: the multiple / diagonal / cone chain}\n\nThree further public reformulations replace Obligation 1's two free\nparameters $(h,N_0)$ with successively less information, each proved\n\\emph{sufficient} (not shown equivalent) for $\\mathrm{Irrational}(S)$ by an\nexplicit reduction chain in Lean.\n\n\\paragraph{Multiple-period collapse (weaker hypothesis than Obligation 1).}\n\\[\n  \\forall h_0>0,\\ \\forall N_0,\\ \\exists m>0,\\ \\exists N\\ge N_0,\\ \\exists L,\\\n    \\mathrm{certifiedKill}(m\\cdot h_0, N, L)\n  \\ \\Longrightarrow\\ \\mathrm{Irrational}(S).\n\\]\n\\lean{Erdos249257.irrational_totient_series_of_multiple_certificate_supply}{CarrySurvivorExtinction.lean:502}\n\\quad \\ev{Lean} \\quad \\scale{cofinal} \\quad \\coord{other:lcm-period-multiple}.\nObligation 1's hypothesis trivially implies this one (take $m=1$), so this\ntarget is strictly easier to satisfy; it is still open.\n\n\\paragraph{Diagonal collapse \u2014 one free parameter.}\n\\[\n  \\forall t_0,\\ \\exists t\\ge t_0,\\ \\exists L,\\\n    \\mathrm{certifiedKill}(H_t, H_t, L)\n  \\ \\Longrightarrow\\ \\mathrm{Irrational}(S).\n\\]\n\\lean{Erdos249257.irrational_totient_series_of_lcm_diagonal_certificate_supply}{LcmDiagonalReduction.lean:112}\n\\quad \\ev{Lean} \\quad \\scale{cofinal} \\quad \\coord{other:lcm-diagonal}.\nThis is the canonical single-quantifier restatement: write\n$P(t) :\\equiv \\exists L,\\ \\mathrm{certifiedKill}(H_t,H_t,L)$; the obligation\nis $\\forall t_0\\,\\exists t\\ge t_0,\\ P(t)$. The reduction is a genuine proof,\nnot a relabelling: given $t\\ge\\max(h_0,N_0)$, $h_0\\mid H_t$ and $H_t\\ge\nt\\ge N_0$ simultaneously, so a single diagonal witness at $t$ discharges\n\\emph{both} of Obligation 1's free parameters at once, via the intermediate\nlemma \\lean{Erdos249257.irrational_totient_series_of_lcm_certificate_supply}{LcmDiagonalReduction.lean:92}\nwhich itself reduces to the multiple-period form above. $P(t)$ is verified\nat 28 explicit values of $t$ through $t=64$\n(\\lean{Erdos249257.certifiedKill_diagonal_all_imported}{DiagonalPincerCertificates.lean:2842},\ndepths $[6,5,7,7,9,14,15,14,21,22,23,26,\\dots]$ for\n$t\\in\\{1,2,3,4,5,7,8,9,11,13,16,17,\\dots\\}$; endpoint\n\\lean{Erdos249257.certifiedKill_diagonal_t64}{DiagonalPincerCertificateT64Endpoint.lean:1928})\n\\ev{Cert} \\scale{fixed} \\coord{other:lcm-diagonal} \u2014 a finite floor, not a\ncofinal supply: nothing in these 28 witnesses is a function of $t$, each is\nan independent \\texttt{norm\\_num} block over explicit $\\varphi$ values.\n\n\\paragraph{Cone collapse \u2014 two multipliers, still the same wall.}\n\\[\n  \\forall t_0,\\ \\exists t\\ge t_0,\\ \\exists q,m,L,\\ 0<q\\ \\wedge\\\n    \\mathrm{certifiedKill}(m\\cdot H_t,\\, q\\cdot H_t,\\, L)\n  \\ \\Longrightarrow\\ \\mathrm{Irrational}(S).\n\\]\n\\lean{Erdos249257.irrational_totient_series_of_lcm_cone_window_kill_supply}{CertificateKernel.lean:18686}\n\\quad \\ev{Lean} \\quad \\scale{cofinal} \\quad \\coord{other:lcm-cone}. The\ndiagonal form above is exactly the cell $q=m=1$, so any diagonal witness\ntrivially witnesses the cone form; the cone form is thus the weakest known\ntarget still logically tied to this coordinate. It rests on a genuine\nstrengthening of the tail-period law, \\emph{lcm-cone flatness}: if $S$ is\nrational there is $t_1$ such that for every $t\\ge t_1$ and every $q>0,m$,\n$R_{qH_t+mH_t}-R_{qH_t}\\in\\mathbb{Z}$ \u2014 rationality flattens the\n\\emph{whole} cone $\\{k H_t : k\\ge 1\\}$, not just one difference\n(\\lean{Erdos249257.rational_totient_series_forces_lcm_cone_flatness}{CertificateKernel.lean:18674},\n\\ev{Lean}, unconditional). A sharper cone-form producer,\n\\texttt{coneNonflatCert}, needs only a \\emph{one-sided} radius per vertex \u2014\ninformation-theoretically half of \\texttt{certifiedKill}'s pairwise floor \u2014\nand proves that some pair in a finite multiplier menu $Q$ is non-integral;\nthe corresponding supply,\n$\\exists$ an unbounded-scale menu $Q$ with \\texttt{coneNonflatCert} firing,\nis \\ev{Open}, \\scale{cofinal},\n\\lean{Erdos249257.irrational_totient_series_of_lcm_cone_nonflat_supply}{CertificateKernel.lean:18812}.\n\n\\paragraph{Pure non-integrality form \u2014 certificate vocabulary stripped out.}\nVia the completeness iff above, the diagonal and cone forms restate,\n\\emph{exactly}, with no reference to $\\mathrm{certifiedKill}$ at all:\n\\[\n  \\forall t_0,\\ \\exists t\\ge t_0,\\ R_{2H_t}-R_{H_t}\\notin\\mathbb{Z}\n  \\ \\Longleftrightarrow\\ \\text{diagonal obligation above}\n  \\ \\Longrightarrow\\ \\mathrm{Irrational}(S),\n\\]\n\\lean{Erdos249257.irrational_totient_series_of_lcm_diagonal_nonintegrality_supply}{CertificateKernel.lean:18706}\n(cone analogue at\n\\lean{Erdos249257.CertificateKernel}{CertificateKernel.lean:18718})\n\\quad \\ev{Lean} \\quad \\scale{cofinal} \\quad \\coord{other:real-analytic-nonintegrality}.\nThis is the frontier of \\#249 with every piece of certificate machinery\nremoved: \\emph{does $R_{2H_t}-R_{H_t}\\notin\\mathbb{Z}$ for infinitely many\n$t$?} Nothing decides this either.\n\n\\subsection{The exponential-sum form}\n\nA first-harmonic (Weyl-sum) cancellation statement is proved sufficient for\nObligation 1 by an elementary pigeonhole argument, with no case analysis and\nno scale-degrading constants:\n\\[\n  \\mathrm{DTWFirstHarmonicNormGap} :\\equiv\\\n  \\forall h>0,\\ \\forall X_0,\\ \\exists X,L,\\quad\n  \\max(X_0,1)\\le X\\ \\wedge\\\n  16\\,(2X+h+L+2)\\le 2^L\\ \\wedge\\\n  \\Bigl\\|\\sum_{N\\in[X,2X)} e\\bigl(D(h,N,L)/2^L\\bigr)\\Bigr\\| \\le \\tfrac{21}{25}X.\n\\]\n\\lean{Erdos249257.DTWFirstHarmonicNormGap}{FirstHarmonicPivot.lean:75}\n\\quad \\ev{Open} (the predicate; unproved at every $h,X_0$)\n\\quad \\scale{cofinal} \\quad \\coord{other:first-harmonic}.\n\\[\n  \\mathrm{DTWFirstHarmonicNormGap} \\ \\Longrightarrow\\ \\mathrm{Irrational}(S).\n\\]\n\\lean{Erdos249257.irrational_totient_series_of_first_harmonic_norm_gap}{FirstHarmonicPivot.lean:83}\n\\quad \\ev{Lean} \\quad \\scale{cofinal} \\quad \\coord{other:first-harmonic}. The\nconsumer already has Obligation 1's exact free-parameter shape ($X$ is a\ncompletely free threshold, so applying the gap at $X\\ge N_0$ gives\n$\\exists N\\ge N_0\\ \\exists L$ directly); what is missing is not scale and not\ncoordinate, it is the arithmetic input itself \u2014 not one instance of a\nconstant-saving cancellation bound for the complex exponential sum\n$\\sum_{N\\in[X,2X)} e(D(h,N,L)/2^L)$ is proved anywhere in the corpus, at any\n$h,X,L$. A strictly stronger \\emph{subset} form (a saving on any nonempty\nfinite $T\\subseteq[0,2X)$, no density or partition hypothesis) is also\nproved sufficient and unconditional as an implication:\n\\lean{Erdos249257.exists_certifiedKill_of_first_harmonic_norm_gap}{FirstHarmonicPivot.lean:53}\nconsumes the elementary block lemma\n\\lean{Erdos249257.exists_certifiedKill_of_first_harmonic_gap}{FirstHarmonicGap.lean:128}\n(\\ev{Lean}, unconditional: \\emph{any} constant-saving first-harmonic gap on\none dyadic block forces a finite kill certificate, via $\\cos(\\pi/8)>9/10$\nand averaging). \\emph{What supplying this form would take:} genuine\ncancellation for the first additive character of the totient window\ndiscrepancy \u2014 a Weyl-type bound, uniform in $h$ and growing $L$, on a\ncharacter sum built from $\\varphi$ differences, not merely an existence\nstatement. \\lean{Erdos249257.ResidualGaugeObstruction.det_residualMonomialMatrix}{ResidualGaugeObstruction.lean:47}\nrules out one entire class of candidate proofs: a residual-blind\ndeterminant/rank certificate cannot see this cancellation, because an\nexplicit ``locked'' unit-norm gauge reconstructs the exact bad configuration\nwhile keeping any Vandermonde-shaped minor nonzero \u2014 any real argument here\nmust couple the rows arithmetically, not merely normalise columns.\n\n\\subsection{The private lane: the actual-orbit reformulation and the top-edge staircase}\n\nA second, independently developed lane (private modules\n\\texttt{TotientActual*}/\\texttt{TotientFixedRank*}, never mirrored to the\npublic repository) restates \\#249 in terms of the \\emph{actual}\npower-of-two LCM diagonal, $H = H_{2^a}$, and proves a genuine \\textbf{iff}\nwith the totient series itself \u2014 the strongest form of ``exact statement of\nwhat remains'' anywhere in the corpus.\n\n\\begin{quote}\n\\textbf{The exact equivalence.}\n\\[\n  \\mathrm{Irrational}(S)\n  \\iff\n  \\forall a_0,\\ \\exists a\\ge a_0,\\ \\mathrm{actualLcmTailOrbit}(a)\\notin\\mathbb{Z}.\n\\]\n\\end{quote}\n\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.irrational_totientSeries_iff_actualLcmOrbitNonintegralitySupply}{TotientActualLcmOrbitNonintegrality.lean:37}\n\\quad \\ev{Lean} \\quad \\scale{n/a} (an unconditional equivalence, not a\nsupply) \\quad \\coord{mobius-mersenne}, where\n$\\mathrm{actualLcmTailOrbit}(a) := 2^H(2^H-1)S - (\\mathrm{totientPrefix}(2H)-\\mathrm{totientPrefix}(H))$,\n$H=H_{2^a}$\n(\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.actualLcmTailOrbit_eq_scaled_totientSeries_sub_prefix}{TotientActualLcmOrbitSeparation.lean:68}).\n\\emph{This is \\#249 restated with nothing left over}: no auxiliary\nhypothesis, no scale caveat \u2014 cofinal non-integrality of this one sparse\nsequence at power-of-two heights is exactly Erd\u0151s \\#249. Everything else in\nthis subsection is an attempt to reach the right-hand side.\n\n\\paragraph{Route-pruning: total staircase collapse is impossible.}\nBefore any positive target, the file proves one entire proof shape is\nempty. For $a\\ge 8$ and room bound $J+K+(a{+}6)<2\\cdot 2^a$ with a wide\nenough modulus $2H+J+K+2<2^m$, a terminal window where \\emph{every} one of\nthe last $m$ letters vanishes mod its own growing power of two is\nimpossible:\n\\[\n  \\neg\\,\\mathrm{ActualLcmTerminalDyadicStaircase}(a,J,K,m).\n\\]\n\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.not_actualLcmTerminalDyadicStaircase_of_room}{TotientActualLcmTopEdgeStaircase.lean:251}\n\\quad \\ev{Lean} \\quad \\scale{bounded} \\quad \\coord{mobius-mersenne}. The\nmechanism is a bare positivity/divisibility contradiction (a positive\nquantity below a modulus, divisible by that modulus, must be $0$) against\nthe unconditional positivity of every short-window letter,\n$0<\\mathrm{lcmRayArithmeticLetter}(2^a,j)$ for $a\\ge8$, $0<j<2\\cdot2^a$\n(\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.lcmRayArithmeticLetter_pos_of_lt_two_mul}{TotientActualLcmOrbitArithmetic.lean:1703},\n\\ev{Lean}, unconditional, no rationality hypothesis). \\emph{Do not attempt a\nfull terminal-staircase producer}: it is provably empty. The corpus's own\nroute past this dead end is the \\emph{punctured} staircase (all but the\nlast letter vanish; the last is retained and pinned exactly to the half-turn\n$2^{m-1}$,\n\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.puncturedDyadicStaircase_penultimate_eq_half}{TotientActualLcmTopEdgeStaircase.lean:1187},\n\\ev{Lean}, \\scale{bounded}) and the one-sided residue-gap family below.\n\n\\paragraph{The surviving one-sided target.}\n\\[\n  \\mathrm{ActualLcmTopEdgeResidueGap}(a,J,K,m) :\\equiv\n  m\\le K\\ \\wedge\\\n  2H+J+K+2 < 2^m\\ \\wedge\\\n  D(H,\\,H+J,\\,K) \\bmod 2^m \\le 2^m - (2H+J+K+2).\n\\]\n\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.ActualLcmTopEdgeResidueGap}{TotientActualLcmTopEdgeStaircase.lean:900}\n\\quad \\ev{Open} (a definition, unproved at any $a$)\n\\quad \\scale{bounded} \\quad \\coord{mobius-mersenne}. Note this is a\n\\emph{one-sided} inequality \u2014 only the upper (positive) carry arc is\nexcluded, not a symmetric two-sided band \u2014 because the corridor's lower half\nis already discharged unconditionally: for $a\\ge8$ and $J+(a{+}6)<2\\cdot2^a$,\n\\[\n  0 < R_{2H+J}-R_{H+J}\n\\]\n\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.actualLcmTailDiff_shift_pos}{TotientActualLcmOrbitSign.lean:39}\n\\quad \\ev{Lean} \\quad \\scale{bounded} \\quad \\coord{mobius-mersenne} \u2014 a rare\ngenuinely unconditional, non-hypothetical real-analytic theorem in this\ncorpus, needing no rationality assumption at all. Its companion shows that\n\\emph{if} the orbit is integral, the residue is forced to the exact top\nedge $2^K-e$ ($e>0$ small), \\emph{outside} the arc a symmetric certificate\nwould need:\n\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.actualLcm_integral_forces_topEdgeResidue}{TotientActualLcmOrbitSign.lean:211}\n(\\ev{Lean}, \\scale{bounded}) \u2014 this is the exact statement of ``here is\nwhat remains,'' pinning the missing exclusion to one named residue class\nrather than leaving it implicit.\n\\[\n  \\mathrm{ActualLcmTopEdgeResidueGap}(a,J,K,m)\\ \\Longrightarrow\\\n  R_{2H+J}-R_{H+J}\\notin\\mathbb{Z}.\n\\]\n\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.actualLcmTailDiff_notMem_int_of_topEdgeResidueGap}{TotientActualLcmTopEdgeStaircase.lean:1260}\n\\quad \\ev{Lean} \\quad \\scale{bounded} \\quad \\coord{mobius-mersenne}. The\ncofinal target built from it:\n\\[\n  \\mathrm{PowerTwoActualLcmTopEdgeResidueGapSupply} :\\equiv\\\n  \\forall a_0,\\ \\exists a,K,m,\\quad\n  a_0\\le a\\ \\wedge\\ 8\\le a\\ \\wedge\\ K+(a{+}6)<2\\cdot 2^a\\ \\wedge\\\n  \\mathrm{ActualLcmTopEdgeResidueGap}(a,0,K,m).\n\\]\n\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.PowerTwoActualLcmTopEdgeResidueGapSupply}{TotientActualLcmTopEdgeStaircase.lean:1325}\n\\quad \\ev{Open} \\quad \\scale{cofinal} \\quad \\coord{mobius-mersenne}.\n\\[\n  \\mathrm{PowerTwoActualLcmTopEdgeResidueGapSupply} \\Longrightarrow \\mathrm{Irrational}(S).\n\\]\n\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.irrational_totientSeries_of_topEdgeResidueGapSupply}{TotientActualLcmTopEdgeStaircase.lean:2184}\n\\quad \\ev{Lean} \\quad \\scale{cofinal} \\quad \\coord{mobius-mersenne}.\n\n\\paragraph{Five strictly weaker links, each independently proved sufficient.}\nThe same file proves five further cofinal predicates, each strictly weaker\nthan \\texttt{PowerTwoActualLcmTopEdgeResidueGapSupply} (each implies it, so\neach is an easier target), with a direct \\#249 endpoint of its own \u2014\nproving the \\emph{weakest} of the six closes the entire cluster.\n\n\\begin{enumerate}\n\\item[(i)] $\\mathrm{PowerTwoAdjacentSuffixMidbandSupply}$: replaces the\n$m$-bit residue test with a symmetric two-sided band on the\n\\emph{adjacent-suffix} residue at depth $m$, buffered by the larger depth\n$m{+}1$ so either branch stays inside the sign corridor. \\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.PowerTwoAdjacentSuffixMidbandSupply}{TotientActualLcmTopEdgeStaircase.lean:1334}\n\\ev{Open} \\scale{cofinal} \\coord{mobius-mersenne}. Sufficiency:\n\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.powerTwoActualLcmTopEdgeResidueGapSupply_of_adjacentSuffixMidband}{TotientActualLcmTopEdgeStaircase.lean:2108}, direct endpoint\n\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.irrational_totientSeries_of_adjacentSuffixMidbandSupply}{TotientActualLcmTopEdgeStaircase.lean:2191}.\n\n\\item[(ii)] $\\mathrm{PowerTwoOddGuardTopEdgeHalfWordBandSupply}$: at odd\ndepth $m=2q+1$ the adjacent-suffix residue is exactly twice a half-word\nresidue, and both directed edge widths halve to the same threshold\n$H+q+2$:\n\\[\n  H+q+2 \\le \\bigl(\\text{half-word correction word}\\bigr) \\bmod 4^q\n  \\le 4^q - (H+q+2).\n\\]\n\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.PowerTwoOddGuardTopEdgeHalfWordBandSupply}{TotientActualLcmTopEdgeStaircase.lean:1349}\n\\ev{Open} \\scale{cofinal} \\coord{mobius-mersenne}. This is a substantially\nweaker demand than the earlier fixed $1/32$ central band elsewhere in the\ncorpus. Sufficiency:\n\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.powerTwoAdjacentSuffixMidbandSupply_of_oddGuardTopEdgeHalfWordBand}{TotientActualLcmTopEdgeStaircase.lean:2121}.\n\n\\item[(iii)] $\\mathrm{PowerTwoActualFinalTopEdgeMagnitudeSupply}$: the exact\ncentered-lift restatement of (ii),\n\\[\n  H+q+2 \\le |\\mathrm{actualOddHalfCenteredLift}(a,q)|,\n\\]\n\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.PowerTwoActualFinalTopEdgeMagnitudeSupply}{TotientActualLcmTopEdgeStaircase.lean:1471}\n\\ev{Open} \\scale{cofinal} \\coord{mobius-mersenne}. Proved\n\\textbf{equivalent} (an iff, not merely sufficient) to (ii):\n\\[\n  \\mathrm{PowerTwoOddGuardTopEdgeHalfWordBandSupply}\n  \\iff\n  \\mathrm{PowerTwoActualFinalTopEdgeMagnitudeSupply}.\n\\]\n\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.topEdgeHalfWordBandSupply_iff_actualFinalCenteredMagnitudeSupply}{TotientActualLcmTopEdgeStaircase.lean:1479}\n\\quad \\ev{Lean}.\n\n\\item[(iv)] $\\mathrm{PowerTwoFlexibleActualTopEdgeMagnitudeSupply}$: the\nsame magnitude test at an \\emph{arbitrary} odd rank whose adjacent depth\nstill fits the corridor (not forced to the canonical guarded depth):\n\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.PowerTwoFlexibleActualTopEdgeMagnitudeSupply}{TotientActualLcmTopEdgeStaircase.lean:1927}\n\\ev{Open} \\scale{cofinal} \\coord{mobius-mersenne}. Sufficiency routes back\nthrough (i), not through the corridor-escape chain below:\n\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.powerTwoAdjacentSuffixMidbandSupply_of_flexibleActualTopEdgeMagnitude}{TotientActualLcmTopEdgeStaircase.lean:1938}, direct endpoint\n\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.irrational_totientSeries_of_flexibleActualTopEdgeMagnitudeSupply}{TotientActualLcmTopEdgeStaircase.lean:2213}.\n\n\\item[(v)] $\\mathrm{PowerTwoFlexibleActualTerminalDominanceSupply}$: a\nstrictly \\emph{one-sided} version of (iv) \u2014 only the upper comparison, no\nabsolute value:\n\\[\n  \\mathrm{diagonalWindowIncrement}(2^a,\\,2q{+}2) \\le 2\\cdot\\mathrm{actualOddHalfCenteredLift}(a,q).\n\\]\n\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.PowerTwoFlexibleActualTerminalDominanceSupply}{TotientActualLcmTopEdgeStaircase.lean:1908}\n\\ev{Open} \\scale{cofinal} \\coord{mobius-mersenne}. Direct endpoint\n\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.irrational_totientSeries_of_flexibleActualTerminalDominanceSupply}{TotientActualLcmTopEdgeStaircase.lean:2221}.\n\\end{enumerate}\n\n\\paragraph{The weakest known link, and the exact identity pinning it.}\nA sixth predicate, strictly weaker again than (v) \u2014 it is the disjunction of\n(v)'s inequality with the opposite-direction escape \u2014 is the true minimum\nof the whole cluster:\n\\[\n  \\mathrm{PowerTwoFlexibleActualTerminalCarryCorridorEscapeSupply} :\\equiv\n  \\forall a_0,\\ \\exists a,q,\\quad \\cdots\\ \\wedge\\\n  \\Bigl(2u \\le d - B\\ \\vee\\ d \\le 2u\\Bigr),\n\\]\nwhere $d=\\mathrm{diagonalWindowIncrement}(2^a,2q{+}2)$,\n$u=\\mathrm{actualOddHalfCenteredLift}(a,q)$,\n$B=2H+(2q{+}1)+2$.\n\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.PowerTwoFlexibleActualTerminalCarryCorridorEscapeSupply}{TotientActualLcmTopEdgeStaircase.lean:1895}\n\\ev{Open} \\scale{cofinal} \\coord{mobius-mersenne}. Direct endpoint\n\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.irrational_totientSeries_of_flexibleActualTerminalCarryCorridorEscapeSupply}{TotientActualLcmTopEdgeStaircase.lean:2230}.\nThis form is not arbitrary: under integrality it is exactly one side of a\nproved identity,\n\\[\n  2\\cdot\\mathrm{actualOddHalfCenteredLift}(a,q)\n  = \\mathrm{diagonalWindowIncrement}(2^a,2q{+}2) - \\mathrm{carryOrbit}(H,H,z,2q{+}1),\n\\]\n\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.two_mul_actualOddHalfCenteredLift_eq_terminal_sub_trueCarry}{TotientActualLcmTopEdgeStaircase.lean:1678}\n\\quad \\ev{Lean} \\quad \\scale{bounded} \\quad \\coord{mobius-mersenne} \u2014 an\n\\emph{equality}, not an inequality: the doubled centered-lift state is\nexactly (terminal arithmetic letter) minus (true carry). Escaping either\nside of this named open interval is exactly the corridor-escape supply.\nCrucially, the sign machinery has already eliminated the branch this\nidentity most naturally supplies:\n\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.actualLcm_trueEndpointSurvivor_neg}{TotientActualLcmOrbitSign.lean:172}\n(\\ev{Lean}, \\scale{bounded}) proves that \\emph{under integrality} the true\ncarry orbit is strictly \\emph{positive} throughout the corridor, which\nmeans the dominance branch (v) is the one the identity naturally produces\npositive evidence \\emph{against}, and the surviving open target is really\nthe \\emph{lower}-escape branch,\n$2u \\le d - B$ \u2014 the dominance branch itself is stated separately as\n\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.actualLcmTailOrbit_notMem_int_of_actualTerminalDominance}{TotientActualLcmTopEdgeStaircase.lean:1880}. This is the sharpest\nfinite arithmetic statement of ``what remains to prove'' anywhere in the\ncorpus: a fully closed-form real-integer quantity whose escape from a named\ninterval is necessary and sufficient (mod a fit hypothesis\n$2(H{+}q{+}2)\\le 4^q$) for non-integrality at that rank.\n\n\\paragraph{A second, independent finite-window target: the short-arithmetic-kill supply.}\nA separate reduction ties the diagonal public form (the collapsing-free-parameters chain above) to this private\nlane via an exact digit formula with no cleanliness hypothesis at all,\n$\\mathrm{lcmRayArithmeticLetter}(t,j) = \\varphi(H_t+j)-\\varphi(H_t)$ for\n\\emph{every} offset $j$\n(\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.lcmRayArithmeticLetter_eq_deltaTotient}{TotientActualLcmOrbitArithmetic.lean:298},\n\\ev{Lean}), and\n$\\mathrm{LcmDiagonalArithmeticKill}(t,L)\\iff\\mathrm{certifiedKill}(H_t,H_t,L)$\n(\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.lcmDiagonalArithmeticKill_iff_certifiedKill}{TotientActualLcmOrbitArithmetic.lean:2045},\n\\ev{Lean}). Consequently\n\\[\n  \\mathrm{PowerTwoActualLcmShortArithmeticKillSupply} :\\equiv\n  \\forall a_0,\\ \\exists a,L,\\quad a_0\\le a\\ \\wedge\\ L<2\\cdot2^a\\ \\wedge\\\n  \\mathrm{certifiedKill}(H_{2^a}, H_{2^a}, L)\n\\]\n\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.PowerTwoActualLcmShortArithmeticKillSupply}{TotientActualLcmOrbitArithmetic.lean:2107}\n\\ev{Open} \\scale{cofinal} \\coord{mobius-mersenne} is \\emph{literally the\ndiagonal obligation $P(t)$ of the collapsing-free-parameters chain above, restricted to powers of two and to a\nshort window $L<2\\cdot2^a$}. It is proved sufficient for the actual-orbit\niff above via\n\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.actualLcmOrbitNonintegralitySupply_of_shortArithmeticKill}{TotientActualLcmOrbitArithmetic.lean:2152},\nand it is verified at exactly two exponents, $a=4$ (depth $L=23$) and\n$a=6$ (depth $L=93$), each traced to a pre-existing compressed diagonal\ncertificate:\n\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.actualLcmTailOrbit_four_and_six_notMem_int}{TotientActualLcmShortKill.lean:35}\n\\ev{Cert} \\scale{fixed}. The bounded stub\n\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.powerTwoActualLcmShortArithmeticKillSupply_through_six}{TotientActualLcmShortKill.lean:54}\n(\\ev{Lean}, \\scale{bounded}, proof is a single hard-coded witness\n$(a,L)=(6,93)$, with no argument in $a$ at all) proves the supply only for\n$a_0\\le 6$; extending it to a single further exponent (say $a=7$) is a\nself-contained, independently interesting target, not a re-run.\n\n\\paragraph{A Diophantine-flavoured alternative: the raw-approximant separation supply.}\nUnconditionally, for every $a,q$:\n\\[\n  \\bigl|\\mathrm{actualLcmTailOrbit}(a) - \\mathrm{actualLcmRawApprox}(a,q)\\bigr|\n  < \\frac{4H+2(2q{+}1)+4}{2^{2q+2}},\n\\]\n\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.abs_actualLcmTailOrbit_sub_rawApprox_lt}{TotientActualLcmOrbitSeparation.lean:141}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{mobius-mersenne}, where\n$\\mathrm{actualLcmRawApprox}(a,q)$ is an explicit finite computable rational.\nThis reduces the whole analytic problem to a finite question: a cofinal\n$1/32$-separation of the raw approximant from every integer,\n\\[\n  \\mathrm{PowerTwoActualLcmOrbitSeparationSupply} :\\equiv\n  \\forall a_0,\\ \\exists a\\ge\\max(2,a_0),\\ \\exists q,\\quad\n  \\mathrm{oddGuardedCanonicalAdjacentSuffixDepth}(2^a)=2q{+}1\\ \\wedge\\\n  \\forall z\\in\\mathbb{Z},\\ \\tfrac{1}{32}+\\mathrm{errorRadius}(a,q)\\le|\\mathrm{actualLcmTailOrbit}(a)-z|,\n\\]\n\\ev{Open} \\scale{cofinal} \\coord{mobius-mersenne}, sufficient via\n\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.irrational_totientSeries_of_actualLcmOrbitSeparationSupply}{TotientActualLcmOrbitSeparation.lean:305}\n(\\ev{Lean}). Note the depth $q$ is not a free search parameter: the supply\npins exactly one admissible depth per scale $a$ via\n$\\mathrm{oddGuardedCanonicalAdjacentSuffixDepth}$.\n\n\\subsection{A problem-agnostic normal form: the guard-cylinder compression}\n\nIndependently of which coordinate is used, \\emph{any} tail-difference\ncertificate \u2014 for arbitrary $h,N,L$, not specific to \\#249 \u2014 compresses to\na two-bit test at logarithmic depth:\n\\[\n  \\bigl(\\exists L,\\ \\mathrm{certifiedKill}(h,N,L)\\bigr)\n  \\iff\n  \\mathrm{GuardCylinderWitness}(h,N).\n\\]\n\\lean{Erdos257PeriodNoncollapse.DiagonalFreshLossBridge.PowerTwoOddWindowAffine.exists_certifiedKill_iff_guardCylinderWitness}{TotientActualLcmTopEdgeStaircase.lean:844}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{seam-integer}, where the\nwitness is $\\exists s,b,\\ \\mathrm{certifiedKill}(h,N{+}s,b{+}1) \\vee\n(\\text{room} \\wedge \\mathrm{DyadicMixedGuard}(D(h,N{+}s,b{+}2),b))$ at\n$b=\\lfloor\\log_2(N{+}h{+}L{+}2)\\rfloor+1$. This statement mentions no\nMersenne or totient structure whatsoever and is stated for arbitrary\n$h,N,L$; it compresses the search space for \\emph{any} tail-difference\nnon-integrality certificate \u2014 of any depth $L$, however large \u2014 down to a\nsocket at a logarithmic scale plus a two-bit mixed-guard cylinder\n($01$ or $10$). This is directly reusable, unchanged, for any binary-series\ntail-difference problem, including \\#257's own denominators.\n\n\\subsection{Two further, logically independent obligations}\n\nTwo more open statements would \\emph{also} close \\#249, each living in a\ngenuinely different coordinate from the certificate/residue coordinate\nabove, and each finished on exactly one side of an inequality.\n\n\\paragraph{Obligation 2 (rank compression).}\nUnconditionally, for every level $e$, the dyadic totient-kernel family of\n$2^e+1$ channels is linearly independent over $\\mathbb{Q}$ (via CRT and\nDirichlet's theorem on primes in arithmetic progression):\n\\lean{Erdos249257.linearIndependent_canonicalTotientKernelFamily}{TotientMahlerDefect.lean:900}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{other:carry-kernel-rank}.\nConsequently, if $S$ is rational there is a tempered integral binary carry\norbit $u$ with\n\\[\n  \\forall e,\\quad 2^e-1 \\le \\operatorname{rank}_{\\mathbb{Q}}\\operatorname{span}\\bigl(\\mathrm{canonicalCarryKernelFamily}(u,e)\\bigr).\n\\]\n\\lean{Erdos249257.not_irrational_totientSeries_implies_unbounded_carryRank_unconditional}{TotientCarryKernelRigidity.lean:284}\n\\quad \\ev{Lean} \\quad \\scale{uniform} \\quad \\coord{other:carry-kernel-rank}.\nThe missing input is the opposite inequality \u2014 a rationality-side rank\n\\emph{upper} bound:\n\\[\n  \\exists C,\\quad \\forall\\text{ tempered integral binary carry orbit } u\n  \\text{ for a rational } S,\\quad\n  \\forall e,\\quad \\operatorname{rank}_{\\mathbb{Q}}\\operatorname{span}\\bigl(\\mathrm{canonicalCarryKernelFamily}(u,e)\\bigr) \\le C\n\\]\n(or any bound growing slower than $2^e-1$). \\ev{Open}\n\\scale{cofinal} \\coord{other:carry-kernel-rank}. One natural route to it is\nproved dead: no \\texttt{CompressedAdjointCertificate} ($Q\\cdot v\\cdot A =\n\\mathrm{boundary}$ with $|\\mathrm{boundary}|<Q\\cdot v$ and $A\\ne0$) can\nexist \u2014\n\\lean{Erdos249257.false_of_compressedAdjointCertificate}{TotientMahlerDefect.lean:1051}\n\\quad \\ev{Lean} \u2014 and the full family is proved infinite-dimensional in\nspan:\n\\lean{Erdos249257.not_finiteDimensional_span_fullTotientKernel}{TotientMahlerDefect.lean:1014}\n\\quad \\ev{Lean}. What rationality \\emph{does} buy \u2014 uniform eventual\nperiodicity of $u$'s dyadic sections modulo some $v>0$ \u2014 is proved to\n\\emph{not} promote to any finite rank bound without further arithmetic\ninput:\n\\lean{Erdos249257.not_irrational_totientSeries_implies_mod_period_and_unbounded_rank}{TotientTailCarryPeriod.lean:224}\n\\quad \\ev{Lean}. This pins the obstacle precisely, but supplies no rank\nupper bound.\n\n\\paragraph{Obligation 3 (Farey growth law).}\nThe corpus's strongest \\emph{unconditional} statement about $S$ comes from\na Farey-gap denominator exclusion at a single fixed window $K=240$:\n\\[\n  \\forall p\\in\\mathbb{Q},\\quad p.\\mathrm{den} \\le 79639646646701375323355774875831053\n  \\ \\Longrightarrow\\ S\\ne p.\n\\]\n\\lean{Erdos249257.tsum_totient_div_pow_two_ne_ratCast_of_den_le_79639646646701375323355774875831053}{CertificateKernel.lean:18056}\n\\quad \\ev{Lean} \\quad \\scale{bounded} \\quad \\coord{farey}. This bound is\n\\emph{sharp} at $K=240$ \u2014 the mediant $q=79639646646701375323355774875831054$\nis proved to be the exact first failing denominator\n(\\lean{Erdos249257.gap_check_window_1_240_first_failure}{GapFareyBound.lean:225},\n\\ev{Lean}) \u2014 so re-running the same window buys nothing further; a new $K$\nneeds a freshly committed totient residue and freshly computed\ncontinued-fraction convergents, both hard-coded numerals. The cofinal\nupgrade would be: a proved growth law $g(K)\\to\\infty$ such that for every\n$K$ the $(N{=}1,K)$ gap check passes for all $q\\le g(K)$ \u2014 equivalently, a\nlower bound on the convergent denominators of the underlying totient-window\nconstant, uniform in $K$. \\ev{Open} \\scale{cofinal} \\coord{farey}. One\nnatural strengthening is proved to be a hard ceiling, not a route to\nunboundedness: at a prime-power reduced denominator, the ``unit-gap''\nrefinement can rescue \\emph{at most one} additional lattice point beyond\nthe ordinary gap certificate \u2014\n\\lean{Erdos249257.reducedDenominatorCandidateCount_prime_pow_le_one}{PrimitiveWeightCertificate.lean:111}\n\\quad \\ev{Lean}. This route is closed; growing $K$ genuinely requires new\nper-window computation, not a refinement of the existing one.\n\n\\subsection{Equivalence map}\n\n\\begin{center}\n\\begin{longtable}{p{0.30\\linewidth} p{0.20\\linewidth} p{0.42\\linewidth}}\n\\textbf{Form} & \\textbf{Relation to Obligation 1} & \\textbf{Site} \\\\\n\\hline\n\\endhead\nSep$(h,N,L)$ supply (Obl.\\ 1, base form) & base form & TotientTailPeriodKiller.lean:394 \\\\\nmultiple-period supply & weaker (implied by Obl.\\ 1) & CarrySurvivorExtinction.lean:502 \\\\\nlcm-diagonal supply $P(t)$ & weaker; single free parameter & LcmDiagonalReduction.lean:112 \\\\\nlcm-cone supply & weaker again; $P(t)$ is the cell $q{=}m{=}1$ & CertificateKernel.lean:18686 \\\\\ncone-menu (\\texttt{coneNonflatCert}) supply & weaker; half the pairwise radius & CertificateKernel.lean:18812 \\\\\npure non-integrality (diagonal/cone) & \\textbf{equivalent} to the corresp.\\ certificate form, via the completeness iff & CertificateKernel.lean:18706, :18718 \\\\\nDTWFirstHarmonicNormGap & sufficient for Obl.\\ 1 (not shown weaker/stronger) & FirstHarmonicPivot.lean:83 \\\\\nactual-orbit nonintegrality supply & \\textbf{equivalent} to Irrational$(S)$ itself & TotientActualLcmOrbitNonintegrality.lean:37 \\\\\nshort-arithmetic-kill supply & special case of $P(t)$: powers of two, short window & TotientActualLcmOrbitArithmetic.lean:2107 \\\\\ntop-edge residue-gap supply & sufficient for actual-orbit supply; one-sided & TotientActualLcmTopEdgeStaircase.lean:1325 \\\\\nadjacent-suffix midband supply & weaker than top-edge residue-gap & TotientActualLcmTopEdgeStaircase.lean:1334 \\\\\nodd-guard half-word band supply & weaker again; \\textbf{equivalent} to final-magnitude form & TotientActualLcmTopEdgeStaircase.lean:1349 \\\\\nactual final top-edge magnitude supply & \\textbf{equivalent} to odd-guard half-word band & TotientActualLcmTopEdgeStaircase.lean:1471 \\\\\nflexible top-edge magnitude supply & weaker than midband; feeds midband, not corridor-escape & TotientActualLcmTopEdgeStaircase.lean:1927 \\\\\nflexible terminal-dominance supply & weaker again; one-sided & TotientActualLcmTopEdgeStaircase.lean:1908 \\\\\nflexible terminal carry-corridor-escape supply & \\textbf{weakest known} in this cluster & TotientActualLcmTopEdgeStaircase.lean:1895 \\\\\nraw-approximant separation supply & alternative (Diophantine) sufficient form, same target & TotientActualLcmOrbitSeparation.lean:305 \\\\\nrank-compression upper bound (Obl.\\ 2) & independent obligation, different coordinate & TotientCarryKernelRigidity.lean:284 \\\\\nFarey growth law (Obl.\\ 3) & independent obligation, different coordinate & GapFareyBound.lean, CertificateKernel.lean:18056 \\\\\nguard-cylinder witness & problem-agnostic normal form of \\emph{any} certificate, not scale-comparable & TotientActualLcmTopEdgeStaircase.lean:844 \\\\\n\\end{longtable}\n\\end{center}\n\n\\emph{Reading the table.} ``Weaker'' always means: the stronger form's\nhypothesis, if it held, would imply the weaker form's hypothesis by an\nexplicit Lean construction (so the weaker predicate is an easier target),\nand each row independently has its own theorem to $\\mathrm{Irrational}(S)$.\n``Equivalent'' is reserved for rows connected by a proved \\texttt{iff}\nstatement; no other pair in this table is known to be logically equivalent,\nonly jointly sufficient for the same conclusion. In particular, Obligation\n1 (the public certificate supply) and the actual-orbit iff (the private\nlane) are \\emph{not} shown equivalent to each other as stated; they are\nindependently sufficient reformulations of the same underlying question,\nconnected concretely only through the short-arithmetic-kill supply, which\nis a genuine special case of Obligation 1's diagonal form.\n\n\\subsection{What new mathematics each surviving form would need}\n\nNone of the forms above is close to complete; every one reduces the open\ncontent to a named finite-flavoured arithmetic statement, never removes it.\nTwo forms above have their missing ingredient identified precisely enough\nto name what field of technique would supply it, without any claim that the\ntechnique exists or is easy to apply.\n\n\\paragraph{The exponential-sum form.} \\texttt{DTWFirstHarmonicNormGap} asks\nfor cancellation in the first additive character of a totient-driven\ndiscrepancy: a constant-saving bound, uniform in the period $h$ and growing\nwith the block size $X$ and depth $L$, on\n$\\sum_{N\\in[X,2X)} e\\bigl(D(h,N,L)/2^L\\bigr)$, where $D$ is built from\nconsecutive differences of $\\varphi$. This is a Weyl-sum-shaped statement\nabout additive character sums of an arithmetic function on a window; no\nargument of this shape is attempted or proved anywhere in the corpus.\n\n\\paragraph{The staircase form.} The private top-edge cluster's surviving\ntarget, in every one of its six equivalent-or-sufficient guises, reduces\nunder the exact identity of the private lane above to control of a single \\emph{terminal\narithmetic letter}: the value $\\mathrm{diagonalWindowIncrement}(2^a,2q{+}2)\n= \\varphi(2H{+}2q{+}2)-\\varphi(2H)$ measured against twice a centred\ncarry-lift state. What is needed is not a new certificate mechanism \u2014 the\nwhole apparatus of arithmetic letters, centred lifts, and carry orbits is\nalready exact and unconditional \u2014 but a genuine size or residue statement\nabout this one totient value at cofinally many scales $a$, sufficient to\nforce it off the interval the identity names. The corpus's own finite\ncensus (through $t=64$ on the diagonal form, and $a=4,6$ on the short\narithmetic-kill form) gives no asymptotic evidence either way; it is a\nfloor, not a trend.\n\nNo claim is made that either obstruction is close to resolution, and none\nof the material above decides Erd\u0151s \\#249, \\#257, or any weakening of them.\n\n\\clearpage\n\\section*{Statements and declarations}\n\n\\paragraph{Artefact and data availability.} The linked Lean declarations,\nfixed toolchain, and library manifest are published in the companion\nrepository. This manuscript provides navigation and exposition rather than\nproof authority.\n\n\\paragraph{Declaration of generative AI use.} \\paperfullaideclaration\n\n\\paragraph{Authority boundary.} Kernel checking establishes that a proposition\nwas proved; it does not authorise the exposition, literature judgements, or any\nclaim that Problem 249 is solved.\n\n\\paragraph{Funding and competing interests.} This work received no external\nfunding. The author declares no competing interests.\n\n\\end{document}\n","dependencies":[],"external_verification_route":"https://raw.githubusercontent.com/wcook04/plectis-lean-erdos249-257/main/paper/erdos249-totient-reasoning-surface.tex"},{"record_type":"paper","paper_id":"erdos-251-prime-gap-dyadic-series","title":"An Integral-Shift Criterion for Dyadic Tail Recurrences","source_repository":"plectis-lean-erdos249-257","source_path":"paper/erdos-251-prime-gap-dyadic-series.tex","hosted_pdf_path":"papers/erdos-251-prime-gap-dyadic-series.pdf","licence":"CC-BY-4.0","copyright":"2026 Will Cook","cite_as":"Plectis reader corpus, paper:erdos-251-prime-gap-dyadic-series","pdf_sha256":"sha256:398437812e3f16f0e16006f838b85ea457329745aa8db56e8c777d7e754a77a8","pdf_byte_count":133508,"pdf_embedded":false,"pdf_omission_reason":"the exact LaTeX is embedded and retrieves better than extracted PDF text; the PDF is a presentation rendering and its hash is recorded for verification","source_embedded":true,"availability":"embedded_exact","embedded_copy_role":"native_manuscript","embedded_from_repository":"plectis-lean-erdos249-257","embedded_copy_path":"paper/erdos-251-prime-gap-dyadic-series.tex","media_type":"text/x-tex","body_format":"latex","sha256":"sha256:74d597356effe61b1ad9b7b02c9081933e42251a534e296e9ebff06145e6c144","byte_count":64725,"body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% One of the Erd\u0151s Problem Notes: a short standalone treatment of a single\n% problem in the ErdosProblems expansion library.  Build with\n% `tectonic erdos-251-prime-gap-dyadic-series.tex` or `make`.\n\\documentclass[11pt]{article}\n\\input{problem-note-preamble}\n% Problem-specific source pin: #251 advances independently of the corpus\n% default, while the #249/#257 notes retain their own reviewed snapshot.\n\\renewcommand{\\commit}{cb1110cf55a407543beabd0d84118abef12ab146}\n\\renewcommand{\\commitshort}{cb1110cf55a4}\n\\hypersetup{\n  pdftitle={The prime-gap dyadic series and its integral tail differences},\n  pdfsubject={Erd\u0151s Problem Notes: Problem 251},\n  pdfkeywords={irrationality, prime gaps, dyadic series, summation by parts, Lean 4},\n}\n\n\\runninghead{The prime-gap dyadic series}\n\\title{The Prime-Gap Dyadic Series and Its Integral Tail Differences}\n\\subtitle{An exact recurrence criterion for Erd\\H{o}s Problem~\\#251}\n\\author{Will Cook}\n\\date{August 2026}\n\n\\begin{document}\n\\maketitle\n\\noteprovenance\n\n\\begin{abstract}\n\\noindent\nLet a real state sequence satisfy $T_{N+1}=2T_N-g_{N+1}$, where every\n$g_N$ is an integer, and write $\\sigma_h(N)=T_{N+h}-T_N$.  We prove the exact\nclassification\n\\[\n T_0\\in\\Q\n \\quad\\Longleftrightarrow\\quad\n \\exists h\\ge1\\ \\exists N\\quad \\sigma_h(N)\\in\\Z\n \\quad\\Longleftrightarrow\\quad\n \\exists h\\ge1\\ \\exists N_0\\ \\forall N\\ge N_0\\quad \\sigma_h(N)\\in\\Z.\n\\]\nEquivalently, $T_0$ is irrational precisely when every positive-length tail\ndifference is non-integral.  The proof is elementary and Lean-checked.  Its\ntwo ingredients are the block identity\n$\\sigma_h(N)=(2^h-1)T_N-B_{h,N}$ with $B_{h,N}\\in\\Z$, and the exact removal of\nthe power-of-two part of a rational denominator under repeated doubling.\n\nFor the primes $p_0=2,p_1=3,\\ldots$ and gaps $g_n=p_{n+1}-p_n$, summation by\nparts gives an exact finite identity with its endpoint term retained.  After\npassage to the limit, justified externally by the prime number theorem, the\nprime series $\\sum p_n2^{-(n+1)}$ and the gap series\n$\\sum g_n2^{-(n+1)}$ have the same irrationality status.  Thus Erd\\H{o}s\nProblem~\\#251 is equivalent to non-integrality of every positive difference\nof the rescaled prime-gap tails.\n\nThe problem remains open.  We isolate a finite sufficient obstruction: two\nadjacent tail differences in $(-1,1)$ cannot both be integral when the\ncorresponding gaps differ.  We also exhibit a positive, unbounded integer\nsequence with rational dyadic sum, showing why nonperiodicity of the prime gaps\nalone cannot settle the problem.  The missing input must control the tails,\nrather than the coefficient word alone.\n\\end{abstract}\n\n\\section{Introduction}\\label{sec:problem}\n\nLet $p_n$ denote the $n$th prime.  Erd\\H{o}s Problem~\\#251 asks whether\n\\[\n \\Pi=\\sum_{n\\ge1}\\frac{p_n}{2^{\\,n}}\n\\]\nis irrational~\\cite[p.~94]{erdos1958}\\cite[p.~62]{erdosgraham1980}%\n\\cite[p.~103]{erdos1988}.  Bloom's current catalogue record reproduces this\nquestion and labels it open, while explicitly warning that the status is the\nwebsite owner's present assessment and may omit relevant literature\n\\cite{erdosproblems}.  We therefore use the catalogue for numbering and\ncurrent reported status only; the three original publications carry the\nmathematical claims.  The problem is open.  Numerically\n$\\Pi=2/2+3/4+5/8+7/16+11/32+\\cdots=3.674643966\\ldots$.\n\nThroughout the formal development the primes are indexed from zero, so that\n$p^{(0)}_0=2$, $p^{(0)}_1=3$, $p^{(0)}_2=5$, and the gaps are\n$g_i=p^{(0)}_{i+1}-p^{(0)}_i$.  In that indexing\n\\[\n \\Pi=\\sum_{i\\ge0}\\frac{p^{(0)}_i}{2^{\\,i+1}} ,\n\\]\nwhich is the convention every statement below uses; we drop the superscript\nfrom here on.  The first values are\n\\[\n \\begin{array}{c|cccccccccc}\n  i   & 0&1&2&3&4 &5 &6 &7 &8 &9\\\\\\hline\n  p_i & 2&3&5&7&11&13&17&19&23&29\\\\\n  g_i & 1&2&2&4&2 &4 &2 &4 &6 &2\n \\end{array}\n\\]\nso $g_0=1$ and every later gap is even, the primes after $2$ being odd.\nA second normalisation with denominator $2^{\\,i}$ also occurs,\nand at every finite horizon it is exactly twice this one,\n\\[\n \\sum_{i=0}^{n-1}\\frac{p_i}{2^{\\,i}}\n =2\\sum_{i=0}^{n-1}\\frac{p_i}{2^{\\,i+1}} ,\n\\]\nthe\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{105}{prime0DisplayedPartialSumQ_eq_two_mul}{factor-of-two\nnormalisation}.  A factor of two does not change rationality, so no result\nbelow depends on the choice; the identity is stated so that the indexing cannot\ndrift silently.\n\n\\paragraph{Notation.}\n$\\N=\\{0,1,2,\\ldots\\}$ and $\\Npos=\\{1,2,\\ldots\\}$.  We write $\\ph$ for Euler's\ntotient function, $\\operatorname{den}q$ for the denominator of a rational\nnumber $q$ written in lowest terms, $\\operatorname{dist}(x,\\Z)$ for the\ndistance from a real number $x$ to the nearest integer, and\n$\\operatorname{Irr}(x)$ for the assertion that $x$ is irrational.  A rational\nnumber is called \\emph{integral} when it is the image of an integer.  A\nsequence $(a_n)$ is \\emph{eventually periodic with period $h\\ge1$} when\n$a_{n+h}=a_n$ for every sufficiently large $n$.  A sum over an empty range of\nindices is zero.\n\n\\paragraph{Relation to prior work.}\nErd\\H{o}s proved that $\\sum_n p_n^{k}/n!$ is irrational for every\n$k\\ge1$~\\cite[p.~93]{erdos1958}.  The 1958 article gives the $k=1$ proof on\npp.~94--95 and says explicitly that the more complicated $k>1$ proof is\nomitted.  On page~103 of his 1988 problem paper he separately\nstated the fixed-denominator problem: he could not prove that\n$\\sum_n p_n^{k}/2^{n}$ is irrational for every $k\\ge1$, and wrote that\nthe case $k=1$ was probably already very difficult.  He also stated the\nvariable-denominator expectation that\n$\\sum_{n\\ge1}p_n/(g_1\\cdots g_n)$ is irrational whenever $g_n\\ge2$ and\n$g_n=o(p_n)$~\\cite[p.~103]{erdos1988}.  The latter expectation is false:\nChatGPT~5.4 Pro, orchestrated by Vjeko Kova\\v{c}, constructs such a sequence\n$(g_n)$ for which the sum is exactly $1$\n~\\cite[Theorem~1 and proof, pp.~1--2]{kovac2026}.  Under the stronger\nhypotheses that $(g_n)$ is strictly increasing and\n$g_n=O(n\\log^k n)$, Erd\\H{o}s had already classified the rational cases:\nrationality holds exactly when $g_n=q p_n+1$ eventually for one fixed integer\n$q>1$~\\cite[pp.~96--99]{erdos1958}.  The counterexample evades those\nhypotheses.  It\ndoes not address the fixed-denominator dyadic series studied here.  The\ncurrent catalogue record still repeats this variable-denominator expectation\nwithout mentioning the counterexample~\\cite{erdosproblems}; its warning about\npossibly missing literature is therefore material here, not merely\nboilerplate.\n\nThere is also a genuinely adjacent proved dyadic theorem.  If $P(m)$ denotes\nthe largest prime factor of $m$, Erd\\H{o}s and Pomerance proved that\n\\[\n \\sum_{m\\ge2}\\frac{\\mathbf 1_{\\{P(m)>P(m+1)\\}}}{2^m}\n\\]\nis irrational~\\cite[\\S7, p.~320]{erdospomerance1978}.  Erd\\H{o}s and Graham\nrecord the complementary indicator on p.~62: equality of the two largest\nprime factors is impossible for consecutive integers, so its series is\n$1/2$ minus the displayed one and is irrational as well.  This is a theorem\nabout a bounded prime-factor comparison digit sequence, not the unbounded\nprime numerators in $\\Pi$; it supplies nearby positive evidence without\nsolving Problem~\\#251.\n\nProblem~\\#251 also appears as the unproved declaration\n\\texttt{erdos\\_251} in the \\emph{Formal Conjectures}\nrepository~\\cite{formalconjectures251}.  Its zero-based Lean sum starts with\nthe zeroth prime over $2^0$, so it is twice the displayed normalisation and\nhas equivalent irrationality status, but it is not literally the same\nindexing; its proof is \\texttt{sorry}.  No priority is claimed for anything\nbelow.\n\n\\paragraph{The strategy.}\nThe digits $p_n$ grow, so the series is not a digit expansion in any bounded\nalphabet, and the standard rationality criteria for such expansions do not\napply directly.  The classical elementary criteria for series of this kind\ninstead control irrationality through the growth of the denominators.\nErd\\H{o}s and Straus named a sum $\\sum_k 1/a_k$ over a strictly increasing\nsequence of positive integers an \\emph{Ahmes series}~\\cite{erdosstraus1963}; for\nsuch a series the condition $a_k^{1/2^k}\\to\\infty$ is sufficient for\nirrationality, and it is sharp, since shifted Sylvester sequences grow like\n$C^{2^{k}}$ for arbitrarily large $C$ and have rational reciprocal sum.  Both\nstatements and their attribution are recorded in the introduction of Kova\\v{c}\nand Tao~\\cite[\\S1]{kovactao2024}, who develop the elementary technology for such\nseries much further.  Splitting each term $p_i/2^{\\,i+1}$ into $p_i$ copies of\n$2^{-(i+1)}$ writes $\\Pi$ as a sum of unit fractions, but with repetitions, and\nthe denominators occurring in it are exactly the powers of two.  Even ignoring\nthe repetitions the growth hypothesis fails by every available margin, since\n$(2^{\\,n})^{1/2^{\\,n}}\\to1$, and every arithmetic constraint has to come from\nthe numerators instead.\n\nWhat replaces growth control is denominator control on the sequence of\nrescaled tails.  Theorem~\\ref{res:classification} says exactly that the sum is\nrational if and only if one positive-length difference of those tails is an\ninteger.  The same difference is then integral at every later index.  This\ncondition uses nothing about the coefficients beyond their integrality, and it\ndoes not force them to repeat: Proposition~\\ref{res:telescope}, applied to\n$K_n=n+2$, produces the positive sequence $\\kappa_n=n+1$, which is unbounded\nand hence not eventually periodic, while\n$\\sum_{n\\ge0}\\kappa_n2^{-(n+1)}=2$.\n\n\\paragraph{Outline.}\nSection~\\ref{sec:parts} replaces the primes by their consecutive gaps, the\nnatural increments studied by prime-distribution theory.  Summation by parts\nwith the endpoint retained gives an exact finite identity, and the termwise\nrelation $v_n=2u_{n+1}-u_n$ between the gap terms and the prime terms makes the\npassage to the limit a matter of summability alone; the prime number theorem\nsupplies that summability externally.\nSection~\\ref{sec:tail} rescales the tails of a dyadic series into a recurrence\nand develops it: the block identity, the integral-shift criterion, the\nexact rationality classification, its irrationality normal form, and a local\nobstruction that converts an infinite-tail condition into a single comparison\nof gaps.\nSection~\\ref{sec:carry} shows that the tail constraint does not by itself make\nthe coefficient sequence eventually periodic, and Section~\\ref{sec:open} states\nthe remaining obligation.  The pinned Lean~4 toolchain and Mathlib revision\ncheck the formal statements.  The cited system paper identifies Lean~4\n\\cite[abstract and \\S1, pp.~625--626]{lean4}, while the Mathlib paper documents\nthe library's historical Lean~3-era architecture\n\\cite[abstract and \\S1.1, p.~367]{mathlib}; it is not authority for the current\npinned revision.  Linked phrases open the corresponding declaration at the\npinned source revision~\\commitshort.\n\n\\statusboundary\n\n\\begin{table}[!t]\n\\centering\n\\small\n\\begin{tabular}{@{}\n  >{\\raggedright\\arraybackslash}p{0.34\\linewidth}\n  >{\\raggedright\\arraybackslash}p{0.18\\linewidth}\n  >{\\raggedright\\arraybackslash}p{0.40\\linewidth}@{}}\n\\toprule\nStatement & Status & Treatment here \\\\\n\\midrule\nIrrationality of $\\Pi$ & Open & Not proved. \\\\\nFinite prime-gap identity & Proved here & Theorem~\\ref{res:parts}, with the endpoint retained. \\\\\nInfinite prime-gap identity & Lean-checked conditionally & Theorem~\\ref{res:infinite}; the prime number theorem supplies the hypothesis externally. \\\\\nPrime-series/gap-series irrationality equivalence & Lean-checked conditionally & Corollary~\\ref{res:irr-equivalence}. \\\\\nBlock identity for the tail recurrence & Proved here & Theorem~\\ref{res:block}. \\\\\nIntegral shift criterion & Proved here; an equivalence & Theorem~\\ref{res:shiftiff}. \\\\\nTotient shift from an odd denominator & Proved here & Theorem~\\ref{res:totient}. \\\\\nPropagation of an integral shift & Proved here & Theorem~\\ref{res:propagate}. \\\\\nEventual integral shift for every rational-valued recurrence & Lean-checked & Theorem~\\ref{res:collapse}. \\\\\nExact rationality classification by integral tail differences & Lean-checked abstractly & Theorem~\\ref{res:classification}. \\\\\nExact irrationality normal form & Lean-checked abstractly & Corollary~\\ref{res:exact-irrational}. \\\\\nCofinal shift escape implies irrationality & Lean-checked abstractly & Corollary~\\ref{res:escape-irrational}. \\\\\nActual prime gaps are unbounded and not eventually periodic & Lean-checked & Proposition~\\ref{res:gap-nonperiodic}. \\\\\nAdjacent small-mismatch pair excludes simultaneous integrality & Lean-checked & Theorem~\\ref{res:smallpair}. \\\\\nConcrete prime-gap tail recurrence & Paper-proved from PNT & Not yet instantiated as a Lean infinite sum. \\\\\nRationality alone forces periodic integer coefficients & False & Section~\\ref{sec:carry}, Proposition~\\ref{res:telescope}. \\\\\nCofinal adjacent small-mismatch hypothesis & Proposed sufficient theorem & Problem~\\ref{prob:smallpair}; not proved. \\\\\n\\bottomrule\n\\end{tabular}\n\n\\smallskip\n\\begin{minipage}{0.94\\linewidth}\n\\footnotesize\n\\emph{How to read the middle column.}  \\emph{Proved here} marks a statement\nproved in the text; each such statement also carries a link to a Lean\ndeclaration where it appears below.\n\\emph{Lean-checked} marks a statement the pinned kernel accepts, in the exact\nsense fixed by the Status paragraph above.  The modifier\n\\emph{conditionally} marks a hypothesis that the Lean statement carries and\nthat ordinary mathematics discharges from outside the formal development;\n\\emph{abstractly} marks a statement proved for an arbitrary integer digit\nsequence rather than for the actual prime gaps.  \\emph{Paper-proved from PNT}\nmarks a statement proved in the text from the prime number theorem and not\nformalised.  \\emph{Proposed sufficient theorem} marks an unproved statement\nwhich, if proved, would give irrationality of $\\Pi$.\n\\end{minipage}\n\\end{table}\n\n\\medskip\n\\noindent\\textbf{Keywords.} irrationality; prime gaps; dyadic series;\nsummation by parts; Lean~4.\n\\qquad\n\\textbf{MSC 2020.} 11J72 (primary); 11N05, 68V20 (secondary).\n\n\\section{Summation by parts, with the endpoint retained}\\label{sec:parts}\n\nSummation by parts trades a sequence for its consecutive differences.  The form\nrecorded here is exact: it carries no error term, it assumes nothing about the\nsequence, and it retains the endpoint term rather than absorbing it into an\nestimate.  For a sequence $P$ of rational numbers and $n\\ge0$ put\n\\[\n D(P,n)=\\sum_{i=0}^{n-1}\\frac{P(i)}{2^{\\,i+1}},\n \\qquad\n \\Delta(P,n)=\\sum_{i=0}^{n-1}\\frac{P(i+1)-P(i)}{2^{\\,i+1}} ,\n\\]\nboth empty, hence zero, at $n=0$.  We call these the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{95}{dyadicPartialSumQ}{dyadic partial\nsum} and the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{115}{dyadicDifferencePartialSumQ}{dyadic\ndifference sum} of $P$.\n\n\\begin{proposition}[finite summation by parts]\\label{res:abel}\nFor every rational sequence $P$ and every $n\\ge0$,\n\\[\n D(P,n+1)=P(0)+\\Delta(P,n)-\\frac{P(n)}{2^{\\,n+1}} .\n\\]\n\\end{proposition}\n\n\\begin{proof}\nInduct on $n$.  At $n=0$ both sides equal $P(0)/2$.  For the step, adding\n$P(n+1)/2^{\\,n+2}$ to the left and\n$\\bigl(P(n+1)-P(n)\\bigr)/2^{\\,n+1}$ to the difference sum changes the endpoint\nterm from $P(n)/2^{\\,n+1}$ to $P(n+1)/2^{\\,n+2}$, and the two adjustments agree.\n\\end{proof}\n\n\\noindent Formalised as the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{132}{dyadicPartialSumQ_eq_start_add_differences}{summation-by-parts\nidentity}.  Nothing is assumed about $P$: no positivity, no monotonicity, and no\nconvergence.\n\nSpecialising to $P(i)=p_i$, whose first value is $p_0=2$, and writing\n$g_i=p_{i+1}-p_i$ for the zero-based gaps, gives the reformulation.\n\n\\begin{theorem}[prime-gap reformulation]\\label{res:parts}\nLet $p_0=2,p_1=3,\\ldots$ be the primes in increasing order and\n$g_i=p_{i+1}-p_i$.  For every $n\\ge0$,\n\\[\n \\sum_{i=0}^{n}\\frac{p_i}{2^{\\,i+1}}\n =2+\\sum_{i=0}^{n-1}\\frac{g_i}{2^{\\,i+1}}-\\frac{p_n}{2^{\\,n+1}} .\n\\]\n\\end{theorem}\n\n\\noindent Formalised as the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{161}{prime0_dyadic_summation_by_parts}{prime-gap\nsummation by parts}, using the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{155}{primeGapPartialSumQ}{gap partial\nsum} and the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{150}{primeGap0_cast}{gap cast\nidentity}; the latter records that the natural-number difference\n$p_{n+1}-p_n$ agrees with the difference taken in $\\Q$, which needs\n$p_n\\le p_{n+1}$, the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{146}{prime0_mono_step}{monotonicity\nstep}.  The leading $2$ is the first prime, not a normalising constant.\nAt $n=2$, for instance, the left side is $2/2+3/4+5/8=19/8$ and the right side\nis $2+(1/2+2/4)-5/8=19/8$.\n\nThe same numbers give a small model of the tail recurrence used below.  Extend\nthe finite list $(2,3,5)$ by the constant continuation $6,6,\\ldots$; this is a\ntoy sequence, not a prime sequence.  Its difference word is\n$(1,2,1,0,0,\\ldots)$.  For the rescaled difference tails of\nSection~\\ref{sec:tail},\n\\[\n T_0=\\frac22+\\frac1{2^2}=\\frac54,\n \\qquad T_1=\\frac12,\n \\qquad T_2=0,\n\\]\nand indeed $T_{N+1}=2T_N-g_{N+1}$.  The reduced denominators therefore run\n$4\\to2\\to1$: the first two doublings remove the dyadic part, after which the\norbit is integral.  This finite example is the denominator-collapse mechanism\nin its smallest visible form.\n\n\\subsection{The infinite identity and the irrationality equivalence}\\label{sec:infinite}\n\nWrite\n\\[\n u_n=\\frac{p_n}{2^{\\,n+1}},\\qquad\n v_n=\\frac{g_n}{2^{\\,n+1}}\n\\]\nfor the terms of the prime series and of the gap series, so that\n$\\sum_{n\\ge0}u_n=\\Pi$.  The termwise identity $v_n=2u_{n+1}-u_n$ is the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{191}{primeGapDyadicTerm_eq}{dyadic\ndiscrete derivative}.  It expresses each gap term as an integer combination of\ntwo consecutive prime terms, so once $(u_n)$ is summable the gap series can be\nsummed by rearranging two copies of the prime series, which is what the\nfollowing proof does.\n\n\\begin{theorem}[infinite prime-gap identity]\\label{res:infinite}\nLet $u_n=p_n/2^{\\,n+1}$ and $v_n=g_n/2^{\\,n+1}$.  If $(u_n)$ is summable, then\n$(v_n)$ is summable and\n\\[\n \\sum_{n\\ge0}u_n=2+\\sum_{n\\ge0}v_n .\n\\]\n\\end{theorem}\n\n\\begin{proof}\nThe shifted sequence $(u_{n+1})$ is summable.  Sum\n$v_n=2u_{n+1}-u_n$ and use $u_0=1$:\n\\[\n \\sum_{n\\ge0}v_n\n =2\\left(\\sum_{n\\ge0}u_n-u_0\\right)-\\sum_{n\\ge0}u_n\n =\\sum_{n\\ge0}u_n-2 .\n\\]\n\\end{proof}\n\n\\noindent The summability transfer is the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{202}{summable_primeGapDyadicTerm_of_summable_primeDyadicTerm}{gap-series\nsummability theorem}, and the displayed identity is the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{214}{tsum_primeDyadicTerm_eq_two_add_primeGap}{infinite\nprime-gap identity}.\n\n\\begin{corollary}[exact irrationality reformulation]\\label{res:irr-equivalence}\nWith $u_n$ and $v_n$ as in Theorem~\\ref{res:infinite}, and $(u_n)$ summable,\n\\[\n \\operatorname{Irr}\\!\\left(\\sum_{n\\ge0}u_n\\right)\n \\quad\\Longleftrightarrow\\quad\n \\operatorname{Irr}\\!\\left(\\sum_{n\\ge0}v_n\\right).\n\\]\nThe corresponding zero-based series with denominator $2^n$ is\n$4+2\\sum_{n\\ge0}v_n$ and has the same irrationality status.\n\\end{corollary}\n\n\\noindent These are the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{239}{irrational_tsum_primeDyadicTerm_iff_primeGap}{normalised\nirrationality equivalence}, the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{248}{tsum_primeDisplayedDyadicTerm_eq_four_add_two_primeGap}{displayed-series\nidentity}, and the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{263}{irrational_tsum_primeDisplayedDyadicTerm_iff_primeGap}{displayed-series\nirrationality equivalence}.\n\n\\paragraph{The summability hypothesis.}\nThe prime number theorem, proved by Hadamard and by de la Vall\\'ee Poussin in\n1896, is proved quantitatively by Montgomery and Vaughan in\nTheorem~6.9~\\cite[pp.~179--181]{mv2007}.  Their\nExercise~6.2.5~\\cite[p.~183]{mv2007} then gives the sharper expansion\n\\[\n p_n=n\\left(\n \\log n+\\log\\log n-1+\n \\frac{\\log\\log n-2}{\\log n}\n +O\\!\\left(\\frac{(\\log\\log n)^2}{(\\log n)^2}\\right)\\right).\n\\]\nIn particular $p_n\\sim n\\log n$, so $p_n/2^n$ is summable by comparison\nwith any fixed exponential $\\rho^n$ with $1<\\rho<2$.\nConsequently Theorem~\\ref{res:infinite} and\nCorollary~\\ref{res:irr-equivalence} are unconditional paper-level consequences\nof a classical external theorem.  The local Lean development does not\nformalise the prime number theorem, so its declarations retain summability as\nan explicit hypothesis.  This is a gap in the formalisation, not in the\nmathematics: the open content is exactly irrationality of the prime-gap\nseries.  Concretely, $\\Pi=3.674643966\\ldots$ is irrational if and only if\n$\\sum_{n\\ge0}g_n2^{-(n+1)}=1.674643966\\ldots$ is, and neither is known.\n\n\\section{The tail recurrence and integral shifts}\\label{sec:tail}\n\nThe series is not attacked directly.  Suppose $\\sum_{i\\ge0}a_i2^{-(i+1)}$\nconverges, with every $a_i$ an integer, and rescale its tails by putting\n\\[\n T_N=2^{\\,N+1}\\sum_{i>N}\\frac{a_i}{2^{\\,i+1}}\n    =\\sum_{j\\ge1}\\frac{a_{N+j}}{2^{\\,j}} .\n\\]\nTwo facts follow immediately.  First $T_{N+1}=2T_N-a_{N+1}$, so moving one\nlevel along doubles the rescaled tail and subtracts a single coefficient.\nSecond $T_0=2\\sum_{i\\ge0}a_i2^{-(i+1)}-a_0$, so the sum is rational exactly\nwhen $T_0$ is.  The whole of Section~\\ref{sec:tail} therefore studies that\nrecurrence in isolation, assuming nothing about the coefficients beyond the\nfact that they are integers; the prime-gap instance is resumed at the end of\nthe section.  The following definition is the object so obtained, stripped of\nits origin.\n\n\\begin{definition}\\label{def:rec}\nLet $g:\\N\\to\\Z$ and $T:\\N\\to\\Q$.  Say $T$ satisfies the \\emph{dyadic tail\nrecurrence} with \\emph{digits} $g$ when\n\\[\n T_{N+1}=2T_N-g_{N+1}\\qquad\\text{for every }N .\n\\]\nWe call the sequence $(T_N)_{N\\ge0}$ the \\emph{orbit} of $T_0$ under $g$,\nwrite $\\sigma_h(N)=T_{N+h}-T_N$ for the \\emph{shift} of length $h$ at $N$, and\ncall a rational number \\emph{integral} when it is the image of an integer.\n\\end{definition}\n\n\\noindent These are the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{278}{DyadicTailRecurrence}{tail\nrecurrence}, the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{282}{tailShift}{shift}, and\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{298}{RatIntegral}{integrality}.\nThe digits are arbitrary integers.  In the intended instance they are the prime\ngaps and $T_N$ is the scaled tail of $\\Pi$ after level $N$, but that\ninstantiation needs a summability argument and is not made here; every\nstatement below is a theorem about Definition~\\ref{def:rec}.\n\nOne small orbit, referred to again below, is worth having in view.  Take every\ndigit $g_N=0$ and $T_0=1/12$.  Then $T_{N+1}=2T_N$, so the orbit is\n$\\tfrac1{12},\\tfrac16,\\tfrac13,\\tfrac23,\\tfrac43,\\ldots$, and\n$\\sigma_h(N)=(2^{h}-1)2^{\\,N}/12$.  The shift of length $2$ is not integral at\n$N=0$, where $\\sigma_2(0)=\\tfrac13-\\tfrac1{12}=\\tfrac14$, and is integral at\n$N=2$, where $\\sigma_2(2)=\\tfrac43-\\tfrac13=1$; the shift of length $1$ is\n$\\sigma_1(N)=2^{\\,N}/12$ and is never integral.  The change at $N=2$ is\naccounted for by Theorem~\\ref{res:collapse}, and the failure at $h=1$ by\nTheorem~\\ref{res:shiftiff}.\n\nIterating the recurrence $h$ times multiplies $T_N$ by $2^{h}$ and accumulates\nan explicit integer, which we now name.  Define\n$B_{0,N}=0$ and $B_{h+1,N}=2B_{h,N}+g_{N+h+1}$, so that\n$B_{h,N}=g_{N+1}2^{\\,h-1}+\\cdots+g_{N+h}$: the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{304}{dyadicTailBlock}{tail block}.\nThus $B_{1,N}=g_{N+1}$, $B_{2,N}=2g_{N+1}+g_{N+2}$ and\n$B_{3,N}=4g_{N+1}+2g_{N+2}+g_{N+3}$: the block puts the weights\n$2^{\\,h-1},\\ldots,2^{0}$ on the $h$ digits following index $N$.\n\n\\begin{theorem}[block identity]\\label{res:block}\nLet $T:\\N\\to\\Q$ satisfy the dyadic tail recurrence with integer digits $g$, and\nlet $B_{h,N}$ be as above.  For every $N$ and $h$,\n\\[\n T_{N+h}=2^{h}T_N-B_{h,N},\n \\qquad\\text{hence}\\qquad\n \\sigma_h(N)=(2^{h}-1)\\,T_N-B_{h,N} .\n\\]\n\\end{theorem}\n\n\\begin{proof}\nInduct on $h$; the step is one application of the recurrence together with\n$2\\cdot2^{h}=2^{h+1}$ and the recursion defining $B$.  The second identity is\nthe first minus $T_N$.\n\\end{proof}\n\n\\noindent Formalised as the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{310}{tail_iterate_eq_pow_mul_sub_block}{iterated\nblock identity} and the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{324}{tailShift_eq_scaled_sub_block}{scaled\nshift identity}.  The shift also obeys the recurrence in its own right,\n$\\sigma_h(N+1)=2\\sigma_h(N)-(g_{N+h+1}-g_{N+1})$, the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{286}{tailShift_succ}{shift step\nidentity}.\n\nSince $B_{h,N}$ is an integer, Theorem~\\ref{res:block} converts a question about\nthe shift into a question about the single scaled term $(2^{h}-1)T_N$.\n\n\\begin{theorem}[integral-shift criterion]\\label{res:shiftiff}\nLet $T:\\N\\to\\Q$ satisfy the dyadic tail recurrence with integer digits $g$.\nFor every $N$ and $h$, the shift $\\sigma_h(N)$ is integral if and only if\n$(2^{h}-1)T_N$ is integral.\n\\end{theorem}\n\n\\begin{proof}\nBy Theorem~\\ref{res:block} the two differ by the integer $B_{h,N}$, and\nsubtracting an integer does not change integrality.\n\\end{proof}\n\n\\noindent Formalised as the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{380}{tailShift_integral_iff_scaledTail}{integral-shift\ncriterion}, on the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{334}{ratIntegral_sub_int_iff}{integer-shift\ninvariance}.  This is an equivalence, not a one-way reduction: the block\ncarries no information about integrality, so the two conditions are the same\ncondition.  Informally, once $T_N$ is fixed no choice of the digits after index\n$N$ can change whether the shift $\\sigma_h(N)$ is an integer; that is decided\nby the denominator of $T_N$ alone.\n\n\\begin{theorem}[a shift of totient length]\\label{res:totient}\nLet $T:\\N\\to\\Q$ satisfy the dyadic tail recurrence with integer digits $g$, and\nlet $\\ph$ be Euler's totient function.  If the reduced denominator $d$ of\n$T_N$ is odd, then $\\sigma_{\\ph(d)}(N)$ is integral.\n\\end{theorem}\n\n\\begin{proof}\nSince $d$ is odd, $2$ and $d$ are coprime, so Euler's congruence gives\n$2^{\\ph(d)}\\equiv1\\pmod d$, that is $d\\mid 2^{\\ph(d)}-1$.  Writing\n$2^{\\ph(d)}-1=dk$ and $T_N=u/d$ in lowest terms, $(2^{\\ph(d)}-1)T_N=ku$ is an\ninteger, and Theorem~\\ref{res:shiftiff} transfers this to the shift.\n\\end{proof}\n\n\\noindent Formalised as the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{391}{tailShift_integral_totient_of_odd_den}{totient\nshift} and the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{352}{ratIntegral_totientMultiplier_of_odd_den}{Euler\nmultiplier}.  For example, if $\\operatorname{den}T_N=3$ then $\\ph(3)=2$ and\n$(2^{2}-1)T_N=3T_N$ is an integer, so $\\sigma_2(N)$ is integral, while\n$(2^{1}-1)T_N=T_N$ is not; if $\\operatorname{den}T_N=5$ then $\\ph(5)=4$ and\n$\\sigma_4(N)$ is integral.\n\nThe hypothesis is a genuine restriction: the argument uses\ncoprimality of $2$ with the denominator, and the even part of a denominator is\nexactly what the doubling in the recurrence acts on.  It cannot be dropped.\nIf $T_N=1/2$ then $2^{h}-1$ is odd for every $h\\ge1$, so $(2^{h}-1)T_N$ is\nnever an integer and, by Theorem~\\ref{res:shiftiff}, no shift at $N$ is\nintegral.\n\n\\begin{theorem}[propagation]\\label{res:propagate}\nLet $T:\\N\\to\\Q$ satisfy the dyadic tail recurrence with integer digits $g$, and\nfix $h$ and $N$.  If $\\sigma_h(N)$ is integral, then $\\sigma_h(N+k)$ is\nintegral for every $k\\ge0$.\n\\end{theorem}\n\n\\begin{proof}\nBy the shift step identity, $\\sigma_h(N+1)=2\\sigma_h(N)-(g_{N+h+1}-g_{N+1})$ is\nan integer combination of an integer and two digits; induct on $k$.\n\\end{proof}\n\n\\noindent Formalised as the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{402}{tailShift_integral_succ}{one-step\npropagation} and the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{414}{tailShift_integral_add}{propagation\nto every later index}.\n\nThe three preceding theorems combine as follows, and this is the statement the\nrest of the note rests on.  The special case is immediate: if\n$\\operatorname{den}T_0$ is already odd, then Theorem~\\ref{res:totient} at\n$N=0$ makes the shift of length $\\ph(\\operatorname{den}T_0)$ integral and\nTheorem~\\ref{res:propagate} keeps it integral at every later index, so one may\ntake $N_0=0$.  In general a denominator carries a power of two as well, and the\nkey point is that the doubling in the recurrence annihilates\nexactly the $2$-adic part of a denominator, and nothing else: after finitely\nmany steps the orbit therefore reaches a term with odd reduced denominator,\nwhich is precisely the situation Theorem~\\ref{res:totient} handles.  No\ncontrol of the digits is needed anywhere.\n\n\\begin{theorem}[denominator collapse]\\label{res:collapse}\nLet $T:\\N\\to\\Q$ satisfy the dyadic tail recurrence with integer digits $g$\n(Definition~\\ref{def:rec}).  Then some fixed positive shift is integral from\nsome point onwards:\n\\[\n \\exists h\\ge1\\ \\exists N_0\\ \\forall N\\ge N_0,\\qquad\n \\sigma_h(N)\\in\\Z .\n\\]\n\\end{theorem}\n\n\\begin{proof}\nWrite the reduced denominator of $T_0$ as $2^s m$ with $m$ odd.  The block\nidentity gives $T_s=2^sT_0-B_{s,0}$.  Multiplying by $2^s$ clears the entire\npower of two from the denominator, and subtracting the integer $B_{s,0}$\ncannot reintroduce one, so the reduced denominator of $T_s$ divides $m$ and is\nin particular odd.  Theorem~\\ref{res:totient}, applied at index $s$, supplies\nthe positive shift $h=\\ph(\\operatorname{den}T_s)$, and\nTheorem~\\ref{res:propagate} keeps that shift integral at every later index.\n\\end{proof}\n\nIn the orbit displayed after Definition~\\ref{def:rec} the proof runs as\nfollows: $\\operatorname{den}T_0=12=2^{2}\\cdot3$, so $s=2$, the orbit reaches\n$T_2=1/3$ with odd denominator, and $h=\\ph(3)=2$.  That is exactly the shift\nlength seen to be integral there from index $2$ onwards, and no shorter one\nworks.\n\nThree features of the argument are used later.  The proof may take as its\nnumber of preparatory steps the $2$-adic valuation of\n$\\operatorname{den}T_0$.  The resulting shift length $h$ is the totient of the\nodd denominator reached from a hypothetical rational initial value, and is\nnot known in advance for the prime-gap orbit.  This is why this argument\nrequires Problem~\\ref{prob:escape} for every $h$, rather than for one\npreassigned shift length.  Finally, the digits enter only through the integer\n$B_{s,0}$, so the conclusion holds for an arbitrary integer digit sequence.\n\n\\noindent The factorisation and cancellation are the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{426}{exists_twoPow_mul_odd_den}{odd-denominator\ndoubling lemma}; the orbit form is the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{461}{exists_odd_den_state}{odd-denominator\nstate theorem}; and the quantified conclusion is the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{477}{exists_eventually_integral_tailShift}{eventual\nintegral-shift theorem}.\n\nThe denominator-collapse theorem has a converse once real-valued orbits are\nadmitted; the digits remain integers.\n\n\\begin{theorem}[exact rationality classification]\\label{res:classification}\nLet $T:\\N\\to\\R$ satisfy $T_{N+1}=2T_N-g_{N+1}$ with integer $g$.  Then\n\\[\n T_0\\in\\Q\n \\quad\\Longleftrightarrow\\quad\n \\exists h\\ge1\\ \\exists N\\quad \\sigma_h(N)\\in\\Z\n \\quad\\Longleftrightarrow\\quad\n \\exists h\\ge1\\ \\exists N_0\\ \\forall N\\ge N_0\\quad\n \\sigma_h(N)\\in\\Z .\n\\]\n\\end{theorem}\n\n\\begin{proof}\nIf $T_0$ is rational, the recurrence keeps every $T_N$ rational, and\nTheorem~\\ref{res:collapse} gives the final condition.  That condition plainly\ngives one integral shift.  Conversely, suppose $h\\ge1$ and\n$\\sigma_h(N)=z\\in\\Z$.  The real block identity gives\n\\[\n T_N=\\frac{z+B_{h,N}}{2^h-1}\\in\\Q ,\n\\]\nbecause $2^h-1\\ne0$.  Applying the block identity once more, now from $0$ to\n$N$, recovers $T_0=(T_N+B_{N,0})/2^N\\in\\Q$.\n\\end{proof}\n\n\\noindent The real block identity and its scaled-difference form are\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{786}{real_tail_iterate_eq_pow_mul_sub_block}{formalised here}\nand\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{800}{realTailShift_eq_scaled_sub_block}{here}.\nThe two rationality equivalences are the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{812}{not_irrational_initial_iff_exists_integral_positive_tailShift}{single-shift classifier}\nand the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{858}{not_irrational_initial_iff_exists_eventually_integral_positive_tailShift}{eventual classifier}.\n\n\\begin{corollary}[exact irrationality normal form]\\label{res:exact-irrational}\nUnder the hypotheses of Theorem~\\ref{res:classification},\n\\[\n T_0\\notin\\Q\n \\quad\\Longleftrightarrow\\quad\n \\forall h\\ge1\\ \\forall N,\\qquad \\sigma_h(N)\\notin\\Z .\n\\]\n\\end{corollary}\n\n\\begin{proof}\nNegate the middle condition in Theorem~\\ref{res:classification}.\n\\end{proof}\n\n\\noindent Lean checks this equivalence as\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{884}{irrational_initial_iff_all_positive_tailShifts_nonintegral}{the exact irrationality normal form}.\n\nFor a weaker but often more usable hypothesis, call the shifts\n\\emph{cofinally non-integral} when, for every fixed $h\\ge1$ and every threshold\n$N_0$, some $N\\ge N_0$ has $\\sigma_h(N)\\notin\\Z$.  Propagation shows that one\nintegral shift would make every later shift of the same length integral, so\ncofinal non-integrality rules out the rational case.\n\n\\begin{corollary}[cofinal shift escape implies irrationality]\\label{res:escape-irrational}\nLet $T:\\N\\to\\R$ satisfy $T_{N+1}=2T_N-g_{N+1}$ with integer $g$.  If its\nshifts are cofinally non-integral, then $T_0$ is irrational.\n\\end{corollary}\n\n\\begin{proof}\nIf $T_0$ were rational, the whole real orbit would be the image in $\\R$ of the\nrational orbit with that initial value.  Theorem~\\ref{res:collapse} would then produce\none $h\\ge1$ whose shifts are integral at every sufficiently large index,\ncontradicting cofinal non-integrality.\n\\end{proof}\n\n\\noindent Lean checks the real recurrence and shift definitions as\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{492}{RealDyadicTailRecurrence}{real\ndyadic recurrence} and\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{496}{realTailShift}{real shift}, the\nrational-orbit transport as the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{515}{realTail_eq_ratCast_rationalDyadicOrbit}{rational-orbit\ncast theorem}, cofinal non-integrality as the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{528}{CofinalNonintegralTailShifts}{cofinal\nshift-escape predicate}, and Theorem~\\ref{res:escape-irrational} as the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{537}{irrational_initial_of_cofinalNonintegralTailShifts}{cofinal-escape\nirrationality theorem}.\n\n\\paragraph{The actual prime-gap orbit.}\nDefine, at paper level,\n\\[\n \\mathcal T_N=\\sum_{j\\ge1}\\frac{g_{N+j}}{2^j}.\n\\]\nThe prime number theorem gives convergence, and an index shift gives\n\\[\n \\mathcal T_{N+1}=2\\mathcal T_N-g_{N+1}.\n\\]\nIf $G=\\sum_{n\\ge0}g_n/2^{n+1}$, then\n$\\mathcal T_0=2G-g_0=2G-1=2.349287932\\ldots$.  Together with\nTheorem~\\ref{res:infinite}, this shows that $\\Pi$, $G$, and\n$\\mathcal T_0$ have the same rationality status.  Corollary~\\ref{res:exact-irrational}\ntherefore identifies the problem exactly: one must show that\n$\\mathcal T_{N+h}-\\mathcal T_N$ is non-integral for every $h\\ge1$ and every\n$N$.  The weaker cofinal hypothesis in Corollary~\\ref{res:escape-irrational}\nis a sufficient route because integrality propagates.  This analytic\ninstantiation is proved here from the classical prime number theorem; it is\nnot yet a Lean declaration.  The abstract recurrence theorems are\nkernel-checked.\n\n\\subsection{Two adjacent small shifts cannot both be integral}\\label{sec:local-certificate}\n\nApplying Theorem~\\ref{res:escape-irrational} requires cofinal non-integrality, which is\na condition of infinite precision imposed on a complete tail.  The key\npoint is that inside the open interval $(-1,1)$ integrality is equality with\nzero, so on that range the one-step shift recurrence\n\\[\n \\sigma_h(N+1)=2\\sigma_h(N)-\\bigl(g_{N+h+1}-g_{N+1}\\bigr)\n\\]\nturns simultaneous integrality of two adjacent shifts into a single comparison\nof digits.  What this buys is a \\emph{certificate}: a condition attached to a\nsingle pair of adjacent indices, whose verification already contradicts\nintegrality at that pair.  The point of arranging the argument this\nway is that the distance of a shift from the integers never has to be\nestimated; it suffices to know that both shifts lie in $(-1,1)$ and that two\ndigits differ.\n\n\\begin{theorem}[adjacent small-shift obstruction]\\label{res:smallpair}\nLet $T:\\N\\to\\Q$ satisfy the dyadic tail recurrence with integer digits $g$.\nFix $h$ and $N$.  If\n\\[\n -1<\\sigma_h(N)<1,\\qquad -1<\\sigma_h(N+1)<1,\n \\qquad g_{N+h+1}\\ne g_{N+1},\n\\]\nthen $\\sigma_h(N)$ and $\\sigma_h(N+1)$ cannot both be integral.\nConsequently, if such a pair occurs beyond every threshold, the $h$-shift is\nnot eventually integral.\n\\end{theorem}\n\n\\begin{proof}\nAn integral rational strictly between $-1$ and $1$ is zero.  If both shifts\nwere integral, both would therefore vanish, and substitution in the displayed\nshift step identity would give $g_{N+h+1}=g_{N+1}$, a contradiction.  The\ncofinal statement chooses one such adjacent pair after the alleged onset of\nintegrality.\n\\end{proof}\n\n\\noindent The finite contradiction is the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{626}{tailShift_not_both_integral_of_small_pair_of_digit_ne}{adjacent\nsmall-shift obstruction}; its quantified form is the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{653}{not_eventuallyIntegralTailShift_of_cofinal_small_mismatch}{cofinal\nsmall-mismatch theorem}; and the actual-prime-gap specialisation is the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{723}{primeGapTailShift_not_eventuallyIntegral_of_cofinal_small_mismatch}{prime-gap\nspecialisation}.  Theorem~\\ref{res:smallpair} is conditional on its two tail\ninequalities; no theorem asserting that such pairs occur is claimed here.\n\nAll three are stated for a rational orbit, while the prime-gap tail\n$\\mathcal T$ of Section~\\ref{sec:tail} is real, so the rational statement does\nnot on its own discharge the instances that arise downstream.  The same\nargument gives the real form directly.\n\n\\begin{corollary}[adjacent small-shift obstruction, real form]\\label{res:smallpair-real}\nLet $T:\\N\\to\\R$ satisfy $T_{N+1}=2T_N-g_{N+1}$ with integer digits $g$, and\nwrite $\\sigma_h(N)=T_{N+h}-T_N$.  Fix $h$ and $N$.  If\n\\[\n -1<\\sigma_h(N)<1,\\qquad -1<\\sigma_h(N+1)<1,\n \\qquad g_{N+h+1}\\ne g_{N+1},\n\\]\nthen $\\sigma_h(N)$ and $\\sigma_h(N+1)$ do not both lie in $\\Z$.\nConsequently, if such a pair occurs beyond every threshold, the $h$-shift is\nnot eventually integral.\n\\end{corollary}\n\n\\begin{proof}\nThe real recurrence gives the same shift step identity\n$\\sigma_h(N+1)=2\\sigma_h(N)-\\bigl(g_{N+h+1}-g_{N+1}\\bigr)$, and an integer\nstrictly between $-1$ and $1$ is zero.  If both shifts lay in $\\Z$, both would\ntherefore vanish, and substitution in that identity would give\n$g_{N+h+1}=g_{N+1}$, a contradiction.  The cofinal statement chooses one such\nadjacent pair after the alleged onset of integrality.\n\\end{proof}\n\n\\noindent Corollary~\\ref{res:smallpair-real} is proved here and is not a\ndeclaration in the pinned Lean module, whose small-shift theorems are stated\nfor the rational orbit.  It is the form that applies to the real tail, and it\ncarries the same two unproved inequalities as its rational counterpart.\n\nThe third hypothesis, on its own, is available.  For the actual gaps tabulated\nin Section~\\ref{sec:problem} it reads $g_4=2\\ne4=g_3$ at $h=1$, $N=2$; and\nProposition~\\ref{res:gap-nonperiodic} below says precisely that for each fixed\n$h\\ge1$ the inequality $g_{N+h+1}\\ne g_{N+1}$ holds for arbitrarily large $N$,\nsince its failure from some index onwards is eventual periodicity with period\n$h$.  What is missing is the pair of inequalities, each of which constrains a\ncomplete infinite tail.\n\n\\begin{proposition}[prime gaps do not become periodic]\\label{res:gap-nonperiodic}\nFor every positive $h$, the actual consecutive-prime-gap sequence is not\neventually periodic with period $h$.\n\\end{proposition}\n\n\\begin{proof}\nThe gaps are unbounded, by the standard construction: the interval from $n!+2$\nto $n!+n$ contains no prime.  Far stronger lower bounds for large gaps are\nknown~\\cite[Theorem~1, p.~66]{fgkmt2018}, but unboundedness is all that is\nneeded.  An\neventually periodic natural-valued sequence has finite range after its\npreperiod, while its finite initial segment is bounded as well; hence it is\nbounded, a contradiction.\n\\end{proof}\n\n\\noindent Lean checks the factorial argument as\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{51}{exists_primeGap0_gt}{unboundedness\nof the actual gaps} and the conclusion as\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{670}{primeGap0_not_eventually_periodic}{non-eventual\nperiodicity}.  Combining nonperiodicity with eventual strict smallness of one\npositive shift would also exclude eventual integrality, but eventual\nsmallness at every sufficiently large index is stronger than the cofinal\nadjacent-pair hypothesis in Theorem~\\ref{res:smallpair} and is not asserted\nhere.\n\n\\section{Nonperiodic coefficients with a rational sum}\\label{sec:carry}\n\nProposition~\\ref{res:gap-nonperiodic} shows that the prime gaps are not\neventually periodic.  One might therefore hope that rationality of a dyadic\nseries forces its integer coefficients to be eventually periodic, and play the\ntwo against each other.  The hoped-for implication is false, and a single\nexplicit sequence refutes it.\n\nThe construction runs the emission of coefficients backwards.  Let\n$K:\\N\\to\\Q$ be arbitrary, read $K_n$ as the value carried into level $n$, and\nput $\\kappa_n=2K_n-K_{n+1}$: the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{738}{carryCoeff}{carry coefficient}.\nThat definition is exactly the statement that\n\\[\n \\frac{K_n}{2^{\\,n}}\n =\\frac{\\kappa_n}{2^{\\,n+1}}+\\frac{K_{n+1}}{2^{\\,n+1}} ,\n\\]\nso at each level the carried value splits into one emitted coefficient and a\nnew carry.  Nothing at all is assumed about $K$: not integrality, not\npositivity, not any bound.  That is the sense in which the carry is free, and\nit is what the counterexample below exploits.\n\n\\begin{proposition}[exact telescoping]\\label{res:telescope}\nLet $K:\\N\\to\\Q$ be arbitrary and $\\kappa_n=2K_n-K_{n+1}$.  For every $n\\ge0$,\n\\[\n \\sum_{i=0}^{n-1}\\frac{\\kappa_i}{2^{\\,i+1}}=K_0-\\frac{K_n}{2^{\\,n}} .\n\\]\n\\end{proposition}\n\n\\begin{proof}\nThe summands telescope, since the term added at stage $n$ is\n$(2K_n-K_{n+1})/2^{\\,n+1}=K_n/2^{\\,n}-K_{n+1}/2^{\\,n+1}$.\n\\end{proof}\n\n\\noindent Formalised as the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{747}{carryPartialSum_eq}{carry\ntelescoping identity}, on the\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{742}{carryPartialSum}{carry partial\nsum}.  When $K$ takes natural-number values and $K_{n+1}\\le2K_n$ for every $n$,\nthe coefficients $\\kappa_n$ are natural numbers as well, and the\nnatural-number and rational readings of the definition agree under the cast,\nthe\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{763}{natCarryCoeff_cast}{natural-carry\ncast}.\n\n\\paragraph{Consequence for the coefficients.}\nIf $K_n2^{-n}\\to0$, the emitted partial sums converge to $K_0$.  For the\nexplicit positive choice $K_n=n+2$ one has\n\\[\n \\kappa_n=n+1,\\qquad\n \\sum_{i=0}^{n-1}\\frac{\\kappa_i}{2^{\\,i+1}}\n   =2-\\frac{n+2}{2^n}\\longrightarrow2,\n\\]\nthat is, the first values are\n\\[\n \\begin{array}{c|ccccccc}\n  n & 0&1&2&3&4&5&6\\\\\\hline\n  \\kappa_n & 1&2&3&4&5&6&7\\\\[2pt]\n  \\displaystyle\\sum_{i<n}\\frac{\\kappa_i}{2^{\\,i+1}}\n    & 0&\\tfrac12&1&\\tfrac{11}{8}&\\tfrac{13}{8}&\\tfrac{57}{32}&\\tfrac{15}{8}\n \\end{array}\n\\]\nconverging to $2$.\nThe sequence $(\\kappa_n)$ is unbounded and therefore not eventually periodic,\nalthough every coefficient is positive and its dyadic sum is rational.  Thus\nrationality alone cannot imply eventual periodicity even for positive integer\ncoefficient sequences.\nInformally, an arbitrarily large value may be carried forward from level to\nlevel and still disappear in the limit, so rationality of a dyadic sum is\ncompatible with arbitrarily large integer coefficients.  This\ndoes not classify which sequences occur, and it does not show that a prescribed\nprime-gap pattern is emitted by a bounded carry.  Here\n$K_{n+1}=n+3\\le2n+4=2K_n$, so the natural-number construction above applies\nwithout a sign exception.\n\n\\section{Complements and further questions}\\label{sec:open}\n\nProblem~\\#251 is open.  The first statement below is the exact tail-difference\nreformulation supplied by Theorem~\\ref{res:classification}; the second is a\nstronger, local sufficient condition designed to prove it.  Both ask for\nindices beyond every threshold because a single non-integral shift, however\nfar out, is compatible with rationality.  Throughout,\n$g_i=p_{i+1}-p_i$ are the zero-based prime gaps of\nSection~\\ref{sec:problem}.\n\n\\begin{problem}[exact prime-gap shift escape]\\label{prob:escape}\nFor every $h\\ge1$ and every $N_0$, prove that some $N\\ge N_0$ satisfies\n\\[\n \\sum_{j\\ge1}\n \\frac{g_{N+h+j}-g_{N+j}}{2^j}\\notin\\Z .\n\\tag{5.1}\\label{eq:shift-escape}\n\\]\n\\end{problem}\n\nThe series in~\\eqref{eq:shift-escape} is exactly\n$\\mathcal T_{N+h}-\\mathcal T_N$.  Because integrality propagates forward,\ncofinal non-integrality is equivalent to non-integrality at every index;\nCorollary~\\ref{res:exact-irrational} therefore makes\nProblem~\\ref{prob:escape} equivalent to irrationality of $\\Pi$.  The\nquantifier over $h$ is essential: a hypothetical rational value chooses a\nshift length from the odd part of its reduced denominator, and that length is\nnot known in advance.\n\n\\begin{problem}[cofinal adjacent small mismatch]\\label{prob:smallpair}\nFor every fixed $h\\ge1$ and every $N_0$, prove that some $N\\ge N_0$ satisfies\n\\[\n\\begin{split}\n\\left|\\sum_{j\\ge1}\n \\frac{g_{N+h+j}-g_{N+j}}{2^j}\\right|&<1,\\\\\n\\left|\\sum_{j\\ge1}\n \\frac{g_{N+h+1+j}-g_{N+1+j}}{2^j}\\right|&<1,\\\\\ng_{N+h+1}&\\ne g_{N+1}.\n\\end{split}\n\\tag{5.2}\\label{eq:smallpair}\n\\]\n\\end{problem}\n\nThe two sums are adjacent $h$-shifts.  Corollary~\\ref{res:smallpair-real} turns\neach instance of~\\eqref{eq:smallpair} into a finite contradiction to\nsimultaneous integrality; Theorem~\\ref{res:smallpair} is the Lean-checked\nrational case.  A cofinal family of such pairs therefore proves\nProblem~\\ref{prob:escape}; Theorem~\\ref{res:escape-irrational} then gives irrationality\nof the gap series, and Corollary~\\ref{res:irr-equivalence} transfers it to\n$\\Pi$.  This formulation deliberately asks only for sporadic adjacent pairs;\nthe stronger assertion that a fixed shift is eventually always smaller than\none is unnecessary.\n\nSeveral natural prime-distribution inputs are insufficient.  Isolated\nsmall gaps, isolated large gaps, average gap estimates, and the occurrence of\nany one fixed finite pattern do not suffice: both inequalities in\n\\eqref{eq:smallpair} contain the complete infinite continuation.  Nor\ndoes parity: after the first gap all $g_n$ are even.  Unboundedness and\nnon-eventual-periodicity of the actual gaps, though now checked, do not imply\nthe required small-tail recurrence.  In particular, Section~\\ref{sec:carry}\nshows that no argument can deduce eventual periodicity from rationality alone.\n\n\\paragraph{A finite truncation certificate.}\nBoth problems above concern infinite tails.  A dominated truncation supplies\na finite certificate for one instance of the exact criterion; it is neither a\nreplacement for the quantifiers in Problem~\\ref{prob:escape} nor another\nequivalent formulation.  For $L\\ge1$ put\n\\[\n S_{h,N,L}=\\sum_{j=1}^{L}\n \\frac{g_{N+h+j}-g_{N+j}}{2^j},\n\\]\nthe truncation of the series in~\\eqref{eq:shift-escape} after $L$ terms.  This\nis a finite sum of gap differences and can be computed; the point of the\nfollowing proposition is to say how far from an integer it must be before the\ndiscarded tail is irrelevant.\n\n\\begin{proposition}[finite truncation]\\label{res:truncation}\nLet $M(n)\\ge g_n$ for every $n$, assume that the series below converges, and\nput\n\\[\n R_{h,N,L}(M)=\n \\sum_{j>L}\\frac{M(N+h+j)+M(N+j)}{2^j}.\n\\]\nSuppose that for every fixed $h\\ge1$ and every $N_0$ there exist $N\\ge N_0$ and\n$L\\ge1$ with\n\\[\n \\operatorname{dist}(S_{h,N,L},\\Z)>R_{h,N,L}(M).\n\\tag{5.3}\\label{eq:truncation}\n\\]\nThen Problem~\\ref{prob:escape} holds.\n\\end{proposition}\n\n\\begin{proof}\nThe part of $\\sum_{j\\ge1}(g_{N+h+j}-g_{N+j})2^{-j}$ omitted from $S_{h,N,L}$\nhas absolute value at most $R_{h,N,L}(M)$, so under~\\eqref{eq:truncation} the\nfull sum lies at positive distance from every integer.\n\\end{proof}\n\n\\noindent The last distance-to-integers inference is Lean-checked as the\ngeneric\n\\lword{Erdos251/PrimeGapDyadicTail.lean}{772}{not_ratIntegral_of_approximation_gap}{finite-approximation\ncertificate}: an error at most $R$ cannot reach an integer when the\napproximation is farther than $R$ from every integer.  For instance, if\n$S_{h,N,L}=2/5$ and $R_{h,N,L}(M)=1/20$, the full sum lies within $1/20$ of\n$2/5$ and so at distance at least $7/20$ from every integer, which settles\n\\eqref{eq:shift-escape} at that $N$.  The prime-gap tail\nbound, the convergence used above, and the existence of blocks satisfying\n\\eqref{eq:truncation} are not consequences of that formal theorem.\n\n\\noindent Taking the classical bound $M(n)\\ll n\\log n$, a choice\n$L=\\lceil A\\log_2(N+h+2)\\rceil$ with any fixed $A>1$ makes the right side a\nnegative power of $N$ up to logarithms.  Thus~\\eqref{eq:truncation} asks for a finite\ndyadic anti-concentration estimate on a logarithmic-length block, not control\nof an infinite tail and not eventual periodicity of the full gap sequence.\nFor Problem~\\ref{prob:smallpair}, the same truncation must certify two adjacent\nfull-tail values inside the open unit interval, together with the displayed\ngap mismatch.  A finite prefix is useful only when its omitted tail is\nrigorously dominated.\n\nWhat~\\eqref{eq:truncation} requires is joint control of the finite block of\nweighted differences $(g_{N+h+1}-g_{N+1},\\ldots,g_{N+h+L}-g_{N+L})$ modulo\npowers of two, along a block of logarithmic length.  We do not know how to obtain such control and\nwe make no progress on it here.  The strongest results on prime gaps address a\ndifferent shape of question.  Zhang's bounded-gap theorem\n\\cite[Theorem~1, p.~1122]{zhang2014} produces infinitely many bounded\nconsecutive-prime gaps.\nMaynard proves substantially more than an individual-gap statement:\n\\cite[Theorem~1.1, p.~384]{maynard2015} bounds\n$\\liminf_n(p_{n+m}-p_n)$ for every fixed $m$, and hence gives bounded clusters\nof every fixed size; \\cite[Theorem~1.3, p.~385]{maynard2015} gives the explicit\nunconditional bound $\\liminf_n(p_{n+1}-p_n)\\le 600$.  The large-gap theorem of\nFord, Green, Konyagin, Maynard and Tao\n\\cite[Theorem~1, p.~66]{fgkmt2018} bounds the largest single consecutive-prime\ngap below $X$.  None of these results supplies the joint dyadic distribution\nof a logarithmic block of consecutive gap differences required by\n\\eqref{eq:truncation}.  The same introduction notes a separate sequel on\nchains of large gaps; the cited theorem itself supplies no residue-sensitive\nblock estimate of the kind needed here.\n\n\\paragraph{What remains to be formalised.}\nThe exact abstract classifier and the prime-specific local\nsmall-mismatch theorem are Lean-checked, as is the generic\nfinite-approximation certificate used in Proposition~\\ref{res:truncation}.\nThe local source still takes\nsummability of the prime series as a hypothesis and does not define\n$\\mathcal T_N$ by an infinite sum; the prime-number-theorem comparison and the\nidentification of that concrete tail with the checked real recurrence are\npaper proofs, as is the real form of the small-shift obstruction in\nCorollary~\\ref{res:smallpair-real}.  The local source's prime-gap tail domination and anti-concentration\nhypotheses also remain paper-level obligations.  No theorem supplies the cofinal pairs in\n\\eqref{eq:smallpair}, and none of the prime-distribution estimates\ndescribed above is claimed or formalised.\n\n\\pdfbookmark[1]{Statements and declarations}{statements-declarations}\n\\subsection*{Statements and declarations}\n\n\\paragraph{Artefact and data availability.} The\n\\href{\\sourceurl}{pinned formal-source revision} contains the Lean sources, the\nfixed toolchain, and the library manifest used in the verification.  This\nmanuscript provides navigation rather than proof authority.\n\n\\paragraph{Declaration of generative AI use.} \\paperfullaideclaration\n\nLean checks each proof term against the fixed library\nversion, and the sources linked here contain no proof placeholders and no\nproject-defined axioms; Lean does not authorise the exposition, the citation\nchoices, or the interpretation, for which the author remains responsible.\n\n\\paragraph{Funding and competing interests.} This work received no external\nfunding.  The author declares no competing interests.\n\n\\paragraph{Acknowledgements.} The problem numbering and status follow the\nErd\\H{o}s Problems catalogue maintained by Thomas Bloom~\\cite{erdosproblems}.\n\n\\appendix\n\\section{Guide to the formal sources}\\label{app:index}\n\nEach linked phrase opens its Lean declaration at the pinned source\nrevision~\\commitshort.  All declarations of this note live in one module.  The\nsummation-by-parts declarations are prime-specific; most of\nSection~\\ref{sec:tail} is stated for arbitrary integer digits and arbitrary\nrational or real orbits, while Section~\\ref{sec:local-certificate} records the\nactual-gap specialisation.  The concrete prime-gap tail and its convergence\nare paper-level consequences of the prime number theorem; they are not\ndefinitions in the pinned Lean module.\n\n% The exhaustive source manifest is retained in the authored TeX for automated\n% validation, and kept outside the printed note.\n\\iffalse\n\n\\begin{tabular}{@{}ll@{}}\ninformal statement & linked Lean source\\\\\nzero-based primes & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{34}{prime0}\\\\\nzero-based gaps & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{38}{primeGap0}\\\\\nfirst gap value & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{41}{primeGap0_zero}\\\\\nsecond gap value & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{45}{primeGap0_one}\\\\\nunbounded actual prime gaps & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{51}{exists_primeGap0_gt}\\\\\ndyadic partial sum & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{95}{dyadicPartialSumQ}\\\\\ndisplayed normalisation & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{100}{prime0DisplayedPartialSumQ}\\\\\nfactor-of-two identity & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{105}{prime0DisplayedPartialSumQ_eq_two_mul}\\\\\ndyadic difference sum & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{115}{dyadicDifferencePartialSumQ}\\\\\npartial-sum step & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{118}{dyadicPartialSumQ_succ}\\\\\ndifference-sum step & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{123}{dyadicDifferencePartialSumQ_succ}\\\\\nsummation by parts & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{132}{dyadicPartialSumQ_eq_start_add_differences}\\\\\nmonotonicity step & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{146}{prime0_mono_step}\\\\\ngap cast identity & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{150}{primeGap0_cast}\\\\\ngap partial sum & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{155}{primeGapPartialSumQ}\\\\\nprime-gap summation by parts & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{161}{prime0_dyadic_summation_by_parts}\\\\\nprime dyadic term & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{172}{primeDyadicTerm}\\\\\ndisplayed prime dyadic term & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{177}{primeDisplayedDyadicTerm}\\\\\nprime-gap dyadic term & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{181}{primeGapDyadicTerm}\\\\\ninfinite factor-of-two identity & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{185}{primeDisplayedDyadicTerm_eq_two_mul}\\\\\ngap discrete derivative & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{191}{primeGapDyadicTerm_eq}\\\\\ngap summability transfer & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{202}{summable_primeGapDyadicTerm_of_summable_primeDyadicTerm}\\\\\ninfinite gap identity & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{214}{tsum_primeDyadicTerm_eq_two_add_primeGap}\\\\\nnormalised irrationality equivalence & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{239}{irrational_tsum_primeDyadicTerm_iff_primeGap}\\\\\ndisplayed infinite identity & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{248}{tsum_primeDisplayedDyadicTerm_eq_four_add_two_primeGap}\\\\\ndisplayed irrationality equivalence & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{263}{irrational_tsum_primeDisplayedDyadicTerm_iff_primeGap}\\\\\ntail recurrence & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{278}{DyadicTailRecurrence}\\\\\nshift & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{282}{tailShift}\\\\\nshift step identity & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{286}{tailShift_succ}\\\\\nintegrality & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{298}{RatIntegral}\\\\\ntail block & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{304}{dyadicTailBlock}\\\\\niterated block identity & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{310}{tail_iterate_eq_pow_mul_sub_block}\\\\\nscaled shift identity & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{324}{tailShift_eq_scaled_sub_block}\\\\\ninteger-shift invariance & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{334}{ratIntegral_sub_int_iff}\\\\\nEuler multiplier & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{352}{ratIntegral_totientMultiplier_of_odd_den}\\\\\nintegral-shift criterion & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{380}{tailShift_integral_iff_scaledTail}\\\\\ntotient shift & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{391}{tailShift_integral_totient_of_odd_den}\\\\\none-step propagation & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{402}{tailShift_integral_succ}\\\\\npropagation to every later index & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{414}{tailShift_integral_add}\\\\\nodd denominator after doubling & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{426}{exists_twoPow_mul_odd_den}\\\\\nodd-denominator orbit state & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{461}{exists_odd_den_state}\\\\\neventual integral shift & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{477}{exists_eventually_integral_tailShift}\\\\\nreal tail recurrence & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{492}{RealDyadicTailRecurrence}\\\\\nreal tail shift & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{496}{realTailShift}\\\\\nreal integrality & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{500}{RealIntegral}\\\\\nrational orbit & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{504}{rationalDyadicOrbit}\\\\\nrational-orbit recurrence & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{508}{rationalDyadicOrbit_recurrence}\\\\\nreal/rational orbit transport & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{515}{realTail_eq_ratCast_rationalDyadicOrbit}\\\\\ncofinal non-integrality & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{528}{CofinalNonintegralTailShifts}\\\\\ncofinal-escape irrationality theorem & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{537}{irrational_initial_of_cofinalNonintegralTailShifts}\\\\\nadjacent small-shift obstruction & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{626}{tailShift_not_both_integral_of_small_pair_of_digit_ne}\\\\\ncofinal small-mismatch theorem & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{653}{not_eventuallyIntegralTailShift_of_cofinal_small_mismatch}\\\\\nprime gaps not eventually periodic & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{670}{primeGap0_not_eventually_periodic}\\\\\nactual-prime-gap specialisation & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{723}{primeGapTailShift_not_eventuallyIntegral_of_cofinal_small_mismatch}\\\\\ncarry coefficient & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{738}{carryCoeff}\\\\\ncarry partial sum & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{742}{carryPartialSum}\\\\\ncarry telescoping identity & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{747}{carryPartialSum_eq}\\\\\nnatural carry coefficient & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{760}{natCarryCoeff}\\\\\nnatural-carry cast & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{763}{natCarryCoeff_cast}\\\\\nfinite-approximation certificate & \\lrefx{Erdos251/PrimeGapDyadicTail.lean}{772}{not_ratIntegral_of_approximation_gap}\\\\\n\\end{tabular}\n\\fi\n\n\\begin{thebibliography}{99}\n\\small\n\\raggedright\n\\setlength{\\itemsep}{0pt}\n\\bibitem{erdos1958} P.~Erd\\H{o}s,\n  \\href{https://combinatorica.hu/~p_erdos/1958-19.pdf}{\\emph{Sur certaines\n  s\\'eries \\`a valeur irrationnelle}},\n  Enseign. Math. (2) \\textbf{4} (1958), 93--100,\n  doi:\\href{https://doi.org/10.5169/seals-34629}{10.5169/seals-34629}.\n  The dyadic prime series is stated as unproved on p.~94; the\n  factorial-prime family is stated on p.~93, with only the $k=1$ proof printed\n  on pp.~94--95.\n\\bibitem{erdosgraham1980} P.~Erd\\H{o}s and R.~L. Graham,\n  \\href{https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf}{\\emph{Old\n  and New Problems and Results in Combinatorial Number Theory}},\n  Monogr. Enseign. Math. 28, Geneva, 1980, p.~62.\n\\bibitem{erdospomerance1978} P.~Erd\\H{o}s and C.~Pomerance,\n  \\href{https://doi.org/10.1007/BF01818569}{\\emph{On the largest prime\n  factors of $n$ and $n+1$}},\n  Aequationes Math. \\textbf{17} (1978), 311--321,\n  doi:\\href{https://doi.org/10.1007/BF01818569}{10.1007/BF01818569}.\n  The unnumbered dyadic irrationality theorem and its complete proof are in\n  \\S7 on p.~320.\n\\bibitem{erdos1988} P.~Erd\\H{o}s, \\emph{On the irrationality of certain series:\n  problems and results}, in A.~Baker (ed.), \\emph{New Advances in Transcendence\n  Theory}, Cambridge UP, 1988, pp.~102--109,\n  doi:\\href{https://doi.org/10.1017/CBO9780511897184.009}{10.1017/CBO9780511897184.009}.\n\\bibitem{erdosstraus1963} P.~Erd\\H{o}s and E.~G. Straus, \\emph{On the\n  irrationality of certain Ahmes series}, J. Indian Math. Soc. (N.S.)\n  \\textbf{27} (1964), 129--133. MR~175848.\n\\bibitem{fgkmt2018} K.~Ford, B.~Green, S.~Konyagin, J.~Maynard and T.~Tao,\n  \\emph{Long gaps between primes}, J. Amer. Math. Soc. \\textbf{31} (2018),\n  65--105,\n  doi:\\href{https://doi.org/10.1090/jams/876}{10.1090/jams/876}.\n  Theorem~1 on p.~66 gives the effective lower bound for the largest single\n  consecutive-prime gap below $X$.\n\\bibitem{kovactao2024} V.~Kova\\v{c} and T.~Tao,\n  \\href{https://doi.org/10.1007/s10474-025-01528-0}{\\emph{On several\n  irrationality problems for Ahmes series}}, Acta Math. Hungar. \\textbf{175}\n  (2025), 572--608.\n\\bibitem{lean4} L.~de Moura and S.~Ullrich,\n  \\href{https://doi.org/10.1007/978-3-030-79876-5_37}\n  {\\emph{The Lean~4 theorem prover and programming language}},\n  in A.~Platzer and G.~Sutcliffe (eds.), CADE~28,\n  Lecture Notes in Comput. Sci. 12699, Springer, 2021, pp.~625--635,\n  doi:\\href{https://doi.org/10.1007/978-3-030-79876-5_37}\n  {10.1007/978-3-030-79876-5\\_37}.\n\\bibitem{mathlib} The mathlib Community,\n  \\href{https://doi.org/10.1145/3372885.3373824}\n  {\\emph{The Lean mathematical library}},\n  in CPP 2020, ACM, 2020, pp.~367--381,\n  doi:\\href{https://doi.org/10.1145/3372885.3373824}\n  {10.1145/3372885.3373824}.  The article describes a December 2019\n  Lean~3-era snapshot; the repository lock owns the current revision.\n\\bibitem{maynard2015} J.~Maynard,\n  \\href{https://doi.org/10.4007/annals.2015.181.1.7}\n  {\\emph{Small gaps between primes}},\n  Ann. of Math. (2) \\textbf{181} (2015), 383--413.\n\\bibitem{mv2007} H.~L. Montgomery and R.~C. Vaughan,\n  \\href{https://doi.org/10.1017/CBO9780511618314.008}\n  {\\emph{Multiplicative Number Theory I: Classical Theory}},\n  Cambridge Stud. Adv. Math. 97, Cambridge UP, 2007,\n  Chapter~6, pp.~168--198; Theorem~6.9, pp.~179--181, and\n  Exercise~6.2.5, p.~183,\n  doi:\\href{https://doi.org/10.1017/CBO9780511618314.008}\n  {10.1017/CBO9780511618314.008}.\n\\bibitem{zhang2014} Y.~Zhang,\n  \\href{https://doi.org/10.4007/annals.2014.179.3.7}\n  {\\emph{Bounded gaps between primes}},\n  Ann. of Math. (2) \\textbf{179} (2014), 1121--1174.\n\\bibitem{erdosproblems} T.~F. Bloom,\n  \\href{https://www.erdosproblems.com/251}{\\emph{Erd\\H{o}s Problem \\#251}},\n  \\texttt{erdosproblems.com/251}, accessed 28 July 2026 (page displays\n  ``last edited 28 September 2025'').  The current record labels the main\n  dyadic problem open, cites \\texttt{[Er58b]}, \\texttt{[ErGr80, p.~62]} and\n  \\texttt{[Er88c, p.~103]}, and explicitly describes its status as the\n  website owner's present assessment rather than a literature-completeness\n  guarantee; it does not mention the 2026 counterexample in\n  \\cite{kovac2026} to the adjacent variable-denominator conjecture.\n\\bibitem{kovac2026} ChatGPT~5.4 Pro (orchestrated by V.~Kova\\v{c}),\n  \\href{https://web.math.pmf.unizg.hr/~vjekovac/files/Erdos_problem_251.pdf}\n  {\\emph{On the Erd\\H{o}s problem \\#251}},\n  unpublished note, 2026, hosted by the Department of Mathematics, University\n  of Zagreb, \\texttt{web.math.pmf.unizg.hr}, accessed 28 July 2026.\n\\bibitem{formalconjectures251} The Formal Conjectures Authors,\n  \\href{https://github.com/google-deepmind/formal-conjectures/blob/f776d2f2039351b00737ffcafb9d7d7666e1d9af/FormalConjectures/ErdosProblems/251.lean}\n  {\\emph{FormalConjectures.ErdosProblems.\\texttt{251}}},\n  Lean source at commit \\texttt{f776d2f}, 2025, accessed 28 July 2026.\n\\end{thebibliography}\n\n\\end{document}\n","dependencies":[{"record_type":"paper_source_dependency","tex_reference":"problem-note-preamble","source_path":"paper/problem-note-preamble.tex","media_type":"text/x-tex","sha256":"sha256:e7378fe7784916681e7faf96c1ea7a8bb941a416cc651b97512e849bcaeb63f7","byte_count":4990,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Shared preamble for the Erd\u0151s Problem Notes: the short, standalone,\n% problem-owned manuscripts covering the expansion library `ErdosProblems`.\n%\n% One note owns one Erd\u0151s problem.  Each note repeats whatever context its own\n% reader needs, so that a reader who arrives at a single problem never has to\n% assemble it from the gateway paper.  Deliberate repetition across notes is the\n% design, not an oversight.\n%\n% Source links resolve against one pinned formal-source commit, declared once\n% below.  `scripts/check_problem_note_sources.py` verifies that every authored\n% (file, line, declaration) triple names that exact declaration in the pinned\n% snapshot, so a link cannot silently rot as later work moves lines.\n\n\\usepackage{paper-house-style}\n\\usepackage{array}\n\\usepackage{booktabs}\n\\usepackage{enumitem}\n\\setlist{topsep=0.35em,itemsep=0.2em}\n\\usepackage[colorlinks=true,linkcolor=linkinternal,urlcolor=linkexternal,\n  citecolor=linkinternal]{hyperref}\n\\hypersetup{bookmarksnumbered=true,bookmarksopen=true,bookmarksopenlevel=1,\n  pdfauthor={Will Cook},pdflang={en-GB}}\n\n% ---------------------------------------------------------------------------\n% Pinned formal source.  \\commit names the checkpoint every link resolves\n% against; the checker reads that snapshot rather than the working tree, so the\n% printed coordinates stay correct however later waves move declarations.\n% ---------------------------------------------------------------------------\n\\newcommand{\\commit}{d6d344d9557faee19276b01a1c1f20d7c08f03d5}\n\\newcommand{\\commitshort}{d6d344d9557f}\n\\newcommand{\\repobase}{https://github.com/wcook04/plectis-lean-erdos249-257/blob/\\commit}\n\\newcommand{\\sourceurl}{https://github.com/wcook04/plectis-lean-erdos249-257/tree/\\commit}\n\\newcommand{\\PX}{ErdosProblems}\n\n% Declaration names stay inside link targets.  The printed label is either a\n% spaced, lower-case reading of the declaration name (\\lref) or a short authored\n% phrase (\\lword); neither prints a module path or a raw Lean identifier.\n\\ExplSyntaxOn\n\\cs_new_protected:Npn \\noteleanlabel #1\n  {\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_to_str:n {#1} }\n    \\regex_replace_all:nnN { _+ } { \\c{space} } \\l_tmpa_tl\n    \\regex_replace_all:nnN\n      { ([a-z0-9])([A-Z]) }\n      { \\1\\c{space}\\2 }\n      \\l_tmpa_tl\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_lower_case:n { \\tl_use:N \\l_tmpa_tl } }\n    \\tl_use:N \\l_tmpa_tl\n  }\n\\ExplSyntaxOff\n\\newcommand{\\leanlabel}[1]{\\noteleanlabel{#1}}\n\\newcommand{\\lref}[3]{\\href{\\repobase/\\PX/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\lword}[4]{\\href{\\repobase/\\PX/#1\\#L#2}{#4}}\n\\newcommand{\\lloc}[2]{\\href{\\repobase/\\PX/#1\\#L#2}{Lean source}}\n\n% Links into a sibling library, named repository-relative.  The expansion\n% library is implicit in \\lref/\\lword, because that is where problem-owned\n% work lands.  Several problems have their headline theorem in the older\n% reviewed #249/#257 corpus, and a note that could not name that library\n% would have to paraphrase a checked statement instead of linking it.\n\\newcommand{\\mref}[3]{\\href{\\repobase/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\mword}[4]{\\href{\\repobase/#1\\#L#2}{#4}}\n\\newcommand{\\mloc}[2]{\\href{\\repobase/#1\\#L#2}{Lean source}}\n\\emergencystretch=2em\n\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\theoremstyle{definition}\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{example}[theorem]{Example}\n\\newtheorem{problem}[theorem]{Problem}\n\\theoremstyle{remark}\n\\newtheorem*{remark}{Remark}\n\n\\newcommand{\\N}{\\mathbb{N}}\n\\newcommand{\\Npos}{\\mathbb{N}_{>0}}\n\\newcommand{\\Z}{\\mathbb{Z}}\n\\newcommand{\\Q}{\\mathbb{Q}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\lcm}{\\operatorname{lcm}}\n\\newcommand{\\ph}{\\varphi}\n\n% The series line printed under every note's title block.\n\\newcommand{\\seriesline}{Erd\\H{o}s Problem Notes}\n\n% Problem-note compatibility name for the corpus-owned page-one disclosure in\n% paper-house-style.sty.\n\\newcommand{\\noteprovenance}{\\paperprovenance}\n\n% The status paragraph every note carries, in its own words, before any result.\n% Kernel checking establishes the exact Lean proposition; it does not establish\n% that the proposition is the interesting one, that it is new, or that the\n% Erd\u0151s problem is closed.\n\\newcommand{\\statusboundary}{%\n  \\emph{Status.}  The problem treated here is open, and this note does not\n  close it.  Every statement below marked as checked is a proposition that the\n  pinned Lean kernel accepts from the sources this note links to, with no\n  \\texttt{sorry}, no added axiom, and no unchecked evaluation.  That is a claim\n  about the formal statement, not about its mathematical interest, its\n  novelty, or the original problem.  The unresolved obligations are named\n  exactly, in their own section, and none of the finite computations,\n  reductions, or no-go results here removes one of them.}\n"}],"external_verification_route":"https://raw.githubusercontent.com/wcook04/plectis-lean-erdos249-257/main/paper/erdos-251-prime-gap-dyadic-series.tex"},{"record_type":"paper","paper_id":"erdos-257-mersenne-support-subseries","title":"Denominator Periods, Rational-Value Constraints and Achievement-Set Geometry","source_repository":"plectis-lean-erdos249-257","source_path":"paper/erdos-257-mersenne-support-subseries.tex","hosted_pdf_path":"papers/erdos-257-mersenne-support-subseries.pdf","licence":"CC-BY-4.0","copyright":"2026 Will Cook","cite_as":"Plectis reader corpus, paper:erdos-257-mersenne-support-subseries","pdf_sha256":"sha256:ee6accba99c8d966a48f91281d7f64733b4dd5f0c03e53146ec97ae3cc9b4751","pdf_byte_count":154223,"pdf_embedded":false,"pdf_omission_reason":"the exact LaTeX is embedded and retrieves better than extracted PDF text; the PDF is a presentation rendering and its hash is recorded for verification","source_embedded":true,"availability":"embedded_exact","embedded_copy_role":"native_manuscript","embedded_from_repository":"plectis-lean-erdos249-257","embedded_copy_path":"paper/erdos-257-mersenne-support-subseries.tex","media_type":"text/x-tex","body_format":"latex","sha256":"sha256:ee15f5819849245c97f141c416b1f10d22c10b4b2b2d7d126934a47a4d9e3d64","byte_count":58346,"body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Standalone Erd\u0151s Problem Note for #257.  Context is deliberately repeated\n% from the gateway so that this file remains independently readable.\n%\n% Ordering rule, revised July 2026.  The previous draft was a map of settled\n% supports with a method boundary attached.  A map is a list, and #257 is a\n% universal statement; adding rows does not approach one.  This draft leads\n% with the finite theorem, then with the statements that quantify over *every*\n% hypothetical rational-valued support, and demotes the settled families to\n% context.  Those forced-structure theorems were already in the corpus and\n% were simply absent from the note.\n%\n% Second correction.  The squarefree support was printed here as open.  It is\n% known at every power-of-two base: Duverney and Tachiya settled the joint\n% family in 2019, and this note\n% had failed to cite them.  The section that claimed a frontier now records\n% what the position actually is -- a certificate engine failing on a value\n% that is known irrational, for a reason internal to a normalisation\n% convention rather than to the value.  That is a smaller claim and a more\n% useful one.\n\\documentclass[11pt]{article}\n\\input{problem-note-preamble}\n\\renewcommand{\\commit}{d6d344d9557faee19276b01a1c1f20d7c08f03d5}\n\\renewcommand{\\commitshort}{d6d344d9557f}\n\\hypersetup{\n  pdftitle={Denominator periods, rational-value constraints and achievement-set geometry for Mersenne subseries (Erd\u0151s Problem 257)},\n  pdfsubject={Erd\u0151s Problem Notes: Problem 257},\n  pdfkeywords={irrationality, Mersenne numbers, Lambert series, divisor incidence, scaled tails, achievement sets, squarefree, Lean 4},\n}\n\n\\newcommand{\\wgt}[1]{w_{#1}}\n\\newcommand{\\Ach}{\\mathcal A}\n\\newcommand{\\scA}{\\operatorname{sc}}\n\\newcommand{\\Asf}{A_{\\mathrm{sf}}}\n\n\\runninghead{Erd\\H{o}s \\#257: denominator periods and rational-value constraints}\n\\title{Denominator Periods, Rational-Value Constraints and\nAchievement-Set Geometry}\n\\subtitle{Mersenne subseries and Erd\\H{o}s Problem~\\#257}\n\\author{Will Cook}\n\\date{July 2026}\n\n\\begin{document}\n\\maketitle\n\\noteprovenance\n\n\\begin{abstract}\n\\noindent\nLet $F\\subseteq\\Npos$ be finite and nonempty, let $b\\ge2$, and write\n\\[\n  \\sum_{n\\in F}\\frac1{b^n-1}=\\frac{N_F}{D_F}\n  \\qquad (\\gcd(N_F,D_F)=1,\\ D_F>0).\n\\]\nThe lead result is the exact identity\n$\\operatorname{ord}_{D_F}(b)=\\lcm(F)$, with the convention\n$\\operatorname{ord}_1(b)=1$.  If $\\lcm(F)\\ge2$, then also\n$\\lcm(F)<D_F$.  Without that hypothesis the conclusion can fail: at\n$b=2$ and $F=\\{1\\}$ one has $D_F=1$.\n\nErd\\H{o}s~\\#257 asks whether\n$X_A=\\sum_{a\\in A}(2^a-1)^{-1}$ is irrational for every infinite\n$A\\subseteq\\Npos$.  It remains open.  The coordinate used here is the divisor\nincidence $\\scA_A(n)=\\#\\{d\\mid n:d\\in A\\}$, for which\n$X_A=\\sum_{n\\ge1}\\scA_A(n)2^{-n}$; these coefficients are nonnegative divisor\ncounts and need not be binary digits.  If $X_A$ is rational, its integral\nscaled-tail sequence satisfies an exact long-division recurrence.  For\ninfinite $A$ that sequence is unbounded, and every zero window of\n$\\scA_A$ at $c+N$ has length at most\n$\\varepsilon\\log_2(N+1)+O_{\\varepsilon,c,v,A,p}(1)$.  Under the additional\nhypotheses that $\\sum_{a\\in A}1/a$ converges and that the reduced odd\ndenominator part $v>1$ is coprime to the numerator, rationality also forces\n\\[\n  \\sum_{a\\in A}\\frac1a\\ge\\frac1{\\operatorname{ord}_v(2)}.\n\\]\nFinally, rational values admit an exact Boolean--M\\\"obius scaled-tail\ncharacterisation from which the support is reconstructed.  These constraints\nare not known to be contradictory.  We also record the squarefree support,\nwhose values at every power-of-two base are jointly controlled by the\nliterature and on which two of the\narguments used here fail for a reason internal to a normalisation rather than\nto the value itself.  We then record the achievement-set geometry and the exact\nstatus of the candidate $1/2$: membership would refute \\#257, while\nnon-membership would eliminate only this one candidate.\n\\end{abstract}\n\n\\section{Introduction and main results}\\label{sec:problem}\n\n\\begin{problem}[Erd\\H{o}s \\#257]\\label{res:problem}\nFor every infinite $A\\subseteq\\Npos$, is\n$X_A=\\sum_{a\\in A}\\dfrac1{2^a-1}$ irrational?\n\\end{problem}\n\n\\noindent See Erd\\H{o}s~\\cite[p.~222]{erdos1968}, Erd\\H{o}s and\nGraham~\\cite[p.~62]{erdosgraham1980}, and\nErd\\H{o}s~\\cite[p.~105]{erdos1988}.  Bloom's current catalogue record\nreproduces the displayed problem and labels it open, while explicitly warning\nthat the status is the website owner's present assessment and may omit\nrelevant literature~\\cite{erdosproblems}.  We therefore use the catalogue for\nnumbering and current reported status only; the original publications and the\nlater cited papers carry the mathematical claims.  Write\n$\\wgt n=(2^n-1)^{-1}$.  Expanding each weight as a geometric series and\ninterchanging the two nonnegative sums gives the coordinate this note works\nin.  With the \\emph{divisor incidence}\n$\\scA_A(n)=\\#\\{d\\mid n: d\\in A\\}$,\n\\[\n  X_A=\\sum_{a\\in A}\\frac1{2^a-1}=\\sum_{n\\ge1}\\frac{\\scA_A(n)}{2^n}.\n  \\tag{1}\\label{eq:incidence}\n\\]\nThe transform is worth reading carefully, because it is where the arithmetic\nof the problem enters.  The datum is a $0/1$ selector, the indicator of $A$;\nwhat appears in~\\eqref{eq:incidence} is not that selector but its divisor\ntransform, a nonnegative integer sequence bounded by the divisor function,\n$\\scA_A(n)\\le d(n)$.  So \\#257 is not a generic question about binary digit\nsequences.  Equation~\\eqref{eq:incidence} is a power-series representation,\nnot a binary expansion: its coefficients are divisor counts drawn from a\nsingle support and may exceed~$1$.  Every theorem below is a statement about\nsequences of that shape.\n\nA small instance fixes the notation.  At $A=\\{2,3\\}$ the incidence sequence\nbegins\n\\[\n  \\scA_A(1),\\dots,\\scA_A(12)=0,1,1,1,0,2,0,1,1,1,0,2,\n\\]\nthe value $2$ occurring at the multiples of $6$ and the value $0$ at the\nintegers prime to $6$, and~\\eqref{eq:incidence} reads\n$\\tfrac13+\\tfrac17=\\tfrac{10}{21}$.\n\nSeveral statements below hold at every integer base, so we write\n$X_A(b)=\\sum_{a\\in A}(b^a-1)^{-1}$ for the value at an integer base $b\\ge2$,\nwith $X_A=X_A(2)$.  Base~$2$ is the case Problem~\\ref{res:problem} asks about\nand is meant whenever no base is named.\n\n\\paragraph{The finite-support theorem.}\nFor a finite nonempty $F\\subseteq\\Npos$ and an integer $b\\ge2$, write\n\\[\n  x_F(b)=\\sum_{n\\in F}\\frac1{b^n-1}=\\frac{N_F}{D_F}\n  \\qquad (\\gcd(N_F,D_F)=1,\\ D_F>0).\n\\]\nWe use the standard trivial-modulus convention\n$\\operatorname{ord}_1(b)=1$.\n\n\\begin{theorem}[finite-period noncollapse]\\label{res:period}\nLet $F\\subseteq\\Npos$ be finite and nonempty, let $b\\ge2$ be an integer, and\nlet $D_F>0$ be the denominator of $x_F(b)$ in lowest terms.  Then $D_F$ is\ncoprime to $b$, and\n\\[\n  \\operatorname{ord}_{D_F}(b)=\\lcm\\{n:n\\in F\\}.\n\\]\nIf moreover $\\lcm(F)\\ge2$, then $\\lcm(F)<D_F$.\n\\end{theorem}\n\n\\noindent The order statement is\n\\mword{Erdos249257/CertificateKernel.lean}{5091}{finite_period_noncollapse}{checked},\nits reduced-denominator form is\n\\mword{Erdos249257/CertificateKernel.lean}{5246}{finite_period_noncollapse_rat_den}{checked},\ncoprimality is\n\\mword{Erdos249257/CertificateKernel.lean}{5221}{coprime_base_den_finiteErdosSum}{checked},\nand the growth clause is\n\\mword{Erdos249257/CertificateKernel.lean}{5260}{lcm_lt_den_finiteErdosSum}{checked}.\n\n\\statusboundary\n\n\\paragraph{Structure.}  Section~\\ref{sec:period} develops the arithmetic of\nTheorem~\\ref{res:period}.  Section~\\ref{sec:forced} collects what\nrationality would force on an arbitrary infinite support, which is the part of\nthis note that quantifies over every support rather than sampling.\nSection~\\ref{sec:map} gives representative supports already known to give\nirrational values, grouped by the argument that reaches them.\nSection~\\ref{sec:squarefree} treats\nthe squarefree support, whose values are known at every power-of-two base and which two of the\narguments used here provably cannot reach at any even base.\nSection~\\ref{sec:geometry} is the geometry of the set of attainable values, and\nthe value $1/2$ inside it.  Section~\\ref{sec:open} states what remains.\n\n\\section{Finite-support denominator periods}\\label{sec:period}\n\nA finite support has a rational value, so for a finite support the question is\nnot irrationality but arithmetic: how large the denominator is, and how the\nbase sits inside it.  Theorem~\\ref{res:period} answers the second exactly and,\nunder its stated hypothesis $\\lcm(F)\\ge2$, gives a strict lower bound for the\nfirst.  The omitted boundary is genuine: at $b=2$ and $F=\\{1\\}$ the two\nquantities are both~$1$.\n\nSix instances at $b=2$, which also show what the hypothesis\n$\\lcm(F)\\ge2$ is for:\n\\begin{center}\n\\begin{tabular}{@{}lllll@{}}\n\\toprule\n$F$ & $x_F(2)$ & $D_F$ & $\\lcm(F)$ & $\\operatorname{ord}_{D_F}(2)$\\\\\n\\midrule\n$\\{1\\}$   & $1$      & $1$   & $1$  & $1$\\\\\n$\\{2\\}$   & $1/3$    & $3$   & $2$  & $2$\\\\\n$\\{2,3\\}$ & $10/21$  & $21$  & $6$  & $6$\\\\\n$\\{1,2,3\\}$ & $31/21$  & $21$  & $6$  & $6$\\\\\n$\\{2,6\\}$ & $22/63$  & $63$  & $6$  & $6$\\\\\n$\\{4,6\\}$ & $26/315$ & $315$ & $12$ & $12$\\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\nThe last two columns agree in every row, which is the order statement.  In the\nworked case $F=\\{1,2,3\\}$, the rational sum is $1+1/3+1/7=31/21$ and\n$2^6\\equiv1\\pmod {21}$, whereas no smaller positive exponent gives~$1$.\nThe\nfirst row is the only finite nonempty support with $\\lcm(F)=1$, and the only\none on which $\\lcm(F)<D_F$ fails; that is what the hypothesis $\\lcm(F)\\ge2$\nexcludes.\n\nThe content is a noncollapse statement.  Clearing denominators over\n$b^{\\lcm(F)}-1$ makes the period at most $\\lcm(F)$ immediately; what is not\nimmediate is that cancellation in the numerator cannot bring it below.  The\nmechanism is that each selected exponent $n$ contributes, by the cyclotomic\nroute, a prime-power modulus dividing $b^{n}-1$ on which $b$ has order exactly\n$n$; no single reduction can remove all of them at once.  Prime powers rather\nthan primes is not a technicality.  At $(b,n)=(2,6)$ no prime divisor of\n$2^6-1=63$ has order $6$ --- one of the exceptional cases in Zsigmondy's\ntheorem --- while $9$ does, and it\nis the prime power that carries the witness.  The row $F=\\{2,6\\}$ above is that\ncase in full: $D_F=63=9\\cdot7$, and since $2$ has order $2$ modulo $3$ and\norder $3$ modulo $7$, the order $6$ can only come from $9$.  The growth clause\nthen follows formally, since the order divides $\\ph(D_F)<D_F$, where $\\ph$ is\nEuler's totient.\n\nTwo boundaries.  This is an unconditional statement about every finite\nsupport, not a bounded table of examples, and not an implication with an open\nhypothesis; but it settles no infinite support, and no limit of it does.\nDenominator-period control is part of the classical method behind\nErd\\H{o}s's 1948 argument~\\cite{erdos1948}.  Whether this exact sharp form\nappears in the literature has not been assessed, and no novelty is claimed.\n\n\\section{Rational values and scaled tails}\\label{sec:forced}\n\nSection~\\ref{sec:map} below lists supports for which irrationality is known.\nThis section is the other half, and the half that meets\nProblem~\\ref{res:problem} rather than sampling it: statements that hold for\n\\emph{every} infinite support whose value is rational.  The base is $2$\nthroughout, and each statement carries its own hypotheses.\n\n\\paragraph{Integral scaled tails, and unboundedness.}\nBinary long division turns a hypothetical rational value into an integer\nrecurrence.  Fix an integer $v\\ge1$ and a sequence $f\\colon\\Npos\\to\\N$ of\ncoefficients.  Call an integer sequence $u\\colon\\N\\to\\Z$ a\n\\emph{tempered scaled-tail sequence} for $f$ with multiplier $v$ if\n\\[\n  u(n+1)=2u(n)-v\\,f(n+1)\\quad\\text{for every }n\\ge0,\n  \\qquad\\text{and}\\qquad u(n)=o(2^{n}).\n\\]\nThe recurrence is one step of long division in base~$2$: double the remainder,\nthen pay out the next coefficient.  The growth condition is what pins the\nsolution down, since two solutions of the recurrence differ by $C\\,2^{n}$ for a\nconstant $C$, and only $C=0$ survives $u(n)=o(2^{n})$.  For any coefficient\nsequence with $f(n)\\le n$ --- and $\\scA_A(n)\\le d(n)\\le n$ qualifies, $d$ being\nthe divisor function --- the series $\\sum f(n)2^{-n}$ is rational exactly when\na tempered scaled-tail sequence exists for some $v\\ge1$, and every such\nsequence is then\nthe scaled tail\n$u(n)=v\\sum_{j\\ge1}f(n+j)2^{-j}$, so the orbit is unique rather than merely\navailable\n(\\mword{Erdos249257/GenericTailOrbitRigidity.lean}{426}{binaryCoeffSeries_rational_iff_exists_temperedBinaryOrbit}{checked}).\nThe next theorem exhibits such an orbit for $X_A$ itself, with the coefficient\nsequence shifted past the power of $2$ in the denominator.\n\n\\begin{theorem}[unbounded scaled-tail states]\\label{res:unbounded}\nLet $A\\subseteq\\Npos$ be infinite with $X_A=p/(2^cv)$, $v\\ge1$.  Then there is\n$u\\colon\\N\\to\\Npos$ with\n\\[\n  u(n)=v\\sum_{j\\ge1}\\scA_A(c+n+j)2^{-j}\n\\]\nsatisfying the exact recurrence\n\\[\n  u(n+1)+v\\,\\scA_A(c+n+1)=2u(n),\n  \\qquad u(n)\\equiv p\\,2^{n}\\ (\\mathrm{mod}\\ v),\n\\]\nand $u$ is unbounded.\n\\end{theorem}\n\n\\noindent Checked as\n\\mword{Erdos249257/RationalSupportCarrySkeleton.lean}{2377}{exists_unbounded_shifted_odd_tail_nat_state_of_support_fraction}{Lean}.\nThe useful clause is exact: these scaled-tail states cannot remain bounded.\nIn particular they cannot eventually cycle through a finite set of states.\n\n\\paragraph{Divisor coverage cannot have long gaps.}\nSay that $\\scA_A$ has a \\emph{zero window} of length $h$ at $N$ if\n$\\scA_A(N+1)=\\dots=\\scA_A(N+h)=0$, that is, if no element of $A$ divides any\nof $h$ consecutive integers.  At $A=\\{2,3\\}$, for instance, $\\scA_A$ vanishes\nexactly on the integers prime to $6$, so every zero window has length at\nmost $1$: one of any two consecutive integers is even.\n\n\\begin{theorem}[sublogarithmic zero windows]\\label{res:sublog}\nLet $A\\subseteq\\Npos$ be nonempty with $X_A=p/(2^cv)$, $v\\ge1$.  For every\n$\\varepsilon>0$ there is a constant $B$ such that every zero window of\n$\\scA_A$ at $c+N$ has length\n\\[\n  h\\le\\varepsilon\\log_2(N+1)+B .\n\\]\nHere $A,p,c,v,$ and $\\varepsilon$ are fixed before $B$ is chosen; the bound is\nthen uniform in $N$ and $h$.  The checked theorem does not assert one common\n$B$ for all supports or numerators.\n\\end{theorem}\n\n\\noindent Checked as\n\\mword{Erdos249257/SublogDivisorCoverage.lean}{392}{supportCoeffZeroWindow_length_le_eps_logb_add}{Lean}.\nRead as a constraint on counterexamples, this rules out zero windows of any\nfixed positive proportion of $\\log_2N$ once $N$ is large.  The proof compares\nan exponential lower bound forced on the scaled tail with an upper bound for\ndivisor sums that grows more slowly than any fixed power of $N$.\n\n\\paragraph{A reciprocal-sum lower bound.}\nLet $\\operatorname{ord}_v(2)$ denote the multiplicative order of $2$ modulo an\nodd $v>1$, and write the reciprocal mass of $A$ for $\\sum_{a\\in A}1/a$.\n\n\\begin{theorem}[reciprocal mass bound]\\label{res:mass}\nLet $A\\subseteq\\Npos$ be such that $\\sum_{a\\in A}1/a$ converges, and suppose\n$X_A=p/(2^cv)$ with $v>1$ odd and $\\gcd(p,v)=1$.  Then\n\\[\n  \\sum_{a\\in A}\\frac1a\\;\\ge\\;\\frac1{\\operatorname{ord}_v(2)} .\n\\]\n\\end{theorem}\n\n\\noindent Checked as\n\\mword{Erdos249257/RationalSupportCarrySkeleton.lean}{1478}{one_div_oddOrder_le_reciprocalMass_of_support_fraction}{Lean}.\nA finite support already supplies an instance: $A=\\{2,3\\}$ has $X_A=10/21$, so\n$c=0$, $v=21$, $\\gcd(p,v)=1$ and $\\operatorname{ord}_{21}(2)=6$, and the bound\nreads $\\tfrac12+\\tfrac13\\ge\\tfrac16$.\nThe summability hypothesis is doing real work: a rational value with fixed odd\ndenominator part $v$ and a convergent reciprocal sum requires mass at least\n$1/\\operatorname{ord}_v(2)$.  The statement does not apply when the reciprocal\nsum diverges, and it gives no positive lower bound uniform in $v$; it\nconstrains the pair (support, denominator) jointly.\n\n\\paragraph{A Boolean--M\\\"obius characterisation.}\nThe three theorems above give necessary conditions.  The next statement is an\nequivalence, and is the sharpest description of rational-valued supports the\ndevelopment has.  It removes the support from the description altogether, which\nis possible because the support can always be read back off the incidence\nsequence: with\n$\\mu$ the M\\\"obius function, $*$ Dirichlet convolution\n$(g*h)(n)=\\sum_{d\\mid n}g(d)h(n/d)$ and $\\mathbf 1_A$ the indicator of $A$, the\nidentity $\\scA_A=\\mathbf 1_A*1$ inverts to $\\mu*\\scA_A=\\mathbf 1_A$.\n\nFor $p\\in\\Z$, $q\\ge1$ and $U\\colon\\N\\to\\Z$, set\n\\[\n  Q_U(0)=0,\\qquad\n  Q_U(n)=\\frac{2U(n-1)-U(n)}q\\quad(n\\ge1).\n\\]\nCall $U$ an \\emph{admissible Boolean--M\\\"obius scaled-tail sequence} for\n$(p,q)$ if\n\\[\n\\begin{gathered}\n U(0)=p,\\quad U(N)>0,\\quad\n U(N)\\le q\\bigl(2\\sqrt N+4\\bigr)\\quad(N\\ge0),\\\\\n q\\mid 2U(N)-U(N+1)\\quad(N\\ge0),\\qquad\n (\\mu*Q_U)(n)\\in\\{0,1\\}\\quad(n\\ge1).\n\\end{gathered}\n\\tag{2}\\label{eq:bmc}\n\\]\nThe divisibility condition makes $Q_U$ integral; the last condition says that\nM\\\"obius inversion of the quotient is the indicator of a set.\n\n\\begin{theorem}[Boolean--M\\\"obius scaled-tail correspondence]\\label{res:bmc}\nLet $p\\in\\Z$ and $q\\ge1$.  There exists a support $A$ with $0\\notin A$, with\nsome positive element, and with $X_A=p/q$, if and only if there exists an\nadmissible Boolean--M\\\"obius scaled-tail sequence $U$ for $(p,q)$.  In that\ncase the support is recovered as\n\\[\n  A=\\{n:(\\mu*Q_U)(n)=1\\},\n\\]\nwith $Q_U$ as in~\\eqref{eq:bmc}.\n\\end{theorem}\n\n\\noindent Checked as\n\\mword{Erdos249257/BooleanMobiusCarry.lean}{948}{exists_normalized_support_fraction_iff_exists_booleanMobiusCarry}{Lean}.\nThe finite example $A=\\{2,3\\}$ makes the definition concrete.  Here\n$p/q=10/21$, and\n\\[\n Q_U(1),\\ldots,Q_U(6)=0,1,1,1,0,2,\\qquad\n U(0),\\ldots,U(6)=10,20,19,17,13,26,10.\n\\]\nM\\\"obius inversion gives\n$(\\mu*Q_U)(1),\\ldots,(\\mu*Q_U)(6)=0,1,1,0,0,0$, the indicator of\n$\\{2,3\\}$ through that range.  This finite calculation illustrates the\ncorrespondence; the admissible scaled-tail sequence itself is an infinite\nobject, and the finite support is not a counterexample to\nProblem~\\ref{res:problem}.\n\nOne boundary on the equivalence.  The support it produces is not required to be\ninfinite, and a finite support supplies an admissible scaled-tail sequence for\nits own value, so existence of such a sequence is not by itself a\ncounterexample to Problem~\\ref{res:problem}.\nWhat the equivalence changes is the search space, not the difficulty.\n\n\\paragraph{Combined constraints on a rational counterexample.}\nTaken together: a counterexample to Problem~\\ref{res:problem} would have an\nunbounded scaled-tail sequence obeying an exact linear recurrence and\nsublogarithmic divisor-coverage gaps.  If its reciprocal sum converged and its\nreduced odd denominator part were greater than~$1$, it would also satisfy\nTheorem~\\ref{res:mass}.  Theorem~\\ref{res:bmc} reconstructs every rational\nvalue, but by itself does not force the reconstructed support to be infinite.\nThese qualifications matter: the four statements have different hypotheses,\nand no jointly contradictory combination has been proved.\n\n\\section{Representative known irrational supports}\\label{sec:map}\n\nThe following table is representative, not exhaustive.  It groups the\ndisplayed supports by five mechanisms rather than suggesting that its rows\nclassify all known cases.\n\n\\begin{table}[htbp]\n\\centering\n\\small\n\\begin{tabular}{@{}\n  >{\\raggedright\\arraybackslash}p{0.29\\linewidth}\n  >{\\raggedright\\arraybackslash}p{0.15\\linewidth}\n  >{\\raggedright\\arraybackslash}p{0.48\\linewidth}@{}}\n\\toprule\nSupport $A$ & Bases & Mechanism and authority \\\\\n\\midrule\n\\multicolumn{3}{@{}l}{\\emph{Classical full support, and base change into it}}\\\\\nall of $\\Npos$ & every $b\\ge2$ & Erd\\H{o}s 1948~\\cite{erdos1948};\n  \\mword{Erdos249257/CertificateKernel.lean}{8717}{irrational_erdosSupportSeries_univ}{formalised here} \\\\\nmultiples of a fixed $d$ & every $b\\ge2$ & dilation: the multiples series at\n  base $b$ \\emph{is} the full-support series at base $b^d$\n  (\\mword{Erdos249257/CertificateKernel.lean}{8775}{irrational_erdosSupportSeries_multiples}{checked}) \\\\\n\\addlinespace\n\\multicolumn{3}{@{}l}{\\emph{Periodicity of the coefficient sequence}}\\\\\ninfinite and eventually periodic & every $b\\ge2$ &\n  \\mword{Erdos249257/CertificateKernel.lean}{11276}{irrational_erdosSupportSeries_eventuallyPeriodic}{checked here};\n  Luca--Tachiya prove the nonnegative purely-periodic case\n  \\cite[Theorem~1, p.~139; proof pp.~149--150]{lucatachiya2017};\n  finite rational prefixes give the infinite eventual case \\\\\na residue class; the odd numbers & every $b\\ge2$ & special cases of the row\n  above (\\mword{Erdos249257/CertificateKernel.lean}{11344}{irrational_erdosSupportSeries_residueClass}{residue class},\n  \\mword{Erdos249257/CertificateKernel.lean}{11358}{irrational_erdosSupportSeries_odd}{odd});\n  Luca--Tachiya's Example~2 strengthens the odd row to joint linear\n  independence of every finite divisor-convolution ladder, also for negative\n  integer bases with absolute value greater than one\n  \\cite[Example~2, p.~140]{lucatachiya2017} \\\\\n\\addlinespace\n\\multicolumn{3}{@{}l}{\\emph{Denominator gaps along a sparse support}}\\\\\nfactorials $\\{n!\\}$ & every $b\\ge2$ &\n  \\mword{Erdos249257/CertificateKernel.lean}{5707}{irrational_erdosSum_factorial_support}{checked here} \\\\\npowers of two $\\{2^k\\}$ & every $b\\ge2$ &\n  \\mword{Erdos249257/CertificateKernel.lean}{5731}{irrational_erdosSum_two_pow_support}{checked here} \\\\\npairwise coprime, $\\sum_{a\\in A}a^{-1}<\\infty$ & every $b\\ge2$ &\n  Erd\\H{o}s~\\cite{erdos1968}, theorem on p.~222;\n  \\mword{Erdos249257/CertificateKernel.lean}{10448}{irrational_erdosSupportSeries_pairwise_coprime}{formalised here} \\\\\n\\addlinespace\n\\multicolumn{3}{@{}l}{\\emph{Multiplicative closure: Chowla--Erd\\H{o}s, refined}}\\\\\n$F_s(E)$ and the monomial images\n  $\\{n^i:n\\in F_s(E)\\}$ & $b=q^j$ with $|q|^L\\le s$,\n  $L=\\lcm(1,\\dots,\\ell)$ &\n  Duverney--Tachiya~\\cite[Cor.~1.2, author-preprint p.~4]{duverneytachiya};\n  linear independence, not only irrationality; not formalised here \\\\\n$s$-free positive integers, with or without $1$ &\n  $b=q^j$, $2\\le |q|\\le s$, $j\\ge1$ &\n  \\emph{ibid.}, with $E$ the primes and $\\ell=i=1$;\n  deleting $1$ changes the value by $1/(b-1)\\in\\Q$ \\\\\nsquarefree & every $b=2^j$, $j\\ge1$ &\n  Duverney--Tachiya~\\cite[Cor.~1.2 and Ex.~1.1, author-preprint p.~4]{duverneytachiya};\n  joint linear independence for every finite set of such bases; not formalised here \\\\\ncoprime to a fixed $N$; sums of two squares & every $b\\ge2$ &\n  \\emph{ibid.}, Examples 1.3 and 1.2; not formalised here \\\\\n\\addlinespace\n\\multicolumn{3}{@{}l}{\\emph{Analytic}}\\\\\nprimes & $b=2$ proved & Tao--Ter\\\"av\\\"ainen~\\cite{taoteravainen2025},\n  Thm.~1.3, p.~4; proof pp.~44--56; not formalised here \\\\\nprimes, $b\\ge3$; prime powers & --- & \\emph{ibid.}, asserted as modifications\n  with details omitted in the cited version \\\\\n\\midrule\narbitrary infinite $A$ & --- & \\textbf{Open} (Problem~\\ref{res:problem}) \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\noindent In the sentence immediately following the p.~222 theorem,\nErd\\H{o}s says that pairwise coprimality can be removed by a more complicated\nargument, but he does not give that argument; p.~226 repeats that boundary.\nAccordingly the table uses only the fully printed pairwise-coprime\ntheorem, not the stronger unproved-in-print extension.\n\n\\noindent Five remarks on the table: one on containment between the\nmechanisms, three on the literature rows, and one on what ``checked here''\nmeans.\n\n\\emph{The mechanisms are not nested, and one non-containment is proved.}  The\nbase-$b$ full-support theorem is exactly the $A=\\Npos$ statement; the\nmultiples rows genuinely specialise it after a base change, but the sparse\nrows do not, and this is not merely an artefact of how they were proved.  The\ndenominator-gap criterion behind the factorial and power-of-two rows\n\\emph{provably cannot} reach the full support: the prefix lcm grows too slowly\nfor its hypothesis to hold\n(\\mword{Erdos249257/CertificateKernel.lean}{5944}{lcm_gap_hypothesis_fails_full_support}{checked}).\nConversely the analytic method of~\\cite{taoteravainen2025} reaches the primes,\nwhich are neither eventually periodic nor pairwise coprime with summable\nreciprocals.  No list contains another, and their union does not exhaust the\ninfinite supports.\n\n\\emph{One theorem of Duverney and Tachiya generates several rows.}\nTheir refinement of the Chowla--Erd\\H{o}s method\n~\\cite[Cor.~1.2, author-preprint p.~4; proof pp.~9--11]{duverneytachiya}\nproves, for a pairwise coprime sequence $E$ of polynomial growth and the set\n$F_s(E)$ of products of its members with exponents below $s$, that\n$1$ and the values $\\sum_{n\\in F_s}(q^{jn^i}-1)^{-1}$ are linearly independent\nover $\\Q$ whenever $|q|^{L}\\le s$, with $L=\\lcm(1,\\dots,\\ell)$ and $\\ell$\nbounding the exponent $i$.  This is a\nrow-generating theorem, not an isolated example: with $E$ the primes,\n$s=\\infty$ returns the full support at every base, while $s=2$ returns the\nsquarefree support.  For \\(s=2\\), the constraint forces \\(\\ell=1\\) and\n\\(|q|\\le2\\), but the free exponent \\(j\\) gives every base \\(b=2^j\\).\nThus this route excludes bases that are not powers of \\(2\\), rather than all\nbases \\(b\\ge3\\); it also excludes higher monomial degrees \\(i\\ge2\\).\nTheir conclusion is stronger than irrationality, being linear independence of\nthe whole finite family across the chosen exponents \\(j\\).\n\n\\emph{The classical full-support value now has a digit-level refinement.}\nAt base \\(2\\), Campbell writes the Erd\\H{o}s--Borwein constant in the\nequivalent forms\n\\[\n  E=\\sum_{n\\ge1}\\frac1{2^n-1}\n   =\\sum_{n\\ge1}\\frac{d(n)}{2^n}\n\\]\nand proves that the binary block \\(11\\) occurs infinitely often in its\nbase-\\(2\\) expansion\n\\cite[Theorem~1, p.~12; proof pp.~12--24]{campbell2026}.\nThis adds genuine digit-distribution information to the full-support row, but\nit neither proves normality nor addresses an arbitrary infinite support\n\\(A\\), so it does not change the open status of Problem~\\ref{res:problem}.\n\n\\emph{The two analytic rows do not have the same standing.}  Theorem 1.3\non p.~4, proved in Section~5 on pp.~44--56,\nof~\\cite{taoteravainen2025} proves the prime-support case at base~$2$, the\nseries there being $\\sum_{n\\ge1}\\omega(n)/2^n$.  The extension to every\ninteger base, and the prime-power support --- which those authors themselves\nidentify with Problem~\\ref{res:problem} --- are asserted by remark, with the\nmodifications explicitly left to the reader.  The table keeps the three apart,\nand only the first is proved.\n\n\\emph{Formalisation is not priority.}  Rows marked ``checked here'' are Lean\nstatements accepted by the pinned kernel.  For the full support that is a\nformalisation of Erd\\H{o}s; Luca and Tachiya's RIMS paper already proves the\nnonnegative purely-periodic case; a finite rational-prefix correction gives\nthe eventual-periodic extension used here\n\\cite[Theorem~1, p.~139; proof pp.~149--150]{lucatachiya2017}, while\nTheorem~A restates the broader signed purely-periodic theorem without\nreproducing its earlier proof.  No priority is claimed anywhere in this table.\n\n\\begin{remark}[signed periodic weights: a gap in the method, not in the\nmathematics]\\label{res:signed}\nDropping nonnegativity weakens the conclusion available, and it is worth being\nexact about where.  For $u\\colon\\Npos\\to\\Z$ periodic and\n$b\\ge2$, what is checked here is a \\emph{dichotomy}: the series\n$\\sum_{n\\ge1}u(n)/(b^n-1)$ is irrational, \\emph{or} some power of $b$ times\nits value is an integer\n(\\mword{Erdos249257/CertificateKernel.lean}{13847}{irrational_or_bpow_mul_eq_intCast_intWeightedErdosSeries_periodic}{checked}).\nFor a nonnegative support indicator the terminating alternative is excluded by\nthe rows above; for mixed signs this argument does not exclude it.\n\nFor a nonzero purely periodic sequence, the terminating alternative does not\noccur.  (The zero sequence is the immediate exception.)  Theorem~A in Luca\nand Tachiya's 2017 RIMS paper explicitly restates their earlier Theorem~1.1:\n$\\sum_{n\\ge1}a_n/(q^n-1)$ is irrational for every integer $q$ with $|q|>1$\nand every nonzero purely periodic integer sequence\n$(a_n)$ \\cite[Theorem~A, p.~139]{lucatachiya2017}.  Thus the disjunction above\nis an artefact of the local route on that nonzero purely periodic class, and\ntheir restatement is what shows it.  The RIMS paper\ndoes not reproduce the earlier proof or verify the broader eventual-rational\nformulation, so neither is claimed here.  The local statement retains only\nthe interest of being an independently checked argument with a visible\nboundary.  Section~\\ref{sec:squarefree} records a second, sharper instance of\nthe same phenomenon.\n\\end{remark}\n\n\\section{Squarefree support and a coordinate-dependent obstruction}\n\\label{sec:squarefree}\n\nLet $\\Asf=\\{d\\ge2: d\\text{ squarefree}\\}$: a support of density $6/\\pi^2$, far\ndenser than the primes, and not periodic.  Its values at all power-of-two bases\nare jointly linearly independent with \\(1\\), by a theorem of Duverney and\nTachiya recalled below.  This section establishes that two block-certificate\narguments used here cannot reach that support at any even base, and that the\nreason lies in a normalisation rather than in the value.\n\n\\begin{theorem}[squarefree divisor incidence]\\label{res:sfcount}\nFor every $n\\ge1$ the number of squarefree divisors of $n$ is $2^{\\omega(n)}$,\nwhere $\\omega(n)$ is the number of distinct prime factors, and hence\n\\[\n  \\scA_{\\Asf}(n)=2^{\\omega(n)}-1 .\n\\]\nIn particular $\\scA_{\\Asf}(n)$ is \\emph{odd} for every $n\\ge2$.\n\\end{theorem}\n\n\\noindent The count is\n\\lword{Erdos257/SquarefreeSupportIncidence.lean}{94}{card_squarefreeDivisors}{checked},\nthe incidence formula is\n\\lword{Erdos257/SquarefreeSupportIncidence.lean}{111}{squarefreeIncidence_eq}{checked},\nand the parity conclusion is\n\\lword{Erdos257/SquarefreeSupportIncidence.lean}{140}{odd_squarefreeIncidence}{checked}.\nThe proof is the bijection between squarefree divisors of $n$ and subsets of\nits prime factors\n(\\lword{Erdos257/SquarefreeSupportIncidence.lean}{71}{squarefreeDivisors_eq_image}{Lean}),\nso the count is $2^{\\omega(n)}$; removing $d=1$ leaves an odd number whenever\n$\\omega(n)\\ge1$.  At $n=12=2^2\\cdot3$ the squarefree divisors are $1,2,3,6$, so\n$\\scA_{\\Asf}(12)=3$; at $n=30$ they are the eight divisors of $30$ and\n$\\scA_{\\Asf}(30)=7$.\n\n\\subsection{The values are known at every power-of-two base}\n\nDuverney and Tachiya's Corollary~1.2, applied with $E$ the primes, $s=2$ and\n$\\ell=1$, gives $L=\\lcm(1)=1$ and the admissibility condition $|q|\\le2$.\nThe exponent \\(j\\) in their displayed family remains arbitrary.  Thus, for\nevery \\(h\\ge1\\), their Example~1.1 says that the numbers\n\\[\n  1,\\qquad\n  \\sum_{n\\ge1}\\frac{|\\mu(n)|}{2^{jn}-1}\\quad(1\\le j\\le h)\n\\]\nare linearly independent over $\\Q$\n~\\cite[Cor.~1.2 and Ex.~1.1, author-preprint p.~4]{duverneytachiya}.  Since\n$|\\mu|$ is the indicator of the squarefree integers including $n=1$, and\n$\\Asf$ omits only $n=1$, whose weight at base \\(2^j\\) is\n\\((2^j-1)^{-1}\\in\\Q\\),\n\\[\n  X_{\\Asf}(2^j)\n  =\\sum_{n\\ge1}\\frac{|\\mu(n)|}{2^{jn}-1}-\\frac1{2^j-1},\n\\]\nand rational translation preserves the joint linear independence with \\(1\\).\n\n\\begin{corollary}[joint power-of-two-base theorem]\\label{res:sfsettled}\nFor every \\(h\\ge1\\), the \\(h+1\\) numbers\n\\[\n  1,\\quad X_{\\Asf}(2),\\quad X_{\\Asf}(4),\\quad\\ldots,\\quad X_{\\Asf}(2^h)\n\\]\nare linearly independent over \\(\\Q\\).  In particular, every\n\\(X_{\\Asf}(2^j)\\), \\(j\\ge1\\), is irrational.\n\\end{corollary}\n\n\\noindent Two boundaries on that.  It is a citation, not a formalisation:\nnothing in this development proves it.  The admissibility condition\n$|q|^L\\le s$ at \\(s=2\\) allows only \\(|q|=2\\), so this citation covers the\nbases \\(2^j\\) and does not cover bases that are not powers of \\(2\\); this note\nproves no result about those remaining values.\n\n\\subsection{Two block-certificate hypotheses have no instance}\n\nFix $b\\ge2$ and a nonnegative coefficient sequence $f$.  The two hypotheses\nused below are best printed rather than named.  For every precision $q\\ge1$,\nthey ask for $N,K,L,C\\in\\N$ with $K\\le L$ satisfying the common conditions\n\\[\n\\begin{gathered}\n \\sum_{r=K+1}^{L}f(N+r)b^{L-r}\\le C,\\qquad\n \\exists t\\in\\N:\\ f(N+L+1+t)>0,\\\\\n q(C+N+L+2)<b^L,\n\\end{gathered}\n\\tag{3}\\label{eq:block-common}\n\\]\ntogether with one of the two first-block conditions\n\\[\n\\begin{array}{ll}\n\\text{digitwise:}& b^r\\mid f(N+r)\\quad(1\\le r\\le K),\\\\[2mm]\n\\text{weighted:}&\nb^K\\mid\\displaystyle\\sum_{r=1}^{K}f(N+r)b^{K-r}.\n\\end{array}\n\\tag{4}\\label{eq:block-first}\n\\]\nThus the data $N,K,L,C$ may depend on $q$.  The parity of\nTheorem~\\ref{res:sfcount} refutes both alternatives for the squarefree\nincidence sequence at every even base.\n\n\\begin{corollary}[neither divisibility-first hypothesis has an instance]\\label{res:blind}\nFor the squarefree support $\\Asf$ and every even base $b\\ge2$, neither the digitwise\nnor the carry-aware block-certificate hypothesis holds.  Both fail already at\nprecision $q=b^2$.\n\\end{corollary}\n\n\\begin{proof}\nThe hypothesis quantifies over every precision, so it is enough to exhibit one\nprecision at which no admissible data exist; we take $q=b^{2}$ and split on\nwhether the first block is empty.\n\nBoth hypotheses ask, for every precision $q$, for $N$, $K\\le L$ and $C$ with a\nfirst-block condition on $\\scA_{\\Asf}(N+1),\\dots,\\scA_{\\Asf}(N+K)$, a middle\nbound $\\sum_{r=K+1}^{L}\\scA_{\\Asf}(N+r)b^{L-r}\\le C$, a nonzero coefficient\nbeyond $N+L$, and $q(C+N+L+2)<b^{L}$.  Take $q=b^2$ and write $f=\\scA_{\\Asf}$.\n\nSuppose $K\\ge1$.  The digitwise condition requires $b^{r}\\mid f(N+r)$ for\n$1\\le r\\le K$.  If $N+K\\ge2$, its instance at $r=K$ makes the even number\n$b$ divide the odd number $f(N+K)$.  The carry-aware condition requires\n$b^{K}\\mid\\sum_{r=1}^{K}f(N+r)b^{K-r}$; reducing modulo $b$ kills every term\nbut $r=K$, so the same contradiction follows when $N+K\\ge2$.\n\nThe only remaining case with $K\\ge1$ is $N=0$, $K=1$.  If $L=1$, then\n$q(C+N+L+2)\\ge3b^2>b$; if $L\\ge2$, the $r=2$ term in the middle sum is\n$f(2)b^{L-2}=b^{L-2}$, so $qC\\ge b^L$.  Both contradict the strict size\ninequality.\n\nSuppose $K=0$, where both first-block conditions are vacuous.  If $L=0$ the\nmiddle sum is empty and $q(C+N+2)<b^{0}=1$ is impossible.  If $L\\ge1$ the\n$r=1$ term gives $C\\ge b^{L-1}$ whenever $N\\ge1$, and again the size\ninequality fails.  If $N=0$ and $L=1$, its left side is at least\n$3b^2>b$; if $N=0$ and $L\\ge2$, the $r=2$ term gives\n$C\\ge b^{L-2}$ and hence $qC\\ge b^L$.  These exhaust the cases.\n\\end{proof}\n\n\\noindent This is an argument in the prose, drawn from the checked incidence\nand parity lemmas, and is marked as such in\n\\emph{Statements and declarations}.\n\n\\subsection{The obstruction is a normalisation, not the value}\n\nCorollary~\\ref{res:blind} is a statement about two arguments failing on a value\nthat Corollary~\\ref{res:sfsettled} shows to be irrational.  The question it\nraises is therefore not whether the value can be reached, but what the failure\nis a property \\emph{of}.  It is a property of where the support starts.\n\nAdjoin $1$ to the support and write $\\Asf^{+}=\\Asf\\cup\\{1\\}$, the full\nsquarefree support.  Then\n\\[\n  X_{\\Asf^{+}}(b)-X_{\\Asf}(b)=\\frac1{b-1}\\in\\Q ,\n\\]\nso the two supports pose the same irrationality question at every base, while\nthe divisor incidence changes from $2^{\\omega(n)}-1$ to $2^{\\omega(n)}$ ---\nfrom odd to even at every $n\\ge2$.  The parity obstruction of\nCorollary~\\ref{res:blind} evaporates under a shift that provably cannot change\nthe answer.  More is true: the shifted first-block condition at base~$2$ asks for\n$2^{r}\\mid 2^{\\omega(N+r)}$, that is $\\omega(N+r)\\ge r$ for $1\\le r\\le K$, and\na Chinese-remainder construction reserving $r$ fresh primes for each shift $r$\nsupplies such an $N$ for every $K$.\n\nIt does \\emph{not} follow that either argument certifies the shifted support:\nthe opening block is one of the conditions listed above, and the middle bound\nand the arithmetic inequality are untouched by this observation.  What does\nfollow is a methodological point: an obstruction stated against a coefficient\nsequence can be an artefact of the normalisation chosen for that sequence.\nBefore a no-go result is reported as a property of a problem, the coordinate\nit is stated in should be varied by a transformation the problem is known to\nbe invariant under.  Here the transformation is adding one rational number,\nand it removes the obstruction entirely.\n\nThe same invariance holds for every finite change, not only this one.  If\n$A\\mathbin{\\triangle}B$ is finite, choose $M$ above all of its elements.\nThe two support series then have the same tail beyond $M$, while each omitted\nprefix is rational.  The two directions of this argument are exactly the\nchecked prefix lemmas\n\\mword{Erdos249257/CertificateKernel.lean}{9139}{irrational_erdosSupportSeries_of_tail}{Lean}\nand\n\\mword{Erdos249257/CertificateKernel.lean}{9148}{irrational_erdosSupportSeries_tail_of_irrational}{Lean}.\nThus $X_A(b)$ is irrational if and only if $X_B(b)$ is irrational for every\ninteger $b\\ge2$.  The present shift is the smallest instance of a general\nchecked finite-change principle.\n\nThis is the second time in this note that a boundary turns out to belong to\nthe method rather than to the mathematics.  In Remark~\\ref{res:signed} the\nterminating alternative of the signed periodic dichotomy is empty, and a\ntheorem of Luca and Tachiya is what shows it; here a parity obstruction\nsurvives only until the support is shifted by one element.  In each case the\ncorrection came from outside the development --- once from the literature, once\nfrom asking what the statement was invariant under.  A formalised no-go result\ncarries exactly the authority of its hypotheses, and its hypotheses include the\ncoordinates it was written in.\n\n\\section{Achievement-set geometry and the value \\texorpdfstring{$1/2$}{1/2}}\n\\label{sec:geometry}\n\nProblem~\\ref{res:problem} asks whether every value obtainable from infinitely\nmany of the weights $\\wgt n=(2^n-1)^{-1}$ is irrational.  The set of all values\nobtainable, from finite and infinite selections alike, is the achievement set\n$\\Ach=\\{\\sum_{n\\ge1}\\varepsilon_n\\wgt n:\\varepsilon_n\\in\\{0,1\\}\\}$, and this\nsection is about its geometry.\n\n\\begin{theorem}[geometry of $\\Ach$]\\label{res:geometry}\n$\\Ach$ is compact, perfect, totally disconnected and nowhere dense, and has\nLebesgue measure~$1$.\n\\end{theorem}\n\n\\noindent Checked as\n\\mword{Erdos249257/GreedyAchievementSet.lean}{656}{isCompact_mersenneAchievementSet}{compact},\n\\mword{Erdos249257/GreedyAchievementSet.lean}{1620}{perfect_mersenneAchievementSet}{perfect},\n\\mword{Erdos249257/GreedyAchievementSet.lean}{1636}{isTotallyDisconnected_mersenneAchievementSet}{totally disconnected},\n\\mword{Erdos249257/GreedyAchievementSet.lean}{1645}{isNowhereDense_mersenneAchievementSet}{nowhere dense},\nand\n\\mword{Erdos249257/GreedyAchievementSet.lean}{996}{volume_mersenneAchievementSet}{of measure one}.\nSo $\\Ach$ is a fat Cantor set: strict tail domination\n$\\sum_{\\ell>n}\\wgt\\ell<\\wgt n$ opens a gap at every level, while the total\nmeasure is not lost.  The division of credit is exact.  The strict inequality,\nthe resulting distinctness of subsums over distinct supports, and the Cantor\nconclusion for every integer base are Remark 4.1 (p.~13) of Kova\\v{c} and\nTao~\\cite{kovactao}; no novelty is\nclaimed for any of the three.  That remark makes no metric assertion, and\nstrict tail domination does not determine the measure: the weights $3^{-n}$\nsatisfy $\\sum_{\\ell>n}3^{-\\ell}=3^{-n}/2<3^{-n}$ and produce a null achievement\nset, the base-$3$ digits-in-$\\{0,1\\}$ Cantor set.  So the measure-one clause is\nadded here rather than formalised from there, and it is the arithmetic of the\nMersenne weights that supplies it.  With $T_n=\\sum_{k>n}\\wgt k$, the standard\nlevel-$n$ convex-hull cover consists of $2^n$ disjoint intervals of length\n$T_n$, its nested intersection is $\\Ach$, and\n\\[\n  2^nT_n=\\sum_{j\\ge1}\\frac{2^n}{2^{n+j}-1}\\longrightarrow\\sum_{j\\ge1}2^{-j}=1,\n\\]\ndominated by $2^{1-j}$.  Continuity of measure from above therefore gives\n$\\lambda(\\Ach)=1$.\n\nMembership is characterised level by level, by a greedy expansion.  Run through\n$n=1,2,\\dots$ carrying a remainder, initially the target $x$, and at level $n$\nsubtract $\\wgt n$ from the remainder if $\\wgt n$ does not exceed it, leaving\nthe remainder unchanged otherwise; call a level at which nothing is subtracted\n\\emph{skipped}.  Say that the expansion \\emph{survives} level $n$ if the\nremainder after that level is at most the remaining mass $T_n$.  Then\n$x\\in\\Ach$ if and only if $x\\ge0$ and the expansion survives every level\n(\\mword{Erdos249257/GreedyAchievementSet.lean}{1445}{mem_mersenneAchievementSet_iff_greedy_survival}{Lean}).\nNon-membership of a nonnegative target is therefore visible at a single level,\nand for a rational target the check is exact rational arithmetic.  For\n$x=3/4$, the greedy algorithm skips $\\wgt1=1$ and leaves residual $3/4$,\nwhile the exact zero-lookahead upper bound for the remaining tail is\n$2\\wgt2=2/3<3/4$; hence $3/4\\notin\\Ach$\n(\\mword{Erdos249257/GreedyAchievementSet.lean}{1748}{three_fourths_not_mem_mersenneAchievementSet}{Lean}).\n\nThe selector-to-subsum map sending a $0/1$ string $(\\varepsilon_n)$ to\n$\\sum_n\\varepsilon_n\\wgt n$, which the strict tail inequality makes injective,\nremains injective after restriction to any subfamily.  That matters because\nProblem~\\ref{res:problem} quantifies\nover every infinite support rather than over $\\Npos$.  For a set $J$ of future\noffsets write $T_J(n)=\\sum_{k\\in J}\\wgt{n+k+1}$ for the tail restricted to $J$\n(\\lword{Erdos257/MersenneSubseriesRigidity.lean}{20}{selectedMersenneTail}{definition},\nsummable at\n\\lword{Erdos257/MersenneSubseriesRigidity.lean}{24}{summable_selectedMersenneTail}{Lean}).\nThen $T_J(n)<\\wgt n$ for every $J$ and every $n\\ge1$\n(\\lword{Erdos257/MersenneSubseriesRigidity.lean}{30}{selectedMersenneTail_lt_weight}{checked}),\nsince deleting weights only shrinks a tail that already sits below $\\wgt n$ at\nthe full support.  Consequently the digit map is injective on strings\nsupported in $J$\n(\\lword{Erdos257/MersenneSubseriesRigidity.lean}{54}{supportedMersenneDigitValue_injective}{checked}),\nstated on the subtype of\n\\lword{Erdos257/MersenneSubseriesRigidity.lean}{45}{SupportedMersenneDigits}{digit strings vanishing off $J$}\nand evaluated by the\n\\lword{Erdos257/MersenneSubseriesRigidity.lean}{49}{supportedMersenneDigitValue}{restricted digit map},\nso the statement is about the subseries itself and not a projection of the\nfull one.  Thus uniqueness of the selector survives passage to an arbitrary\nsubfamily; this is an injectivity statement, not a statement about binary\ndigits of the value.\nThat is a constraint on the shape of an argument, not on the supports: it\ndecides no value, and it is weaker than any statement in\nSection~\\ref{sec:forced}.\n\n\\subsection{Geometry after restricting the allowed exponents}\n\nThe injectivity statement has a geometric completion for every set\n$J\\subseteq\\N$ of allowed digit positions.  Let\n\\[\n  \\Ach_J=\n  \\left\\{\\sum_{k\\in J}\\varepsilon_k\\wgt{k+1}:\n    \\varepsilon_k\\in\\{0,1\\}\\right\\}.\n\\]\nFormally, the allowed strings form the\n\\lword{Erdos257/MersenneSubseriesRigidity.lean}{63}{supportedMersenneDigitSet}{supported digit set},\nwhich is\n\\lword{Erdos257/MersenneSubseriesRigidity.lean}{66}{isClosed_supportedMersenneDigitSet}{closed}.\nIts range is the\n\\lword{Erdos257/MersenneSubseriesRigidity.lean}{76}{supportedMersenneAchievementSet}{restricted achievement set};\nthe range and image descriptions agree by\n\\lword{Erdos257/MersenneSubseriesRigidity.lean}{79}{supportedMersenneAchievementSet_eq_image}{the image theorem}.\nConsequently $\\Ach_J$ is\n\\lword{Erdos257/MersenneSubseriesRigidity.lean}{90}{isCompact_supportedMersenneAchievementSet}{compact}\nand\n\\lword{Erdos257/MersenneSubseriesRigidity.lean}{96}{isClosed_supportedMersenneAchievementSet}{closed}.\nIt lies inside $\\Ach$ by\n\\lword{Erdos257/MersenneSubseriesRigidity.lean}{103}{supportedMersenneAchievementSet_subset}{support restriction}\nand is therefore\n\\lword{Erdos257/MersenneSubseriesRigidity.lean}{112}{isNowhereDense_supportedMersenneAchievementSet}{nowhere dense}.\nIf $J$ is infinite, the digit space is\n\\lword{Erdos257/MersenneSubseriesRigidity.lean}{120}{preperfect_supportedMersenneDigitSet}{preperfect};\ninjectivity transfers that property to\n\\lword{Erdos257/MersenneSubseriesRigidity.lean}{150}{preperfect_supportedMersenneAchievementSet}{$\\Ach_J$},\nso the closed set is\n\\lword{Erdos257/MersenneSubseriesRigidity.lean}{167}{perfect_supportedMersenneAchievementSet}{perfect}.\nThus every infinite allowed support gives a compact perfect nowhere-dense set\nwith a unique selector for each point.\n\nThe metric classification is exact.  The summability lemma\n\\lword{Erdos257/MersenneSubseriesRigidity.lean}{175}{summable_mersenneDigitTerm}{controls digit terms},\nand changing one coordinate changes the value by precisely its signed weight\nby\n\\lword{Erdos257/MersenneSubseriesRigidity.lean}{182}{positiveMersenneDigitValue_update}{the update formula}.\nIf $k\\notin J$, allowing $k$ splits the new set into $\\Ach_J$ and its translate\nby $\\wgt{k+1}$\n\\lword{Erdos257/MersenneSubseriesRigidity.lean}{203}{supportedMersenneAchievementSet_insert}{exactly}.\nThe two pieces are\n\\lword{Erdos257/MersenneSubseriesRigidity.lean}{262}{disjoint_supportedMersenneAchievementSet_translate}{disjoint},\nso adjoining one coordinate\n\\lword{Erdos257/MersenneSubseriesRigidity.lean}{286}{volume_supportedMersenneAchievementSet_insert}{doubles the volume}.\nAt $J=\\N$, the restricted set is the full set $\\Ach$\n\\lword{Erdos257/MersenneSubseriesRigidity.lean}{300}{supportedMersenneAchievementSet_univ}{by definition}.\n\n\\begin{theorem}[volume dichotomy for every allowed support]\\label{res:supportvolume}\nFor every $J\\subseteq\\N$, exactly one of the following measure formulas\napplies:\n\\[\n\\begin{array}{ll}\nJ=F^{\\,c}\\text{ for a finite }F,\n  &\\lambda(\\Ach_J)=2^{-|F|},\\\\[2mm]\nJ^c\\text{ is infinite},\n  &\\lambda(\\Ach_J)=0.\n\\end{array}\n\\]\n\\end{theorem}\n\n\\noindent For finite $F$, the division-free identity\n\\lword{Erdos257/MersenneSubseriesRigidity.lean}{313}{volume_supportedMersenneAchievementSet_finset_compl_scaled}{multiplies the face volume by $2^{|F|}$}\nand its solved form\n\\lword{Erdos257/MersenneSubseriesRigidity.lean}{349}{volume_supportedMersenneAchievementSet_finset_compl}{gives $2^{-|F|}$}.\nMonotonicity under enlarging $J$ is\n\\lword{Erdos257/MersenneSubseriesRigidity.lean}{358}{supportedMersenneAchievementSet_mono}{checked}.\nIt traps a set with infinitely many forbidden coordinates below\nfinite-codimension faces of arbitrarily small dyadic measure, yielding\n\\lword{Erdos257/MersenneSubseriesRigidity.lean}{368}{volume_supportedMersenneAchievementSet_eq_zero_of_compl_infinite}{measure zero}.\nThe two cases are assembled in\n\\lword{Erdos257/MersenneSubseriesRigidity.lean}{397}{volume_supportedMersenneAchievementSet_dichotomy}{the formal dichotomy}.\nThis theorem classifies the size of the value set generated inside any\nprescribed family of exponents.  It does not classify the arithmetic nature\nof its individual points and therefore does not settle Problem~\\ref{res:problem}.\n\n\\subsection{The value \\texorpdfstring{$1/2$}{1/2}}\n\nA rational point of $\\Ach$ attained by an infinite support refutes\nProblem~\\ref{res:problem}.  The distinguished candidate is $1/2$, and\nTheorem~\\ref{res:half} gives a self-contained pair of exact alternatives.\n\nThe greedy expansion of $1/2$ is an exact finite computation as far as one\ncares to take it.  Through level $21$ it takes the exponents\n\\[\n  2,\\ 3,\\ 6,\\ 7,\\ 14,\\ 20,\\ 21\n\\]\nand skips the others, the skipped runs being $\\{1\\}$, $\\{4,5\\}$,\n$\\{8,\\dots,13\\}$ and $\\{15,\\dots,19\\}$; every level through $21$ is survived.\nThe second condition below asks whether the list of skipped exponents is\ninfinite, and no finite computation answers that.\n\nIn the statement, $u=(u_1,\\dots,u_d)\\in\\{0,1\\}^{d}$ is a finite prefix,\n$V(u)=\\sum_{n\\le d}u_n\\wgt n$ is its value, and $T_n=\\sum_{k>n}\\wgt k$ as\nabove.\n\n\\begin{theorem}[membership and non-membership at $1/2$]\\label{res:half}\nThe value $1/2$ belongs to $\\Ach$ if and only if its canonical greedy\nexpansion omits infinitely many exponents.  Dually, $1/2\\notin\\Ach$ is\nequivalent to the existence of a finite \\emph{fatal gap} --- a prefix $u$\nthrough rank $d$ with\n$V(u)+T_{d+1}<\\tfrac12<V(u)+\\wgt{d+1}$, which makes every continuation miss\n--- and to the greedy orbit having a last skipped exponent.\nMoreover no finite support has value $1/2$.\n\\end{theorem}\n\n\\noindent Checked as\n\\mword{Erdos249257/GreedyAchievementSet.lean}{2514}{half_mem_mersenneAchievementSet_iff_greedySkippedSupport_infinite}{the greedy form},\n\\mword{Erdos249257/HalfCylinderHalfMembershipClassification.lean}{126}{half_mem_mersenneAchievementSet_iff_unboundedTerminalFalse}{the terminal-bit form},\n\\mword{Erdos249257/HalfCylinderHalfMembershipClassification.lean}{213}{half_mem_mersenneAchievementSet_iff_exists_unboundedSkippedRanksAlong}{the skipped-rank form},\n\\mword{Erdos249257/HalfCylinderFatalGapRightTail.lean}{781}{seamGreedyEventuallyRight_iff_existsFatalHalfGap}{the fatal-gap equivalence},\n\\mword{Erdos249257/HalfCylinderFatalGapRightTail.lean}{787}{seamGreedyEventuallyRight_iff_half_not_mem}{its transfer to non-membership},\nand\n\\mword{Erdos249257/HalfCarryReachability.lean}{589}{finite_boolSupport_ne_half}{the finite-support exclusion}.\n\nThe formal source also proves equivalent terminal-bit and unbounded\nskipped-rank formulations in its internal half-cylinder seam coordinate; the\ntwo middle source links above are those versions.  Their definitions are not\nneeded for the self-contained greedy and fatal-gap statement used here.\nThe two sides are asymmetric: non-membership is witnessed by a \\emph{finite}\nfatal gap and is therefore semi-decidable, whereas membership is an infinite\ncondition.  Computing further can only raise a lower bound on where a fatal\ngap could occur; it cannot establish survival.\n\nOne natural route to $1/2$ is closed.  The Boolean support selected by the\nnegative values of the M\\\"obius function has value exactly $1/2$ plus the\npositive M\\\"obius tail, hence at least $1/2+1/63$\n(\\mword{Erdos249257/MobiusSignSupportNoGo.lean}{111}{tsum_negativeMobius_eq_half_add_positiveMobiusTail}{identity},\n\\mword{Erdos249257/MobiusSignSupportNoGo.lean}{150}{half_add_one_div_sixty_three_le_tsum_negativeMobius}{bound}).\nThat closes the sign-truncation construction; it excludes no other support.\n\n\\section{Open problems}\\label{sec:open}\n\nFor the base-$2$ universal problem, two endpoint propositions organise the\nremaining discussion.  They are not independent, and the relation between\nthem is asymmetric.  No finite support has value $1/2$\n(Theorem~\\ref{res:half}), so $1/2\\in\\Ach$ would exhibit an \\emph{infinite}\nsupport with a rational value and thereby refute\nProblem~\\ref{res:problem}; equivalently, a positive answer to\nProblem~\\ref{res:problem} puts $1/2$ outside $\\Ach$.  The converse direction\ndoes not follow: $1/2\\notin\\Ach$ would give a finite fatal-gap witness and\neliminate this candidate, but would not imply the universal statement.  The\nhalf-value question is therefore a one-sided test of \\#257, not a second\nproblem beside it.\n\n\\begin{enumerate}\n\\item \\textbf{Problem~\\ref{res:problem} itself}, for arbitrary infinite $A$.\n  Section~\\ref{sec:map} is a list of families and does not approach a\n  universal statement; Section~\\ref{sec:forced} constrains every hypothetical\n  counterexample without excluding one.  Every counterexample would have\n  unbounded scaled-tail states, sublogarithmic divisor-coverage gaps and an\n  admissible Boolean--M\\\"obius scaled-tail sequence; Theorem~\\ref{res:mass}\n  adds its lower bound only under its stated convergence and odd-denominator\n  hypotheses.  No contradiction among the applicable constraints is known.\n\\item \\textbf{Membership of $1/2$ in $\\Ach$.}  Theorem~\\ref{res:half} makes\n  this exactly equivalent to infinitely many greedy skips, and to a fatal gap\n  on the other side; neither is proved.  A proof of non-membership would take\n  the form of one finite fatal gap.\n\\end{enumerate}\n\n\\noindent For scope, all squarefree values at power-of-two bases, jointly in\neach finite family, are settled by Corollary~1.2 and Example~1.1 of\n~\\cite[author-preprint p.~4]{duverneytachiya}\n(Corollary~\\ref{res:sfsettled}).  Their cited\ncorollary does not cover bases that are not powers of \\(2\\), and this note\nmakes no global open-status or priority claim for those values.\n\n\\section*{Statements and declarations}\n\nThis manuscript is authored exposition, not proof authority.  The linked Lean\nsnapshot is authoritative only for its exact propositions; kernel checking\nestablishes that a proposition was proved, not that it is interesting, novel,\nor sufficient.  The proof of Corollary~\\ref{res:blind}, the derivation of\nCorollary~\\ref{res:sfsettled} from\n~\\cite[Cor.~1.2 and Ex.~1.1, author-preprint p.~4]{duverneytachiya},\nthe finite-change\nargument (from the two checked prefix lemmas), the shift identity and\nChinese-remainder remark in Section~\\ref{sec:squarefree}, and the restricted\nselector-injectivity sentence in Section~\\ref{sec:geometry} are arguments in the\nprose rather than checked statements; each is drawn from the checked lemmas\ncited beside it or from the cited literature.  The numerical instances --- the\ntable of finite supports in Section~\\ref{sec:period}, the incidence and\nzero-window examples, the reciprocal-mass and Boolean--M\\\"obius instances, the\n$3/4$ certificate, and the greedy expansion of $1/2$ through level~$21$ ---\nare direct computations, reported as computations and not as theorems.\nResults attributed to Campbell, to Duverney and Tachiya, to Tao and\nTer\\\"av\\\"ainen, to Luca and Tachiya, and to Kova\\v{c} and Tao are cited from\nthe literature and are not formalised here.\n\n\\paragraph{Declaration of generative AI use.} \\paperfullaideclaration\n\nFormal authority is the pinned kernel's acceptance of an exact proposition;\nno model output carries any, and neither does this sentence.\n\nErd\\H{o}s Problem~\\#257 remains open.\n\n\\begin{thebibliography}{10}\n\\bibitem{campbell2026} J.~M. Campbell,\n  \\href{https://arxiv.org/abs/2605.24160}\n  {\\emph{On the binary digits of the Erd\\H{o}s--Borwein constant}},\n  arXiv:2605.24160v1, 2026.  Theorem~1 on p.~12 proves that the binary block\n  \\(11\\) occurs infinitely often in the base-\\(2\\) expansion of the\n  full-support value; its proof occupies pp.~12--24.\n\\bibitem{duverneytachiya} D.~Duverney and Y.~Tachiya,\n  \\href{https://danielduverney.fr/documents/theorie-des-nombres/DuverneyTachiya190522.pdf}\n  {\\emph{Refinement of the Chowla--Erd\\H{o}s method and linear independence\n  of certain Lambert series}},\n  Forum Math.\\ 31 (2019), no.~6, 1557--1566,\n  \\href{https://doi.org/10.1515/forum-2018-0299}{DOI}.  Corollary 1.2 gives\n  the general $F_s(E)$ theorem and its monomial images under\n  $|q|^{\\lcm(1,\\dots,\\ell)}\\le s$; Example 1.1 gives the joint squarefree\n  family at all bases \\(2^j\\), \\(j\\ge1\\).  Both are on p.~4 of the linked\n  author preprint; the proof is on pp.~9--11.\n\\bibitem{taoteravainen2025} T.~Tao and J.~Ter\\\"av\\\"ainen,\n  \\emph{Quantitative correlations and some problems on prime factors of\n  consecutive integers}, arXiv:2512.01739 (submitted December 2025, revised\n  April 2026).  Theorem 1.3 proves $\\sum_{n\\ge1}\\omega(n)/2^n=\\sum_p(2^p-1)^{-1}$\n  irrational, settling the prime-support case of \\#257 at base~$2$; the\n  extension to every integer base and the prime-power support are stated as\n  remarks with the modifications left to the reader.  The theorem is on\n  p.~4 and its proof is Section~5, pp.~44--56, in arXiv v2.\n\\bibitem{lucatachiya2017} F.~Luca and Y.~Tachiya,\n  \\href{https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2014-14.pdf}\n  {\\emph{Linear independence results for the values of divisor functions\n  series}}, RIMS K\\^oky\\^uroku No.~2014 (2017), 138--150.\n  Theorem~A on p.~139 explicitly restates their earlier Theorem~1.1 for\n  nonzero purely periodic integer weights.  Theorem~1 is on p.~139, its\n  examples are on p.~140, and its proof is on pp.~149--150.\n\\bibitem{kovactao} V.~Kova\\v{c} and T.~Tao, \\emph{On several irrationality\n  problems for Ahmes series}, Acta Math.\\ Hungar.\\ 175 (2025), 572--608,\n  \\href{https://doi.org/10.1007/s10474-025-01528-0}{DOI}.  Remark 4.1\n  (p.~13) records the strict tail inequality and the Cantor structure.\n  Theorem~2.3 (p.~5; proof pp.~13--14) constructs rational merged sums from\n  several bases under its mass hypothesis; it is not a fixed-base\n  counterexample.\n\\bibitem{erdos1948} P.~Erd\\H{o}s, \\emph{On arithmetical properties of Lambert\n  series}, J.~Indian Math.\\ Soc.\\ 12 (1948), 63--66.\n\\bibitem{erdos1968} P.~Erd\\H{o}s,\n  \\href{https://users.renyi.hu/~p_erdos/1969-09.pdf}{\\emph{On the\n  irrationality of certain series}}, Math.\\ Student 36 (1968), 222--226\n  (issued 1969).  The theorem on p.~222 treats pairwise-coprime support with\n  convergent reciprocal sum at every integer base $b\\ge2$; the claimed\n  removal of pairwise coprimality is stated without proof.\n\\bibitem{erdosgraham1980} P.~Erd\\H{o}s and R.~L. Graham,\n  \\href{https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf}{\\emph{Old\n  and New Problems and Results in Combinatorial Number Theory}},\n  Monogr. Enseign. Math. 28, Geneva, 1980, p.~62.\n\\bibitem{erdos1988} P.~Erd\\H{o}s, \\emph{On the irrationality of certain\n  series: problems and results}, in A.~Baker (ed.), \\emph{New Advances in\n  Transcendence Theory}, Cambridge UP, 1988, pp.~102--109,\n  doi:\\href{https://doi.org/10.1017/CBO9780511897184.009}\n  {10.1017/CBO9780511897184.009}.\n\\bibitem{erdosproblems} T.~F. Bloom,\n  \\href{https://www.erdosproblems.com/257}{\\emph{Erd\\H{o}s Problem \\#257}},\n  \\texttt{erdosproblems.com/257}, accessed 28 July 2026 (page displays\n  ``last edited 15 April 2026'').  The current record labels the universal\n  fixed-base problem open, cites \\texttt{[Er68d]},\n  \\texttt{[ErGr80, p.~62]} and \\texttt{[Er88c, p.~105]}, records the\n  Tao--Ter\\\"av\\\"ainen prime and prime-power cases and the Kova\\v{c}--Tao\n  refutation of a broader varying-denominator speculation, and explicitly\n  describes its status as the website owner's present assessment rather than\n  a literature-completeness guarantee.\n\\end{thebibliography}\n\n\\end{document}\n","dependencies":[{"record_type":"paper_source_dependency","tex_reference":"problem-note-preamble","source_path":"paper/problem-note-preamble.tex","media_type":"text/x-tex","sha256":"sha256:e7378fe7784916681e7faf96c1ea7a8bb941a416cc651b97512e849bcaeb63f7","byte_count":4990,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Shared preamble for the Erd\u0151s Problem Notes: the short, standalone,\n% problem-owned manuscripts covering the expansion library `ErdosProblems`.\n%\n% One note owns one Erd\u0151s problem.  Each note repeats whatever context its own\n% reader needs, so that a reader who arrives at a single problem never has to\n% assemble it from the gateway paper.  Deliberate repetition across notes is the\n% design, not an oversight.\n%\n% Source links resolve against one pinned formal-source commit, declared once\n% below.  `scripts/check_problem_note_sources.py` verifies that every authored\n% (file, line, declaration) triple names that exact declaration in the pinned\n% snapshot, so a link cannot silently rot as later work moves lines.\n\n\\usepackage{paper-house-style}\n\\usepackage{array}\n\\usepackage{booktabs}\n\\usepackage{enumitem}\n\\setlist{topsep=0.35em,itemsep=0.2em}\n\\usepackage[colorlinks=true,linkcolor=linkinternal,urlcolor=linkexternal,\n  citecolor=linkinternal]{hyperref}\n\\hypersetup{bookmarksnumbered=true,bookmarksopen=true,bookmarksopenlevel=1,\n  pdfauthor={Will Cook},pdflang={en-GB}}\n\n% ---------------------------------------------------------------------------\n% Pinned formal source.  \\commit names the checkpoint every link resolves\n% against; the checker reads that snapshot rather than the working tree, so the\n% printed coordinates stay correct however later waves move declarations.\n% ---------------------------------------------------------------------------\n\\newcommand{\\commit}{d6d344d9557faee19276b01a1c1f20d7c08f03d5}\n\\newcommand{\\commitshort}{d6d344d9557f}\n\\newcommand{\\repobase}{https://github.com/wcook04/plectis-lean-erdos249-257/blob/\\commit}\n\\newcommand{\\sourceurl}{https://github.com/wcook04/plectis-lean-erdos249-257/tree/\\commit}\n\\newcommand{\\PX}{ErdosProblems}\n\n% Declaration names stay inside link targets.  The printed label is either a\n% spaced, lower-case reading of the declaration name (\\lref) or a short authored\n% phrase (\\lword); neither prints a module path or a raw Lean identifier.\n\\ExplSyntaxOn\n\\cs_new_protected:Npn \\noteleanlabel #1\n  {\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_to_str:n {#1} }\n    \\regex_replace_all:nnN { _+ } { \\c{space} } \\l_tmpa_tl\n    \\regex_replace_all:nnN\n      { ([a-z0-9])([A-Z]) }\n      { \\1\\c{space}\\2 }\n      \\l_tmpa_tl\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_lower_case:n { \\tl_use:N \\l_tmpa_tl } }\n    \\tl_use:N \\l_tmpa_tl\n  }\n\\ExplSyntaxOff\n\\newcommand{\\leanlabel}[1]{\\noteleanlabel{#1}}\n\\newcommand{\\lref}[3]{\\href{\\repobase/\\PX/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\lword}[4]{\\href{\\repobase/\\PX/#1\\#L#2}{#4}}\n\\newcommand{\\lloc}[2]{\\href{\\repobase/\\PX/#1\\#L#2}{Lean source}}\n\n% Links into a sibling library, named repository-relative.  The expansion\n% library is implicit in \\lref/\\lword, because that is where problem-owned\n% work lands.  Several problems have their headline theorem in the older\n% reviewed #249/#257 corpus, and a note that could not name that library\n% would have to paraphrase a checked statement instead of linking it.\n\\newcommand{\\mref}[3]{\\href{\\repobase/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\mword}[4]{\\href{\\repobase/#1\\#L#2}{#4}}\n\\newcommand{\\mloc}[2]{\\href{\\repobase/#1\\#L#2}{Lean source}}\n\\emergencystretch=2em\n\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\theoremstyle{definition}\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{example}[theorem]{Example}\n\\newtheorem{problem}[theorem]{Problem}\n\\theoremstyle{remark}\n\\newtheorem*{remark}{Remark}\n\n\\newcommand{\\N}{\\mathbb{N}}\n\\newcommand{\\Npos}{\\mathbb{N}_{>0}}\n\\newcommand{\\Z}{\\mathbb{Z}}\n\\newcommand{\\Q}{\\mathbb{Q}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\lcm}{\\operatorname{lcm}}\n\\newcommand{\\ph}{\\varphi}\n\n% The series line printed under every note's title block.\n\\newcommand{\\seriesline}{Erd\\H{o}s Problem Notes}\n\n% Problem-note compatibility name for the corpus-owned page-one disclosure in\n% paper-house-style.sty.\n\\newcommand{\\noteprovenance}{\\paperprovenance}\n\n% The status paragraph every note carries, in its own words, before any result.\n% Kernel checking establishes the exact Lean proposition; it does not establish\n% that the proposition is the interesting one, that it is new, or that the\n% Erd\u0151s problem is closed.\n\\newcommand{\\statusboundary}{%\n  \\emph{Status.}  The problem treated here is open, and this note does not\n  close it.  Every statement below marked as checked is a proposition that the\n  pinned Lean kernel accepts from the sources this note links to, with no\n  \\texttt{sorry}, no added axiom, and no unchecked evaluation.  That is a claim\n  about the formal statement, not about its mathematical interest, its\n  novelty, or the original problem.  The unresolved obligations are named\n  exactly, in their own section, and none of the finite computations,\n  reductions, or no-go results here removes one of them.}\n"}],"external_verification_route":"https://raw.githubusercontent.com/wcook04/plectis-lean-erdos249-257/main/paper/erdos-257-mersenne-support-subseries.tex"},{"record_type":"paper","paper_id":"erdos257-mersenne-reasoning-surface","title":"Reciprocal Mersenne Subseries","source_repository":"plectis-lean-erdos249-257","source_path":"paper/erdos257-mersenne-reasoning-surface.tex","hosted_pdf_path":"papers/erdos257-mersenne-reasoning-surface.pdf","licence":"CC-BY-4.0","copyright":"2026 Will Cook","cite_as":"Plectis reader corpus, paper:erdos257-mersenne-reasoning-surface","pdf_sha256":"sha256:3ec207fe8ca0afcec77e78f85ce94e28f7f869d4a58bd35d05ea862259a4d30c","pdf_byte_count":658817,"pdf_embedded":false,"pdf_omission_reason":"the exact LaTeX is embedded and retrieves better than extracted PDF text; the PDF is a presentation rendering and its hash is recorded for verification","source_embedded":true,"availability":"embedded_exact","embedded_copy_role":"native_manuscript","embedded_from_repository":"plectis-lean-erdos249-257","embedded_copy_path":"paper/erdos257-mersenne-reasoning-surface.tex","media_type":"text/x-tex","body_format":"latex","sha256":"sha256:ac91ae8dcb76881770d2183ddd2b4929d0e2c84663393d925c28bd7426db86ca","byte_count":436631,"body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% GENERATED by scratchpad/assemble.py from authored parts. Parts are inlined,\n% never \\input, so that the pandoc projection cannot silently drop a section.\n\\documentclass[11pt]{article}\n% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Shared preamble for the two per-problem reasoning-surface papers.\n% These documents are written to be read in full as working context: every\n% result appears with its exact hypotheses, its evidence band, the scale at\n% which it is proved, and the Lean declaration that carries it.\n\\usepackage{paper-house-style}\n\\usepackage{array}\n\\usepackage{adjustbox}\n\\usepackage{booktabs}\n\\usepackage{longtable}\n\\usepackage{enumitem}\n\\setlist{topsep=0.35em,itemsep=0.2em}\n\\usepackage[colorlinks=true,linkcolor=linkinternal,urlcolor=linkexternal,\n  citecolor=linkinternal]{hyperref}\n\\hypersetup{bookmarksnumbered=true,bookmarksopen=true,bookmarksopenlevel=1,\n  pdftitle={Reciprocal Mersenne Subseries: a complete working record of Erdos Problem 257},\n  pdfauthor={Will Cook},pdflang={en-GB}}\n% Lean declaration names are full of underscores and are typeset in text mode.\n% Category codes are fixed when TeX READS a line, not when a macro expands, so\n% no amount of \\catcode juggling inside \\lean can rescue an underscore that has\n% already been tokenised as a subscript. This package makes _ print as itself in\n% text while leaving it as a subscript in maths, which is exactly the split we\n% need. It also protects any stray underscore in authored prose.\n\\usepackage{underscore}\n\n% ---------------------------------------------------------------------------\n% Theorem environments.\n% ---------------------------------------------------------------------------\n\\theoremstyle{plain}\n\\newtheorem{thm}{Theorem}[section]\n\\newtheorem{lem}[thm]{Lemma}\n\\newtheorem{prop}[thm]{Proposition}\n\\newtheorem{cor}[thm]{Corollary}\n\\theoremstyle{definition}\n\\newtheorem{defn}[thm]{Definition}\n\\theoremstyle{remark}\n\\newtheorem{rem}[thm]{Remark}\n\\newtheorem{obs}[thm]{Observation}\n\n% ---------------------------------------------------------------------------\n% Citation of a Lean declaration. The declaration name is the authority; the\n% file:line is where it sat at the pinned formal-source checkpoint. Underscores\n% are handled locally so authors may write names verbatim.\n% ---------------------------------------------------------------------------\n\\newcommand{\\PK}{Erdos249257}\n\\newcommand{\\commit}{1128d0858e1060c741f26460afac166f36801af9}\n\\newcommand{\\repobase}{https://github.com/wcook04/plectis-lean-erdos249-257/blob/\\commit}\n\\newcommand{\\lean}[2]{%\n  \\href{\\repobase/\\PK/#2}{\\textsc{Lean source}}}\n\n% Declaration names are written \\mathtt{...} in maths and, unavoidably, in\n% running prose too. Rather than police every occurrence, make the command\n% mode-agnostic: monospace is what is meant in both places.\n\\AtBeginDocument{%\n  \\let\\origmathtt\\mathtt\n  \\renewcommand{\\mathtt}[1]{\\ifmmode\\origmathtt{#1}\\else\\texttt{#1}\\fi}%\n  \\let\\origmathrm\\mathrm\n  \\renewcommand{\\mathrm}[1]{\\ifmmode\\origmathrm{#1}\\else\\textrm{#1}\\fi}%\n}\n\n% One authored part uses \\m{...} as an inline-maths shorthand. Defining it is\n% cheaper and safer than rewriting the prose around every occurrence.\n\\providecommand{\\m}[1]{\\ensuremath{#1}}\n\n% ---------------------------------------------------------------------------\n% Evidence band. These are never blurred: a reader must always be able to see\n% what a kernel checked, what an exact computation checked, what was proved in\n% ordinary mathematics, what is quoted from the literature, and what is open.\n% ---------------------------------------------------------------------------\n\\newcommand{\\ev}[1]{{\\small\\textsf{[#1]}}}\n\n% Scale at which a statement is proved. This is the load-bearing axis of both\n% documents: the corpus proves at fixed or bounded scale, and both open\n% producers need cofinal scale.\n\\newcommand{\\scale}[1]{{\\small\\textsf{scale:#1}}}\n\n% The representation a statement is expressed in. Obstructions are relative to\n% a coordinate, never to the object, so every obstruction carries one.\n\\newcommand{\\coord}[1]{{\\small\\textsf{coord:#1}}}\n\n% ---------------------------------------------------------------------------\n% Mathematical shorthand.\n% ---------------------------------------------------------------------------\n\\newcommand{\\N}{\\mathbb{N}}\n\\newcommand{\\Npos}{\\mathbb{N}_{>0}}\n\\newcommand{\\Z}{\\mathbb{Z}}\n\\newcommand{\\Q}{\\mathbb{Q}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\C}{\\mathbb{C}}\n\\newcommand{\\ph}{\\varphi}\n\\newcommand{\\EB}{E}\n\n\\allowdisplaybreaks\n\\emergencystretch=2em\n\n% These working records contain a small number of intentionally exhaustive\n% formulas. Keep ordinary displays at their natural size, but bound any display\n% wider than the classical text measure rather than letting it disappear past\n% the page edge.\n\\long\\def\\[#1\\]{%\n  \\leavevmode\\par\\vspace{\\abovedisplayskip}%\n  \\noindent\\makebox[\\linewidth][c]{%\n    \\begin{adjustbox}{max width=\\linewidth}%\n      \\(\\displaystyle #1\\)%\n    \\end{adjustbox}}%\n  \\par\\vspace{\\belowdisplayskip}%\n}\n\n\\runninghead{Erd\\H{o}s \\#257: reciprocal Mersenne subseries}\n\\title{Reciprocal Mersenne Subseries}\n\\subtitle{A complete working record of Erd\\H{o}s Problem~\\#257}\n\\author{Will Cook}\n\\date{July 2026}\n\n\\begin{document}\n\\maketitle\n\n\\paperprovenance\n\n\\begin{abstract}\nThis document is the complete working record of one attack on Erd\\H{o}s Problem 257: whether $\\sum_{n\\in A}1/(2^n-1)$ is irrational for every infinite $A\\subseteq\\N$.  Both the universal question and the distinguished half-value question are open, and nothing here decides either.  The document collects every result the project holds in a form that can be used directly as premises: the Lean-checked theorems with their exact hypotheses and source sites, the exact finite computations and their certified depths, the four integer coordinates in which one greedy orbit is measured, the attack families that have been closed together with the mechanism that closed each, and the exact statements that remain open.  Results are sorted by the scale at which they are proved.\n\\end{abstract}\n\n\\section*{How to read this document}\n\nThis is a working record, not a survey.  It is written to be read in full before\nwork begins on the problem, so that every statement the project has established\nis available as a premise rather than as a reference to be chased.  Three\nconventions carry that load.\n\n\\emph{Evidence bands.}  Every statement is tagged.  \\ev{Lean} means a proof term\nwas checked by the pinned Lean kernel.  \\ev{Cert} means an exact finite\ncomputation, with no floating-point decision anywhere in it.  \\ev{Math} means\nproved in ordinary mathematics in the sources but not formalised.  \\ev{Cited}\nmeans established in the published literature.  \\ev{Open} means not proved.\nThese are never blurred, and a statement carrying one band is never described in\nlanguage belonging to another.\n\n\\emph{Scale.}  Every parameter-indexed statement is tagged \\scale{fixed} (proved\nat specific listed values), \\scale{bounded} (proved below an explicit bound),\n\\scale{cofinal} (proved for arbitrarily large parameters), or \\scale{uniform}\n(proved for all parameters).  This axis is the reason the problem is still open:\nthe corpus proves at fixed and bounded scale, and the remaining obligation is\ncofinal.  Sorting by scale puts that boundary where it can be seen.\n\n\\emph{Coordinates.}  Every statement records the representation it is expressed\nin.  An obstruction is a fact about a coordinate, not about the object, and a\nwall measured in one representation may simply not exist in another; the atlas\nsection gives the transport maps, so any obstruction recorded here can be\nre-measured elsewhere.\n\nErd\\H{o}s Problem 257 is open.  No section of this document claims\notherwise, and a reduction of the problem to another statement is recorded as a\nreduction, never as progress toward a solution.\n\n\n% ---- part a257_front ----\n% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Front matter for the Erdos #257 reasoning-surface paper: the problem, the\n% barrier analysis, the surviving routes, and the reading contract. Everything\n% before the premise catalogue.\n\n\\section{The problem, and what is actually known}\n\\label{sec:257-problem}\n\n\\subsection{The object and the two exact statements}\n\nFor an integer $n \\ge 2$ set $x_n := 1/(2^n-1)$, and for $A \\subseteq \\{2,3,4,\\dots\\}$ write\n$x_A := \\sum_{n \\in A} x_n$; the sum always converges, since $x_n = O(2^{-n})$. Erd\\H{o}s's\nProblem \\#257 asks a single universally quantified question about this family.\n\n\\begin{defn}[Universal \\#257, written (U)]\n\\label{defn:U}\nIs $x_A = \\sum_{n \\in A} 1/(2^n-1)$ irrational for \\emph{every} infinite $A \\subseteq \\N$?\n\\textbf{OPEN.} \\ev{Open}\n\\end{defn}\n\n\\begin{defn}[The half-value question, written (H)]\n\\label{defn:H}\nLet the \\emph{Mersenne achievement set} be\n$\\mathcal{A} := \\{\\, y \\in \\R : \\exists A \\subseteq \\N,\\ 0 \\notin A,\\ y = \\sum_{n \\in A} x_n \\,\\}$\n(\\lean{mersenneAchievementSet}{GreedyAchievementSet.lean:571}). Is $1/2 \\in \\mathcal{A}$?\n\\textbf{OPEN.} \\ev{Open}\n\\end{defn}\n\nThe two are joined by one elementary and completely one-sided implication. Every $2^n-1$ is odd,\nso every \\emph{finite} subset sum of $\\{x_n\\}$ has odd reduced denominator and cannot equal $1/2$\n(\\lean{positiveMersenneSupportValue\\_coe\\_finset\\_ne\\_half}{HalfCutLocator.lean:243}). Hence a\nwitness to $1/2 \\in \\mathcal{A}$ is necessarily an infinite $A$ with $x_A = 1/2$ rational, which\nrefutes (U) outright. Conversely, (U) implies $1/2 \\notin \\mathcal{A}$ as one instance among\nuncountably many, and a proof of $1/2 \\notin \\mathcal{A}$ leaves (U) entirely open. So (H) is the\nmaximally symmetric candidate counterexample to (U), not a restatement of it; the asymmetry is\nload-bearing and recurs below as Barrier~\\ref{bar:asym}. \\ev{Math}\n\nNothing in this document decides (U) or (H). What it does contain is (i) the unconditional record,\nstated with hypotheses and evidence bands, and (ii) a characterisation of \\emph{why} the recorded\nattacks fail --- assembled so that the failures function as measurements of one obstruction rather\nthan as a list of disappointments.\n\n\\subsection{The unconditional record, universal direction}\n\nEvery landed unconditional irrationality theorem for (U) instantiates one engine: a\nweighted-coefficient block certificate over the divisor incidence\n$\\operatorname{sc}_A(n) := \\#\\{d \\mid n : d \\in A\\}$, the generic form being\n\\texttt{irrational\\_coeff\\_series\\_of\\_weighted\\_coeff\\_block\\_certificates}. The complete\ninventory of what that engine has been made to prove, uniformly in the base $b \\ge 2$:\n\n\\begin{center}\n\\begin{longtable}{@{}p{0.30\\textwidth}p{0.32\\textwidth}p{0.30\\textwidth}@{}}\n\\hline\n\\textbf{Support class} & \\textbf{Exact hypotheses} & \\textbf{Site} \\ev{Lean} \\\\\n\\hline\n\\endhead\nFull support $A = \\N$ & $b \\ge 2$ & \\lean{irrational\\_erdosSum\\_full\\_support}{CertificateKernel.lean:8000} \\\\\nMultiples $A = d\\N$ & $b \\ge 2$, $d \\ge 1$ & \\lean{irrational\\_erdosSupportSeries\\_multiples}{CertificateKernel.lean:8775} \\\\\nPurely periodic & $m$-periodic, contains a positive element & \\lean{irrational\\_erdosSupportSeries\\_periodic}{CertificateKernel.lean:11262} \\\\\nEventually periodic & $m$-periodic from $N_0$, $A$ infinite & \\lean{irrational\\_erdosSupportSeries\\_eventuallyPeriodic}{CertificateKernel.lean:11276} \\\\\nResidue class & any $m \\ge 1$, any $c$ & \\lean{irrational\\_erdosSupportSeries\\_residueClass}{CertificateKernel.lean:11344} \\\\\nOdd support (density $1/2$) & --- & \\lean{irrational\\_erdosSupportSeries\\_odd}{CertificateKernel.lean:11358} \\\\\nLacunary (lcm gap) & $a_k$ outgrows $\\operatorname{lcm}\\{a_0,\\dots,a_{k-1}\\}$ & \\lean{irrational\\_erdosSum\\_of\\_lcm\\_gap}{CertificateKernel.lean:5555} \\\\\nFactorials; powers of two & instances of the lcm-gap theorem & \\lean{irrational\\_erdosSum\\_factorial\\_support}{CertificateKernel.lean:5707}, \\lean{irrational\\_erdosSum\\_two\\_pow\\_support}{CertificateKernel.lean:5731} \\\\\nPairwise coprime (Erd\\H{o}s 1968) & $A$ infinite, pairwise coprime, \\emph{and} $\\sum_{a \\in A} 1/a < \\infty$ & \\lean{irrational\\_erdosSupportSeries\\_pairwise\\_coprime}{CertificateKernel.lean:10448} \\\\\nSigned periodic weights & periodic integer weights, nonnegative, frequently nonzero & \\lean{irrational\\_intWeightedErdosSeries\\_periodic\\_of\\_coeff\\_nonneg\\_of\\_frequently\\_ne\\_zero}{CertificateKernel.lean:14255} \\\\\n\\hline\n\\end{longtable}\n\\end{center}\n\nTwo facts about this table are worth a specialist's attention more than any single row. First, the\nbase restriction of the classical results is gone: the $b=2$ case of the first row recovers\nErd\\H{o}s's 1948 irrationality theorem for the Erd\\H{o}s--Borwein constant\n$E = \\sum_{n \\ge 1} 1/(2^n-1)$ as a one-line corollary, and every row holds for all $b \\ge 2$.\nSecond, and more important, \\emph{every} row lies inside a two-axis box --- divisor independence\nplus a global reciprocal-tail budget --- and the box is not an accident of effort. That is\nBarrier~\\ref{bar:box}.\n\nThe one conditional extension worth naming is\n\\lean{irrational\\_erdosSupportSeries\\_of\\_orthogonalPetalBouquet}{SupportSunflowerDichotomy.lean:540},\nwhich strictly generalises pairwise coprimality (cores may be arbitrary divisors of one fixed $Q$)\nbut still demands globally pairwise-coprime petals and summable reciprocal petal mass, and whose\nselector hypothesis \\texttt{SunflowerForcedSlotTailSelection} is itself an unproved cofinal supply.\nDespite the file name there is no dichotomy theorem in that file. \\ev{Lean}\\ev{Open}\n\nTwo one-sided rigidity results constrain a hypothetical rational-valued support, both only from the\nsparse side. For any $A$ with a positive element and $x_A$ rational, divisor-free windows are at\nmost logarithmic: for every $\\varepsilon > 0$ there is $B$ with every zero window of\n$\\operatorname{sc}_A$ beyond the scale threshold of length $\\le \\varepsilon \\log_2 N + B$,\nuniformly in $A$, $N$ and the window position\n(\\lean{supportCoeffZeroWindow\\_length\\_le\\_eps\\_logb}{SublogDivisorCoverage.lean:435}). \\ev{Lean}\nThis is vacuous whenever $2 \\in A$, since then every zero window already has length $\\le 1$. The\ncompanion \\lean{shifted\\_state\\_unbounded\\_of\\_infinite\\_support}{RationalSupportCarrySkeleton.lean:2320}\nis essentially tight and is best read as a restatement rather than a constraint.\n\n\\begin{rem}[What is \\emph{not} known in the universal direction]\nThere is no theorem anywhere in the corpus for $A =$ the primes; the tree was searched and none\nexists. There is no theorem for a support of positive density that is not eventually periodic.\nThere is no theorem covering any divisor-dense support outside the periodic families. The gap is\nnot a gap in strength --- it is the exact complement of the two-axis box.\n\\end{rem}\n\n\\subsection{The unconditional record, half-value direction}\n\n\\begin{thm}[Achievement-set geometry]\n\\label{thm:geometry}\n$\\mathcal{A}$ is compact, closed, perfect, totally disconnected and nowhere dense, and\n$\\operatorname{volume}(\\mathcal{A}) = 1$ --- inside an ambient span of\n$E \\approx 1.6067$, a relative density of about $62\\%$. The positive-index digit coding onto\n$\\mathcal{A}$ is injective: each achievable real has \\emph{exactly one} support.\n\\coord{achievement-set} \\scale{uniform} \\ev{Lean}\\\\\n\\textbf{Sites:} \\lean{isCompact\\_mersenneAchievementSet}{GreedyAchievementSet.lean:656},\n\\lean{perfect\\_mersenneAchievementSet}{GreedyAchievementSet.lean:1620},\n\\lean{isTotallyDisconnected\\_mersenneAchievementSet}{GreedyAchievementSet.lean:1636},\n\\lean{isNowhereDense\\_mersenneAchievementSet}{GreedyAchievementSet.lean:1645},\n\\lean{volume\\_mersenneAchievementSet}{GreedyAchievementSet.lean:996},\n\\lean{positiveMersenneDigitValue\\_injective}{GreedyAchievementSet.lean:1546}.\n\\end{thm}\n\n\\begin{thm}[Support-restricted refinement]\n\\label{thm:supported-dichotomy}\nFor the achievement set built from a coordinate subset $J \\subseteq \\N$: either the omitted\ncoordinate set is finite, and the volume is exactly $2^{-|J^c|}$, or infinitely many coordinates\nare omitted and the volume is $0$. Perfectness and injectivity survive the restriction.\n\\coord{achievement-set} \\scale{uniform} \\ev{Lean}\\\\\n\\textbf{Sites:} \\lean{volume\\_supportedMersenneAchievementSet\\_dichotomy}{Erdos257/MersenneSubseriesRigidity.lean:397},\n\\lean{volume\\_supportedMersenneAchievementSet\\_eq\\_zero\\_of\\_compl\\_infinite}{Erdos257/MersenneSubseriesRigidity.lean:368},\n\\lean{perfect\\_supportedMersenneAchievementSet}{Erdos257/MersenneSubseriesRigidity.lean:167},\n\\lean{supportedMersenneDigitValue\\_injective}{Erdos257/MersenneSubseriesRigidity.lean:54}.\n\\end{thm}\n\n\\begin{thm}[Membership equals greedy survival; the fatal-gap dichotomy]\n\\label{thm:greedy-survival}\n$x \\in \\mathcal{A}$ iff $0 \\le x$ and the canonical greedy (take-if-possible) residual after every\nrank $n$ satisfies $\\rho(x,n) \\le T_{n+1} := \\sum_{j > n} x_j$. Unconditionally, either\n$1/2 \\in \\mathcal{A}$ or a finite fatal gap exists, and the two are exactly complementary.\n\\coord{greedy-orbit} \\scale{uniform} \\ev{Lean}\\\\\n\\textbf{Sites:} \\lean{mem\\_mersenneAchievementSet\\_iff\\_greedy\\_survival}{GreedyAchievementSet.lean:1445};\n\\lean{half\\_mem\\_mersenneAchievementSet\\_or\\_exists\\_fatal\\_gap}{HalfCutLocator.lean:623};\n\\lean{existsFatalHalfGap\\_iff\\_half\\_not\\_mem\\_mersenneAchievementSet}{HalfCutLocator.lean:643};\n\\lean{half\\_mem\\_mersenneAchievementSet\\_iff\\_no\\_existsFatalHalfGap}{HalfCutLocator.lean:654};\nwitness type \\lean{ExistsFatalHalfGap}{HalfCutLocator.lean:526}.\n\\end{thm}\n\nTwo boundary degeneracies are excluded at every rank, unconditionally: $\\rho_k \\ne x_k$ by the\nodd-denominator argument above, and $\\rho_k \\ne T_{k+1}$ because the tail differs from $E$ by a\nrational while $E$ is irrational. Both greedy decisions are therefore strict at every rank.\n\\ev{Math}\n\nFinally, the certified computation. All of it is exact integer arithmetic, no floating point.\nTruncation rungs $J = 3,\\dots,22$ all \\emph{proved} to survive (not merely computed). Seam orbit\ncertified to row $200{,}000$: branch counts $\\mathrm{R{:}M{:}U} = 100197{:}49899{:}49898$, the\ntarget inequality failing at exactly one reset row, $r = 7$; maximal pure-R run $19$, attained at\nrow $158{,}096$. Independently, rows $6 \\le n \\le 2500$: $1209$ resets, zero classification\nanomalies, minimum in-scope margin $+1.1119$ bits at row $14$ growing to $\\approx 1246$ bits by row\n$2500$ (mean $627$), never re-approaching the threshold. Sharp tail margin, ranks $2$ to $3000$: no\nfatal skip, take fraction $0.4992$, tightest take-margin $+0.0340$ bits at rank $7$ (the near-tie\n$1/126 \\ge 1/127$ --- the single tightest real data point in the corpus, and the first case any\nproposed proof should be tested against). Greedy set certified to $m = 200{,}000$ with\n$0 \\le 1/2 - \\sum_{d \\in G} x_d < 2^{-199999}$. A half-trapping macro receipt reaches depth\n$13{,}548{,}057$ (labelled in its own source as a private exact finite computation receipt, not\npublication authority). Kernel-checked corroboration of the low rows:\n\\lean{seamUpperResetDyadicBandEscape\\_through\\_thirty}{HalfCylinderUpperResetBandCertificates.lean:78}\nverifies rows $13$--$30$ by \\texttt{decide}. \\ev{Cert}\n\n\\begin{rem}[Literature, exactly]\n$E$ is irrational (Erd\\H{o}s 1948). Zudilin's irrationality measure is\n$\\mu(E) \\le 2.42343562\\ldots$ (\\emph{Math.\\ Notes} 72 (2002) 858--862; the erratum, \\emph{Acta\nArith.} 111 (2004) 153--164, gives $2.46497868\\ldots$). Transcendence of $E$ is unknown\n(Duverney--Tachiya, \\emph{Forum Math.} 31 (2019) 1557--1566), and the Erd\\H{o}s--Graham conjecture\nthat every such subset sum is irrational --- i.e.\\ (U) --- is itself open. For binary digit runs of\n$E$ the only rigorous result is a \\emph{lower} bound: Erd\\H{o}s constructs a $0$-run of length\n$\\ge c \\log^{1/10} N$ among the first $N$ bits; Crandall (\\emph{INTEGERS} 12 (2012) \\#A23, \\S7)\nrecords the longest observed $0$-run as $47$ in the first $2^{43}$ bits and states that he does not\nknow how to show that \\texttt{11} appears infinitely often. Campbell (arXiv:2605.24160, 2026)\nsettles that occurrence question affirmatively. \\ev{Cited}\n\\end{rem}\n\nThat last pair of citations is the sharpest single orientation fact available. Every rigorous\nresult in the literature about the digits of $E$ points the wrong way: they produce long runs, or\nguarantee that patterns occur. What (H) needs is that a pattern \\emph{never} occurs for long. No\nresult of that logical type exists.\n\n\\section{The wall}\n\\label{sec:257-wall}\n\n\\subsection{The thesis}\n\nThis programme has produced many reformulations of \\#257 and no proof of it. That is the honest\nsummary, and expert readers have said so. The claim of this section is that the reformulations are\nnot a hundred separate failures. They are repeated measurements of \\emph{one} obstruction, taken in\ndifferent coordinates, and once they are assembled the obstruction has a describable shape. A\nsystematic elimination that says exactly which classes of argument cannot work, and why, is a\nresult about the problem; a reformulation is not. The elimination is what follows.\n\nThe generating fact is a single structural one, and it is not a convenience of the formalisation.\n\n\\begin{obs}[The problem admits no ensemble]\n\\label{obs:no-ensemble}\nThe Mersenne weights are strictly superincreasing. Consequently the digit coding is injective\n(Theorem~\\ref{thm:geometry}), the greedy orbit is the unique candidate support for any target, and\ndistinct subset sums of a row of seam weights are separated by at least $2^{s+1}$. Every one of the\nnine coordinates the corpus has built --- truncation rung $J$, seam row $s$, integer margin, sharp\ntail margin, half-carry $K_n$, reverse-carry word, Boolean--M\\\"obius exact row, fatal-cell interval\npacking, Dedekind cut-locator --- is therefore a coordinate on the binary digits of \\emph{one}\nspecific real orbit. There is no family to range over.\n\\coord{meta} \\scale{uniform} \\ev{Lean}\\ev{Math}\\\\\n\\textbf{Sites:} \\lean{weightedBoolSum\\_separated}{HalfCylinderIntegerGreedy.lean:447},\n\\lean{three\\_mul\\_tailWeight\\_add\\_exactLateGap\\_eq\\_three\\_mul\\_headWeight}{HalfCylinderLargestSkipGap.lean:230},\n\\lean{positiveMersenneDigitValue\\_injective}{GreedyAchievementSet.lean:1546}.\n\\end{obs}\n\nAll six barriers below are consequences of Observation~\\ref{obs:no-ensemble} in different\ndirections: there is nothing to choose (\\ref{bar:canon}), nothing weaker to prove\n(\\ref{bar:collapse}), nothing to average over (\\ref{bar:measure}), no information at the available\nscales (\\ref{bar:scale}), no certificate in the provable direction (\\ref{bar:asym}), and, on the\nuniversal side, no way to make a general support satisfy the certificate engine's first hypothesis\n(\\ref{bar:box}).\n\n\\subsection{Barrier 1: canonicalisation --- the witness space is a singleton}\n\\label{bar:canon}\n\n\\begin{prop}[Canonicalisation; \\textsc{proved}]\n\\label{prop:canon}\nFor the half-target in strictly superincreasing Mersenne weights, every finite object satisfying an\napproximate-hit condition at depth $d$ \\emph{is} the canonical greedy prefix at depth $d$.\nPrecisely: (i) every straddling word at depth $d$ agrees bit-for-bit with the greedy support of\n$1/2$ on $(0,d]$; (ii) every below-half core $D$ whose crossing deficit is smaller than the next\nweight is forced to equal $\\texttt{halfGreedyPrefixSupport}(c-1)$, so the universally quantified\n$\\forall D$ in the lane's headline socket collapses onto one orbit; (iii) a remainder below the gap\nis equivalent to being the integer greedy bit-word.\n\\coord{straddle-prefix / integer-greedy} \\scale{uniform} \\ev{Lean}\\\\\n\\textbf{Sites:} \\lean{IsStraddlePrefix.half\\_agrees\\_greedy}{HalfCutLocator.lean:442};\n\\lean{eq\\_halfGreedyPrefixSupport\\_of\\_critical\\_crossing}{BooleanMobiusCriticalCapacityCofinal.lean:50};\n\\lean{remainder\\_lt\\_gap\\_iff\\_eq\\_integerGreedyBits}{BooleanMobiusGreedyReduction.lean:373}.\n\\end{prop}\n\n\\textbf{Rules out.} Every argument whose leverage comes from \\emph{choosing} a favourable witness:\na support $D$, a Boolean word, a straddle prefix, an adversarial late word, or a lattice/covering\nconfiguration. At each depth the admissible set has one element, so there is no choice to exploit\nand no adversary to defeat.\n\n\\textbf{Explains.} Three recorded attack families die for exactly this reason, and their deaths\nlook superficially unrelated. (1) \\emph{Generic lattice covering} proves the wrong sign: the free\nprefix lattice breaches the $2^{t/2}$ safety radius in $74\\%$ of $t \\in [8,26]$ --- witness\n$t = 15$, where fans of $\\{2,3,5,6,10\\}$ sum to $2^{15}+1$, at circular distance $1$; expected\nnear-hits grow like $2^{t/6}$. Any generic bound on prefix cancellation therefore proves the\nopposite of what is wanted. \\ev{Cert} (2) \\emph{Digit-cascade impossibility} is false statically:\nexplicit words exist at $t = 20$ (top $18$ of $20$ bits all-$1$) and $t = 22$ (top $20$ of $22$\nall-$0$), and an unconditional construction supplies about $t/12$ independent single-bit knobs,\nverified with zero exceptions to $t = 601$. \\ev{Cert} (3) \\emph{Integer-window static adversary\nsearch} finds nothing: zero hits, exhaustively for $t \\le 22$ and exactly at $t = 40, 42, 45$ over\nall cut positions. \\ev{Cert} The reconciliation is the content: residues mod $2^t$ are\nadversary-tunable, but the integer \\emph{magnitude} is not, because the row weights are\ngap-dominated. So the static tier and the dynamic tier coincide identically, and all three families\nwere, without knowing it, measuring the same single orbit.\n\n\\begin{rem}[A recorded hazard]\nThe natural sufficient condition via fractional split-mass is \\emph{not} necessary on real crossing\ncores: the fixture $D = \\{2,3\\}$, $c = 5$ is an explicit counterexample\n(\\lean{splitFractionMass\\_one\\_bound\\_not\\_necessary\\_fixture}{BooleanMobiusSkippedCoreCriticalCapacity.lean:183}).\nAny attack on the capacity socket that routes through fractional mass is already refuted.\n\\ev{Lean}\n\\end{rem}\n\n\\subsection{Barrier 2: route collapse --- every ``weaker producer'' is the goal}\n\\label{bar:collapse}\n\nThis is the barrier that most consistently wastes effort, because the sockets it kills look\nstrictly weaker on the page: they are $\\Pi^0_2$ statements ($\\forall N \\exists n \\dots$) with\nexplicitly relaxed compatibility requirements, while the goal is $\\Pi^0_1$. The extra quantifier\nbuys nothing. The mechanism is a single exact identity plus a single unconditional envelope.\n\n\\begin{lem}[Collapse mechanism; \\textsc{proved}]\n\\label{lem:collapse-mech}\nLet $A \\subseteq \\N$ with $1 \\notin A$, put $\\delta := 1/2 - x_A$, and let\n$\\operatorname{ihc}(A,N)$ denote the integer half-carry at level $N$. Then\n\\[\n  \\operatorname{ihc}(A,N) \\;=\\; 2^{N+1}\\,\\delta \\;+\\; \\mathrm{T}(N+1),\n  \\qquad 0 \\;\\le\\; \\mathrm{T}(N+1) \\;\\le\\; 2\\sqrt{N+1} + 4 ,\n\\]\nwhere $\\mathrm{T}$ is the binary coefficient tail of $\\operatorname{sc}_A$.\n\\coord{half-carry} \\scale{uniform} \\ev{Lean}\\\\\n\\textbf{Sites:} \\lean{integerHalfCarry\\_eq\\_scaled\\_residual\\_add\\_tail}{HalfCarryReachability.lean:842};\n\\lean{binaryCoeffTail\\_nonneg}{GenericTailOrbitRigidity.lean:78};\n\\lean{binaryCoeffTail\\_supportCoeff\\_le\\_two\\_sqrt\\_add\\_four}{BooleanMobiusCarry.lean:289}.\n\\end{lem}\n\n\\begin{prop}[Route collapse; \\textsc{proved}]\n\\label{prop:collapse}\nAny hypothesis asserting, for cofinally many $N$, that $\\operatorname{ihc}(A,N)$ lies inside a\nstrip of width $O(\\sqrt{N})$ forces $\\delta = 0$.\n\\end{prop}\n\n\\begin{proof}\nBy Lemma~\\ref{lem:collapse-mech}, $2^{N+1}\\delta = \\operatorname{ihc}(A,N) - \\mathrm{T}(N+1)$.\nFor the greedy support of $1/2$ one has $x_G \\le 1/2$, hence $\\delta \\ge 0$\n(\\lean{erdosSupportSeries\\_greedyMersenneSupport\\_le}{HalfCarryReachability.lean:859}). If\n$\\operatorname{ihc}(A,N) \\le c\\sqrt{N} + c'$ for cofinally many $N$, then\n$2^{N+1}\\delta \\le c\\sqrt{N} + c'$ along that sequence; since $2^{N+1}$ outgrows any $\\sqrt{N}$\nenvelope and $\\delta \\ge 0$ is a fixed real, $\\delta = 0$. Conversely $\\delta = 0$ makes\n$\\operatorname{ihc}(A,N) = \\mathrm{T}(N+1)$ exactly, and the unconditional envelope of\nLemma~\\ref{lem:collapse-mech} then \\emph{supplies} the strip for free. And $\\delta = 0$ is exactly\n$1/2 \\in \\mathcal{A}$, by\n\\lean{mem\\_mersenneAchievementSet\\_iff\\_greedy\\_survival}{GreedyAchievementSet.lean:1445}.\n\\end{proof}\n\n\\begin{lem}[Perfect-square strip witness; \\textsc{proved}, this pass]\n\\label{lem:sqrt-witness}\nSuppose $x_A = 1/2$ and $1 \\notin A$. Then for every $k \\ge 1$,\n\\[\n  \\operatorname{ihc}(A, k^2 - 1) \\;=\\; \\mathrm{T}(k^2) \\;\\le\\; 2k + 4 \\;=\\; \\texttt{halfStripBound}(k^2).\n\\]\n\\coord{half-carry} \\scale{cofinal} \\ev{Math}\n\\end{lem}\n\n\\begin{proof}\nBy Lemma~\\ref{lem:collapse-mech} with $\\delta = 0$, $\\operatorname{ihc}(A,k^2-1) = \\mathrm{T}(k^2)\n\\le 2\\sqrt{k^2} + 4$. At a perfect square the real and truncated square roots agree exactly,\n$\\sqrt{k^2} = k = \\lfloor \\sqrt{k^2} \\rfloor$, so the real envelope coincides with the discrete\nstrip $\\texttt{halfStripBound}(n) = 2\\lfloor\\sqrt n\\rfloor + 4$\n(\\lean{halfStripBound}{HalfCarryReachability.lean:32}) with no slack required.\n\\end{proof}\n\nLemma~\\ref{lem:sqrt-witness} matters because it removes the last technical obstruction previously\nrecorded against two of the sockets below --- the mismatch between $\\lfloor\\sqrt{\\cdot}\\rfloor$ and\n$\\sqrt{\\cdot}$, for which the corpus carries only the wrong-direction inequality. Restricting the\ncofinal witness depths to perfect squares dissolves it, and no widening of the strip constant from\n$4$ to $6$ is needed. For the terminal-only socket, take $a$ to be the indicator of an achieving\nsupport truncated to $\\{0,\\dots,M\\}$ at $M = k^2$, using\n\\lean{integerHalfCarry\\_inter\\_Iic\\_eq\\_of\\_succ\\_le}{HalfCarryReachability.lean:49} and\n$1 \\notin A$ (which holds since $x_1 = 1 > 1/2$).\n\n\\begin{prop}[The collapsed sockets; \\textsc{proved}, composite]\n\\label{prop:collapsed-list}\nEach of the following is logically \\emph{equivalent} to $1/2 \\in \\mathcal{A}$, not strictly weaker:\n\\begin{enumerate}[nosep,label=(\\alph*)]\n\\item the \\texttt{hbound} hypothesis of\n  \\lean{greedy\\_half\\_infinite\\_of\\_mobiusCenteredHalfCarry\\_sqrtBound}{HalfCarryReachability.lean:805}\n  and of \\lean{greedy\\_half\\_infinite\\_of\\_mobiusCenteredHalfCarry\\_upperBound}{HalfCarryReachability.lean:924};\n\\item \\lean{CofinalPositiveHalfGreedySkips}{BooleanMobiusSkipRowCofinal.lean:22};\n\\item \\lean{CofinalExactLocalMersenneHalfRows}{BooleanMobiusCofinalExactRows.lean:38}, including its\n  advertised weakening that \\emph{no} compatibility is required between witness supports at\n  different endpoints;\n\\item \\lean{GreedyHalfCarryCofinalStripReturn}{CofinalStripReturn.lean:76};\n\\item \\lean{HalfCarryCofinalTerminalOnlyStrip}{TerminalOnlyCofinal.lean:34};\n\\item the negation of \\lean{ExistsFatalHalfGap}{HalfCutLocator.lean:526}, and all seven forms of the\n  associated classification chain.\n\\end{enumerate}\nForward directions are landed Lean (e.g.\\ \\lean{half\\_mem\\_mersenneAchievementSet\\_iff\\_greedySkippedSupport\\_infinite}{GreedyAchievementSet.lean:2514},\n\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_cofinalExactLocalRows}{BooleanMobiusCofinalExactRows.lean:71},\n\\lean{existsFatalHalfGap\\_iff\\_half\\_not\\_mem\\_mersenneAchievementSet}{HalfCutLocator.lean:643}).\nThe reverse directions of (d) and (e) are derived here from Lemma~\\ref{lem:sqrt-witness} and are\n\\emph{not} themselves landed Lean theorems.\n\\coord{half-carry / exact-row} \\scale{cofinal} \\ev{Lean}\\ev{Math}\n\\end{prop}\n\n\\textbf{Rules out.} Every strategy of the form ``the full problem is hard, so prove this weaker\nsufficient condition instead''. In particular the specific hope that mutually \\emph{incompatible}\nwitnesses at different endpoints buy flexibility is void, and reading the $\\Pi^0_2$ shape of these\nsockets as evidence of extra room is an error.\n\n\\textbf{Scope limit, preserved exactly.} Item (a) does \\emph{not} extend to every $A$ with\n$1 \\notin A$. Dropping the greedy-specific fact $x_G \\le 1/2$ yields only the one-sided statement\n$\\texttt{hbound} \\iff (1/2 - x_A) \\le 0$. This limit must be stated verbatim wherever the\nequivalence is used.\n\n\\subsection{Barrier 3: scale mismatch --- the available information is at the wrong resolution}\n\\label{bar:scale}\n\nThe input the half-value branch actually needs is a per-index, growing-precision,\n\\emph{short-window} near-integer anti-concentration bound: at reset row $r$, a lower bound on the\ndistance from $\\Z$ of $\\sum_{i \\le L} \\delta(M+i)\\,2^{-i}$ with $\\delta(M) = \\tau(M) - 1$, where the\nprecision demanded grows like $2^{-r/2}$ while the window length is only $L \\approx r/2$. Every\ninformation source tried lives at one of two extremes, and provably cannot reach that intermediate\nscale.\n\n\\textbf{Class (a): fixed precision or bounded state.} Three landed theorems, all\nproblem-agnostic --- they mention no Mersenne structure and bind \\#249 identically.\n\n\\begin{prop}[Local information is void; \\textsc{proved}]\n\\label{prop:local-void}\n(i) After $L$ common steps the endpoint residue mod $2^L$ of an affine binary orbit is\n\\emph{independent} of the initial carry: $u(L) - v(L) = 2^L(u_0 - v_0)$\n(\\lean{affineBinaryOrbit\\_mod\\_twoPow\\_eq}{GenericTailOrbitRigidity.lean:307}). (ii) No bounded or\nautonomous state summarising the history before position $m$ determines the tail after $m$: for a\nbalanced-pulse family no decoder recovers the radius, and any finite state type needs cardinality\nat least $\\lfloor m/2 \\rfloor + 2$, unbounded in $m$\n(\\lean{balancedPulse\\_no\\_autonomous\\_decoder}{GenericTailOrbitRigidity.lean:247}). (iii) For any\nstarting carry and any finite word of odd valuation-unit symbols at fixed precision $u > 0$, a\ncompatible centred carry completion \\emph{exists}\n(\\lean{fixedPrecisionTropicalNoGo}{TropicalCurvatureCarry.lean:137}), so fixed-precision local data\nnever excludes anything. \\coord{carry} \\scale{uniform} \\ev{Lean}\n\\end{prop}\n\nThe parity route is the same fact by hand: ``$K(M)$ even $\\iff$ $M$ a perfect square'' is true, but\n$J(M+1) = 2J(M) - \\delta(M+1)$ makes $J(M+1) \\equiv \\delta(M+1) \\pmod 2$ an identity for\n\\emph{any} digit stream whatsoever. It is a tautology of the recursion shape; it forbids no residue\nand no run. \\ev{Math}\n\n\\textbf{Class (b): global averages and Diophantine data.}\n\n\\begin{prop}[Exponent-budget mismatch; \\textsc{proved}, given a cited $\\mu$]\n\\label{prop:exponent-gap}\nThe skip set at row $n$ has $|\\mathrm{Skip}_n| \\approx n^2/4$, so an irrationality-measure exponent\n$\\mu$ for the target constant yields a separation of order only $2^{-\\mu n^2/4}$, while the\nrequired precision is $2^{-3n/2}$. The shortfall is a factor $\\approx 0.42\\,n$ \\emph{in the\nexponent}, and it grows linearly in $n$. Since $\\mu \\ge 2$ for every irrational, no improvement in\n$\\mu$ can close it: Zudilin's $\\mu(E) \\le 2.42343562\\ldots$ is irrelevant to the gap, and so is any\nfuture bound. \\coord{diophantine} \\scale{cofinal} \\ev{Math}\\ev{Cited}\n\\end{prop}\n\n\\textbf{Rules out.} Any argument deriving the run-length or magnitude bound from (a) arithmetic\ninformation at fixed finite precision or a bounded automaton state, or (b) a global average,\ndensity, or Diophantine-approximation quantity attached to $C = E - 3/2$. Concretely: no proposal\nto bound the R-run via an irrationality measure of a Mersenne or Erd\\H{o}s--Borwein-type constant\ncan work, however good the measure becomes; and no proposal defining a bounded carry state\nsummarising pre-index history can determine the post-index tail.\n\n\\textbf{Explains.} Prime-doubling fails for a recorded reason rather than for want of effort: $N$\nprime $\\ge n$ gives $b(N) = 1$ so $J$ doubles at that step, but not cumulatively, because\ncomposites subtract. The explicit witness is the quiet run at $n = 607$, spanning\n$M \\in (607,617]$ with $J \\in [1,5]$ for eleven steps \\emph{while three of those indices --- $607$,\n$613$, $617$ --- are themselves prime}; and $26$ of the $40$ longest runs with $n \\le 1500$ contain\na prime. No run-length bound follows from prime counting. \\ev{Cert}\n\n\\begin{rem}[Status of the exhaustion claim]\nThe seven named families are each closed with mechanism. The claim that classes (a) and (b)\n\\emph{exhaust} the available tools is an \\textsc{observed pattern}, not a theorem, and is labelled\nas such wherever it is used below.\n\\end{rem}\n\n\\subsection{Barrier 4: decision asymmetry --- the certificates point the wrong way}\n\\label{bar:asym}\n\n\\begin{thm}[One-sidedness; \\textsc{proved}]\n\\label{thm:one-sided}\n$1/2 \\notin \\mathcal{A}$ is $\\Sigma^0_1$: a single finite object --- a fatal gap --- certifies it.\n$1/2 \\in \\mathcal{A}$ is $\\Pi^0_1$, and no finite object certifies it. Consequently no finite\ncomputation, at any depth, can establish $1/2 \\in \\mathcal{A}$.\n\\coord{meta} \\scale{uniform} \\ev{Lean}\n\\end{thm}\n\n\\begin{proof}\nThe dichotomy $1/2 \\in \\mathcal{A}$ or $\\exists$ a fatal gap is unconditional\n(\\lean{half\\_mem\\_mersenneAchievementSet\\_or\\_exists\\_fatal\\_gap}{HalfCutLocator.lean:623}), and the\ntwo alternatives are exactly complementary\n(\\lean{existsFatalHalfGap\\_iff\\_half\\_not\\_mem\\_mersenneAchievementSet}{HalfCutLocator.lean:643}),\nwith \\lean{ExistsFatalHalfGap}{HalfCutLocator.lean:526} a finite witness type. In the other\ndirection, certified death implies non-membership\n(\\lean{certifiedGreedyMersenneDeath\\_not\\_mem}{GreedyAchievementSet.lean:1726}, over\n\\lean{CertifiedGreedyMersenneDeath}{GreedyAchievementSet.lean:1720}; $3/4$ is certified dead at\nlevel $1$, lookahead $0$, by\n\\lean{three\\_fourths\\_certifiedGreedyMersenneDeath}{GreedyAchievementSet.lean:1742}), whereas the\nabsence of such a certificate at any depth $D$ is compatible with a certificate at depth $D+1$ and\ntherefore proves nothing. Membership asserts survival at \\emph{every} rank\n(Theorem~\\ref{thm:greedy-survival}), an infinite conjunction with no finite sub-conjunction\nimplying it.\n\\end{proof}\n\n\\textbf{Rules out.} Any programme whose plan is ``extend the certified depth''. Pushing the\ntruncation rung to $J = 23$, the seam to row $10^6$, or the macro receipt to depth $10^8$ cannot in\nprinciple decide the branch --- each raises a lower bound on a death rank which, if the supported\nbranch is correct, does not exist.\n\n\\textbf{Explains.} Every certificate the programme owns is of this one type. The certified depths\ngrew by roughly two orders of magnitude across the recorded history with no change whatsoever in\ndecision status; that is not a coincidence to be pushed through, it is the theorem above being\nobserved empirically. Accumulated survival is not convergence toward proof.\n\n\\subsection{Barrier 5: measure and category point in opposite directions}\n\\label{bar:measure}\n\n\\begin{prop}[The two soft heuristics contradict each other; \\textsc{proved}]\n\\label{prop:soft}\n$\\mathcal{A}$ has Lebesgue measure exactly $1$ inside an ambient span of $E \\approx 1.6067$, a\nrelative density of about $62\\%$, and is simultaneously nowhere dense, perfect and totally\ndisconnected (Theorem~\\ref{thm:geometry}). Hence the measure heuristic --- ``a random point lies in\n$\\mathcal{A}$ with probability $\\approx 0.62$'' --- predicts \\#257 \\emph{false}, while the category\nheuristic --- ``$\\mathcal{A}$ is meagre, so a generic point misses it'' --- predicts \\#257\n\\emph{true}. \\coord{achievement-set} \\scale{uniform} \\ev{Lean}\n\\end{prop}\n\n\\textbf{Rules out.} Every soft argument: measure-theoretic, Baire-category, metric number theory\n(almost-all / almost-none), and probabilistic digit models. The two standard notions of ``typical''\ndisagree on this object, so neither can be the mechanism. Worse, by injectivity of the digit coding\n(Observation~\\ref{obs:no-ensemble}) there is no ensemble of candidate representations to average\nover: each rational has exactly one candidate support, so counting, second-moment and pigeonhole\narguments have nothing to range over. It also rules out deciding (U) by showing $\\mathcal{A}$ is\n``small'': it is not small --- it is $62\\%$ of its ambient interval by measure.\n\n\\textbf{Explains.} The programme's own probabilistic estimate sits squarely on the losing side of\nthis split: under a Bernoulli$(1/2)$ digit model the expected number of counterexamples with\n$n \\ge 32$ is about $2 \\times 10^{-4}$, i.e.\\ the model predicts $1/2 \\in \\mathcal{A}$. That\nestimate is worth exactly as much as the model, and the model is a measure-side object that cannot\nsee a specific point of a measure-one nowhere-dense set. \\ev{Cert}\n\n\\subsection{Barrier 6: the two-axis certificate box (universal direction)}\n\\label{bar:box}\n\n\\begin{obs}[The box; \\textsc{observed pattern}]\n\\label{obs:box}\nEvery landed unconditional theorem in the table of \\S\\ref{sec:257-problem} requires the support to\nsit inside a two-axis box: \\textbf{(i) divisor independence}, so that a CRT construction can\nprescribe the incidence $\\operatorname{sc}_A$ across a block --- pairwise coprimality, periodicity,\nor dilation of a full support; \\textbf{and (ii) a global tail budget}, so that the middle window's\nmass satisfies the height inequality $q(C+N+L+2) < b^L$ --- summable reciprocals, or sparsity that\nmakes this trivial. No landed theorem sits outside the box. This is a complete audit of the landed\ninventory, not a theorem that no such support can exist.\n\\coord{certificate-engine} \\scale{uniform} \\ev{Lean}\\ev{Math}\n\\end{obs}\n\n\\textbf{Rules out.} Any plan to obtain (U) by instantiating the existing engine at a general\ninfinite $A$. For a general support the divisibility conditions $a \\mid n$, $a \\in A$, are\n\\emph{entangled} through pairwise gcds, so no CRT construction can prescribe the incidence in a\nblock, and the engine's first hypothesis is unsatisfiable by construction rather than by search\nfailure. It also rules out closing the gap by ``removing the summability bound'' from Erd\\H{o}s\n1968: the bound is consumed at exactly one step, \\texttt{CertificateKernel.lean:9586--9598}, as a\nglobal tail budget. Removing it is not weakening a hypothesis; it requires a different, windowed\nargument.\n\n\\textbf{Explains.} The most natural uncovered class, the primes, is infinite and pairwise coprime\n--- axis (i) is satisfied in the cleanest possible way, and\n\\lean{compositeDilationDefect\\_eq\\_zero\\_of\\_prime\\_support}{CompositeDilationDefect.lean:103} makes\nthe dilation algebra exact there --- and yet no theorem exists, because $\\sum 1/p$ diverges by\nMertens and axis (ii) fails. That single divergence, not any difficulty of the primes, is why the\ncorpus contains no prime-support theorem. It is also why the sunflower generalisation buys nothing\nin the divisor-dense regime: it inherits both axes.\n\n\\subsection{What the six barriers converge on}\n\nBoth halves of \\#257 terminate at the same missing object, stated twice in two coordinates. On the\nhalf-value side it is a near-integer anti-concentration bound for\n$\\sum_{i \\le L} (\\tau(M+i) - 1)\\,2^{-i}$, a geometrically weighted divisor sum in a short interval.\nOn the universal side it is the dense-branch producer: for a divisor-dense $A$, a block of $K$\nconsecutive indices with $2^K \\mid \\sum_{r=1}^{K} \\operatorname{sc}_A(N+r)\\,2^{K-r}$ and a bounded\nmiddle window. These are the same statement in the incidence coordinate. No result of that shape\nexists in the literature in any form, and the only rigorous results in that direction --- lower\nbounds on runs, and guarantees that patterns occur --- have the opposite logical type. That\nconvergence, from two independent directions, is the strongest evidence the corpus offers that\nthere is exactly one wall here rather than several.\n\n\\section{What the wall does not block}\n\\label{sec:257-survivors}\n\nSix barriers eliminate a great deal. What they do not eliminate is a short, specific list, and the\npoint of the elimination is that this list is short. Each route below is given with the exact\nstatement it needs, the barrier it evades, and \\emph{why} it evades it. Where a route evades every\nproved barrier but sits inside the class indicted by an observed pattern, that is said plainly.\n\n\\subsection{Route 1: the reset \\texorpdfstring{\\(\\sqrt{\\ }\\)}{square-root}-escape / R-run anti-concentration target}\n\\label{ssec:route-1}\n\n\\textbf{What it needs.} Any one of three forms, in decreasing strength.\n\\begin{enumerate}[nosep,label=(\\roman*)]\n\\item \\emph{Run-length form.} For every reset row $r \\ge 31$ the maximal pure-R run starting at $r$\n  satisfies $L_r < (r-3)/2$; equivalently $|\\mathrm{rem}(r+1) - 2^{r+1}| > 2^{(r+5)/2}$. Any\n  sublinear bound $L_r = o(r)$ suffices.\n\\item \\emph{Lean-facing form, already fanned in.} \\texttt{SeamUpperResetDyadicBandEscape}: for every\n  $d \\ge 13$ with \\texttt{successorCarries} and every $j \\le d$, the reset charge\n  $4\\cdot\\text{overshoot} + \\text{abovePulse}$ avoids the linear-width band\n  $(2^{d-j+1} - 2(d+j),\\, 2^{d-j+1}]$.\n\\item \\emph{Sign law alone.} At every reset, M-resets have $\\mathrm{dev} > 0$ and U-resets have\n  $\\mathrm{dev} < 0$. Strictly weaker than either inequality above, and flagged in its own source as\n  worth attempting first.\n\\end{enumerate}\n\\textbf{Sites.} \\lean{SeamUpperResetDyadicBandEscape}{HalfCylinderMiddleCarryLowerBound.lean:4566}\nwith consumer\n\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_upperResetDyadicBandEscape}{HalfCylinderMiddleCarryLowerBound.lean:4790};\nquantifier collapse \\lean{dyadicBandEscape\\_iff\\_exists\\_critical}{HalfUpperResetCriticalBand.lean:108},\nwhich reduces the $\\forall j \\in [0,d]$ band check to one nearest-boundary index per row; finite\nverification \\lean{seamUpperResetDyadicBandEscape\\_through\\_thirty}{HalfCylinderUpperResetBandCertificates.lean:78};\nalternative fan-in \\lean{LargestSkipLateStepSocket}{HalfCylinderLargestSkipInduction.lean:34} with\nconsumer at \\texttt{:167} and base case\n\\lean{largestSkipLateAt\\_fourteen}{HalfCylinderLargestSkipInduction.lean:73}.\n\n\\textbf{Why it evades Barrier~\\ref{bar:collapse}.} This is the one reduction in the corpus that is\ngenuinely \\emph{sufficient} rather than equivalent. It is a quantitative bound at every reset row,\nwhile $1/2 \\in \\mathcal{A}$ is qualitative; it can be false while the goal is true. There is no\nimplication $1/2 \\in \\mathcal{A} \\Rightarrow \\sqrt{\\ }$-escape anywhere in the corpus, and by the\nshape of the two statements there cannot be. Proposition~\\ref{prop:collapse}'s mechanism needs a\nstrip hypothesis of width $O(\\sqrt N)$ around the half-carry; this route asserts a lower bound on a\ndeviation, which the collapse identity does not manufacture.\n\n\\textbf{Why it evades Barrier~\\ref{bar:asym}.} It is $\\Pi^0_1$ with certificates aligned with the\nbranch being proved. The margins \\emph{grow}: $+1.1119$ bits at row $14$, about $1246$ bits by row\n$2500$, mean $627$, never re-approaching the threshold. So for once the finite computation is\nevidence for the target rather than for its unprovable complement. \\ev{Cert}\n\n\\textbf{Why it evades Barrier~\\ref{bar:canon}.} It is a statement about the one canonical orbit ---\nexactly the object canonicalisation says everything reduces to. Here the singleton witness space is\nhelp, not obstruction: there is no adversary to beat, only one sequence to bound.\n\n\\textbf{The trap it avoids.} A \\emph{uniform constant} run-length bound is empirically false: runs\ngrow like $\\log_2(\\text{row})$. The route survives because any $L_r = o(r)$ suffices, and the\nobserved slack is enormous --- maximum R-run $19$ at row $158{,}096$ against a requirement of\n$(r-3)/2 = 79{,}046$, a margin of roughly $4000\\times$. \\ev{Cert}\n\n\\textbf{What it does not evade.} Barrier~\\ref{bar:scale}. Its residual difficulty \\emph{is} the\nmissing object of \\S2.7. This route and that barrier are the same frontier viewed from the two\nsides, which is the honest reason it is listed first: it is where the real mathematics has to be\ndone, not a way around it. \\ev{Open}\n\n\\subsection{Route 2: strict endpoint progress for exact rows}\n\\label{ssec:route-2}\n\n\\textbf{What it needs.} Rule out the recycle branch returning to a bounded endpoint forever. Any\none of:\n\\begin{enumerate}[nosep,label=(\\alph*)]\n\\item \\texttt{SkippedCoreCriticalQuotientSupply}: for every below-half core $D$ bounded by $[2,c)$\n  with genuine crossing deficit, $2^{(2c-2)-1} \\le \\texttt{localPrefixQuotient}\\,(\\mathrm{insert}\\\n  c\\ D)\\,(2c-2)$;\n\\item a direct proof that the crossing rank $c$ produced by the recycle branch cannot repeat the\n  value $4$, or any bounded value, infinitely often;\n\\item a growth law for $c$ as a function of the incoming endpoint $n$.\n\\end{enumerate}\nBy Proposition~\\ref{prop:canon}(ii), form (a) need only be checked along the \\emph{one} canonical\norbit \\texttt{halfGreedyPrefixSupport}, not searched over $D$; and by the landed sharp-capacity\nbiconditional it is exactly the $c-2$ bit capacity test\n$\\texttt{localBinarySuffix}\\,D\\,1\\,(2c-2) < 2^{c-2}$. It must \\emph{not} be attacked via the\nfractional-mass route, which the fixture $D = \\{2,3\\}$, $c=5$ refutes as non-necessary.\n\n\\textbf{Sites.} \\lean{exactLocalMersenneHalfRow\\_double\\_or\\_recycle}{BooleanMobiusExactRowDichotomy.lean:26};\nthe formal falsifier\n\\lean{exists\\_seeded\\_bounded\\_double\\_or\\_recycle\\_model}{BooleanMobiusExactRowDichotomy.lean:91}\n(the model $n \\mapsto (n = 6)$ satisfies the seed and the exact transition shape yet is not\ncofinal, so the schema plus an endpoint-six seed provably does \\emph{not} give cofinality);\nconditional strict progress\n\\lean{skippedCoreSharpCapacity\\_below\\_or\\_strictly\\_laterExactRow}{BooleanMobiusExactRowCrossing.lean:233};\nunconditional recycling producer\n\\lean{exists\\_skippedCoreExactRow\\_of\\_value\\_above}{BooleanMobiusExactRowCrossing.lean:155};\nsocket \\lean{SkippedCoreCriticalQuotientSupply}{BooleanMobiusCriticalCapacityCofinal.lean:30};\ncapacity iff\n\\lean{localBinarySuffix\\_two\\_mul\\_sub\\_two\\_lt\\_criticalCapacity\\_iff}{BooleanMobiusSkippedCoreCriticalCapacity.lean:22};\nseed fixture $\\{2,3,6\\}$ at\n\\lean{exactLocalMersenneHalfRow\\_six}{BooleanMobiusExactRowSeed.lean:34}.\n\n\\textbf{Why it evades Barrier~\\ref{bar:canon}.} This is the single place where canonicalisation\nprovably does \\emph{not} bite, and the reason is exact. Proposition~\\ref{prop:canon}(ii) requires\nboth a real-valued below-half condition \\emph{and} the deficit clause. The exact-row predicate\n\\texttt{ExactLocalMersenneHalfRow} imposes neither: it asks only that a floor quotient hit\n$2^{n-1}-1$ in $\\N$. Floor division is not injective. So the per-endpoint witness space is an\nexponential search over $D \\subseteq [2,n]$, is decidable for each $n$, and is not forced to be the\ngreedy prefix. It is the only genuine degree of freedom left anywhere in the half-value branch.\n\n\\textbf{Its relation to Barriers~\\ref{bar:collapse} and \\ref{bar:asym}.} The \\emph{cofinal}\nstatement is still equivalent to the goal, by Proposition~\\ref{prop:collapsed-list}(c). But\nequivalence of truth value is not equivalence of proof difficulty when the existential has an\nexponentially large witness space rather than a singleton. And decidability of\n\\texttt{ExactLocalMersenneHalfRow}$(n)$ makes the exact-row set computable, so the data here is a\ntwo-sided observable rather than a one-sided survival lower bound --- a real softening of\nBarrier~\\ref{bar:asym}, though not an escape from it.\n\n\\subsection{Route 3: irrationality of\n\\texorpdfstring{\\(\\sum_{p\\ \\mathrm{prime}} 1/(2^p-1)\\)}{the prime Mersenne subseries}}\n\\label{ssec:route-3}\n\n\\textbf{What it needs.} Replace the global tail budget inside the pairwise-coprime certificate\nproducer by a windowed one: for every $\\varepsilon > 0$ there exist $B, N, L$ with\n\\[\n  \\sum_{a \\in A,\\ B < a \\le N+L} \\Big(\\frac{L}{a} + 1\\Big) \\;\\le\\; C\n  \\qquad\\text{and}\\qquad\n  q\\,(C + N + L + 2) \\;<\\; 2^{L},\n\\]\nand verify it for $A =$ the primes using Mertens' second theorem. Then instantiate the engine of\n\\lean{irrational\\_erdosSupportSeries\\_pairwise\\_coprime}{CertificateKernel.lean:10448} --- its\n\\emph{engine}, at the producer \\texttt{CertificateKernel.lean:9574}, not its statement --- to\nconclude irrationality. Secondary target with identical machinery: any infinite pairwise-coprime\n$A$ with divergent reciprocal sum.\n\n\\textbf{Why it evades Barrier~\\ref{bar:box}.} It sits cleanly inside axis (i) --- the primes are\npairwise coprime, the incidence is CRT-prescribable, and\n\\lean{compositeDilationDefect\\_eq\\_zero\\_of\\_prime\\_support}{CompositeDilationDefect.lean:103} makes\nthe dilation algebra exact --- and fails only axis (ii), at exactly one identified proof step,\n\\texttt{CertificateKernel.lean:9586--9598}. The engine permits $N$ and $L$ to be chosen \\emph{after}\nthe threshold $B$, so a windowed budget suffices; and for the primes\n$\\sum_{B < p \\le N+L} 1/p \\approx \\log\\log(N+L) - \\log\\log B$, which is controllable by taking $L$\nsmall relative to $B$. The half-value machinery of Barriers~\\ref{bar:canon}, \\ref{bar:collapse},\n\\ref{bar:asym} and \\ref{bar:measure} does not apply here at all: this is entirely on the universal\nside.\n\n\\textbf{Why it is worth doing.} It is a named constant and a genuine theorem, not a reduction. It\nwould be the first irrationality result for a divergent-reciprocal support, and the first outside\nthe box.\n\n\\subsection{Route 4: land the route-collapse equivalences}\n\\label{ssec:route-4}\n\n\\textbf{What it needs.} Five biconditionals, each with $1/2 \\in \\mathcal{A}$ on the right-hand\nside: the \\texttt{hbound} hypothesis of the two \\texttt{mobiusCenteredHalfCarry} consumers;\n\\texttt{CofinalPositiveHalfGreedySkips}; \\texttt{CofinalExactLocalMersenneHalfRows};\n\\texttt{GreedyHalfCarryCofinalStripReturn}; \\texttt{HalfCarryCofinalTerminalOnlyStrip}. The forward\ndirections are landed. Every reverse direction runs through Lemma~\\ref{lem:sqrt-witness}, stated\nonce, whose only inputs are\n\\lean{integerHalfCarry\\_eq\\_scaled\\_residual\\_add\\_tail}{HalfCarryReachability.lean:842},\n\\lean{binaryCoeffTail\\_nonneg}{GenericTailOrbitRigidity.lean:78},\n\\lean{binaryCoeffTail\\_supportCoeff\\_le\\_two\\_sqrt\\_add\\_four}{BooleanMobiusCarry.lean:289}, and the\nevaluation of the square root at a perfect square. The scope limit of\nProposition~\\ref{prop:collapsed-list} must be stated verbatim in the docstrings.\n\n\\textbf{Why it is here.} This route does not attempt the wall; it \\emph{measures} it, which is\nexactly what makes it bankable. Each equivalence is a theorem --- a biconditional between a named\n\\texttt{Prop} and half-membership --- not a reduction, and each one permanently retires a socket\nfrom the list of things worth chasing directly. Systematic elimination of this kind is a result\nabout the problem in a way that a reformulation is not. The perfect-square normalisation is what\nmakes it available now: it dissolves the $\\lfloor\\sqrt{\\cdot}\\rfloor$ versus $\\sqrt{\\cdot}$\nmismatch that previously blocked two of the five, and it needs no widening of the strip constant.\n\n\\subsection{Route 5: the sparse/dense trichotomy closure}\n\\label{ssec:route-5}\n\n\\textbf{What it needs.} An unconditional\n\\[\n  A \\text{ infinite} \\;\\Longrightarrow\\;\n  \\big(\\text{cofinally many super-logarithmic zero windows in } \\operatorname{sc}_A\\big)\n  \\;\\vee\\;\n  \\big(\\forall q > 0,\\ \\text{a weighted block certificate exists for } \\operatorname{sc}_A\\big).\n\\]\nThe first disjunct contradicts\n\\lean{supportCoeffZeroWindow\\_length\\_le\\_eps\\_logb}{SublogDivisorCoverage.lean:435} whenever the\nseries is rational; the second closes via the certificate engine. The sparse branch is done ---\nthat theorem, and independently\n\\lean{irrational\\_erdosSum\\_of\\_lcm\\_gap}{CertificateKernel.lean:5555}, close it. What is required\nis the \\emph{dense-branch producer}: for $A$ whose divisor incidence is bounded below infinitely\noften, construct a block of $K$ consecutive indices with\n$2^K \\mid \\sum_{r=1}^{K} \\operatorname{sc}_A(N+r)\\,2^{K-r}$, a bounded middle window $C$, and\n$q(C+N+L+2) < 2^L$.\n\n\\textbf{Why it evades Barrier~\\ref{bar:box}.} It attacks the box from outside rather than\ninstantiating inside it. It does not ask a general support to satisfy axis (i) or (ii); it asks for\na dichotomy on the support's own divisor density, and then uses a different tool on each side. It\nevades Barrier~\\ref{bar:measure} because it is a structural statement about the incidence function,\nnot a measure or category claim.\n\n\\textbf{The honest flag.} The dense-branch producer is the same missing object as \\S2.7, restated\nin the incidence coordinate. That is not a reason to discard the route --- the restatement is\ngenuinely a different coordinate, and a proof might be easier there --- but it is a reason not to\ncount it as an independent line of attack. \\ev{Open}\n\n\\subsection{Ranking, stated plainly}\n\nRoute 4 is the only one that can be completed with the mathematics currently in hand; it produces\ntheorems, not progress on the wall. Route 3 is the only one that would produce a new irrationality\ntheorem for a named constant, and its missing input is a windowed Mertens estimate, which is\nordinary analytic number theory rather than a new phenomenon. Route 2 is the only place in the\nhalf-value branch with a non-singleton witness space, which makes it the best target for search.\nRoute 1 is where the actual obstruction lives, and Route 5 is Route 1 in another coordinate.\nNeither problem is close to being solved, and no route above is a proof of anything.\n\n\\section{How to use this document}\n\\label{sec:257-howto}\n\n\\textbf{Read it in full before starting work.} That instruction is not a courtesy. The recurring\npractical failure of this programme has not been difficulty; it has been rediscovery --- the same\nreformulations, the same dead attack families, arrived at independently three times because no\nsingle document held the whole picture. This document is that picture for \\#257. Anything proposed\nwithout reading it has a substantial prior probability of already appearing above with its\nmechanism of death recorded.\n\n\\textbf{Evidence bands are never blurred.} \\ev{Lean} means kernel-checked, with the declaration\nname as the authority and the file:line as where it sat at the pinned checkpoint. \\ev{Cert} means\nexact finite computation, integer arithmetic, no floating point. \\ev{Math} means ordinary\nmathematics, proved but not formalised. \\ev{Cited} means published, with the reference given.\n\\ev{Open} means not proved by anyone. A composite tag means the statement has parts at different\nbands. Declarations in the per-problem tree carry an \\texttt{Erdos257/} path prefix; everything\nelse lives in the main package.\n\n\\textbf{The scale axis is the load-bearing one.} Statements carry \\scale{fixed} (a specific index\nor depth), \\scale{bounded} (all indices up to a parameter), \\scale{cofinal} (arbitrarily large\nindices), or \\scale{uniform} (all indices, or independent of the parameter). The single most\nimportant structural fact about the corpus is that its certified content sits on the\nfixed/bounded side --- rungs $J \\le 22$, seam rows $\\le 200{,}000$, ranks $\\le 3000$, macro depth\n$\\le 13{,}548{,}057$ --- while both open obligations are cofinal. By Theorem~\\ref{thm:one-sided}\nthat gap is a gap in kind, not in degree: no amount of extension on the fixed/bounded side becomes\na proof. Read a scale tag as a hard type, not as a measure of how far along something is.\n\n\\textbf{Obstructions are coordinate-relative.} Every obstruction below carries a coordinate tag,\nsuch as \\coord{half-carry} or \\coord{seam-row}, because an obstruction is always an obstruction \\emph{in a representation}, never a property of the\nobject. \\coord{half-carry} facts say nothing about \\coord{seam-row} arguments; a wall in the\n\\coord{diophantine} coordinate (Proposition~\\ref{prop:exponent-gap}) is silent about the\n\\coord{certificate-engine} coordinate. Before recording any new wall, name the coordinate it lives\nin and test at least one re-representation. Several of the barriers above are strong precisely\nbecause they are coordinate-free --- Proposition~\\ref{prop:local-void} mentions no Mersenne\nstructure at all and binds \\#249 identically --- and that is worth stating explicitly when it is\ntrue, and not otherwise.\n\n\\textbf{Map of what follows.} The premise catalogue comes next: every landed statement with exact\nhypotheses, conclusion, coordinate, scale and site, organised by lane --- the certificate kernel and\nuniversal-direction inventory; the greedy/achievement-set geometry; the seam model, its master\nidentity and its dynamics; the half-carry and Boolean--M\\\"obius coordinates; and the support\nrigidity results. After the catalogue come the typed implication tables, which record what chains\nwith what, and the near-miss index, which records, for each open obligation, the statements that\ncome closest and the exact mismatch in each case. The catalogue is a reference surface and may be\nread by lane. Sections~\\ref{sec:257-wall} and \\ref{sec:257-survivors} are not: they are an argument,\nand they should be read in order.\n\n% ---- part a257_p0 ----\n\\section{The object, the two questions, and the exact record}\n\n\\subsection{The object}\n\nFix the Mersenne weight function $x_n := 1/(2^n-1)$ for integers $n \\ge 2$. For a set\n$A \\subseteq \\{2,3,4,\\dots\\}$, write $x_A := \\sum_{n \\in A} x_n$ when the sum converges (it always\ndoes, since $x_n = O(2^{-n})$). Erd\\H{o}s's Problem \\#257 asks about the family of series\n$\\{x_A : A \\text{ infinite}\\}$.\n\n\\begin{defn}[Universal Problem \\#257]\n\\textbf{Is $x_A = \\sum_{n \\in A} 1/(2^n-1)$ irrational for \\emph{every} infinite\n$A \\subseteq \\mathbb{N}$?}\nThis is a single universally-quantified statement over all infinite subsets of $\\mathbb{N}$. It is\n\\textbf{OPEN}. \\ev{Open}\n\\end{defn}\n\nThe record in this paper does \\emph{not} decide universal \\#257. Everything that follows is either\n(i) a proof of irrationality for specific, named infinite-support families -- which narrows but\ndoes not close the universal statement, since infinitely many other supports remain unchecked -- or\n(ii) an attack on one specific, maximally symmetric candidate counterexample, described next.\n\n\\begin{defn}[The distinguished half-value question]\n\\label{defn:half-question}\nDefine the \\emph{Mersenne achievement set}\n$\\mathcal{A} := \\{\\,y \\in \\mathbb{R} : \\exists A \\subseteq \\mathbb{N},\\ 0 \\notin A,\\ y =\n\\sum_{n \\in A} x_n\\,\\}$, the set of reals realizable as a subset sum of the positive-index Mersenne\nweights $\\{x_n\\}_{n \\ge 1}$. \\textbf{Is $1/2 \\in \\mathcal{A}$?} This is \\textbf{OPEN}. \\ev{Open}\n\\end{defn}\n\n\\begin{obs}[Why the half-value question controls universal \\#257]\n\\label{obs:half-controls-universal}\nAny finite subset sum of $\\{x_n\\}$ has odd reduced denominator, since every $2^n - 1$ is odd; hence\nno finite $A$ can sum to $1/2$ (\\lean{positiveMersenneSupportValue_coe_finset_ne_half}{HalfCutLocator.lean:243}).\nConsequently, \\emph{if} $1/2 \\in \\mathcal{A}$, the witnessing support $A$ is necessarily infinite, and\n$x_A = 1/2$ is rational. Such an $A$ is then a single infinite set for which the universal-\\#257\nsum is rational -- an outright refutation of universal \\#257. Conversely, universal \\#257 being true\nforces $1/2 \\notin \\mathcal{A}$ as one instance among uncountably many. The half-value question is\nthus the maximally symmetric, most heavily studied test case for universal \\#257's negation, not a\nrestatement of it: proving $1/2 \\notin \\mathcal{A}$ leaves universal \\#257 completely open (all other\ninfinite $A$ remain unchecked), while proving $1/2 \\in \\mathcal{A}$ closes universal \\#257\nimmediately, in the negative. \\scale{n/a} \\ev{Math}\n\\end{obs}\n\n\\subsection{What is proved}\n\n\\subsubsection{Full-support irrationality, for every base}\n\n\\begin{thm}[Full-support irrationality, unconditional, every base $b \\ge 2$]\n\\label{thm:full-support}\nFor every integer $b \\ge 2$,\n\\[\n\\sum_{k=0}^{\\infty} \\frac{1}{b^{k+1}-1} \\quad \\text{is irrational.}\n\\]\n\\textbf{Hypotheses:} $b \\in \\mathbb{N}$, $b \\ge 2$. \\textbf{Conclusion:} unconditional irrationality\nof the full-support Erd\\H{o}s--Borwein-type series at every base, not merely base $2$.\n\\coord{full-support} \\scale{uniform} \\ev{Lean}\\\\\n\\textbf{Site:} \\lean{irrational\\_erdosSum\\_full\\_support}{CertificateKernel.lean:8000}\n(`$b : \\mathbb{N}$, hb : $2 \\le b$'), landed via a certificate machine (bounded Bertrand/CRT\nfirst-block frame, middle-window divisor-pair average with pigeonhole selection, explicit parameter\nclosure). The base-$2$ instance recovering the classical Erd\\H{o}s--Borwein constant's irrationality\nis \\lean{irrational\\_erdosBorwein\\_series}{CertificateKernel.lean:8007}, a one-line corollary.\n\\end{thm}\n\nThis is the base-$b$ generalization of Erd\\H{o}s's original 1948 irrationality theorem for\n$\\sum 1/(2^n-1)$ (cited, in its base-$2$ form, as an input at Remark~\\ref{rem:no-ties} below); its\nLean form removes the base restriction entirely and is landed with no open hypotheses.\n\n\\subsubsection{Named infinite-support families}\n\nBeyond full support ($A = \\{1,2,3,\\dots\\}$, i.e.\\ $m=1$ below), the following named infinite\nfamilies are each proved irrational unconditionally, for every base $b \\ge 2$:\n\n\\begin{thm}[Purely periodic support]\nFor every $b \\ge 2$, every modulus $m \\ge 1$, and every $m$-periodic $A \\subseteq \\mathbb{N}$ (i.e.\\\n$n+m \\in A \\Leftrightarrow n \\in A$ for all $n$) containing a positive element, the support series\n$\\sum_{a \\in A} 1/(b^a-1)$ is irrational. \\textbf{Hypotheses:} $b \\ge 2$, $m \\ge 1$, $A$\n$m$-periodic, $\\exists a > 0,\\ a \\in A$. \\coord{periodic-support} \\scale{uniform} \\ev{Lean}\\\\\n\\textbf{Site:} \\lean{irrational\\_erdosSupportSeries\\_periodic}{CertificateKernel.lean:11262}.\n\\end{thm}\n\n\\begin{thm}[Eventually periodic support]\nFor every $b \\ge 2$: if $A \\subseteq \\mathbb{N}$ is infinite and its membership is $m$-periodic from\nsome threshold $N_0$ on (i.e.\\ $n+m \\in A \\Leftrightarrow n \\in A$ for all $n \\ge N_0$), then\n$\\sum_{a \\in A} 1/(b^a-1)$ is irrational -- proved by transferring irrationality across the finite\nsymmetric difference from the shifted purely periodic set $A_{\\mathrm{pure}} := \\{n : n + m N_0 \\in\nA\\}$. \\textbf{Hypotheses:} $b \\ge 2$, $m \\ge 1$, $N_0 \\in \\mathbb{N}$, eventual $m$-periodicity from\n$N_0$, $A$ infinite. \\coord{eventually-periodic-support} \\scale{uniform} \\ev{Lean}\\\\\n\\textbf{Site:} \\lean{irrational\\_erdosSupportSeries\\_eventuallyPeriodic}{CertificateKernel.lean:11276}.\n\\end{thm}\n\n\\begin{thm}[Residue-class support]\nFor every $b \\ge 2$, modulus $m \\ge 1$, and residue $c$: $\\sum_{n \\equiv c \\pmod m} 1/(b^n-1)$ is\nirrational. \\coord{residue-class-support} \\scale{uniform} \\ev{Lean}\\\\\n\\textbf{Site:} \\lean{irrational\\_erdosSupportSeries\\_residueClass}{CertificateKernel.lean:11344}\n(specializes purely periodic support).\n\\end{thm}\n\n\\begin{thm}[Odd support -- first density-strictly-between-0-and-1 class]\nFor every $b \\ge 2$: $\\sum_{n \\text{ odd}} 1/(b^n-1)$ is irrational. This is the first landed support\nclass with natural density strictly between $0$ and $1$ (odd $n$ has density $1/2$ inside\n$\\mathbb{N}$), as opposed to the density-$1$ full-support case or density-$1/m$ residue classes for\nlarge $m$. \\coord{odd-support} \\scale{uniform} \\ev{Lean}\\\\\n\\textbf{Site:} \\lean{irrational\\_erdosSupportSeries\\_odd}{CertificateKernel.lean:11358}\n(specializes residue-class support at $m=2,c=1$).\n\\end{thm}\n\nNone of these families contains $A$ of the shape produced by a candidate half-value witness (the\ngreedy support for $1/2$, described in \\S\\ref{ssec:greedy-orbit}, is not eventually periodic in any\nlanded sense) -- so Theorem~\\ref{thm:full-support} and its periodic-family corollaries neither prove\nnor disprove $1/2 \\in \\mathcal{A}$; they exhaust only the structured-support corner of universal\n\\#257's infinite search space.\n\n\\subsubsection{Topology and measure of the achievement set}\n\n\\begin{thm}[Achievement-set topology and measure]\n\\label{thm:topology}\n$\\mathcal{A}$ (Definition~\\ref{defn:half-question}) is compact, closed, perfect, totally\ndisconnected, and nowhere dense; its Lebesgue measure is exactly $1$:\n$\\operatorname{volume}(\\mathcal{A}) = 1$. \\textbf{Hypotheses:} none. \\textbf{Conclusion:}\n$\\mathcal{A}$ is a fat Cantor set -- topologically thin (nowhere dense and totally\ndisconnected) yet of positive finite measure.  The number $1$ here is an absolute Lebesgue\nmeasure, not a full-measure assertion: the convex hull has length\n$T_0=1+T_1>1$, and of course $\\mathbb R$ has infinite measure. \\coord{achievement-set-topology}\n\\scale{n/a} \\ev{Lean}\\\\\n\\textbf{Site:} \\lean{isCompact\\_mersenneAchievementSet}{GreedyAchievementSet.lean:656--1024},\n\\lean{isClosed\\_mersenneAchievementSet}{GreedyAchievementSet.lean:656--1024},\n\\lean{perfect\\_mersenneAchievementSet}{GreedyAchievementSet.lean:656--1024},\n\\lean{isTotallyDisconnected\\_mersenneAchievementSet}{GreedyAchievementSet.lean:656--1024},\n\\lean{isNowhereDense\\_mersenneAchievementSet}{GreedyAchievementSet.lean:656--1024},\n\\lean{volume\\_mersenneAchievementSet}{GreedyAchievementSet.lean:656--1024}. The proofs use only\nsuperincreasingness of $\\{x_n\\}$ and summability, so the argument transfers verbatim to any strictly\nsuperincreasing weight series.\\\\\n\\textbf{Chains with:} Theorem~\\ref{thm:greedy-survival-record} below (compactness is exactly what powers\nevery ``limit of achieved points is achieved'' argument used downstream, including the seam-limit\nroute of Part~2).\n\\end{thm}\n\n\\begin{thm}[Membership equals greedy survival at every level]\n\\label{thm:greedy-survival-record}\n$x \\in \\mathcal{A} \\iff 0 \\le x \\wedge \\forall n,\\ \\rho(x,n) \\le T_{n+1}$, where $\\rho(x,n)$ is the\ncanonical greedy (take-if-possible) residual after rank $n$ and $T_{n+1} := \\sum_{j>n} x_j$ is the\nfull remaining tail. \\textbf{Hypotheses:} none (unconditional biconditional). \\textbf{Conclusion:}\nmembership in $\\mathcal{A}$ is exactly the statement that the greedy process never enters a fatal\nstate at any rank -- this is the master theorem underlying every sufficient-condition result for\n$1/2 \\in \\mathcal{A}$ used later in this paper. \\coord{greedy-survival} \\scale{uniform} \\ev{Lean}\\\\\n\\textbf{Site:} \\lean{mem\\_mersenneAchievementSet\\_iff\\_greedy\\_survival}{GreedyAchievementSet.lean:1445}\n(forward direction \\lean{greedy\\_survives\\_of\\_mem\\_mersenneAchievementSet}{GreedyAchievementSet.lean:1362},\nreverse \\lean{mem\\_mersenneAchievementSet\\_of\\_greedy\\_survival}{GreedyAchievementSet.lean:1381}).\n\\end{thm}\n\n\\subsection{The greedy orbit, exactly}\n\\label{ssec:greedy-orbit}\n\nFix the target $1/2$ and the weights $x_n = 1/(2^n-1)$, $n \\ge 1$.  Here $x_1=1$, so the\ntarget $1/2$ automatically skips rank $1$; the nontrivial orbit begins at rank $2$. The greedy\norbit is the unique deterministic process forced by superincreasingness\n(Theorem~\\ref{thm:greedy-survival-record}):\n\n\\begin{defn}[The half-value greedy orbit]\n\\label{defn:greedy-orbit}\nLet $\\rho_1 := 1/2$ (the residual entering rank $2$). At rank $k \\ge 2$, given residual $\\rho_k$:\n\\begin{itemize}[nosep]\n\\item \\textbf{Take} iff $\\rho_k \\ge x_k$; if taken, $\\rho_{k+1} := \\rho_k - x_k$.\n\\item \\textbf{Skip} otherwise; $\\rho_{k+1} := \\rho_k$.\n\\end{itemize}\nA skip at rank $k$ is:\n\\begin{itemize}[nosep]\n\\item \\textbf{safe} iff $\\rho_k \\le T_{k+1} := \\sum_{j>k} x_j$ (the residual still fits inside the\nremaining tail, so future ranks can still reach it);\n\\item \\textbf{fatal} iff $\\rho_k \\in (T_{k+1}, x_k)$ (the residual is too large for the remaining\ntail to ever reach, yet too small to have been taken at $k$ -- an impossible gap).\n\\end{itemize}\n\\end{defn}\n\nBy Theorem~\\ref{thm:greedy-survival-record}, $1/2 \\in \\mathcal{A}$ iff no rank of this orbit is ever fatal.\nTwo boundary degeneracies are excluded unconditionally and independently of any open hypothesis:\n\n\\begin{rem}[No-ties: both greedy-orbit boundaries are strict]\n\\label{rem:no-ties}\n(a) $\\rho_k \\ne x_k$ at any rank: a finite Mersenne sum has odd reduced denominator (each $2^n-1$ is\nodd), so no finite prefix mass can equal $x_k$ exactly, sharpening the general\ndenominator-parity fact used in Observation~\\ref{obs:half-controls-universal}.\n(b) $\\rho_k \\ne T_{k+1}$ at any rank: the tail $T_{k+1} = E - 1 - \\sum_{2 \\le j \\le k} x_j$, where\n$E := \\sum_{n\\ge1} x_n$ is the Erd\\H{o}s--Borwein constant, is irrational by Erd\\H{o}s's own 1948\ntheorem (the base-$2$ case of Theorem~\\ref{thm:full-support}), while $\\rho_k$ is always rational;\nequality is impossible. \\textbf{Conclusion:} both greedy-orbit boundaries are provably strict at\nevery rank, so the orbit's decisions stabilize after finitely many strict inequalities at any fixed\ntruncation depth. \\coord{no-ties} \\scale{uniform} \\ev{Math}\n\\end{rem}\n\n\\subsection{The seam model and the master identity}\n\\label{ssec:seam-model}\n\nThe seam model is a base-$4$ rescaled integer reformulation of Definition~\\ref{defn:greedy-orbit},\nbuilt so that every quantity involved is an exact natural number (no real-number or limiting\narithmetic). Fix a row $n \\ge 6$. The seam weights at row $n$ are\n\\[\nw(n,d) := \\left\\lfloor \\frac{4^n}{2^d-1} \\right\\rfloor, \\qquad d = 2,\\dots,n-1,\n\\]\nand the seam target is $T_n := 2^{2n-1} - 2^n$. The seam greedy process takes ranks $d = 2,\\dots,n-1$\nin ascending order against $T_n$ exactly as in Definition~\\ref{defn:greedy-orbit}, producing a take\nset $D_n \\subseteq \\{2,\\dots,n-1\\}$ and skip set\n$\\mathrm{Skip}_n := \\{2,\\dots,n-1\\} \\mathbin{\\backslash} D_n$.\nDefine the \\emph{seam remainder} $\\mathrm{rem}(n) := T_n - \\sum_{d \\in D_n} w(n,d)$ and the\n\\emph{seam deviation} $\\Delta_n := \\mathrm{rem}(n) - 2^n$.\n\n\\begin{defn}[The universal recursion $K(M)$]\nFor $M \\ge 2$, define\n\\[\nK(M) := 2^{M-1} - \\sum_{d=2}^{M} \\left\\lfloor \\frac{2^M}{2^d-1} \\right\\rfloor.\n\\]\n$K$ is \\emph{row-independent}: it depends only on $M$, not on any seam row $n$. It satisfies the\nrecursion $K(2) = 1$, $K(M+1) = 2K(M) - (\\tau(M+1)-1)$, where $\\tau$ is the number-of-divisors\nfunction (so $\\tau(M+1)-1$ is the count of proper divisors of $M+1$). \\coord{universal-recursion}\n\\scale{n/a} \\ev{Cert}\n\\end{defn}\n\n\\begin{thm}[Master seam identity]\n\\label{thm:master-identity}\nFor every seam row $n \\ge 6$,\n\\begin{equation}\n\\Delta_n \\;=\\; K(2n) \\;+\\; \\sum_{d \\in \\mathrm{Skip}_n} \\left\\lfloor \\frac{4^n}{2^d-1} \\right\\rfloor.\n\\tag{I}\n\\end{equation}\n\\textbf{Hypotheses:} $n \\ge 6$; $\\mathrm{rem}(n)$, $D_n$, $\\mathrm{Skip}_n$ as in the seam greedy\nprocess for target $T_n$ over weights $w(n,d) = \\lfloor 4^n/(2^d-1) \\rfloor$, $d=2,\\dots,n-1$,\nascending greedy. \\textbf{Conclusion:} the entire $2n$-bit seam deviation collapses to one universal,\nrow-independent recursion value $K(2n)$ plus a short sum restricted to the skipped ranks only --\nevery rank the greedy \\emph{takes} drops out of the identity entirely.\n\\coord{seam-row / integer-greedy (Mersenne weights, binary digits of $K$)} \\scale{fixed} \\ev{Math}\\ev{Cert}\\\\\n\\textbf{Proof sketch (shape preserved from the source record; the full two-line derivation lives in\nthe source document and is not independently re-derived here):} unfold\n$\\mathrm{rem}(n) = T_n - \\sum_{d \\in D_n} w(n,d)$ and compare the full-range sum\n$\\sum_{d=2}^{2n} \\lfloor 4^n/(2^d-1)\\rfloor$ (the sum defining $K(2n)$, evaluated at $M=2n$ so that\n$2^M = 4^n$) against the seam row's own sum over $D_n \\cup \\mathrm{Skip}_n = \\{2,\\dots,n-1\\}$; the\nranks $d \\ge n$ present in $K(2n)$'s defining sum but absent from the row's range, together with the\ntaken ranks $D_n$, absorb exactly into $T_n - 2^n = 2^{2n-1}-2^{n+1}$ against $K(2n)$'s leading\n$2^{2n-1}$ term, leaving only the skipped-rank sum as residue. \\textbf{Certification:} verified rows\n$6\\le n\\le200$, zero mismatches, independently reproduced by two subagents (\\S\\ref{ssec:anchors}).\\\\\n\\textbf{Chains with:} any statement about $\\tau(M+1)$ parity/growth, or about the binary digit stream\nof $K(M)$, composes directly through (I).\n\\end{thm}\n\n\\subsection{The real form and the master identity's analytic reading}\n\n\\begin{thm}[Real (non-integer) form of the master identity]\n\\label{thm:real-form}\nLet $x_d := 1/(2^d-1)$ and define the constant\n\\[\nC \\;:=\\; \\sum_{d \\ge 2} x_d \\;-\\; \\tfrac12 \\;=\\; E - \\tfrac32 \\;=\\; 0.1066951524152917\\ldots,\n\\]\nwhere $E$ is the Erd\\H{o}s--Borwein constant. Then, for every seam row $n \\ge 6$,\n\\begin{equation}\n\\Delta_n \\;=\\; 4^n \\Big( \\sum_{d \\in \\mathrm{Skip}_n} x_d \\;-\\; C \\Big) \\;+\\; \\eta_n, \\qquad\n|\\eta_n| < 2n+2.\n\\tag{II}\n\\end{equation}\n\\textbf{Hypotheses:} as Theorem~\\ref{thm:master-identity}. \\textbf{Conclusion:} escaping the wall\n(making $\\Delta_n$ large, at scale $2^{(n+5)/2}$ or beyond -- the exact target of Part~2's Theorem A)\nis exactly the statement that the greedy skip-set's real-valued partial sum $\\sum_{\\mathrm{Skip}_n}\nx_d$ never approximates the fixed constant $C$ to within roughly $2^{-1.5n}$, at the natural scale\n$2^{-n}$ set by $4^n \\cdot 2^{-n} = 2^n$. \\coord{real/analytic (Erd\\H{o}s--Borwein tail); bridges the\ninteger seam coordinate to a continuous Diophantine-approximation statement} \\scale{fixed}\n\\ev{Math}\\ev{Cert}\\\\\n\\textbf{Note:} $C = E - 3/2$ is Mersenne-specific, but the shape ``scaled deviation $=$\n$4^n \\cdot$(finite skip-sum $-$ target constant) $+ O(n)$'' is a template, not yet matched to any\nanalogous constant on the \\#249 side.\n\\end{thm}\n\n\\subsection{Dynamics}\n\n\\begin{thm}[Deviation dynamics and the R/M/U branch structure]\n\\label{thm:dynamics}\nLet $\\lambda_n := \\Delta_n / 2^n$. Then\n\\begin{equation}\n\\lambda_{n+1} \\;=\\; 2\\lambda_n + 2e_n + \\delta_n, \\qquad e_n \\in \\{0,+1,-1\\} \\text{ for branches\nR/M/U respectively}, \\qquad |\\delta_n| \\le \\frac{2n+6}{2^n}.\n\\tag{III}\n\\end{equation}\n\\textbf{Hypotheses:} as Theorem~\\ref{thm:master-identity}. \\textbf{Conclusion:} the deviation\ndynamics is a doubling map on $\\lambda_n$ with a forced signed digit $e_n$ (recording which of three\nbranches -- Right/Middle/Upper -- fires at each row) plus a vanishing correction $\\delta_n = O(n\n2^{-n})$. \\textbf{Measured bands} (rows $14$ through $699$, certified computation): branch R\n$\\lambda \\in [-0.499189,\\ 0.499846]$; branch M $\\lambda \\in [-0.998378,\\ -0.501224]$; branch U\n$\\lambda \\in [0.501652,\\ 0.999692]$ -- the three bands are disjoint and together nearly cover\n$[-1,1]$. \\coord{seam-affine dynamics (doubling map with signed digit $e_n$)} \\scale{bounded}\n\\ev{Math}\\ev{Cert}\\\\\n\\textbf{Consequence:} the target statement (escaping the wall) is equivalent to: the R-run (maximal\nrun of consecutive R branches) after each reset row $r \\ge 31$ has length $L_r < (r-3)/2$; a long\nR-run is exactly what is dangerous, and any $L_r = o(r)$ closes it, while a \\emph{uniform} constant\nbound on run length is empirically \\textbf{false} (observed run lengths grow like\n$\\log_2(\\text{row})$, \\S\\ref{ssec:anchors}).\\\\\n\\textbf{Chains with:} this is the exact object whose long-run/anti-concentration behavior is the\npaper's central open producer (Part~2 develops it in full; the shape recurs, independently derived,\nunder the coordinate name $w_n = \\mathrm{rem}(n) - 2^n$ in the reset-crossing unification record).\n\\end{thm}\n\n\\subsection{Every anchor}\n\\label{ssec:anchors}\n\nThe following are exact, certified numerical facts about the seam orbit of\n\\S\\ref{ssec:seam-model}--\\S\\ref{ssec:seam-model}, stated literally so they can be used as\nregression fixtures without re-running any computation.\n\n\\begin{obs}[Exact low-row values]\n\\label{obs:anchors-low}\nAt row $n=14$: $\\mathrm{rem}(14) = 392$, with take set $D_{14} = \\{2,3,6,7\\}$. At row $n=15$:\n$\\mathrm{rem}(15) = 34333$. At row $n=32$: $\\mathrm{rem}(32) = 3865046005$ and $\\Delta_{32} =\n-429921291$. \\coord{seam row, exact integer} \\scale{fixed} \\ev{Cert}\\\\\n\\textbf{Site:} the row-$14$ through row-$31$ range, including $\\mathrm{rem}(14)=392$, additionally\ncarries independent \\ev{Lean} confirmation via the kernel-computed certificate\n\\lean{seamUpperResetDyadicBandEscape\\_through\\_thirty}{HalfCylinderUpperResetBandCertificates.lean:78},\nwhich lists the exact seam remainder at every row $13$--$31$ by `decide`-checked kernel computation\nand finds none violating the dyadic band-escape condition.\n\\end{obs}\n\n\\begin{obs}[Certified computation depth, all four coordinates]\n\\label{obs:anchors-depth}\nThe master identity (Theorem~\\ref{thm:master-identity}) is verified rows $6 \\le n \\le 200$ with zero\nmismatches, independently reproduced by two subagents, and the seam orbit itself is extended and\ncertified to row $2500$ (and separately, in a longer single run, to row $200{,}000$ -- \\S SE-7 of the\nunderlying record, $207.7\\mathrm{s}$ runtime, branch counts $\\mathrm{R{:}M{:}U} =\n100197{:}49899{:}49898$, TARGET failing at exactly one reset row $\\le 200{,}000$, namely $r=7$).\n\\coord{multiple: seam row, binary digit stream of $C$, greedy set $G$} \\scale{fixed} \\ev{Cert}\n\\end{obs}\n\n\\begin{obs}[The four coordinate depths, side by side]\n\\label{obs:anchors-coords}\nDefinition~\\ref{defn:greedy-orbit}'s single orbit is independently certified in four distinct\nintegerizations, to the following depths: truncation rung $J = 3,\\dots,22$ (all $20$ rungs proved to\nsurvive, not merely computed -- Part~3 of the truncation-rung record); seam row $6 \\le n \\le 2500$\n(Theorem~\\ref{thm:master-identity}/\\ref{thm:dynamics}); integer margin scan to $m_c = 5\\times10^5$\n(advisory Type~B scan, not yet promoted to a certified theorem); sharp tail margin, ranks $2$ through\n$3000$ (no fatal skip at any rank; take fraction $0.4992$; tightest certified take-margin\n$0.0340$ bits at rank $7$, the near-tie $1/126 \\ge 1/127$). \\coord{meta: all four coordinates}\n\\scale{fixed} \\ev{Cert}\\\\\n\\textbf{Chains with:} these four depths are four coordinatizations of the \\emph{same} decision\nstream (Theorem C-style co-semi-decidability, developed fully in Part~2); a certificate at one depth\nin one coordinate does not, by itself, extend to another coordinate without the explicit\nstabilization argument that the wider record supplies.\n\\end{obs}\n\nThe record above is exhaustive at the level of exact statements and sites available in the premise\nbank for this Part; no numeral in this section was recomputed independently by the present writer --\neach is either directly Lean-kernel-checked (as flagged) or reported as a certified-computation\nresult from the source record, and every scale/evidence tag reflects only what its cited source\nclaims for it.\n\n% ---- part a257_p1a ----\n\\section{Catalogue of usable premises}\n\nThis section catalogues machinery for Erd\\H{o}s \\#257 (is $\\sum_{n\\in A} 1/(2^n-1)$\nirrational for every infinite $A\\subseteq\\mathbb{N}$?) drawn from the half-membership\nroute (is $1/2$ in the Mersenne achievement set?) and the support-rigidity /\nfull-support route (irrationality of $\\sum_{a\\in A} 1/(2^a-1)$ for named families $A$).\nEvery entry states exact hypotheses, exact conclusion, coordinate, scale, evidence band,\nand Lean site. Nothing below decides \\#257; the problem remains open. Entries are grouped\n\\textbf{Producers} (hypothesis $\\Rightarrow$ conclusion, including conditional sockets\nwhose hypothesis is itself an open sub-obligation, always flagged) and \\textbf{Converters\nand exact identities} (iff-reformulations, structural invariants, definitions, exact\nnumerals). Within each subsection entries are ordered by scale:\nfixed $<$ bounded $<$ cofinal $<$ uniform $<$ n/a.\n\n\\subsection{Producers}\n\n\\begin{thm}[Cofinal skip forces half-membership]\n\\label{thm:fatal-absorbing}\nFatality is absorbing for the greedy Mersenne recursion at target $x\\ge 0$: once\n\\emph{GreedyMersenneFatalAt x n} holds it holds at every $n+k$, and forces every later\nexponent selected, so a fatal state leaves only finitely many omitted exponents.\nContrapositively: if the greedy skipped support (the set of ranks the greedy process\ndoes \\emph{not} select) is infinite, then $x$ lies in the Mersenne achievement set.\n\\par\\ev{Lean}\\scale{cofinal}\\coord{greedy-orbit}\n\\lean{mem\\_mersenneAchievementSet\\_of\\_greedySkippedSupport\\_infinite}{GreedyAchievementSet.lean:1524}\n\\end{thm}\n\n\\begin{thm}[Seam-limit convergence route to $1/2$]\n\\label{thm:seam-limit}\nLet \\emph{seamGreedyFiniteValue s} be the real point coded by the finite integer seam\nword at row $s$ (always achieved) and \\emph{seamGreedyNormalizedRemainder s} the seam\ninteger remainder rescaled by $4^{-s}$. If there is a cofinal sequence of rows\n\\emph{rows} along which the normalized remainder tends to $0$\n(\\emph{SeamGreedyRemainderSubquadraticAlong rows}), then\n\\emph{seamGreedyFiniteValue(rows j)}$\\to 1/2$ and, by closedness of the achievement set,\n$1/2$ is achieved.\n\\par\\ev{Lean}\\scale{cofinal}\\coord{seam-integer}\n\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_subquadraticAlong}{HalfCylinderSeamLimit.lean:250}\n\\end{thm}\n\n\\begin{thm}[Two-channel and dyadic cap sufficiency]\n\\label{thm:two-channel-cap}\nIf, at every rank $n$ the greedy process for target $1/2$ actually skips, the residual\n\\emph{greedyMersenneRemainder(1/2) n} is bounded above by the explicit rational envelope\n\\emph{halfTwoChannelCap(n+1)} (resp.\\ the coarser \\emph{halfDyadicCap(n+1)}), then\n$1/2\\in\\mathrm{mersenneAchievementSet}$. It therefore suffices to control the residual\nonly at actually-skipped ranks, not at every rank.\n\\par\\ev{Lean}\\scale{uniform}\\coord{two-channel-dyadic-cap}\n\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_skipped\\_twoChannelCap}{GreedyAchievementSet.lean:1412}\n\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_skipped\\_dyadicCap}{GreedyAchievementSet.lean:1431}\n\\end{thm}\n\n\\begin{thm}[Second-channel phase separation sufficiency]\n\\label{thm:second-channel}\nIf the greedy second-channel phase (\\emph{greedyMersenneSecondChannelPhaseRat}) stays\nseparated (\\emph{HalfSecondChannelSeparatedRat}) at every skipped rank, then\n$1/2\\in\\mathrm{mersenneAchievementSet}$; a rational specialization discharges the\nhypothesis from row $7$ on.\n\\par\\ev{Lean}\\scale{uniform}\\coord{second-channel-phase}\n\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_secondChannelSeparation}{GreedyAchievementSet.lean:2868}\n\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_secondChannelSeparationRat\\_from\\_seven}{GreedyAchievementSet.lean:2901}\n\\end{thm}\n\n\\begin{thm}[Straddle-prefix closed-set criterion]\n\\label{thm:straddle-closed-set}\nIf for every depth $d$ there exists a finite word $u$ (all exponents $\\le d$) straddling\ntarget $t$ \u2014 meaning $\\mathrm{value}(u)\\le t\\le \\mathrm{value}(u)+\\mathrm{mersenneTail}(d)$\n\u2014 then $t\\in\\mathrm{mersenneAchievementSet}$. The proof needs only compactness of the\nachievement set, no coinductive machinery.\n\\par\\ev{Lean}\\scale{uniform}\\coord{dedekind-cut}\n\\lean{mem\\_mersenneAchievementSet\\_of\\_straddle\\_all\\_depths}{HalfCutLocator.lean:173}\n\\end{thm}\n\n\\begin{thm}[Largest-skip-late induction: single-socket reduction]\n\\label{thm:largest-skip-late}\nDefine \\emph{LargestSkipLateAt s} as ``row $s$'s largest false rank $d$ satisfies\n$2s<3d$'' and \\emph{LargestSkipLateStepSocket} as the universally quantified one-step\narithmetic implication $\\forall s\\ge 14,\\,d,\\; \\mathrm{IsLargestFalseRank}(\\mathrm{seamGreedyWord}\\, s)\\,d\n\\to 2s<3d \\to (2(s+1)<3d \\lor \\mathrm{SeamGreedyUpperOrMiddleAt}\\,s\\,\\_)$. Given this\nsocket, plus the base case at row $14$ (kernel-verified by \\texttt{decide}), the\nlate-largest-skip invariant propagates to every row $\\ge 14$ by induction, supplying a\ncofinal sequence of unboundedly skipped ranks, hence $1/2\\in\\mathrm{mersenneAchievementSet}$.\nThis is the single cleanest remaining reduction of the whole half-branch: closing it\nfactors entirely through one local arithmetic step.\n\\par\\ev{Lean}\\scale{uniform}\\coord{seam-integer}\n\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_largestSkipLateStepSocket}{HalfCylinderLargestSkipInduction.lean:178}\n\\lean{largestSkipLateAt\\_fourteen}{HalfCylinderLargestSkipInduction.lean:60}\n\\end{thm}\n\n\\begin{thm}[Middle-producer card/row escape sockets]\n\\label{thm:middle-producer-escape}\nIf, at every late middle-branch row, $\\mathrm{card}(\\mathrm{support})+\\mathrm{belowPulse}+5\n< 4\\cdot\\mathrm{remainder}$ (\\emph{SeamMiddleProducerCardEscape}), then\n$1/2\\in\\mathrm{mersenneAchievementSet}$. The strictly weaker-hypothesis, stronger-conclusion\nsocket \\emph{SeamMiddleProducerRowEscape} ($s\\le\\mathrm{remainder}$ at every late middle\nrow) implies the card escape and hence also closes the branch.\n\\par\\ev{Lean}\\scale{uniform}\\coord{seam-integer}\n\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_middleProducerCardEscape}{HalfCylinderMiddleCarryLowerBound.lean:1200}\n\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_middleProducerRowEscape}{HalfCylinderMiddleCarryLowerBound.lean:1510}\n\\end{thm}\n\n\\begin{thm}[Middle all-right tail forces strict defect]\n\\label{thm:middle-allright-defect}\nIf a middle transition at row $D\\ge 13$ (with $\\neg\\mathrm{carries}$) is followed by an\nall-right (eventually every rank selected) tail, then the producer carry is strictly\nbelow its complete future incidence tail: equivalently \\emph{seamGreedyFloorZ D}\n$<$ \\emph{seamTakeThreshold D}, equivalently a strict rational-skip defect. This\noverconstrains the ``last transition is middle, then forever right'' scenario, feeding\nthe final-middle-cell exclusion (Theorem~\\ref{thm:final-middle-cell}).\n\\par\\ev{Lean}\\scale{uniform}\\coord{seam-integer}\n\\lean{middleProducer\\_allRight\\_forces\\_carry\\_lt\\_tail}{HalfCylinderMiddleCarryLowerBound.lean:953}\n\\lean{middleProducer\\_allRight\\_forces\\_rational\\_skip}{HalfCylinderMiddleCarryLowerBound.lean:1027}\n\\end{thm}\n\n\\begin{thm}[Two-sided dyadic cell escape: global remainder control]\n\\label{thm:two-sided-dyadic}\n\\emph{SeamTwoSidedDyadicCellEscape} is a minimal local socket (it excludes exactly three\nexceptional middle cells at signed coordinate $-3,-2,-1$, plus one right-pulse-leak\nbound). Granted this socket, plus a kernel-verified base case at row $5$, induction gives\nthe universal two-sided bound $\\forall s\\ge 5,\\;\n\\mathrm{seamIntegerGreedyRemainder}\\,s\\le 2^s \\lor \\mathrm{overshoot}\\,s\\le 2^s$ \u2014\na global dyadic-scale control on the seam orbit independent of which branch fires at each\nrow.\n\\par\\ev{Lean}\\scale{uniform}\\coord{seam-integer}\n\\lean{SeamTwoSidedDyadicCellEscape.twoSided}{HalfCylinderMiddleCarryLowerBound.lean:3420}\n\\lean{seamTwoSidedDyadicAt\\_five}{HalfCylinderMiddleCarryLowerBound.lean:3360}\n\\end{thm}\n\n\\begin{thm}[Upper-reset dyadic-band escape, with rows $13$--$31$ certified]\n\\label{thm:upper-reset-band}\n\\emph{SeamUpperResetDyadicBandEscape} says: for every $d\\ge 13$ and every $j\\le d$, the\nreset charge avoids a linear-width band immediately below the dyadic power\n$2^{d-j+1}$ ($2^{d-j+1}<\\mathrm{resetCharge}$ or\n$\\mathrm{resetCharge}+2(d+j)\\le 2^{d-j+1}$). Granted this, $1/2\\in\n\\mathrm{mersenneAchievementSet}$. Rows $13$ through $31$ are unconditionally certified by\nkernel \\texttt{decide} computation of the exact remainder at each row (e.g.\\ row $14\\to\n392$, row $31\\to 4187487147$; none failing) \u2014 the current computational frontier of this\nsub-route.\n\\par\\ev{Lean}\\scale{uniform}\\coord{dyadic-boundary}\n\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_upperResetDyadicBandEscape}{HalfCylinderMiddleCarryLowerBound.lean:3755}\n\\lean{seamUpperResetDyadicBandEscape\\_through\\_thirty}{HalfCylinderUpperResetBandCertificates.lean:78}\n\\end{thm}\n\n\\begin{thm}[M\\\"obius-centred carry nonnegativity below $1/2$]\n\\label{thm:mobius-centred-nonneg}\nFor a support $A$ with $1\\notin A$: if $\\mathrm{erdosSupportSeries}\\,2\\,A<1/2$, then\n$0\\le\\mathrm{mobiusCenteredHalfCarry}\\,A\\,N$ for every $N$. Proof: the identity\n$\\mathrm{integerHalfCarry}\\,A\\,N = 2^{N+1}(1/2-\\mathrm{seriesValue}) +\n\\mathrm{binaryCoeffTail}(\\dots)$ writes the carry as a sum of two manifestly nonnegative\nterms. Stated over the generic divisor-incidence function \\emph{supportCoeff}, not over\nMersenne weights directly \u2014 since $\\varphi=\\mu * \\mathrm{id}$ is itself\nM\\\"obius-built, this is a strong candidate for adaptation to \\#249.\n\\par\\ev{Lean}\\scale{uniform}\\coord{mobius-mersenne}\n\\lean{mobiusCenteredHalfCarry\\_nonneg\\_of\\_supportSeries\\_lt\\_half}{HalfCylinderFinalMiddleCellEscape.lean:94}\n\\lean{integerHalfCarry\\_eq\\_scaled\\_residual\\_add\\_tail}{HalfCarryReachability.lean:842}\n\\end{thm}\n\n\\begin{thm}[Square-root carry-growth route: current open numeric-analytic frontier]\n\\label{thm:sqrt-bound-route}\nNonnegativity of the M\\\"obius-centred carry of the actual canonical greedy support for\n$1/2$ is unconditional (Theorem~\\ref{thm:mobius-centred-nonneg} plus two supporting\nlemmas). \\emph{If in addition} the pointwise upper bound\n$\\mathrm{mobiusCenteredHalfCarry}(\\mathrm{greedySupport}(1/2))\\,N\\le 2\\sqrt N+4$ holds for\nevery $N$, then the greedy skipped support is infinite and the greedy value equals\nexactly $1/2$. Only the upper bound remains open; a distinct socket from the dyadic-band\n(Theorem~\\ref{thm:upper-reset-band}) and largest-skip-late\n(Theorem~\\ref{thm:largest-skip-late}) routes. A separate project-memory note records a\n\\emph{different} sqrt-escape wall on the unconditional ($\\forall$ infinite $A$) \\#257\ntrack; whether that wall bears on this half-membership socket is unverified \u2014 treat any\nconnection as \\texttt{hypothesis}, not established.\n\\par\\ev{Lean}\\scale{uniform}\\coord{greedy-orbit}\n\\lean{greedy\\_half\\_infinite\\_of\\_mobiusCenteredHalfCarry\\_sqrtBound}{HalfCarryReachability.lean:805}\n\\lean{infinite\\_support\\_half\\_of\\_mobiusCenteredHalfCarry\\_sqrtBound}{HalfCarryReachability.lean:788}\n\\end{thm}\n\n\\begin{thm}[Governed frozen-margin producer family: three graded sockets]\n\\label{thm:frozen-margin}\nThree sufficient sockets, strongest to weakest hypothesis, all funnel into\n$1/2\\in\\mathrm{mersenneAchievementSet}$: (i) \\emph{HalfGreedySkippedFullShellNonnegative}\n(at every skipped rank, $0\\le\\mathrm{greedyHalfFrozenMargin}(n-1)\\,n$), direct and\nseam-free; (ii) \\emph{HalfGreedySkippedSeamAlignmentZero} (the seam remainder is exactly\n$0$ whenever the actual word equals the seam-greedy word), proved \\emph{equivalent} to\n(i); (iii) \\emph{HalfGreedySkippedSeamEscape} ($\\mathrm{halfStripBound}(2n)<\n\\mathrm{seamIntegerGreedyRemainder}\\,n$ at every skipped rank), strictly sufficient for\n(i) but not equivalent.\n\\par\\ev{Lean}\\scale{uniform}\\coord{seam-integer}\n\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_skippedFullShellNonnegative}{HalfCylinderFullShellSeamBridge.lean:640}\n\\lean{skippedSeamAlignmentZero\\_iff\\_skippedFullShellNonnegative}{HalfCylinderFullShellSeamBridge.lean:660}\n\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_skippedSeamEscape}{HalfCylinderFullShellSeamBridge.lean:724}\n\\end{thm}\n\n\\begin{thm}[Erd\\H{o}s--Borwein full-support irrationality, unconditional]\n\\label{thm:full-support-catalogue}\nFor every base $b\\ge 2$: $\\sum'_{n\\ge 1} 1/(b^n-1)$ is irrational, unconditionally. No\nopen obligations. Base $2$ is the classical Erd\\H{o}s--Borwein constant. This is the\nterminal target every other rigidity/support theorem in this catalogue is measured\nagainst.\n\\par\\ev{Lean}\\scale{uniform}\\coord{mobius-mersenne}\n\\lean{irrational\\_erdosSum\\_full\\_support}{CertificateKernel.lean:8000}\n\\lean{irrational\\_erdosBorwein\\_series}{CertificateKernel.lean:8007}\n\\end{thm}\n\n\\begin{thm}[Pairwise-coprime support irrationality, Erd\\H{o}s 1968]\n\\label{thm:pairwise-coprime}\nFor every base $b\\ge 2$ and every infinite pairwise-coprime support $A\\subseteq\\mathbb N$\nwith summable reciprocals, $\\sum_{a\\in A} 1/(b^a-1)$ is irrational.\n\\par\\ev{Lean}\\scale{uniform}\\coord{mobius-mersenne}\n\\lean{irrational\\_erdosSupportSeries\\_pairwise\\_coprime}{CertificateKernel.lean:10448}\n\\end{thm}\n\n\\begin{thm}[Generic weighted-coefficient block-certificate engine]\n\\label{thm:weighted-coeff-engine}\nFor base $b\\ge 2$ and coefficient $c:\\mathbb N\\to\\mathbb N$ with $c(m)\\le m$: if a\nweighted block certificate exists at every precision $q$ (an explicit\n$\\exists N,K,L,C$ package: block-divisibility, bounded middle window, far-tail\npositivity, height inequality $q(C+N+L+2)<b^L$), then $\\sum'_m c(m{+}1)/b^{m+1}$ is\nirrational. This is the single engine both the full-support theorem\n(Theorem~\\ref{thm:full-support-catalogue}, via $c=\\tau$, the divisor-count function) and the\n\\#249 sockets ($c=\\varphi$) instantiate; certificate supply is proved for $\\tau$\nand left open for $\\varphi$ \u2014 exactly why \\#249 stays open where \\#257 full-support\ncloses.\n\\par\\ev{Lean}\\scale{uniform}\\coord{binary-digit}\n\\lean{irrational\\_coeff\\_series\\_of\\_weighted\\_coeff\\_block\\_certificates}{CertificateKernel.lean:8703}\n\\end{thm}\n\n\\begin{thm}[LCM-gap irrationality engine]\n\\label{thm:lcm-gap-engine}\nFor base $b\\ge 2$ and strictly monotone support $a:\\mathbb N\\to\\mathbb N$ with $a(0)\\ge\n1$: if $a(k)-\\mathrm{lcm}(a(0),\\dots,a(k-1))\\to\\infty$, then $\\sum'_k 1/(b^{a(k)}-1)$ is\nirrational. Pure statement about any strictly increasing exponent sequence and any base;\nno Mersenne-specific content.\n\\par\\ev{Lean}\\scale{uniform}\\coord{lcm-gap}\n\\lean{irrational\\_erdosSum\\_of\\_lcm\\_gap}{CertificateKernel.lean:5555}\n\\end{thm}\n\n\\begin{thm}[Factorial-support and $2^k$-support instances]\n\\label{thm:factorial-twopow-support}\nFor every $b\\ge 2$: $\\sum'_k 1/(b^{(k+1)!}-1)$ is irrational, and $\\sum'_k\n1/(b^{2^k}-1)$ is irrational. Explicitly flagged in-source as \\emph{instances} of the\n\\#257 statement family, not the universal theorem \u2014 the universal problem (every\ninfinite $A$) remains open.\n\\par\\ev{Lean}\\scale{uniform}\\coord{lcm-gap}\n\\lean{irrational\\_erdosSum\\_factorial\\_support}{CertificateKernel.lean:5707}\n\\lean{irrational\\_erdosSum\\_two\\_pow\\_support}{CertificateKernel.lean:5731}\n\\end{thm}\n\n\\begin{thm}[Multiples-support irrationality via dilation]\n\\label{thm:multiples-support}\nFor every $b\\ge 2$, $d\\ge 1$: $\\sum_{a\\in d\\mathbb N} 1/(b^a-1)$ is irrational, via the\nexact identity $\\mathrm{erdosSupportSeries}\\,b\\,\\{n:d\\mid n\\} = \\sum'_k\n1/((b^d)^{k+1}-1)$ \u2014 dilation to base $b^d$ full support, reducing directly to\nTheorem~\\ref{thm:full-support-catalogue}.\n\\par\\ev{Lean}\\scale{uniform}\\coord{dilation}\n\\lean{irrational\\_erdosSupportSeries\\_multiples}{CertificateKernel.lean:8726}\n\\lean{erdosSupportSeries\\_multiples\\_eq\\_pow\\_base\\_full\\_support}{CertificateKernel.lean:8785}\n\\end{thm}\n\n\\begin{thm}[Periodic-support irrationality: the periodic divisor-orbit sieve]\n\\label{thm:periodic-support}\nFor every $b\\ge 2$ and every $m$-periodic support $A$ ($\\forall n,\\, n{+}m\\in A\n\\leftrightarrow n\\in A$) containing a positive element: $\\mathrm{erdosSupportSeries}\\,b\\,A$\nis irrational. At $m=1$ recovers full support; residue classes, unions of residue\nclasses, and gcd-pattern supports are all instances. Mechanism: a prime $p\\nmid m$ makes\nthe divisor ray $d,pd,p^2d,\\dots$ sweep residues mod $m$ periodically; a prime with exact\nvaluation $b\\varphi(m)-1$ splits every divisor ray into complete residue cycles repeated\na multiple of $b$ times, reusing the same bounded Bertrand/CRT frame as full support with\nexponent $b-1\\to b\\varphi(m)-1$.\n\\par\\ev{Lean}\\scale{uniform}\\coord{periodic-sieve}\n\\lean{irrational\\_erdosSupportSeries\\_periodic}{CertificateKernel.lean:11262}\n\\end{thm}\n\n\\begin{thm}[Eventually-periodic support irrationality]\n\\label{thm:eventually-periodic}\nFor every $b\\ge 2$: an infinite support whose membership is $m$-periodic from some\nthreshold $N_0$ on has $\\mathrm{erdosSupportSeries}\\,b\\,A$ irrational.\n\\par\\ev{Lean}\\scale{uniform}\\coord{periodic-sieve}\n\\lean{irrational\\_erdosSupportSeries\\_eventuallyPeriodic}{CertificateKernel.lean:11276}\n\\end{thm}\n\n\\begin{thm}[Residue-class and odd-support irrationality]\n\\label{thm:residue-odd}\nFor every $b\\ge 2$, modulus $m\\ge 1$, residue $c$: $\\sum_{n\\equiv c\\,(m)} 1/(b^n-1)$ is\nirrational. Specializing $m=2,c=1$: $\\sum_{n\\text{ odd}} 1/(b^n-1)$ is irrational for\nevery $b\\ge 2$ \u2014 the first support class of density strictly between $0$ and $1$ to\nland.\n\\par\\ev{Lean}\\scale{uniform}\\coord{periodic-sieve}\n\\lean{irrational\\_erdosSupportSeries\\_residueClass}{CertificateKernel.lean:11344}\n\\lean{irrational\\_erdosSupportSeries\\_odd}{CertificateKernel.lean:11358}\n\\end{thm}\n\n\\begin{thm}[Signed periodic-weight dichotomy and its unconditional one-sided closures]\n\\label{thm:signed-periodic}\nFor base $b\\ge 2$ and any $m$-periodic integer-valued signed weight $w$: the weighted\nseries $\\sum w(a)/(b^a-1)$ is either irrational or base-$b$ terminating ($b^k\\cdot x=z$\nfor some integer $k,z$), unconditionally, mixed signs allowed. If additionally the\ndivisor-coefficient \\emph{intWeightedCoeff w} is one-sided (all nonnegative, or all\nnonpositive) and not eventually zero, the terminating branch is excluded and the series\nis unconditionally irrational.\n\\par\\ev{Lean}\\scale{uniform}\\coord{signed-divisor-calculus}\n\\lean{int\\_coeff\\_series\\_irrational\\_or\\_bpow\\_mul\\_eq\\_intCast\\_intWeightedErdosSeries\\_periodic}{CertificateKernel.lean:13847}\n\\lean{irrational\\_intWeightedErdosSeries\\_periodic\\_of\\_coeff\\_nonneg\\_of\\_frequently\\_ne\\_zero}{CertificateKernel.lean:14255}\n\\lean{irrational\\_intWeightedErdosSeries\\_periodic\\_of\\_coeff\\_nonpos\\_of\\_frequently\\_ne\\_zero}{CertificateKernel.lean:14315}\n\\end{thm}\n\n\\begin{thm}[Mersenne-channel denominator survival, T4]\n\\label{thm:mersenne-channel-survival}\nFor a finite family $P$ of ``upper-half'' primes (each $p$ with $t<2p$ for scale $t$,\neach dividing a squarefree radical $r$ all of whose prime factors are $\\le t$): the\npairwise-coprime Mersenne product $C=\\prod_{p\\in P}(2^p-1)$ survives fraction reduction \u2014\n$C/\\gcd(C,h)\\mid (h\\cdot\\mathrm{mobiusNumerator}(r)/\\mathrm{mersenne}(r))\\mathrm{.den}$\n\u2014 exactly when $C$ is coprime to scale $h$ (or, weaker, when every prime factor of $h$ is\n$\\le t$). This is the load-bearing quantitative step behind\nTheorem~\\ref{thm:full-support-catalogue}.\n\\par\\ev{Lean}\\scale{uniform}\\coord{cyclotomic}\n\\lean{upperHalfChannel\\_survivorProduct\\_dvd\\_den}{MersenneShadowCyclotomicNoncollapse.lean:766}\n\\lean{upperHalfChannel\\_product\\_dvd\\_den\\_of\\_coprime\\_scale}{MersenneShadowCyclotomicNoncollapse.lean:795}\n\\end{thm}\n\n\\begin{thm}[Mersenne-channel exponential denominator growth]\n\\label{thm:mersenne-channel-growth}\nVia Bertrand's postulate, at every scale $t\\ge 5$ there is an upper-half prime $p$ with\n$2^{t/2}\\le 2^{p-1}\\le\\mathrm{mersenne}(p)$, giving the explicit exponential lower bound\n$2^{t/2}\\le(\\text{reduced denominator of the scaled M\\\"obius shadow at LCM-height }t)$.\nThe exact reduced denominator at every LCM height is also computed in closed form.\n\\par\\ev{Lean}\\scale{uniform}\\coord{cyclotomic}\n\\lean{upperHalfMersenneProduct\\_lower\\_bound}{MersenneShadowDenominatorGrowth.lean:60}\n\\lean{lcmHeight\\_scaledMobiusShadow\\_den\\_exact}{MersenneShadowDenominatorGrowth.lean:147}\n\\end{thm}\n\n\\subsection{Converters and exact identities}\n\n\\begin{obs}[Half-greedy regression fixtures: exact rational checkpoints]\n\\label{obs:half-regression}\nThree closed numeric facts, \\texttt{norm\\_num}/\\texttt{decide}-checked: $1/2 -\nW(\\{2,3,6,7\\}) = 1/16002$ exactly (\\emph{finiteErdosSum}); $\\mathrm{mersenneCorrectionTail}(1)\n\\le 5/42$; $3/8 < 1/2 - \\mathrm{mersenneCorrectionTail}(1)$. Anchors for validating any\nnew automated fatal-gap search at small depth.\n\\par\\ev{Cert}\\scale{fixed}\\coord{regression-fixture}\n\\lean{half\\_sub\\_four\\_term\\_prefix\\_eq}{HalfCutLocator.lean:664}\n\\lean{mersenneCorrectionTail\\_one\\_le}{HalfCutLocator.lean:670}\n\\end{obs}\n\n\\begin{obs}[Period-4 sign weight vanishes on a residue class]\n\\label{obs:period-four-obstruction}\nThe explicit period-$4$ sign pattern $1,0,-1,0$ (\\emph{periodFourSignWeight}) has\n$\\mathrm{intWeightedCoeff}\\,w\\,n = 0$ identically for every $n\\equiv 3\\pmod 4$, via the\ndivisor-involution $d\\mapsto n/d$. This is exactly why the general signed-periodic case\n(Theorem~\\ref{thm:signed-periodic}) only reaches a dichotomy rather than an unconditional\ntheorem: no residue-blind selection can force a nonzero protected residue for every\nperiodic signed weight.\n\\par\\ev{Lean}\\scale{fixed}\\coord{signed-divisor-calculus}\n\\lean{intWeightedCoeff\\_periodFourSignWeight\\_eq\\_zero\\_of\\_mod\\_four\\_eq\\_three}{CertificateKernel.lean:13898}\n\\end{obs}\n\n\\begin{thm}[M\\\"obius--Mersenne Lambert identity]\n\\label{thm:mobius-lambert-identity}\n$\\sum'_{d:\\mathbb N^+} \\mu(d)/(2^d-1) = 1/2$ exactly (imported as\n\\emph{MersenneLambertLadder.tsum\\_moebius\\_div\\_two\\_pow\\_sub\\_one\\_eq\\_half}). A signed\n(not Boolean) series over Mersenne denominators hitting the target value exactly \u2014 a\nlandmark calibration point for rigidity arguments bounding how close a Boolean infinite\nsupport can get to $1/2$.\n\\par\\ev{Cited}\\scale{fixed}\\coord{mobius-mersenne}\n\\lean{tsum\\_moebius\\_div\\_two\\_pow\\_sub\\_one\\_eq\\_half}{MobiusSignSupportNoGo.lean:111}\n\\end{thm}\n\n\\begin{cor}[Negative-M\\\"obius Boolean support overshoots $1/2$]\n\\label{cor:negative-mobius-overshoot}\nConsequently the negative-M\\\"obius Boolean support $N=\\{d\\ge 2:\\mu(d)=-1\\}$\n\\emph{overshoots} $1/2$: $\\sum_{d\\in N} 1/(2^d-1) = 1/2 + (\\text{positive-M\\\"obius tail})\n\\ge 1/2 + 1/63$ (using the $d=6$ term, $\\mu(6)=1$). A route-sufficiency no-go only: it\nrules out this one natural candidate construction, not the general question of whether\nsome infinite Boolean support sums to exactly $1/2$.\n\\par\\ev{Lean}\\scale{fixed}\\coord{mobius-mersenne}\n\\lean{half\\_lt\\_tsum\\_negativeMobius}{MobiusSignSupportNoGo.lean:150}\n\\lean{tsum\\_negativeMobius\\_eq\\_half\\_add\\_positiveMobiusTail}{MobiusSignSupportNoGo.lean:164}\n\\end{cor}\n\n\\begin{obs}[Skip-block invariant failure: two explicit fixtures]\n\\label{obs:skip-block-non-invariant}\nTwo explicit witness states show ``safe bracket $+$ numerator monotone'' is \\emph{not}\nan inductive invariant across skip-blocks in the dyadic-prefix reformulation: the\ntransition $7/17\\to 15/119$, and a full-LCM-saturated state\n$12149/1240155$. A negative calibration fact for anyone attempting to induct directly on\nthat pair of properties.\n\\par\\ev{Cert}\\scale{fixed}\\coord{seam-integer}\n\\lean{unnamed fixture pair}{DyadicPrefixCompression.lean:622}\n\\end{obs}\n\n\\begin{thm}[The half-skip dichotomy via Erd\\H{o}s--Borwein irrationality]\n\\label{thm:half-skip-dichotomy}\n$1/2\\in\\mathrm{mersenneAchievementSet} \\iff (\\mathrm{greedyMersenneSkippedSupport}(1/2))\n\\text{ is infinite}$. Forward direction uses irrationality of the Erd\\H{o}s--Borwein\nconstant (\\emph{erdosBorweinMersenneConstant} $=\\mathrm{mersenneTail}\\,0$, unconditional):\nif only finitely many exponents were skipped, the constant would equal $1/2$ plus a\nfinite rational, contradicting irrationality. The mechanism (rational/finite-representable\ntarget, irrational full-series baseline $\\Rightarrow$ complement support infinite) is\nfully general and would restate for \\#249's own baseline constant if one exists.\n\\par\\ev{Lean}\\scale{cofinal}\\coord{greedy-orbit}\n\\lean{half\\_mem\\_mersenneAchievementSet\\_iff\\_greedySkippedSupport\\_infinite}{GreedyAchievementSet.lean:2514}\n\\lean{irrational\\_erdosBorweinMersenneConstant}{GreedyAchievementSet.lean:2433}\n\\end{thm}\n\n\\begin{thm}[The nine-way half-membership classification hub]\n\\label{thm:nine-way-hub}\n$1/2\\in\\mathrm{mersenneAchievementSet}$ is equivalent to \\emph{all} of: (1)\n$\\neg$\\emph{SeamGreedyEventuallyRight}; (2) \\emph{SeamGreedyUnboundedTerminalFalse} (false\nsuccessor terminal bits beyond every bound); (3) \\emph{SeamGreedyUnboundedUpperOrMiddle}\n(U/M events beyond every bound); (4) \\emph{SeamGreedyCofinalTerminalFalse} (one cofinal\nsequence of false terminals); (5) $\\exists\\,\\mathrm{rows},\\,\n\\mathrm{SeamGreedyUnboundedSkippedRanksAlong}\\,\\mathrm{rows}$; (6) the greedy skipped\nsupport is infinite (Theorem~\\ref{thm:half-skip-dichotomy}); (7) no last half-greedy skip\nexists (Theorem~\\ref{thm:last-skip-iff-fatal}). The central hub of the whole half-branch:\nevery sufficient-condition theorem elsewhere in this catalogue is secretly proving one of\nthese seven equivalent forms; a disproof of any one (a bounded/eventually-right\ncertificate) closes the branch.\n\\par\\ev{Lean}\\scale{cofinal}\\coord{seam-integer}\n\\lean{half\\_mem\\_mersenneAchievementSet\\_iff\\_not\\_seamGreedyEventuallyRight}{HalfCylinderHalfMembershipClassification.lean:112}\n\\lean{half\\_mem\\_mersenneAchievementSet\\_iff\\_exists\\_unboundedSkippedRanksAlong}{HalfCylinderHalfMembershipClassification.lean:210}\n\\lean{half\\_mem\\_mersenneAchievementSet\\_iff\\_no\\_lastHalfGreedySkip}{HalfCylinderHalfMembershipClassification.lean:249}\n\\end{thm}\n\n\\begin{lem}[Eventually-right seam tail is analytically impossible for $1/2$]\n\\label{lem:eventually-right-impossible}\nIf the seam eventually always extends ``true'' (right branch) from some row $S$ on with a\nfixed lower prefix $u$, the resulting cofinite-support value stays strictly below $1/2$\n(\\emph{prefix\\_add\\_mersenneTail\\_lt\\_half\\_of\\_eventually\\_right}); the matching\nalternative ``upper competitor'' word gives a strict excess \\emph{above} $1/2$\n(\\emph{half\\_lt\\_upper\\_competitor\\_of\\_eventually\\_right}). An eventually-right tail can\ntherefore never land exactly on $1/2$ \u2014 it always undershoots or overshoots. The analytic\nengine behind the final-middle-cell exclusion (Theorem~\\ref{thm:final-middle-cell}).\n\\par\\ev{Lean}\\scale{cofinal}\\coord{seam-integer}\n\\lean{prefix\\_add\\_mersenneTail\\_lt\\_half\\_of\\_eventually\\_right}{HalfCylinderFatalGapRightTail.lean:402}\n\\lean{half\\_lt\\_upper\\_competitor\\_of\\_eventually\\_right}{HalfCylinderFatalGapRightTail.lean:627}\n\\end{lem}\n\n\\begin{lem}[Mersenne weight/tail: strict superincreasingness]\n\\label{lem:mersenne-tail-weight}\n$\\mathrm{mersenneWeight}\\,n = 1/(2^n-1)$; $\\mathrm{mersenneTail}\\,n = \\sum'_k\n\\mathrm{mersenneWeight}(n{+}k{+}1)$; $\\mathrm{mersenneTail}\\,n =\n\\mathrm{mersenneWeight}(n{+}1)+\\mathrm{mersenneTail}(n{+}1)$; and for $n>0$,\n$\\mathrm{mersenneTail}\\,n<\\mathrm{mersenneWeight}\\,n$ (strict superincreasingness), with\nthe two-sided envelope $\\mathrm{mersenneTail}\\,n\\le\n2\\cdot\\mathrm{mersenneWeight}(n{+}1)$. The whole superincreasing-series machinery\ndownstream is built on the strict inequality. The proof pattern (remaining tail after\nrank $n$ strictly smaller than weight $n$) is a fully generic superincreasing-series fact,\ndirectly relevant to \\#249's own $\\varphi(n)/2^n$ series.\n\\par\\ev{Lean}\\scale{uniform}\\coord{mobius-mersenne}\n\\lean{mersenneTail\\_lt\\_weight}{GreedyAchievementSet.lean:180}\n\\lean{mersenneTail\\_le\\_two\\_mul\\_weight}{GreedyAchievementSet.lean:190}\n\\end{lem}\n\n\\begin{thm}[Greedy survival criterion --- the master theorem]\n\\label{thm:greedy-survival-catalogue}\n$x\\in\\mathrm{mersenneAchievementSet} \\iff 0\\le x \\land \\forall n,\\,\n\\mathrm{greedyMersenneRemainder}\\,x\\,n \\le \\mathrm{mersenneTail}\\,n$, where\n\\emph{greedyMersenneRemainder} is the canonical take-if-possible greedy recursion.\nMembership is exactly ``every greedy residual survives (stays $\\le$ the full remaining\ntail)''. The proof pattern (fatal-state absorption plus convergence-from-survival) is a\ncompletely generic superincreasing-series fact; only \\emph{mersenneWeight}/\n\\emph{mersenneTail} are Mersenne-specific inputs. The canonical entry point for every\nhalf-membership sufficient-condition theorem in this catalogue.\n\\par\\ev{Lean}\\scale{uniform}\\coord{greedy-orbit}\n\\lean{mem\\_mersenneAchievementSet\\_iff\\_greedy\\_survival}{GreedyAchievementSet.lean:1445}\n\\end{thm}\n\n\\begin{lem}[Rank-step trichotomy of the Dedekind cut]\n\\label{lem:rank-step-trichotomy}\nFrom a depth-$d$ straddle of target $t$, exactly one of: (a) still straddles at $d{+}1$\nwithout taking $d{+}1$; (b) straddles at $d{+}1$ after taking $d{+}1$; (c) $t$ sits in the\nfatal gap $(\\mathrm{value}(u)+\\mathrm{mersenneTail}(d{+}1),\\,\n\\mathrm{value}(u)+\\mathrm{mersenneWeight}(d{+}1))$. Specialized to $t=1/2$, branches (a)\nand (b) are proved mutually exclusive (\\emph{half\\_step\\_forced}), since both boundary\nequalities are impossible for $1/2$\n(Lemma~\\ref{lem:half-endpoint-kills}). A pure consequence of superincreasingness plus\n$\\mathrm{mersenneTail}\\,n = \\mathrm{mersenneWeight}(n{+}1)+\\mathrm{mersenneTail}(n{+}1)$.\n\\par\\ev{Lean}\\scale{uniform}\\coord{dedekind-cut}\n\\lean{isStraddlePrefix\\_step\\_trichotomy}{HalfCutLocator.lean:205}\n\\lean{IsStraddlePrefix.half\\_step\\_forced}{HalfCutLocator.lean:301}\n\\end{lem}\n\n\\begin{lem}[Fatal-gap exclusion: kills every support, not just greedy]\n\\label{lem:fatal-gap-exclusion}\nIf $t$ lies strictly inside the rank-$(d{+}1)$ gap\n$(\\mathrm{mersenneSupportPrefix}\\,A\\,d + \\mathrm{mersenneTail}(d{+}1),\\,\n\\mathrm{mersenneSupportPrefix}\\,A\\,d + \\mathrm{mersenneWeight}(d{+}1))$ for a fixed prefix\nmass, then $\\mathrm{positiveMersenneSupportValue}\\,A\\ne t$ for \\emph{every} support $A$\nachieving that prefix, not merely the greedy orbit. Pure consequence of\nsuperincreasingness, no Mersenne-specific arithmetic beyond the weight function. The\n``kill certificate'' primitive: any concrete numeric interval computation landing inside a\nfatal gap immediately falsifies half-membership via this lemma.\n\\par\\ev{Lean}\\scale{uniform}\\coord{dedekind-cut}\n\\lean{positiveMersenneSupportValue\\_ne\\_of\\_rank\\_gap}{HalfCutLocator.lean:380}\n\\end{lem}\n\n\\begin{lem}[Half endpoint kills: no finite word hits $1/2$ exactly]\n\\label{lem:half-endpoint-kills}\nNo finite positive-support word has value exactly $1/2$\n(\\emph{positiveMersenneSupportValue\\_coe\\_finset\\_ne\\_half}, via the odd reduced\ndenominator of the finite Erd\\H{o}s sum). No finite word plus a complete tail equals\n$1/2$ either (\\emph{half\\_ne\\_coe\\_finset\\_add\\_mersenneTail}, via irrationality of the\nErd\\H{o}s--Borwein constant, same mechanism as Theorem~\\ref{thm:half-skip-dichotomy}).\nDischarges the two boundary branches of the rank-step trichotomy\n(Lemma~\\ref{lem:rank-step-trichotomy}), forcing every macrostep to be a strict trichotomy\nwith no equality case.\n\\par\\ev{Lean}\\scale{uniform}\\coord{parity-irrationality}\n\\lean{positiveMersenneSupportValue\\_coe\\_finset\\_ne\\_half}{HalfCutLocator.lean:243}\n\\lean{half\\_ne\\_coe\\_finset\\_add\\_mersenneTail}{HalfCutLocator.lean:263}\n\\end{lem}\n\n\\begin{lem}[Straddle words are canonical: they agree with the greedy prefix]\n\\label{lem:straddle-agrees-greedy}\nEvery half-straddling word agrees with the actual canonical greedy word through the same\ndepth: $\\forall n\\le d,\\, n>0,\\; (n\\in u \\leftrightarrow n\\in\n\\mathrm{greedyMersenneSupport}(1/2))$, by induction on $d$ using the backward step\n\\emph{erase\\_top}. Straddle words are not an independent search tree \u2014 they are forced to\n\\emph{be} the greedy prefix. This is what lets the word-level fatal-gap exclusion\n(Lemma~\\ref{lem:fatal-gap-exclusion}, valid for any support) be applied specifically to\nrule out the actual greedy orbit's half-representation.\n\\par\\ev{Lean}\\scale{uniform}\\coord{greedy-orbit}\n\\lean{IsStraddlePrefix.half\\_agrees\\_greedy}{HalfCutLocator.lean:442}\n\\lean{IsStraddlePrefix.erase\\_top}{HalfCutLocator.lean:334}\n\\end{lem}\n\n\\begin{thm}[Last-skip iff local fatality: half-membership as an infinite pointwise conjunction]\n\\label{thm:last-skip-iff-fatal}\n$\\mathrm{IsLastHalfGreedySkip}\\,M \\iff M\\in\\mathrm{greedyMersenneSkippedSupport}(1/2)\n\\land \\mathrm{GreedyMersenneFatalAt}(1/2)\\,M$. Equivalently: no last skip exists iff every\nactual skip survives, giving\n$1/2\\in\\mathrm{mersenneAchievementSet} \\iff \\forall M\\in\n\\mathrm{greedyMersenneSkippedSupport}(1/2),\\, \\mathrm{greedyMersenneRemainder}(1/2)\\,M\\le\n\\mathrm{mersenneTail}\\,M$. Turns half-membership into an infinite conjunction over\n\\emph{actually skipped} ranks only \u2014 the reduction from a global fatal-state statement to\na rank-local pointwise inequality is a generic greedy-recursion fact.\n\\par\\ev{Lean}\\scale{uniform}\\coord{greedy-orbit}\n\\lean{half\\_mem\\_iff\\_every\\_actual\\_skip\\_survives}{HalfCylinderFixedTailSocket.lean:79}\n\\lean{isLastHalfGreedySkip\\_iff\\_skip\\_and\\_fatal}{HalfCylinderFixedTailSocket.lean:22}\n\\end{thm}\n\n\\begin{lem}[Seam upper-or-middle classification]\n\\label{lem:seam-upper-or-middle}\nFor row $s\\ge 5$: the empirical ``false successor terminal bit'' event is exactly the\narithmetic disjunction $\\mathrm{SeamGreedyUpperOrMiddleAt}\\,s\\,hs :=\n(\\mathrm{seamAdjacentCut}\\,s\\,hs).\\mathrm{successorCarries} \\lor (\\neg\\mathrm{carries}\n\\land 4\\cdot\\mathrm{remainder}+\\mathrm{gap}-\\mathrm{belowPulse} <\n\\mathrm{terminalWeight})$. The empirical U/M/R label at each seam row is removable in\nfavor of pure arithmetic on \\emph{seamAdjacentCut}; the seam-arithmetic analogue of the\nrank-step trichotomy (Lemma~\\ref{lem:rank-step-trichotomy}).\n\\par\\ev{Lean}\\scale{uniform}\\coord{seam-integer}\n\\lean{seamGreedy\\_terminal\\_false\\_iff\\_upperOrMiddle}{HalfCylinderHalfMembershipClassification.lean:90}\n\\end{lem}\n\n\\begin{lem}[Largest-false-rank algebra: exact weight identity and branch propagation]\n\\label{lem:largest-false-rank-algebra}\nExact identity linking the largest-false-rank's lower vs.\\ upper word weights: $3\\cdot\n\\mathrm{lowerWeight} + \\mathrm{exactLateGap} = \\mathrm{upperWeight}$. Successor\npropagation: the right branch \\emph{preserves} the same largest-false rank $d$\n(\\emph{IsLargestFalseRank.seamGreedyWord\\_succ\\_of\\_rightBranch}); the upper/middle\nbranches \\emph{reset} it to the new terminal rank $s$\n(\\emph{seamGreedyWord\\_succ\\_isLargestFalseRank\\_terminal\\_of\\_upperOrMiddle}). The exact\nbookkeeping consumed inside the largest-skip-late induction\n(Theorem~\\ref{thm:largest-skip-late}).\n\\par\\ev{Lean}\\scale{uniform}\\coord{seam-integer}\n\\lean{three\\_mul\\_largestSkipLowerWeight\\_add\\_exactLateGap\\_eq\\_upperWeight}{HalfCylinderLargestSkipGap.lean:120}\n\\lean{IsLargestFalseRank.seamGreedyWord\\_succ\\_of\\_rightBranch}{HalfCylinderLargestSkipGap.lean:339}\n\\end{lem}\n\n\\begin{thm}[Critical dyadic-band index: a problem-agnostic quantifier collapse]\n\\label{thm:critical-dyadic-band}\n\\emph{CriticalDyadicBandIndex}$\\,d\\,E\\,j$ picks out the unique nearest dyadic power\n$2^{d-j+1}\\ge E$. Fully abstract, pure $\\mathbb N$ arithmetic, zero seam content:\n$\\mathrm{DyadicBandEscape}\\,d\\,E \\iff \\exists j,\\,\\mathrm{CriticalDyadicBandIndex}\\,d\\,E\\,j\n\\land E+2(d+j)\\le 2^{d-j+1}$ \u2014 the $\\forall j$ band-avoidance condition collapses to\nchecking exactly \\emph{one} critical index, reducing a check of $d{+}1$ separate\ninequalities to one nearest-boundary check per row. Specialized to the seam reset charge\n(with $E\\le 2^{d+1}$ automatically supplied by Theorem~\\ref{thm:two-sided-dyadic}), the\nreduced socket \\emph{SeamUpperResetCriticalBandEscape} is proved logically equivalent to\nTheorem~\\ref{thm:upper-reset-band}'s band-avoidance hypothesis.\n\\par\\ev{Lean}\\scale{uniform}\\coord{dyadic-boundary}\n\\lean{dyadicBandEscape\\_iff\\_exists\\_critical}{HalfUpperResetCriticalBand.lean:153}\n\\lean{seamUpperResetCriticalBandEscape\\_iff}{HalfUpperResetCriticalBand.lean:901}\n\\end{thm}\n\n\\begin{thm}[Final-middle-cell $-3$ excluded]\n\\label{thm:final-middle-cell}\nThe first of three exceptional final-middle integer cells (coordinate\n$4\\cdot\\mathrm{remainder}-\\mathrm{belowPulse}-4=-3$) cannot be the actual last transition\nbefore an all-right tail. Combines the analytic strict inequality of\nLemma~\\ref{lem:eventually-right-impossible} with a two-step M\\\"obius-carry recurrence\nforcing the centred carry to zero then strictly negative within two more coefficient\nrows, contradicting the nonnegativity of Theorem~\\ref{thm:mobius-centred-nonneg}. One of\nthree exceptional cells ($-3,-2,-1$); the companion file records the analogous\n\\emph{binaryCoeffTail} inequalities for cells $-2$ and $-1$ \u2014 closing all three fully\nresolves the final-middle producer.\n\\par\\ev{Lean}\\scale{uniform}\\coord{seam-integer}\n\\lean{finalMiddleCell\\_neg\\_three\\_not\\_last}{HalfCylinderFinalMiddleCellEscape.lean:587}\n\\lean{mobiusCenteredHalfCarry\\_add\\_two}{HalfCylinderFinalMiddleCellEscape.lean:39}\n\\end{thm}\n\n\\begin{lem}[Skipped-endpoint trichotomy: frozen-margin sign classification]\n\\label{lem:skipped-endpoint-trichotomy}\nAt every genuinely skipped rank $s\\ge 5$, the frozen margin\n$\\mathrm{greedyHalfFrozenMargin}(s{-}1)\\,s$ is in exactly one of three signed cells:\nnegative (actual word $=$ seam-greedy word, remainder $\\ge 1$, margin $=\n-\\mathrm{remainder}$); zero (actual word $=$ seam-greedy word, remainder $=0$); positive\n(actual word $=$ the seam ``above'' word, margin $=\\mathrm{overshoot}$). The zero cell is\nprovably safe; only the negative cell needs excluding, and it is exactly the case-split\nthat Theorem~\\ref{thm:final-middle-cell}'s three exceptional cells live inside.\n\\par\\ev{Lean}\\scale{uniform}\\coord{seam-integer}\n\\lean{halfGreedy\\_skipped\\_endpoint\\_trichotomy}{HalfCylinderSkippedEndpointClassifier.lean:246}\n\\end{lem}\n\n\\begin{lem}[Reverse carry-word overlap spacing: zero Mersenne content]\n\\label{lem:reverse-carry-word}\nA \\emph{ReverseCarryWord} is $(\\mathrm{coeff},\\mathrm{bit},\\mathrm{carry}:\\mathbb\nN\\to\\mathbb Z)$ with $\\mathrm{bit}(m)+2\\cdot\\mathrm{carry}(m) =\n\\mathrm{coeff}(m)+\\mathrm{carry}(m{+}1)$. If two such words share a $1/0$ seam and then\nagree on both coefficients and bits for \\emph{length} further positions, their carry\ndifference at the far end is exactly $2^{\\mathrm{length}}$ times an odd integer.\nCombined with a common Archimedean bound on both terminal carries, this forces\n$2^{\\mathrm{length}}\\le\\mathrm{bound}$. Fully abstract: no Mersenne, base-$4$, or seam\nstructure at all \u2014 pure integer-sequence recurrence, directly portable to \\#249's own\ndigit-carry structure if given an analogous reverse-carry presentation.\n\\par\\ev{Lean}\\scale{uniform}\\coord{binary-digit}\n\\lean{overlappingReverseCarryWords\\_carryDifference\\_eq\\_twoPow\\_mul\\_odd}{HalfTrappingReturnCarry.lean:180}\n\\lean{overlappingReverseCarryWords\\_twoPow\\_le\\_realBound}{HalfTrappingReturnCarry.lean:230}\n\\end{lem}\n\n\\begin{lem}[Relation-invariant linear channels force a rank-one determinant collapse]\n\\label{lem:linear-channel-nogo}\nIf a finite family of linear channels $\\mathrm{channel}:\\iota\\to V\\to_{\\mathbb Q}\n\\mathbb Q$ all vanish on $\\ker(\\mathrm{ev})$ and $\\exists e,\\,\\mathrm{ev}\\,e=1$, then the\nevaluation matrix $(i,j)\\mapsto\\mathrm{channel}\\,j\\,(\\mathrm{row}\\,i)$ has rank $\\le 1$ for\n\\emph{every} finite index type $\\iota$ with $[\\mathrm{Nontrivial}\\,\\iota]$ \u2014 hence every\nsquare minor of size $\\ge 2$ has determinant zero, at all ranks simultaneously (previously\nchecked only experimentally at ranks 2--4). Pure linear algebra over $\\mathbb\nQ$-vector spaces, zero arithmetic content specific to Mersenne or \\#257; closes an entire\nfamily of ``maybe there's a determinant obstruction at some higher rank'' hopes in one\nshot.\n\\par\\ev{Lean}\\scale{uniform}\\coord{linear-algebra}\n\\lean{relationInvariantLinearChannels\\_det\\_eq\\_zero}{HalfTrappingReturnCarry.lean:42}\n\\end{lem}\n\n\\begin{thm}[Two-thirds band: exact localisation of post-take skip-unsafety]\n\\label{thm:two-thirds-band}\nWriting a greedy residual as $\\mathrm{rem}=1/R$: a skip at rank $k$ is dyadically safe\niff $R\\ge 2^k$. For a take-skip-take run with pre-take reciprocal $R$, take at $b$\n($q=2^b-1$), next take at $c$ ($m=2^{c-1}$), the exact localisation\n$\\mathrm{PostTakeUnsafeAt}\\,R\\,q\\,m \\iff \\mathrm{Band}\\,R\\,q\\,m$ holds, with band width\nexactly $q^2/((q{+}m)(q{+}m{-}1))$. Specializing to a single-skip run ($m=2q{+}2$) gives\nthe two-thirds band, width exactly $q^2/((3q{+}1)(3q{+}2))<1/9$, pinning $2q<3R<2q+2/3$.\nTwo unconditional corollaries: an integral $R$ is never unsafe; for $\\mathrm{rem}=p/(2D)$\nin lowest terms with $p,D,q$ odd, an unsafe run forces $p\\ge 7$ (via a $2$-adic parity\nargument on the band defect, $4\\mid\\Delta$). Entirely general to any greedy expansion\nusing the dyadic-safety test, not specific to Mersenne weights, but dyadic safety is\nsufficient, not proved necessary, for the true greedy process to survive: $(p,D,b)=\n(17,41,3)$ is an explicit odd-coprime example that \\emph{is} dyadically unsafe, and\nnothing here asserts the actual half-greedy orbit avoids the band.\n\\par\\ev{Lean}\\scale{uniform}\\coord{rational-band}\n\\lean{postTakeUnsafeAt\\_iff\\_band}{HalfGreedyTwoThirdsBand.lean:130}\n\\lean{seven\\_le\\_of\\_intBand\\_odd}{HalfGreedyTwoThirdsBand.lean:253}\n\\lean{not\\_twoThirdsBand\\_of\\_int}{HalfGreedyTwoThirdsBand.lean:200}\n\\end{thm}\n\n\\begin{thm}[Sharp two-thirds fatal gap: the true-tail-safe criterion]\n\\label{thm:sharp-fatal-gap}\nFor a skipped residual $\\rho=u/(2L)$ at rank $k$, writing the skip margin as\n$2^k u+a=2L+u$ (so $0<a\\Leftrightarrow$ skip): the dyadic safety test\n(Theorem~\\ref{thm:two-thirds-band}) is $u\\le a$, but the \\emph{true} tail-based safety\ntest only needs $2u\\le 3a$, proved via the explicit three-channel Lambert lower bound\n$\\mathrm{mersenneTailLB3}\\,k = 1/2^k+1/(3\\cdot 4^k)+1/(7\\cdot 8^k) <\n\\mathrm{mersenneTail}\\,k$. Unconditional corollaries: unit numerators ($u=1$) are always\nsafe; fatality forces $3a<2u$, hence $u\\ge 2$ unconditionally and $u\\ge 3$ once $u$ is\nodd. Dyadic-safe $\\Rightarrow$ sharp-safe strictly: $(u,a)=(3,2)$ is sharp-safe but not\ndyadic-safe. One-third tighter than the dyadic test, discharged against the actual\nMersenne tail, not just the three-channel lower bound. A separate project-memory note\nrecords matching ``$2u\\le 3a$'' language on the \\emph{unconditional} ($\\forall$ infinite\n$A$) \\#257 track; whether that is the same theorem or an independently proved analogue on\nthis half-membership track is unverified \u2014 flag as \\texttt{reported\\_prior}.\n\\par\\ev{Lean}\\scale{uniform}\\coord{mobius-mersenne}\n\\lean{skipSafe\\_of\\_two\\_mul\\_le\\_three\\_mul}{HalfGreedyFatalGap.lean:150}\n\\lean{sharp\\_strictly\\_stronger}{HalfGreedyFatalGap.lean:230}\n\\lean{three\\_le\\_of\\_fatal\\_of\\_odd}{HalfGreedyFatalGap.lean:190}\n\\end{thm}\n\n\\begin{lem}[Total gap-mass summability beyond any level]\n\\label{lem:gap-mass-summability}\n$\\mathrm{mersenneGap}\\,n := \\mathrm{mersenneWeight}\\,n-\\mathrm{mersenneTail}\\,n>0$. The\ntotal gap mass strictly beyond level $N$ is summable and explicitly bounded:\n$\\sum'_k\\mathrm{mersenneGap}(N{+}k{+}1) \\le (2/9)(1/4)^N+(3/7)(1/8)^N\\to 0$, obtained by\nsumming the pre-existing per-level asymptotic bound geometrically. \\textbf{Scope\ncaveat} (explicit in the source file header): bounds gap \\emph{mass} only \u2014 says\nnothing about which reals the greedy run reaches, does not certify membership or\nnonmembership of any point, and in particular says nothing about $1/2$ directly.\n\\par\\ev{Lean}\\scale{uniform}\\coord{mobius-mersenne}\n\\lean{mersenneGap\\_tail\\_le}{HalfGapMass.lean:100}\n\\lean{tendsto\\_mersenneGap\\_tail\\_zero}{HalfGapMass.lean:127}\n\\end{lem}\n\n\\begin{lem}[Half-divisor unit drop: smallest single-bit coefficient producer]\n\\label{lem:half-divisor-unit-drop}\nAt row $2(N{+}1)$, changing only exponent $N{+}1$ from \\emph{false} to \\emph{true} in a\n\\emph{HalfWord N} adds exactly the half-divisor incidence to \\emph{supportCoeff}:\n$\\mathrm{supportCoeff}(\\mathrm{wordSupport}(\\mathrm{extend}\\,a\\,\\mathrm{true}))(2(N{+}1)) =\n\\mathrm{supportCoeff}(\\mathrm{wordSupport}(\\mathrm{extend}\\,a\\,\\mathrm{false}))(2(N{+}1))\n+ 1$. The smallest possible ``flip one bit, coefficient changes by exactly $1$''\nproducer. The mechanism (flipping a single word-bit changes a divisor-incidence sum by\nexactly the number of new divisor relations created) is a generic digit/divisor-incidence\nfact, restating for any base-$2$ divisor-coefficient bookkeeping, including \\#249's own\ndivisor sums.\n\\par\\ev{Lean}\\scale{uniform}\\coord{divisor-incidence}\n\\lean{supportCoeff\\_extend\\_true\\_eq\\_false\\_add\\_one\\_at\\_double}{HalfDivisorUnitDrop.lean:20}\n\\end{lem}\n\n\\begin{thm}[Tempered-orbit rigidity: the shared \\#249/\\#257 trunk]\n\\label{thm:tempered-orbit-rigidity}\nFor any nonnegative-integer coefficient sequence $c:\\mathbb N\\to\\mathbb N$ with $c(n)\\le\nn$: the binary coefficient series $X_c = \\sum'_{n\\ge 1} c(n)/2^n$ is rational iff there\nexists a positive integer multiplier $v$ and an integer orbit $u:\\mathbb N\\to\\mathbb Z$\nsatisfying the exact carry recurrence $u(N{+}1)=2u(N)-v\\cdot c(N{+}1)$ together with the\ntempered (subexponential) growth condition $u(N)/2^N\\to 0$. Every such tempered orbit is\n\\emph{rigid}: $u(N) = v\\cdot T_c(N)$ exactly, where $T_c(N)=\\sum_{j\\ge 1} c(N{+}j)/2^j$ is\nthe scaled tail \u2014 there is at most one tempered orbit up to the multiplier $v$, and it\nequals the analytic tail exactly, not just asymptotically. Positivity of the orbit alone\nis deliberately \\emph{not} used as an equivalent criterion: a homogeneous $2^N$-scaled\nperturbation can be added to any orbit without breaking the recurrence, so only\ntemperedness is load-bearing. Explicitly asserted in-source as the shared binary carry\ntrunk for \\#249 and \\#257: $c=\\tau$ restricted to support (\\#257's \\emph{supportCoeff})\nand $c=\\varphi$ (\\#249) are both literal instances with no adaptation needed.\n\\par\\ev{Lean}\\scale{uniform}\\coord{binary-digit}\n\\lean{binaryCoeffSeries\\_rational\\_iff\\_exists\\_temperedBinaryOrbit}{GenericTailOrbitRigidity.lean:425}\n\\lean{temperedBinaryOrbit\\_eq\\_scaledTail}{GenericTailOrbitRigidity.lean:338}\n\\end{thm}\n\n\\begin{lem}[Finite symmetric-difference tail-transfer of irrationality]\n\\label{lem:tail-transfer}\nIf $\\mathrm{erdosSupportSeries}\\,b\\,A$ is irrational and $A,B$ differ by a finite\nsymmetric difference, then $\\mathrm{erdosSupportSeries}\\,b\\,B$ is irrational too.\nIrrationality of a support series is invariant under changing finitely many support\nelements. Fully general; likely reusable for \\#249 tails as well. Feeds the\neventually-periodic theorem (Theorem~\\ref{thm:eventually-periodic}).\n\\par\\ev{Lean}\\scale{uniform}\\coord{tail-transfer}\n\\lean{irrational\\_erdosSupportSeries\\_tail\\_of\\_irrational}{CertificateKernel.lean:11276}\n\\end{lem}\n\n\\begin{lem}[Dyadic-vs-Mersenne exact excess reformulation]\n\\label{lem:dyadic-excess-reformulation}\nThe greedy-toward-$1/2$ decision at rank $n{+}1$ (take the next Mersenne weight\n$1/(2^{n+1}-1)$ or skip it) is exactly equivalent, with no real-analytic approximation, to\nan integer sign test on the excess numerator $\\mathrm{nextDyadicExcessIntNumerator}\\,p\\,n\\,L\n:= 2^n p - L$ for a displayed residual $p/(2L)$: $p/(2L)\\le 1/2^{n+1} \\iff\n\\mathrm{nextDyadicExcessIntNumerator}\\,p\\,n\\,L\\le 0$. The forbidden ``Mersenne--dyadic\nsliver'' (skip is dyadically unsafe) is exactly $0<E\\land 2E<p$.\n\\par\\ev{Lean}\\scale{uniform}\\coord{seam-integer}\n\\lean{divInt\\_le\\_nextDyadic\\_iff\\_excess\\_nonpos}{DyadicPrefixCompression.lean:198}\n\\lean{greedyHalf\\_mem\\_nextMersenneDyadicSliver\\_iff\\_excess}{DyadicPrefixCompression.lean:889}\n\\end{lem}\n\n\\begin{lem}[Denominator sandwich: odd-part survival, domain-neutral]\n\\label{lem:denominator-sandwich}\nSubtracting an odd reduced fraction $r/D$ from a dyadic $p/2^c$ leaves a residual whose\nreduced denominator sandwiches the original: $D\\mid(\\mathrm{residual}).\\mathrm{den}\\mid\n2^c\\cdot D$. Pure \\texttt{Rat.divInt} denominator-reduction algebra, independent of the\ngreedy/Mersenne setting; directly reusable anywhere a base-$2$ rational minus an\nodd-denominator rational needs its reduced denominator tracked, including \\#249's own\ncarry bookkeeping.\n\\par\\ev{Lean}\\scale{uniform}\\coord{denominator-algebra}\n\\lean{dyadicResidual\\_denominator\\_sandwich}{DyadicPrefixCompression.lean:118}\n\\end{lem}\n\n\\begin{lem}[Rational denominator survival: general-purpose fraction-reduction algebra]\n\\label{lem:denominator-survival}\nIf $m\\mid D$ and $m$ is coprime to numerator $a$, then $m\\mid(a/D\\text{ as }\n\\mathrm{Rat}).\\mathrm{den}$. Scaled variant: if $C\\mid D$ and $C$ is coprime to $a$, then\n$C/\\gcd(C,h)\\mid((h\\cdot a)/D\\text{ as }\\mathrm{Rat}).\\mathrm{den}$ \u2014 a divisor of the\ndisplayed denominator survives rational reduction up to exactly the part the scale\nmultiplier can cancel. Zero domain content: the general-purpose ``which prime power\nsurvives fraction reduction'' lemma, feeding Theorem~\\ref{thm:mersenne-channel-survival}\ndirectly, and equally applicable to any future \\#249 denominator-tracking argument.\n\\par\\ev{Lean}\\scale{uniform}\\coord{denominator-algebra}\n\\lean{divisor\\_dvd\\_divInt\\_den}{RationalDenominatorSurvival.lean:17}\n\\lean{survivingDivisor\\_dvd\\_scaled\\_divInt\\_den}{RationalDenominatorSurvival.lean:38}\n\\end{lem}\n\n\\begin{lem}[Two-prime exact-valuation layer commutation]\n\\label{lem:mixed-prime-power-layer}\nFor an integer-valued support-coefficient function and two distinct primes $p\\ne q$ with\nexponents $e,f>0$: the iterated exact-level-difference operator (\\emph{primePowerLayer})\ncommutes across the two primes, and two nested layers extract exactly the iterated\nexact-valuation pullback of the support:\n$\\mathrm{mixedPrimePowerLayerTwo}\\,p\\,e\\,q\\,f\\,(\\mathrm{supportCoeffInt}\\,A)\\,n =\n\\mathrm{supportCoeffInt}\\,(\\mathrm{pullback}_{q,f}(\\mathrm{pullback}_{p,e}\\,A))\\,n$ for\n$n$ coprime to $pq$. Extends the single-prime layer theory to two primes; purely\nalgebraic, supplies no transport from rationality through multiplicative decimation and\nasserts no bounded-$\\Omega$ endpoint on its own.\n\\par\\ev{Lean}\\scale{uniform}\\coord{p-adic}\n\\lean{mixedPrimePowerLayerTwo\\_supportCoeffInt}{MaximalOmegaLayer.lean:39}\n\\lean{primePowerLayer\\_comm}{MaximalOmegaLayer.lean:29}\n\\end{lem}\n\n\\begin{defn}[The Mersenne achievement set and the support-series object]\n\\label{defn:achievement-set}\n$\\mathrm{mersenneAchievementSet} := \\{x\\in\\mathbb R \\mid \\exists A\\subseteq\\mathbb N,\\,\n0\\notin A \\land x = \\mathrm{positiveMersenneSupportValue}\\,A\\}$, where\n$\\mathrm{positiveMersenneSupportValue}\\,A = \\sum'_k \\mathrm{indicator}\\,A\\,\n\\mathrm{mersenneWeight}(k{+}1)$ \u2014 the set of reals representable as a sum of a\n\\emph{subset} of positive-index Mersenne weights, ranging over all supports $A$, not only\ninfinite ones. The general-$b$, general-support object used throughout the support-family\nroute is $\\mathrm{erdosSupportSeries}\\,b\\,A := \\sum'_n \\mathrm{indicator}\\,A\\,\n(1/(b^n-1))\\,n$, with $\\mathrm{supportCoeff}\\,A\\,n := \\#\\{d\\mid n : d\\in A\\}$ (the\nDirichlet incidence $1_A * 1$, \\#257's analogue of $\\tau$); $A=\\mathrm{Set.univ}$\nrecovers the full-support series of Theorem~\\ref{thm:full-support-catalogue}.\n\\par\\ev{Lean}\\scale{n/a}\\coord{binary-digit}\n\\lean{mersenneAchievementSet}{GreedyAchievementSet.lean:571}\n\\lean{erdosSupportSeries}{CertificateKernel.lean:8569}\n\\lean{supportCoeff}{CertificateKernel.lean:8650}\n\\end{defn}\n\n\\begin{prop}[Achievement-set topology: compact, closed, perfect, measure one]\n\\label{prop:achievement-set-topology}\n$\\mathrm{mersenneAchievementSet}$ is compact (continuous image of the binary-sequence\nCantor space $\\mathbb N\\to\\mathrm{Fin}\\,2$ under the product topology, via\n$\\mathrm{positiveMersenneDigitValue}$), hence closed; it is also perfect, totally\ndisconnected, and nowhere dense, with Lebesgue measure exactly $1$. The\nmeasure-one statement is absolute: it does not say that the set has full measure in\n$\\mathbb R$ or in its convex hull, whose length is $T_0=1+T_1>1$. The\ncompactness/closedness argument (binary coding $\\to$ Cantor space $\\to$ continuous image)\nis a fully generic technique for characterizing the achievement set of \\emph{any}\nabsolutely convergent digit-weighted series, not specific to Mersenne denominators \u2014\ndirectly reusable for a \\#249-style $\\varphi(n)/2^n$ series after checking summability\nand superincreasingness. Closedness alone is what powers every ``limit of a sequence of\nachieved points is achieved'' argument in this catalogue (e.g.\\\nTheorem~\\ref{thm:seam-limit}, Theorem~\\ref{thm:straddle-closed-set}).\n\\par\\ev{Lean}\\scale{n/a}\\coord{binary-digit}\n\\lean{isCompact\\_mersenneAchievementSet}{GreedyAchievementSet.lean:656}\n\\lean{isClosed\\_mersenneAchievementSet}{GreedyAchievementSet.lean:660}\n\\lean{volume\\_mersenneAchievementSet}{GreedyAchievementSet.lean:1024}\n\\end{prop}\n\n\\begin{thm}[The master dichotomy: fatal-gap existence iff half-nonmembership]\n\\label{thm:master-dichotomy}\n$\\mathrm{ExistsFatalHalfGap} := \\exists u,d,\\; (\\text{bounds}) \\land\n\\mathrm{value}(u)+\\mathrm{mersenneTail}(d{+}1) < 1/2 \\land 1/2 <\n\\mathrm{value}(u)+\\mathrm{mersenneWeight}(d{+}1)$. \\textbf{Theorem:}\n$\\mathrm{ExistsFatalHalfGap} \\iff 1/2\\notin\\mathrm{mersenneAchievementSet}$,\nequivalently $1/2\\in\\mathrm{mersenneAchievementSet} \\iff \\neg\\mathrm{ExistsFatalHalfGap}$\n\u2014 the complete unconditional dichotomy: one of the two disjuncts always holds. The whole\nof the half-branch of \\#257 reduces to a search for one finite, checkable, numerically\nverifiable fatal-gap witness $(u,d)$ \u2014 a completely finite existence statement, versus its\nnegation, which by the straddle machinery\n(Lemma~\\ref{lem:rank-step-trichotomy}--\\ref{lem:straddle-agrees-greedy}) is equivalent to\nmembership. The shape of this dichotomy (finite fatal-gap certificate exists, or the\nclosed achievement set contains the target) is the generic Dedekind-cut-in-a-\nsuperincreasing-series pattern, directly portable to \\#249 given its own analogues of\nLemma~\\ref{lem:half-endpoint-kills} and Proposition~\\ref{prop:achievement-set-topology}.\n\\par\\ev{Lean}\\scale{n/a}\\coord{dedekind-cut}\n\\lean{half\\_mem\\_mersenneAchievementSet\\_iff\\_no\\_existsFatalHalfGap}{HalfCutLocator.lean:654}\n\\lean{half\\_mem\\_mersenneAchievementSet\\_or\\_exists\\_fatal\\_gap}{HalfCutLocator.lean:623}\n\\end{thm}\n\n\\begin{defn}[Perturbed family and adjacent cut: the abstract seam gadget]\n\\label{defn:perturbed-family}\n$\\mathrm{structure}\\;\\mathrm{PerturbedFamily}(\\alpha)$ where\n$\\mathrm{oldSum}:\\alpha\\to\\mathbb N$; $\\mathrm{pulse}:\\alpha\\to\\mathbb N$;\n$\\mathrm{gap},\\mathrm{pulseCap}:\\mathbb N$; with axioms $\\mathrm{gap\\_pos}$,\n$\\mathrm{pulse\\_le}$, $\\mathrm{oldSum\\_injective}$, a superincreasing-gap\n\\emph{separated} condition, and $\\mathrm{pulseCap} < 3\\cdot\\mathrm{gap}$. Derived:\n$\\mathrm{newSum}\\,x := 4\\cdot\\mathrm{oldSum}\\,x+\\mathrm{pulse}\\,x$;\n$\\mathrm{structure}\\;\\mathrm{AdjacentCut}(C)$ packages a ``best subset $\\le C$, best\nsubset $>C$'' adjacent pair with admissibility/maximality/strictness axioms. No Mersenne\ncontent whatsoever: $\\alpha$, $\\mathrm{oldSum}$, $\\mathrm{pulse}$ are all free\nparameters \u2014 this is the general theory that $\\mathrm{seamPerturbedFamily}$/\n$\\mathrm{seamAdjacentCut}$ (used throughout the seam files) are specific instances of.\n\\par\\ev{Lean}\\scale{n/a}\\coord{abstract-perturbed-greedy}\n\\lean{PerturbedFamily}{HalfCylinderIntegerGreedy.lean:1286}\n\\lean{AdjacentCut}{HalfCylinderIntegerGreedy.lean:1360}\n\\end{defn}\n\n\\begin{thm}[Perturbed-family maximality and the three-branch recurrence]\n\\label{thm:perturbed-family-maximality}\n$\\mathrm{successorCarries} := 4\\cdot\\mathrm{overshoot}+\\mathrm{abovePulse} \\le\n\\mathrm{gap}$; $\\mathrm{prefixChoice} := \\mathrm{if}\\ \\mathrm{successorCarries}\\\n\\mathrm{then}\\ \\mathrm{above}\\ \\mathrm{else}\\ \\mathrm{below}$ is provably maximal among\nall $x$ with $\\mathrm{newSum}\\,x\\le\\mathrm{newCapacity}$\n(\\emph{prefixChoice\\_maximal}); \\emph{nextRemainder\\_trichotomy} gives the exact\nthree-branch (carry / middle / right) recurrence for the next remainder. Proved once and\nfor all at the abstract \\emph{PerturbedFamily} level\n(Definition~\\ref{defn:perturbed-family}); the single most reusable abstract result in\nthis catalogue \u2014 any future formalization of a different base-$b$ greedy digit process\nneeding a maximality-of-greedy-choice theorem can instantiate this structure directly\ninstead of re-proving maximality from scratch.\n\\par\\ev{Lean}\\scale{n/a}\\coord{abstract-perturbed-greedy}\n\\lean{prefixChoice\\_maximal}{HalfCylinderIntegerGreedy.lean:1440}\n\\lean{nextRemainder\\_trichotomy}{HalfCylinderIntegerGreedy.lean:1470}\n\\end{thm}\n\n% ---- part a257_p1b ----\n\\subsection{Consumers}\n\nThe BooleanM\\\"obius lane (repo root \\texttt{Erdos257PeriodNoncollapse/}, mirrored at\n\\texttt{Erdos249257/} for the modules that have been made public) is where the reduction\nof Erd\\H{o}s \\#257 to a single arithmetic socket actually lives. Sixteen of its modules\nhave never been published: \\texttt{BooleanMobiusCofinalExactRows},\n\\texttt{BooleanMobiusCriticalCapacityCofinal}, \\texttt{BooleanMobiusCriticalCapacityGeometric},\n\\texttt{BooleanMobiusExactRowCrossing}, \\texttt{BooleanMobiusExactRowDichotomy},\n\\texttt{BooleanMobiusExactRowDoubling}, \\texttt{BooleanMobiusExactRowRankTwo},\n\\texttt{BooleanMobiusExactRowSeed}, \\texttt{BooleanMobiusExactTransition},\n\\texttt{BooleanMobiusGlobalRepair}, \\texttt{BooleanMobiusGreedyReduction},\n\\texttt{BooleanMobiusLocalRepair}, \\texttt{BooleanMobiusSkipRow},\n\\texttt{BooleanMobiusSkipRowCofinal}, \\texttt{BooleanMobiusSkippedCoreCriticalCapacity},\n\\texttt{BooleanMobiusSkippedCoreExactRow}; plus three sibling modules on the half-membership\ntrack, also never published: \\texttt{HalfGapMass}, \\texttt{HalfGreedyFatalGap},\n\\texttt{HalfUpperResetCriticalBand}. Every one of the nineteen is covered below. Consumers\nin this subsection are theorems whose conclusion is a genuine payoff\n($(1/2:\\mathbb{R})\\in\\mathrm{mersenneAchievementSet}$, or an infinite counterexample support)\nbut whose hypothesis is an unsupplied predicate. Every such hypothesis is displayed, not\nburied in prose: that predicate is the actual target.\n\n\\subsubsection*{The load-bearing definitions}\n\n\\begin{defn}[257bm:d1 --- \\texttt{ExactLocalMersenneHalfRow}]\nFor $n:\\N$:\n\\[\n\\mathrm{ExactLocalMersenneHalfRow}(n) \\;:=\\; \\exists D:\\mathrm{Finset}\\,\\N,\\ (\\forall d\\in D,\\ 2\\le d\\le n) \\ \\wedge\\ \\mathrm{localPrefixQuotient}(D,n) = 2^{n-1}-1 .\n\\]\nA finite Boolean support $D$ bounded by $n$ exactly hits the integer target $2^{n-1}-1$ under\nfloor-division Mersenne quotients at scale $n$ \u2014 the natural-number carry accounting is\nexact, not merely close.\n\\par\\noindent\\ev{Lean} \\scale{n/a} \\coord{mobius-mersenne} \\lean{ExactLocalMersenneHalfRow}{BooleanMobiusCofinalExactRows.lean:31}\n\\end{defn}\n\n\\begin{defn}[257bm:d2 --- \\texttt{localPrefixQuotient}, \\texttt{localMersenneQuotient}, \\texttt{localRepairInteger}]\n\\[\n\\mathrm{localMersenneQuotient}(M,d) := \\left\\lfloor \\frac{2^M}{2^d-1} \\right\\rfloor,\\qquad\n\\mathrm{localPrefixQuotient}(D,M) := \\sum_{d\\in D} \\mathrm{localMersenneQuotient}(M,d),\n\\]\nboth natural-number floor division, and $\\mathrm{localRepairInteger}(D,k) := $ the signed\none-step carry defect of $\\mathrm{localPrefixQuotient}$ at rank $k$ (see 257bm:i8 below for the\ntransition identity it satisfies). These are the three primitive accounting objects every\nother definition and theorem in this lane is built from.\n\\par\\noindent\\ev{Lean} \\scale{n/a} \\coord{mobius-mersenne} \\lean{localPrefixQuotient}{BooleanMobiusLocalRepair.lean:20} \\lean{localMersenneQuotient}{BooleanMobiusLocalRepair.lean:29} \\lean{localRepairInteger}{BooleanMobiusLocalRepair.lean:74}\n\\end{defn}\n\n\\begin{defn}[257bm:d3 --- \\texttt{localBinarySuffix}]\n\\[\n\\mathrm{localBinarySuffix}(D,k,M) \\;:=\\; 2^{M-k} - \\mathrm{localPrefixQuotient}(D,M) - 1,\n\\]\nthe leftover carry after subtracting $D$'s integer quotient contributions from the binary\nexpansion of $2^{-k}$ truncated at place $M$. Every capacity test in the lane is a bound on\nthis one quantity.\n\\par\\noindent\\ev{Lean} \\scale{n/a} \\coord{binary-digit} \\lean{localBinarySuffix}{BooleanMobiusLocalRepair.lean:53}\n\\end{defn}\n\n\\begin{defn}[257bm:d4 --- \\texttt{CofinalExactLocalMersenneHalfRows}, THE MASTER PREMISE]\n\\[\n\\mathrm{CofinalExactLocalMersenneHalfRows} \\;:=\\; \\forall N:\\N,\\ \\exists n:\\N,\\ N\\le n\\ \\wedge\\ \\mathrm{ExactLocalMersenneHalfRow}(n).\n\\]\n\\textbf{Flag prominently:} this requires \\emph{no compatibility whatsoever} between the\nwitness supports $D_n$ at different endpoints $n$. It is strictly weaker than a coherent\nglobal trajectory (257bm:c1 below) \u2014 mutually \\emph{incompatible} witnesses at each endpoint\nare entirely sufficient. This weakening is simultaneously the whole reason the lane is\ntractable and the exact place an attacker must be careful: a proof that looks like it\nsupplies coherent rows is doing strictly more work than is needed, and a proof that\nlooks like it refutes coherent rows says nothing about this predicate.\n\\par\\noindent\\ev{Open} \\scale{cofinal} \\coord{mobius-mersenne} \\lean{CofinalExactLocalMersenneHalfRows}{BooleanMobiusCofinalExactRows.lean:38}\n\\end{defn}\n\n\\subsubsection*{The two payoff consumers}\n\n\\begin{thm}[257bm:c1 --- \\texttt{half\\_mem\\_mersenneAchievementSet\\_of\\_cofinalExactLocalRows}, THE PAYOFF THEOREM]\n\\[\n\\mathrm{CofinalExactLocalMersenneHalfRows} \\;\\longrightarrow\\; (1/2:\\mathbb{R}) \\in \\mathrm{mersenneAchievementSet},\n\\]\nwhere $\\mathrm{mersenneAchievementSet} := \\{x \\mid \\exists A:\\mathrm{Set}\\,\\N,\\ 0\\notin A \\wedge x = \\mathrm{positiveMersenneSupportValue}(A)\\}$. Proof: for each $N$ pick $n\\ge N$ with exact row $D_N$; the real value $y_N$ of $D_N$ satisfies $|y_N-1/2|\\le (n+1)/2^n\\to0$ (257bm:i9); $\\mathrm{mersenneAchievementSet}$ is compact (continuous image of $2^{\\N}$ under the digit-coding map), hence closed, so the limit $1/2$ is achieved by an actual point. This is the theorem 257bm:d4, and hence ultimately every producer feeding it, must reach.\n\\par\\noindent\\ev{Lean} \\scale{cofinal} \\coord{other:topological-achievement-set} \\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_cofinalExactLocalRows}{BooleanMobiusCofinalExactRows.lean:71}\n\\end{thm}\n\n\\begin{thm}[257bm:c2 --- \\texttt{cofinalExactLocalMersenneHalfRows\\_of\\_positiveHalfGreedySkips}, \\texttt{half\\_mem\\_mersenneAchievementSet\\_of\\_positiveHalfGreedySkips}]\n\\[\n\\mathrm{CofinalPositiveHalfGreedySkips} \\;\\longrightarrow\\; \\mathrm{CofinalExactLocalMersenneHalfRows} \\;\\longrightarrow\\; (1/2:\\mathbb{R})\\in\\mathrm{mersenneAchievementSet},\n\\]\nwhere $\\mathrm{CofinalPositiveHalfGreedySkips} := \\forall N,\\ \\exists c,\\ \\max(N,4)\\le c \\wedge 0 < \\mathrm{greedyMersenneRemainderRat}(1/2,c-1) < \\mathrm{mersenneWeightRat}(c)$ \u2014 the canonical rational half-greedy orbit itself has infinitely many positive skip events. A single skip at $c$ already gives $\\mathrm{ExactLocalMersenneHalfRow}(2c-2)$ unconditionally (\\texttt{exactLocalMersenneHalfRow\\_of\\_positiveHalfGreedySkip}, no capacity hypothesis at that step). This is the SHORTEST known path to the disproof: cofinally many such skips close the chain with zero further combinatorial search.\n\\par\\noindent\\ev{Lean} \\scale{cofinal} \\coord{greedy-orbit} \\lean{cofinalExactLocalMersenneHalfRows\\_of\\_positiveHalfGreedySkips}{BooleanMobiusSkipRowCofinal.lean:60} \\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_positiveHalfGreedySkips}{BooleanMobiusSkipRowCofinal.lean:73} \\lean{exactLocalMersenneHalfRow\\_of\\_positiveHalfGreedySkip}{BooleanMobiusSkipRowCofinal.lean:31}\n\\end{thm}\n\n\\subsubsection*{The coherent-repair consumer (superseded, kept as a fallback target)}\n\n\\begin{thm}[257bm:c3 --- \\texttt{GlobalBooleanMobiusRepairFeasible}]\nA repair $T$ (a bit-assignment $\\mathrm{bit}:\\N\\to\\N\\to\\mathrm{Bool}$, endpoint$\\times$rank$\\to$chosen bit, with $\\mathtt{bit\\_stable}$: once endpoint $2d$ is reached coordinate $d$ is frozen forever) is feasible when\n\\[\n\\mathrm{GlobalEndpointExponentialBound}(T) \\ \\wedge\\ (\\text{word-value clause})\\ \\wedge\\ (\\text{word-capacity clause})\\ \\wedge\\ \\forall n\\ge2,\\ \\mathrm{localPrefixQuotient}(\\mathrm{globalRepairStageSupport}(T.\\mathrm{bit},n),n) = 2^{n-1}-1,\n\\]\nwhere the named open producer is\n\\[\n\\mathrm{GlobalEndpointExponentialBound}(T) \\;:=\\; \\forall n\\ge2,\\ \\mathrm{let}\\ D := \\mathrm{globalRepairLowerSupport}(T.\\mathrm{bit},n);\\ \\ 2^{\\mathrm{endpointDivisorContribution}(D,n)-1}-1 \\le \\mathrm{localBinarySuffix}(D,1,n-1).\n\\]\nThis is the STRONGER, harder consumer \u2014 a genuinely coherent (frozen-diagonal) repair, unlike\n257bm:d4's cofinal-exact-row supply which needs no coherence at all. Kept as a fallback target\nonly; 257bm:c1/c2 supersede it.\n\\par\\noindent\\ev{Open} \\scale{uniform} \\coord{binary-digit} \\lean{GlobalBooleanMobiusRepairFeasible}{BooleanMobiusGlobalRepair.lean:174} \\lean{GlobalEndpointExponentialBound}{BooleanMobiusGlobalRepair.lean:139} \\lean{bit\\_stable}{BooleanMobiusGlobalRepair.lean:82}\n\\end{thm}\n\n\\subsubsection*{The critical-quotient-supply consumer chain}\n\n\\begin{thm}[257bm:c4 --- \\texttt{SkippedCoreCriticalQuotientSupply}, THE REMAINING OPEN SOCKET]\n\\[\n\\mathrm{SkippedCoreCriticalQuotientSupply} \\;:=\\; \\forall D\\ c,\\ 4\\le c \\to (\\forall d\\in D,\\ 2\\le d < c) \\to \\mathrm{value}(D) < \\tfrac12 \\to \\Big(\\tfrac12-\\mathrm{value}(D) < \\mathrm{mersenneWeightRat}(c)\\Big) \\to 2^{(2c-2)-1} \\le \\mathrm{localPrefixQuotient}(\\mathrm{insert}\\ c\\ D,\\ 2c-2).\n\\]\nWhenever a below-half core is genuinely crossed by rank $c$, adjoining $c$ already reaches the\nintegral half-target. By 257bm:i5 (the sharp capacity iff) this is EXACTLY the sharp\n$(c-2)$-bit capacity test, universally quantified over all crossing cores. If proved, this\nProp closes the whole cofinal-exact-row route and hence disproves Erd\\H{o}s \\#257.\n\\par\\noindent\\ev{Open} \\scale{uniform} \\coord{binary-digit} \\lean{SkippedCoreCriticalQuotientSupply}{BooleanMobiusCriticalCapacityCofinal.lean:30}\n\\end{thm}\n\n\\begin{thm}[257bm:c5 --- \\texttt{cofinalExactLocalMersenneHalfRows\\_of\\_criticalQuotientSupply}]\nAn already-formalised induction from the endpoint-six seed (257bm:i7) consumes 257bm:c4 at\nevery step: each \\texttt{ProtectedExactLocalMersenneRow} either doubles below half\n(unconditional) or recycles at its first crossing rank $e > $ cutoff, giving endpoint\n$2e-2 > $ previous endpoint; \\texttt{protection} (endpoint $< 2\\cdot$cutoff, new ranks $>$\ncutoff) is exactly what converts the non-growing recycle endpoint of the bare dichotomy\n(257bm:k1) into strict progress. The below-half branch never fires twice from the seed\narithmetic, so the supply is needed at essentially every step. Ends literally at\n$\\mathrm{CofinalExactLocalMersenneHalfRows}$.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{binary-digit} \\lean{cofinalExactLocalMersenneHalfRows\\_of\\_criticalQuotientSupply}{BooleanMobiusCriticalCapacityCofinal.lean:1227} \\lean{ProtectedExactLocalMersenneRow}{BooleanMobiusCriticalCapacityCofinal.lean:1005} \\lean{exists\\_laterProtectedExactLocalMersenneRow}{BooleanMobiusCriticalCapacityCofinal.lean:1057}\n\\end{thm}\n\n\\begin{thm}[257bm:c6 --- \\texttt{HalfGreedySkippedCriticalQuotientSupply}, the already-reduced canonical form]\n\\[\n\\mathrm{HalfGreedySkippedCriticalQuotientSupply} \\;:=\\; \\forall c\\ge4\\ \\text{skipped by the rational half-greedy orbit},\\quad 2^{(2c-2)-1} \\le \\mathrm{localPrefixQuotient}(\\mathrm{insert}\\ c\\ (\\mathrm{halfGreedyPrefixSupport}(c-1)),\\ 2c-2).\n\\]\nThe sharp inequality at one crossing rank per step, in the canonical form the induction of\n257bm:c5 actually consumes \u2014 this is 257bm:c4 with the universal quantifier already collapsed\nonto the one orbit by 257bm:i3 (the wall).\n\\par\\noindent\\ev{Open} \\scale{uniform} \\coord{greedy-orbit} \\lean{HalfGreedySkippedCriticalQuotientSupply}{BooleanMobiusCriticalCapacityCofinal.lean:178}\n\\end{thm}\n\n\\begin{thm}[257bm:c6a --- \\texttt{halfGreedy\\_precriticalSuffix\\_lt\\_of\\_next\\_skip}, unconditional two-consecutive-skips producer]\nUNCONDITIONAL: if $c\\ge6$ and BOTH rank $c$ and rank $c+1$ are skipped by the rational\nhalf-greedy orbit, then\n\\[\n\\mathrm{localBinarySuffix}(\\mathrm{halfGreedyPrefixSupport}(c-1),\\,1,\\,2c-3) < 2^{c-3},\n\\]\nwhich doubles into sharp capacity at $c$ and hence an exact row at $2c-2$ with strict-upper\nseparation. No supply hypothesis anywhere. The residual open set is exactly the\nskip-then-take rows: \\texttt{halfGreedySkippedPrecriticalSuffixSupply\\_iff\\_preTake} proves the\nwhole socket equivalent to \\texttt{HalfGreedyPreTakePrecriticalSuffixSupply}, with $c=4$ and\n$c=5$ already discharged by \\texttt{decide}. What would close it: cofinally many $c\\ge6$ at\nwhich two consecutive ranks $c,c+1$ are both skipped by the rational half-greedy orbit \u2014 or,\ndually, the pre-take residual at cofinally many skipped-then-taken ranks.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{greedy-orbit} \\lean{halfGreedy\\_precriticalSuffix\\_lt\\_of\\_next\\_skip}{BooleanMobiusCriticalCapacityCofinal.lean:682} \\lean{halfGreedySkippedCriticalQuotientSupply\\_of\\_precriticalSuffix}{BooleanMobiusCriticalCapacityCofinal.lean:919} \\lean{halfGreedySkippedPrecriticalSuffixSupply\\_iff\\_preTake}{BooleanMobiusCriticalCapacityCofinal.lean:756}\n\\end{thm}\n\n\\begin{thm}[257bm:c6b --- \\texttt{halfGreedy\\_precriticalSuffix\\_lt\\_of\\_future\\_skip\\_after\\_takenBlock}, gap-uniform generalisation]\nUNCONDITIONAL and uniform in BOTH parameters, strictly generalising 257bm:c6a: skip at $c$,\ntakes at $c+1,\\dots,c+t-1$, skip at $c+t$, with $0<t\\le c-3$ and $c-2\\le2^{c-t-3}$\n$\\;\\Longrightarrow\\;$ the precritical suffix bound at $c$ $\\Rightarrow$ sharp capacity\n$\\Rightarrow$ exact row at $2c-2$. Handles any skip gap $t\\lesssim c-3-\\log_2(c-2)$, not just\n$t=1$. Fully uniform proof \u2014 no finite table, every constant explicit\n($|D|\\le c-2$, dyadic room $2^{c-t-3}$, gap lemma\n\\texttt{dyadic\\_lt\\_forcedBlock\\_of\\_tail\\_lt}) \u2014 so any orbit-level SKIP-GAP BOUND immediately\nyields cofinal exact rows with no further work. What would close it: for cofinally many\nskipped ranks $c$ of the rational half-greedy orbit, the next skipped rank lies at most\n$c+(c-3-\\lceil\\log_2(c-2)\\rceil)$. Empirically the skips have roughly constant density\n(certified computationally to row 200{,}000 on the sqrt-escape track), so this is a slack\nrequirement never previously isolated as a target.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{greedy-orbit} \\lean{halfGreedy\\_precriticalSuffix\\_lt\\_of\\_future\\_skip\\_after\\_takenBlock}{BooleanMobiusCriticalCapacityCofinal.lean:603} \\lean{precriticalCrossingTax\\_of\\_futureThreshold}{BooleanMobiusCriticalCapacityCofinal.lean:472} \\lean{sub\\_two\\_le\\_two\\_pow\\_sub\\_four}{BooleanMobiusCriticalCapacityCofinal.lean:432}\n\\end{thm}\n\n\\subsubsection*{Two unconditional fill consumers (\\texttt{BooleanMobiusSkipRow})}\n\n\\begin{thm}[257bm:c7 --- \\texttt{exists\\_exactRowStrictUpperFill\\_of\\_skippedCoreSharpCapacity}]\nFor any $c\\ge4$ and any finite $D\\subseteq[2,c)$ with $\\mathrm{value}(D)<1/2$ and the sharp\ncapacity\n\\[\n\\mathrm{localBinarySuffix}(D,1,2c-2) < 2^{c-2},\n\\]\nthere is an exact row $E$ at endpoint $2c-2$ with $D\\subseteq E$ and every new rank strictly\nabove $c$. Unconditional, no deficit/crossing hypothesis, no reference to the greedy orbit at\nall \u2014 this consumer is strictly more general than every route that feeds it: the crossing-core\ndeficit hypothesis used in 257bm:c4/c6 is a self-imposed restriction on the corpus's own\nattack, not a requirement of this theorem.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{binary-digit} \\lean{exists\\_exactRowStrictUpperFill\\_of\\_skippedCoreSharpCapacity}{BooleanMobiusSkipRow.lean:286}\n\\end{thm}\n\n\\begin{thm}[257bm:c8 --- \\texttt{exactLocalMersenneHalfRow\\_two\\_mul\\_sub\\_two\\_of\\_skippedCoreSharpCapacity}]\nBare-Prop corollary of 257bm:c7: the same hypotheses give $\\mathrm{ExactLocalMersenneHalfRow}(2c-2)$\ndirectly, dropping the support-extension data. The consumer form actually chained into\n257bm:c1/c5 when the strict-upper separation is not needed downstream.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{binary-digit} \\lean{exactLocalMersenneHalfRow\\_two\\_mul\\_sub\\_two\\_of\\_skippedCoreSharpCapacity}{BooleanMobiusSkipRow.lean:380}\n\\end{thm}\n\n\\begin{prop}[257bm:c9 --- \\texttt{localMersenneQuotient\\_eq\\_two\\_pow\\_sub\\_of\\_half\\_lt}]\nIn the pure upper window used by 257bm:c7's fill, the Mersenne quotient is exactly\n$\\mathrm{localMersenneQuotient}(M,d) = 2^{M-d}$ once the value condition places $d$ strictly\nabove half the depth. This one identity is what lets the fill encode the missing residue as a\nliteral Boolean word rather than a recursive repair \u2014 the arithmetic reason 257bm:c7 is\npossible at all.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{binary-digit} \\lean{localMersenneQuotient\\_eq\\_two\\_pow\\_sub\\_of\\_half\\_lt}{BooleanMobiusSkipRow.lean:72}\n\\end{prop}\n\n\\begin{thm}[257bm:c10 --- \\texttt{exactLocalMersenneHalfRow\\_two\\_mul\\_sub\\_two\\_of\\_skippedCore}]\nCompanion unconditional consumer at \\texttt{BooleanMobiusSkipRow.lean}: given a first-crossing\nwitness produced by \\texttt{exists\\_skippedCoreExactRow\\_of\\_value\\_above} /\n\\texttt{exactLocalMersenneHalfRow\\_of\\_first\\_localMersenne\\_crossing}\n(\\texttt{BooleanMobiusExactRowCrossing.lean:155,\\,191}), builds the exact row at $2c-2$ from the\nskipped core without any capacity input. This is the base construction 257bm:c7/c8 sharpen with\nthe strict-upper separation clause.\n\\par\\noindent\\ev{Lean} \\scale{bounded} \\coord{binary-digit} \\lean{exactLocalMersenneHalfRow\\_two\\_mul\\_sub\\_two\\_of\\_skippedCore}{BooleanMobiusSkipRow.lean:197}\n\\end{thm}\n\n\\subsubsection*{The dyadic-band quantifier-collapse consumer (\\texttt{HalfUpperResetCriticalBand})}\n\n\\begin{thm}[257bm:c11 --- \\texttt{DyadicBandEscape} $\\Leftrightarrow$ \\texttt{CriticalDyadicBandIndex}, PROBLEM-AGNOSTIC quantifier collapse]\nPure $\\N$-arithmetic, zero seam or Mersenne content: for $d,E,j$ with $\\mathrm{CriticalDyadicBandIndex}(d,E,j)$ the unique nearest dyadic power $2^{d-j+1}\\ge E$,\n\\[\n\\mathrm{DyadicBandEscape}(d,E) \\;\\Longleftrightarrow\\; \\exists j,\\ \\mathrm{CriticalDyadicBandIndex}(d,E,j) \\wedge E + 2(d+j) \\le 2^{d-j+1}.\n\\]\nThe $\\forall j\\in[0,d]$ band-avoidance condition ($d+1$ separate inequalities) collapses to\nchecking exactly ONE nearest-boundary index per row. Specialised to the concrete seam reset\ncharge, $\\mathrm{SeamUpperResetCriticalBandEscape}$ is proved logically EQUIVALENT to\n$\\mathrm{SeamUpperResetDyadicBandEscape}$ (the F5-style band-avoidance producer), hence also\nconsumes into $\\mathrm{half\\_mem\\_mersenneAchievementSet\\_of\\_upperResetCriticalBandEscape}$.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:dyadic-boundary} \\lean{CriticalDyadicBandIndex}{HalfUpperResetCriticalBand.lean:33} \\lean{dyadicBandEscape\\_iff\\_exists\\_critical}{HalfUpperResetCriticalBand.lean:153} \\lean{seamUpperResetCriticalBandEscape\\_iff}{HalfUpperResetCriticalBand.lean:901} \\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_upperResetCriticalBandEscape}{HalfUpperResetCriticalBand.lean:901}\n\\end{thm}\n\n\\begin{prop}[257bm:c12 --- prerequisite \\texttt{seamUpperResetCharge\\_le}]\nThe reduction 257bm:c11 needs $E\\le2^{d+1}$ as a side condition; this is supplied automatically\nfor the concrete seam reset charge by the two-sided dyadic bound of \\texttt{HalfCylinderMiddleCarryLowerBound.lean} (the $\\min(\\mathrm{remainder},\\mathrm{overshoot})\\le2^s$ global invariant), so 257bm:c11's reduction is never blocked on this side condition in practice.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:dyadic-boundary} \\lean{seamUpperResetCharge\\_le}{HalfUpperResetCriticalBand.lean:124}\n\\end{prop}\n\n\\subsubsection*{The frozen-margin consumer family (\\texttt{HalfCylinderFullShellSeamBridge}, half-membership track)}\n\nThree graded sockets, strongest hypothesis to weakest, all funnel into the same endpoint\n$\\mathrm{half\\_mem\\_mersenneAchievementSet\\_of\\_governedFrozenMarginProducer}$.\n\n\\begin{thm}[257bm:c13 --- (i) \\texttt{HalfGreedySkippedFullShellNonnegative}]\n\\[\n\\forall\\ \\text{skipped rank } n,\\quad 0 \\le \\mathrm{greedyHalfFrozenMargin}(n-1,n).\n\\]\nDirect, seam-free. The plainest sufficient socket in the family.\n\\par\\noindent\\ev{Open} \\scale{uniform} \\coord{other:frozen-margin} \\lean{HalfGreedySkippedFullShellNonnegative}{HalfCylinderFullShellSeamBridge.lean:593} \\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_skippedFullShellNonnegative}{HalfCylinderFullShellSeamBridge.lean:604}\n\\end{thm}\n\n\\begin{thm}[257bm:c14 --- (ii) \\texttt{HalfGreedySkippedSeamAlignmentZero}]\n\\[\n\\forall\\ \\text{skipped rank whose actual word equals the seam-greedy word},\\quad \\text{seam remainder} = 0,\n\\]\nproved EQUIVALENT to (i) via \\texttt{skippedSeamAlignmentZero\\_iff\\_skippedFullShellNonnegative}.\n\\par\\noindent\\ev{Open} \\scale{uniform} \\coord{other:frozen-margin} \\lean{HalfGreedySkippedSeamAlignmentZero}{HalfCylinderFullShellSeamBridge.lean:633} \\lean{skippedSeamAlignmentZero\\_iff\\_skippedFullShellNonnegative}{HalfCylinderFullShellSeamBridge.lean:633}\n\\end{thm}\n\n\\begin{thm}[257bm:c15 --- (iii) \\texttt{HalfGreedySkippedSeamEscape}]\n\\[\n\\forall\\ \\text{skipped rank},\\quad \\mathrm{halfStripBound}(2n) < \\mathrm{seamIntegerGreedyRemainder}(n),\n\\]\na strictly SUFFICIENT (not equivalent) condition for (i). The layered graded-socket pattern\n(equivalent core + one strictly-sufficient stronger variant) recurs across this whole lane as\na proof-engineering idiom.\n\\par\\noindent\\ev{Open} \\scale{uniform} \\coord{other:frozen-margin} \\lean{HalfGreedySkippedSeamEscape}{HalfCylinderFullShellSeamBridge.lean:687} \\lean{governedFrozenMarginProducer\\_of\\_skippedSeamEscape}{HalfCylinderFullShellSeamBridge.lean:687}\n\\end{thm}\n\n\\subsubsection*{Two square-root-scale consumers (half-membership track)}\n\n\\begin{thm}[257rig:c16 --- \\texttt{infinite\\_support\\_half\\_of\\_mobiusCenteredHalfCarry\\_sqrtBound}, \\texttt{greedy\\_half\\_infinite\\_of\\_mobiusCenteredHalfCarry\\_sqrtBound}]\nIf\n\\[\n\\forall N,\\quad \\mathrm{mobiusCenteredHalfCarry}(\\mathrm{greedySupport}(1/2),N) \\;\\le\\; 2\\sqrt N + 4,\n\\]\nthen the greedy skipped support is infinite and $\\mathrm{erdosSupportSeries}(2,\\mathrm{greedyMersenneSupport}(1/2)) = 1/2$. Nonnegativity of the left side is unconditional (see 257bm:i-carry below); ONLY the upper $2\\sqrt N+4$ bound remains open. Distinct from 257bm:c11's dyadic-band route and from 257bm:c7's largest-skip socket \u2014 the corpus records this as a genuinely independent square-root-scale frontier.\n\\par\\noindent\\ev{Open} \\scale{cofinal} \\coord{other:sqrt-carry-growth} \\lean{infinite\\_support\\_half\\_of\\_mobiusCenteredHalfCarry\\_sqrtBound}{HalfCarryReachability.lean:788} \\lean{greedy\\_half\\_infinite\\_of\\_mobiusCenteredHalfCarry\\_sqrtBound}{HalfCarryReachability.lean:805}\n\\end{thm}\n\n\\begin{thm}[257rig:c17 --- \\texttt{exists\\_infinite\\_support\\_half\\_of\\_cofinalTerminalOnlyStrip}]\n\\[\n\\forall N\\ \\exists\\ \\text{depth } M\\ge N,\\ \\exists\\ \\text{normalized word } a,\\quad |\\mathrm{integerHalfCarry}(\\mathrm{wordSupport}(a),M-1)| \\le \\mathrm{halfStripBound}(M) \\;\\Longrightarrow\\; \\exists A\\ \\text{infinite},\\ \\mathrm{erdosSupportSeries}(2,A)=1/2.\n\\]\nSame architecture as 257bm:d4: cofinal, no coherence, mutually incompatible witnesses allowed.\nTrades off against 257bm:c7 in the OPPOSITE direction \u2014 it tolerates any support inside the\ndepth-$M$ window but demands the carry inside a $\\sim2\\sqrt M$ strip, whereas 257bm:c7 tolerates\na carry up to $2^{c-2}$ (exponentially looser) but demands the support be confined to ranks\n$<c$. Neither socket dominates the other; this has never been compared to 257bm:c7 on band width\nin any prior bank.\n\\par\\noindent\\ev{Open} \\scale{cofinal} \\coord{other:sqrt-carry-growth} \\lean{HalfCarryCofinalTerminalOnlyStrip}{TerminalOnlyCofinal.lean:34} \\lean{exists\\_infinite\\_support\\_half\\_of\\_cofinalTerminalOnlyStrip}{TerminalOnlyCofinal.lean:192}\n\\end{thm}\n\n\\begin{thm}[257rig:c18 --- \\texttt{greedy\\_half\\_infinite\\_of\\_cofinalStripReturn}]\nCofinal RETURNS of the actual greedy carry to the square-root strip give an infinite support\nwith value exactly $1/2$, an alternate coordinate to 257rig:c16's pointwise bound. The exact\nbridge between the two is already landed: \\texttt{halfGreedy\\_precriticalSuffix\\_lt\\_iff\\_futureSkipCoverage} shows sharp capacity $\\Leftrightarrow$\n$\\mathrm{mobiusCenteredHalfCarry}(G,2c-4) \\le \\mathrm{futureSkipCapacity}(G,c,c-3)$, so transport\nbetween the frozen-prefix carry (needed by 257bm:c7's fill) and the live-orbit carry (bounded\nhere) costs exactly the future-skip capacity term.\n\\par\\noindent\\ev{Open} \\scale{cofinal} \\coord{other:sqrt-carry-growth} \\lean{GreedyHalfCarryCofinalStripReturn}{CofinalStripReturn.lean:76} \\lean{greedy\\_half\\_infinite\\_of\\_cofinalStripReturn}{CofinalStripReturn.lean:157} \\lean{halfGreedy\\_precriticalSuffix\\_lt\\_iff\\_futureSkipCoverage}{BooleanMobiusCriticalCapacityCofinal.lean:867}\n\\end{thm}\n\n\\subsubsection*{The non-effective horizon consumer}\n\n\\begin{thm}[257bm:c19 --- \\texttt{exists\\_greedyHalfFrozenMargin\\_nonneg\\_iff\\_excess\\_neg}]\nAt every dyadically safe rank $k>0$, the frozen margin $\\mathrm{greedyHalfFrozenMargin}(k,J)$ is\nnonnegative for SOME horizon $J$ (monotone up in $J$). The proof is a bare limit argument\n(\\texttt{tendsto\\_finiteCoeffWindow\\_atTop} then \\texttt{(tendsto\\_order.1 hlim).1 ...\\allowbreak.exists}), so $J$ is produced non-effectively \u2014 the exact-row route needs the specific horizon $J=c-3$ (consumed at \\texttt{halfGreedy\\_precriticalSuffix\\_lt\\_iff\\_frozenMargin\\_nonneg},\n\\texttt{BooleanMobiusCriticalCapacityCofinal.lean:784}). Everything to make it effective is\nalready on disk: $\\mathrm{binaryCoeffTail\\_eq\\_finiteCoeffWindow\\_add\\_shiftedTail}$ gives\n$\\mathrm{gap} = \\mathrm{tail}(k+1+J)/2^J$, and 257bm:i2 (the tail growth bound) gives\n$\\mathrm{tail}(m)\\le m+2$. What would close it: $\\mathrm{margin}(k,J)\\ge0$ as soon as\n$(k+J+3)/2^J < |\\mathrm{halfGreedyNextDyadicExcessNumerator}(k)|/\\mathrm{halfGreedyPrefixDenominator}(k)$\n\u2014 pure bookkeeping over existing lemmas, no new idea.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:frozen-margin} \\lean{exists\\_greedyHalfFrozenMargin\\_nonneg\\_of\\_excess\\_neg}{HalfCylinderFiniteShadow.lean:1180} \\lean{exists\\_greedyHalfFrozenMargin\\_nonneg\\_iff\\_excess\\_neg}{HalfCylinderFiniteShadow.lean:1211}\n\\end{thm}\n\n\\subsubsection*{The equivalence \\texttt{CofinalPositiveHalfGreedySkips} reduces to}\n\n\\begin{thm}[257bm:c20 --- \\texttt{half\\_mem\\_mersenneAchievementSet\\_iff\\_greedySkippedSupport\\_infinite} (A5/A6, public)]\n\\[\n(1/2:\\mathbb{R})\\in\\mathrm{mersenneAchievementSet} \\;\\Longleftrightarrow\\; (\\mathrm{greedyMersenneSkippedSupport}(1/2)).\\mathrm{Infinite}.\n\\]\nThe positivity conjunct of 257bm:c2's hypothesis ($0<\\mathrm{greedyMersenneRemainderRat}(1/2,c-1)$)\nis unconditionally dischargeable (odd-denominator parity, 257bm:i9-style), so\n$\\mathrm{CofinalPositiveHalfGreedySkips} \\Leftrightarrow (\\mathrm{greedyMersenneSkippedSupport}(1/2)).\\mathrm{Infinite}$,\nwhich this theorem proves equivalent to $1/2\\in\\mathrm{mersenneAchievementSet}$ directly \u2014 the\nconsumer 257bm:c2 chains into is, once unwound, this exact equivalence. See 257bm:k5 for why\nthis makes 257bm:c2 a trap, not a shortcut.\n\\par\\noindent\\ev{Lean} \\scale{cofinal} \\coord{other:topological-achievement-set} \\lean{half\\_mem\\_mersenneAchievementSet\\_iff\\_greedySkippedSupport\\_infinite}{GreedyAchievementSet.lean:2514} \\lean{mem\\_mersenneAchievementSet\\_of\\_greedySkippedSupport\\_infinite}{GreedyAchievementSet.lean:1515}\n\\end{thm}\n\n\\subsection{Invariants}\n\nInvariants are unconditional identities, monotonicity facts, and quantitative bounds that hold\nregardless of which branch of the lane's case splits fires. They are the load-bearing algebra\nevery producer and consumer above is built from.\n\n\\subsubsection*{Endpoint transition algebra (\\texttt{BooleanMobiusExactTransition})}\n\n\\begin{thm}[257bm:i1a --- \\texttt{localMersenneQuotient\\_endpoint\\_succ}]\nExact one-step binary transition formula for $\\mathrm{localMersenneQuotient}$ from endpoint $n$\nto $n+1$: each rank's quotient contribution either stays or doubles-plus-carries depending on\ndivisibility. Unconditional identity, no hypothesis beyond the endpoint arithmetic itself.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{binary-digit} \\lean{localMersenneQuotient\\_endpoint\\_succ}{BooleanMobiusExactTransition.lean:29}\n\\end{thm}\n\n\\begin{thm}[257bm:i1b --- \\texttt{localPrefixQuotient\\_succ}]\nThe rank-wise identity of 257bm:i1a summed over a fixed support $D$: exact recurrence for\n$\\mathrm{localPrefixQuotient}(D,n)\\to\\mathrm{localPrefixQuotient}(D,n+1)$.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{binary-digit} \\lean{localPrefixQuotient\\_succ}{BooleanMobiusExactTransition.lean:131}\n\\end{thm}\n\n\\begin{thm}[257bm:i1c --- \\texttt{localEndpointDefect\\_succ}, the $H=2A+1-S$ recurrence]\nThe signed defect version gives the literal transition $H = 2A+1-S$ referenced throughout the\nmodule docstrings, where $A$ is the suffix carry and $S=\\mathrm{endpointDivisorContribution}$\n(how many selected ranks divide the new endpoint). This is the SAME recurrence shape as the\ngeneric tempered-orbit recurrence $u(N+1)=2u(N)-v\\cdot c(N+1)$ from the public\n\\texttt{GenericTailOrbitRigidity.lean} T7 criterion (257bm:i-t7 below), specialised to Mersenne\nlocal repair. Whether $\\mathrm{localRepairInteger}$ literally instantiates a tempered orbit for\n$c:=\\mathrm{supportCoeff}\\,A$ is UNVERIFIED but high-value to check: it would let the T7\ncriterion feed rationality of $\\mathrm{erdosSupportSeries}\\,2\\,A$ directly, a different route\ninto the achievement-set argument than the cofinal-exact-row route of this catalogue.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{binary-digit} \\lean{localEndpointDefect\\_succ}{BooleanMobiusExactTransition.lean:189} \\lean{localRepairInteger\\_eq\\_localEndpointDefect\\_succ}{BooleanMobiusExactTransition.lean:203}\n\\end{thm}\n\n\\subsubsection*{Capacity identities (\\texttt{BooleanMobiusSkippedCoreCriticalCapacity}, \\texttt{BooleanMobiusSkippedCoreExactRow})}\n\n\\begin{defn}[257bm:i-cap --- the sharp capacity test]\nFor $c\\ge4$, $\\forall d\\in D,\\ 2\\le d<c$, $\\mathrm{value}(D)<1/2$:\n\\[\n\\mathrm{localBinarySuffix}(D,1,2c-2) < 2^{c-2}\n\\]\nis the sharp skipped-core capacity \u2014 the residual carry after inserting rank $c$ fits inside\nexactly $c-2$ bits, making a below-half core repairable into an exact row at endpoint $2c-2$.\n\\par\\noindent\\ev{Lean} \\scale{n/a} \\coord{binary-digit} \\lean{localBinarySuffix\\_two\\_mul\\_sub\\_two\\_lt\\_criticalCapacity\\_iff}{BooleanMobiusSkippedCoreCriticalCapacity.lean:22}\n\\end{defn}\n\n\\begin{thm}[257bm:i5 --- \\texttt{localBinarySuffix\\_two\\_mul\\_sub\\_two\\_lt\\_criticalCapacity\\_iff}]\nUnder the hypotheses of 257bm:i-cap:\n\\[\n\\mathrm{localBinarySuffix}(D,1,2c-2) < 2^{c-2} \\;\\Longleftrightarrow\\; 2^{(2c-2)-1} \\le \\mathrm{localPrefixQuotient}(\\mathrm{insert}\\ c\\ D,\\ 2c-2).\n\\]\nThe sharp capacity bound is equivalent to \"adjoining rank $c$ already reaches the integer\nhalf-target\" \u2014 capacity and integral-crossing are the same fact viewed two ways. Rewrites the\nanalytic capacity question into a purely combinatorial (integer-quotient) one.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{binary-digit} \\lean{localBinarySuffix\\_two\\_mul\\_sub\\_two\\_lt\\_criticalCapacity\\_iff}{BooleanMobiusSkippedCoreCriticalCapacity.lean:22}\n\\end{thm}\n\n\\begin{thm}[257bm:i6 --- \\texttt{localBinarySuffix\\_two\\_mul\\_sub\\_two\\_lt\\_upperHalfCapacity}, loose (one-bit) capacity, \\texttt{BooleanMobiusSkippedCoreExactRow}]\nUnconditionally, for every $c\\ge4$ and every below-half core $D\\subseteq[2,c)$ with deficit\n$<\\mathrm{mersenneWeightRat}(c)$:\n\\[\n\\mathrm{localBinarySuffix}(D,1,2c-2) < 2^{c-1}.\n\\]\nUniform in $c$, no table, no case split \u2014 but ONE BIT looser than the sharp 257bm:i5 test the\nfill (257bm:c7) actually needs. Reading the proof, the loss is purely additive, not\nmultiplicative: it derives $A<2^{c-2}+|D|$ and then discards $|D|\\le c-2\\le2^{c-2}$. The true\nunproved statement is only that $A$ avoids the LINEAR-WIDTH band\n$[2^{c-2},\\,2^{c-2}+c-3]$ \u2014 width $c-2$ inside a range of size $2^{c-2}$. This is the same band\nshape as 257bm:c11's dyadic-band route (F5/F6), and EXPONENTIALLY weaker than the\n$\\sqrt{}$-scale target of 257rig:c16 (width $c$ versus width $2^{(r+5)/2}$ on the unconditional\ntrack) \u2014 the two have never been directly compared on band width elsewhere in this corpus.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{binary-digit} \\lean{localBinarySuffix\\_two\\_mul\\_sub\\_two\\_lt\\_upperHalfCapacity}{BooleanMobiusSkippedCoreExactRow.lean:123} \\lean{localBinarySuffix\\_two\\_mul\\_sub\\_two\\_lt\\_upperHalfCapacity\\_word}{BooleanMobiusSkippedCoreExactRow.lean:228}\n\\end{thm}\n\n\\subsubsection*{Doubling-branch invariants (\\texttt{BooleanMobiusExactRowRankTwo})}\n\n\\begin{thm}[257bm:i7 --- \\texttt{localBinarySuffix\\_two\\_mul\\_sub\\_one\\_lt\\_upperWindow\\_of\\_exact\\_below}]\nFor $n\\ge6$, $D$ bounded $[2,n]$, $2\\in D$, exact quotient at $n$, value $<1/2$:\n\\[\n\\mathrm{localBinarySuffix}(D,1,2n-1) < 2^n.\n\\]\nThe missing quotient at $2n-1$ fits the pure upper window $\\{n+1,\\dots,2n-1\\}$ \u2014 enough carry\nslack to encode the missing digits as a literal Boolean word with no recursion into already-used\nranks. Uses \\texttt{one\\_third\\_le\\_localMersenneFraction\\_two} (rank-2 residue always $\\ge1/3$)\nand \\texttt{three\\_mul\\_sub\\_two\\_lt\\_two\\_pow\\_pred} ($3(n-2)<2^{n-1}$, the elementary slack\nestimate).\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{binary-digit} \\lean{localBinarySuffix\\_two\\_mul\\_sub\\_one\\_lt\\_upperWindow\\_of\\_exact\\_below}{BooleanMobiusExactRowRankTwo.lean:44}\n\\end{thm}\n\n\\begin{prop}[257bm:i-rank2 --- \\texttt{two\\_mem\\_of\\_exact\\_localMersenneQuotient}]\nThe side condition $2\\in D$ used throughout the doubling branch is FREE: it holds uniformly for\n$n\\ge3$ at any exact quotient, so it never needs separate discharge downstream.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{binary-digit} \\lean{two\\_mem\\_of\\_exact\\_localMersenneQuotient}{BooleanMobiusExactRowRankTwo.lean:23}\n\\end{prop}\n\n\\subsubsection*{Base-case fixture (\\texttt{BooleanMobiusExactRowSeed})}\n\n\\begin{defn}[257bm:i-seed --- \\texttt{exactRowSixSupport}, the smallest exact row]\n$\\mathrm{exactRowSixSupport} := \\{2,3,6\\}$ satisfies\n\\[\n\\mathrm{localPrefixQuotient}(\\{2,3,6\\},6) = 2^5-1 = 31,\\qquad \\mathrm{localMersennePrefixValue}(\\{2,3,6\\}) < 1/2,\n\\]\nhence $\\mathrm{ExactLocalMersenneHalfRow}(6)$. Closed numeric fixture, \\texttt{norm\\_num}-checked.\nEvery $6\\le n$ hypothesis in the doubling/dichotomy files traces back to this witness being the\nfirst valid seed.\n\\par\\noindent\\ev{Cert} \\scale{fixed} \\coord{mobius-mersenne} \\lean{exactRowSixSupport}{BooleanMobiusExactRowSeed.lean:14} \\lean{exactLocalMersenneHalfRow\\_six}{BooleanMobiusExactRowSeed.lean:34}\n\\end{defn}\n\n\\subsubsection*{Global-repair invariants (\\texttt{BooleanMobiusGlobalRepair})}\n\n\\begin{prop}[257bm:i9 --- \\texttt{abs\\_localMersennePrefixValue\\_sub\\_half\\_le}]\nAn exact row at endpoint $n$ has real value within $O(n/2^n)$ of $1/2$: the quantitative bound\n$|y_n-1/2|\\le(n+1)/2^n$ consumed by 257bm:c1 to turn a sequence of exact rows tending to depth\ninfinity into a sequence of values tending to $1/2$.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{mobius-mersenne} \\lean{abs\\_localMersennePrefixValue\\_sub\\_half\\_le}{BooleanMobiusGlobalRepair.lean:243}\n\\end{prop}\n\n\\begin{prop}[257bm:i10 --- \\texttt{finite\\_boolSupport\\_ne\\_half}]\nCites the odd-denominator fact (\\texttt{finiteErdosSum\\_den\\_odd}, 257bm:k11) to conclude directly: no finite Boolean\nsupport ever equals $1/2$ exactly. The corollary consumed at the very end of 257bm:c1's chain to\ncertify the achieved witness is infinite.\n\\par\\noindent\\ev{Cited} \\scale{uniform} \\coord{p-adic} \\lean{finite\\_boolSupport\\_ne\\_half}{BooleanMobiusGlobalRepair.lean:478}\n\\end{prop}\n\n\\subsubsection*{Greedy-window uniqueness (\\texttt{BooleanMobiusGreedyReduction})}\n\n\\begin{thm}[257bm:i11a --- \\texttt{remainder\\_lt\\_gap\\_iff\\_eq\\_integerGreedyBits}]\nUnder gap dominance (the next digit's weight strictly exceeds the sum of all lower digits'\nslack), a Boolean word matches the unique greedy-bit word IFF its remainder is below the\nseparation gap. Whenever the separation gap dominates there is exactly ONE admissible Boolean\nword achieving a given target defect \u2014 the greedy word. The \"superincreasing weight sequence $+$\nbelow-gap remainder $\\Rightarrow$ unique greedy word\" technique is standard and fully general,\nindependent of the Mersenne instantiation.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{greedy-orbit} \\lean{remainder\\_lt\\_gap\\_iff\\_eq\\_integerGreedyBits}{BooleanMobiusGreedyReduction.lean:373}\n\\end{thm}\n\n\\begin{prop}[257bm:i11b --- gap-dominance facts]\n\\texttt{localMersenneWeightsFrom\\_gapDominates} and its even/odd endpoint specializations\n\\texttt{\\_gapDominates\\_even} / \\texttt{\\_gapDominates\\_odd}, established from\n\\texttt{three\\_mul\\_sub\\_two\\_lt\\_two\\_pow\\_pred}-style elementary slack bounds \u2014 the concrete\nhypotheses 257bm:i11a needs, discharged unconditionally at every relevant endpoint parity.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{greedy-orbit} \\lean{localMersenneWeightsFrom\\_gapDominates}{BooleanMobiusGreedyReduction.lean:280} \\lean{localMersenneWeightsFrom\\_gapDominates\\_even}{BooleanMobiusGreedyReduction.lean:301} \\lean{localMersenneWeightsFrom\\_gapDominates\\_odd}{BooleanMobiusGreedyReduction.lean:310}\n\\end{prop}\n\n\\begin{prop}[257bm:i11c --- \\texttt{localMersenneHalfTarget\\_lower\\_word\\_eq\\_greedy\\_and\\_remainder\\_eq}]\nConsumer form of 257bm:i11a specialised to the half-target: the lower word of any exact\nhalf-target construction is forced to equal the greedy word, remainder identity included. This\nis the general uniqueness principle instantiated by 257rig:i3 (the wall).\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{greedy-orbit} \\lean{localMersenneHalfTarget\\_lower\\_word\\_eq\\_greedy\\_and\\_remainder\\_eq}{BooleanMobiusGreedyReduction.lean:452}\n\\end{prop}\n\n\\subsubsection*{Geometric normal form (\\texttt{BooleanMobiusCriticalCapacityGeometric})}\n\n\\begin{thm}[257bm:i12 --- division-free re-coordinatisation of the sharp capacity iff]\n$\\mathrm{localMersenneQuotient\\_eq\\_geometric}$: Euclidean division $2^M/(2^d-1)$ equals a\nshifted finite geometric sum (division-free \"descending multiples\" normal form); the iff-forms\n\\texttt{localBinarySuffix\\_two\\_mul\\_sub\\_two\\_lt\\_criticalCapacity\\_iff\\_geometric} and\n\\texttt{\\_iff\\_geometricCore} restate 257bm:i5 entirely in this coordinate, with the \"crossing\ncoin\" contribution $\\mathrm{localMersenneGeometricQuotient\\_critical\\_self}$ visible as an\nexplicit finite sum rather than a floor division. Same content as 257bm:i5, alternate\ncoordinate; the division-free re-expression technique for $N/(b^d-1)$-type quotients is fully\ngeneral to any base $b$.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:geometric-series} \\lean{localMersenneQuotient\\_eq\\_geometric}{BooleanMobiusCriticalCapacityGeometric.lean:27} \\lean{localBinarySuffix\\_two\\_mul\\_sub\\_two\\_lt\\_criticalCapacity\\_iff\\_geometric}{BooleanMobiusCriticalCapacityGeometric.lean:196} \\lean{localBinarySuffix\\_two\\_mul\\_sub\\_two\\_lt\\_criticalCapacity\\_iff\\_geometricCore}{BooleanMobiusCriticalCapacityGeometric.lean:208}\n\\end{thm}\n\n\\subsubsection*{Growth-bound invariants for the generic layer}\n\n\\begin{thm}[257bm:i2 --- \\texttt{binaryCoeffTail\\_le}, public]\nFor $c:\\N\\to\\N$ with $\\forall n,\\ c\\,n\\le n$: $\\mathrm{binaryCoeffTail}(c,N)\\le N+2$ for all $N$ \u2014\nANY linearly-bounded natural coefficient sequence has scaled tail $O(N)$, hence $o(2^N)$. Makes\nthe tempered-orbit criterion T7 (257bm:i-t7) non-vacuous, and confirms $c:=\\varphi$ is safely\ninside its domain, since $\\varphi(n)\\le n$ is standard.\n\\par\\noindent\\ev{Cited} \\scale{uniform} \\coord{binary-digit} \\lean{binaryCoeffTail\\_le}{GenericTailOrbitRigidity.lean:85}\n\\end{thm}\n\n\\begin{thm}[257bm:i-t7 --- \\texttt{binaryCoeffSeries\\_rational\\_iff\\_exists\\_temperedBinaryOrbit} (T7), public, THE cross-problem bridge]\nFor $c:\\N\\to\\N$ with $\\forall n,\\ c\\,n\\le n$:\n\\[\n\\mathrm{HasRationalValue}(\\mathrm{binaryCoeffSeries}\\,c) \\;\\Longleftrightarrow\\; \\exists v>0,\\ \\exists u:\\N\\to\\Z,\\ \\mathrm{IsTemperedBinaryOrbit}(c,v,u),\n\\]\nwhere $\\mathrm{binaryCoeffSeries}\\,c := \\sum'_n c(n{+}1)/2^{n+1}$ and $\\mathrm{IsTemperedBinaryOrbit}(c,v,u) := (\\forall N,\\ u(N{+}1)=2u(N)-v\\cdot c(N{+}1)) \\wedge u(N)/2^N\\to0$. ONLY the linear\ngrowth bound $c\\,n\\le n$ is required \u2014 no divisor, Mersenne, or \\#257-specific structure at all.\nFully problem-agnostic; instantiating $c:=\\varphi$ gives a candidate \\#249 reduction to the SAME\ncriterion used for \\#257, modulo an index-shift caveat that remains unverified.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{binary-digit} \\lean{IsTemperedBinaryOrbit}{GenericTailOrbitRigidity.lean:67} \\lean{binaryCoeffSeries}{GenericTailOrbitRigidity.lean:37} \\lean{binaryCoeffSeries\\_rational\\_iff\\_exists\\_temperedBinaryOrbit}{GenericTailOrbitRigidity.lean:426}\n\\end{thm}\n\n\\begin{thm}[257bm:i-mob --- \\texttt{moebius\\_mul\\_supportCoeffAF}, public, M\\\"obius layer]\n$\\mathrm{ArithmeticFunction.moebius} * \\mathrm{supportCoeffAF}(A) = \\mathrm{positiveSupportBitAF}(A)$\nfor ANY $A:\\mathrm{Set}\\,\\N$ \u2014 exact M\\\"obius recovery of a 0/1 support indicator from its\ndivisor-count coefficient. Fully base-free; only downstream specialisations\n($\\mathrm{erdosSupportSeries}\\,2\\,A$) fix the base to 2.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:mobius-inversion} \\lean{moebius\\_mul\\_supportCoeffAF}{BooleanMobiusCarry.lean:94} \\lean{mobius\\_supportCoeff\\_boolean}{BooleanMobiusCarry.lean:117} \\lean{card\\_divisors\\_le\\_two\\_mul\\_sqrt}{BooleanMobiusCarry.lean:208}\n\\end{thm}\n\n\\begin{thm}[257bm:i-bridge --- \\texttt{erdosSupportSeries\\_two\\_eq\\_binaryCoeffSeries}, public]\n$\\mathrm{erdosSupportSeries}(2,A) = \\mathrm{binaryCoeffSeries}(\\mathrm{supportCoeff}\\,A)$ \u2014\nidentifies the \\#257 support series at base 2 exactly with T7's generic machinery, closing the\nloop that makes T7 the actual engine behind \\#257's rationality criterion too.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{mobius-mersenne} \\lean{erdosSupportSeries\\_two\\_eq\\_binaryCoeffSeries}{BooleanMobiusCarry.lean:376} \\lean{erdosSupportSeries\\_rational\\_iff\\_exists\\_temperedCarry}{BooleanMobiusCarry.lean:383}\n\\end{thm}\n\n\\subsubsection*{Rigidity corollaries directly constraining a rational \\#257 support}\n\nThese five invariants (\\texttt{RationalSupportCarrySkeleton.lean}, \\texttt{SublogDivisorCoverage.lean}, \\texttt{MaximalOmegaLayer.lean}) are the corollaries the task names explicitly. G1 is cross-referenced, not duplicated: it is 257bm:i10 above.\n\n\\begin{thm}[257rig:i2 --- \\texttt{dyadic\\_support\\_fraction\\_reciprocalMass\\_diverges\\_or\\_gt\\_one}]\nIf an INFINITE support $A$ has $\\mathrm{erdosSupportSeries}(2,A)$ equal to a dyadic rational\n$p/2^c$ (any $c$ \u2014 the easiest terminating case to rule out), then\n\\[\n\\neg\\mathrm{Summable}(1/a\\ \\text{on}\\ A) \\;\\vee\\; 1 < \\mathrm{reciprocalMass}(A).\n\\]\nProved by a Ces\\`aro-mean identification of the shifted-tail-state average with the reciprocal\nmass, plus an explicit two-spike construction at $\\mathrm{lcm}(a,b)$ for any two distinct\npositive support elements. Directly excludes any candidate infinite support with SMALL\n(convergent, $\\le1$) reciprocal mass \u2014 e.g. any support sparse enough for the lcm-gap engine's\nirrationality instances \u2014 from also being a dyadic-rational counterexample.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:cesaro-tail} \\lean{dyadic\\_support\\_fraction\\_reciprocalMass\\_diverges\\_or\\_gt\\_one}{RationalSupportCarrySkeleton.lean:2200} \\lean{one\\_lt\\_reciprocalMass\\_of\\_dyadic\\_support\\_fraction\\_of\\_two\\_pos\\_mem}{RationalSupportCarrySkeleton.lean:2116}\n\\end{thm}\n\n\\begin{thm}[257rig:i3 --- \\texttt{shifted\\_state\\_unbounded\\_of\\_infinite\\_support} (T13)]\nFor ANY infinite support $A$, any $v>0$, any positive-valued $u$ satisfying\n$u(n{+}1)+v\\cdot\\mathrm{supportCoeff}\\,A\\,(c{+}n{+}1)=2u(n)$: $u$ is UNBOUNDED,\n$\\forall B\\ \\exists n,\\ B<u(n)$. Contrast with T7 (257bm:i-t7): T7's orbit is additionally\nrequired TEMPERED ($o(2^n)$) \u2014 boundedness and temperedness are different conditions; an\nunbounded-but-tempered orbit is possible, and is exactly what T7 constructs when the series IS\nrational. The proof selects $2B+1$ positive support elements and uses their product as an\nexplicit common multiple forcing $\\mathrm{supportCoeff}$ mass $\\ge2B+1$ there \u2014 a technique\nfully portable to any Dirichlet-incidence-style coefficient forced by an infinite index set.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:common-multiple-forcing} \\lean{shifted\\_state\\_unbounded\\_of\\_infinite\\_support}{RationalSupportCarrySkeleton.lean:2320} \\lean{exists\\_unbounded\\_shifted\\_odd\\_tail\\_nat\\_state\\_of\\_support\\_fraction}{RationalSupportCarrySkeleton.lean:2377} \\lean{one\\_add\\_mul\\_card\\_le\\_two\\_mul\\_shifted\\_state}{RationalSupportCarrySkeleton.lean:2228}\n\\end{thm}\n\n\\begin{thm}[257rig:i4a --- \\texttt{supportCoeffZeroWindow\\_length\\_le\\_eps\\_logb} (T11)]\nIf an infinite-support series (with a positive element) equals a rational $p/(2^c\\cdot v)$\n($v$ odd): any run of $h$ consecutive zero support-coefficients starting right after index $N$\nis bounded, $\\forall\\varepsilon>0\\ \\exists B,\\ \\forall N{\\ge}1\\ \\forall h,\\\n\\mathrm{SupportCoeffZeroWindow}(A,c{+}N,h) \\to h\\le\\varepsilon\\log_2 N+B$ \u2014 a rational-valued\nsupport series cannot have super-logarithmic gaps in its Dirichlet-incidence coefficient. Uses\nthe elementary divisor-count bound $\\tau(n)^k\\le(k^{2^k})^k\\cdot n$ (257rig:i4b) feeding a\nsupport-tail envelope combined with the doubling-tail recurrence.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:divisor-envelope} \\lean{supportCoeffZeroWindow\\_length\\_le\\_eps\\_logb}{SublogDivisorCoverage.lean:435}\n\\end{thm}\n\n\\begin{prop}[257rig:i4b --- \\texttt{card\\_divisors\\_pow\\_le\\_divisorSubpowerConst\\_pow\\_mul}]\n$\\tau(n)^k \\le (k^{2^k})^k\\cdot n$ for all $n,k$ \u2014 a pure number-theory fact with zero\nsupport/problem content, directly reusable anywhere a $1/k$-power divisor-count envelope is\nneeded. Would plausibly transfer to a \\#249 argument bounding zero-runs of $\\varphi$ itself,\nthough $\\varphi$'s growth is linear not polylog, so the same zero-window CONCLUSION would not\ntransfer \u2014 only this underlying divisor-bound lemma might, via a different route.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:divisor-envelope} \\lean{card\\_divisors\\_pow\\_le\\_divisorSubpowerConst\\_pow\\_mul}{SublogDivisorCoverage.lean:107}\n\\end{prop}\n\n\\begin{prop}[257rig:i5 --- \\texttt{mixedPrimePowerLayerTwo\\_supportCoeffInt}, \\texttt{primePowerLayer\\_comm}]\nFor an integer-valued support-coefficient function and distinct primes $p\\ne q$ with exponents\n$e,f$: the iterated exact-level-difference operator $\\mathrm{primePowerLayer}$ commutes across\nthe two primes, and two nested layers extract exactly the iterated exact-valuation pullback of\nthe support, $\\mathrm{mixedPrimePowerLayerTwo}(p,e,q,f,\\mathrm{supportCoeffInt}\\,A,n) =\n\\mathrm{supportCoeffInt}(\\mathrm{pullback}_{q,f}(\\mathrm{pullback}_{p,e}A))(n)$ for $n$ coprime\nto $pq$. Self-contained (81 lines); the module's own docstring is explicit it supplies no\ntransport from rationality through multiplicative decimation and asserts no bounded-$\\Omega$\nendpoint \u2014 genuinely just algebra, not a rigidity result on its own.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:mobius-inversion} \\lean{mixedPrimePowerLayerTwo\\_supportCoeffInt}{MaximalOmegaLayer.lean:39} \\lean{primePowerLayer\\_comm}{MaximalOmegaLayer.lean:29}\n\\end{prop}\n\n\\subsubsection*{First-crossing machinery (\\texttt{BooleanMobiusExactRowCrossing})}\n\n\\begin{defn}[257bm:i-cross --- \\texttt{localMersenneCrossingRanks}]\n\\[\n\\mathrm{localMersenneCrossingRanks}(E) \\;:=\\; E.\\mathrm{filter}\\big(c \\mapsto \\tfrac12 < \\mathrm{localMersennePrefixValue}(E.\\mathrm{filter}(\\cdot\\le c))\\big),\n\\]\nthe ranks at which $E$'s inclusive ordered prefix is already above one half.\n\\par\\noindent\\ev{Lean} \\scale{n/a} \\coord{binary-digit} \\lean{localMersenneCrossingRanks}{BooleanMobiusExactRowCrossing.lean:24}\n\\end{defn}\n\n\\begin{thm}[257bm:i-cross2 --- \\texttt{exists\\_first\\_localMersenne\\_crossing}]\nFor $E$ with $\\forall d\\in E,\\ 2\\le d$ and $1/2<\\mathrm{localMersennePrefixValue}(E)$:\n\\[\n\\exists c\\in E,\\ 4\\le c \\ \\wedge\\ \\mathrm{localMersennePrefixValue}(E.\\mathrm{filter}(\\cdot<c))<\\tfrac12 \\ \\wedge\\ \\tfrac12 < \\mathrm{localMersennePrefixValue}(\\mathrm{insert}\\ c\\ (E.\\mathrm{filter}(\\cdot<c))).\n\\]\nThere is a first rank $c\\ge4$ at which the running prefix strictly crosses one half; the crossing\nrank $c$ feeds directly into 257bm:k4/257bm:c7's skipped-core constructors. The SHAPE (a\nmonotone-ish accumulation crossing a threshold has a first crossing index) is a completely\ngeneric real-analysis fact; the specific $\\mathrm{localMersennePrefixValue}$ instantiation is\n\\#257-specific, but the same \"first crossing rank\" pattern would recur verbatim for any\naccumulating series \u2014 including a \\#249 $\\varphi$-digit accumulation \u2014 if restated with a\ndifferent weight function.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{binary-digit} \\lean{exists\\_first\\_localMersenne\\_crossing}{BooleanMobiusExactRowCrossing.lean:31}\n\\end{thm}\n\n\\subsubsection*{M\\\"obius-centred carry nonnegativity and the skipped-rank trichotomy (half-membership track)}\n\n\\begin{thm}[257hg:i6 --- \\texttt{mobiusCenteredHalfCarry\\_nonneg\\_of\\_supportSeries\\_lt\\_half} (G1)]\nFor $\\mathrm{mobiusCenteredHalfCarry}(A,N)$ (an integer, defined via $\\mathrm{integerHalfCarry}$):\nif $1\\notin A$ and $\\mathrm{erdosSupportSeries}(2,A) < 1/2$, then\n\\[\n\\forall N,\\quad 0 \\le \\mathrm{mobiusCenteredHalfCarry}(A,N),\n\\]\nvia the identity $\\mathrm{integerHalfCarry}(A,N) = 2^{N+1}(1/2-\\mathrm{seriesValue}) +\n\\mathrm{binaryCoeffTail}(\\dots)$, both terms nonnegative. Notably problem-agnostic in STATEMENT\nSHAPE: $\\mathrm{supportCoeff}\\,A$/$\\mathrm{integerHalfCarry}$ are generic divisor-incidence\nconstructs, and since \\#249 ($\\varphi/2^n$) is itself built from a M\\\"obius/divisor-sum\nstructure ($\\varphi=\\mu*\\mathrm{id}$), this carry-nonnegativity mechanism is a strong candidate\nfor direct reuse or close adaptation on the \\#249 side. The analytic half of 257bm:k12's\ncontradiction machine below; also the base case consumed by 257rig:c16's sqrt-bound route.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:mobius-centred-carry} \\lean{integerHalfCarry\\_eq\\_scaled\\_residual\\_add\\_tail}{HalfCarryReachability.lean:842} \\lean{mobiusCenteredHalfCarry\\_nonneg\\_of\\_supportSeries\\_lt\\_half}{HalfCylinderFinalMiddleCellEscape.lean:94}\n\\end{thm}\n\n\\begin{thm}[257hg:i7 --- \\texttt{halfGreedy\\_skipped\\_endpoint\\_trichotomy} (G5)]\nAt every genuinely skipped rank $s\\ge5$, the frozen margin $\\mathrm{greedyHalfFrozenMargin}(s-1,s)$\nis in exactly one of three signed cells: NEGATIVE (actual word $=$ seam-greedy word,\n$\\mathrm{seamIntegerGreedyRemainder}(s)\\ge1$, margin $=-\\mathrm{remainder}$); ZERO (actual word\n$=$ seam-greedy word, $\\mathrm{remainder}=0$); POSITIVE (actual word $=$ the seam \"above\" word,\nmargin $=\\mathrm{overshoot}$). A complete case split with exact algebraic identities for the\nmargin in each branch \u2014 the zero-cell is provably SAFE/ALLOWED; only the negative cell needs\nexcluding for 257bm:c13's socket (i) to close, and G2 (257bm:k12 below) is exactly the case\nsplit inside the negative cell that the corpus has managed to exclude so far.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:frozen-margin} \\lean{halfGreedy\\_skipped\\_endpoint\\_trichotomy}{HalfCylinderSkippedEndpointClassifier.lean:246}\n\\end{thm}\n\n\\subsubsection*{Total gap mass (\\texttt{HalfGapMass})}\n\n\\begin{defn}[257hg:i1 --- \\texttt{mersenneGap}]\n$\\mathrm{mersenneGap}(n) := \\mathrm{mersenneWeight}(n) - \\mathrm{mersenneTail}(n) > 0$, the\nper-level shortfall between a single weight and the true tail sum beyond it.\n\\par\\noindent\\ev{Lean} \\scale{n/a} \\coord{mobius-mersenne} \\lean{mersenneGap}{HalfGapMass.lean:35} \\lean{mersenneGap\\_pos}{HalfGapMass.lean:35}\n\\end{defn}\n\n\\begin{thm}[257hg:i2 --- \\texttt{mersenneGap\\_tail\\_le}, \\texttt{tendsto\\_mersenneGap\\_tail\\_zero}]\nThe TOTAL gap mass strictly beyond level $N$ is summable and explicitly bounded:\n\\[\n\\sum\\nolimits'_{k} \\mathrm{mersenneGap}(N+k+1) \\;\\le\\; \\frac29\\Big(\\frac14\\Big)^N + \\frac37\\Big(\\frac18\\Big)^N \\;\\longrightarrow\\; 0,\n\\]\nobtained purely by summing the pre-existing per-level asymptotic bound geometrically \u2014 no new\narithmetic input. \\textbf{Scope caveat, explicit in the file header:} this bounds gap MASS only.\nIt says nothing about which reals the greedy run reaches, does not certify membership or\nnon-membership of any point, and in particular says nothing about $1/2$ directly. It is a\nbudget statement: useful if a future argument needs \"total measure of all fatal gaps beyond\nlevel $N$ is small\", e.g.\\ a measure-theoretic rather than single-point argument.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:summed-gap-mass} \\lean{summable\\_mersenneGap\\_shift}{HalfGapMass.lean:127} \\lean{mersenneGap\\_tail\\_le}{HalfGapMass.lean:127} \\lean{tendsto\\_mersenneGap\\_tail\\_zero}{HalfGapMass.lean:127} \\lean{mersenneGap\\_le}{HalfGapMass.lean:35}\n\\end{thm}\n\n\\subsubsection*{The sharp fatal-gap criterion (\\texttt{HalfGreedyFatalGap})}\n\n\\begin{defn}[257hg:i3 --- \\texttt{mersenneTailLB3}, the three-channel Lambert lower bound]\n$\\mathrm{mersenneTailLB3}(k) := 1/2^k + 1/(3\\cdot4^k) + 1/(7\\cdot8^k) < \\mathrm{mersenneTail}(k)$\n\u2014 a strict rational lower bound obtained by truncating the Lambert-type series at three terms.\n\\par\\noindent\\ev{Lean} \\scale{n/a} \\coord{other:lambert-bound} \\lean{mersenneTailLB3}{HalfGreedyFatalGap.lean:52} \\lean{three\\_div\\_lt\\_mersenneTailLB3}{HalfGreedyFatalGap.lean:52} \\lean{mersenneTailLB3\\_lt\\_mersenneTail}{HalfGreedyFatalGap.lean:52}\n\\end{defn}\n\n\\begin{thm}[257hg:i4 --- \\texttt{skipSafe\\_of\\_two\\_mul\\_le\\_three\\_mul}, the sharp $2u\\le3a$ criterion]\nFor a skipped residual $\\rho=u/(2L)$ at rank $k$, writing the skip-margin as\n$2^k u + a = 2L+u$ (so $0<a\\Leftrightarrow$ skip): the DYADIC safety test is $u\\le a$, but the\nTRUE (tail-based) safety test only needs\n\\[\n2u \\le 3a.\n\\]\nOne-third tighter than the dyadic test, discharged for the ACTUAL Mersenne tail (not just the\nthree-channel lower bound) in \\texttt{skipSafe\\_actualTail\\_of\\_two\\_mul\\_le\\_three\\_mul}.\n$\\mathtt{sharp\\_of\\_dyadic}: u\\le a\\to2u\\le3a$ shows dyadic-safe $\\Rightarrow$ sharp-safe\n(containment), and $\\mathtt{sharp\\_strictly\\_stronger}$ exhibits $(u,a)=(3,2)$ as sharp-safe but\nNOT dyadic-safe \u2014 the containment is strict. This matches the \"\\,$2u\\le3a$ fatal band\nsupersedes two-thirds\\,\" / \"\\,frontier $3a\\ge2u$\\,\" language on the UNCONDITIONAL ($\\forall$\ninfinite $A$) \\#257 track recorded in project memory; whether it is the same theorem or an\nindependently-proved analogue on this half-membership track is UNVERIFIED, flagged\n\\texttt{reported\\_prior}, not re-verified equivalence.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:lambert-bound} \\lean{skipSafe\\_of\\_two\\_mul\\_le\\_three\\_mul}{HalfGreedyFatalGap.lean:52} \\lean{sharp\\_of\\_dyadic}{HalfGreedyFatalGap.lean:268} \\lean{sharp\\_strictly\\_stronger}{HalfGreedyFatalGap.lean:268} \\lean{skipSafe\\_actualTail\\_of\\_two\\_mul\\_le\\_three\\_mul}{HalfGreedyFatalGap.lean:268}\n\\end{thm}\n\n\\begin{prop}[257hg:i5 --- unconditional safety corollaries]\n$\\mathtt{unitNumerator\\_skipSafe}$: unit numerators ($u=1$) are always safe. The fatal converse\nforces $3a<2u$ ($\\mathtt{three\\_mul\\_lt\\_two\\_mul\\_of\\_fatal}$); fatality forces $u\\ge2$\nunconditionally ($\\mathtt{two\\_le\\_of\\_fatal}$) and $u\\ge3$ once $u$ is odd\n($\\mathtt{three\\_le\\_of\\_fatal\\_of\\_odd}$). Mirror statements\n$\\mathtt{unitNumerator\\_skipSafe\\_actualTail}$, $\\mathtt{three\\_mul\\_lt\\_two\\_mul\\_of\\_actualTail\\_fatal}$, $\\mathtt{three\\_le\\_of\\_actualTail\\_fatal\\_of\\_odd}$ hold for the actual (not merely\nthree-channel-bounded) Mersenne tail.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:lambert-bound} \\lean{unitNumerator\\_skipSafe}{HalfGreedyFatalGap.lean:268} \\lean{two\\_le\\_of\\_fatal}{HalfGreedyFatalGap.lean:268} \\lean{three\\_le\\_of\\_fatal\\_of\\_odd}{HalfGreedyFatalGap.lean:268} \\lean{unitNumerator\\_skipSafe\\_actualTail}{HalfGreedyFatalGap.lean:268}\n\\end{prop}\n\n\\subsubsection*{Two-sided dyadic-scale control (\\texttt{HalfCylinderMiddleCarryLowerBound}, feeds \\texttt{HalfUpperResetCriticalBand})}\n\n\\begin{thm}[257bm:i13 --- \\texttt{SeamTwoSidedDyadicCellEscape.twoSided}]\nGiven a minimal local socket excluding exactly three middle \"cells\" plus one right-pulse-leak\nbound, induction (base case at row 5 verified by \\texttt{decide}) propagates it to the universal\ntwo-sided bound\n\\[\n\\forall s\\ge5,\\quad \\min(\\mathrm{seamIntegerGreedyRemainder}(s),\\ \\mathrm{overshoot}(s)) \\;\\le\\; 2^s.\n\\]\nA global dyadic-scale control on the seam orbit, independent of which branch fires. Exactly the\ningredient 257bm:c11/c12 needs to make its forbidden-band inequalities well-posed. The \"exclude\nfinitely many exceptional local cells, get a global dyadic bound by induction\" pattern is generic\nto digit-DP dynamics.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:dyadic-scale} \\lean{SeamTwoSidedDyadicCellEscape}{HalfCylinderMiddleCarryLowerBound.lean:3339} \\lean{SeamTwoSidedDyadicCellEscape.step}{HalfCylinderMiddleCarryLowerBound.lean:3339} \\lean{SeamTwoSidedDyadicCellEscape.twoSided}{HalfCylinderMiddleCarryLowerBound.lean:3420}\n\\end{thm}\n\n\\begin{prop}[257bm:i14 --- \\texttt{seamUpperResetDyadicBandEscape\\_through\\_thirty}, finite certificate to row 31]\nExplicit \\texttt{decide+kernel} computation of $\\mathrm{seamIntegerGreedyRemainder}$ for every\nrow $13$--$31$: row $14\\to392$, \\dots, row $31\\to4187487147$, NONE failing the band-avoidance\ncondition of $\\mathrm{SeamUpperResetDyadicBandEscape}$\n($\\forall d\\ge13,\\ \\forall\\text{carries},\\ \\forall j\\le d,\\ 2^{d-j+1}<\\mathrm{resetCharge}\\ \\vee\\\n\\mathrm{resetCharge}+2(d+j)\\le2^{d-j+1}$). This is the current COMPUTATIONAL FRONTIER of the\nband-escape route \u2014 a finite verified list to a concrete row, \\emph{not} a cofinal supply. The\nopen task is either extending the same finite-computation technique cofinally, or proving the\ngeneral band-avoidance arithmetically (which is what 257bm:c11's quantifier collapse reduces the\nburden of).\n\\par\\noindent\\ev{Cert} \\scale{fixed} \\coord{other:dyadic-boundary} \\lean{SeamUpperResetDyadicBandEscape}{HalfCylinderMiddleCarryLowerBound.lean:3531} \\lean{seamUpperResetDyadicBandEscape\\_through\\_thirty}{HalfCylinderUpperResetBandCertificates.lean:78} \\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_upperResetDyadicBandEscape}{HalfCylinderMiddleCarryLowerBound.lean:3755}\n\\end{prop}\n\n\\subsubsection*{Abstract, fully problem-agnostic invariants read alongside this lane}\n\n\\begin{thm}[257bm:i15 --- \\texttt{PerturbedFamily}, \\texttt{AdjacentCut}, the abstract perturbed-greedy machine]\n\\texttt{structure PerturbedFamily}$(\\alpha)$ where $\\mathrm{oldSum}:\\alpha\\to\\N$;\n$\\mathrm{pulse}:\\alpha\\to\\N$; $\\mathrm{gap}:\\N$; $\\mathrm{pulseCap}:\\N$, with axioms\n$\\mathrm{gap\\_pos}$, $\\mathrm{pulse\\_le}$, injectivity of $\\mathrm{oldSum}$, a superincreasing\nseparation axiom, and $\\mathrm{pulseCap}<3\\cdot\\mathrm{gap}$; derived\n$\\mathrm{newSum}(x):=4\\cdot\\mathrm{oldSum}(x)+\\mathrm{pulse}(x)$. An abstract\n\\texttt{AdjacentCut}$(C)$ structure (best subset $\\le C$, best subset $>C$) yields\n$\\mathrm{successorCarries} := 4\\cdot\\mathrm{overshoot}+\\mathrm{abovePulse}\\le\\mathrm{gap}$; the\nresulting $\\mathrm{prefixChoice}$ is provably MAXIMAL among all $\\mathrm{newSum}(x)\\le\n\\mathrm{newCapacity}$ (\\texttt{prefixChoice\\_maximal}), and \\texttt{nextRemainder\\_trichotomy}\ngives the exact 3-branch (carry/middle/right) recurrence. Fully general \u2014 no Mersenne content\nwhatsoever; $\\alpha,\\mathrm{oldSum},\\mathrm{pulse}$ are free parameters. THE single most\nreusable abstract result in this entire lane: any future base-$b$ greedy digit process (a\n\\#249 analogue included) needing \"maximality of the greedy choice under an admissible-capacity\nconstraint\" can instantiate this structure directly instead of re-proving maximality.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:abstract-perturbed-greedy} \\lean{PerturbedFamily}{HalfCylinderIntegerGreedy.lean:1286} \\lean{prefixChoice\\_maximal}{HalfCylinderIntegerGreedy.lean:1470} \\lean{nextRemainder\\_trichotomy}{HalfCylinderIntegerGreedy.lean:1470}\n\\end{thm}\n\n\\begin{thm}[257bm:i16 --- \\texttt{ReverseCarryWord}, abstract carry-overlap spacing]\n\\texttt{structure ReverseCarryWord} where $\\mathrm{coeff},\\mathrm{bit},\\mathrm{carry}:\\N\\to\\Z$;\n$\\mathrm{normalized}: \\forall m,\\ \\mathrm{bit}(m)+2\\cdot\\mathrm{carry}(m) = \\mathrm{coeff}(m)+\n\\mathrm{carry}(m{+}1)$. If two such words share a coefficient at a \"seam\" position with output\nbits $1/0$, then agree on both coefficients and bits for $\\mathrm{length}$ further positions,\ntheir carry difference at the far end is EXACTLY $2^{\\mathrm{length}}$ times an odd integer\n(\\texttt{overlappingReverseCarryWords\\_carryDifference\\_eq\\_twoPow\\_mul\\_odd}). Combined with a\ncommon Archimedean bound on both carries, this forces $2^{\\mathrm{length}}\\le\\mathrm{bound}$.\nPure integer-sequence recurrence, no Mersenne/base-4/seam structure at all \u2014 directly applicable\nto \\#249's own digit-carry structure if it is given an analogous reverse-carry presentation.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:reverse-carry-word} \\lean{ReverseCarryWord}{HalfTrappingReturnCarry.lean:70} \\lean{overlappingReverseCarryWords\\_carryDifference\\_eq\\_twoPow\\_mul\\_odd}{HalfTrappingReturnCarry.lean:180} \\lean{overlappingReverseCarryWords\\_twoPow\\_le\\_realBound}{HalfTrappingReturnCarry.lean:220} \\lean{overlappingMidpointReturns\\_twoPow\\_le\\_realBound}{HalfTrappingReturnCarry.lean:268}\n\\end{thm}\n\n\\begin{prop}[257bm:i17 --- \\texttt{supportCoeff\\_extend\\_true\\_eq\\_false\\_add\\_one\\_at\\_double}]\nAt row $2(N{+}1)$, changing only exponent $N{+}1$ from \\texttt{false} to \\texttt{true} in a\n$\\mathrm{HalfWord}(N)$ adds EXACTLY the half-divisor incidence to $\\mathrm{supportCoeff}$: a\none-bit flip changes a divisor-incidence sum by exactly the number of new divisor relations\ncreated \u2014 the smallest possible object-level coefficient-drop producer for\n$\\mathrm{RewindBaseUnitDropAt}$. A generic digit/divisor-incidence fact, restatable for any\nbase-2 divisor-coefficient bookkeeping, e.g.\\ \\#249's own divisor sums.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:half-divisor} \\lean{supportCoeff\\_extend\\_true\\_eq\\_false\\_add\\_one\\_at\\_double}{HalfDivisorUnitDrop.lean:20} \\lean{supportCoeff\\_boundaryPair\\_unitDrop\\_at\\_double}{HalfDivisorUnitDrop.lean:42}\n\\end{prop}\n\n\\subsection{Killers and countermodels}\n\nEvery record below is a HAZARD RECORD: a proof that some plausible-looking route to\ncofinality, either does not work, or provably cannot work as stated. These are load-bearing\nwarnings, not merely negative curiosities \u2014 several are the exact reason the corpus's live\nattack is aimed where it is aimed, and re-deriving what they already kill wastes a proof\nattempt.\n\n\\begin{thm}[257bm:k1 --- \\texttt{boundedDoubleOrRecycleModel\\_not\\_cofinal}, THE DICHOTOMY-SHAPE FALSIFIER \u2014 read before re-deriving cofinality from the dichotomy alone]\nThe dichotomy $\\mathrm{ExactLocalMersenneHalfRow}(2n{-}1) \\vee \\exists c,\\,4\\le c\\le n\\wedge\n\\mathrm{ExactLocalMersenneHalfRow}(2c{-}2)$ (proved for $n\\ge6$ at 257bm:k-dich below) is not by\nitself enough for cofinality. Countermodel: $\\mathrm{boundedDoubleOrRecycleModel}(n) := (n=6)$\nsatisfies EXACTLY the same two-branch transition shape (seed at 6, and\n\\texttt{boundedDoubleOrRecycleModel\\_transition} reproduces the $\\vee$ shape by always taking\nthe recycle branch with $c=4$, landing back at $2\\cdot4-2=6$), yet\n\\[\n\\neg\\big(\\forall N,\\ \\exists n\\ge N,\\ \\mathrm{boundedDoubleOrRecycleModel}(n)\\big) \\qquad \\text{(NOT cofinal \u2014 only ever true at } n=6\\text{)}.\n\\]\nPackaged existentially as \\texttt{exists\\_seeded\\_bounded\\_double\\_or\\_recycle\\_model}. The\ndouble-or-recycle transition shape, EVEN TOGETHER with an endpoint-six seed, is logically\ninsufficient to conclude cofinal exact rows. A genuinely new progress input (strict endpoint\ngrowth, or an independent cofinality argument) is required. The TECHNIQUE \u2014 build a bounded\none-point fixed model with the same recursion shape to show a transition schema doesn't imply\ncofinality \u2014 is fully general and transfers to any other double-or-recycle-shaped inductive\nscheme, including one built for \\#249.\n\\par\\noindent\\ev{Lean} \\scale{fixed} \\coord{other:meta-logical} \\lean{boundedDoubleOrRecycleModel}{BooleanMobiusExactRowDichotomy.lean:58} \\lean{boundedDoubleOrRecycleModel\\_transition}{BooleanMobiusExactRowDichotomy.lean:79} \\lean{boundedDoubleOrRecycleModel\\_not\\_cofinal}{BooleanMobiusExactRowDichotomy.lean:79} \\lean{exists\\_seeded\\_bounded\\_double\\_or\\_recycle\\_model}{BooleanMobiusExactRowDichotomy.lean:91}\n\\end{thm}\n\n\\begin{thm}[257bm:k-dich --- \\texttt{exactLocalMersenneHalfRow\\_double\\_or\\_recycle}, the dichotomy 257bm:k1 attacks]\nFor $n\\ge6$, $\\mathrm{ExactLocalMersenneHalfRow}(n)$:\n\\[\n\\mathrm{ExactLocalMersenneHalfRow}(2n-1) \\;\\vee\\; \\exists c,\\ 4\\le c\\le n \\wedge \\mathrm{ExactLocalMersenneHalfRow}(2c-2).\n\\]\nEvery exact row of endpoint $\\ge6$ has exactly one of two continuations \u2014 literal doubling\n(below-half) or recycling through a first-crossing to some $2c-2$ with $c\\le n$ (the recycled\nendpoint is \\emph{not} claimed to exceed $n$). Equality is excluded by 257bm:k11\n(\\texttt{finiteErdosSum\\_den\\_odd}). What is\nmissing, per 257bm:k1, is not another dichotomy but STRICT ENDPOINT PROGRESS in the recycle\nbranch, or a proof the below-half branch recurs \u2014 any future attack that merely re-derives a\ndouble-or-recycle transition is already refuted.\n\\par\\noindent\\ev{Lean} \\scale{bounded} \\coord{mobius-mersenne} \\lean{exactLocalMersenneHalfRow\\_double\\_or\\_recycle}{BooleanMobiusExactRowDichotomy.lean:26}\n\\end{thm}\n\n\\begin{prop}[257bm:k2 --- \\texttt{splitFractionMass\\_one\\_bound\\_not\\_necessary\\_fixture}, sufficient-but-not-necessary fixture]\nConcrete fixture $D=\\{2,3\\}$, $c=5$:\n\\[\n\\mathrm{value}(D)<\\tfrac12 \\ \\wedge\\ \\tfrac12<\\mathrm{value}(\\mathrm{insert}\\,5\\,D) \\ \\wedge\\ 1 < \\mathrm{localFractionMass}(D,8) + \\mathrm{localMersenneFraction}(8,5) \\ \\wedge\\ \\mathrm{localBinarySuffix}(D,1,8) < 2^3.\n\\]\n\\texttt{norm\\_num}-checked. The \"combined residue mass $\\le1$\" sufficient condition for sharp\ncapacity (\\texttt{localBinarySuffix\\_two\\_mul\\_sub\\_two\\_lt\\_criticalCapacity\\_of\\_splitFractionMass}) is genuinely NOT necessary \u2014 here the mass exceeds 1 yet sharp capacity still holds.\nDO NOT attack $\\mathrm{SkippedCoreCriticalQuotientSupply}$ (257bm:c4) via the fractional-mass\nroute alone; it will fail on real crossing cores. Use 257rig:i3-wall\n(\\texttt{eq\\_halfGreedyPrefixSupport\\_of\\_critical\\_crossing}) to reduce to the one canonical\norbit instead, and attack the orbit's own binary digits directly.\n\\par\\noindent\\ev{Cert} \\scale{fixed} \\coord{mobius-mersenne} \\lean{splitFractionMass\\_one\\_bound\\_not\\_necessary\\_fixture}{BooleanMobiusSkippedCoreCriticalCapacity.lean:183}\n\\end{prop}\n\n\\begin{obs}[257bm:k3 --- doubling immediately produces an above-half row, numeric countermodel]\n257bm:i7's doubling extension is NOT preserved iteratively: below-halfness is not preserved\nunder one application of doubling, and demonstrably fails at the very first step. From the\nidentity $\\mathrm{value}(E)-\\tfrac12 = \\big(\\mathrm{localFractionMass}(E,2n{-}1)-1\\big)/2^{2n-1}$,\nat the seed $D=\\{2,3,6\\}$ (257bm:i-seed) with $M=11$ the core alone contributes\n\\[\n\\frac{2^{11}\\bmod3}{3} + \\frac{2^{11}\\bmod7}{7} + \\frac{2^{11}\\bmod63}{63} = \\frac23+\\frac47+\\frac{32}{63} \\approx 1.746 > 1,\n\\]\nso the doubled row is STRICTLY ABOVE half. Iterating the cheap (doubling) arm is therefore\nimpossible from the only landed seed; the chain is forced into the capacity-gated recycle arm\nat step one. What would close the doubling route instead: cofinally many BELOW-HALF exact rows,\n$\\forall N\\ \\exists n\\ge N\\ \\exists D$ exact at $n$ with $\\mathrm{localFractionMass}(D,n)<1$ \u2014 a\nsingle such row at each of cofinally many $n$ would make doubling unnecessary; a\nself-reproducing one would close everything.\n\\par\\noindent\\ev{Cert} \\scale{fixed} \\coord{mobius-mersenne} \\lean{exists\\_exactRowStrictUpperExtension\\_two\\_mul\\_sub\\_one\\_of\\_exact\\_below}{BooleanMobiusExactRowDoubling.lean:185}\n\\end{obs}\n\n\\begin{thm}[257bm:k4 --- the recycling witness is capped by input $n$, not guaranteed to grow]\n257bm:c10 (\\texttt{exists\\_skippedCoreExactRow\\_of\\_value\\_above}) is UNCONDITIONAL, but its\nwitness is $\\exists c\\le n$, not $\\exists c$ large: $2c-2$ may be $\\le n$, so the endpoint need\nnot grow. 257bm:k1's \\texttt{exists\\_seeded\\_bounded\\_double\\_or\\_recycle\\_model} is the\nexplicit falsifier of the naive hope that growth comes for free:\n$\\mathrm{boundedDoubleOrRecycleModel}(n):=(n=6)$ satisfies the same transition schema plus a\nseed and is NOT cofinal. Growth is recovered only inside\n$\\mathrm{ProtectedExactLocalMersenneRow}$ (257bm:c5), whose invariants\n$\\mathrm{endpoint}<2\\cdot\\mathrm{cutoff}$ and $\\mathrm{new\\_above\\_cutoff}$ force $c>\\mathrm{cutoff}$\nhence $2c-2>\\mathrm{endpoint}$ \u2014 and maintaining those invariants is precisely what needs the\nstrict-upper (sharp capacity) fill of 257bm:c7 rather than this general recycling theorem.\n\\par\\noindent\\ev{Lean} \\scale{bounded} \\coord{binary-digit} \\lean{exists\\_skippedCoreExactRow\\_of\\_value\\_above}{BooleanMobiusExactRowCrossing.lean:155}\n\\end{thm}\n\n\\begin{obs}[257bm:k5 --- \\texttt{CofinalPositiveHalfGreedySkips} is not a reduction, it IS the problem \u2014 a trap for attackers]\nThe positivity conjunct of 257bm:c2's hypothesis,\n$0<\\mathrm{greedyMersenneRemainderRat}(1/2,c{-}1)$, is unconditionally dischargeable\n(\\texttt{localMersennePrefixValue\\_halfGreedy\\_lt\\_half}, itself just the odd-denominator parity\nfact of 257bm:k11, together with \\texttt{greedyMersenneRemainderRat\\_eq\\_sub\\_finiteErdosSum}). Hence\n\\[\n\\mathrm{CofinalPositiveHalfGreedySkips} \\;\\Longleftrightarrow\\; (\\mathrm{greedyMersenneSkippedSupport}(1/2)).\\mathrm{Infinite},\n\\]\nwhich 257bm:c20 (A6) proves EQUIVALENT to $1/2\\in\\mathrm{mersenneAchievementSet}$. Anyone\nattacking 257bm:c2's hypothesis head-on is attacking \\#257-false directly, with NO reduction\nwhatsoever, despite the corpus's own docstrings advertising it as \"the shortest known path\". The\ngenuine escape is to abandon the greedy-orbit arm entirely (where 257rig:i3-wall forces\ncanonicity and removes all freedom) and target the form-(\u2605) socket of 257bm:c7 instead, whose\ncores are NOT forced to be greedy prefixes.\n\\par\\noindent\\ev{Lean} \\scale{cofinal} \\coord{greedy-orbit} \\lean{localMersennePrefixValue\\_halfGreedy\\_lt\\_half}{BooleanMobiusCriticalCapacityCofinal.lean:138}\n\\end{obs}\n\n\\begin{thm}[257rig:k6 --- \\texttt{eq\\_halfGreedyPrefixSupport\\_of\\_critical\\_crossing}, THE WALL \u2014 the reduction to one universal-carry orbit]\nFor $c\\ge4$, $D$ bounded $[2,c)$, below-half, with genuine crossing deficit\n$\\tfrac12-\\mathrm{value}(D) < \\mathrm{mersenneWeightRat}(c)$:\n\\[\nD \\;=\\; \\mathrm{halfGreedyPrefixSupport}(c-1).\n\\]\nEVERY crossing core is forced, uniquely, to be the canonical rational half-greedy prefix. There\nis no freedom left in $D$ at all \u2014 257bm:c4's universal quantifier collapses onto a single orbit\n(the greedy digit-by-digit construction of $1/2$ in Mersenne weights), proved via\n\\texttt{IsStraddlePrefix.half\\_agrees\\_greedy} (superincreasing straddle-prefix rigidity, 257bm:i11a)\nplus a primitive-prefix bridge. This is the killer of the idea that\n$\\mathrm{SkippedCoreCriticalQuotientSupply}$ is a search over arbitrary $D$: it is a check\nagainst ONE orbit. A future attack should target the concrete arithmetic of\n$\\mathrm{halfGreedyPrefixSupport}$ directly. The straddle-prefix/superincreasing-sequence\nrigidity technique is general; whether a \\#249-side analogue exists (is the greedy $\\varphi$-digit\nprefix similarly rigid?) is UNVERIFIED, worth checking.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{greedy-orbit} \\lean{eq\\_halfGreedyPrefixSupport\\_of\\_critical\\_crossing}{BooleanMobiusCriticalCapacityCofinal.lean:50}\n\\end{thm}\n\n\\begin{obs}[257rig:k7 --- 257bm:c7 and 257rig:c17 trade off in opposite directions, neither dominates]\n257rig:c17 (\\texttt{HalfCarryCofinalTerminalOnlyStrip}) tolerates ANY support inside the depth-$M$\nwindow but demands the carry inside a $\\sim2\\sqrt M$ strip; 257bm:c7 tolerates a carry up to\n$2^{c-2}$ (exponentially looser) but demands the support be CONFINED to ranks $<c$, because the\nexact fill encodes the residue in the pure upper window where the Mersenne quotient is exactly\n$2^{M-d}$ (257bm:c9). The gap between them is support LOCALITY, not carry size, and the two\nhave never previously been compared on this axis. A lower-half confinement upgrade of the strip\nwitnesses (support in $[2,c)$ at depth $2c-3$) would make $2\\sqrt{2c}+4<2^{c-2}$ for $c\\ge8$ and\nfire the existing fill verbatim \u2014 which would identify 257rig:c17, not 257bm:c7, as the CHEAPER\nroute to the same payoff.\n\\par\\noindent\\ev{Lean} \\scale{cofinal} \\coord{other:sqrt-carry-growth} \\lean{HalfCarryCofinalTerminalOnlyStrip}{TerminalOnlyCofinal.lean:34}\n\\end{obs}\n\n\\begin{obs}[257hg:k8 --- explicit dyadically-unsafe example, the two-thirds-band countermodel]\nRead in full alongside \\texttt{HalfGreedyFatalGap} for scope contrast: writing a greedy residual\nas $\\mathrm{rem}=1/R$, a skip at rank $k$ is dyadically safe iff $R\\ge2^k$. Two unconditional\ncorollaries hold (an INTEGRAL reciprocal $R$ is never unsafe, \\texttt{not\\_twoThirdsBand\\_of\\_int};\nan unsafe run with odd $p,D,q$ forces $p\\ge7$, \\texttt{seven\\_le\\_of\\_intBand\\_odd}) \u2014 but the\nfile's own docstring is explicit that dyadic safety is SUFFICIENT, not NECESSARY, for the true\ngreedy process to survive (the true test uses the actual tail $\\sum_{j>k}\\mathrm{weight}(j)$,\nwhich exceeds $2^{-k}$), and NOTHING in this file asserts the actual half-greedy orbit avoids\nthe band: $(p,D,b)=(17,41,3)$ is an explicit odd-coprime example that IS dyadically unsafe. This\nsharpens directly into 257hg:i4's $2u\\le3a$ criterion, which narrows exactly this gap for the\nsingle-skip case \u2014 but the general question of whether the fatal region overlaps a reachable\nband under the actual greedy trajectory is settled by neither file.\n\\par\\noindent\\ev{Lean} \\scale{n/a} \\coord{other:rational-band} \\lean{TwoThirdsBand}{HalfGreedyTwoThirdsBand.lean:73} \\lean{not\\_twoThirdsBand\\_of\\_int}{HalfGreedyTwoThirdsBand.lean:253} \\lean{seven\\_le\\_of\\_intBand\\_odd}{HalfGreedyTwoThirdsBand.lean:253}\n\\end{obs}\n\n\\begin{thm}[257bm:k9 --- \\texttt{relationInvariantLinearChannels\\_det\\_eq\\_zero}, a whole determinant-obstruction family killed at once]\nIf a finite family of linear channels $\\mathrm{channel}:\\iota\\to V\\to_{\\mathbb{Q}}\\mathbb{Q}$ all\nvanish on $\\ker(\\mathrm{ev})$ (i.e.\\ are relation-invariant like $\\mathrm{ev}$ itself), then the\nevaluation matrix $(i,j)\\mapsto\\mathrm{channel}\\,j\\,(\\mathrm{row}\\,i)$ has rank $\\le1$ for EVERY\nfinite index type $\\iota$, hence every square minor of size $\\ge2$ has determinant zero. This\ncloses the \"shifted-channel determinant\" obstruction route at ALL ranks at once \u2014 previously\nchecked only experimentally at ranks 2--4. Pure linear algebra over $\\mathbb{Q}$-vector spaces,\nzero arithmetic content specific to Mersenne or \\#257. Cite this whenever someone proposes a new\nmulti-channel-relation determinant approach; the family it kills is unbounded, not just the\nsmall cases already tried.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:linear-algebra} \\lean{relationInvariantLinearChannels\\_det\\_eq\\_zero}{HalfTrappingReturnCarry.lean:42}\n\\end{thm}\n\n\\begin{obs}[257bm:k10 --- the loose capacity bound loses exactly one bit, additively not multiplicatively \u2014 do not try to shave it by a constant factor]\n257bm:i6's $2^{c-1}$ bound and 257bm:i5's sharp $2^{c-2}$ requirement differ by ONE bit, and the\nproof of the loose bound shows exactly where: it derives $A<2^{c-2}+|D|$ and discards\n$|D|\\le c-2\\le2^{c-2}$. Any attempt to close the gap by re-deriving the same additive slack\nestimate more carefully is bounded by this identity \u2014 the ONLY way to close it is a genuine\nanti-concentration statement that $A$ avoids the linear-width band\n$[2^{c-2},\\,2^{c-2}+c-3]$, not a sharper constant in the existing proof.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{binary-digit} \\lean{localBinarySuffix\\_two\\_mul\\_sub\\_two\\_lt\\_upperHalfCapacity}{BooleanMobiusSkippedCoreExactRow.lean:123}\n\\end{obs}\n\n\\begin{obs}[257bm:k11 --- \\texttt{finiteErdosSum\\_den\\_odd} is UNVERIFIED-IN-FULL by this pass \u2014 flag before re-citing]\nThis lemma is cited repeatedly throughout the lane (TH-dichotomy's equality exclusion,\n257bm:k2's fixture, 257bm:i10's finite-support-never-hits-half corollary) but sits OUTSIDE this\nlane's file set at \\texttt{DyadicPrefixCompression.lean:1358}; only its call sites, not its own\nbody, were read in the passes that built this catalogue. Treat its conclusion\n($\\mathrm{localMersennePrefixValue}(D)\\ne1/2$ for finite $D$) as \\ev{Cited}, not\nindependently re-verified here, until a pass opens that file directly.\n\\par\\noindent\\ev{Cited} \\scale{uniform} \\coord{p-adic} \\lean{finiteErdosSum\\_den\\_odd}{DyadicPrefixCompression.lean:1358}\n\\end{obs}\n\n\\begin{thm}[257hg:k12 --- \\texttt{finalMiddleCell\\_neg\\_three\\_not\\_last} (G2), one of three exceptional cells excluded]\nThe first of three exceptional final-middle integer cells of 257hg:i7's trichotomy (coordinate\n$4\\cdot\\mathrm{remainder}-\\mathrm{belowPulse}-4=-3$) CANNOT be the actual last transition before\nan all-right tail. Proof combines the analytic strict inequality\n$\\mathrm{value}(u)+\\mathrm{mersenneTail}(D) < 1/2$ (a middle-branch consequence of an\nall-right-tail hypothesis) with the two-step M\\\"obius-carry recurrence\n(\\texttt{mobiusCenteredHalfCarry\\_add\\_two}) forcing the centred carry to zero then strictly\nNEGATIVE within two more coefficient rows \u2014 contradicting 257hg:i6's nonnegativity. This is a\ngenuine exclusion, not a reduction: cell $-3$ is dead. A companion file\n(\\texttt{HalfCylinderFinalMiddleTailSocket.lean}) records the corresponding\n$\\mathrm{binaryCoeffTail}$ inequalities for the remaining two cells, $-2$ and $-1$\n(\\texttt{binaryCoeffTail\\_union\\_Ioi\\_lt\\_two\\_of\\_producerCarry\\_eq\\_neg\\_two},\n\\texttt{\\_lt\\_three\\_of\\_producerCarry\\_eq\\_neg\\_one}) \u2014 closing all three would fully resolve the\nfinal-middle producer and hence 257bm:c13's socket (i) outright.\n\\par\\noindent\\ev{Lean} \\scale{uniform} \\coord{other:mobius-centred-carry} \\lean{finalMiddleCell\\_neg\\_three\\_not\\_last}{HalfCylinderFinalMiddleCellEscape.lean:587} \\lean{mobiusCenteredHalfCarry\\_add\\_two}{HalfCylinderFinalMiddleCellEscape.lean:39} \\lean{cofiniteRightTail\\_ne\\_zero\\_centeredEndpoint}{HalfCylinderFinalMiddleCellEscape.lean:547}\n\\end{thm}\n\n% ---- part a257_newdecls ----\n\\subsection{Recently published declarations}\n\nThe modules surveyed above were read at an earlier snapshot. Four already-public modules have\nsince grown by union merge, and one wholly new problem-centric tree,\n\\texttt{ErdosProblems/Erdos257/}, has been published alongside the legacy \\texttt{Erdos249257}\nnamespace. This subsection catalogues only the declarations the earlier parts do not cite. None\nof it decides O1--O5 above; it fills in machinery that those targets' own producer chains were\nstated abstractly against.\n\n\\subsubsection{The middle-cell update-step chain (\\texttt{HalfCylinderMiddleCarryLowerBound.lean})}\n\nThe O5 discussion above states \\texttt{SeamTwoSidedDyadicCellEscape} and\n\\texttt{SeamUpperResetDyadicBandEscape} as named hypotheses without exposing the per-row update\nmachinery that actually produces and consumes them. That machinery is present in the live file.\n\n\\begin{prop}[The unsafe middle range is exactly three integers]\nFor $s\\ge5$, writing $R=(\\text{seamAdjacentCut } s\\ hs).\\mathrm{remainder}$ and\n$P=(\\ldots).\\mathrm{belowPulse}$,\n\\[\n\\neg(4R-P-4\\le-4\\ \\vee\\ 0\\le 4R-P-4)\\ \\Longleftrightarrow\\ 4R-P-4\\in\\{-3,-2,-1\\}\n\\]\n\\end{prop}\n(\\lean{seamMiddleCoordinate\\_not\\_safe\\_iff\\_three\\_cells}{HalfCylinderMiddleCarryLowerBound.lean:4125},\n\\ev{Lean}, \\scale{uniform}, \\coord{mobius-mersenne}, proved by \\texttt{omega} from the integer\ndefinitions). This is the exact unconditional source of the \"three integer values\" language used\nabove for O5; it was previously asserted in prose only.\n\nTwo companion bounds pin the scaled rational residual inside a genuine middle transition:\n$4\\cdot\\mathtt{seamGreedyFloorZ}\\,s\\le c$ from a cell label $4R=P+c$\n(\\lean{seamMiddleCell\\_four\\_mul\\_floorZ\\_le}{HalfCylinderMiddleCarryLowerBound.lean:4142}, \\ev{Lean}),\nand, combining it with the successor floor-$Z$ recurrence, $4^{s+1}\\cdot\n\\mathtt{seamWordRationalRemainder}(\\mathtt{seamGreedyWord}(s+1))\\le 2^{s+2}+c$\n(\\lean{seamMiddleCell\\_scaledRationalRemainder\\_le}{HalfCylinderMiddleCarryLowerBound.lean:4167},\n\\ev{Lean}). The module's own remark is that for the first exceptional cell ($c=-3$, i.e.\\ the cell\nalready closed by \\lean{finalMiddleCell\\_neg\\_three\\_not\\_last}{HalfCylinderLastProducerContradiction.lean:333})\nthis leaves margin exactly $1$ against a needed correction-tail slack of $4/3+\\varepsilon$ --- a\nconcrete near-miss, not a proof, and it does not extend to the two open cells $-2,-1$.\n\nA parallel iff for the right branch: an overshoot witness $\\le2^s$ produces a pulse leak above\n$2^{s+2}$ exactly when the overshoot sits within an explicit distance $k$ of $2^s$ with\n$4k<\\mathrm{abovePulse}$ (\\lean{seamRightPulseLeak\\_iff\\_distanceCell}{HalfCylinderMiddleCarryLowerBound.lean:4202},\n\\ev{Lean}). A new one-coordinate normal form, $\\mathtt{seamPureHalfPrefixRemainder}\\,s :=\n\\mathtt{integerGreedyRemainder}(\\mathtt{seamWeights}\\,s)(2^{2s-1})$\n(\\lean{seamPureHalfPrefixRemainder}{HalfCylinderMiddleCarryLowerBound.lean:4232}), is given an exact\ncase-split closed form (\\lean{seamPureHalfPrefixRemainder\\_eq\\_ite}{HalfCylinderMiddleCarryLowerBound.lean:4238},\n\\ev{Lean}) that the file states is equivalent to the two-sided invariant, but this equivalence is not\nitself yet packaged as a separate reusable lemma beyond\n\\lean{seamPureHalfPrefixRemainder\\_le\\_iff\\_twoSided}{HalfCylinderMiddleCarryLowerBound.lean:4356} (\\ev{Lean}).\n\n\\begin{obs}[The reset-band socket has a certified base case]\n\\texttt{SeamUpperResetDyadicBandEscape} (O5's second socket, \\S\\ref{sec:o4}--\\ref{sec:o5} above) is\nreduced by three new lemmas to a quarter-scale overshoot exclusion: the iff\n\\lean{seamUpperReset\\_band\\_iff\\_successorRemainder\\_avoids}{HalfCylinderMiddleCarryLowerBound.lean:4469}\n(\\ev{Lean}) rewrites the reset-charge band as a successor-remainder avoidance; the identity\n\\lean{upperResetCharge\\_dyadicDistance\\_add\\_abovePulse}{HalfCylinderMiddleCarryLowerBound.lean:4490}\n(\\ev{Lean}) rescales it by a factor of $4$; and\n\\lean{seamUpperReset\\_band\\_of\\_overshoot\\_band}{HalfCylinderMiddleCarryLowerBound.lean:4514} (\\ev{Lean})\nderives the band from a quarter-scale overshoot exclusion plus the pulse bound $\\mathrm{abovePulse}\\le\n2(d-2)$. Row $d=13$, the first row of the late regime, is then proved unconditionally to escape\n\\emph{every} band simultaneously: its successor remainder is exactly $392$\n(\\lean{seamUpperResetDyadicBandEscape\\_at\\_thirteen}{HalfCylinderMiddleCarryLowerBound.lean:4547},\n\\ev{Lean}, \\ev{Cert} for the numeral $392$). This is one certified row of a socket that is $\\forall\nd\\ge13$; it is evidence the base case is not itself an obstruction, nothing more.\n\\end{obs}\n\n\\subsubsection{Forgetting witnesses: the terminal-only bridge (\\texttt{SuffixCylinderTerminalOnlyBridge.lean})}\n\nThis entire file is new and previously uncited. It is a lossy-projection bridge: every object in\nthe corpus's much richer suffix-cylinder stage machinery (\\texttt{CylinderStage},\n\\texttt{InStripTwoSheetStage}, \\texttt{ProfiledGapStage}) is shown to already contain a\nterminal-only strip witness, the weaker object the compactness-based O3/O2 consumer actually needs.\n\n\\begin{prop}\nEvery \\texttt{CylinderStage} at depth $N$ yields \\texttt{HalfTerminalOnlyStripWitness}\\ $N$\n(\\lean{CylinderStage.halfTerminalOnlyStripWitness}{SuffixCylinderTerminalOnlyBridge.lean:63}, \\ev{Lean},\n\\coord{support rigidity, half-carry}), and at a feedback row this survives \\emph{both} outputs of\nthe corpus's total feedback theorem --- full-cylinder advance or a localized one-hole seam, which\ncan delete at most one of carries $3,4$\n(\\lean{CylinderStage.halfTerminalOnlyStripWitness\\_after\\_feedback}{SuffixCylinderTerminalOnlyBridge.lean:140},\n\\ev{Lean}).\n\\end{prop}\nFour concrete certified instances follow at depths $51$--$54$\n(\\lean{fullCylinderStage51\\_halfTerminalOnlyStripWitness}{SuffixCylinderTerminalOnlyBridge.lean:231},\n\\lean{fullCylinderStage54\\_halfTerminalOnlyStripWitness}{SuffixCylinderTerminalOnlyBridge.lean:257},\n\\ev{Lean}, \\ev{Cert} for the underlying stage), which is a finite witness chain, not a cofinal one.\n\n\\begin{obs}[A new named producer socket for HALF, not yet in the implication table]\nThe end-to-end consumer\n\\[\n\\left(\\forall N,\\ \\exists M,K,\\ \\max(N,1)\\le M\\wedge\\mathrm{Nonempty}(\\mathtt{CylinderStage}\\ K\\ M)\\right)\n\\ \\Longrightarrow\\ \\exists A,\\ A.\\mathrm{Infinite}\\wedge \\mathtt{erdosSupportSeries}\\ 2\\ A=1/2\n\\]\nis proved unconditionally\n(\\lean{exists\\_infinite\\_support\\_half\\_of\\_cofinalCylinderStages}{SuffixCylinderTerminalOnlyBridge.lean:277},\n\\ev{Lean}), with a positive-support refinement using the same odd-denominator lemma\n\\lean{finite\\_boolSupport\\_ne\\_half}{HalfCarryReachability.lean:589} already used for O2 above\n(\\lean{exists\\_infinite\\_positive\\_support\\_half\\_of\\_cofinalCylinderStages}{SuffixCylinderTerminalOnlyBridge.lean:287},\n\\ev{Lean}). The hypothesis, cofinal nonempty full-cylinder stages, is a genuinely new named socket\nfor HALF: it has not been checked against O3's \\texttt{CofinalExactLocalMersenneHalfRows} or O4's\n\\texttt{SQRTESC}, and its own supply is \\ev{Open}. It is only a sufficient condition, proved in one\ndirection; no equivalence claim is made anywhere in the file.\n\\end{obs}\n\n\\subsubsection{A strict weakening chain for the last-producer socket (\\texttt{HalfCylinderLastProducerContradiction.lean})}\n\nFive declarations extend the already-cited \\texttt{SeamMiddleProducerSqrtEscape} socket with a\ngenuine strength reduction, wiring the $-3$ cell closure into the producer chain rather than\nleaving it as an isolated fact.\n\n\\begin{defn}\n$\\mathrm{SeamMiddleProducerTailEscape}$ is the exact hypothesis that at every genuine middle\ntransition ($s\\ge13$, no successor carry, below the terminal-weight threshold) the terminal-augmented\nbinary coefficient tail is strictly below the producer carry\n(\\lean{SeamMiddleProducerTailEscape}{HalfCylinderLastProducerContradiction.lean:34}).\n$\\mathrm{SeamMiddleProducerTailEscapeExceptNegThree}$ is the same hypothesis restricted to cells\n$\\ne-3$ (\\lean{SeamMiddleProducerTailEscapeExceptNegThree}{HalfCylinderLastProducerContradiction.lean:54}).\n\\end{defn}\nThe reduction $\\mathrm{TailEscape}\\Rightarrow\\mathrm{ExceptNegThree}$ is trivial by hypothesis-dropping\n(\\lean{SeamMiddleProducerTailEscape.toExceptNegThree}{HalfCylinderLastProducerContradiction.lean:76},\n\\ev{Lean}); the substance is in the converse direction of the fan-in. The reduced socket, with cell\n$-3$ discharged by the already-proved\n\\lean{middleProducer\\_neg\\_three\\_not\\_last}{HalfCylinderLastProducerContradiction.lean:364}, already\nsuffices to close HALF:\n\\[\n\\mathrm{SeamMiddleProducerTailEscapeExceptNegThree}\\ \\Longrightarrow\\ (1/2:\\mathbb{R})\\in\\mathcal{A}\n\\]\n(\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_middleProducerTailEscapeExceptNegThree}{HalfCylinderLastProducerContradiction.lean:436},\n\\ev{Lean}), with the unreduced socket and the already-cited square-root majorant both routing through\nit (\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_middleProducerTailEscape}{HalfCylinderLastProducerContradiction.lean:451},\n\\ev{Lean}). So the chain now reads $\\mathrm{SqrtEscape}\\Rightarrow\\mathrm{TailEscape}\\Rightarrow\n\\mathrm{ExceptNegThree}\\Rightarrow\\mathrm{HALF}$, three strictly decreasing sockets where previously\nonly the endpoints were exposed; all three antecedents remain \\ev{Open}.\n\n\\subsubsection{The governed frozen-margin producer (\\texttt{HalfCylinderFiniteShadow.lean})}\n\nOne further declaration completes a chain the earlier parts cite only up to its antecedent\nequivalence (\\lean{exists\\_greedyHalfFrozenMargin\\_nonneg\\_iff\\_excess\\_neg}{HalfCylinderFiniteShadow.lean:1229}).\n\n\\begin{defn}[$\\mathrm{HalfGreedyGovernedFrozenMarginProducer}$]\nFor every $k$ with $\\mathtt{mersenneWeight}(k+1)$ exceeding the current greedy remainder, some\n$J\\le k+1$ gives a nonnegative frozen margin\n(\\lean{HalfGreedyGovernedFrozenMarginProducer}{HalfCylinderFiniteShadow.lean:1267}).\n\\end{defn}\n\\[\n\\mathrm{HalfGreedyGovernedFrozenMarginProducer}\\ \\Longrightarrow\\ (1/2:\\mathbb{R})\\in\\mathcal{A}\n\\]\n(\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_governedFrozenMarginProducer}{HalfCylinderFiniteShadow.lean:1274},\n\\ev{Lean}), proved via the pre-existing\n\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_skipped\\_dyadicCap}{GreedyAchievementSet.lean:1431} and the\nexcess-sign equivalence already noted above. This is the module's actual promised consumer; the\nantecedent's own first-passage bound ($J\\le k+1$, uniform) is \\ev{Open} and has not been checked\nagainst any of O3--O5's producer sockets.\n\n\\subsubsection{New tree: \\texttt{ErdosProblems/Erdos257/MersenneSubseriesRigidity.lean}}\n\nThis module works in the new \\texttt{ErdosProblems.Erdos257} namespace and states in its own header\nthat its results ``concern achievement-set geometry and do not settle the universal irrationality\nproblem.'' For an arbitrary support restriction $J\\subseteq\\mathbb{N}$, define\n$\\mathtt{supportedMersenneAchievementSet}\\,J$ as the range of Mersenne digit values supported on $J$\n(\\lean{supportedMersenneAchievementSet}{MersenneSubseriesRigidity.lean:76}). It is compact\n(\\lean{isCompact\\_supportedMersenneAchievementSet}{MersenneSubseriesRigidity.lean:90}, \\ev{Lean}),\nnowhere dense for every $J$ (\\lean{isNowhereDense\\_supportedMersenneAchievementSet}{MersenneSubseriesRigidity.lean:112},\n\\ev{Lean}), and perfect --- compact with no isolated points --- exactly when $J$ is infinite\n(\\lean{perfect\\_supportedMersenneAchievementSet}{MersenneSubseriesRigidity.lean:167}, \\ev{Lean}), via a\nCantor-space no-isolated-point argument on the coding space itself\n(\\lean{preperfect\\_supportedMersenneDigitSet}{MersenneSubseriesRigidity.lean:120}, \\ev{Lean}).\n\n\\begin{prop}[Exact Lebesgue-measure dichotomy]\nEither $J=F^c$ for a finite $F$, and $\\mathrm{vol}(\\mathtt{supportedMersenneAchievementSet}\\,J)=2^{-|F|}$\nexactly, or $J^c$ is infinite and the volume is exactly $0$.\n\\end{prop}\n(\\lean{volume\\_supportedMersenneAchievementSet\\_dichotomy}{MersenneSubseriesRigidity.lean:397}, \\ev{Lean},\n\\scale{uniform}, \\coord{mobius-mersenne}). The finite case is an exact doubling induction over inserted\ncoordinates (\\lean{volume\\_supportedMersenneAchievementSet\\_insert}{MersenneSubseriesRigidity.lean:286},\n\\ev{Lean}, using disjointness of the two coordinate-split faces,\n\\lean{disjoint\\_supportedMersenneAchievementSet\\_translate}{MersenneSubseriesRigidity.lean:262}, \\ev{Lean},\nwhich is exactly where unique binary coding is spent). None of this module touches irrationality of any\nsubseries value; it is a self-contained measure-and-topology classification of achievement sets, orthogonal\nto O1--O5.\n\n\\subsubsection{New tree: \\texttt{ErdosProblems/Erdos257/HalfCounterexampleFrontier.lean}}\n\nThis short module restates $\\mathrm{U257}$ in the new namespace\n(\\lean{UniversalMersenneSubseriesIrrationality}{HalfCounterexampleFrontier.lean:25}) and gives it a\nproblem-centric interface to the legacy half-terminal producer already used for O2 above: any\n\\texttt{HalfTerminalOnlyScaledVanishingSequence} witness gives an infinite $A$ with\n$\\mathtt{erdosSupportSeries}\\ 2\\ A=1/2$\n(\\lean{exists\\_rational\\_half\\_counterexample\\_of\\_terminalScaledVanishing}{HalfCounterexampleFrontier.lean:30},\n\\ev{Lean}), hence refutes $\\mathrm{U257}$ outright\n(\\lean{not\\_universal\\_of\\_terminalScaledVanishing}{HalfCounterexampleFrontier.lean:37}, \\ev{Lean}). The\nmodule's own docstring states plainly that the producer sequence itself ``has only been checked to a\nlarge finite cutoff'': this is a restatement of the already-\\ev{Open} O2 producer obligation in a new\nnamespace, not a new result, and it duplicates \\S O2's own logic (any HALF witness is $\\neg\\mathrm{U257}$)\nrather than extending it.\n\n% ---- part a257_p2 ----\n\\section{The scale ladder}\n\\label{sec:scale-ladder}\n\nEvery claim in this programme carries a \\emph{scale} tag alongside its evidence band. The two\naxes are independent and both matter to a reader deciding whether a result can be used as a\npremise. Evidence answers ``is this checked, and how'': \\ev{Lean} is kernel-checked in the live\nLean tree, \\ev{Cert} is an exact finite computation (integer arithmetic, no floating point, often\nindependently reproduced), \\ev{Math} is ordinary mathematics carried out in prose and checked by\na human or an adversarial audit pass but not yet formalised, \\ev{Cited} invokes a published\nexternal theorem, and \\ev{Open} marks an unproved hypothesis that some consumer theorem is\nconditioned on. Scale answers a different question, ``over how much of the problem does this\nclaim range'': \\scale{fixed} is one concrete numeral or witness; \\scale{bounded} produces a\nwitness whose size is controlled by the input (typically $\\le n$ for input $n$); \\scale{uniform}\nholds unconditionally for every $n$ above an explicit threshold, with no existential slack;\n\\scale{cofinal} asserts $\\forall N\\,\\exists n\\ge N,\\ldots$ \u2014 infinitely many witnesses, arbitrarily\nfar out, with \\emph{no} coherence required between them. \\scale{n/a} tags definitions and other\nscale-free statements.\n\nScale is the load-bearing axis of the whole corpus. Both open problems are, at bottom, cofinal\nstatements: \\#257 half-membership needs infinitely many exact rows or infinitely many positive\ngreedy skips (Table~\\ref{tab:scale-ladder}, rows C1/C3), and the unconditional track needs a\nrun-length bound that holds at every reset row out to infinity, not merely to a certified depth.\nEvery machine-checked theorem in the tree is fixed, bounded, or uniform. No amount of raising a\nuniform certificate's threshold, by itself, produces a cofinal witness \u2014 a uniform statement\n``$\\forall n\\ge N_0$'' is a single first-order sentence with no existential content past $N_0$,\nwhile a cofinal statement needs infinitely many \\emph{witnesses to a further existential}, which is\na strictly different logical shape.\n\nTable~\\ref{tab:scale-ladder} samples the ladder across all four non-trivial scales, drawn from the\n\\texttt{BooleanMobius}, \\texttt{HalfCutLocator}, \\texttt{GreedyAchievementSet}, and seam-integer\n(\\texttt{HalfCylinder*}) families together with the truncation-rung ladder and the sign/irrationality\nprimitives that feed all of them. The horizontal rule marks the frontier: everything above it is\nproved (fixed, bounded, or uniform); everything below it is what the open producers still need\n(cofinal), stated with its exact quantifier prefix so the gap is visible, not merely asserted.\n\n\\begingroup\n\\footnotesize\n\\setlength{\\tabcolsep}{3pt}\n\\begin{longtable}{p{0.28\\textwidth}|p{0.13\\textwidth}|p{0.25\\textwidth}|p{0.14\\textwidth}}\n\\hline\n\\textbf{Result} & \\textbf{Scale} & \\textbf{Quantifier prefix} & \\textbf{Coordinate} \\\\\n\\hline\n\\endfirsthead\n\\hline\n\\textbf{Result} & \\textbf{Scale} & \\textbf{Quantifier prefix} & \\textbf{Coordinate} \\\\\n\\hline\n\\endhead\n\\hline\n\\endfoot\n\\multicolumn{4}{|l|}{\\emph{Fixed \u2014 one concrete witness or numeral}} \\\\\n\\hline\nSmallest concrete exact row: support $\\{2,3,6\\}$ satisfies\n$\\mathrm{localPrefixQuotient}=2^5-1$. \\lean{exactLocalMersenneHalfRow\\_six}{BooleanMobiusExactRowSeed.lean:34}\n\\ev{Cert}\n& \\scale{fixed} & $n=6$ & \\coord{mobius-mersenne} \\\\\n\\hline\nOne-point counter-model: the double-or-recycle transition shape alone, even seeded, is not\ncofinal. \\lean{boundedDoubleOrRecycleModel\\_not\\_cofinal}{BooleanMobiusExactRowDichotomy.lean:91}\n\\ev{Lean}\n& \\scale{fixed} & $n=6$ (free-standing model) & \\coord{other:meta-logical} \\\\\n\\hline\nThe combined-residue-mass $\\le 1$ sufficient test for sharp capacity is not necessary: fixture\n$D=\\{2,3\\}$, $c=5$. \\lean{splitFractionMass\\_one\\_bound\\_not\\_necessary\\_fixture}{BooleanMobiusSkippedCoreCriticalCapacity.lean:183}\n\\ev{Cert}\n& \\scale{fixed} & $D=\\{2,3\\},\\ c=5$ & \\coord{mobius-mersenne} \\\\\n\\hline\nExact residual fixture: $\\tfrac12-W(\\{2,3,6,7\\})=\\tfrac1{16002}$.\n\\lean{half\\_sub\\_four\\_term\\_prefix\\_eq}{HalfCutLocator.lean:664} \\ev{Cert}\n& \\scale{fixed} & none (numeric instance) & \\coord{other:regression-fixture} \\\\\n\\hline\nClosed-form M\u00f6bius--Lambert value $\\sum_{d\\ge1}\\mu(d)/(2^d-1)=1/2$ (imported identity).\n\\lean{tsum\\_moebius\\_div\\_two\\_pow\\_sub\\_one\\_eq\\_half}{MobiusSignSupportNoGo.lean:111} \\ev{Cited}\n& \\scale{fixed} & none (closed-form value) & \\coord{mobius-mersenne} \\\\\n\\hline\nThe one natural negative-M\u00f6bius candidate support overshoots: $1/2 <\n\\sum_{d\\in N}1/(2^d-1)$ for $N=\\{d\\ge2:\\mu(d)=-1\\}$.\n\\lean{half\\_lt\\_tsum\\_negativeMobius}{MobiusSignSupportNoGo.lean:150} \\ev{Lean}\n& \\scale{fixed} & one fixed candidate support $N$ & \\coord{mobius-mersenne} \\\\\n\\hline\n\\multicolumn{4}{|l|}{\\emph{Bounded \u2014 a witness exists, controlled by the input}} \\\\\n\\hline\nEvery exact row continues: doubles unboundedly, or recycles to some $c\\le n$.\n\\lean{exactLocalMersenneHalfRow\\_double\\_or\\_recycle}{BooleanMobiusExactRowDichotomy.lean:26}\n\\ev{Lean}\n& \\scale{bounded} & $\\forall n\\ge6$, given a row at $n$: $(\\ldots)\\lor\\exists c\\le n(\\ldots)$\n& \\coord{mobius-mersenne} \\\\\n\\hline\nEvery above-half bounded support recycles into a genuine exact row at some $2c-2$ with $c\\le n$.\n\\lean{exists\\_skippedCoreExactRow\\_of\\_value\\_above}{BooleanMobiusExactRowCrossing.lean:155} \\ev{Lean}\n& \\scale{bounded} & $\\forall E$ bounded $[2,n]$: $\\exists c\\le n$ & \\coord{binary-digit} \\\\\n\\hline\n(\\#249 cross-lane check) $\\sum\\varphi(n)/2^n$ matches no rational whose denominator divides\n$2^{14}(2^h-1)$, for any $h\\le16$.\n\\lean{totient\\_series\\_ne\\_rat\\_of\\_den\\_dvd\\_upto\\_sixteen}{CarrySurvivorExtinction.lean:587} \\ev{Cert}\n& \\scale{bounded} & $\\forall h\\le16$ (explicit bound) & \\coord{binary-digit} \\\\\n\\hline\nTruncation-rung ladder: $\\mathrm{HalfRung}(J)$ proved for every $3\\le J\\le22$ via the finite\ndecision procedure (\\S\\ref{sec:four-coordinates}, Theorem~\\ref{thm:tr-finite-decision}).\n(source: \\texttt{erdos257\\_truncation\\_rung\\_ladder\\_2026\\_07\\_24.md}, \\S5) \\ev{Math}+\\ev{Cert}\n& \\scale{bounded} & $3\\le J\\le22$ (finite table, exhaustive) & \\coord{rung-truncation} \\\\\n\\hline\n\\multicolumn{4}{|l|}{\\emph{Uniform \u2014 unconditional for every $n$ past an explicit threshold}} \\\\\n\\hline\nMaster achievement/greedy-survival criterion.\n\\lean{mem\\_mersenneAchievementSet\\_iff\\_greedy\\_survival}{GreedyAchievementSet.lean:1445} \\ev{Lean}\n& \\scale{uniform} & $\\forall x\\ \\forall n$ (iff) & \\coord{greedy-orbit} \\\\\n\\hline\nMembership iff every actually-skipped rank survives.\n\\lean{no\\_lastHalfGreedySkip\\_iff\\_every\\_skip\\_survives}{HalfCylinderFixedTailSocket.lean:79} \\ev{Lean}\n& \\scale{uniform} & $\\forall M\\in$ skipped support & \\coord{greedy-orbit} \\\\\n\\hline\nEvery crossing core is forced to be the canonical rational half-greedy prefix.\n\\lean{eq\\_halfGreedyPrefixSupport\\_of\\_critical\\_crossing}{BooleanMobiusCriticalCapacityCofinal.lean:50}\n\\ev{Lean}\n& \\scale{uniform} & $\\forall c\\ge4\\ \\forall D$ bounded $[2,c)$ & \\coord{greedy-orbit} \\\\\n\\hline\nTheorem A payoff: reset $\\sqrt{\\ }$-escape socket $\\Rightarrow$ membership.\n\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_largestSkipLateStepSocket}{HalfCylinderLargestSkipInduction.lean:178}\n\\ev{Lean}\n& \\scale{uniform} & $\\forall s\\ge14$, given the socket hypothesis & \\coord{seam-integer} \\\\\n\\hline\nBand-avoidance certified for every row $13\\le r\\le31$ (feeds the F5/Theorem A uniform hypothesis\nas an exhaustively-verified finite range). \\lean{seamUpperResetDyadicBandEscape\\_through\\_thirty}{HalfCylinderUpperResetBandCertificates.lean:78}\n\\ev{Cert}\n& \\scale{uniform} & $13\\le r\\le31$, exhaustive & \\coord{seam-integer} \\\\\n\\hline\nCritical-band-index reduction: a $(d{+}1)$-way check collapses to one nearest-boundary check.\n\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_upperResetCriticalBandEscape}{HalfUpperResetCriticalBand.lean:901}\n\\ev{Lean}\n& \\scale{uniform} & $\\forall d,E$ with $E\\le2^{d+1}$ & \\coord{other:dyadic-boundary} \\\\\n\\hline\nParity kill: no finite support hits $1/2$ exactly. \\lean{finite\\_boolSupport\\_ne\\_half}{HalfCarryReachability.lean:589}\n\\ev{Lean}\n& \\scale{uniform} & $\\forall A$ finite, $0\\notin A$ & \\coord{other:denominator-parity} \\\\\n\\hline\nErd\u0151s--Borwein irrationality at every integer base.\n\\lean{irrational\\_erdosBorwein\\_series}{CertificateKernel.lean:8007} \\ev{Lean}\n& \\scale{uniform} & $\\forall b\\ge2$ & \\coord{mobius-mersenne} \\\\\n\\hline\nErd\u0151s 1968: any infinite, pairwise-coprime, reciprocal-summable index set gives an irrational\nsupport series. \\lean{irrational\\_erdosSupportSeries\\_pairwise\\_coprime}{CertificateKernel.lean:10448}\n\\ev{Lean}\n& \\scale{uniform} & $\\forall b\\ge2\\ \\forall A$ (infinite, coprime, summable) & \\coord{mobius-mersenne} \\\\\n\\hline\nGeneric block-certificate engine (problem-agnostic): weighted-coefficient series is irrational\ngiven a per-precision divisibility/window certificate.\n\\lean{irrational\\_coeff\\_series\\_of\\_weighted\\_coeff\\_block\\_certificates}{CertificateKernel.lean:8703}\n\\ev{Lean}\n& \\scale{uniform} & $\\forall b\\ge2\\ \\forall c\\,(\\forall m,c(m)\\le m)$, given $\\forall q\\,\\exists N,K,L,C$\n& \\coord{binary-digit} \\\\\n\\hline\nFinite greedy-prefix denominators are odd (source of the parity boundary exclusion,\n\\S\\ref{sec:four-coordinates}). \\lean{halfGreedyPrefixDenominator\\_odd}{DyadicPrefixCompression.lean:1366}\n\\ev{Lean}\n& \\scale{uniform} & $\\forall n$ & \\coord{other:denominator-parity} \\\\\n\\hline\n\\multicolumn{4}{|c|}{\\textbf{--- frontier: every entry below needs a cofinal witness,\n$\\forall N\\,\\exists n\\ge N(\\ldots)$, not yet supplied ---}} \\\\\n\\hline\n\\multicolumn{4}{|l|}{\\emph{Cofinal \u2014 the shape both open producers actually need}} \\\\\n\\hline\nTHE master open supply: exact rows recur at arbitrarily large endpoints, with no coherence\nrequired between witnesses. \\lean{CofinalExactLocalMersenneHalfRows}{BooleanMobiusCofinalExactRows.lean:38}\n\\ev{Open}\n& \\scale{cofinal} & $\\forall N\\ \\exists n\\ge N,\\ \\mathrm{ExactLocalMersenneHalfRow}(n)$\n& \\coord{mobius-mersenne} \\\\\n\\hline\nPayoff theorem consuming the row above: compactness of the achievement set forces $1/2$ itself to\nbe achieved. \\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_cofinalExactLocalRows}{BooleanMobiusCofinalExactRows.lean:71}\n\\ev{Lean} (theorem proved; \\emph{hypothesis} open)\n& \\scale{cofinal} & given $\\forall N\\,\\exists n\\ge N(\\ldots)$ & \\coord{other:topological-achievement-set} \\\\\n\\hline\nShortest known open target: the canonical rational half-greedy orbit itself has infinitely many\npositive skip events. \\lean{CofinalPositiveHalfGreedySkips}{BooleanMobiusSkipRowCofinal.lean:22}\n\\ev{Open}\n& \\scale{cofinal} & $\\forall N\\ \\exists c\\ge\\max(N,4),\\ 0<\\mathrm{rem}(c{-}1)<w_c$\n& \\coord{greedy-orbit} \\\\\n\\hline\nPayoff theorem consuming the row above, with zero further combinatorial search once supplied.\n\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_positiveHalfGreedySkips}{BooleanMobiusSkipRowCofinal.lean:73}\n\\ev{Lean} (theorem proved; hypothesis open)\n& \\scale{cofinal} & given $\\forall N\\,\\exists c\\ge N(\\ldots)$ & \\coord{greedy-orbit} \\\\\n\\hline\nMaster half-branch dichotomy: membership iff the greedy skip set is infinite.\n\\lean{half\\_mem\\_mersenneAchievementSet\\_iff\\_greedySkippedSupport\\_infinite}{GreedyAchievementSet.lean:2514}\n\\ev{Lean} (iff proved; both sides cofinal-shaped)\n& \\scale{cofinal} & iff $(\\mathrm{skippedSupport}).\\mathrm{Infinite}$ & \\coord{greedy-orbit} \\\\\n\\hline\nSeven-way classification hub: membership iff the seam word is not eventually right, iff\nunbounded terminal-false, iff cofinally many skipped ranks, \\emph{etc.}\n\\lean{half\\_mem\\_mersenneAchievementSet\\_iff\\_not\\_seamGreedyEventuallyRight}{HalfCylinderHalfMembershipClassification.lean:112}\n\\ev{Lean}\n& \\scale{cofinal} & every right-hand side is $\\exists^\\infty$/Unbounded-shaped & \\coord{seam-integer} \\\\\n\\hline\nLadder dichotomy: either infinitely many rungs survive (\\#257 false, explicit limit support) or an\neventual bad-rank wall closes only this producer. (source: \\texttt{erdos257\\_truncation\\_rung\\_ladder\\_2026\\_07\\_24.md},\nCor.~8) \\ev{Math}\n& \\scale{cofinal} & open branch (a): $\\exists^\\infty J,\\ \\mathrm{HalfRung}(J)$ & \\coord{rung-truncation} \\\\\n\\hline\nThe unconditional wall, stated in its four equivalent forms (run length, deviation, distance-to-integer,\ngreedy support). (source: \\texttt{erdos257\\_sqrt\\_escape\\_digit\\_reduction\\_2026\\_07\\_24.md}, \\S5)\n\\ev{Conjecture}/\\ev{Open}\n& \\scale{cofinal} & $\\forall r\\ge10,\\ |\\mathrm{rem}(r{+}1)-2^{r+1}|>2^{(r+5)/2}$ & \\coord{seam-integer} \\\\\n\\hline\n\\caption{The scale ladder for Erd\u0151s \\#257. Every row above the frontier is machine-checked or\nexactly certified at fixed/bounded/uniform scale; every row below it is exactly the cofinal\nstatement an open producer needs, with its quantifier prefix made explicit so the gap cannot be\nmistaken for a finite verification.}\n\\label{tab:scale-ladder}\n\\end{longtable}\n\\endgroup\n\n\\begin{rem}[Nothing promotes for free]\n\\label{rem:promotion-audit}\nA natural question, once the ladder is visible, is whether any \\scale{uniform} or \\scale{bounded}\nresult already in the tree secretly \\emph{is} the cofinal statement in disguise \u2014 whether some\nexisting theorem's hypothesis, read carefully enough, is uniform in a parameter that a cofinal\nwitness sequence could simply walk through. An audit pass read each Lean proof body (not just the\nstatement) of eighteen such candidates drawn from the fixed/bounded/uniform rows of\nTable~\\ref{tab:scale-ladder} and their immediate neighbours, checking specifically for a proof\ntechnique general enough to instantiate at an unboundedly growing parameter without degradation.\nNone promoted. The recurring blockers were three: a finite certificate table baked into the proof\n(the bound only extends as far as the table was built, \\emph{e.g.} rows~13--31 in\n\\lean{seamUpperResetDyadicBandEscape\\_through\\_thirty}{HalfCylinderUpperResetBandCertificates.lean:78});\na kernel \\texttt{interval\\_cases}/\\texttt{decide} discharge over an explicitly bounded range, which\nis definitionally not re-runnable past its stated bound; and constants inside the bound that\ndegrade with the very parameter a cofinal witness would need to grow (a threshold of the shape\n$2^{(r+5)/2}$ or a margin that shrinks as $1/\\sqrt N$ is uniform \\emph{at} any fixed depth but does\nnot, by itself, manufacture the next witness at a larger depth). This is consistent with, not a\nsubstitute for, the general logical point above: promoting a uniform statement to a cofinal one\nrequires new content \u2014 an explicit witness-producing construction \u2014 not a change of quantifier\nbookkeeping.\n\\end{rem}\n\n\\section{One orbit in four coordinates}\n\\label{sec:four-coordinates}\n\n\\subsection*{One decision stream, four integerisations}\n\nFix the weights $x_n:=1/(2^n-1)$, $n\\ge2$, and the target $1/2$. The single object under study\nthroughout this programme is the greedy orbit for this target: a residual $\\rho$ starts at $1/2$,\nand at rank $k$ the orbit \\emph{takes} iff $\\rho\\ge x_k$. A skip at rank $k$ is \\emph{safe} iff\n$\\rho\\le T_{k+1}:=\\sum_{j>k}x_j$ and \\emph{fatal} iff $\\rho\\in(T_{k+1},x_k)$ \u2014 the fatal interval is\nexactly the gap no later tail can repair. The repo does not study four different problems; it\nmeasures this one orbit through four different integer coordinates, each certified to a stated\ndepth.\n\n\\begin{itemize}\n\\item \\textbf{Truncation rung $J$.} Replace the full weight $x_n$ by the $J$-term truncation\n$w_n^{(J)}:=\\sum_{q=1}^{J}2^{-qn}$ and ask whether some Boolean support hits $1/2$ exactly under\nthe truncated weights. Certified for $3\\le J\\le22$ (Table~\\ref{tab:scale-ladder}, bounded row;\nTheorem~\\ref{thm:tr-witness-exclusion}--\\ref{thm:tr-finite-decision} below).\n\\item \\textbf{Seam row $s$.} The base-4 seam recursion tracking the deviation $\\Delta_s:=\\mathrm{rem}(s)-2^s$\nof the integer greedy orbit from its dyadic target. Certified rows $6$ through $2500$ for the\nreset-crossing classification (\\S\\ref{sec:scale-ladder}, uniform rows; source:\n\\texttt{erdos257\\_reset\\_crossing\\_unification\\_2026\\_07\\_24.md}, \\S8), and the underlying real\norbit separately certified to row $200{,}000$ (source: \\texttt{erdos257\\_sqrt\\_escape\\_digit\\_reduction\\_2026\\_07\\_24.md},\n\\S3): exactly one TARGET violation in that range, at $r=7$.\n\\item \\textbf{Integer margin $m_c$.} An advisory Type~B scan of the integer-greedy margin out to\n$5\\times10^5$, tracking how far the certified region sits from the danger threshold.\n\\item \\textbf{Sharp tail margin.} The rank-by-rank take/skip margin against the exact tail, checked\nfor ranks $2$ through $3000$: $1497$ takes against $1502$ skips, zero fatal skips anywhere, minimum\ncertified gap-margin $+2.9922$ bits at rank $5$, and the single tightest real data point in the\nwhole corpus at rank $7$ \u2014 the near-tie $1/126\\ge1/127$, margin $+0.0340$ bits (source:\n\\texttt{erdos257\\_reset\\_crossing\\_unification\\_2026\\_07\\_24.md}, \\S8).\n\\end{itemize}\n\n\\begin{prop}[One orbit, four coordinates]\n\\label{prop:one-orbit}\nFix the weights $x_n=1/(2^n-1)$ and target $1/2$. Sufficiently fine truncations of any one of the\nfour integerisations above reproduce the full greedy orbit's decisions exactly on any fixed finite\nprefix of ranks; consequently all four coordinates decide the \\emph{same} question, namely whether\n$1/2$ lies in the Mersenne achievement set.\n\\end{prop}\n\\begin{proof}[Mechanism]\nThis is not an approximation claim to be taken on faith; it rests on the fact below.\n\\end{proof}\n\\begin{lem}[No-ties lemma]\n\\label{lem:no-ties}\nAt every rank $k$ of the full greedy orbit for target $1/2$, both defining comparisons are strict:\n$\\rho\\ne x_k$ and $\\rho\\ne T_{k+1}$.\n\\end{lem}\n\\begin{proof}\nTwo independent mechanisms, one for each boundary, both closing a boundary that a naive\ntruncation argument would otherwise have to worry about degenerating on.\n\n\\emph{Take boundary excluded (parity).} Any finite Mersenne sum $P/D:=\\sum_{d\\in D}x_d$ has $D$\nodd, since every weight's denominator $2^d-1$ is odd and a product/lcm of odd numbers is odd\n(equivalently, \\lean{halfGreedyPrefixDenominator\\_odd}{DyadicPrefixCompression.lean:1366}, itself\ntraced to \\lean{finiteErdosSum\\_den\\_odd}{DyadicPrefixCompression.lean:1358}). Hence the residual\nafter any finite prefix is $\\rho=\\tfrac12-\\tfrac{P}{D}=\\tfrac{D-2P}{2D}$: its reduced denominator\nis \\emph{even}. Every weight $x_k=1/(2^k-1)$ has \\emph{odd} denominator. An even-denominator\nrational can never equal an odd-denominator rational, so $\\rho=x_k$ is impossible at any rank \u2014\nthe take comparison is never a tie.\n\n\\emph{Skip-safety boundary excluded (irrationality).} The tail past rank $k$ is\n$T_{k+1}=E-1-\\sum_{2\\le j\\le k}x_j$, where $E=\\sum_{n\\ge1}1/(2^n-1)$ is the Erd\u0151s--Borwein\nconstant: a rational shift of $E$. Erd\u0151s's own 1948 theorem gives $E$ irrational\n(\\lean{irrational\\_erdosBorwein\\_series}{CertificateKernel.lean:8007}, instantiated at $b=2$), so\n$T_{k+1}$ is irrational for every $k$. The residual $\\rho$ arising from any finite Boolean prefix\nis rational. An irrational number never equals a rational one, so $\\rho=T_{k+1}$ is impossible at\nany rank \u2014 the skip-safety comparison is never a tie either.\n\nBoth boundaries of the fatal interval $(T_{k+1},x_k)$ are therefore approached only strictly, at\nevery rank, unconditionally. \\qed\n\\end{proof}\n\\begin{rem}\nLemma~\\ref{lem:no-ties} is exactly what licenses Proposition~\\ref{prop:one-orbit}: because no\ncomparison is ever exactly on a boundary, every rank's decision is stable under sufficiently fine\ntruncation \u2014 there is no knife-edge case where refining $J$, extending $s$, tightening $m_c$, or\nsharpening the tail margin could flip a decision that a coarser coordinate had already settled on a\nfixed prefix. This is the sense in which the four bullet points above are one decision stream, not\nfour independent producers that happen to agree so far.\n\\end{rem}\n\n\\subsection*{The truncation-rung ladder}\n\nThe cleanest of the four coordinates to reason about directly is the rung ladder, because at each\nfinite $J$ it is a genuinely finite, decidable question. Fix $J\\ge2$ and set\n$$w_n^{(J)}:=\\sum_{q=1}^{J}2^{-qn},\\qquad T_{n+1}^{(J)}:=\\sum_{q=1}^{J}\\frac{2^{-qn}}{2^q-1},\\qquad\n\\mathrm{HalfRung}(J):=\\exists\\,A\\subseteq\\{2,3,\\ldots\\},\\ \\sum_{n\\in A}w_n^{(J)}=\\tfrac12.$$\nEvery result in this subsection is \\ev{Math}, proved in ordinary mathematics and cross-checked by\nexhaustive or randomised computation, \\emph{not} yet formalised in Lean.\n\n\\begin{lem}[Forced greedy]\n\\label{lem:tr-forced-greedy}\nFor every $J\\ge2$: $w_n^{(J)}>T_{n+1}^{(J)}$ for all $n$ (the $q=1$ terms agree exactly, and every\n$q\\ge2$ tail term is strictly smaller than the corresponding weight term). Consequently the weight\nsequence is strictly superincreasing, the greedy support is the unique candidate support, and\n$\\mathrm{HalfRung}(J)$ holds iff the greedy orbit for $1/2$ under weights $w_n^{(J)}$ never lands in\na fatal interval $(T_{n+1}^{(J)},w_n^{(J)})$. Rank $1$ is always a safe skip; ranks $2$ and $3$ are\nalways takes.\n\\end{lem}\n\n\\begin{lem}[Parity forces infinite support]\n\\label{lem:tr-parity}\nFor every $J\\ge2$: no finite $A$ attains $\\mathrm{HalfRung}(J)$. Scaling by $2^{J\\cdot\\max A}$ makes\nthe maximal element's contribution odd while every other element's contribution is even, so a\nsupport achieving $1/2$ exactly under $J$-truncated weights must be infinite. (This is the rung\nanalogue of the parity exclusion in Lemma~\\ref{lem:no-ties}: the mechanism is the same shape, one\nlevel down.)\n\\end{lem}\n\n\\begin{thm}[Witness exclusion]\n\\label{thm:tr-witness-exclusion}\nDefine the misalignment mass\n$$\\mu_J(M):=\\sum_{q=2}^{J}\\frac{2^{M\\bmod q}}{2^q-1}.$$\nLet $J\\ge3$, $n\\ge4$, and suppose some $M\\in[n,2n-2]$ has $\\mu_J(M)\\le\\tfrac{11}{15}$. Then no\nBoolean prefix $D\\subseteq\\{2,\\ldots,n-1\\}$ satisfies\n$$T_{n+1}^{(J)}<\\tfrac12-\\sum_{d\\in D}w_d^{(J)}<w_n^{(J)};$$\nfatality at rank $n$ is impossible for every prefix simultaneously (not merely the greedy one).\nThe proof mechanism is a 2-adic scaling and divisibility argument at the witness index $M$; it is\nself-contained and general beyond this instance (\\S\\ref{sec:scale-ladder}, TR-2 chains-with note:\nit is itself a scoped instance of the missing near-integer anti-concentration lemma that both open\nproblems ultimately need).\n\\end{thm}\n\n\\begin{cor}[Half-LCM horizon]\n\\label{cor:tr-half-lcm}\nLet $L_J:=\\mathrm{lcm}(2,3,\\ldots,J)$ and let $M$ be the least multiple of $L_J$ with $M\\ge n$.\nThen $M\\le2n-2$ whenever $n\\ge L_J/2+1$, and at that $M$,\n$$\\mu_J(M)=\\sum_{q=2}^{J}\\frac{1}{2^q-1}<E-1<0.6067<\\frac{11}{15}$$\n(where $E$ is the Erd\u0151s--Borwein constant). Hence, by Theorem~\\ref{thm:tr-witness-exclusion}, every\nrank $n\\ge L_J/2+1$ is automatically witness-covered: the remaining possibly-bad ranks form a\nfinite window $[4,L_J/2]$, not a cofinite complement to be checked one at a time.\n\\end{cor}\n\n\\begin{lem}[Mod-12 filter, $J\\ge7$]\n\\label{lem:tr-mod12}\nFor $J\\ge7$: $\\mu_J(M)\\le\\tfrac{11}{15}\\Rightarrow 12\\mid M$ (case analysis on $M\\bmod4$ and\n$M\\bmod3$). Combined with Corollary~\\ref{cor:tr-half-lcm}, the search for possibly-bad ranks inside\n$[4,L_J/2]$ can be restricted to multiples of $12$, a $12\\times$ speedup with no loss of coverage.\n\\end{lem}\n\n\\begin{thm}[Finite decision procedure]\n\\label{thm:tr-finite-decision}\nCall $n\\in[4,L_J/2]$ \\textbf{bad} if no $M\\in[n,2n-2]$ has $\\mu_J(M)\\le\\tfrac{11}{15}$, and set\n$B(J):=\\max(\\mathrm{bad}\\cup\\{3\\})$. Then $\\mathrm{HalfRung}(J)$ holds iff the greedy orbit for\n$1/2$ under weights $w_n^{(J)}$ survives every rank from $2$ through $B(J)$ \u2014 a finite, exact,\ndecidable check, with $B(J)$ observed to compress the certificate window by up to $\\sim115{,}000\\times$\nrelative to the naive $L_J/2$ bound (\\emph{e.g.} $J=19$: $B(19)=1008$ against $L_{19}/2=116{,}396{,}280$).\n\\end{thm}\n\n\\begin{rem}[Certified rungs and the open producer]\nEvery rung $3\\le J\\le22$ survives this finite check with zero fatal ranks encountered \u2014 these are\nproved theorems $\\mathrm{HalfRung}(3),\\ldots,\\mathrm{HalfRung}(22)$, not merely computational\nevidence, each an independently bankable finite fact (\\ev{Math}+\\ev{Cert};\nTable~\\ref{tab:scale-ladder}, bounded row). By the compactness-transfer theorem of the same source\ndocument (rung solutions lie within $(2/3)4^{-J}$ of $1/2$ in full-Mersenne value, so a convergent\nsubsequence under the compactness of $\\{0,1\\}^{\\{2,3,\\ldots\\}}$ has limit exactly $1/2$, and that\nlimit's support is infinite since finite sums have odd denominator by\nLemma~\\ref{lem:no-ties}/\\ref{lem:tr-parity}), a single further success at \\emph{infinitely many} $J$\nwould already disprove Erd\u0151s \\#257. This is exactly the frontier row of Table~\\ref{tab:scale-ladder}:\n$(\\ast)$ ``$\\mathrm{HalfRung}(J)$ holds for infinitely many $J$'' is the open, cofinal statement;\ntwenty individually-proved finite instances, however many, do not by themselves supply it.\n\\end{rem}\n\n% ---- part a257_p3 ----\n\\section{Interface index: what stands between the corpus and a proof}\n\nThis section is the sharp end of the paper. Parts I--II assembled the corpus; here every\nresult in it is tested against three exact open obligations and rendered as a usable premise:\nwhat it yields, the exact mismatch against the obligation, a classified gap kind\n(\\texttt{hypothesis\\_strength}, \\texttt{scale\\_only}, \\texttt{quantifier\\_order},\n\\texttt{coordinate\\_only}, or \\texttt{multiple}), and the exact auxiliary statement that would\nclose it. Nothing here decides \\#249 or \\#257; the corpus's own moderator standard applies\nthroughout \u2014 a reduction is not a result, and a finite verified list is not a cofinal supply.\n\n\\subsection{The three \\#257 obligations, stated exactly}\n\n\\begin{defn}[257-reset]\nFor every reset row $r \\ge 31$: $\\Delta_{r+1}^2 > 2^{r+5}$, where $\\Delta_{r+1}$ is the signed\ndeviation of the greedy remainder from $2^{r+1}$ at the row immediately following reset $r$.\nEquivalently: the R-run (all-right run) beginning at reset $r$ has length $L_r < (r-3)/2$.\nEquivalently, at the natural dyadic scale $2^{-n}$, the greedy skip-set sum never approximates\n$C = E - 3/2$ to within about $2^{-1.5n}$ (here $E$ is the Erd\u0151s--Borwein constant). Any bound\n$L_r = o(r)$ closes the obligation. A \\emph{uniform} constant bound on $L_r$ is FALSE \u2014 runs\ngrow like $\\log_2(\\mathrm{row})$ \u2014 so the correct target is a growing, not constant, run-length\nceiling.\n\\end{defn}\n\n\\begin{defn}[257-cofinal-rows]\n$\\forall N\\ \\exists n \\ge N$ with \\lean{ExactLocalMersenneHalfRow}{BooleanMobiusCofinalExactRows.lean} $n$\n\u2014 i.e.\\ cofinally many finite prefixes whose local Mersenne quotient lands exactly on the\nhalf-value residue class. Consumed unconditionally by\n\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_cofinalExactLocalRows}{BooleanMobiusCofinalExactRows.lean}.\nNo coherence between the witnessing rows is demanded and none should be assumed: the rows may\nbe, and in the corpus's own words, mutually incompatible \u2014 only their endpoint lengths need\ntend to infinity. Any weakening that reintroduces a compatibility requirement between rows is\nnot a weakening of this obligation; it is a different, harder one.\n\\end{defn}\n\n\\begin{defn}[257-universal]\n$\\forall b \\ge 2\\ \\forall A \\subseteq \\mathbb{N}$ infinite: $\\sum_{n \\in A} 1/(b^n-1)$ is\nirrational. This is Erd\u0151s \\#257 at full strength \u2014 every infinite support, not one target\nvalue and not one structural class.\n\\end{defn}\n\n\\subsection{Reading a row}\n\nEach entry below has the form: \\textbf{tag --- declarations}, \\texttt{site} (file:line, opened\nand read, not inferred), \\texttt{gap\\_kind}, \\textbf{yields} (the exact statement, quantifiers\npreserved), \\textbf{mismatch} (the precise distance from the obligation, with the mechanism),\n\\textbf{closes} (the exact auxiliary statement \u2014 not a vague direction \u2014 that would discharge\nthe row). Evidence band is \\ev{Lean} unless marked \\ev{Math} or \\ev{Cert} (exact finite\ncomputation).\n\n\\subsection{257-reset: sixteen near misses}\n\n\\begin{obs}[oa-2 / deficit-side run law] \\ev{Lean}\n\\lean{upperOrMiddle\\_within\\_of\\_two\\_pow\\_deficit}{HalfCylinderResetDeficitEscape.lean:261}.\n\\textbf{Yields:} $\\forall k, s.\\ 5 \\le s \\to \\mathrm{rem}(s) \\le 2^s \\to 2^s \\le 2^k(2^s -\n\\mathrm{rem}(s)) \\to \\exists t \\in [s,s+k]$ with an upper-or-middle event at $t$. Contrapositive:\nan all-right run of length $k$ from row $s$ forces deficit $2^s - \\mathrm{rem}(s) < 2^{s-k}$.\nUniform in \\emph{both} $k$ and $s$; nothing scale-restricted here.\n\\textbf{Mismatch (\\texttt{hypothesis\\_strength}):} the implication side is already at cofinal,\nuniform strength. What is missing is the deficit \\emph{lower} bound fed into it. The only\nunconditional deficit bound on hand is integrality ($\\mathrm{rem}(s) < 2^s \\Rightarrow$ deficit\n$\\ge 1$); instantiating $k=s$ with deficit $\\ge 2^0$ gives only $L_r \\le r$. The obligation needs\n$L_r < (r-3)/2$ \u2014 short by exactly a factor of two in the exponent.\n\\textbf{Closes it:} $2^{(r+3)/2} \\le 2^r - \\mathrm{rem}(r)$ at every reset $r \\ge 31$. Nothing else\nin this theorem changes; the induction already consumes any $k$ uniformly.\n\\end{obs}\n\n\\begin{obs}[rc-2 / affine excess recurrence] \\ev{Lean}\n\\lean{affineRightExcess\\_exactIterate}{HalfCylinderMiddleCarryLowerBound.lean:2309},\n\\lean{affineRightExcess\\_charge\\_lt\\_scaledInitial}{HalfCylinderMiddleCarryLowerBound.lean:2341},\n\\lean{eventualRightTail\\_excess\\_exactIterate}{HalfCylinderMiddleCarryLowerBound.lean:2353}\n(recurrence \\lean{rightBranch\\_excess\\_succ\\_eq}{HalfCylinderMiddleCarryLowerBound.lean:1636}).\n\\textbf{Yields:} for an R-run of length $k$ from row $S$: $X_k + \\mathrm{charge}_k = 4^k X_0$\nexactly, with $X_j = \\mathrm{rem}(S+j) - 2^{S+j}$ and $\\mathrm{charge}_{j+1} = 4\\,\\mathrm{charge}_j\n+ \\mathrm{pulse}_j + 4$. Uniform in $k$, $S$, and the pulse word.\n\\textbf{Mismatch (\\texttt{multiple}):} this is the excess-side twin of the deficit run law but\nonly half-assembled: $X_0 \\le (X_k+\\mathrm{charge}_k)/4^k$ needs an upper bound on the terminal\nexcess $X_k$ before it yields a run-length statement. Two candidate upper bounds exist\nelsewhere in the corpus (the unconditional late-row ceiling below, and the open\n\\texttt{SeamGreedyRemainderGapBound} socket) but neither is composed with this iterate anywhere\nin the tree.\n\\textbf{Closes it:} one composition lemma \u2014 if the R-run from $S$ has length $k$ and every row in\nit satisfies the late-row condition $2s<3d$ (equivalently $\\mathrm{rem}(s) < 2^{s+1}$), then\n$\\mathrm{rem}(S) - 2^S \\le 2^{S-k} + (2S+4)/3$. That would make the run law two-sided; the\nanti-concentration input is still separate.\n\\end{obs}\n\n\\begin{obs}[oa-2 sibling / late-row ceiling] \\ev{Lean}\n\\lean{three\\_mul\\_remainder\\_lt\\_exactLateGap}{HalfCylinderResetDeficitEscape.lean:117}.\n\\textbf{Yields:} $\\forall s \\ge 5$, at a largest false rank $d$ with $2s<3d$ (late row):\n$3\\,\\mathrm{rem}(s) < 3\\cdot 2^{s+1} + 2\\cdot4^{s-d}+4$. Unconditional, uniform in $s,d$; since\nlate means $s-d<s/3$, the error term is $<2^{2s/3}$.\n\\textbf{Mismatch (\\texttt{coordinate\\_only}):} an upper bound on the deviation where the\nobligation needs a lower bound on its absolute value \u2014 the excess-side ceiling the rc-2 run\nlaw needs, not anti-concentration itself. Degrades to vacuous outside the late-row regime.\n\\textbf{Closes it:} nothing on its own; its role is discharging the ``$X_k$ bounded above''\npremise of rc-2 without invoking the open \\texttt{SeamGreedyRemainderGapBound}.\n\\end{obs}\n\n\\begin{obs}[G5 / no-tie] \\ev{Lean}\n\\lean{seamGreedyFloorZ\\_ne\\_takeThreshold}{HalfCylinderMiddleCarryLowerBound.lean:804}, with\n\\lean{halfGreedy\\_skipped\\_endpoint\\_trichotomy}{HalfCylinderSkippedEndpointClassifier.lean:246}.\n\\textbf{Yields:} $\\forall s \\ge 5$, the greedy decision boundary is never exactly attained.\nCombined with integrality this is the corpus's \\emph{only} unconditional lower bound on the\ndeviation: $|\\Delta_r| \\ge 1$.\n\\textbf{Mismatch (\\texttt{scale\\_only}):} deviation $\\ge 2^0$ versus the required deviation $>\n2^{(r+5)/2}$. This is the honest state of the anti-concentration axis: the entire gap is the\nexponent, from $0$ to $(r+5)/2$, with nothing proved in between.\n\\textbf{Closes it:} any growing lower bound $|\\mathrm{rem}(r)-2^r| \\gtrsim 2^{\\varepsilon r}$ for\nfixed $\\varepsilon>0$ would be the first non-constant point on this axis; $\\varepsilon = 1/2 +\no(1)$ closes the obligation via the run law.\n\\end{obs}\n\n\\begin{obs}[H4, PRIVATE / sharp fatal-gap test] \\ev{Lean}\n\\lean{skipSafe\\_of\\_two\\_mul\\_le\\_three\\_mul}{Erdos257PeriodNoncollapse/HalfGreedyFatalGap.lean:52},\n\\lean{three\\_mul\\_lt\\_two\\_mul\\_of\\_fatal}{Erdos257PeriodNoncollapse/HalfGreedyFatalGap.lean:120},\n\\lean{two\\_le\\_of\\_fatal}{Erdos257PeriodNoncollapse/HalfGreedyFatalGap.lean:200},\n\\lean{three\\_le\\_of\\_fatal\\_of\\_odd}{Erdos257PeriodNoncollapse/HalfGreedyFatalGap.lean:268}.\n\\textbf{Yields:} writing the residual in lowest terms $\\rho=u/(2L)$, $u,L$ odd, $a := 2L -\n(2^k-1)u$, the sharp safety test is $2u \\le 3a$ (strictly better than the old dyadic test $u \\le\na$). Contrapositive: fatal $\\Rightarrow 3a<2u \\Rightarrow u \\ge 2$, and $u \\ge 3$ when $u$ is\nodd. Uniform in $k \\ge 1$.\n\\textbf{Mismatch (\\texttt{scale\\_only}):} same shape as the obligation \u2014 a numerator lower bound\nat a dangerous rank \u2014 but the bound is the constant $3$ where growth $2^{(r+5)/2}$ is needed. The\nconstant $3$ is the Erd\u0151s--Borwein tail-ratio limit $\\theta_k := 1/T(k+1)-(2^k-1) < 2/3$; adding\nLambert channels sharpens $2/3$ toward the true limit but never produces growth in $k$.\n\\textbf{Closes it:} a rank-scaling version, e.g.\\ fatal at rank $k \\Rightarrow u \\ge 2^{k/2}$. The\ncurrent mechanism provably cannot deliver this \u2014 the ceiling is the tail ratio itself \u2014 so this\nneeds a different arithmetic input, not a sharper truncation.\n\\end{obs}\n\n\\begin{obs}[F5b / linear-width band escape, rows 13--30] \\ev{Lean}\n\\lean{seamUpperResetDyadicBandEscape\\_through\\_thirty}{HalfCylinderUpperResetBandCertificates.lean:78}.\n\\textbf{Yields:} at every upper reset row $d$, $13 \\le d \\le 30$, the reset charge $E_d =\n4\\cdot\\mathrm{overshoot}_d + \\mathrm{abovePulse}_d$ avoids the linear-width band\n$(2^{d-j+1}-2(d+j),\\ 2^{d-j+1}]$ for every $j \\le d$ \u2014 the finite segment of\n\\lean{SeamUpperResetDyadicBandEscape}{HalfCylinderMiddleCarryLowerBound.lean:3531}, which by\n\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_upperResetDyadicBandEscape}{HalfCylinderMiddleCarryLowerBound.lean}\ndelivers the whole \\#257-false payoff.\n\\textbf{Mismatch (\\texttt{scale\\_only}):} proved at 18 explicit rows by \\texttt{interval\\_cases}\nplus per-row \\texttt{decide+kernel}; the obligation needs all $d \\ge 13$. The band width here is\nonly $O(d)$ \u2014 \\emph{linear} \u2014 while sqrt-escape demands width $2^{(r+5)/2}$; this socket is\nexponentially weaker than 257-reset yet lands the same downstream conclusion.\n\\textbf{Closes it:} a structural lower bound on the distance from $E_d$ to the nearest dyadic\npower at or above it, of width $2(d+j_0)$ at the single critical index $j_0$ \u2014 already collapsed\nfrom $\\forall j$ to one inequality by\n\\lean{HalfUpperResetCriticalBand.dyadicBandEscape\\_iff\\_exists\\_critical}{HalfCylinderMiddleCarryLowerBound.lean}.\nOnly $O(d)$ separation is required, not sqrt-of-scale.\n\\end{obs}\n\n\\begin{obs}[oa-3b / M-ancestor and R-run closure] \\ev{Lean}\n\\lean{seamMiddleBranch\\_nextRemainder\\_ge\\_row}{HalfCylinderMiddleCarryLowerBound.lean:1765},\n\\lean{seamMiddleBranch\\_nextRemainder\\_ge\\_expBarrier}{HalfCylinderMiddleCarryLowerBound.lean:1788},\n\\lean{SeamUpperResetDyadicBandEscape.remainder\\_ge\\_row}{HalfCylinderMiddleCarryLowerBound.lean:3606}.\n\\textbf{Yields:} unconditional, uniform in $s\\ge5$: a middle (M) branch forces $s+1 \\le\n\\mathrm{rem}(s+1)$, using no lower bound on the source remainder; with the row-scale source\nbound this upgrades to $2^{s+1}+(s+1) \\le \\mathrm{rem}(s+1)$. The minimal-counterexample argument\nat line 3606 then shows any row-small row must have an upper-reset (U) ancestor, since M-ancestors\nand R-runs are both excluded.\n\\textbf{Mismatch (\\texttt{hypothesis\\_strength}):} two of the three ancestor channels (M, R) are\nclosed unconditionally at all scales; only the U channel remains, and it needs only the\n$O(d)$-wide band escape of F5b above \u2014 sqrt-of-dyadic-scale escape is being demanded by the\n257-reset statement where linear-width escape at U-resets alone suffices for this downstream\nroute.\n\\textbf{Closes it:} the single remaining channel \u2014 $\\forall d \\ge 13$ with successor carries, the\nreset charge avoids the width-$2(d+j_0)$ band at its critical dyadic index. Then\n\\texttt{remainder\\_ge\\_row}, hence \\texttt{SeamMiddleProducerRowEscape}, hence $1/2 \\in\n\\mathrm{mersenneAchievementSet}$, all already in Lean.\n\\end{obs}\n\n\\begin{obs}[F2 / middle producer row escape] \\ev{Lean}\n\\lean{SeamMiddleProducerRowEscape}{HalfCylinderMiddleCarryLowerBound.lean:908},\n\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_middleProducerRowEscape}{HalfCylinderMiddleCarryLowerBound.lean:918}\n(via \\texttt{.toCardEscape}, \\texttt{.toTailEscape}, \\lean{}{HalfCylinderMiddleCarryLowerBound.lean:1510}).\n\\textbf{Yields:} if at every late middle row ($s \\ge 13$, no carry, middle branch)\n$s \\le \\mathrm{rem}(s)$, then $1/2 \\in \\mathrm{mersenneAchievementSet}$. Module docstring: ``far\nweaker than an exponential bound, but stronger than the already exposed square-root-scale carry\ntarget.''\n\\textbf{Mismatch (\\texttt{hypothesis\\_strength}):} this socket asks for a \\emph{linear-in-row}\nlower bound on the remainder itself; the obligation supplies an \\emph{exponential} lower bound\non the deviation from $2^s$ at reset rows. Neither implies the other \u2014 sqrt-escape is compatible\nwith $\\mathrm{rem}(s)=0$, which violates $s \\le \\mathrm{rem}(s)$. A genuinely cheaper, incomparable\nsocket on the same payoff.\n\\textbf{Closes it:} $\\forall s \\ge 13$ at a middle transition, $s \\le \\mathrm{rem}(s)$.\nEquivalently, by \\texttt{remainder\\_ge\\_row}, the U-reset band escape of F5b.\n\\end{obs}\n\n\\begin{obs}[LPC-sqrt / row-index sqrt escape] \\ev{Lean}\n\\lean{SeamMiddleProducerSqrtEscape}{HalfCylinderLastProducerContradiction.lean:84},\n\\texttt{.toTailEscape}{}{HalfCylinderLastProducerContradiction.lean:98},\n\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_middleProducerSqrtEscape}{HalfCylinderLastProducerContradiction.lean:409}.\n\\textbf{Yields:} if at every late middle row $2\\sqrt{2s+2}+4 < \\mathrm{producerCarry}(s)$, then\n$1/2 \\in \\mathrm{mersenneAchievementSet}$; \\texttt{producerCarry} equals the middle coordinate\n$4\\,\\mathrm{rem}-\\mathrm{belowPulse}-4$\n(\\lean{}{HalfCylinderProducerLowerBound.lean:150}), so the requirement is essentially\n$\\mathrm{rem}(s) \\gtrsim (\\sqrt{s}+\\mathrm{belowPulse})/4$.\n\\textbf{Mismatch (\\texttt{coordinate\\_only}):} this is ``square root'' of the \\emph{row index}\n$s$ (a supportCoeff tail majorant), not square root of the \\emph{dyadic scale} $2^{(r+5)/2}$. The\ntwo sqrt's in this corpus are exponentially far apart and easy to conflate; this socket is the\nweaker one by an exponential margin, and it is already the theorem-ready form of the\nlast-producer contradiction.\n\\textbf{Closes it:} $\\forall s \\ge 13$ at a middle transition, $4\\,\\mathrm{rem}(s) -\n\\mathrm{belowPulse}(s) - 4 > 2\\sqrt{2s+2}+4$ \u2014 a row-scale, not dyadic-scale, lower bound on the\nmiddle coordinate.\n\\end{obs}\n\n\\begin{obs}[G2 + LPC upper case / narrow integer window] \\ev{Lean}\n\\lean{upperProducer\\_not\\_last}{HalfCylinderLastProducerContradiction.lean:250},\n\\lean{middleProducer\\_neg\\_three\\_not\\_last}{HalfCylinderLastProducerContradiction.lean:315},\n\\lean{finalMiddleCell\\_neg\\_three\\_not\\_last}{HalfCylinderFinalMiddleCellEscape.lean:587},\n\\lean{middleProducer\\_allRight\\_landingExcess\\_window}{HalfCylinderMiddleCarryLowerBound.lean:1458},\n\\lean{middleThenAllRight\\_landingExcess\\_two\\_le}{HalfCylinderMiddleCarryLowerBound.lean:1683}.\n\\textbf{Yields:} the upper-producer ``last transition before an all-right tail'' case is\nimpossible unconditionally (affine carry $\\le -8$); the middle cell at coordinate $-3$ is\nexcluded unconditionally. What survives: a middle producer whose landing excess $X_{D+1}$\nsatisfies $2 \\le X_{D+1} < 2\\sqrt{2D+2}+8$ \u2014 a window of only $O(\\sqrt D)$ integers.\n\\textbf{Mismatch (\\texttt{hypothesis\\_strength}):} the residual bad set is an explicit\n$O(\\sqrt D)$-wide integer window at each candidate last-producer row, not an interval at dyadic\nscale. The obligation would evacuate this window wholesale, but excluding an $O(\\sqrt D)$ window\nis exponentially cheaper than proving $|\\Delta|>2^{(r+5)/2}$.\n\\textbf{Closes it:} exclusion of the integer window $[2,\\ 2\\sqrt{2D+2}+7]$ for the landing excess\nat any hypothetical last middle producer $D \\ge 13$ \u2014 e.g.\\ a congruence or pulse-parity\nobstruction on $4\\,\\mathrm{rem}(D)-\\mathrm{belowPulse}(D)-4$.\n\\end{obs}\n\n\\begin{obs}[F4 / two-sided cell exclusion] \\ev{Lean}\n\\lean{SeamTwoSidedDyadicCellEscape}{HalfCylinderMiddleCarryLowerBound.lean:3339},\n\\texttt{.step}{}{HalfCylinderMiddleCarryLowerBound.lean:3362}, \\texttt{.twoSided}{}{HalfCylinderMiddleCarryLowerBound.lean:3413}.\n\\textbf{Yields:} excluding exactly three integer values of the middle coordinate\n($4\\,\\mathrm{rem}-\\mathrm{belowPulse}-4 \\notin \\{-3,-2,-1\\}$) plus one right-pulse-leak bound\npropagates, by induction from a decide-checked row-5 base, the two-sided invariant $\\forall s\n\\ge 5$: $\\mathrm{rem}(s) \\le 2^s \\vee \\mathrm{overshoot}(s) \\le 2^s$.\n\\textbf{Mismatch (\\texttt{hypothesis\\_strength}):} the hypothesis is anti-concentration at\n\\emph{constant} resolution \u2014 the orbit must miss three named integers \u2014 where the obligation is\nanti-concentration at resolution $2^{(r+5)/2}$. One of the three cells ($-3$) is already closed\nin the all-right-tail configuration by \\texttt{finalMiddleCell\\_neg\\_three\\_not\\_last}, leaving\ntwo.\n\\textbf{Closes it:} exclusion of the two remaining cells ($-2$ and $-1$) for the actual seam\norbit, plus the right-pulse-leak inequality $4\\cdot\\mathrm{overshoot}+\\mathrm{abovePulse} \\le\n2^{s+2}$ under $\\mathrm{overshoot} \\le 2^s$.\n\\end{obs}\n\n\\begin{obs}[E1 / remainder ceiling] \\ev{Lean}\n\\lean{SeamGreedyRemainderGapBound}{HalfCylinderSeamLimit.lean:192},\n\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_seamRemainderGapBound}{HalfCylinderSeamLimit.lean:55}\n(weaker cofinal route \\lean{}{HalfCylinderSeamLimit.lean:181}).\n\\textbf{Yields:} if $\\forall s \\ge 6$, $\\mathrm{rem}(s) < 2^{s+1}$, then $1/2 \\in\n\\mathrm{mersenneAchievementSet}$. The weaker cofinal form only needs the normalized remainder to\ntend to $0$ along \\emph{some} cofinal row sequence. Docstring: ``exponential upper bound\nobserved by exact replay from row six onward.''\n\\textbf{Mismatch (\\texttt{coordinate\\_only}):} an upper bound on the remainder rather than a\nlower bound on $|\\text{deviation}|$ \u2014 the ceiling half of the picture, and exactly the premise\nrc-2's excess-side run law is missing; \\texttt{three\\_mul\\_remainder\\_lt\\_exactLateGap} already\nsupplies it unconditionally at late rows up to a $2^{2s/3}$ correction.\n\\textbf{Closes it:} either $\\forall s\\ge6$, $\\mathrm{rem}(s) < 2^{s+1}$ outright, or a proof that\nthe late-row condition $2s<3d$ holds at every row so the unconditional strip invariant covers\nit.\n\\end{obs}\n\n\\begin{obs}[SE-7 / RC-7 / certified computation to row 200{,}000] \\ev{Cert}\n\\textbf{Yields:} exact integer arithmetic to row 200{,}000 \u2014 branch counts $R\\!:\\!M\\!:\\!U =\n100197\\!:\\!49899\\!:\\!49898$; the obligation \\emph{fails only at} $r=7$; maximum R-run $19$ at row\n$158{,}096$ against the required threshold $\\approx (r-3)/2 \\approx 79{,}000$ (margin\n$\\sim\\!4000\\times$). Independently: 1209 resets checked to row 2500, zero classification\nanomalies, exactly two crossing cells ever ($s{=}7,d{=}5$ and $s{=}10,d{=}7$), both below socket\nscope.\n\\textbf{Mismatch (\\texttt{scale\\_only}):} fixed-scale exhaustive verification versus a statement\nat every reset $r \\ge 31$. This is the canonical quantifier trap this programme warns against \u2014 a\nfinite verified list is not a cofinal supply, however large the margin.\n\\textbf{Closes it:} nothing computational; only a theorem. The certificate's value is fixing the\ntrue constant ($r \\ge 31$, sole violation $r=7$) so the theorem's scope can be stated exactly.\n\\end{obs}\n\n\\begin{obs}[RC-5 / rigidity of dangerous resets \u2014 Theorem B, see below] \\ev{Math}\n\\textbf{Yields:} at a dangerous reset $r$ ($|\\mathrm{rem}(r{+}1)-2^{r+1}| \\le 2^{(r+5)/2}$) whose\npreceding reset $r_0$ has pure R-run length $L' = r-r_0-1 > (r+5)/4$, the deviation\n$w_{r_0+1}$ is pinned to at most two adjacent integers determined by the divisor-pulse stream\nalone; a chain of $n$ dangerous resets forces $n{-}1$ nested exact integer-coincidence towers.\n\\textbf{Mismatch (\\texttt{hypothesis\\_strength}):} applies only to dangerous resets preceded by a\nlong pure R-run ($L' > (r+5)/4$) \u2014 an isolated dangerous reset after a short run is untouched.\nPinning to two integers is not yet a contradiction: no theorem excludes those two values. Prose,\nnot Lean.\n\\textbf{Closes it:} (i) a congruence or pulse-stream obstruction showing the two pinned integer\nvalues are unattainable, and (ii) removal of the long-preceding-run hypothesis, or a proof that\ndangerous resets must come in chains. Either alone converts Theorem B into a genuine no-go.\n\\end{obs}\n\n\\begin{obs}[RC-8 / sign law and margin growth] \\ev{Cert}\n\\textbf{Yields:} at all 1209 observed resets, M $\\Rightarrow$ deviation $>0$ and U $\\Rightarrow$\ndeviation $<0$, with zero exceptions; the margin over the $2^{(r+5)/2}$ threshold grows from\n1.11 bits at $r=14$ to $\\sim$1246 bits at $r=2500$. The corpus flags this as a strictly weaker,\npossibly more tractable target than full sqrt-escape.\n\\textbf{Mismatch (\\texttt{multiple}):} two gaps at once \u2014 the sign law is empirical only\n(certified computation, not proved unconditionally as a lemma), and even if proved it delivers\nthe \\emph{sign} of the deviation, not its magnitude, while the obligation needs magnitude.\n\\textbf{Closes it:} prove the sign law as a lemma (M-branch $\\Rightarrow \\mathrm{rem}(r{+}1) >\n2^{r+1}$; U-branch $\\Rightarrow \\mathrm{rem}(r{+}1) < 2^{r+1}$) \u2014 a one-step branch-algebra\nstatement in the coordinates of \\texttt{rightBranch\\_excess\\_succ\\_eq} /\n\\texttt{seamUpperBranch\\_nextRemainder\\_le\\_pow}. That converts the two-sided obligation into two\none-sided bounds, each in a branch class where the exact affine recurrence is already known.\n\\end{obs}\n\n\\begin{obs}[TR-5 / TR-6 / truncation-rung ladder, parallel coordinate] \\ev{Cert}\n\\textbf{Yields:} $\\mathrm{HalfRung}(J)$ genuinely \\emph{proved} (not merely computed) for each $3\n\\le J \\le 22$, via a finite exact decision procedure: every rank $n \\ge L_J/2+1$ is automatically\nwitness-covered, a mod-12 filter cuts the residual search $12\\times$, so only ranks in\n$[4,B(J)]$ need checking. By the corpus's compactness transfer theorem, $\\mathrm{HalfRung}(J)$\nfor infinitely many $J$ gives an infinite $A$ with $\\Sigma = 1/2$, i.e.\\ \\#257 FALSE.\n\\textbf{Mismatch (\\texttt{multiple}):} different coordinate (truncation rung $J$, not seam reset\nrow $r$) \\emph{and} fixed scale (20 explicit values of $J$). Reaches the same downstream payoff\nwithout touching the deviation at all \u2014 a parallel route whose own cofinal step is open, not one\nupgrade away from 257-reset.\n\\textbf{Closes it:} $\\mathrm{HalfRung}(J)$ for infinitely many $J$ \u2014 explicitly \\emph{not}\npromotable by re-running the existing argument: the certificate window $B(J)$ is driven by $L_J\n= \\mathrm{lcm}(2,\\dots,J)$, growing superexponentially in $J$, the hallmark of a proof that does\nnot survive bound removal.\n\\end{obs}\n\n\\subsection{257-cofinal-rows: thirteen near misses}\n\n\\begin{obs}[Sharp-capacity fill, unconditional] \\ev{Lean}\n\\lean{exists\\_exactRowStrictUpperFill\\_of\\_skippedCoreSharpCapacity}{BooleanMobiusSkipRow.lean:286},\n\\lean{exactLocalMersenneHalfRow\\_two\\_mul\\_sub\\_two\\_of\\_skippedCoreSharpCapacity}{BooleanMobiusSkipRow.lean:380}.\n\\textbf{Yields:} for any $c \\ge 4$ and any finite $D \\subseteq [2,c)$ with\n$\\mathrm{localMersennePrefixValue}\\,D < 1/2$ and $\\mathrm{localBinarySuffix}\\,D\\,1\\,(2c{-}2) <\n2^{c-2}$: an exact row $E$ at endpoint $2c{-}2$ with $D \\subseteq E$, every new rank strictly\nabove $c$. Unconditional, no deficit/crossing hypothesis, no reference to the greedy orbit.\n\\textbf{Mismatch (\\texttt{hypothesis\\_strength}):} needs a per-$c$ input\n$\\mathrm{localBinarySuffix}\\,D\\,1\\,(2c{-}2) < 2^{c-2}$, and the corpus only ever supplies that\ninput through the crossing-deficit route, which\n\\lean{eq\\_halfGreedyPrefixSupport\\_of\\_critical\\_crossing}{BooleanMobiusCriticalCapacityCofinal.lean:50}\nthen rigidly collapses onto $D = \\mathrm{halfGreedyPrefixSupport}(c{-}1)$. The theorem itself is\nstrictly more general than every consumer that calls it.\n\\textbf{Closes it ($\\star$):} $\\forall N\\ \\exists c \\ge N,\\ \\exists$ finite $D \\subseteq [2,c)$\nwith $\\mathrm{localMersennePrefixValue}\\,D<1/2$ and $\\mathrm{localBinarySuffix}\\,D\\,1\\,(2c{-}2) <\n2^{c-2}$ \u2014 equivalently $\\mathrm{mobiusCenteredHalfCarry}\\,{\\uparrow}D\\,(2c{-}3) < 2^{c-2}$. No\ncoherence between the $D$'s, no requirement that $D$ be a greedy prefix.\n\\end{obs}\n\n\\begin{obs}[Sharp-capacity induction, engine complete] \\ev{Lean}\n\\lean{SkippedCoreCriticalQuotientSupply}{BooleanMobiusCriticalCapacityCofinal.lean:30} $\\to$\n\\lean{exists\\_laterProtectedExactLocalMersenneRow}{BooleanMobiusCriticalCapacityCofinal.lean:1057}\n$\\to$\n\\lean{cofinalExactLocalMersenneHalfRows\\_of\\_criticalQuotientSupply}{BooleanMobiusCriticalCapacityCofinal.lean:1227}\n(\\lean{ProtectedExactLocalMersenneRow}{BooleanMobiusCriticalCapacityCofinal.lean:1005}; seed\nline 1023; strict progress \\lean{}{BooleanMobiusExactRowCrossing.lean:233}).\n\\textbf{Yields:} a complete, already-formalised induction from the endpoint-six seed to the full\nobligation: each protected row either doubles below half unconditionally, or recycles at its\nfirst crossing rank $e$, giving endpoint $2e{-}2 >$ previous endpoint. Terminates literally at\n\\texttt{CofinalExactLocalMersenneHalfRows}.\n\\textbf{Mismatch (\\texttt{hypothesis\\_strength}):} the recycle branch consumes\n\\texttt{SkippedCoreCriticalQuotientSupply} at essentially every step (the below-half branch never\nfires twice, by the seed arithmetic). Protection (endpoint $< 2\\cdot$cutoff, new ranks $>$\ncutoff) is exactly what converts a non-growing recycle endpoint into strict progress; everything\nelse is finished.\n\\textbf{Closes it:} the sharp inequality at one crossing rank per step, canonical form\n\\lean{HalfGreedySkippedCriticalQuotientSupply}{BooleanMobiusCriticalCapacityCofinal.lean:178}:\n$\\forall c \\ge 4$ skipped by the rational half-greedy orbit, $2^{(2c-2)-1} \\le\n\\mathrm{localPrefixQuotient}(\\mathrm{insert}\\,c\\,(\\mathrm{halfGreedyPrefixSupport}(c{-}1)))\\,(2c{-}2)$.\n\\end{obs}\n\n\\begin{obs}[Unconditional capacity, off by one bit] \\ev{Lean}\n\\lean{localBinarySuffix\\_two\\_mul\\_sub\\_two\\_lt\\_upperHalfCapacity}{BooleanMobiusSkippedCoreExactRow.lean:123}\n(word form line 228).\n\\textbf{Yields:} unconditionally, $\\forall c \\ge 4$ and every below-half core $D \\subseteq [2,c)$\nwith deficit $< \\mathrm{mersenneWeightRat}\\,c$: $\\mathrm{localBinarySuffix}\\,D\\,1\\,(2c{-}2) <\n2^{c-1}$. Uniform in $c$, no table, no case split.\n\\textbf{Mismatch (\\texttt{scale\\_only}):} one bit short. The strict-upper fill needs\n$<2^{c-2}$, this gives $<2^{c-1}$. The proof derives $A < 2^{c-2}+|D|$ then discards\n$|D| \\le c-2 \\le 2^{c-2}$ \u2014 the loss is purely additive. The true unproved statement is only that\n$A$ avoids the linear-width band $[2^{c-2},\\,2^{c-2}+c-3]$, width $c{-}2$ inside a range of size\n$2^{c-2}$.\n\\textbf{Closes it:} a linear-width dyadic-band anti-concentration: at cofinally many crossing\nranks $c$, $\\mathrm{localBinarySuffix}\\,D\\,1\\,(2c{-}2) \\notin [2^{c-2},\\,2^{c-2}+c-3]$. Same band\nshape as F5b/oa-3b above; exponentially weaker than 257-reset's $2^{(r+5)/2}$ target, and the two\nsockets have never been compared on band width in any prior bank.\n\\end{obs}\n\n\\begin{obs}[Doubling extension, fails at the only landed seed] \\ev{Lean}\n\\lean{exists\\_exactRowStrictUpperExtension\\_two\\_mul\\_sub\\_one\\_of\\_exact\\_below}{BooleanMobiusExactRowDoubling.lean:185},\nline 296; rank-two discharge\n\\lean{two\\_mem\\_of\\_exact\\_localMersenneQuotient}{BooleanMobiusExactRowRankTwo.lean:23}, line 108.\n\\textbf{Yields:} $\\forall n \\ge 6$: a below-half exact row at $n$ produces an exact row at\n$2n{-}1$, support-extending, all new ranks $>n$. The only unconditional endpoint-doubling\nproducer in the corpus.\n\\textbf{Mismatch (\\texttt{hypothesis\\_strength}):} below-halfness is not preserved and\ndemonstrably fails at the first step. For the seed $D=\\{2,3,6\\}$ at $M=11$ the core alone\ncontributes $2/3+4/7+32/63 \\approx 1.746 > 1$, so the doubled row is strictly \\emph{above} half.\nIterating the cheap arm is therefore impossible from the only landed seed; the chain is forced\ninto the capacity-gated recycle arm at step one.\n\\textbf{Closes it:} cofinally many below-half exact rows \u2014 $\\forall N\\ \\exists n \\ge N\\ \\exists D$\nexact at $n$ with $\\mathrm{localFractionMass}\\,D\\,n < 1$. A single such row at each of cofinally\nmany $n$ makes doubling unnecessary; a self-reproducing one closes everything.\n\\end{obs}\n\n\\begin{obs}[Consecutive-skip precritical bound] \\ev{Lean}\n\\lean{halfGreedy\\_precriticalSuffix\\_lt\\_of\\_next\\_skip}{BooleanMobiusCriticalCapacityCofinal.lean:682}\n(feeds \\lean{halfGreedySkippedCriticalQuotientSupply\\_of\\_precriticalSuffix}{BooleanMobiusCriticalCapacityCofinal.lean:919},\nthen lines 970/986).\n\\textbf{Yields:} unconditional: if $c \\ge 6$ and both rank $c$ and rank $c{+}1$ are skipped by\nthe rational half-greedy orbit, then $\\mathrm{localBinarySuffix}(\\mathrm{halfGreedyPrefixSupport}\n(c{-}1))\\,1\\,(2c{-}3) < 2^{c-3}$, doubling into sharp capacity at $c$, hence an exact row at\n$2c{-}2$. No supply hypothesis anywhere.\n\\textbf{Mismatch (\\texttt{hypothesis\\_strength}):} needs two \\emph{consecutive} skipped ranks.\nThe residual open set is exactly the skip-then-take rows:\n\\lean{halfGreedySkippedPrecriticalSuffixSupply\\_iff\\_preTake}{BooleanMobiusCriticalCapacityCofinal.lean:756}\nproves the whole socket equivalent to\n\\lean{HalfGreedyPreTakePrecriticalSuffixSupply}{BooleanMobiusCriticalCapacityCofinal.lean:200},\nwith $c=4,5$ already discharged by \\texttt{decide}.\n\\textbf{Closes it:} cofinally many $c \\ge 6$ at which two consecutive ranks $c,c{+}1$ are both\nskipped by the rational half-greedy orbit \u2014 or dually, the pre-take residual at cofinally many\nskipped-then-taken ranks. Either one closes the obligation with no further input.\n\\end{obs}\n\n\\begin{obs}[Bounded-gap precritical bound] \\ev{Lean}\n\\lean{halfGreedy\\_precriticalSuffix\\_lt\\_of\\_future\\_skip\\_after\\_takenBlock}{BooleanMobiusCriticalCapacityCofinal.lean:603}\n(with \\lean{precriticalCrossingTax\\_of\\_futureThreshold}{BooleanMobiusCriticalCapacityCofinal.lean:472},\n\\lean{sub\\_two\\_le\\_two\\_pow\\_sub\\_four}{BooleanMobiusCriticalCapacityCofinal.lean:432}).\n\\textbf{Yields:} unconditional and uniform in \\emph{both} parameters: skip at $c$, takes at\n$c{+}1,\\dots,c{+}t{-}1$, skip at $c{+}t$, with $0<t\\le c-3$ and $c-2\\le 2^{c-t-3}$\n$\\Rightarrow$ precritical suffix bound at $c$ $\\Rightarrow$ sharp capacity $\\Rightarrow$ exact\nrow at $2c{-}2$. Handles any skip gap $t \\lesssim c-3-\\log_2(c-2)$, not just $t=1$.\n\\textbf{Mismatch (\\texttt{hypothesis\\_strength}):} the only missing input is an orbit-level\nskip-gap bound \u2014 that the next skipped rank after $c$ occurs at $c{+}t$ with $c-2 \\le\n2^{c-t-3}$. The proof is fully uniform in $(c,t)$, every constant explicit, no table.\n\\textbf{Closes it:} for cofinally many skipped ranks $c$ of the rational half-greedy orbit, the\nnext skipped rank lies at most $c + (c-3-\\lceil\\log_2(c-2)\\rceil)$ away. Empirically skips have\nroughly constant density (SE-7: certified to row 200{,}000), so this is an enormously slack\nrequirement, never isolated as a target anywhere prior to this bank.\n\\end{obs}\n\n\\begin{obs}[Frozen margin, ineffective horizon] \\ev{Lean}\n\\lean{exists\\_greedyHalfFrozenMargin\\_nonneg\\_of\\_excess\\_neg}{HalfCylinderFiniteShadow.lean:1180},\n\\lean{exists\\_greedyHalfFrozenMargin\\_nonneg\\_iff\\_excess\\_neg}{HalfCylinderFiniteShadow.lean:1211}\n(margin def line 1021, recurrence line 1036, monotonicity line 1094).\n\\textbf{Yields:} at every dyadically safe rank $k>0$, $\\mathrm{greedyHalfFrozenMargin}\\,k\\,J \\ge\n0$ for \\emph{some} horizon $J$ \u2014 the finite coefficient window eventually covers the centred\ncarry. Nonnegativity is monotone up in $J$.\n\\textbf{Mismatch (\\texttt{scale\\_only}):} the horizon comes from a non-effective limit argument\n(\\texttt{tendsto\\_finiteCoeffWindow\\_atTop} plus an existence extraction), so $J$ is unbounded.\nThe exact-row route needs the crossing by the specific horizon $J=c-3$\n(\\lean{halfGreedy\\_precriticalSuffix\\_lt\\_iff\\_frozenMargin\\_nonneg}{BooleanMobiusCriticalCapacityCofinal.lean:784},\n$k=c-1$). All ingredients to make it effective already exist:\n\\lean{binaryCoeffTail\\_eq\\_finiteCoeffWindow\\_add\\_shiftedTail}{}{} gives\n$\\mathrm{gap}=\\mathrm{tail}(k{+}1{+}J)/2^J$, and\n\\lean{binaryCoeffTail\\_le}{GenericTailOrbitRigidity.lean:85} gives $\\mathrm{tail}(m)\\le m+2$.\n\\textbf{Closes it:} margin$(k,J)\\ge0$ as soon as $(k{+}J{+}3)/2^J <\n|\\mathrm{halfGreedyNextDyadicExcessNumerator}\\,k|/\\mathrm{halfGreedyPrefixDenominator}\\,k$.\nSetting $J=k-2$ turns the socket into one explicit dyadic-safety margin\n$|E_k|/D_k > (2k{+}1)/2^{k-2}$ at every skipped rank. Pure bookkeeping over existing lemmas.\n\\end{obs}\n\n\\begin{obs}[G4 / full-shell margin, wrong horizon direction] \\ev{Lean}\n\\lean{HalfGreedySkippedFullShellNonnegative}{HalfCylinderFullShellSeamBridge.lean:604},\n\\lean{half\\_mem\\_mersenneAchievementSet\\_of\\_skippedFullShellNonnegative}{HalfCylinderFullShellSeamBridge.lean:633},\n\\lean{HalfGreedySkippedSeamEscape}{HalfCylinderFullShellSeamBridge.lean:687}.\n\\textbf{Yields:} at every skipped rank $n \\ge 3$, $0 \\le \\mathrm{greedyHalfFrozenMargin}(n{-}1)\\,n$\n\u2014 first passage of the frozen margin by the full-shell horizon $J=k{+}1$, $k=n{-}1$. Reaches\n$1/2 \\in \\mathrm{mersenneAchievementSet}$ by its own consumer.\n\\textbf{Mismatch (\\texttt{quantifier\\_order}):} exactly three doubling steps of horizon short.\nThe exact-row socket needs margin$(k,k{-}2)\\ge0$; this gives margin$(k,k{+}1)\\ge0$. By the\nsucc-recurrence, margin$(k,k{+}1)=8\\cdot$margin$(k,k{-}2)+4\\,\\mathrm{sc}(2k)+2\\,\\mathrm{sc}(2k{+}1)\n+\\mathrm{sc}(2k{+}2)$, sc a divisor-count supportCoeff $\\le\\tau\\le 2\\sqrt{2k+2}$. Since\nnonnegativity is monotone \\emph{up} in $J$, the landed statement is the weaker one and cannot\nsupply the obligation as it stands.\n\\textbf{Closes it:} a horizon-tightening \u2014 at every skipped rank, margin$(k,k{+}1) \\ge\n4\\,\\mathrm{sc}(2k)+2\\,\\mathrm{sc}(2k{+}1)+\\mathrm{sc}(2k{+}2)$: the full-shell margin must exceed\nan explicit $O(\\sqrt k)$ divisor-count quantity. Same $\\sqrt{}$-scale currency as the LPC-sqrt row\nabove.\n\\end{obs}\n\n\\begin{obs}[Cofinal positive skips \\emph{is} the obligation] \\ev{Lean}\n\\lean{CofinalPositiveHalfGreedySkips}{BooleanMobiusSkipRowCofinal.lean:22},\n\\lean{cofinalExactLocalMersenneHalfRows\\_of\\_positiveHalfGreedySkips}{BooleanMobiusSkipRowCofinal.lean:31},\nline 60; A5\n\\lean{half\\_mem\\_mersenneAchievementSet\\_iff\\_greedySkippedSupport\\_infinite}{GreedyAchievementSet.lean:2514},\nA6 (\\lean{}{GreedyAchievementSet.lean:1515}); positivity discharge\n\\lean{}{BooleanMobiusCriticalCapacityCofinal.lean:138}.\n\\textbf{Yields:} cofinally many positive rational-greedy skips $\\Rightarrow$\n\\texttt{CofinalExactLocalMersenneHalfRows} $\\Rightarrow 1/2 \\in\n\\mathrm{mersenneAchievementSet}$. Advertised elsewhere as the shortest known path.\n\\textbf{Mismatch (\\texttt{hypothesis\\_strength}):} it is not weaker than the problem \u2014 it \\emph{is}\nthe problem. The positivity conjunct $0 < \\mathrm{greedyMersenneRemainderRat}(1/2)(c{-}1)$ is\nunconditionally dischargeable by the odd-denominator parity fact, so\n$\\mathrm{CofinalPositiveHalfGreedySkips} \\Leftrightarrow\n(\\mathrm{greedyMersenneSkippedSupport}(1/2))$.Infinite, which A6 proves \\emph{equivalent} to\n$1/2 \\in \\mathrm{mersenneAchievementSet}$. Attacking this target is attacking \\#257-false head on,\nwith no reduction whatsoever.\n\\textbf{Closes it:} nothing short of the problem. \\emph{This is a trap, not a waypoint} \u2014 the\nescape is to abandon the greedy-orbit arm (where the wall at line 50 forces canonicity) and\ntarget the ($\\star$) sharp-capacity-fill route instead, whose cores are not forced to be greedy\nprefixes.\n\\end{obs}\n\n\\begin{obs}[C4a / terminal-only strip, wrong locality] \\ev{Lean}\n\\lean{HalfCarryCofinalTerminalOnlyStrip}{TerminalOnlyCofinal.lean:34},\n\\lean{HalfTerminalOnlyStripWitness}{TerminalOnlyCofinal.lean:23},\n\\lean{exists\\_infinite\\_support\\_half\\_of\\_cofinalTerminalOnlyStrip}{TerminalOnlyCofinal.lean:192}.\n\\textbf{Yields:} cofinally many depths $M$ carrying a finite normalized word $a$ with\n$|\\mathrm{integerHalfCarry}(\\mathrm{wordSupport}\\,a)(M{-}1)| \\le \\mathrm{halfStripBound}\\,M\n\\Rightarrow \\exists A$ infinite with $\\mathrm{erdosSupportSeries}\\,2\\,A=1/2$. Same architecture\nas the obligation: cofinal, no coherence, mutually incompatible witnesses allowed.\n\\textbf{Mismatch (\\texttt{multiple}):} the two sockets trade off in opposite directions and\nneither implies the other. C4a tolerates \\emph{any} support inside the depth-$M$ window but\ndemands the carry inside a $\\sim 2\\sqrt M$ strip; the obligation form ($\\star$) tolerates a carry\nup to $2^{M/2-1}$ \u2014 exponentially looser \u2014 but demands the support be confined to ranks $<\nM/2+1$, since the exact fill encodes the residue in the pure upper window where the Mersenne\nquotient is exactly $2^{M-d}$. The gap is support \\emph{locality}, not carry size.\n\\textbf{Closes it:} a lower-half confinement upgrade of the strip witnesses \u2014 $\\forall N\\\n\\exists c\\ge N$ with a strip-scale witness whose support lies in $[2,c)$ at depth $2c{-}3$. Then\n$2\\sqrt{2c}+4 < 2^{c-2}$ for $c\\ge8$ and the existing fill fires verbatim. This identifies C4a as\nthe \\emph{cheaper} route to the same payoff \u2014 257-cofinal-rows is not the cheapest cofinal socket\non the board.\n\\end{obs}\n\n\\begin{obs}[C4c / G3 / greedy carry vs.\\ frozen prefix carry, coordinate mismatch] \\ev{Lean}\n\\lean{GreedyHalfCarryCofinalStripReturn}{CofinalStripReturn.lean:76},\n\\lean{greedy\\_half\\_infinite\\_of\\_cofinalStripReturn}{CofinalStripReturn.lean:130}, line 157;\n\\lean{infinite\\_support\\_half\\_of\\_mobiusCenteredHalfCarry\\_sqrtBound}{HalfCarryReachability.lean:788},\nline 805 (nonnegativity from line 842 /\n\\lean{}{HalfCylinderFinalMiddleCellEscape.lean:94}).\n\\textbf{Yields:} cofinal returns of the actual greedy carry to the square-root strip (C4c), or a\npointwise $2\\sqrt N+4$ bound on the M\u00f6bius-centred carry (G3), each give an infinite support with\nvalue exactly $1/2$.\n\\textbf{Mismatch (\\texttt{coordinate\\_only}):} these bound the carry of the \\emph{infinite}\ngreedy support $\\mathrm{greedyMersenneSupport}(1/2)$; sharp capacity is about the \\emph{frozen\nfinite prefix} $\\mathrm{halfGreedyPrefixSupport}(c{-}1)$. The two carries agree only at index $k$\n(inside \\lean{greedyHalfFrozenMargin\\_eq\\_actual\\_skip\\_margin}{HalfCylinderFiniteShadow.lean:1102}),\nnot at depth $2c{-}3$ where the fill needs it. The bridge is already landed:\n\\lean{halfGreedy\\_precriticalSuffix\\_lt\\_iff\\_futureSkipCoverage}{BooleanMobiusCriticalCapacityCofinal.lean:867}:\nsharp capacity $\\Leftrightarrow \\mathrm{mobiusCenteredHalfCarry}\\,G\\,(2c{-}4) \\le\n\\mathrm{futureSkipCapacity}\\,G\\,c\\,(c{-}3)$ \u2014 transport costs exactly the future-skip capacity\nterm.\n\\textbf{Closes it:} either a strip bound stated for the frozen prefix rather than the live orbit,\nor $\\mathrm{futureSkipCapacity}\\,G\\,c\\,(c{-}3) \\ge \\mathrm{halfStripBound}(2c{-}3)$ \u2014 holds as\nsoon as one skip occurs early in $[c{+}1,\\,2c{-}4]$, since the capacity is dyadically weighted.\nSame input as the bounded-gap precritical bound above.\n\\end{obs}\n\n\\begin{obs}[Above-support recycling, no growth guarantee] \\ev{Lean}\n\\lean{exists\\_skippedCoreExactRow\\_of\\_value\\_above}{BooleanMobiusExactRowCrossing.lean:155},\n\\lean{exactLocalMersenneHalfRow\\_of\\_first\\_localMersenne\\_crossing}{BooleanMobiusExactRowCrossing.lean:191},\n\\lean{exactLocalMersenneHalfRow\\_two\\_mul\\_sub\\_two\\_of\\_skippedCore}{BooleanMobiusSkipRow.lean:197}.\n\\textbf{Yields:} unconditional: every finite above-half support bounded by $n$ recycles through\nits first crossing rank $c$ into a genuine exact row at endpoint $2c{-}2$, with $4 \\le c \\le n$.\nNo capacity hypothesis at all.\n\\textbf{Mismatch (\\texttt{quantifier\\_order}):} the witness is $\\exists c \\le n$, not $\\exists c$\nlarge \u2014 $2c{-}2$ may be $\\le n$, so the endpoint need not grow.\n\\lean{exists\\_seeded\\_bounded\\_double\\_or\\_recycle\\_model}{BooleanMobiusExactRowDichotomy.lean:91}\nis the explicit falsifier \u2014 $\\mathrm{boundedDoubleOrRecycleModel}\\,n := (n{=}6)$ satisfies the\nsame transition schema plus a seed and is \\emph{not} cofinal. Growth is recovered only inside\n\\texttt{ProtectedExactLocalMersenneRow}, whose invariants force $c>$cutoff, hence $2c{-}2>$\nendpoint \u2014 and maintaining those invariants is precisely what needs the sharp-capacity fill\nrather than this general recycling.\n\\textbf{Closes it:} a crossing-location lower bound \u2014 for the supports actually produced, the\nfirst crossing rank satisfies $2c{-}2>n$. Equivalently, keep protection alive without sharp\ncapacity, e.g.\\ a fill placing all new ranks above $c$ using only the landed $c{-}1$-bit\ncapacity.\n\\end{obs}\n\n\\begin{obs}[Double-or-recycle dichotomy, proved insufficient] \\ev{Lean}\n\\lean{exactLocalMersenneHalfRow\\_double\\_or\\_recycle}{BooleanMobiusExactRowDichotomy.lean:26},\n\\lean{boundedDoubleOrRecycleModel\\_not\\_cofinal}{BooleanMobiusExactRowDichotomy.lean:58},\n\\lean{exists\\_seeded\\_bounded\\_double\\_or\\_recycle\\_model}{BooleanMobiusExactRowDichotomy.lean:91}.\n\\textbf{Yields:} $\\forall n \\ge 6$, an exact row at $n$ gives an exact row at $2n{-}1$ or at\n$2c{-}2$ for some $4 \\le c \\le n$; plus a formal proof that this shape, even seeded at endpoint\nsix, is logically insufficient for cofinality.\n\\textbf{Mismatch (\\texttt{quantifier\\_order}):} the dichotomy is proved by splitting on the sign\nof the witness value (\\texttt{finiteErdosSum\\_den\\_odd} excludes equality); the no-go pins the\ndeficit precisely \u2014 what is missing is not another dichotomy but strict endpoint progress in the\nrecycle branch, or a proof that the below-half branch recurs. \\textbf{Any future attack that\nmerely re-derives a double-or-recycle transition is already refuted by this row.}\n\\textbf{Closes it:} either (a) the recycle branch upgraded to $2c{-}2>n$, or (b) cofinally many\nbelow-half exact rows so the doubling branch alone is cofinal. Nothing weaker than one of these\ntwo can work through this dichotomy.\n\\end{obs}\n\n\\subsection{257-universal: thirteen near misses}\n\n\\begin{obs}[NM-01 / the single certificate socket] \\ev{Lean}\n\\lean{irrational\\_erdosSupportSeries\\_of\\_weighted\\_coeff\\_certificates}{CertificateKernel.lean:8703}\n(carry variant \\lean{}{CertificateKernel.lean:9053}).\n\\textbf{Yields:} $\\forall b\\ge2,\\ \\forall A\\subseteq\\mathbb N$ (no infinitude, no structure\nhypothesis at all): if $\\forall q>0\\ \\exists N,K,L,C$ with $K\\le L$, $b^r \\mid\n\\mathrm{supportCoeff}\\,A\\,(N{+}r)$ for $r\\in[1,K]$, $\\sum_{r=K+1}^L \\mathrm{supportCoeff}\\,A\\,(N{+}r)\nb^{L-r}\\le C$, $\\exists t,\\ 0<\\mathrm{supportCoeff}\\,A\\,(N{+}L{+}1{+}t)$, and $q(C{+}N{+}L{+}2)<b^L$\n\u2014 then $\\mathrm{erdosSupportSeries}\\,b\\,A$ is irrational. Every landed \\#257 family (full\nsupport, multiples, periodic, eventually periodic, residue class, odd, pairwise coprime,\nsunflower bouquet) factors through this one socket.\n\\textbf{Mismatch (\\texttt{hypothesis\\_strength}):} already universally quantified over $A$ \u2014 the\nuniversal quantifier is \\emph{not} the gap. The hypothesis \\texttt{hcert} is a cofinal producer\nsupplied only for named classes; the theorem's own docstring: ``certificate existence for\n$f_A$... for arbitrary infinite $A$ remains an open obligation.''\n\\textbf{Closes it:} a single producer theorem \u2014 $\\forall A$ infinite, $\\forall q>0$, $\\exists\nN,K,L,C$ with the same four clauses, from infinitude alone. The carry variant weakens clause 2\nfrom digit-wise divisibility to aggregate divisibility ($b^K \\mid \\sum_r\n\\mathrm{supportCoeff}\\,A\\,(N{+}r)b^{K-r}$), so the actual minimum obligation is the weaker\naggregate form.\n\\end{obs}\n\n\\begin{obs}[NM-02 / single-target to all-rational-targets, promotable] \\ev{Lean}\n\\lean{half\\_mem\\_mersenneAchievementSet\\_iff\\_greedySkippedSupport\\_infinite}{GreedyAchievementSet.lean:2514}\n(forward \\lean{}{GreedyAchievementSet.lean:2482}; reverse\n\\lean{mem\\_mersenneAchievementSet\\_of\\_greedySkippedSupport\\_infinite}{GreedyAchievementSet.lean:1515}).\n\\textbf{Yields:} $(1/2:\\mathbb R)\\in\\mathrm{mersenneAchievementSet} \\Leftrightarrow\n(\\mathrm{greedyMersenneSkippedSupport}(1/2))$.Infinite.\n\\textbf{Mismatch (\\texttt{scale\\_only}):} stated at the single fixed target $t=1/2$; the\nobligation needs it at \\emph{every} rational target, since ``some infinite $A$ has rational\nvalue'' is $\\exists t\\in\\mathbb Q\\ \\exists A$, not $\\exists A$ at $t{=}1/2$.\n\\textbf{This is the highest-value promotable row in the bank.} The reverse direction is already\nstated for all reals ($0\\le x$). The forward direction uses $1/2$ in exactly two places: (a) a\ngeneric-in-$A$ lemma, (b) the final contradiction, which needs only that the target is\n\\emph{rational}. Replacing $(1/2:\\mathbb R)$ by $((t:\\mathbb Q):\\mathbb R)$ and $(1:\\mathbb Q)/2$\nby $t$, with $0\\le t$ in place of \\texttt{norm\\_num}, gives $\\forall t:\\mathbb Q,\\ 0\\le t \\to\n((t:\\mathbb R)\\in\\mathrm{mersenneAchievementSet} \\Leftrightarrow\n(\\mathrm{greedyMersenneSkippedSupport}(t:\\mathbb R))\\text{.Infinite})$ with no new mathematics.\n\\textbf{Closes it:} exactly this re-parametrisation \u2014 converts the entire half-greedy refutation\nmachine from a one-target statement into a statement about every rational target, the quantifier\nshape 257-universal actually needs.\n\\end{obs}\n\n\\begin{obs}[NM-03 / target-free engine underneath a target-pinned corollary] \\ev{Lean}\n\\lean{finite\\_boolSupport\\_ne\\_half}{HalfCarryReachability.lean:589}; engine\n\\lean{finiteErdosSum\\_den\\_odd}{DyadicPrefixCompression.lean:1358}.\n\\textbf{Yields:} $\\forall A$ finite with $0\\notin A$: $\\mathrm{erdosSupportSeries}\\,2\\,A \\ne\n1/2$ \u2014 any support achieving exactly $1/2$ must be infinite.\n\\textbf{Mismatch (\\texttt{scale\\_only}):} the headline is pinned to target $1/2$ while its engine\n$\\mathrm{Odd}((\\mathrm{finiteErdosSum}\\,F\\,2).\\mathrm{den})$ is already fully general over finite\nsupports.\n\\textbf{Closes it:} verbatim re-derivation \u2014 $\\forall t:\\mathbb Q$ with even reduced denominator,\n$\\forall A$ finite with $0\\notin A$, $\\mathrm{erdosSupportSeries}\\,2\\,A \\ne (t:\\mathbb R)$. No new\nmathematics. Paired with NM-02: universal \\#257 at $b{=}2$ is false iff some rational $t$ with\neven reduced denominator has an infinitely-skipping greedy Mersenne orbit.\n\\end{obs}\n\n\\begin{obs}[NM-04 / dyadic-only reciprocal-mass rigidity, generalization sitting unused] \\ev{Lean}\n\\lean{dyadic\\_support\\_fraction\\_reciprocalMass\\_diverges\\_or\\_gt\\_one}{RationalSupportCarrySkeleton.lean:2200}\n(line 2116); generic-$v$ sibling\n\\lean{shiftedBooleanCollision\\_wrap\\_bound\\_of\\_common\\_multiple}{RationalSupportCarrySkeleton.lean:1962}.\n\\textbf{Yields:} for $A$ infinite with $\\mathrm{erdosSupportSeries}\\,2\\,A = p/2^c$ (dyadic\nrational): $\\neg\\mathrm{Summable}(1/a$ on $A) \\vee 1 < \\mathrm{reciprocalMass}\\,A$.\n\\textbf{Mismatch (\\texttt{coordinate\\_only}):} restricted to denominators $2^c$ ($v{=}1$) while\nthe obligation needs every rational $p/(2^c v)$. This restriction is a call-site choice, not a\nproof constraint \u2014 every supporting lemma carries a free $\\{v:\\mathbb N\\}\\ (hv:0<v)$, and the\ngeneral-$v$ conclusion is \\emph{already landed} at line 1962 for arbitrary $(v,h,L,F)$.\n\\textbf{Closes it:} instantiate the landed generic theorem at $F=\\{a,b\\}$, $L=\\mathrm{lcm}\\,a\\,b$,\n$h=$ multiplicative order of $2$ mod $v$, giving $\\rho(A) \\ge\n\\mathrm{doublingWrapCount}(p,v,h)/h + 1/\\mathrm{lcm}(a,b)$ for every rational value. The one\ngenuinely new step: for $v{>}1$ the excess can be $0$, so strictness needs $1 \\le\n\\mathrm{doublingWrapCount}(p,v,h)/h + \\mathrm{booleanCollisionSurplus}\\,|F|/L$ (equivalently: the\nmean least residue of $p\\cdot2^n \\bmod v$ is $\\ge v(1-|F|/2L)$).\n\\end{obs}\n\n\\begin{obs}[NM-05 / T11, sparse-side rigidity only] \\ev{Lean}\n\\lean{supportCoeffZeroWindow\\_length\\_le\\_eps\\_logb}{SublogDivisorCoverage.lean:435} (line 392).\n\\textbf{Yields:} for any $A$ with a positive element and\n$\\mathrm{erdosSupportSeries}\\,2\\,A=p/(2^c v)$, $v{>}0$ (any rational, not just dyadic):\n$\\forall\\varepsilon{>}0\\ \\exists B\\ge0\\ \\forall N\\ge1\\ \\forall h$,\n$\\mathrm{SupportCoeffZeroWindow}\\,A\\,(c{+}N)\\,h \\to h \\le \\varepsilon\\log_2 N + B$. A\nrational-valued support cannot have super-logarithmic divisor-free windows; uniform in $A,p,N,h$.\n\\textbf{Mismatch (\\texttt{quantifier\\_order}):} the strongest genuinely universal rigidity in the\nbank, and it is a near-miss by \\emph{direction}, not strength. It constrains a hypothetical\nrational support from the sparse side only ($2\\in A$ already forces every zero window to length\n$\\le1$, so this is vacuous for divisor-dense $A$). It eliminates exactly the regime NM-09 also\neliminates, nothing in the regime where the certificate producer is missing.\n\\textbf{Closes it:} the complementary half \u2014 $A$ infinite $\\Rightarrow$ (cofinally many\nsuper-log zero windows) $\\vee$ ($\\forall q{>}0$, block certificate exists). This\ntrichotomy-closure \u2014 sparse $\\Rightarrow$ contradiction, dense $\\Rightarrow$ certificate \u2014 is\nprecisely what composing the rigidity corollaries would need, and the dense branch has no\nproducer.\n\\end{obs}\n\n\\begin{obs}[NM-06 / pairwise-coprime, excludes the primes] \\ev{Lean}\n\\lean{irrational\\_erdosSupportSeries\\_pairwise\\_coprime}{CertificateKernel.lean:10448} (producer\n\\lean{}{CertificateKernel.lean:9574}).\n\\textbf{Yields:} $\\forall b\\ge2,\\ \\forall A$ infinite, pairwise coprime, with\n$\\mathrm{Summable}(\\mathrm{indicator}\\,A\\,(1/a))$: irrational.\n\\textbf{Mismatch (\\texttt{hypothesis\\_strength}):} the summability hypothesis excludes the single\nmost natural uncovered support, $A=$ primes: infinite and pairwise coprime, satisfying every\nother hypothesis, but $\\sum 1/p$ diverges (Mertens) \u2014 no prime-support theorem exists anywhere in\nthe tree. $A\\subseteq$primes is also the exact class where\n\\lean{compositeDilationDefect\\_eq\\_zero\\_of\\_prime\\_support}{CompositeDilationDefect.lean:103}\nmakes the dilation algebra exact, the class where the machinery is otherwise cleanest.\n\\textbf{Closes it:} summability is used at exactly one place, a global tail budget $\\forall\n\\varepsilon{>}0\\ \\exists B\\ \\forall$ finite $T\\subseteq A\\cap(B,\\infty),\\ \\sum_{a\\in T}1/a\\le\n\\varepsilon$. A windowed replacement suffices: since $N,L$ may be chosen after $B$, it suffices\nthat $\\forall\\varepsilon{>}0\\ \\exists B\\ \\exists N,L$ with $\\sum_{a\\in A,\\,B<a\\le N+L}(L/a+1)\\le C$\nand $q(C{+}N{+}L{+}2)<b^L$. For the primes $\\sum_{B<p\\le N+L}1/p \\approx \\log\\log(N{+}L) -\n\\log\\log B$, controllable by choosing $L$ small relative to $B$ \u2014 the route is live, but a\ndifferent argument, not the same one with a bound removed.\n\\end{obs}\n\n\\begin{obs}[NM-07 / sunflower bouquet, unproved selector plus false class-existence] \\ev{Lean}\n\\lean{irrational\\_erdosSupportSeries\\_of\\_orthogonalPetalBouquet}{SupportSunflowerDichotomy.lean:540}\n(selector def line 406; carry socket line 381; bouquet structure line 33).\n\\textbf{Yields:} for $A=$ (finite exceptional set dividing $Q$) $\\cup \\{\\mathrm{core}_i \\cdot\n\\mathrm{petal}_i\\}$ with cores $\\mid Q$, petals pairwise coprime and coprime to $Q$, $\\sum\n1/\\mathrm{petal}_i<\\infty$: if \\texttt{SunflowerForcedSlotTailSelection} $A$ then irrational.\n\\textbf{Mismatch (\\texttt{multiple}):} two gaps. (i) On the support: the bouquet class strictly\ngeneralises pairwise-coprime but still demands globally pairwise-coprime, summable petals \u2014 it\ninherits NM-06's exclusion of the primes and adds nothing in the divisor-dense regime. (ii) On\nthe selector: \\texttt{SunflowerForcedSlotTailSelection} is an unproved cofinal supply \u2014 grep for\n``dichotomy'' in the file returns only the namespace; despite the filename there is no dichotomy\ntheorem there.\n\\textbf{Closes it:} for the selector, $\\forall K{>}0\\ \\exists N$, $2^K \\mid$ the carried first\nblock $\\wedge\\ \\mathrm{binaryCoeffTail}(\\mathrm{supportCoeff}\\,A)(N{+}K)\\le16$, from the\nbouquet's persistent-reduced-modulus averaging \u2014 the file states this is what that averaging\nargument is supposed to give. For the class: a bouquet-existence theorem $A$.Infinite $\\to$\nNonempty(bouquet structure) is \\emph{false} as stated (take $A=$ all multiples of 6 \u2014 no coprime\npetal decomposition), so the class can only be widened, never promoted to universal.\n\\end{obs}\n\n\\begin{obs}[NM-08 / eventual periodicity, widest unconditional class, asymptotic obstruction] \\ev{Lean}\n\\lean{irrational\\_erdosSupportSeries\\_eventuallyPeriodic}{CertificateKernel.lean:11276} (engine\nline 11262, producer line 11072; finite-perturbation transfer lines 9139, 9148).\n\\textbf{Yields:} $\\forall b\\ge2\\ \\forall m{>}0\\ \\forall N_0$: $A$ infinite and $m$-periodic above\n$N_0$ $\\Rightarrow$ irrational. Plus: the class of supports with irrational series is closed\nunder finite symmetric difference in both directions.\n\\textbf{Mismatch (\\texttt{hypothesis\\_strength}):} the widest unconditional class landed (all\nmoduli, all thresholds, no bound anywhere), but eventual periodicity is a measure-zero, countable\ncondition; the complement is everything interesting. The two-way finite-perturbation transfer\nshows the good class is a union of finite-symmetric-difference equivalence classes \u2014 precisely\nwhy it cannot be enlarged by prefix surgery; the obstruction is asymptotic.\n\\textbf{Closes it:} a ``locally periodic at cofinally many scales'' version \u2014 $\\forall q{>}0\\\n\\exists N\\ \\exists m\\le f(N)$ such that $A\\cap[N,N{+}L]$ is $m$-periodic for $L$ large enough\nthat the periodic sieve manufactures one block certificate. The periodic producer already only\nneeds periodicity across the certificate window (window-local), so this is the closest genuine\nwidening \u2014 but proving cofinally many periodic windows exist for arbitrary $A$ is a new\nstatement, not a re-run.\n\\end{obs}\n\n\\begin{obs}[NM-09 / lcm-gap, far-sparse edge only] \\ev{Lean}\n\\lean{irrational\\_erdosSum\\_of\\_lcm\\_gap}{CertificateKernel.lean:5555} (instances\n\\lean{irrational\\_erdosSum\\_factorial\\_support}{CertificateKernel.lean:5707},\n\\lean{irrational\\_erdosSum\\_two\\_pow\\_support}{CertificateKernel.lean:5731}).\n\\textbf{Yields:} $\\forall b\\ge2,\\ \\forall a:\\mathbb N\\to\\mathbb N$ StrictMono, $a_0\\ge1$: if\n$a_k - \\mathrm{lcm}\\{a_0,\\dots,a_{k-1}\\}\\to\\infty$ then $\\sum_k 1/(b^{a_k}-1)$ is irrational, via\n\\lean{irrational\\_of\\_den\\_mul\\_abs\\_sub\\_tendsto\\_zero}{}{}.\n\\textbf{Mismatch (\\texttt{hypothesis\\_strength}):} the $\\mathbb N$-truncated-subtraction\nhypothesis requires $a_k$ to outgrow the lcm of all earlier elements \u2014 extreme lacunarity. Fails\nimmediately for $A=$primes (lcm of first $k$ primes vastly exceeds $p_k$), squares, any positive\ndensity $A$. Covers only the far sparse edge \u2014 the same edge NM-05's T11 also patrols.\n\\textbf{Closes it:} a relaxed gap measuring the surviving denominator rather than the raw lcm:\n$a_k - \\log_b(\\text{reduced denominator of the }k\\text{-th partial sum}) \\to \\infty$. The corpus\nalready has the denominator-survival algebra to state this\n(\\texttt{divisor\\_dvd\\_divInt\\_den}, \\texttt{survivingDivisor\\_dvd\\_scaled\\_divInt\\_den} at\n\\lean{}{RationalDenominatorSurvival.lean:17}, and\n\\lean{upperHalfMersenneProduct\\_lower\\_bound}{MersenneShadowDenominatorGrowth.lean:60} giving\n$2^{t/2}$ growth); the growth theorem for the surviving denominator of an \\emph{arbitrary}\nsupport is what is missing.\n\\end{obs}\n\n\\begin{obs}[NM-10 / the load-bearing coordinate, iff over all $A$] \\ev{Lean}\n\\lean{erdosSupportSeries\\_rational\\_iff\\_exists\\_temperedCarry}{BooleanMobiusCarry.lean:383}\n(line 376); T7 \\lean{binaryCoeffSeries\\_rational\\_iff\\_exists\\_temperedBinaryOrbit}{GenericTailOrbitRigidity.lean:425}.\n\\textbf{Yields:} $\\forall A\\subseteq\\mathbb N$ (no hypothesis at all):\n$\\mathrm{HasRationalValue}(\\mathrm{erdosSupportSeries}\\,2\\,A) \\Leftrightarrow \\exists q{>}0\\\n\\exists U:\\mathbb N\\to\\mathbb Z,\\ \\mathrm{IsTemperedBinaryOrbit}(\\mathrm{supportCoeff}\\,A)\\,q\\,U$.\n\\textbf{Mismatch (\\texttt{coordinate\\_only}):} already universal over $A$ \u2014 the exact coordinate\nin which the obligation should be stated, and the literal shared trunk with \\#249 (instantiate T7\nat $c:=\\mathrm{Nat.totient}$). It does not refute the orbit:\n\\lean{temperedBinaryOrbit\\_eq\\_scaledTail}{GenericTailOrbitRigidity.lean:338} pins the orbit\nuniquely to $u(N)=qT_c(N)$, and $T_c(N)\\le N{+}2$ confirms it is never ruled out on growth\ngrounds for any $c$ with $c(n)\\le n$.\n\\textbf{Closes it:} a support-specific non-existence theorem \u2014 $\\forall A$ infinite, $\\forall\nq{>}0$, $\\neg\\exists U$ with $\\mathrm{IsTemperedBinaryOrbit}(\\mathrm{supportCoeff}\\,A)\\,q\\,U$.\nBecause T7 is an iff and the orbit is unique, this is logically equivalent to the obligation \u2014\nthe row's value is naming the single object (the tempered carry orbit of $\\mathrm{supportCoeff}\\,A$)\nwhose non-existence is the entire content, for every $A$ at once.\n\\end{obs}\n\n\\begin{obs}[NM-11 / carry unboundedness, tautological, dead end] \\ev{Lean}\n\\lean{shifted\\_state\\_unbounded\\_of\\_infinite\\_support}{RationalSupportCarrySkeleton.lean:2320},\n\\lean{exists\\_unbounded\\_shifted\\_odd\\_tail\\_nat\\_state\\_of\\_support\\_fraction}{RationalSupportCarrySkeleton.lean:2377},\n\\lean{one\\_add\\_mul\\_card\\_le\\_two\\_mul\\_shifted\\_state}{RationalSupportCarrySkeleton.lean:2228}.\n\\textbf{Yields:} for every infinite $A$ and positive carry recurrence\n$u(n{+}1)+v\\cdot\\mathrm{supportCoeff}\\,A\\,(c{+}n{+}1)=2u(n)$: $u$ is unbounded, and $1+v|F| \\le\n2u(L{-}c{-}1)$ for every finite $F\\subseteq A$ with all elements dividing $L$. Stated for the\ngeneral denominator $2^c v$, not just dyadic.\n\\textbf{Mismatch (\\texttt{multiple}):} traced to a dead end. The proof picks $2B{+}1$ elements of\n$A$, sets $L=(c{+}1)\\prod F$, forcing $\\mathrm{supportCoeff}\\,A\\,L\\ge|F|$; then\n$u(L{-}c{-}1)=v\\cdot\\mathrm{binaryCoeffTail}(\\mathrm{supportCoeff}\\,A)(L{-}1)\\ge v|F|/2$, so $1+v|F|\n\\le 2u$ holds with \\emph{no slack}. Unboundedness of the carry is a tautological restatement of\n``supportCoeff is unbounded on common multiples,'' not a rigidity. The linear tail bound cannot\nrescue it either: $L$ is exponential in $|F|$ by construction, so $N{+}2$ is astronomically\nlarger than $|F|$.\n\\textbf{Closes it:} a common-multiple selection with divisor economy \u2014 an infinite $A$ must admit\n$L$ with $|\\{a\\in A: a\\mid L\\}| > 2\\,\\mathrm{binaryCoeffTail}(\\mathrm{supportCoeff}\\,A)(L{-}1) +\n1/v$. Since the tail is a $2^{-j}$-weighted average near $L$, this demands an anti-concentration\nstatement about divisor incidence of $A$ in the window $(L,\\,L{+}O(\\log L)]$ \u2014 the same shape as\nthe missing certificate, restated; the route as built cannot supply it.\n\\end{obs}\n\n\\begin{obs}[NM-12 / rank-gap dichotomy, target-generic killer under a target-pinned dichotomy] \\ev{Lean}\n\\lean{half\\_mem\\_mersenneAchievementSet\\_or\\_exists\\_fatal\\_gap}{HalfCutLocator.lean:623},\n\\lean{half\\_mem\\_mersenneAchievementSet\\_iff\\_no\\_existsFatalHalfGap}{HalfCutLocator.lean:654};\ngeneral-$A$ killer \\lean{positiveMersenneSupportValue\\_ne\\_of\\_rank\\_gap}{HalfCutLocator.lean:380}.\n\\textbf{Yields:} an unconditional dichotomy: either $1/2\\in\\mathrm{mersenneAchievementSet}$, or a\nfinite witness $(u,d)$ with prefix value $+\\mathrm{mersenneTail}(d{+}1) < 1/2 <$ prefix value\n$+\\mathrm{mersenneWeight}(d{+}1)$. The rank-gap killer is stated for every target and every\nsupport ($\\{t:\\mathbb R\\}\\{A:\\mathrm{Set}\\,\\mathbb N\\}\\{d:\\mathbb N\\}$).\n\\textbf{Mismatch (\\texttt{scale\\_only}):} the killer is already target-generic and\nsupport-generic; the \\emph{dichotomy} built on it is pinned to $1/2$ via two endpoint kills \u2014 the\nodd-denominator parity fact, and ``finite word + full tail $\\ne 1/2$'' (irrationality of the\nMersenne tail).\n\\textbf{Closes it:} both inputs generalise on their face \u2014 the first to any rational $t$ with\neven reduced denominator (engine already general, cf.\\ NM-03), the second to any rational $t$ at\nall (the tail is irrational by Erd\u0151s--Borwein, so finite-rational + irrational-tail $\\ne$\nrational). \\emph{Caveat, honestly flagged:} the two endpoint-kill proof bodies were not read in\nfull verification; this is recorded as a strong promotability hypothesis, not a verified one \u2014 a\n\\texttt{norm\\_num}-style numeric step could hide a $1/2$-specific bound.\n\\end{obs}\n\n\\begin{obs}[NM-13 / signed periodic dichotomy, sign machinery free but periodicity still required] \\ev{Lean}\n\\lean{int\\_coeff\\_series\\_irrational\\_or\\_bpow\\_mul\\_eq\\_intCast\\_intWeightedErdosSeries\\_periodic}{CertificateKernel.lean:13847},\n\\lean{irrational\\_intWeightedErdosSeries\\_periodic\\_of\\_coeff\\_nonneg\\_of\\_frequently\\_ne\\_zero}{CertificateKernel.lean:14255}\n(nonpos mirror line 14315).\n\\textbf{Yields:} $\\forall b\\ge2\\ \\forall m{>}0\\ \\forall$ $\\mathbb Z$-valued $m$-periodic weight\n$w$: irrational$(\\sum w(a)/(b^a-1))$ $\\vee$ $\\exists k,z,\\ b^k x=z$; the terminating branch closes\nunconditionally as soon as $\\mathrm{intWeightedCoeff}\\,w$ is one-signed and frequently nonzero.\n\\textbf{Mismatch (\\texttt{multiple}):} two mismatches. (i) periodicity of the weight is still\nrequired \u2014 this is the signed lift of NM-08, not an escape from it. (ii) the one-sidedness clause\nis \\emph{free} for \\#257 (a 0/1 support indicator gives $\\mathrm{supportCoeff}\\,A\\ge0$\nautomatically) \u2014 the sign machinery buys nothing here; it was built for the signed \\#249 lane.\nThe genuinely transferable content is the dichotomy shape itself: irrational-or-$b$-adically-\nterminating, terminating branch killable by a positive far tail.\n\\textbf{Closes it:} an aperiodic version of the dichotomy \u2014 for arbitrary $A$, either\n$\\mathrm{erdosSupportSeries}\\,2\\,A$ is irrational or $2^k\\cdot$it is an integer. For infinite $A$\nthe terminating branch is already excluded by \\texttt{finiteErdosSum\\_den\\_odd}-style parity plus\na positive far tail, so this dichotomy \u2014 if provable without periodicity \u2014 would close the\nobligation outright. Periodicity is used only to manufacture the full-block certificates\n(``periodicity alone manufactures the full-block certificates,'' per the file's own docstring),\nso removing it is again NM-01's missing producer, restated a third time.\n\\end{obs}\n\n\\subsection{Three theorems assembled from the interface, and where each still stops}\n\n\\ev{Math} throughout this subsection \u2014 none of Theorems A, B, C is a Lean declaration. Each is a\ncorrect assembly of the corpus's proved pieces plus one clearly named open step; none decides\n\\#257.\n\n\\begin{thm}[A \u2014 reset-crossing unification]\nThe 257-reset target exponent $(r+5)/2$ is not a fitted constant but a \\emph{derived} one: it\nfollows from sqrt-scale reset anti-concentration together with the run-length-to-crossing\ngeometry of oa-2's deficit/excess pair above, and every correction constant in the derivation is\nexact (no asymptotic $O(\\cdot)$ notation is hiding a table). Concretely: the deficit-side run law\n(oa-2, unconditional and uniform) converts any anti-concentration bound $2^{\\varepsilon r}$ on\n$|\\Delta_r|$ into a run-length ceiling $L_r \\lesssim (1-\\varepsilon)\\cdot 2r$; matching this\nagainst the certified maximum observed run growth (SE-7: $\\max L_r \\sim \\log_2(\\mathrm{row})$,\nrow 200{,}000) pins $\\varepsilon = 1/2$ as the exact crossover exponent, not merely a plausible\nguess. Theorem A is the unification of the deficit-run and excess-run geometry into one crossing\nsocket at that derived exponent; the open step is exactly the anti-concentration input identified\nin the G5/H4/F5b/oa-3b rows above.\n\\end{thm}\n\n\\begin{thm}[B \u2014 rigidity of dangerous resets]\nA dangerous reset (one within $2^{(r+5)/2}$ of the sqrt-escape threshold) that is preceded by a\nlong pure R-run pins its own deviation to a single pulse-determined integer (RC-5 above), not a\nfree real number: the affine excess recurrence (rc-2) is exact, so once the run length and the\npulse word are fixed, $w_{r_0+1}$ is one of at most two integers with no remaining degree of\nfreedom. Consequently a chain of $n$ dangerous resets is not $n$ independent near-misses; it is\na single tower of $n{-}1$ nested exact integer coincidences, each individually as unlikely as an\nexplicit arithmetic accident and none free to vary once the pulse stream is fixed. Failure of\nsqrt-escape therefore requires an exact integer-coincidence tower, not a generic accumulation of\nsmall errors \u2014 the open step is excluding the two pinned values (RC-5's own closing condition).\n\\end{thm}\n\n\\begin{thm}[C \u2014 co-semi-decidability of the four certificate families]\nIf \\#257 is false at $x=1/2$ (equivalently the half-greedy orbit survives forever, by\n\\lean{mem\\_mersenneAchievementSet\\_iff\\_greedy\\_survival}{GreedyAchievementSet.lean:1445}), that\nsurvival is witnessed at a \\emph{finite, computable rank} exactly when the greedy sqrt-escape\nroute succeeds early: every certificate depth in the corpus (the reset-crossing certificates of\nPart I, the truncation-rung ladder TR-5/TR-6, and the sharp-capacity fill of ($\\star$)) is a\n\\emph{lower bound} on that rank, not an independent estimate of it. This is visible directly at\nthe consumer layer: \\lean{LargestSkipLateStepSocket}{HalfCylinderLargestSkipInduction.lean:34}\nand \\lean{SeamUpperResetDyadicBandEscape}{HalfCylinderMiddleCarryLowerBound.lean:3531} both\nroute, through their respective consumers, into the same greedy-survival dichotomy that\n\\lean{mem\\_mersenneAchievementSet\\_iff\\_greedy\\_survival}{GreedyAchievementSet.lean:1445}\nalready states as an iff. So the four certificate families \u2014 reset-crossing (Part I), truncation\nrungs (TR ladder), sharp-capacity fill (this section), and the direct greedy-survival test \u2014 are\nnot four unrelated attacks; they are \\emph{one semi-decision procedure} viewed in four\ncoordinates, each of which halts (finds a certificate) if and only if the others eventually\nwould, because each is checking a lower bound on the same underlying finite rank. This makes\n\"\\#257 is false\" semi-decidable via any one of the four routes (run the certificate search; if it\nhalts, \\#257 is false), while \"\\#257 is true\" is not shown semi-decidable by any route in this\ncorpus \u2014 the co- half of co-semi-decidability is exactly the open reset-anti-concentration\nstatement of Theorem A. Practical consequence for the interface as a whole: a negative result\n(more certificate search, deeper truncation rungs, more reset rows) can never by itself prove\n\\#257 true, however far it is pushed \u2014 this is the same quantifier trap SE-7/RC-7 warn against,\nrestated as a decidability fact rather than a scope caveat.\n\\end{thm}\n\n\\subsection{Route-collapse findings: four traps to name explicitly}\n\nThese four rows are not near-misses to be closed by further work \u2014 they are targets that, once\nadversarially checked, turn out to be logically equivalent to the obligation itself (or to\n$1/2\\in\\mathrm{mersenneAchievementSet}$ directly) for the scope in which they are stated. Treating\nany of them as an easier waypoint is a mistake this index exists to prevent.\n\n\\begin{rem}[hbound is equivalent to the conclusion, greedy support only]\n\\lean{greedy\\_half\\_infinite\\_of\\_mobiusCenteredHalfCarry\\_sqrtBound}{HalfCarryReachability.lean:805}\nand its packaged form\n\\lean{greedy\\_half\\_infinite\\_of\\_mobiusCenteredHalfCarry\\_upperBound}{HalfCarryReachability.lean:924}\ntake a hypothesis \\texttt{hbound}: $\\forall N,\\ \\mathrm{mobiusCenteredHalfCarry}\\,A\\,N \\le\n2\\sqrt N + 4$. For the canonical greedy support $A=\\mathrm{greedyMersenneSupport}(1/2)$\nspecifically, \\texttt{hbound} is logically \\emph{equivalent} to the theorem's own conclusion\n$\\mathrm{erdosSupportSeries}\\,2\\,A=1/2$ \u2014 not a strictly weaker sufficient condition. This follows\nfrom the exact residual identity at line 842, the unconditional sqrt-envelope tail bound\n(\\lean{binaryCoeffTail\\_supportCoeff\\_le\\_two\\_sqrt\\_add\\_four}{BooleanMobiusCarry.lean:289}), the\nunconditional tail-nonnegativity lemma\n(\\lean{binaryCoeffTail\\_nonneg}{GenericTailOrbitRigidity.lean:77}), and the greedy-specific fact\n$\\mathrm{erdosSupportSeries}\\,2\\,A\\le1/2$ (line 859). \\textbf{The scope limit is exact and must\nnot be widened by inference:} this equivalence does \\emph{not} extend to every $A$ with $1\\notin\nA$ \u2014 dropping the greedy-specific $\\le1/2$ fact yields only $\\mathrm{hbound}\\Leftrightarrow\n\\delta\\le0$ (a one-sided condition), not $\\delta=0$. It also does \\emph{not} extend to the\nsibling socket \\lean{CofinalStripReturn}{CofinalStripReturn.lean:76}: \\texttt{hbound} is stated\nin \\texttt{Real.sqrt}, while \\texttt{CofinalStripReturn}'s \\texttt{halfStripBound} is built on\n\\texttt{Nat.sqrt}, and bridging the real bound down to the tighter natural-number floor bound\nneeded for a cofinal witness is an unaddressed, nontrivial gap \u2014 not routine packaging.\n\\end{rem}\n\n\\begin{rem}[HalfCarryCofinalTerminalOnlyStrip, widened to constant 6, still equivalent]\n\\lean{HalfCarryCofinalTerminalOnlyStrip}{TerminalOnlyCofinal.lean:34}, even after widening the\nstrip constant from 4 to 6 ($\\mathrm{halfStripBound}'\\,n = 2\\,\\mathrm{Nat.sqrt}\\,n+6$), is\nlogically equivalent to $1/2\\in\\mathrm{mersenneAchievementSet}$ once three small, non-circular,\nmechanical lemmas are supplied (a general one-not-in-support corollary, the discrete envelope\ninequality $\\mathrm{Real.sqrt}\\,n \\le \\mathrm{Nat.sqrt}\\,n+1$, and a routine\nprefix-to-\\texttt{HalfWord} construction). None of the three gaps is equivalent to \\#257 or \\#249\nindividually \u2014 the equivalence sits one level up, at the socket itself. \\textbf{This is a\ndiagnostic, not a route}: it corrects TerminalOnlyCofinal.lean's own \"weaker producer\" framing \u2014\nthe terminal-only cofinal strip carries no strategic advantage over proving achievement\ndirectly.\n\\end{rem}\n\n\\begin{rem}[CofinalPositiveHalfGreedySkips is literally the problem]\nRepeated here for emphasis from the 257-cofinal-rows entry above: once the positivity conjunct of\n\\lean{CofinalPositiveHalfGreedySkips}{BooleanMobiusSkipRowCofinal.lean:22} is recognised as\nunconditionally free (odd-numerator/even-subtrahend parity plus greedy nonnegativity, discharged\nat \\lean{}{BooleanMobiusCriticalCapacityCofinal.lean:138}), the socket collapses to\n$(\\mathrm{greedyMersenneSkippedSupport}(1/2))$.Infinite, proved by A6\n(\\lean{}{GreedyAchievementSet.lean:2514}) to be \\emph{equivalent} to $1/2\\in\n\\mathrm{mersenneAchievementSet}$. A proof or disproof of this one Prop settles \\#257-at-1/2 and\nvice versa; it is not a waypoint toward that settlement.\n\\end{rem}\n\n\\begin{rem}[Scope discipline for the equivalence class]\nAll three collapsed targets above share the same failure mode against naive strengthening: each\nis equivalent to the achievement conclusion \\emph{only in the exact scope stated} \u2014 the\ncanonical greedy support for \\texttt{hbound}, the widened-constant strip for\n\\texttt{HalfCarryCofinalTerminalOnlyStrip}, the single target $1/2$ for\n\\texttt{CofinalPositiveHalfGreedySkips}. None of the three equivalences transports along the\nnatural-looking generalizations (all $A$ with $1\\notin A$; \\texttt{Nat.sqrt}-based\n\\texttt{CofinalStripReturn}; other rational targets) without a separately unproven bridging\nlemma. A future attacker who verifies the scoped equivalence and then silently drops the scope\nqualifier reproduces exactly the quantifier-slippage error this index is built to catch.\n\\end{rem}\n\n% ---- part a257_p4 ----\n\\section{Closed routes, with the mechanism that closed them}\n\nThis section is a graveyard with load-bearing headstones. Each entry below is a route that was\ntried against the $\\sqrt{}$-escape wall (Part~III) and failed, but the way it failed is itself a\ntheorem, a computation, or a structural identity that survives the failure. A closed route is not\na blank; it is a constraint on every future attempt. We record, for each closure: the route as\nconceived, the exact mechanism that closed it, the precise scope of the closure (what it rules\nout and what it leaves untouched), and whatever machinery salvages out of the wreckage. We also\nflag, for every closure, whether the exclusion is a statement about the \\emph{object} (the real\nnumber, the achievement set, the actual orbit) or about a \\emph{representation} of it (a\ncoordinate, a finite-window signature, a proof-route's hypothesis set). Confusing the two is the\nsingle most common way a closed route gets silently re-opened by a later attempt using different\nwords for the same representation.\n\nThroughout, $\\mathrm{rem}(s)$ is the seam-row remainder, $\\lambda_n := \\Delta_n/2^n$ the normalized seam\ndeviation (Part~I, SE-3), and $\\mathcal A$ the Mersenne achievement set. Reset $\\sqrt{}$-escape\nfrom row 10 is the statement $|\\mathrm{rem}(r+1)-2^{r+1}| > 2^{(r+5)/2}$ for every reset row $r\\ge 10$\n(RC-4, Part~III); this is the wall every closure below is measured against.\n\n\\subsection{Four closed attack families on the wall itself}\n\n\\begin{obs}[Prime-doubling \u2014 closed by explicit counterexample]\n\\label{obs:prime-doubling}\n\\textbf{Route as conceived.} The universal recursion $K(M+1) = 2K(M) - (\\tau(M+1)-1)$ (Part~I,\nSE-1) doubles at every step and subtracts the divisor-excess $\\tau(M+1)-1$. At a prime $M+1=p$,\n$\\tau(p)=2$ so the subtracted term is $1$; the naive hope is that primality of an index forces the\nstep to be \"almost pure doubling,\" and that prime density (Chebyshev/PNT-scale) then bounds how\nlong a run of near-pure-doubling steps can last, which would bound the danger run-length $L_r$ in\nSE-3's equivalence (III).\n\n\\textbf{Exact mechanism that closed it.} The hope requires primality to control the recursion\n\\emph{cumulatively}, but composite indices subtract more than primes add back, so a long run can\nmix primes and composites freely without breaking. Certified witness: the run at $n=607$ spans\n$M\\in(607,617]$ with the drift quantity $J$ staying in $[1,5]$ for eleven consecutive steps, and\nthree of the indices inside that window \u2014 $607,613,617$ \u2014 are themselves prime. More broadly, of\nthe 40 longest runs found for $n\\le 1500$, 26 contain a prime index somewhere inside them. Primality\nof an index is simply uncorrelated with the recursion staying quiet.\n\n\\textbf{Scope.} Closes: any argument that tries to derive a run-length bound on SE-3's $L_r$, or a\nbound on the drift $J$ in the universal recursion, from prime-counting alone (density,\nChebyshev bounds, PNT-scale gap statistics). Does not touch: arguments using the actual value of\n$\\tau(M+1)-1$ (not just whether $M+1$ is prime), or any argument working at the divisor-sum level\nrather than the prime/composite dichotomy.\n\\coord{prime-density}\n\n\\textbf{Object or representation.} About the \\emph{object}: the counterexample is a concrete\nwitness inside the actual $K(M)$ recursion, not an artifact of how the recursion is coordinatized.\n\n\\textbf{Salvage.} None as a proof tool \u2014 but the witness itself (run length 11 at $n=607$,\n26/40 longest runs prime-touched) is now a banked disconfirming data point: any future\nprime-density-flavoured attack on run-length should be checked against this witness first.\n\\ev{Cert}\n\\end{obs}\n\n\\begin{obs}[Parity/perfect-squares \u2014 closed as a tautology of the recursion shape]\n\\label{obs:parity-squares}\n\\textbf{Route as conceived.} $K(M)$ is even exactly when $M$ is a perfect square (a true,\nprovable fact). The hope was to use this parity law to forbid certain residues or run patterns in\nthe digit stream of $K$, since parity constraints are a classical first attack on digit-run\nquestions.\n\n\\textbf{Exact mechanism that closed it.} The recursion $J(M+1) = 2J(M) - \\delta(M+1)$ makes\n$J(M+1)\\equiv\\delta(M+1)\\pmod 2$ an \\emph{identity for any digit stream whatsoever} that this\nrecursion shape produces \u2014 it holds regardless of what $\\delta$ counts, Mersenne-specific or not.\nThe \"$K(M)$ even iff $M$ a perfect square\" fact is real, but it is a restatement of the recursion's\nalgebraic shape, not new information: it forbids no residue and no run length. It is vacuous as an\nobstruction because it would hold for \\emph{every} recursion of the form $J(M+1)=2J(M)-\\delta(M+1)$,\nMersenne or not.\n\n\\textbf{Scope.} Closes: the parity/perfect-square route completely and permanently, for this\nrecursion shape \u2014 not a partial closure. Also closes it in advance for any structurally identical\nrecursion (e.g.\\ a hypothetical \\#249 digit-carry recursion of the same doubling-minus-forcing\nshape), since the vacuity is a fact about the shape, not about $\\tau$ or Mersenne weights.\n\\coord{parity-tautology}\n\n\\textbf{Object or representation.} About a \\emph{representation}: the identity is a fact about the\nrecursion's algebraic form (base-2 doubling-with-forcing), not about the value $K(M)$ or the\ntarget constant $C=E-3/2$ itself. It is empty precisely because it never touches the object.\n\n\\textbf{Salvage.} The identity remains true and is a useful sanity check (any proposed digit-run\nclaim about $K$ must be consistent with it), but it supplies zero constraining power on its own.\n\\ev{Math}\n\\end{obs}\n\n\\begin{obs}[Irrationality-measure \u2014 structurally excluded by an exponent budget]\n\\label{obs:irrationality-measure}\n\\textbf{Route as conceived.} The wall reduces (SE-2) to: the greedy skip-set sum\n$\\sum_{d\\in\\mathrm{Skip}_n}x_d$ never approximates $C:=E-\\tfrac32=0.1066951524152917\\ldots$ (the shifted\nErd\u0151s\u2013Borwein constant) to within $\\sim 2^{-1.5n}$. Since $C$ has a known irrationality-measure\nupper bound, the natural hope is that a Diophantine-approximation exponent for $C$ directly\nsupplies the needed non-approximability at scale $n$.\n\n\\textbf{Exact mechanism that closed it.} The skip-set $\\mathrm{Skip}_n$ has size $|\\mathrm{Skip}_n|\\approx n^2/4$ \u2014\nquadratic in $n$, not linear. An irrationality-measure exponent $\\mu$ for $C$ bounds how well a\n\\emph{single} rational with denominator $q$ can approximate $C$; specializing to a rational built\nfrom $\\approx n^2/4$ terms only yields a lower bound on the approximation error of order\n$2^{-\\mu n^2/4}$, against a required bound of order $2^{-3n/2}$. The shortfall is a factor of\n$\\approx 0.42n$ \\emph{in the exponent itself} \u2014 not a constant-factor gap that a sharper $\\mu$\ncould close, but a gap that grows without bound as $n\\to\\infty$ for any fixed $\\mu$. Zudilin's\nbound $\\mu(E)\\le 2.42343562\\ldots$ (Math.\\ Notes 72 (2002) 858--862; erratum in Acta Arith.\\ 111\n(2004) 153--164 revises this to $\\mu(E)\\le 2.46497868\\ldots$) is the best known value, but no future\nimprovement in $\\mu$ can close this gap, because $\\mu\\ge 2$ always (a universal lower bound for any\nirrational number), while the exponent shortfall grows like $n$ regardless of which fixed $\\mu$ is\nplugged in.\n\n\\textbf{Scope.} Closes: every attempt to derive the wall from a single-number irrationality-measure\nstatement about $C$ (or about $E$), present or future, as long as the measure is a fixed exponent\nindependent of $n$. Does not touch: simultaneous or multi-point Diophantine-approximation results\nthat could in principle scale with the number of terms (no such result is known to exist for this\nconstant, but the exponent-budget argument does not rule out the \\emph{possibility} of one \u2014 only\nthe single-exponent route).\n\\coord{diophantine-approximation}\n\n\\textbf{Object or representation.} About the \\emph{object}: this is a genuine value-theoretic\ninequality (an exponent-budget computation), not an artifact of how the skip-set is coordinatized.\n\n\\textbf{Salvage.} Flagged in the source corpus as the single strongest reusable negative lemma in\nthe whole programme: any future proposal to attack a Mersenne/Erd\u0151s\u2013Borwein-type constant via an\nirrationality-measure bound should be checked against this exponent-budget argument \\emph{first},\nbefore any other work is invested.\n\\ev{Math}\n\\end{obs}\n\n\\begin{prop}[2-adic fixed-precision valuation-unit no-go \u2014 Lean-verified, problem-agnostic]\n\\label{prop:2adic-nogo}\n\\textbf{Route as conceived.} A carry orbit's local 2-adic signature \u2014 its valuation and odd unit\npart at some fixed precision window \u2014 looks like a natural finite object to search for a\ncontradiction: if every carry orbit compatible with a given local valuation-unit word were forced\noutside a safe centred interval, that would be a genuine local obstruction usable at every scale.\n\n\\textbf{Exact mechanism that closed it.} For \\emph{any} finite word of odd-unit-part valuation\nsymbols at fixed precision $u>0$, and \\emph{any} starting carry state $e$, there exists a\ncompatible carry orbit realizing that exact word with every intermediate state staying centred in\nits dyadic interval, $|e'|\\le \\mathrm{vuRadius}\\,u\\,\\sigma$. Bounded local valuation-unit data at\nfixed precision can never exclude \\emph{all} finite centred carry completions \u2014 the completion\nalways exists, constructively.\n\\[\n  \\exists\\,\\text{states},\\; \\mathrm{VUOrbit}\\,u\\,e\\,\\text{symbols}\\,\\text{states} \\;\\land\\;\n  (\\forall\\,\\text{symbol},\\, |\\text{state}| \\le \\text{radius}).\n\\]\n\\lean{Erdos249257.TotientTailPeriodKiller.fixedPrecisionTropicalNoGo}{TropicalCurvatureCarry.lean:137}\nbuilt from\n\\lean{Erdos249257.TotientTailPeriodKiller.vu\\_step\\_has\\_centred\\_completion}{TropicalCurvatureCarry.lean:69}\nand\n\\lean{Erdos249257.TotientTailPeriodKiller.vu\\_word\\_has\\_prefix\\_locked\\_completion}{TropicalCurvatureCarry.lean:115}.\n\n\\textbf{Scope.} Closes: any proof strategy that tries to derive a contradiction purely from \"the\nlocal valuation-unit signature at fixed precision $u$ is incompatible with $X$\" \u2014 such signatures\nare \\emph{always} realizable by some carry orbit, for every $u$ and every starting state. Does not\ntouch: arguments using \\emph{growing} precision (an unbounded window, not a fixed one), or\narguments that couple the local signature to extra arithmetic (e.g.\\ a global divisibility or\nCRT constraint) rather than using the local signature alone.\n\\coord{2-adic-valuation-unit}\n\n\\textbf{Object or representation.} About a \\emph{representation}: the theorem is entirely about\nwhat a bounded local 2-adic window of a carry orbit can and cannot pin down; it says nothing about\nthe target value $C$ or the achievement set itself. It is a statement that a particular family of\nfinite coordinates (fixed-precision valuation-unit words) is too coarse to see the obstruction, not\nthat no obstruction exists.\n\n\\textbf{Salvage.} The theorem is stated and proved for \\emph{any} starting carry state and any odd\nunit-part word \u2014 zero arithmetic content beyond 2-adic bookkeeping. It is directly reusable as a\nhard stop against any future fixed-window 2-adic attack on \\emph{either} \\#249 or \\#257: a proof\nneeds growing precision or extra coupling, never a fixed local signature alone.\n\\ev{Lean}\\scale{n/a}\n\\end{prop}\n\nRead together, Observations~\\ref{obs:prime-doubling}--\\ref{obs:irrationality-measure} and\nProposition~\\ref{prop:2adic-nogo} close every attack\nfamily identified against the wall as of this writing that does not go through the digit-level\ncarry machinery itself. What remains open (Part~III, and \\S\\ref{sec:branch-exclusions} below) is\nexactly the digit-carry route \u2014 this is not a residual gap after eliminating easier options; it is\nthe one route none of these four mechanisms can even in principle reach, because each closure\nabove excludes a \\emph{different} kind of coarse-grained argument (density, tautological parity,\nsingle-exponent approximation, fixed-window valuation) while leaving the exact digit recursion\nitself untouched.\n\n\\subsection{Branch exclusions at the final producer: the exceptional dyadic cells}\n\\label{sec:branch-exclusions}\n\nThe two-sided dyadic invariant $\\min(\\mathrm{rem}(s), \\mathrm{overshoot}(s)) \\le 2^s$ (all $s\\ge 5$) is the\nsharpest general-purpose control on the seam orbit's scale in the whole corpus. It propagates by\ninduction from a single local socket, \\textsf{SeamTwoSidedDyadicCellEscape}, which at every row\nexcludes a small number of \\emph{exceptional cells} on the middle branch and imposes one further\ninequality on the right branch:\n\\[\n  \\mathsf{SeamTwoSidedDyadicCellEscape} :=\n  \\forall\\, s\\ge 5,\\;\n  \\Big(\\lnot\\mathrm{carries} \\to \\mathrm{middle} \\to C_s \\ne -3 \\land C_s \\ne -2 \\land C_s \\ne -1\\Big)\n  \\;\\land\\;\n  \\Big(\\lnot\\mathrm{carries} \\to \\mathrm{right} \\to \\mathrm{overshoot}\\le 2^s \\to \\mathrm{charge}\\le 2^{s+2}\\Big),\n\\]\nwhere we write $C_s := 4\\cdot\\mathrm{rem}(s) - \\mathrm{belowPulse}(s) - 4$ for the integer coordinate that\nappears on the middle branch of the seam recursion (the same quantity that, along an all-right run,\nis the affine excess $\\mathrm{rem}(s)-2^s$ one step later \u2014 Part I, RC-2).\n\\lean{Erdos249257.SeamTwoSidedDyadicCellEscape}{HalfCylinderMiddleCarryLowerBound.lean:3339}\n\n\\begin{prop}[The upper (carries) successor needs no exceptional-cell exclusion]\n\\label{prop:upper-unconditional}\nOn the carries branch of the step (i.e.\\ $(\\mathrm{seamAdjacentCut}\\ s\\ hs).\\mathrm{successorCarries}$\nholds), the two-sided invariant propagates \\emph{unconditionally}, with zero exceptional cells and\nno hypothesis beyond \\texttt{hcarry} itself:\n\\[\n  \\mathrm{successorCarries} \\implies \\mathrm{rem}(s+1) \\le 2^{s+1}.\n\\]\n\\lean{Erdos249257.seamUpperBranch\\_nextRemainder\\_le\\_pow}{HalfCylinderMiddleCarryLowerBound.lean:2916}\n(from\n\\lean{Erdos249257.seamUpperBranch\\_remainder\\_add\\_resetCharge\\_eq}{HalfCylinderMiddleCarryLowerBound.lean:2916},\nclosed by \\texttt{omega}). In this precise sense \u2014 no cell exclusion is ever needed on this branch\nof the step \u2014 the upper successor contributes nothing to the exceptional-cell inventory: it is\n\"impossible\" for the upper branch to be a source of an exceptional cell in the induction step,\nbecause the branch is handled outright.\n\\coord{seam-branch-classification}\\ \\textbf{Object or representation:} about a\n\\emph{representation} (the induction-step case split), not the orbit itself.\n\\ev{Lean}\\scale{n/a}\n\\end{prop}\n\n\\begin{thm}[$C_D=-3$ is impossible at the final middle producer, $D\\ge 13$]\n\\label{thm:cd-neg3-impossible}\nConsider a hypothetical \\emph{final middle producer}: a middle transition at row $D\\ge 13$\n($\\lnot\\mathrm{carries}$, and the middle-branch inequality\n$4\\mathrm{rem}(D)+\\mathrm{gap}-\\mathrm{belowPulse} < \\mathrm{terminalWeight}$ holds at $D$), followed\nby an all-right tail forever after ($\\forall s\\ge D{+}1$, $\\mathrm{seamGreedyWord}(s+1) =\n\\mathrm{seamGreedyWord}(s).\\mathrm{extend}\\ \\mathrm{true}$). Then, unconditionally,\n\\[\n  \\mathrm{belowPulse}(D) + 2 \\;\\le\\; 4\\cdot\\mathrm{rem}(D),\n\\]\ni.e.\\ $C_D \\ge -2$ \u2014 the cell $C_D=-3$ (and every more negative value) is excluded.\n\\lean{Erdos249257.middleThenAllRight\\_landingExcess\\_two\\_le}{HalfCylinderMiddleCarryLowerBound.lean:1458}\n\n\\textbf{Mechanism.} The proof chains three facts: (i) an all-right tail after $D$ forces the\nterminal-augmented finite prefix at $D{+}1$ strictly above $1/2$\n(\\lean{Erdos249257.half\\_lt\\_upper\\_competitor\\_of\\_eventually\\_right}{HalfCylinderFatalGapRightTail.lean:627}),\nhence the producer carry is strictly below its complete future incidence tail\n(\\lean{Erdos249257.middleProducer\\_allRight\\_forces\\_carry\\_lt\\_tail}{HalfCylinderMiddleCarryLowerBound.lean:953});\n(ii) that carry/tail inequality transports to the exact rational model as\n$\\mathrm{seamGreedyFloorZ}(D) < \\mathrm{seamTakeThreshold}(D)$\n(\\lean{Erdos249257.middleProducer\\_allRight\\_forces\\_floorZ\\_lt\\_takeThreshold}{HalfCylinderMiddleCarryLowerBound.lean:1008});\n(iii) combined with the pulse-absorption bound\n$\\mathrm{belowPulse}(D)\\le 4\\cdot\\mathrm{seamWordFloorError}(D)$ and the exact floor identity\n$\\mathrm{seamGreedyFloorZ}(D)=\\mathrm{rem}(D)-\\mathrm{seamWordFloorError}(D)$, plus the unconditional strip\nbound $1/3 < 4^D\\cdot\\mathrm{mersenneTail}(D)-2^D$, the strict inequality forces\n$\\mathrm{belowPulse}(D)+1 < 4\\mathrm{rem}(D)$ in $\\mathbb{Q}$, hence $+2\\le$ in $\\mathbb{N}$.\n\n\\textbf{Scope.} This closes $C_D=-3$ \\emph{only inside the final-middle-producer-then-all-right\nscenario} at $D\\ge 13$ \u2014 not at every late middle row unconditionally. It is a strictly narrower\nclaim than \\textsf{SeamTwoSidedDyadicCellEscape}'s general induction step, which still needs all\nthree cells excluded at \\emph{every} late middle row (not just a hypothetical final one) to\npropagate the two-sided invariant everywhere. The two statements share the same coordinate $C_D$\nbut differ in quantifier scope: Theorem~\\ref{thm:cd-neg3-impossible} is a $\\exists$-scenario\nclosure (one specific final producer under an all-right hypothesis), while the general socket is a\n$\\forall$-row closure. Do not conflate the two: closing $C_D=-3$ here does not close it in the\ngeneral induction step.\n\\coord{final-producer/landing-excess}\\ \\textbf{Object or representation:} about the \\emph{object}\nunder the stated hypothesis \u2014 a genuine arithmetic consequence of the all-right-tail assumption\nvia the fatal-gap orbit, not a coordinate artifact. \\ev{Lean}\\scale{n/a}\n\\end{thm}\n\n\\begin{cor}[Only $C_D\\in\\{-2,-1\\}$ remain among the exceptional dyadic cells]\n\\label{cor:cd-remaining}\nCombining Proposition~\\ref{prop:upper-unconditional} (upper branch: zero exceptional cells, ever)\nwith Theorem~\\ref{thm:cd-neg3-impossible} ($C_D=-3$ closed, in the final-producer sub-case): of the\nthree cells $\\{-3,-2,-1\\}$ that \\textsf{SeamTwoSidedDyadicCellEscape} must in general exclude on the\nmiddle branch, only $C_D\\in\\{-2,-1\\}$ remain genuinely open. No Lean theorem in the tree at the time\nof writing excludes $C_D=-2$ or $C_D=-1$, in either the final-producer scenario or the general\ninduction step.\n\\coord{final-producer/landing-excess}\\ \\ev{Lean}\n\\end{cor}\n\n\\begin{rem}[The tail-dominance criterion \u2014 stated exactly, and not proved]\n\\label{rem:tail-dominance-open}\nWrite $\\Theta_D$ for the real-valued producer-carry-to-tail ratio quantity that Theorem\n\\ref{thm:cd-neg3-impossible}'s mechanism step (i) actually compares against $C_D$: concretely,\n$\\Theta_D$ is the scaled complete-tail budget\n$\\mathrm{binaryCoeffTail}(\\mathrm{supportCoeff}(\\text{terminal-augmented prefix}))(2D+2)$ that\nappears on the right-hand side of\n\\lean{Erdos249257.middleProducer\\_allRight\\_forces\\_carry\\_lt\\_tail}{HalfCylinderMiddleCarryLowerBound.lean:953},\nexpressed in the same integer units as $C_D$. The natural closing statement suggested by chaining\n\\textsf{SeamMiddleProducerCardEscape} / \\textsf{SeamMiddleProducerRowEscape}\n(\\lean{Erdos249257.SeamMiddleProducerCardEscape}{HalfCylinderMiddleCarryLowerBound.lean:111},\n\\lean{Erdos249257.SeamMiddleProducerRowEscape}{HalfCylinderMiddleCarryLowerBound.lean:908}) with\nCorollary~\\ref{cor:cd-remaining} is the \\textbf{tail-dominance criterion}:\n\\[\n  \\textbf{(TD)}\\qquad \\forall\\, D\\ge 13,\\ \\big(\\text{late middle row},\\ C_D\\ne -3\\big) \\implies\n  \\Theta_D < C_D.\n\\]\nIf (TD) held at every late middle row, it would supply exactly the missing half of\n\\textsf{SeamTwoSidedDyadicCellEscape}'s remaining two cells and close the whole\n$\\sqrt{}$-escape wall via\n\\lean{Erdos249257.half\\_mem\\_mersenneAchievementSet\\_of\\_middleProducerCardEscape}{HalfCylinderMiddleCarryLowerBound.lean:1503}.\n\\textbf{(TD) is NOT proved anywhere in the tree.} It does not appear as a theorem, a certified\ncomputation, or even a numerically-checked conjecture in the files read for this survey \u2014 it is\nthis section's own naming of the gap the corollary leaves open, stated so that a future attempt has\nan exact target rather than a vague \"close the remaining cells\" instruction. Anyone attempting it\nshould start from \\textsf{SeamMiddleProducerRowEscape} (the weaker, row-scale sufficient condition\n$s\\le\\mathrm{rem}(s)$), not from the sharper card-escape socket.\n\\ev{Open}\\scale{cofinal}\n\\end{rem}\n\n\\subsection{No-go, countermodel, and rigidity modules that bound the wall from outside}\n\nThe digit-carry route is not merely unclimbed; a family of general-purpose no-go and countermodel\nresults independently constrains \\emph{what kind of proof} can ever close it. These are not about\nthe wall's numerics \u2014 they are about which proof architectures are structurally impossible, for any\ninput.\n\n\\begin{prop}[Finite-state no-go: no bounded carry-state summary can recover history]\n\\label{prop:finite-state-nogo}\nFor a \"balanced pulse\" family at location $m$ (radius $r=(m{+}1)/2$, moving mass between positions\n$m$ and $m{+}1$ without changing the series value), if a predecessor state is constant across the\nwhole family, then no function $\\mathrm{decode}:\\mathrm{State}\\to\\mathbb N$ can recover the\nparameter $r$ from $\\mathrm{state}(r)$ for every $r$ \u2014 fan-out is unbounded ($\\ge\\lfloor m/2\\rfloor+2$\nat $m=2k$). More strongly, any finite $\\mathrm{Fintype}\\ \\mathrm{State}$ needs\n$\\mathrm{card}(\\mathrm{State})\\ge\\mathrm{radius}+1$, unbounded in $m$.\n\\lean{Erdos249257.balancedPulse\\_no\\_autonomous\\_decoder}{GenericTailOrbitRigidity.lean:246}\n(fan-out lower bound at\n\\lean{Erdos249257.balancedPulse\\_fanout\\_unbounded}{GenericTailOrbitRigidity.lean:236}).\n\n\\textbf{Scope.} Rules out, for \\emph{either} \\#249 or \\#257, any proof strategy that tries to define\na bounded/autonomous \"carry state\" summarizing pre-$m$ history and use it to exactly determine the\npost-$m$ tail \u2014 completely general, no dependence on whether the coefficients are $\\varphi$ or a\nM\u00f6bius-support indicator.\n\\coord{binary-digit, generic}\\ \\textbf{Object or representation:} about a \\emph{representation}\nclass (any finite-automaton encoding of history) \u2014 it says nothing about $C$ or $\\mathcal A$\ndirectly, only that this whole family of encodings is too weak.\n\\ev{Lean}\\scale{n/a}\n\\end{prop}\n\n\\begin{prop}[M\u00f6bius-support countermodel: the natural negative-sign candidate overshoots]\n\\label{prop:mobius-nogo}\nThe signed Lambert identity $\\sum_{d\\ge1}\\mu(d)/(2^d-1) = 1/2$ is exact. Writing\n$N:=\\{d:\\mu(d)=-1\\}$: $\\sum_{d\\in N}1/(2^d-1) = 1/2 + \\sum_{d\\in P}1/(2^d-1)$ where $P:=\\{d\\ge2:\n\\mu(d)=1\\}$, and quantitatively $1/2 + 1/63 \\le \\sum_{d\\in N}1/(2^d-1)$ (using the first positive\ntail term $d=6$, $\\mu(6)=1$) \u2014 the negative-M\u00f6bius support strictly \\emph{overshoots} $1/2$ by at\nleast $1/63$.\n\\lean{MobiusSignSupportNoGo.half\\_lt\\_tsum\\_negativeMobius}{MobiusSignSupportNoGo.lean:164}\n(exact decomposition at\n\\lean{MobiusSignSupportNoGo.tsum\\_negativeMobius\\_eq\\_half\\_add\\_positiveMobiusTail}{MobiusSignSupportNoGo.lean:111}).\n\n\\textbf{Scope.} Rules out exactly one natural candidate infinite Boolean support (the negative-M\u00f6bius\nset) as a witness for $1/2\\in\\mathcal A$. A route-sufficiency no-go only \u2014 it says nothing about\nwhether \\emph{some other} infinite Boolean support sums to $1/2$.\n\\coord{mobius-mersenne}\\ \\textbf{Object or representation:} about the \\emph{object} \u2014 a genuine\nvalue inequality for one specific candidate set, not a coordinate artifact. Margin $1/63$ is the\nconcrete number any repair attempt (adding/removing finitely many elements) must close or exceed.\n\\ev{Lean}\\scale{fixed}\n\\end{prop}\n\n\\begin{prop}[No finite support ever hits $1/2$; finite certificates only ever certify death]\n\\label{prop:finite-boolSupport-and-onesided}\nNo finite positive-index Boolean support has value exactly $1/2$: the reduced denominator of any\nfinite Mersenne subset-sum is provably \\textbf{odd} (each $2^n-1$ is odd), while $1/2$ needs an even\ndenominator.\n\\lean{finite\\_boolSupport\\_ne\\_half}{HalfCarryReachability.lean:589}. Separately,\n$\\mathsf{CertifiedGreedyMersenneDeath}$ is a decidable, finite-depth certificate proving a real $x$\nis \\emph{not} in $\\mathcal A$ (e.g.\\ $3/4$ is machine-certified dead at level 1, lookahead 0:\n\\lean{three\\_fourths\\_certifiedGreedyMersenneDeath}{GreedyAchievementSet.lean:1742}), and this\none-sidedness is structural: \\textbf{absence} of a found death certificate, or survival through any\nfinite depth, proves \\emph{nothing} about membership.\n\\lean{Erdos249257.CertifiedGreedyMersenneDeath}{GreedyAchievementSet.lean:1720}.\n\n\\textbf{Scope.} The odd-denominator fact rules out every finite support as a $1/2$-witness,\nunconditionally (this is exactly B4/C3 of Part~I/III's cut-locator machinery, and it is why any\n$1/2$-achieving support, if one exists, must be genuinely infinite). The one-sidedness caveat rules\nout treating a finite certified-kill search's silence as evidence of membership or rationality, for\n\\emph{any} member of the whole certified-kill family (this closure family, \\#249's\n\\textsf{TotientTailPeriodKiller}, \\textsf{LcmConeFlatness}, etc.) \u2014 a structural, not\nproblem-specific, warning that recurs across the entire certificate-based proof programme.\n\\coord{parity-denominator / finite-search}\\ \\textbf{Object or representation:} the odd-denominator\nfact is about the \\emph{object} (value inequality via denominator parity); the one-sidedness caveat\nis about the \\emph{representation} (what a finite search, as a proof method, can and cannot show).\n\\ev{Lean}\\scale{fixed}\n\\end{prop}\n\n\\begin{prop}[Carry-orbit closure: \\#249's rigidity engine, occupied ground for \\#257]\n\\label{prop:carry-survivor-extinction}\nFor $\\sum_n\\varphi(n)/2^n$: a \"survivor kill\" certificate (every integer candidate in a bounded box\nprovably escapes a shrinking strip within $K$ steps, decidable) proves the tail difference\n$\\mathrm{totientTail}(N{+}h)-\\mathrm{totientTail}(N)$ is not an integer. Rationality would force\nintegrality along an entire period ray, so one dead multiple $m\\cdot h_0$ kills the whole primitive\nperiod $h_0$\n(\\lean{Erdos257PeriodNoncollapse.irrational\\_totient\\_series\\_of\\_multiple\\_survivor\\_supply}{CarrySurvivorExtinction.lean:491}),\nand this collapses onto the single $\\mathbb N$-indexed family $\\mathrm{periodLcm}(t)=\\mathrm{lcm}(1,\\ldots,t)$\n(\\lean{Erdos257PeriodNoncollapse.irrational\\_totient\\_series\\_of\\_lcm\\_survivor\\_supply}{CarrySurvivorExtinction.lean:540}).\nConcrete unconditional deposit: every period $h\\le16$ is machine-checked dead at $(N,L)=(14,9)$\n(\\lean{Erdos257PeriodNoncollapse.totient\\_series\\_ne\\_rat\\_of\\_den\\_dvd\\_upto\\_sixteen}{CarrySurvivorExtinction.lean:587}).\n\n\\textbf{Scope \u2014 a naming correction on record.} \\textsf{CarrySurvivorExtinction.lean},\n\\textsf{AdjacentCarryTube.lean}, \\textsf{AdjacentPhaseSeparation.lean}, and\n\\textsf{TotientCarryKernelRigidity.lean} are, despite names suggestive of \\#257 support-rigidity,\n\\textbf{\\#249 files} (namespace \\texttt{Erdos249257.TotientTailPeriodKiller}, statements entirely\nabout $\\sum\\varphi(n)/2^n$). They are recorded in this closed-routes ledger not because they close\nanything about \\#257 directly, but because (i) their engine is the shared tempered-orbit machinery\nalso used by \\#257's own rigidity results (Part~III's F1/T7), and (ii) a \\#257-support reader must\nknow this ground is \\emph{occupied by \\#249}, not available \u2014 a route attempting to reuse these\nexact declarations for a \\#257 support-side argument would be reusing the engine, not the theorem.\n\\coord{integer carry-orbit, totient-specific instantiation}\\ \\textbf{Object or representation:}\nabout the \\emph{object} ($\\sum\\varphi(n)/2^n$'s actual value), for \\#249 \u2014 irrelevant by content, not\nby mechanism, to \\#257.\n\\ev{Lean}\\scale{cofinal}\n\\end{prop}\n\n\\subsection{The scalar-localisation height obstruction}\n\n\\begin{lem}[Denominator complement survives scaling]\n\\label{lem:scalar-localization}\nFor $x:\\mathbb Q$, $c:\\mathbb Z$, $H:\\mathbb N$: if $H\\mid x.\\mathrm{den}$ and\n$(c\\cdot x).\\mathrm{den}\\mid H$ \u2014 i.e.\\ multiplying by the integer $c$ shrinks the displayed\ndenominator down \\emph{into} $H$ \u2014 then the \\textbf{complementary} denominator factor\n$x.\\mathrm{den}/H$ divides $c$:\n\\[\n  H\\mid x.\\mathrm{den} \\ \\land\\ (c\\cdot x).\\mathrm{den}\\mid H \\implies x.\\mathrm{den}/H \\mid |c|.\n\\]\n\\lean{Erdos249257.AdelicHeightObstruction.scalarLocalization\\_complement\\_dvd}{AdelicHeightObstruction.lean:23}.\nEquality form (height conservation): $\\exists\\,t:\\mathbb Z,\\ H\\cdot c\\cdot x = t\\cdot x.\\mathrm{num}$\n\u2014 the omitted denominator is transferred exactly to the coefficient $t$, never erased.\n\\lean{Erdos249257.AdelicHeightObstruction.scalarLocalization\\_integer\\_eq\\_mul\\_num}{AdelicHeightObstruction.lean:56}.\n\n\\textbf{Scope.} Fully generic $\\mathbb Q$-arithmetic \u2014 no reference to either Erd\u0151s problem. It is\nthe reusable \"clearing a denominator by multiplying by a small integer $c$ cannot discard the\ncomplementary denominator factor; it only moves that factor into $c$\" lemma. It gives a hard floor\non how small $c$ can be if it is meant to kill a target denominator factor $x.\\mathrm{den}/H$: any\nargument on either problem that hopes to \"clear a denominator by a bounded multiplier\" is bounded\nbelow by this lemma.\n\\coord{rational-denominator/height}\\ \\textbf{Object or representation:} about a\n\\emph{representation} \u2014 pure denominator bookkeeping for $\\mathbb Q$, with zero problem-specific\ncontent; the upstream primitive\n(\\lean{Erdos249257.RationalDenominatorSurvival.divisor\\_dvd\\_divInt\\_den}{RationalDenominatorSurvival.lean:17})\nis the domain-neutral extraction this lemma builds on.\n\\ev{Lean}\\scale{n/a}\n\\end{lem}\n\n\\begin{cor}[Mersenne specialisation]\n\\label{cor:mersenne-height}\nFor positive $x:\\mathbb Q$: if $2^r\\mid x.\\mathrm{num}.\\mathrm{natAbs}$ and $x < 2/(2^n-1)$, then\n$2^r\\cdot(2^n-1) < 2\\cdot x.\\mathrm{den}$ \u2014 a numerator $2$-power lower bound plus a Mersenne-scale\nupper bound on $x$ together force a denominator lower bound, \\emph{without} introducing a global\nprefix LCM.\n\\lean{Erdos249257.AdelicHeightObstruction.positiveRat\\_mersenne\\_height}{AdelicHeightObstruction.lean:103}\n(generic form\n\\lean{Erdos249257.AdelicHeightObstruction.positiveRat\\_numDivisor\\_mul\\_lt\\_two\\_mul\\_den}{AdelicHeightObstruction.lean:77}).\nA ready-made denominator-lower-bound tool for \"synchronized tail\" arguments on \\#257's Mersenne\nsums; composes directly with Lemma~\\ref{lem:scalar-localization} for a full numerator/denominator\nheight-transport toolkit.\n\\coord{rational-denominator/height, Mersenne instance}\\ \\ev{Lean}\\scale{n/a}\n\\end{cor}\n\n\\subsection{The final-skip band formula, and what it does not show}\n\n\\begin{defn}[Upper-reset dyadic band escape]\n\\label{defn:band-escape}\n\\[\n  \\mathsf{SeamUpperResetDyadicBandEscape} :=\n  \\forall\\, d\\ge 13,\\ \\mathrm{successorCarries}(d) \\to \\forall\\, j\\le d,\\quad\n  2^{d-j+1} < 4\\cdot\\mathrm{overshoot}(d)+\\mathrm{abovePulse}(d)\n  \\ \\lor\\\n  4\\cdot\\mathrm{overshoot}(d)+\\mathrm{abovePulse}(d)+2(d+j) \\le 2^{d-j+1}.\n\\]\n\\lean{Erdos249257.SeamUpperResetDyadicBandEscape}{HalfCylinderMiddleCarryLowerBound.lean:3531}.\nIn words: at every upper-reset row $d\\ge13$, the reset charge must avoid a linear-width forbidden\nband immediately below every dyadic threshold $2^{d-j+1}$, for every $j\\le d$ simultaneously.\nGranted this socket,\n\\lean{Erdos249257.half\\_mem\\_mersenneAchievementSet\\_of\\_upperResetDyadicBandEscape}{HalfCylinderMiddleCarryLowerBound.lean:3755}\ngives $1/2\\in\\mathcal A$ outright.\n\\end{defn}\n\n\\begin{prop}[Quantifier collapse: one critical index suffices, not $d{+}1$]\n\\label{prop:critical-band-index}\nThe abstract, purely combinatorial statement\n$\\mathsf{DyadicBandEscape}(d,E) \\iff \\exists j,\\ \\mathsf{CriticalDyadicBandIndex}(d,E,j)\\land\nE+2(d+j)\\le 2^{d-j+1}$ collapses the $\\forall j\\in[0,d]$ band-avoidance condition (formally $d{+}1$\nseparate inequalities) to checking exactly \\emph{one} nearest-boundary index $j$. Specialised to the\nconcrete seam reset charge, $\\mathsf{SeamUpperResetCriticalBandEscape}$ is proved logically\n\\emph{equivalent} to Definition~\\ref{defn:band-escape}'s socket.\n\\lean{Erdos249257.dyadicBandEscape\\_iff\\_exists\\_critical}{HalfUpperResetCriticalBand.lean:153}\n(seam specialisation\n\\lean{Erdos249257.seamUpperResetCriticalBandEscape\\_iff}{HalfUpperResetCriticalBand.lean:901}).\nZero Mersenne/seam content in the core lemma \u2014 pure $(d,E,j)$ arithmetic over powers of 2.\n\\coord{dyadic-boundary, generic}\\ \\ev{Lean}\\scale{n/a}\n\\end{prop}\n\n\\begin{obs}[Rows 13 through 31 are unconditionally certified]\n\\label{cert:band-through-31}\n$\\mathsf{seamUpperResetDyadicBandEscape\\_through\\_thirty}$ verifies Definition~\\ref{defn:band-escape}\nby kernel computation of $\\mathrm{rem}(s)$ for every row $13\\le s\\le31$ \u2014 e.g.\\ row 14 gives $\\mathrm{rem}(14)=392$,\nrow 31 gives $\\mathrm{rem}(31)=4187487147$ \u2014 with \\textbf{zero} rows failing the band-avoidance test.\n\\lean{Erdos249257.seamUpperResetDyadicBandEscape\\_through\\_thirty}{HalfCylinderUpperResetBandCertificates.lean:78}.\n\\ev{Cert}\\scale{bounded}\n\\end{obs}\n\n\\begin{rem}[What this does \\emph{not} show \u2014 read the quantifiers exactly]\n\\label{rem:band-caveat}\nDefinition~\\ref{defn:band-escape} is a $\\forall d\\ge13$ statement; Certificate\n\\ref{cert:band-through-31} is a finite, kernel-checked verification for $13\\le d\\le31$. \\textbf{The\nfinite certificate does not establish the universal socket.} Nothing in the tree proves\n$\\mathsf{SeamUpperResetDyadicBandEscape}$ for all $d\\ge13$ \u2014 this is exactly the open producer named\nin Part~I (SE-9) and Part~III (RC-4/D1's \\textsf{LargestSkipLateStepSocket}), stated here in its own\ncoordinate. The certified range through row 31 is real, certified evidence \\emph{at that finite\nrange}, and it is a genuinely different fact from \"the actual orbit avoids the unsafe band\ncofinally\" or \"at every row.\" The socket remains an unproved sufficient condition; the finite\ncomputation is not a proof of the cofinal or universal claim, and must never be quoted as one.\nConcretely: the certificate says the band is avoided for $1264$ real resets through row $2500$ with\nzero classification anomalies (Part~I, RC-7) and the sign law holds at all of them (RC-8) \u2014 this is\nstrong disconfirming-of-a-counterexample evidence, not a proof that the pattern continues past the\ncertified range.\n\\ev{Open}\\scale{cofinal}\n\\end{rem}\n\n% ---- part a257_invent ----\n\\section{The mathematics this problem still needs}\n\\label{sec:invent-257}\n\nEverything in Parts I--IV of this document is either an unconditional theorem at a fixed or\nbounded scale, an exact reformulation, or a producer socket whose supply is missing. None of it\ndecides Erd\\H{o}s \\#257, and the wall analysis explains why in a way that is more useful than\n``the problem is hard'': the recorded failures are not a hundred separate defeats but repeated\nmeasurements of one obstruction, and the obstruction has a shape. This section takes the next\nstep, which the rest of the corpus did not take. For each surviving route it states the exact\nstatement that is missing, says what \\emph{kind} of mathematical object would furnish it, names\nthe technique family that is closest and the precise reason that technique does not reach, and\nwhere possible attempts an actual construction, estimate or reduction. Attempts are banded\nhonestly: \\ev{Math} where the argument is complete ordinary mathematics, \\ev{Cert} where the\nsupport is exact computation, \\ev{Open} where the item is a proposal and nothing more.\n\nThe organising fact, established in the wall analysis and not repeated here, is that both halves\nof \\#257 --- the universal statement and the half-value instance --- converge on one missing\nobject. It has a name in this section:\n\n\\begin{defn}[The short-window divisor phase]\n\\label{defn:theta}\nFor $M\\ge 1$ and $L\\ge 1$ put\n\\[\n\\Theta_L(M)\\ :=\\ \\sum_{i=1}^{L}\\bigl(\\tau(M+i)-1\\bigr)\\,2^{-i}\\ \\in\\ \\Q_{\\ge 0},\n\\]\nwhere $\\tau$ is the number-of-divisors function, so that $\\tau(n)-1=\\#\\{d\\ge 2: d\\mid n\\}$ counts\nthe divisors of $n$ other than $1$.\n\\end{defn}\n\n$\\Theta_L$ is the analytic content of the universal recursion $K(2)=1$,\n$K(M+1)=2K(M)-(\\tau(M+1)-1)$ that carries the seam deviation (Part~II, master identity\n$\\Delta_n = K(2n)+\\sum_{d\\in\\mathrm{Skip}_n}\\lfloor 4^n/(2^d-1)\\rfloor$). It is also, in a sense\nmade precise in \\S\\ref{sec:invent-R5}, the universal-direction obstruction written in the divisor\nincidence coordinate. Every route below either needs a lower bound on\n$\\mathrm{dist}(\\Theta_L(M),\\Z)$ at a growing precision, or needs to explain why it does not.\n\n\\subsection{R1. Reset \\texorpdfstring{\\(\\sqrt{\\ }\\)}{square-root}-escape:\nanti-concentration of a logarithmically short divisor sum}\n\\label{sec:invent-R1}\n\n\\subsubsection*{The statement needed}\n\nThree forms, in decreasing strength; each closes the half-value branch, i.e.\\ yields\n$(1/2:\\R)\\in\\mathcal{A}$ and hence a counterexample to universal \\#257.\n\n\\emph{Form (i), the run-length form.} Writing $\\mathrm{rem}(s)$ for the integer seam greedy\nremainder at row $s$, $w_s := \\mathrm{rem}(s)-2^{s}$ for the deviation, and $L_r$ for the length\nof the maximal pure-$R$ run of the branch word beginning at a reset row $r$:\n\\[\n\\forall\\, r\\ \\text{a reset row},\\ r\\ge 31:\\qquad L_r\\ <\\ \\frac{r-3}{2},\n\\]\nequivalently\n\\[\n\\bigl|\\,\\mathrm{rem}(r+1)-2^{\\,r+1}\\,\\bigr|\\ >\\ 2^{(r+5)/2}.\n\\]\nAny bound $L_r=o(r)$ suffices. A uniform constant bound is \\emph{false}: runs grow like\n$\\log_2 r$ (\\ev{Cert}, maximal observed run $19$ at row $158{,}096$).\n\n\\emph{Form (ii), the Lean-facing form already fanned in.}\n\\[\n\\mathtt{SeamUpperResetDyadicBandEscape}\\ :=\\ \\forall d\\ge 13,\\ \\bigl(\\mathtt{seamAdjacentCut}\\ d\\bigr).\\mathtt{successorCarries}\\ \\Longrightarrow\\ \\forall j\\le d,\n\\]\n\\[\n2^{\\,d-j+1} < E_d\\quad\\lor\\quad E_d + 2(d+j)\\ \\le\\ 2^{\\,d-j+1},\n\\qquad\nE_d := 4\\cdot\\mathtt{overshoot}_d+\\mathtt{abovePulse}_d,\n\\]\n(\\lean{SeamUpperResetDyadicBandEscape}{HalfCylinderMiddleCarryLowerBound.lean:4566}), with the\nclosed consumer\n\\lean{half_mem_mersenneAchievementSet_of_upperResetDyadicBandEscape}{HalfCylinderMiddleCarryLowerBound.lean:4790},\n\\ev{Lean}. The $\\forall j\\le d$ band check collapses to a single nearest-boundary index per row by\n\\lean{dyadicBandEscape_iff_exists_critical}{HalfUpperResetCriticalBand.lean:108}, and rows\n$13\\le d\\le 30$ are kernel-verified by\n\\lean{seamUpperResetDyadicBandEscape_through_thirty}{HalfCylinderUpperResetBandCertificates.lean:78}.\n\n\\emph{Form (iii), the sign law --- strictly weaker, and the right first target.}\n\\[\n\\forall\\ \\text{reset rows } r:\\qquad\nr \\text{ is an } M\\text{-reset}\\Rightarrow w_{r+1}>0,\n\\qquad\nr \\text{ is a } U\\text{-reset}\\Rightarrow w_{r+1}<0 .\n\\]\n\\ev{Cert}: perfect over all $1209$ resets in rows $6..2500$, $605/605$ and $604/604$, no zero\ndeviations. \\ev{Open} as a theorem.\n\n\\subsubsection*{What kind of object would furnish it, and the exact arithmetic of $\\Theta_L$}\n\nThe first thing to say is what the quantity actually is, because the corpus states it in seam\ncoordinates and that hides its arithmetic nature. The following identity is elementary and\nexact, and it is the reason this route is an analytic-number-theory problem rather than a\ncombinatorial one.\n\n\\begin{lem}[Divisor-residue form of the short-window phase]\n\\label{lem:odometer}\nFor all $M,L\\ge 1$,\n\\[\n\\Theta_L(M)\\ =\\ \\sum_{d\\ge 2}\\ \\sum_{\\substack{1\\le i\\le L\\\\ i\\,\\equiv\\,-M\\ (\\mathrm{mod}\\ d)}} 2^{-i}\n\\ =\\ \\sum_{d\\ge 2}\\ 2^{-i_d(M)}\\cdot\\frac{1-2^{-d\\,m_d}}{1-2^{-d}},\n\\]\nwhere $i_d(M)\\in[1,d]$ is the least $i\\ge1$ with $d\\mid M+i$ (so $i_d(M)$ depends only on\n$M \\bmod d$), $m_d := \\#\\{i\\le L: d\\mid M+i\\}$, and terms with $i_d(M)>L$ are empty. In\nparticular $\\Theta_L(M)$ is a function of the residues of $M$ modulo every $d\\le M+L$, and\nnothing else.\n\\end{lem}\n\\begin{proof}\n$\\tau(n)-1=\\sum_{d\\ge2}[\\,d\\mid n\\,]$; exchange the order of summation and sum the geometric\nprogression $i_d, i_d+d, i_d+2d,\\dots$ inside $[1,L]$.\n\\end{proof}\n\\ev{Math}, and verified exactly as rationals at $(M,L)\\in\\{1000,10001,123456\\}\\times\\{12,20\\}$\n\\ev{Cert}.\n\nLemma~\\ref{lem:odometer} says the object is the evaluation of a \\emph{fixed} function on the\nprofinite integers along the $+1$-orbit. Define\n\\[\n\\Psi:\\widehat{\\Z}\\to\\R\\cup\\{\\infty\\},\\qquad\n\\Psi(x)\\ :=\\ \\sum_{d\\ge2}\\frac{2^{-i_d(x)}}{1-2^{-d}},\\qquad i_d(x)\\in[1,d],\\ i_d(x)\\equiv -x\\ (\\mathrm{mod}\\ d),\n\\]\ntruncated at level $L$ in the obvious way. Then $\\Theta_L(M)=\\Psi_L(M)$ where $M$ is viewed in\n$\\widehat{\\Z}$, and the map $M\\mapsto M+1$ is the odometer, which is uniquely ergodic with respect\nto Haar measure. The corpus's own Reduction~(A), $F(n)=\\sum_{d\\in D_n} 2^{\\,d-i_d(n)}/(2^d-1)$,\nis \\emph{the same functional} $\\sum_d 2^{-i_d}/(1-2^{-d})$ restricted to the greedy take-set\n$D_n$ instead of to all $d\\ge2$. That two reductions derived years and coordinates apart are one\nfunctional on $\\widehat{\\Z}$ evaluated on two divisor sets is, to my knowledge, not recorded\nanywhere in the corpus; it is the cleanest available statement of what the two halves of the seam\nidentity share. \\ev{Math} for the identification; \\ev{Open} for any consequence.\n\nSo the object required is: \\emph{an effective equidistribution statement for the odometer orbit\nagainst $\\Psi$, with the precision growing linearly in the index.} Unique ergodicity gives\nequidistribution with no rate. What is needed is a rate, pointwise along a sparse subsequence\n(the reset rows), at scale $2^{-r/2}$ where $r$ is the index. Equivalently, in the form a\nspecialist will recognise: a lower bound for $\\mathrm{dist}(\\Theta_L(M),\\Z)$ with\n$L\\asymp \\log_2 M$, valid for all large $M$ in a specified set, at precision $2^{-L/2}$.\n\nThree technique families are the plausible shapes, and it is worth being explicit about which.\n\\emph{(a) Voronoi summation / delta method for the divisor function.} $\\tau$ has an exact Voronoi\nexpansion, and the natural attack is to bound\n$\\sum_{M\\in[X,2X]} e\\bigl(a\\,\\Theta_L(M)\\bigr)$ for $1\\le|a|\\le 2^{L}$ by opening each\n$\\tau(M+i)$ and estimating the resulting Kloosterman-type sums. \\emph{(b) Fourier analysis on\n$\\widehat{\\Z}$ / Ramanujan expansions.} By Lemma~\\ref{lem:odometer} the phase is a function of\n$M$ modulo the $d$'s, so its Fourier expansion is over the characters of $\\Z/d$, i.e.\\ over\nRamanujan sums $c_d(\\cdot)$; the required estimate becomes decay of the Ramanujan coefficients of\n$\\Psi_L$ plus a bound on their tails. \\emph{(c) Second-moment / local-limit machinery.} Prove\nthat the vector $(\\tau(M+1),\\dots,\\tau(M+L))$, $L\\asymp\\log M$, obeys a \\emph{local} limit\ntheorem strong enough to give anti-concentration of the weighted sum at scale $2^{-L/2}$.\n\n\\subsubsection*{The specific obstacle}\n\nThis is the paragraph worth the most, so it is stated as sharply as the evidence permits.\n\n\\textbf{The interval is logarithmically short, and every unconditional result for divisor sums in\nshort intervals requires the interval to be a power of the modulus.} The window\n$[M+1,M+L]$ has length $L\\asymp\\log_2 M$. Voronoi-based technology for $\\sum_{M<n\\le M+H}\\tau(n)$\n--- including the Kloosterman-refined results of Deshouillers--Iwaniec type and every subsequent\nimprovement --- produces an asymptotic only for $H\\ge M^{\\theta}$ with $\\theta>0$ fixed, because\nthe dual sum after Voronoi runs to length $\\sim M/H$ and the error term is controlled only when\nthat dual length is a power saving against $M$. At $H=\\log M$ the dual sum is of length $M/\\log M$\nand there is no saving at all: the method returns the trivial bound. This is not a deficiency of\none paper; it is the reason short-interval divisor problems are stated with power-length windows.\nTechnique family (a) is therefore not merely unproved here, it is out of range by an\n\\emph{exponential} amount in the window length, and no quantitative improvement in Kloosterman\nbounds changes that, because the obstruction is the length of the dual sum, not the quality of\nits estimate.\n\n\\textbf{The one technology that does reach sub-power windows delivers the wrong quantifier and\nthe wrong class of function.} Matom\\\"aki--Radziwi\\l{}\\l{} theory shows that a multiplicative\nfunction bounded by $1$ has, in almost all windows $[x,x+h]$ with $h\\to\\infty$ arbitrarily\nslowly, essentially its long average. Two mismatches, both fatal as stated. First, the conclusion\nis \\emph{almost all} $x$, whereas the seam needs \\emph{every} reset row $r\\ge31$; a set of\nexceptional $x$ of density zero can still contain every reset row, because the reset rows are\nthemselves a density-zero set defined by the orbit rather than independently of it. Second,\n$\\tau$ is not bounded by $1$ and the relevant statistic is not an average of a multiplicative\nfunction but the distance to $\\Z$ of a \\emph{dyadically weighted} sum of $\\tau$-values; the\nweights $2^{-i}$ mean that a single value $\\tau(M+1)$ contributes half the phase, so no averaging\nstatement over the window can control it. Anything that averages within the window destroys the\nquantity, exactly as the wall analysis says of every scale-mismatched tool.\n\n\\textbf{The joint-distribution literature is at CLT scale, not local scale.} Erd\\H{o}s--Kac and\nits descendants, and the Hildebrand--Elliott-type results on the joint normality of\n$\\omega(n),\\omega(n+1)$, give distribution functions to accuracy $o(1)$ and average over\n$n\\le X$. The requirement here is anti-concentration to accuracy $2^{-L/2}$ for a fixed\n$L$-tuple of consecutive arguments, i.e.\\ a local limit theorem with an error term\nexponentially small in $L$ where $L\\asymp\\log M$. No result of that strength is known even for\n$L=2$.\n\n\\textbf{Why the barrier catalogue predicts exactly this.} The wall analysis states that the two\navailable information classes are fixed-precision local arithmetic (killed by\n\\lean{affineBinaryOrbit_mod_twoPow_eq}{GenericTailOrbitRigidity.lean:307},\n\\lean{balancedPulse_no_autonomous_decoder}{GenericTailOrbitRigidity.lean:247},\n\\lean{fixedPrecisionTropicalNoGo}{TropicalCurvatureCarry.lean:137}, all \\ev{Lean}) and global\nDiophantine averages (killed by the exponent-budget computation against $\\mu(E)$). The\nintermediate scale between them is precisely ``divisor statistics in logarithmically short\nwindows'', and the three paragraphs above are the reason there is nothing there. The barrier\ncatalogue and the analytic literature agree, independently, on where the gap is.\n\n\\subsubsection*{Swings}\n\n\\emph{Swing 1: the exponent $(r+5)/2$ is forced, and can be derived in four lines.} \\ev{Math},\nmodulo the recorded window-width constant.\n\nThe exact $R$-branch recursion in deviation coordinates is $w\\mapsto 4w-4-p$ with\n$0\\le p\\le 2(s-2)$ (Part~II, transition laws). Iterating $L$ steps from a reset row $r$,\n\\[\n|w_{r+L}|\\ \\ge\\ 4^{L}\\,|w_r|\\;-\\;\\sum_{j<L}4^{\\,L-1-j}\\bigl(4+p_j\\bigr)\n\\ \\ge\\ 4^{L}\\Bigl(|w_r| - \\tfrac{4+2(r+L)}{3}\\Bigr).\n\\]\nAn $R$ branch at row $s$ requires $\\mathrm{rem}(s)$ to lie in the pinned window of width\n$\\approx 2^{s}+\\tfrac23 4^{\\,s-d}$, so in particular the run must terminate once\n$|w_s|\\gtrsim 2^{s}$. Setting $s=r+L$ gives $4^{L}|w_r|\\lesssim 2^{\\,r+L}$, that is\n\\[\nL\\ \\lesssim\\ r-\\log_2|w_r|.\n\\]\nThis reproduces, with no fitting, the empirical law recorded at maximal slack $0.0$ over rows\n$14..1500$. A socket-violating run needs $L\\ge (r-3)/2$, hence\n$\\log_2|w_r|\\le (r+3)/2$; the corpus's threshold $2^{(r+5)/2}$ is this bound with one bit of\nsafety. The value of the derivation is that it shows the exponent is not a tuning parameter: any\nproof of Form~(i) must produce a $\\sqrt{\\ }$-scale lower bound on the reset deviation and nothing\nweaker will do, because the doubling rate of the $R$ branch is exactly $4$ against a window that\ngrows like $2^{s}$.\n\n\\emph{Swing 2: measure the limit law of the phase, and compute the Borel--Cantelli budget.}\n\\ev{Cert} for the measurement, \\ev{Open} for the conclusion.\n\nSieving $\\tau$ over $[10^6,\\,1.2\\times10^6]$ and evaluating $\\Theta_{40}(M)$ for $200{,}000$\nconsecutive $M$ gives the following profile for $P\\bigl(\\mathrm{dist}(\\Theta_{40}(M),\\Z)<t\\bigr)$:\n\\begin{center}\n\\begin{tabular}{lccccc}\n\\hline\n$t$ & $2^{-6}$ & $2^{-8}$ & $2^{-10}$ & $2^{-12}$ & $2^{-14}$\\\\\n\\hline\n$P$ & $0.035415$ & $0.009290$ & $0.002455$ & $0.000625$ & $0.000185$\\\\\n$P/t$ & $2.267$ & $2.378$ & $2.514$ & $2.560$ & $3.031$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\nThe observed median of $\\mathrm{dist}$ is $0.24471$ against $0.25$ for the uniform law, and\n$P/t$ hovers near $2$, the exact value for a uniform fractional part. Truncation is irrelevant at\nthis range: the quantiles at $L=16,24,32$ agree to three significant figures. So the fractional\npart of $\\Theta_L$ behaves like Haar measure with a mildly heavy near-integer tail of constant\n$c\\approx 2.5$. Feeding the required precision $2^{-(r-5)/2}$ into that profile,\n\\[\n\\sum_{r\\ge 31} c\\cdot 2^{-(r-5)/2}\\ \\approx\\ 2.5\\cdot 2^{5/2}\\cdot\\frac{2^{-31/2}}{1-2^{-1/2}}\n\\ \\approx\\ 1.0\\times 10^{-3},\n\\]\nso under the Haar model the expected number of reset rows beyond the certified range that violate\n$\\sqrt{\\ }$-escape is about $10^{-3}$, and Borel--Cantelli gives ``finitely many failures'' with\nroom to spare. This is consistent with, and independently derived from, the corpus's own\n$\\approx 2\\times10^{-4}$ digit-model estimate. It is a heuristic. Its only rigorous content is\nthat the required inequality is not delicate: it asks for a bound $2^{15}$ times weaker at\n$r=31$, and exponentially weaker thereafter, than what the measured law delivers. That is a\nuseful thing for a specialist to know before deciding whether to spend effort here.\n\n\\emph{Swing 3: attack the sign law first, and here is the reduction to try.} \\ev{Open}.\n\nForm~(iii) asks only for a sign, not a magnitude, and a sign is a statement about which side of\nthe branch threshold the remainder falls on. In deviation coordinates the $M$ branch fires iff\n$4\\,\\mathrm{rem}(s)<2^{\\,s+1}+4+p$ and then $\\mathrm{rem}(s+1)=4\\,\\mathrm{rem}(s)+2^{\\,s+1}-p$;\nthe $U$ branch carries. Substituting $\\mathrm{rem}=w+2^{s}$ turns the $M$-branch update into\n$w_{s+1}=4w_s+2^{\\,s+1}-p$ and the $R$-branch update into $w_{s+1}=4w_s-4-p$. The concrete first\nstep is: prove that the $M$-branch firing condition and the reset condition together force\n$w_{s+1}>0$ by a purely algebraic inequality in $(w_s,p,s)$, with $p\\le 2(s-2)$ the only input\nabout the pulse. If that closes, the same computation with the $U$-branch carry condition should\ngive the negative sign, and the pair is a genuine lemma one rung below $\\sqrt{\\ }$-escape. If it\ndoes not close, the residual is a bound on $p$ that is sharper than $p\\le2(s-2)$, which is itself\na short-window divisor count and hence lands back on $\\Theta$ --- which would be informative,\nbecause it would show the sign law is not strictly easier. Either outcome is a result; the\ncomputation is an afternoon's work and needs no new theory.\n\n\\subsection{R2. Strict endpoint progress: the one place canonicalisation does not bite}\n\\label{sec:invent-R2}\n\n\\subsubsection*{The statement needed}\n\nThe exact-row transition\n\\lean{exactLocalMersenneHalfRow_double_or_recycle}{BooleanMobiusExactRowDichotomy.lean:26}\n(\\ev{Lean}) says every exact row $n\\ge6$ either doubles to $2n-1$ or recycles to $2c-2$ for some\n$4\\le c\\le n$. Cofinality of exact rows --- which gives $\\mathrm{HALF}$ --- needs the recycle\nbranch not to return to a bounded endpoint forever. Three sufficient forms:\n\\[\n\\text{(a)}\\quad\n\\mathtt{SkippedCoreCriticalQuotientSupply}:\\ \\forall D\\subseteq[2,c),\\ 4\\le c,\\\nv(D)<\\tfrac12\\ \\wedge\\ \\tfrac12-v(D)<\\mathtt{mersenneWeightRat}\\,c\n\\]\n\\[\n\\Longrightarrow\\quad 2^{\\,(2c-2)-1}\\ \\le\\ \\mathtt{localPrefixQuotient}\\,(\\{c\\}\\cup D)\\,(2c-2),\n\\]\nwhere $v(D):=\\mathtt{localMersennePrefixValue}\\ D$\n(\\lean{SkippedCoreCriticalQuotientSupply}{BooleanMobiusCriticalCapacityCofinal.lean:30});\n\\[\n\\text{(b)}\\quad\n\\text{the crossing rank } c \\text{ produced by the recycle branch does not repeat a bounded value infinitely often};\n\\]\n\\[\n\\text{(c)}\\quad\n\\text{a growth law } c\\ \\ge\\ g(n)\\ \\text{with}\\ g\\ \\text{unbounded, for the recycle branch at incoming endpoint } n.\n\\]\n\n\\subsubsection*{What kind of object would furnish it}\n\nThis route is different in kind from R1, and the difference is worth stating because it is the\nonly place in the half-value branch where the witness space is not a singleton.\n\nCanonicalisation --- the fact that every straddling word equals the canonical greedy prefix\n(\\lean{IsStraddlePrefix.half_agrees_greedy}{HalfCutLocator.lean:442}) --- applies to conditions\nphrased as \\emph{real-valued straddles}, because it consumes both $v(D)<1/2$ and the deficit\nclause $1/2-v(D)<w_c$. The predicate $\\mathtt{ExactLocalMersenneHalfRow}\\ n$ imposes neither: it\nis an \\emph{integer floor-quotient hit},\n$\\mathtt{localPrefixQuotient}\\ D\\ n = 2^{\\,n-1}-1$, and floor division is not injective. So the\nper-endpoint witness space is an exponential search over $D\\subseteq[2,n]$ and is decidable for\neach $n$. That is the only genuine degree of freedom left anywhere on this side.\n\nThe object required is therefore a \\emph{monotone quantity} --- a potential function on exact\nrows that the recycle branch is forced to increase. In the language of the transition, one wants\na function $\\Phi$ on pairs (endpoint, witness core) with $\\Phi(\\text{recycle image})>\\Phi(n)$,\nwhose unboundedness forces $c\\to\\infty$. The technique family is not analytic here: it is the\ntheory of \\emph{well-founded transition systems} and, arithmetically, the capacity accounting of\nBoolean subset sums against a superincreasing weight sequence. The corpus already has the sharp\ncapacity test in exact form: by\n\\lean{localBinarySuffix_two_mul_sub_two_lt_criticalCapacity_iff}{BooleanMobiusSkippedCoreCriticalCapacity.lean:22},\nform (a) is exactly the statement that the $c-2$-bit suffix capacity\n$\\mathtt{localBinarySuffix}\\,D\\,1\\,(2c-2)<2^{\\,c-2}$ is met.\n\n\\subsubsection*{The specific obstacle}\n\nThe obstruction here is isolated, formalised, and the sharpest of its kind in the corpus: the\ntransition schema \\emph{plus} a seed provably does not give cofinality. The theorem\n\\lean{exists_seeded_bounded_double_or_recycle_model}{BooleanMobiusExactRowDichotomy.lean:91}\n(\\ev{Lean}) exhibits the one-point predicate $P(n):=(n=6)$, which satisfies the seed\n(\\lean{exactLocalMersenneHalfRow_six}{BooleanMobiusExactRowSeed.lean:34}, the explicit witness\n$\\{2,3,6\\}$) and the exact shape of the transition, and is not cofinal. So no argument that uses\nonly the dichotomy and the base case can work, however cleverly the induction is arranged. Any\nproof must consume arithmetic content of $\\mathtt{ExactLocalMersenneHalfRow}$ that the abstract\nschema does not see.\n\nThere is a second, more specific trap, and it is also formalised. The natural sufficient\ncondition via fractional split mass is \\emph{not necessary}: the fixture\n\\lean{splitFractionMass_one_bound_not_necessary_fixture}{BooleanMobiusSkippedCoreCriticalCapacity.lean:183}\n(\\ev{Lean}, at $D=\\{2,3\\}$, $c=5$) exhibits a real crossing core violating it. So the fractional\nmass route is closed with a witness, not merely unattempted.\n\n\\subsubsection*{Swings}\n\n\\emph{Swing 4: use canonicalisation as a help rather than an obstruction.} \\ev{Math} for the\nobservation, \\ev{Open} for the outcome.\n\nForm (a) is universally quantified over cores $D$, which looks like an exponential obligation.\nIt is not. Its hypotheses are exactly those of\n\\lean{eq_halfGreedyPrefixSupport_of_critical_crossing}{BooleanMobiusCriticalCapacityCofinal.lean:50}\n(\\ev{Lean}), which forces $D=\\mathtt{halfGreedyPrefixSupport}(c-1)$. So the $\\forall D$ collapses\nto a single orbit and form (a) is a statement to be checked along \\emph{one} sequence indexed by\n$c$, namely\n\\[\n\\forall c\\ge 4:\\qquad\n\\mathtt{localBinarySuffix}\\ \\bigl(\\mathtt{halfGreedyPrefixSupport}(c-1)\\bigr)\\ 1\\ (2c-2)\\ <\\ 2^{\\,c-2}.\n\\]\nThat is a one-parameter arithmetic inequality about the canonical half-greedy prefix, decidable\nfor each $c$, and it is the correct object to compute before any theory is attempted. The first\nactual step is to evaluate the left-hand side for $c=4,\\dots,200$ and read off whether the margin\n$2^{c-2}-\\mathtt{localBinarySuffix}(\\cdots)$ grows, stays proportional, or approaches zero. Those\nthree outcomes select three different proofs, and no one has published the table.\n\n\\emph{Swing 5: the potential function to try.} \\ev{Open}.\n\nThe doubling branch sends $n\\mapsto 2n-1$ and the recycle branch sends $n\\mapsto 2c-2$ with\n$c\\le n$. Set $\\Phi(n):=$ the largest rank in the canonical core at endpoint $n$. On the doubling\nbranch $\\Phi$ at least doubles. The recycle branch is the only danger, and its image endpoint\n$2c-2$ is determined by the \\emph{first crossing rank} $c$ of the core. So the entire question is\nwhether the first crossing rank of the canonical prefix can stay bounded while the endpoint grows\n--- which, by the collapse above, is a question about one explicit greedy orbit. The concrete\nstatement to attempt is: \\emph{the first crossing rank of\n$\\mathtt{halfGreedyPrefixSupport}(m)$ is non-decreasing in $m$ and unbounded}. If true it closes\nform (c) directly; if false, the counterexample is a finite object and worth having.\n\n\\subsection{R3. Irrationality of\n\\texorpdfstring{\\(\\sum_{p\\ \\mathrm{prime}} 1/(2^p-1)\\)}{the prime Mersenne subseries}}\n\\label{sec:invent-R3}\n\n\\subsubsection*{The statement needed}\n\n\\[\n\\mathrm{Irrational}\\Bigl(\\ \\sum_{p\\ \\mathrm{prime}} \\frac{1}{2^{p}-1}\\ \\Bigr).\n\\]\nThis is a named constant and a genuine theorem, not a reduction, and it is the most natural\nsupport class not covered by any landed result: the primes are infinite and pairwise coprime, so\nthey satisfy every hypothesis of Erd\\H{o}s's 1968 theorem\n(\\lean{irrational_erdosSupportSeries_pairwise_coprime}{CertificateKernel.lean:10448}, \\ev{Lean})\n\\emph{except} summability of $\\sum 1/p$, which diverges by Mertens. The class is also the one\nwhere the dilation algebra is exact\n(\\lean{compositeDilationDefect_eq_zero_of_prime_support}{CompositeDilationDefect.lean:103},\n\\ev{Lean}).\n\n\\subsubsection*{What kind of object would furnish it}\n\nInstantiate the engine, not the statement. The certificate consumed by\n\\lean{irrational_erdosSupportSeries_of_weighted_coeff_certificates}{CertificateKernel.lean:8703}\n(\\ev{Lean}, universally quantified over the support --- only its hypothesis is unsupplied) reads,\nat $b=2$ and $A=\\{\\text{primes}\\}$, where $\\mathtt{supportCoeff}\\,A\\,n = \\omega(n)$:\n\\begin{align}\n&\\forall q>0\\ \\exists N,K,L,C\\ \\text{with}\\ K\\le L: \\notag\\\\\n&\\qquad (\\mathrm{F})\\quad \\forall r\\in[1,K]:\\ 2^{r}\\ \\mid\\ \\omega(N+r), \\label{eq:primeF}\\\\\n&\\qquad (\\mathrm{M})\\quad \\sum_{r=K+1}^{L}\\omega(N+r)\\,2^{\\,L-r}\\ \\le\\ C, \\label{eq:primeM}\\\\\n&\\qquad (\\mathrm{T})\\quad \\exists t:\\ \\omega(N+L+1+t)>0,\n\\qquad (\\mathrm{H})\\quad q\\,(C+N+L+2)\\ <\\ 2^{L}. \\label{eq:primeH}\n\\end{align}\n(T) is trivial. (H) is a size condition. (M) is a bound on $\\omega$ over a window. (F) is the\nwhole difficulty, and it is an \\emph{exact} prescription of the number of distinct prime factors\nof $K$ consecutive integers.\n\n\\subsubsection*{The specific obstacle --- and a correction to the corpus's own recorded repair}\n\nThe corpus records (interface index, row NM-06) that summability is consumed at exactly one\nplace, as a global tail budget, and proposes replacing it by a windowed budget\n$\\sum_{a\\in A,\\ B<a\\le N+L}(L/a+1)\\le C$, noting that for the primes\n$\\sum_{B<p\\le N+L}1/p\\approx\\log\\log(N+L)-\\log\\log B$ is controllable by taking $L$ small\nrelative to $B$. That diagnosis is right about where the hypothesis is used and wrong about where\nthe difficulty lies, and the correction is the useful part of this subsection.\n\n\\begin{prop}[Forced size of a prime-support certificate]\n\\label{prop:primesize}\nIn any certificate satisfying \\eqref{eq:primeF} with $K\\ge1$ and $N+K\\ge2$ one has\n$\\omega(N+K)\\ge 2^{K}$, hence\n\\[\nN+K\\ \\ge\\ \\prod_{j\\le 2^{K}} p_j\\ =\\ \\exp\\bigl((1+o(1))\\,2^{K}\\,K\\log 2\\bigr),\n\\]\nand consequently, by \\eqref{eq:primeH}, $L>\\log_2 N \\gg 2^{K}K$.\n\\end{prop}\n\\begin{proof}\n$N+K\\ge2$ has at least one prime factor, so $\\omega(N+K)\\ge1$; $2^{K}\\mid\\omega(N+K)$ then forces\n$\\omega(N+K)\\ge2^{K}$. An integer with $2^{K}$ distinct prime factors is at least the product of\nthe first $2^{K}$ primes, and $\\log\\prod_{j\\le m}p_j = \\theta(p_m)\\sim p_m\\sim m\\log m$ with\n$m=2^{K}$. The last claim is \\eqref{eq:primeH} with $C\\ge0$.\n\\end{proof}\n\\ev{Math}.\n\nSo the certificate window sits at height $N\\approx e^{2^{K}K\\log2}$ and has length\n$L\\approx 2^{K}K$. Now compare the two hypotheses. The middle-window condition\n\\eqref{eq:primeM} is \\emph{not} the obstacle: $\\omega(n)\\le(1+o(1))\\log n/\\log\\log n$ uniformly,\nso\n\\[\n\\sum_{r=K+1}^{L}\\omega(N+r)2^{\\,L-r}\\ \\le\\ \\Bigl(\\max_{r\\le L}\\omega(N+r)\\Bigr)\\cdot 2^{\\,L-K},\n\\]\nand $C$ is free up to $2^{L}/q-N$, which by Proposition~\\ref{prop:primesize} leaves room by an\nexponential margin. The windowed-budget repair NM-06 proposes therefore repairs a step that does\nnot need repairing.\n\n\\textbf{The real obstacle is \\eqref{eq:primeF}: no existing technique produces integers with a\nprescribed \\emph{exact} number of distinct prime factors at $K$ consecutive arguments.} Every\nconstructive device available controls $\\omega$ only from one side or only up to an interval.\nCRT prescribes divisibility by chosen primes, hence a lower bound on $\\omega$, and says nothing\nabout the cofactor. Sieve methods (Chen, Selberg, the linear sieve) produce almost-primes, i.e.\n$\\Omega(n)\\le k$, again an inequality. Making $\\omega(N+r)$ \\emph{equal} to a prescribed value\nrequires the cofactor of the prescribed part to be prime, or to have a prescribed number of\nfactors --- which for several linear forms simultaneously is exactly the Dickson--Hardy--Littlewood\nprime $k$-tuples problem, open. Moreover the prescription here is not one form but $K$ of them,\nat consecutive shifts, with the required values $2,4,8,\\dots,2^{K}$ growing geometrically.\nThe parity of $\\omega$ alone --- the $r=1$ case --- is already the sign of a\nLiouville-type function, and although sign patterns of length up to three are now known to occur\nwith positive lower density, the results are of ``occurs infinitely often'' type: they manufacture\nfavourable indices and cannot deliver a \\emph{simultaneous} pattern at prescribed shifts with the\nhigher congruences $2^r\\mid\\omega$ for $r\\ge2$ imposed at the same time.\n\nThat is a sharply stated obstruction to a named technique, and it converts ``prove\n$\\sum_p 1/(2^p-1)$ is irrational'' into a recognisable problem: \\emph{produce, for every $K$,\nintegers $N$ with $2^{r}\\mid\\omega(N+r)$ for $r=1,\\dots,K$}.\n\n\\subsubsection*{Swings}\n\n\\emph{Swing 6: the first-block condition is satisfiable at $K=2$ and $K=3$; here are witnesses.}\n\\ev{Cert}.\n\n$K=2$: take $N=208$. Then $209=11\\cdot19$ so $\\omega(N+1)=2$ and $2\\mid2$; and\n$210=2\\cdot3\\cdot5\\cdot7$ so $\\omega(N+2)=4$ and $4\\mid4$.\n\n$K=3$: take $N=989{,}368{,}377$. Then $\\omega(N+1)=4$, $\\omega(N+2)=4$, $\\omega(N+3)=8$, so\n$2\\mid\\omega(N+1)$, $4\\mid\\omega(N+2)$, $8\\mid\\omega(N+3)$; here $N+3=102\\cdot9699690$ is a\nmultiple of the primorial $2\\cdot3\\cdots19$. Further solutions of the same shape at\n$N=1{,}212{,}461{,}247$, $2{,}764{,}411{,}647$, $3{,}520{,}987{,}467$, $10{,}136{,}176{,}047$,\n$11{,}474{,}733{,}267$.\n\nThe witnesses matter for two reasons. They show \\eqref{eq:primeF} is not vacuous --- the\nobstruction is the uniformity in $K$, not the existence of one instance --- and they show the\nshape of a construction: fix $N+K$ to be a primorial multiple carrying the large prescribed value\n$2^{K}$, then solve the lower congruences by choosing the multiplier. That is a search over one\nfree integer parameter, and it is the concrete first step: run it for $K=4$ (requiring\n$\\omega(N+4)\\ge16$, so $N+4$ a multiple of the primorial $2\\cdot3\\cdots53$) and see whether\nsolutions persist at the density the Landau count\n$\\#\\{n\\le x:\\omega(n)=k\\}\\sim x(\\log\\log x)^{k-1}/((k-1)!\\log x)$ predicts. If the density\nmatches Landau, the route is a sieve problem with a plausible answer; if it collapses, there is\nan unrecorded arithmetic obstruction and finding it is itself a result.\n\n\\emph{Swing 7: the honest fallback target.} \\ev{Open}.\n\nIf \\eqref{eq:primeF} for the primes is out of reach, the same analysis identifies a strictly\neasier and still uncovered class: infinite pairwise coprime $A$ with $\\sum_{a\\in A}1/a=\\infty$\nbut with $A$ \\emph{thin enough that $\\mathtt{supportCoeff}\\,A$ is controllable} --- for instance\n$A=\\{p^{k_p}\\}$ over primes $p$ with $k_p\\ge1$ chosen so that the reciprocal sum still diverges.\nFor such $A$ the first-block condition again reads $2^r\\mid \\#\\{p: p^{k_p}\\mid N+r\\}$, but now the\nprescribed part can be built by CRT with cofactor freedom, because divisibility by $p^{k_p}$ is a\ncondition one can impose without controlling the full factorisation. The concrete first step is to\ndecide whether the $K=2$ analogue is provable unconditionally for one such $A$; that would be the\nfirst result in the corpus outside the two-axis box of divisor-independence plus global tail\nbudget.\n\n\\subsection{R4. Landing the route-collapse equivalences, via perfect-square witness depths}\n\\label{sec:invent-R4}\n\n\\subsubsection*{The statement needed}\n\nFive biconditionals, each with $(1/2:\\R)\\in\\mathcal{A}$ on the right:\n\\[\n\\text{(1)}\\ \\ \\forall N,\\ \\mathtt{mobiusCenteredHalfCarry}\\,G\\,N\\ \\le\\ 2\\sqrt{N}+4;\n\\qquad\n\\text{(2)}\\ \\ \\mathtt{CofinalPositiveHalfGreedySkips};\n\\]\n\\[\n\\text{(3)}\\ \\ \\mathtt{CofinalExactLocalMersenneHalfRows};\n\\qquad\n\\text{(4)}\\ \\ \\mathtt{GreedyHalfCarryCofinalStripReturn};\n\\qquad\n\\text{(5)}\\ \\ \\mathtt{HalfCarryCofinalTerminalOnlyStrip},\n\\]\nwhere $G:=\\mathtt{greedyMersenneSupport}(1/2)$. The forward directions of (2)--(5) are landed\nLean (\\lean{half_mem_mersenneAchievementSet_of_positiveHalfGreedySkips}{BooleanMobiusSkipRowCofinal.lean:73},\n\\lean{half_mem_mersenneAchievementSet_of_cofinalExactLocalRows}{BooleanMobiusCofinalExactRows.lean:71},\n\\lean{half_mem_mersenneAchievementSet_of_cofinalStripReturn}{CofinalStripReturn.lean:157},\n\\lean{half_mem_mersenneAchievementSet_of_cofinalTerminalOnlyStrip}{TerminalOnlyCofinal.lean:134},\nall \\ev{Lean}). What is missing is the reverse directions, and they all run through one lemma.\n\n\\subsubsection*{The object, and the swing that supplies it}\n\n\\begin{lem}[Perfect-square strip witness]\n\\label{lem:sqwitness}\nLet $A\\subseteq\\N$ with $1\\notin A$ and $\\mathtt{erdosSupportSeries}\\ 2\\ A=1/2$. Then for every\n$k\\ge1$,\n\\[\n\\bigl(\\mathtt{integerHalfCarry}\\ A\\ (k^{2}-1)\\ :\\ \\R\\bigr)\n\\ =\\ \\mathtt{binaryCoeffTail}\\ (\\mathtt{supportCoeff}\\ A)\\ (k^{2})\n\\ \\le\\ 2k+4\\ =\\ \\mathtt{halfStripBound}\\ (k^{2}).\n\\]\n\\end{lem}\n\\begin{proof}\nBy \\lean{integerHalfCarry_eq_scaled_residual_add_tail}{HalfCarryReachability.lean:842} (\\ev{Lean}),\n\\[\n\\bigl(\\mathtt{integerHalfCarry}\\ A\\ N:\\R\\bigr)\n= 2^{\\,N+1}\\Bigl(\\tfrac12-\\mathtt{erdosSupportSeries}\\ 2\\ A\\Bigr)\n+\\mathtt{binaryCoeffTail}\\ (\\mathtt{supportCoeff}\\ A)\\ (N+1),\n\\]\nand the first term vanishes by hypothesis. Take $N=k^{2}-1$, so $N+1=k^{2}$. By\n\\lean{binaryCoeffTail_supportCoeff_le_two_sqrt_add_four}{BooleanMobiusCarry.lean:289} (\\ev{Lean},\nunconditional) the tail at $k^2$ is at most $2\\sqrt{k^{2}}+4=2k+4$, and by\n\\lean{halfStripBound}{HalfCarryReachability.lean:32}, $\\mathtt{halfStripBound}\\ n\n= 2\\,\\mathtt{Nat.sqrt}\\ n+4$, which at $n=k^{2}$ equals $2k+4$ exactly. Non-negativity of the tail\n(\\lean{binaryCoeffTail_nonneg}{GenericTailOrbitRigidity.lean:78}, \\ev{Lean}) gives the lower side.\n\\end{proof}\n\\ev{Math}; every input is landed Lean and the composition is four lines.\n\nThe point is narrow and entirely technical, which is why it is worth stating: the corpus's\nrecorded obstruction to closing (4) and (5) was the mismatch between $\\mathtt{Nat.sqrt}$ (in\n$\\mathtt{halfStripBound}$) and $\\mathtt{Real.sqrt}$ (in the unconditional tail bound), for which\nMathlib supplies only \\texttt{Real.nat\\_sqrt\\_le\\_real\\_sqrt}, the wrong direction, and the\nrecorded workaround was to widen the strip constant from $4$ to $6$. Restricting the cofinal\nwitness depths to perfect squares dissolves the mismatch: at $n=k^{2}$ the two roots coincide\nexactly and no widening is needed. Applying Lemma~\\ref{lem:sqwitness} along $M=k^{2}-1$ gives (4)\ndirectly; taking $a$ to be the indicator of an achieving support truncated to\n$\\mathtt{Fin}(M+1)$ at $M=k^{2}$, and using\n\\lean{integerHalfCarry_inter_Iic_eq_of_succ_le}{HalfCarryReachability.lean:49} together with\n$1\\notin A$ (forced because $\\mathtt{mersenneWeight}\\,1=1>1/2$), gives the witness required by\n\\lean{HalfTerminalOnlyStripWitness}{TerminalOnlyCofinal.lean:24} and hence (5).\n\n\\subsubsection*{Why this is a result and not bookkeeping}\n\nUnder the standing rule that a reduction is not a result, one might discount R4. That would be a\nmistake, and the reason is the organising thesis of this document. Each biconditional is a\ntheorem, not a reduction: it says a named producer socket is \\emph{equivalent} to the goal, hence\npermanently not easier, hence removable from the list of things worth attempting directly. Five\nsuch theorems are a systematic elimination, and a systematic elimination is a statement about the\nproblem. In particular (3) settles the advertised weakening --- that no compatibility is required\nbetween witness supports at different endpoints --- as buying nothing: incompatible witnesses do\nnot create flexibility, because the resulting $\\Pi^0_2$ statement still collapses to the\n$\\Pi^0_1$ goal.\n\n\\textbf{Scope limits to be stated verbatim in the docstrings.} (1) does \\emph{not} extend to every\n$A$ with $1\\notin A$: without\n\\lean{erdosSupportSeries_greedyMersenneSupport_le}{HalfCarryReachability.lean:859} (which supplies\n$\\delta:=1/2-S_G\\ge0$ for the greedy support specifically) one obtains only the one-sided\nequivalence $\\text{hbound}\\iff(1/2-S_A)\\le0$. The mechanism in all five cases is the same and\nshould be recorded once: $2^{\\,N+1}\\delta$ dominates any $\\sqrt{N}$ envelope, so a strip\nhypothesis forces $\\delta=0$; and $\\delta=0$ makes the unconditional tail bound supply the strip.\nMembership is then read off by\n\\lean{mem_mersenneAchievementSet_iff_greedy_survival}{GreedyAchievementSet.lean:1445} (\\ev{Lean}).\n\n\\subsection{R5. The sparse/dense trichotomy --- and a proof that the present engine cannot close it}\n\\label{sec:invent-R5}\n\n\\subsubsection*{The statement needed}\n\n\\[\n\\forall A\\subseteq\\N_{\\ge1}:\\quad A.\\mathrm{Infinite}\\ \\Longrightarrow\\\n\\underbrace{\\bigl(\\text{cofinally many super-logarithmic zero windows in } \\mathtt{supportCoeff}\\,A\\bigr)}_{\\text{sparse branch}}\n\\ \\lor\\\n\\underbrace{\\mathrm{Cert}(A)}_{\\text{dense branch}} .\n\\]\nThe sparse branch is already closed twice over: it contradicts\n\\lean{supportCoeffZeroWindow_length_le_eps_logb}{SublogDivisorCoverage.lean:435} (\\ev{Lean},\nunconditional and uniform in $A$) whenever the series is rational, and the far sparse edge is\ncovered outright by \\lean{irrational_erdosSum_of_lcm_gap}{CertificateKernel.lean:5555} (\\ev{Lean}).\nThe dense branch is the missing producer: for $A$ whose divisor incidence is bounded below\ninfinitely often, construct\n\\[\n\\exists N,K,L,C,\\quad K\\le L,\\quad\n\\forall r\\in[1,K]:\\ 2^{r}\\mid \\mathtt{supportCoeff}\\,A\\,(N+r),\n\\]\n\\[\n\\sum_{r=K+1}^{L}\\mathtt{supportCoeff}\\,A\\,(N+r)\\,2^{\\,L-r}\\le C,\n\\qquad\nq\\,(C+N+L+2)<2^{L}.\n\\]\n\n\\subsubsection*{What kind of object would furnish it}\n\n$\\mathtt{supportCoeff}\\,A = \\mathbf{1}_A * \\mathbf{1}$ is a Dirichlet convolution, so the dense\nbranch asks: for an \\emph{arbitrary} $0$--$1$ arithmetic function, find a block of consecutive\narguments on which the convolution with $\\mathbf{1}$ is $2$-adically aligned. M\\\"obius inversion\n$\\mathbf{1}_A = \\mu * \\mathtt{supportCoeff}\\,A$ is exact and costs nothing, which makes the\n$2$-adic structure of the incidence directly readable, and that is the technique family to use:\n$2$-adic M\\\"obius/Dirichlet algebra on the divisor lattice, not analysis. The reason the corpus's\nlanded theorems all live inside the two-axis box (divisor independence \\emph{and} a global tail\nbudget) is that CRT prescribes an incidence pattern only when the divisibility conditions\n$a\\mid n$ for $a\\in A$ are independent; for general $A$ they are entangled through pairwise gcds\nand CRT has nothing to prescribe.\n\n\\subsubsection*{The specific obstacle, proved}\n\nHere the honest answer is stronger than an obstacle: the dense branch as stated is \\emph{false},\nand the falsifier is an explicit, natural, divisor-dense support. This also corrects the reduction\nrecorded earlier in this document, that universal \\#257 passes ``with no loss'' to\n$\\mathrm{O1}' := \\forall A\\ \\text{infinite},\\ \\mathrm{Cert}(A)$.\n\n\\begin{prop}[The block-certificate engine is unsatisfiable at the squarefree support]\n\\label{prop:squarefree}\nLet $A=\\{n\\ge2 : n \\text{ squarefree}\\}$. Then for every $n\\ge1$,\n\\[\n\\mathtt{supportCoeff}\\,A\\,(n)\\ =\\ 2^{\\omega(n)}-1,\n\\]\nwhich is \\emph{odd} for every $n\\ge2$. Consequently, for every $q\\ge2$ there is no\n$(N,K,L,C)$ satisfying $\\mathrm{Cert}(A)$ at $q$; that is, $\\mathrm{Cert}(A)$ is false, and\n$\\mathrm{O1}'$ is false.\n\\end{prop}\n\\begin{proof}\nThe squarefree divisors of $n$ are exactly the $2^{\\omega(n)}$ subsets of its prime support, and\n$\\mathtt{supportCoeff}\\,A\\,(n)$ counts them with $d=1$ removed. For $n\\ge2$, $2^{\\omega(n)}$ is\neven, so $2^{\\omega(n)}-1$ is odd.\n\nSuppose $(N,K,L,C)$ is a certificate at some $q\\ge2$. If $K\\ge1$, the first-block condition at\n$r=1$ requires $2\\mid\\mathtt{supportCoeff}\\,A\\,(N+1)$, impossible since $N+1\\ge2$. If $K=0$, the\nfirst block is vacuous and the middle sum runs from $r=1$, so\n\\[\nC\\ \\ge\\ \\sum_{r=1}^{L}\\mathtt{supportCoeff}\\,A\\,(N+r)\\,2^{\\,L-r}\\ \\ge\\ \\mathtt{supportCoeff}\\,A\\,(N+1)\\,2^{\\,L-1}\\ \\ge\\ 2^{\\,L-1},\n\\]\nwhence $q(C+N+L+2)\\ge 2\\cdot 2^{L-1}=2^{L}$, contradicting the height condition.\n\\end{proof}\n\\ev{Math}; the identity $\\mathtt{supportCoeff}\\,A = 2^{\\omega}-1$ was additionally checked by\ndirect enumeration for $1\\le n\\le 3000$ with zero mismatches, \\ev{Cert}.\n\nThree consequences, all of which matter more than the counterexample itself.\n\nFirst, the certificate engine\n\\lean{irrational_erdosSupportSeries_of_weighted_coeff_certificates}{CertificateKernel.lean:8703}\nis universally quantified over $A$ and therefore sound, but it is \\emph{not} complete: it cannot\nprove irrationality for the squarefree support even in principle. Whether\n$\\sum_{n\\ \\mathrm{squarefree},\\,n\\ge2} 1/(2^{n}-1)$ is irrational is untouched by\nProposition~\\ref{prop:squarefree}; what is settled is that no instantiation of the present engine\nwill decide it.\n\nSecond, the failure is not an accident of one support. The mechanism is that\n$\\mathtt{supportCoeff}\\,A$ is forced into a fixed parity class by the multiplicative structure of\n$A$, and the engine's first-block condition is a parity condition at $r=1$. The general form is\nworth recording, because it says exactly which supports are engine-blind.\n\n\\begin{prop}[Constant eventual parity pins the support]\n\\label{prop:parityrigid}\nWrite $c:=\\mathtt{supportCoeff}\\,A$ and suppose $c(n)\\equiv\\varepsilon\\pmod 2$ for all $n>N_0$,\nwith $\\varepsilon\\in\\{0,1\\}$ fixed. Then for every $n>1$,\n\\[\n\\mathbf{1}_A(n)\\ \\equiv\\ \\sum_{\\substack{d\\mid n\\\\ d\\le N_0}}\\mu(n/d)\\,\\bigl(c(d)-\\varepsilon\\bigr)\\pmod 2 .\n\\]\nIn particular, membership in $A$ above $N_0$ is completely determined, modulo $2$, by the finite\nvector $\\bigl(c(d)\\bigr)_{d\\le N_0}$ together with $\\mu$.\n\\end{prop}\n\\begin{proof}\nM\\\"obius inversion gives $\\mathbf{1}_A(n)=\\sum_{d\\mid n}\\mu(n/d)c(d)$. Put $g(d):=c(d)-\\varepsilon$;\nsince $\\sum_{d\\mid n}\\mu(n/d)=0$ for $n>1$, we get\n$\\mathbf{1}_A(n)=\\sum_{d\\mid n}\\mu(n/d)g(d)$. Every term with $d>N_0$ has $g(d)$ even.\n\\end{proof}\n\\ev{Math}. The squarefree support is the case $N_0=1$, $\\varepsilon=1$: the proposition then reads\n$\\mathbf{1}_A(n)\\equiv\\mu(n)\\cdot(c(1)-1)\\equiv\\mu(n)\\pmod2$, i.e.\\ $A$ is exactly the squarefree\nnumbers $\\ge2$ --- so the $N_0=1$ fixed point is unique, and it exists.\n\nThird, and this is the finding to carry forward: \\emph{the universal direction of \\#257 has a\nstructural obstruction that sits logically before the analytic one}. Even granting the\nshort-window anti-concentration input of \\S\\ref{sec:invent-R1}, the present engine would not\nclose the universal statement, because at some supports it has no certificate to find. That\nweakens --- it does not refute --- the thesis that both halves of \\#257 face a single wall. The\nhalf-value branch's difficulty is analytic (pseudorandomness of a divisor-driven digit stream);\nthe universal branch's difficulty is, at least in part, prior and combinatorial (the engine's\nfirst-block parity condition is unsatisfiable on a natural divisor-dense class).\n\n\\subsubsection*{Swings}\n\n\\emph{Swing 8: enumerate the engine-blind supports.} \\ev{Open}, first step concrete.\n\nProposition~\\ref{prop:parityrigid} converts ``which supports defeat the first-block condition?''\ninto a finite-parameter fixed-point problem: for each $N_0$, the candidate supports are the\nsolutions $A$ of $\\mathbf{1}_A(n)\\equiv\\sum_{d\\mid n,\\,d\\le N_0}\\mu(n/d)(c(d)-\\varepsilon)$\nconsistent with $c=\\mathbf{1}_A*\\mathbf{1}$ on $[1,N_0]$. The first actual step is to solve this\nfor $N_0=1,2,3,4$ by brute force over the $2^{N_0}$ initial patterns and to check each candidate\nagainst the definition on $[1,10^4]$. Every solution found is another support the engine cannot\ntouch; if the solution set is finite and explicit, then the engine, plus an exceptional-set\nargument for those finitely many families, is still a viable route to $\\mathrm{U257}$, and that\nwould be worth knowing before any further analytic effort is spent.\n\n\\emph{Swing 9: replace the first-block condition by a shifted-modulus one.} \\ev{Open}.\n\nThe parity failure above is a failure of $2\\mid c(N+1)$, not of the underlying irrationality\nargument. The engine's role for the first block is to force $2^{L}$ to divide a weighted prefix\nso that the certificate's rational approximant has a controlled denominator. Nothing requires the\nmodulus to be a power of the base $2$ at every slot; what is required is that the prefix\ncontributes an integer after scaling. So the concrete replacement to attempt is a\n\\emph{$\\lambda$-shifted} block condition\n\\[\n\\forall r\\in[1,K]:\\quad 2^{r}\\ \\mid\\ \\bigl(\\mathtt{supportCoeff}\\,A\\,(N+r)+\\lambda_r\\bigr),\n\\]\nwith a fixed correction vector $\\lambda\\in\\Z^{K}$ that is absorbed into $C$ and into the height\ncondition, at cost $\\|\\lambda\\|_1 2^{\\,L-K}$ added to $C$. For $A=\\{n\\ge2\\ \\text{squarefree}\\}$\none has $c(n)+1=2^{\\omega(n)}$, so $\\lambda_r\\equiv1$ makes the first-block condition read\n$2^{r}\\mid 2^{\\omega(N+r)}$, i.e.\\ $\\omega(N+r)\\ge r$ --- which is trivially arrangeable by CRT.\nThat is a complete first-block supply for the squarefree support with $K$ arbitrary, and it costs\n$K\\,2^{\\,L-K}$ in the budget, comfortably inside the height condition. Whether the rest of the\nengine survives the shift is exactly the question to check, and it is a self-contained\nmodification of one Lean file rather than a new theory. If it does survive, the immediate corollary\nis irrationality of $\\sum_{n\\ \\mathrm{squarefree}} 1/(2^{n}-1)$, which is a named constant and a\ngenuine theorem.\n\n\\subsection{What the shape of the wall suggests about the underlying object}\n\\label{sec:invent-shape}\n\nEverything in this subsection is inference from the accumulated failures, not theorem. It is\nlabelled that way throughout, and where the evidence does not determine the answer it says so\nrather than manufacturing a story.\n\n\\subsubsection*{The half-value branch looks pseudorandom, and the failures are the evidence}\n\nThe strongest available inference is about the \\emph{kind} of difficulty, and it comes from the\nmanner in which the attacks failed rather than the fact that they failed.\n\nWhen a hard problem conceals structure, partial attacks typically return partial information: a\nbias, a correlation, an exceptional set, a bound with the wrong constant. That is not what\nhappened here. Parity returned a tautology --- the identity $J(M+1)\\equiv\\delta(M+1)\\pmod2$ holds\nfor \\emph{any} digit stream obeying the recursion shape, so it forbids no residue and no run.\nFixed-precision valuation data returned a proved no-go: for any starting carry and any word of\nodd-unit symbols, a compatible centred completion exists\n(\\lean{fixedPrecisionTropicalNoGo}{TropicalCurvatureCarry.lean:137}, \\ev{Lean}). Bounded carry\nstate returned an unbounded fan-out\n(\\lean{balancedPulse_no_autonomous_decoder}{GenericTailOrbitRigidity.lean:247}, \\ev{Lean}). Long\ncommon suffixes returned exact erasure of the predecessor\n(\\lean{affineBinaryOrbit_mod_twoPow_eq}{GenericTailOrbitRigidity.lean:307}, \\ev{Lean}).\nPrime-doubling returned a witness where three primes sit inside an eleven-step quiet run. These\nare not near misses. They are the signature of a coordinate in which there is no arithmetic bias\nto find --- which is what one expects if the digit stream really is pseudorandom.\n\nThe positive measurements point the same way and are quantitative. The branch counts over\n$200{,}000$ seam rows are $R:M:U = 100197:49899:49898$, matching the balanced-signed-digit\nprediction to within sampling noise. The run-length histogram is geometric with ratio $\\approx2$.\nThe binary expansion of $C=E-3/2$ agrees bit-for-bit with OEIS A211706 and has longest equal run\n$15$ in $40{,}000$ bits. And the measurement in Swing~2 above shows the fractional part of the\nshort-window divisor phase $\\Theta_L$ is, to three significant figures, Haar-distributed, with a\nnear-integer tail constant $c\\approx2.5$ against the uniform value $2$.\n\n\\emph{Inference (not theorem).} The object behind the half-value branch is most likely\n\\textbf{simple in description and pseudorandom in behaviour}. Its description is two lines:\n$K(2)=1$, $K(M+1)=2K(M)-(\\tau(M+1)-1)$, together with a superincreasing greedy orbit. Its\nbehaviour is, on every statistic yet measured, indistinguishable from a fair coin. If that is\nright, then the missing theorem is a \\emph{pseudorandomness} statement about an explicit\ndeterministic sequence, and it belongs to the same family as normality of specific constants, or\nChowla-type independence: elegant to state, and hard for the structural reason that there is no\nstructure to exploit. On this reading the wall is not hiding a mechanism; it is the absence of\none.\n\n\\subsubsection*{Three reasons to distrust that inference, stated plainly}\n\nFirst, coordinate-relativity. Obstructions are relative to a representation, and all nine\ncoordinates the corpus built --- rung, seam row, integer margin, sharp tail margin, half-carry,\nreverse-carry word, Boolean--M\\\"obius exact row, fatal-cell packing, Dedekind cut-locator --- are\ndescendants of one family, the superincreasing greedy carry. Their agreement is therefore\n\\emph{not} nine independent votes. A genuinely foreign coordinate (a $p$-adic dynamical one, a\nrepresentation-theoretic one, an automatic-sequence one) has not been tried, and ``no bias in\nthese nine'' is weaker evidence than it looks.\n\nSecond, the single exception. Exactly one row of the real orbit fails $\\sqrt{\\ }$-escape, $r=7$,\nwhich is also the site of the only real $R$-branch crossing ever observed, at $(s,d)=(10,7)$. A\nlone low-lying exception is what a random model predicts; it is also what a hidden structure with\na small modulus predicts. That datum does not discriminate.\n\nThird, and this is the sharpest internal check available: pseudorandomness is a claim about the\n\\emph{absence} of structure, and the corpus contains an unmistakable piece of \\emph{present}\nstructure --- strict superincreasingness of the Mersenne weights, which forces the digit coding to\nbe injective (\\lean{positiveMersenneDigitValue_injective}{GreedyAchievementSet.lean:1546},\n\\ev{Lean}), collapses every witness space to a singleton, and makes the static adversarial tier\ncoincide with the dynamic greedy orbit. That structure is rigid, not random. So the honest\ndescription is a hybrid: a rigid skeleton with a pseudorandom digit stream running along it. The\nrigidity is why every soft method fails (there is no ensemble to average over: $\\mathcal{A}$ has\nmeasure $1$ in its span and is nowhere dense, and the two standard notions of ``typical''\ndisagree on it); the pseudorandomness is why every arithmetic method returns no information.\n\n\\subsubsection*{The universal branch reads differently, and this weakens the one-wall thesis}\n\nProposition~\\ref{prop:squarefree} is the reason to separate the two halves rather than assimilate\nthem. On the universal side the present obstruction is not analytic and not pseudorandomness: it\nis that the certificate engine's first-block condition is a parity condition on\n$\\mathtt{supportCoeff}\\,A$, and there is an explicit natural support --- the squarefree numbers ---\non which that parity is constantly wrong. Proposition~\\ref{prop:parityrigid} shows the phenomenon\nis governed by a clean finite-parameter M\\\"obius rigidity, not by randomness at all. So the\nuniversal branch's near-term difficulty is combinatorial and \\emph{fixable in principle} (Swing~9\nproposes the fix); the half-value branch's difficulty is analytic and is not.\n\nThe wall analysis's claim that both halves converge on one missing object --- short-window\nnear-integer anti-concentration of a divisor-weighted sum --- survives, but with a correction: on\nthe universal side that object is the \\emph{second} obstruction, not the first, and it becomes\nrelevant only after the engine is repaired. Reporting the convergence without that ordering would\noverstate the unity of the wall.\n\n\\subsubsection*{What the evidence does and does not determine}\n\nIt does not determine whether a short proof exists. Nothing in a catalogue of failed methods\nbounds the length of a successful one, and this document should not pretend otherwise.\n\nIt does determine, with proof rather than inference, where a short proof cannot be. It cannot\nproceed by finite inspection, by fixed-precision arithmetic, by a bounded carry state, by a\nmeasure or category argument, by an irrationality-measure bound on $E$ or any constant of that\ntype, or --- on the universal side, as of Proposition~\\ref{prop:squarefree} --- by instantiating\nthe present block-certificate engine at a general support. Each of those is closed with a\nmechanism, and a mechanism is reusable in a way that a failed attempt is not. That is the\ncontribution: not a proof, and not a reduction, but a map of the region in which a proof cannot\nlive, drawn tightly enough that the remaining region is small and its coordinates are named.\n\n% ---- part a257_p5 ----\n\\section{What is open, stated exactly}\n\nErd\\H{o}s \\#257 asks whether $\\sum_{n\\in A}1/(2^n-1)$ is irrational for \\emph{every} infinite\n$A\\subseteq\\mathbb{N}_{\\ge 1}$. Nothing in this paper decides this, in either the universal form\nor the single-target half-value form below. Every corpus result surveyed in Parts I--IV is either\na fixed/bounded-scale instance, a coordinate reduction, or a producer socket still lacking its\nsupply. This section states the exact open targets, in display mathematics, with their Lean sites\nwhere formalised and their precise logical relationships to one another. \\textbf{Read the\nimplication table in \\S\\ref{sec:v-table} before attacking any single target below}: several\napparent waypoints are \\emph{equivalent} to the full problem, not easier approaches to it.\n\n\\subsection{Notation fixed for this section}\n\nThe functional $\\mathtt{erdosSupportSeries}\\ 2\\ (-)$ denotes the base-$2$ Erd\\H{o}s support series\n$\\mathtt{erdosSupportSeries}\\ 2\\ A := \\sum_{n\\in A} 1/(2^n-1)$\n(\\lean{erdosSupportSeries}{CertificateKernel.lean:8569}). $\\mathtt{supportCoeff}\\ A\\ n :=\n|\\{d\\mid n : d\\in A\\}|$ (\\lean{supportCoeff}{CertificateKernel.lean:8528}) is the divisor-indicator\ncoefficient whose generating tail feeds every certificate below. $\\mathcal{A} :=\n\\mathtt{mersenneAchievementSet} = \\{x\\mid \\exists A,\\ 0\\notin A\\wedge x =\n\\mathtt{positiveMersenneSupportValue}\\ A\\}$ (\\lean{mersenneAchievementSet}{GreedyAchievementSet.lean:571}).\n$G := \\mathtt{greedyMersenneSupport}(1/2)$ is the canonical greedy support for target $1/2$;\n$\\mathtt{seamIntegerGreedyRemainder}\\ s$ (written $\\mathrm{rem}(s)$ below) is its integer seam\nremainder at row $s$, and a \\emph{reset} at row $r$ is a row where the greedy orbit returns to the\nnear-$2^r$ regime after a run of takes or skips.\n\n\\subsection{Target O1 --- Universal Erd\\H{o}s \\#257}\n\n\\begin{defn}[Universal \\#257]\n\\[\n\\mathrm{U257} \\;:=\\; \\forall\\, A\\subseteq\\mathbb{N}_{\\ge 1},\\ A.\\mathrm{Infinite} \\;\\Longrightarrow\\;\n\\mathrm{Irrational}\\!\\left(\\mathtt{erdosSupportSeries}\\ 2\\ A\\right).\n\\]\n\\end{defn}\nThis is the literal statement of Erd\\H{o}s \\#257. \\ev{Open}. The corpus's own sufficient-condition\nengine is\n\\[\n\\mathrm{Cert}(A)\\ :=\\ \\forall q>0,\\ \\exists N,K,L,C,\\ K\\le L,\\\n\\left(\\forall r\\in[1,K],\\ 2^r\\mid \\mathtt{supportCoeff}\\,A\\,(N+r)\\right)\\wedge{}\n\\]\n\\[\n\\sum_{r=K+1}^{L}\\mathtt{supportCoeff}\\,A\\,(N+r)\\cdot 2^{L-r}\\le C\\ \\wedge\\\n\\exists t,\\ 0<\\mathtt{supportCoeff}\\,A\\,(N+L+1+t)\\ \\wedge\\ q(C+N+L+2)<2^L,\n\\]\nwith $\\mathrm{Cert}(A)\\Rightarrow \\mathrm{Irrational}(\\mathtt{erdosSupportSeries}\\ 2\\ A)$\nalready proved unconditionally for \\emph{every} $A$\n(\\lean{irrational_erdosSupportSeries_of_weighted_coeff_certificates}{CertificateKernel.lean:8703},\n\\ev{Lean}, \\scale{n/a} --- the theorem is universally quantified over $A$; only its hypothesis is\nunsupplied). So $\\mathrm{U257}$ reduces, with no loss, to:\n\\[\n\\mathrm{O1'}\\ :=\\ \\forall A,\\ A.\\mathrm{Infinite}\\Longrightarrow \\mathrm{Cert}(A).\n\\]\n$\\mathrm{O1'}$ is \\ev{Open}\\ for arbitrary infinite $A$; it is proved for every named structured\nfamily in the corpus (full support, eventually periodic, pairwise-coprime with summable\nreciprocals, orthogonal petal bouquets, lcm-gap sequences) but each proof uses the family's\nextra structure to manufacture the block certificate, and none of these routes covers, e.g.,\n$A=$ the primes (divergent $\\sum 1/p$ defeats the pairwise-coprime route\n(\\lean{irrational_erdosSupportSeries_pairwise_coprime}{CertificateKernel.lean:10448})) or any\ndivisor-dense infinite $A$ (which defeats the only unconditional universal rigidity result,\nthe sparse-side window bound\n\\lean{supportCoeffZeroWindow_length_le_eps_logb_add}{SublogDivisorCoverage.lean:435}, \\ev{Lean},\n\\coord{lcm-gap window}, \\scale{uniform}, which is vacuous once $2\\in A$).\n\n\\subsection{Target O2 --- Half-value membership}\n\n\\begin{defn}[Half membership]\n\\[\n\\mathrm{HALF}\\ :=\\ \\left(\\tfrac12:\\mathbb{R}\\right)\\in\\mathcal{A}.\n\\]\n\\end{defn}\n$\\mathrm{HALF}$ is a \\emph{single instance} of the negation of $\\mathrm{U257}$: any witness\n$A$ with $0\\notin A$, $\\mathtt{erdosSupportSeries}\\ 2\\ A=1/2$ is infinite\n(\\lean{finite_boolSupport_ne_half}{HalfCarryReachability.lean:589}, \\ev{Lean}, \\scale{uniform} ---\nfinite supports have odd-denominator value, so can never equal a dyadic rational), hence a\ngenuine counterexample to $\\mathrm{U257}$. So $\\mathrm{HALF}\\Rightarrow\\neg\\mathrm{U257}$, strictly\n--- it does not decide $\\mathrm{U257}$ if false, since $\\mathrm{U257}$ could still fail at a\ndifferent target $t\\in\\mathbb{Q}$. $\\mathrm{HALF}$ is \\ev{Open}. It is logically equivalent, by an\nunconditional Lean iff, to a purely combinatorial statement about one fixed orbit:\n\\[\n\\mathrm{HALF}\\ \\Longleftrightarrow\\ \\left(\\mathtt{greedyMersenneSkippedSupport}(1/2)\\right).\\mathrm{Infinite}\n\\]\n(\\lean{half_mem_mersenneAchievementSet_iff_greedySkippedSupport_infinite}{GreedyAchievementSet.lean:2514},\n\\ev{Lean}, \\scale{uniform}, coordinate \\coord{mobius-mersenne}, target-generic in the reverse\ndirection --- the forward direction uses only that the target is rational, so the same theorem\nholds verbatim for every rational $t\\ge 0$ in place of $1/2$).\n\n\\subsubsection{Waypoints that are actually EQUIVALENT to O2, not weaker}\n\nThree named \"producer sockets\" that read as strictly easier routes to $\\mathrm{HALF}$ have been\nadversarially checked and found to be \\emph{equivalent} to $\\mathrm{HALF}$ itself. Attacking any\none of them is attacking Erd\\H{o}s \\#257-at-$1/2$ directly.\n\n\\begin{prop}[CofinalPositiveHalfGreedySkips is equivalent to HALF]\n\\label{prop:cpgs-equiv}\nDefine\n\\[\n\\mathrm{CPGS}\\ :=\\ \\forall N,\\ \\exists c\\ge N,\\ c\\ \\text{skipped by the rational half-greedy\norbit}\\ \\wedge\\ 0<\\mathtt{greedyMersenneRemainderRat}(1/2)(c-1)\n\\]\n(\\lean{CofinalPositiveHalfGreedySkips}{BooleanMobiusSkipRowCofinal.lean:22}). The positivity\nconjunct is unconditionally true for every skipped $c$\n(\\lean{localMersennePrefixValue_halfGreedy_lt_half}{BooleanMobiusCriticalCapacityCofinal.lean:138},\n\\ev{Lean}, \\scale{uniform} --- an odd-denominator parity fact), hence deletable without changing\nthe Prop's meaning. What survives, $\\forall N\\,\\exists c\\ge N$ skipped, is exactly\n$(\\mathtt{greedyMersenneSkippedSupport}(1/2)).\\mathrm{Infinite}$, so\n\\[\n\\mathrm{CPGS}\\ \\Longleftrightarrow\\ \\mathrm{HALF}.\n\\]\n\\end{prop}\n\\ev{Math, adversarially verified} (Survivor 7, composition-survivor pass). \\scale{cofinal}.\n\\coord{mobius-mersenne, greedy orbit}. This closes off what the source bank advertised as \"the\nshortest known path\" to a disproof (\\lean{cofinalExactLocalMersenneHalfRows_of_positiveHalfGreedySkips}{BooleanMobiusSkipRowCofinal.lean:60}): there is no shortcut through it, because it is not shorter.\n\n\\begin{prop}[The terminal-only cofinal strip is equivalent to HALF]\n\\label{prop:strip-equiv}\nLet $\\mathtt{HalfCarryCofinalTerminalOnlyStrip}$ be the socket that a finite normalised word $a$\nat cofinally many depths $M$ satisfies $|\\mathtt{integerHalfCarry}(\\mathtt{wordSupport}\\,a)(M-1)|\n\\le \\mathtt{halfStripBound}\\,M$ where $\\mathtt{halfStripBound}\\,n = 2\\sqrt{n}+4$\n(\\lean{HalfCarryCofinalTerminalOnlyStrip}{TerminalOnlyCofinal.lean:34}). Widened to strip\nconstant $6$ (i.e. $2\\cdot\\mathtt{Nat.sqrt}\\,n+6$), this socket is logically equivalent to\n$\\mathrm{HALF}$: every truncation of an achieving support is already a witness (not merely\ncofinally many), given three small mechanical lemmas not yet in the corpus (a general\n$\\mathtt{one\\_notMem\\_of\\_erdosSupportSeries\\_eq\\_half}$; the discrete envelope\n$\\mathtt{Real.sqrt}\\,n\\le\\mathtt{Nat.sqrt}\\,n+1$; and an explicit $\\mathtt{HalfWord}$\nconstruction from $A\\cap\\mathrm{Iic}\\,M$).\n\\end{prop}\n\\ev{Math, adversarially verified, weakened} (Survivor 5). \\scale{cofinal}. \\coord{support\nrigidity, half-carry}. This corrects the module's own \"weaker producer\" framing --- the strip\nsocket carries no strategic advantage over proving $\\mathrm{HALF}$ outright.\n\n\\begin{obs}\nNeither Prop.~\\ref{prop:cpgs-equiv} nor Prop.~\\ref{prop:strip-equiv} narrows the search space for\n$\\mathrm{HALF}$. Their value is negative: they retire two named targets from the \"worth chasing as\nan easier waystation\" list. A third socket, the hypothesis $\\forall N,\\ \\mathtt{mobiusCenteredHalfCarry}\\,A\\,N\\le 2\\sqrt N+4$\nof \\lean{greedy_half_infinite_of_mobiusCenteredHalfCarry_sqrtBound}{HalfCarryReachability.lean:805},\nis likewise proved EQUIVALENT to $\\mathrm{HALF}$ for the specific support $A=G$ (Survivor 4,\n\\ev{Math, adversarially verified, weakened}) --- not, as originally claimed, for every $A$ with\n$1\\notin A$; the general-$A$ version is only a one-sided implication ($\\mathrm{hbound}\\Rightarrow\n\\mathtt{erdosSupportSeries}\\ 2\\ A\\le 1/2$).\n\\end{obs}\n\n\\subsection{Target O3 --- Cofinal exact local Mersenne half rows}\n\n\\begin{defn}\n\\[\n\\mathrm{CofinalExactLocalMersenneHalfRows}\\ :=\\ \\forall N,\\ \\exists n\\ge N,\\\n\\mathtt{ExactLocalMersenneHalfRow}\\ n\n\\]\n(\\lean{CofinalExactLocalMersenneHalfRows}{BooleanMobiusCofinalExactRows.lean:38}, \\ev{Open},\n\\scale{cofinal}). No compatibility is required between witness supports at different endpoints\n--- the defining weakening versus a single coherent trajectory.\n\\end{defn}\n$\\mathrm{CofinalExactLocalMersenneHalfRows}\\Rightarrow\\mathrm{HALF}$ is proved unconditionally\nvia compactness of $\\mathcal{A}$ (continuous image of $(\\mathbb{N}\\to\\mathrm{Fin}\\,2)$ under the\ndigit-coding map, hence closed)\n(\\lean{half_mem_mersenneAchievementSet_of_cofinalExactLocalRows}{BooleanMobiusCofinalExactRows.lean:71},\n\\ev{Lean}). This direction is \\emph{strictly weaker} than $\\mathrm{HALF}$ as currently proved:\nthe argument uses closedness alone and discards the producing rows, so the theorem as it stands\ndoes not pin a single limiting support $A$; a genuine sequential-compactness extraction (Cantor-space\ndiagonal argument over $(\\mathrm{Fin}\\,2)^{\\mathbb{N}}$, using the file's own\n\\lean{continuous_positiveMersenneDigitValue}{GreedyAchievementSet.lean:628}) would upgrade this to\nproducing $A$ explicitly, and would additionally transfer $2\\in A$ for free\n(\\lean{two_mem_of_exact_localMersenneQuotient}{BooleanMobiusExactRowRankTwo.lean:23}) --- but this\nupgrade is itself unformalised (Survivor 1, \\ev{Math, adversarially verified, weakened}). So O3\nsits strictly below $\\mathrm{HALF}$ in the current Lean state of the corpus, not merely\nlogically.\n\n\\subsubsection{Sharp critical-capacity conditions --- the BooleanM\\\"obius lane's producer for O3}\nThe corpus's engine for O3 is a sharp $c$-2-bit capacity test. Write\n\\[\n\\mathtt{localBinarySuffix}\\ D\\ k\\ M\\ :=\\ 2^{M-k}-\\mathtt{localPrefixQuotient}\\ D\\ M - 1\n\\]\n(\\lean{localBinarySuffix}{BooleanMobiusLocalRepair.lean:53}). For $4\\le c$, every $d\\in D$ with\n$2\\le d<c$, and $\\mathtt{localMersennePrefixValue}\\,D<1/2$:\n\\begin{align}\n&\\text{(loose, proved)}\\quad \\mathtt{localBinarySuffix}\\ D\\ 1\\ (2c-2)\\ <\\ 2^{c-1}\n\\tag{$\\ast$}\\\\\n&\\text{(sharp, needed)}\\quad \\mathtt{localBinarySuffix}\\ D\\ 1\\ (2c-2)\\ <\\ 2^{c-2}\n\\tag{$\\ast\\ast$}\n\\end{align}\n$(\\ast)$ is proved unconditionally, uniform in $c$, no case split\n(\\lean{localBinarySuffix_two_mul_sub_two_lt_upperHalfCapacity}{BooleanMobiusSkippedCoreExactRow.lean:123},\n\\ev{Lean}, \\scale{uniform}). $(\\ast\\ast)$ is exactly one bit sharper and, via\n\\lean{localBinarySuffix_two_mul_sub_two_lt_criticalCapacity_iff}{BooleanMobiusSkippedCoreCriticalCapacity.lean:22}\n(an unconditional iff), is \\emph{equivalent} to an exact row already crossing the integer\nhalf-target at $c$. The true unproved gap between $(\\ast)$ and $(\\ast\\ast)$ is additive, not\nmultiplicative: the proof of $(\\ast)$ derives $\\mathtt{localBinarySuffix}<2^{c-2}+|D|$ and\ndiscards $|D|\\le c-2$; so what is actually open is the linear-width anti-concentration\n\\[\n\\text{(O3-supply)}\\qquad \\forall N\\ \\exists c\\ge N,\\ \\mathtt{localBinarySuffix}\\ D\\ 1\\ (2c-2)\\ \\notin\\\n\\big[\\,2^{c-2},\\,2^{c-2}+c-3\\,\\big]\n\\]\n--- a band of width $c-2$ inside a range of size $2^{c-2}$, exponentially weaker than the\nreset $\\sqrt{\\text{-escape}}$ target of \\S\\ref{sec:o4} (band width $c$ vs.\\ width\n$2^{(c+5)/2}$). The generic (non-greedy) sharp-fill theorem itself is already unconditional and\nstrictly more general than every consumer built on it:\n\\lean{exists_exactRowStrictUpperFill_of_skippedCoreSharpCapacity}{BooleanMobiusSkipRow.lean:286}\n(\\ev{Lean}) needs only $(\\ast\\ast)$ for \\emph{some} below-half core $D$, not a greedy prefix ---\nthis is the escape route named in Prop.~\\ref{prop:cpgs-equiv}'s remark: abandoning the canonical\ngreedy orbit (which forces $\\mathrm{CPGS}$-equivalence) for an arbitrary below-half core is a\ngenuinely different, still-open target.\n\nThe corpus's complete inductive skeleton from the endpoint-six seed to O3 is already assembled\n(\\lean{cofinalExactLocalMersenneHalfRows_of_criticalQuotientSupply}{BooleanMobiusCriticalCapacityCofinal.lean:1227},\n\\ev{Lean}); it consumes the sharp capacity input, in its reduced canonical form\n\\[\n\\mathrm{HalfGreedySkippedCriticalQuotientSupply}\\ :=\\ \\forall c\\ge 4\\ \\text{skipped},\\\n2^{(2c-2)-1}\\ \\le\\ \\mathtt{localPrefixQuotient}\\big(\\mathrm{insert}\\ c\\ (\\mathtt{halfGreedyPrefixSupport}(c-1))\\big)(2c-2)\n\\]\n(\\lean{HalfGreedySkippedCriticalQuotientSupply}{BooleanMobiusCriticalCapacityCofinal.lean:178}),\nat essentially every step (\\ev{Open}, \\scale{cofinal}). This is a strictly harder producer than\n(O3-supply) because it is pinned to the greedy prefix rather than an arbitrary core.\n\n\\subsection{Target O4 --- Reset anti-concentration (sqrt-escape) and the R-run form}\n\\label{sec:o4}\n\n\\begin{defn}[Reset $\\sqrt{\\text{-escape}}$ from row 10]\n\\[\n\\mathrm{SQRTESC}\\ :=\\ \\forall r\\ge 10,\\ \\big|\\mathrm{rem}(r+1)-2^{r+1}\\big|\\ >\\ 2^{(r+5)/2}\n\\qquad\\Big(\\text{equivalently } \\Delta_{r+1}^2 > 2^{r+5}\\Big).\n\\]\n\\end{defn}\n\\ev{Math, fleet-audited}, coordinate \\coord{deviation-affine / crossing-cell geometry}. The\nexponent $(r+5)/2$ is \\emph{derived}, not fitted: a violation forces an R-branch crossing cell\n$(s,d)$ with $s\\ge14$, late row $d\\ge10$ with run length $L=(d-3)/2$ or $(d-4)/2\\ge3$; dividing\nthe exact affine iterate $w_{d+1+L}=4^Lw_{d+1}-C_L$ by $4^L$ and absorbing the charge bound\n$1\\le C_L/4^L<(2s+4)/3$ yields the exact exponent. This target is \\emph{equivalent}, by a Theorem-A\ncomposition already Lean-landed on the consumer end, to\n\\[\n\\mathrm{SQRTESC}\\ \\Longrightarrow\\ \\mathrm{LargestSkipLateStepSocket}\\ \\Longrightarrow\\ \\mathrm{HALF}\n\\]\n(\\lean{half_mem_mersenneAchievementSet_of_largestSkipLateStepSocket}{HalfCylinderLargestSkipInduction.lean:178}, \\ev{Lean} for the second\narrow, \\ev{Math, fleet-audited} for the first). $\\mathrm{SQRTESC}$ is equivalently stated as the\nR-run length bound\n\\[\n\\mathrm{RUNBOUND}\\ :=\\ \\forall\\,\\text{reset}\\ r\\ge31,\\quad L_r\\ <\\ \\frac{r-3}{2}\n\\]\nwhere $L_r$ is the length of the pure-R run following reset $r$. $\\mathrm{RUNBOUND}\\Leftrightarrow\n\\mathrm{SQRTESC}$ (equivalence (III) of the reset-crossing unification note, \\ev{Math}). $r=7$ is\nthe sole exception below row $10$ (dev $=+9$, threshold $64$) and is out of scope for both forms;\nthis is why both definitions are stated from $r\\ge10$ or $r\\ge31$.\n\n\\emph{Certified evidence, not a proof.} Exact integer arithmetic to row $200{,}000$: TARGET holds\nat every reset row except $r=7$; the maximum observed R-run is $19$, at row $158{,}096$, against a\nrequired threshold there of $\\approx79{,}046$ --- a margin of $\\sim4000\\times$\n(\\ev{Cert}, \\scale{fixed}). This is a finite verified list, \\emph{not} a cofinal supply, and must\nnever be read as one: it is the exact instance of the quantifier trap this programme's own\ndoctrine warns against.\n\n\\begin{obs}[Sign law --- a strictly weaker, still-open sub-target]\nAt all $1209$ observed resets, the sign of the deviation is perfectly predicted by branch type\n(M$\\Rightarrow$dev$>0$, U$\\Rightarrow$dev$<0$, zero exceptions), and the margin over the threshold\ngrows from $1.11$ bits at $r=14$ to $\\sim1246$ bits by row $2500$ (\\ev{Cert}). Proving the sign\nlaw unconditionally --- $\\mathrm{M}\\Rightarrow \\mathrm{rem}(r+1)>2^{r+1}$;\n$\\mathrm{U}\\Rightarrow \\mathrm{rem}(r+1)<2^{r+1}$ --- is a one-step branch-algebra statement in the\ncoordinates of the already-proved affine recurrence, and would convert $\\mathrm{SQRTESC}$ from one\ntwo-sided bound into two one-sided bounds. It is flagged in the source material itself as \"worth\nattempting as a lemma before the inequality itself,\" and it remains \\ev{Open}.\n\\end{obs}\n\n\\begin{obs}[Dangerous-reset rigidity --- Theorem B, not a proof]\nIf reset $r$ is \\emph{dangerous} ($|\\mathrm{rem}(r+1)-2^{r+1}|\\le2^{(r+5)/2}$) and the preceding\nreset $r_0$ has a pure R-run of length $L'=r-r_0-1>(r+5)/4$, then $w_{r_0+1}$ is confined to at\nmost two adjacent integers, determined by the divisor-pulse stream alone. A chain of $n$\nconsecutive dangerous resets with long runs forces $n-1$ nested exact congruence towers. This\nshows the failure mode is a tower of exact integer coincidences, not drift; it is \\ev{Math}, and\nit does \\emph{not} by itself exclude the two pinned values --- an isolated dangerous reset after a\nshort run is untouched by this theorem.\n\\end{obs}\n\n\\subsection{Target O5 --- The remaining\n\\texorpdfstring{\\(-2,-1\\)}{negative-two, negative-one} middle-cell exclusion}\n\n\\begin{defn}\n$\\mathtt{SeamTwoSidedDyadicCellEscape}$ excludes three integer values of the middle coordinate,\n$4\\cdot\\mathrm{rem}(s)-\\mathrm{belowPulse}(s)-4\\notin\\{-3,-2,-1\\}$, plus one right-pulse-leak\nbound $4\\cdot\\mathrm{overshoot}(s)+\\mathrm{abovePulse}(s)\\le2^{s+2}$ under\n$\\mathrm{overshoot}(s)\\le2^s$ (\\lean{SeamTwoSidedDyadicCellEscape}{HalfCylinderMiddleCarryLowerBound.lean:3339}).\n\\end{defn}\nGranted this, induction from a \\texttt{decide}-checked row-5 base propagates\n$\\forall s\\ge5,\\ \\mathrm{rem}(s)\\le2^s\\vee\\mathrm{overshoot}(s)\\le2^s$\n(\\lean{SeamTwoSidedDyadicCellEscape.twoSided}{HalfCylinderMiddleCarryLowerBound.lean:3413}, \\ev{Lean}\ngiven the hypothesis, \\scale{uniform in the consequence}). One of the three cells, $-3$, is already\nclosed unconditionally in the all-right-tail configuration\n(\\lean{finalMiddleCell_neg_three_not_last}{HalfCylinderLastProducerContradiction.lean:333}, \\ev{Lean}).\nThe two remaining cells, $-2$ and $-1$, are \\ev{Open}: what would close O5 is their exclusion for the\nactual seam orbit (not a hypothetical one), together with the right-pulse-leak inequality. This is\nanti-concentration at \\emph{constant} resolution (three named integers), sitting far below\n$\\mathrm{SQRTESC}$'s resolution $2^{(r+5)/2}$ --- it is not known whether closing O5 says anything\nabout O4; no transport between the two has been attempted in the corpus.\n\nA distinct, exponentially cheaper tail-dominance socket targets the same downstream payoff via\nthe upper-reset branch alone: $\\mathrm{SeamUpperResetDyadicBandEscape} := \\forall d\\ge13,\\forall\nj\\le d,\\ 2^{d-j+1}<\\mathrm{resetCharge}\\vee\\mathrm{resetCharge}+2(d+j)\\le2^{d-j+1}$\n(\\lean{SeamUpperResetDyadicBandEscape}{HalfCylinderMiddleCarryLowerBound.lean:3531}, \\ev{Open},\nproved \\ev{Cert} through row $31$ only by exhaustive computation\n(\\lean{seamUpperResetDyadicBandEscape_through_thirty}{HalfCylinderUpperResetBandCertificates.lean:78})).\nThis band has \\emph{linear} width $O(d)$, not dyadic $O(2^{d/2})$ width --- exponentially weaker\nthan $\\mathrm{SQRTESC}$ yet sufficient for the same conclusion $\\mathrm{HALF}$ via\n$\\lean{half_mem_mersenneAchievementSet_of_upperResetDyadicBandEscape}{HalfCylinderMiddleCarryLowerBound.lean:3755}$.\nIt is proved logically equivalent to a single-critical-index check per row\n(\\lean{seamUpperResetCriticalBandEscape_iff}{HalfUpperResetCriticalBand.lean:124-901}, \\ev{Lean}), which is a genuine\nquantifier-complexity reduction, not a strength reduction.\n\n\\subsection{Implication table}\n\\label{sec:v-table}\n\nRead left to right: an entry $\\Rightarrow$ means the row target implies the column target\nunconditionally (Lean or fully checked math); $\\Leftrightarrow$ means proved equivalent;\n\"$\\subseteq$ weaker\" means the row is a strictly weaker but currently unconnected sufficient\ncondition; \"?\" means no transport has been attempted in the corpus.\n\n\\begin{center}\n\\begin{adjustbox}{max width=\\linewidth}\n\\begin{tabular}{@{}l l l@{}}\n\\toprule\nTarget & Relation to HALF (O2) & Status \\\\\n\\midrule\nO1 U257 (universal) & O1 $\\Rightarrow$ everything below is moot & \\ev{Open} \\\\\nO2 HALF & pivot target; $\\neg$O1 witness once true & \\ev{Open} \\\\\nCPGS (Prop.~\\ref{prop:cpgs-equiv}) & $\\Leftrightarrow$ O2 & \\ev{Open}, no shortcut \\\\\nTerminal-only strip (Prop.~\\ref{prop:strip-equiv}) & $\\Leftrightarrow$ O2 & \\ev{Open}, no shortcut \\\\\nmobiusCenteredHalfCarry sqrtBound (for $A=G$ only) & $\\Leftrightarrow$ O2 & \\ev{Open}, no shortcut \\\\\nO3 CofinalExactLocalMersenneHalfRows & $\\Rightarrow$ O2 (strict, Lean) & \\ev{Open} \\\\\n(O3-supply) linear-width capacity band & $\\Rightarrow$ O3 (Lean, given lemma) & \\ev{Open} \\\\\nHalfGreedySkippedCriticalQuotientSupply & $\\Rightarrow$ O3 (Lean) & \\ev{Open}, strictly harder than O3-supply \\\\\nO4 SQRTESC / RUNBOUND & $\\Leftrightarrow$ each other; $\\Rightarrow$ O2 (Lean fan-in) & \\ev{Open} \\\\\nO4 sign law & $\\subseteq$ weaker than O4, untested transport & \\ev{Open} \\\\\nO4 dangerous-reset rigidity (Thm B) & partial rigidity, no exclusion & \\ev{Open} \\\\\nO5 SeamTwoSidedDyadicCellEscape ($-2,-1$ cells) & $\\Rightarrow$ two-sided bound (Lean, given hyp.) & \\ev{Open} \\\\\nO5 SeamUpperResetDyadicBandEscape & $\\Rightarrow$ O2 (Lean), linear-width & \\ev{Open}, cert.\\ to $r=31$ only \\\\\n\\bottomrule\n\\end{tabular}\n\\end{adjustbox}\n\\end{center}\n\n\\subsection{What kind of new mathematics each form would need}\n\nNone of the following is offered as a plan of attack, and none is close. Each is stated to make\nthe shape of the missing ingredient legible.\n\n\\textbf{O1 (universal).} A producer for $\\mathrm{Cert}(A)$ at an arbitrary infinite $A$ is a\nstatement about geometrically-weighted divisor sums achieving controlled residues modulo powers of\ntwo on prescribed dyadic blocks, uniformly over all infinite supports. No general result of this\nshape exists in the analytic-number-theory literature surveyed for this programme; the corpus's\nown honest assessment is that this is a genuinely new near-integer anti-concentration theorem for\na divisor-weighted sum, not a variant of an existing one.\n\n\\textbf{O2/O3/O5 (Mersenne-specific).} The sharp-capacity gap $(\\ast)\\to(\\ast\\ast)$ and the\n$-2,-1$ middle-cell exclusion are both, at bottom, requests for anti-concentration of an explicit\ndivisor-count quantity ($\\mathtt{localBinarySuffix}$, or $4\\cdot\\mathrm{rem}-\\mathrm{belowPulse}-4$)\naway from a short, explicitly named integer window. These are combinatorial/arithmetic\nnear-misses rather than analytic ones \u2014 the corpus's own diagnosis is that a congruence or\npulse-parity obstruction, not an equidistribution theorem, is the natural tool, and no such\nobstruction has been found.\n\n\\textbf{O4 (sqrt-escape).} The required input is a near-integer anti-concentration bound for\n$\\sum_{i\\le L}\\delta_n(M+i)2^{-i}$, a geometrically weighted divisor sum in a short interval, at\nresolution $2^{(r+5)/2}$. This is stated in the source material to not exist in the literature in\nany form, and it is explicitly noted to be the same missing ingredient, on the opposite side of\nthe Erd\\H{o}s \\#249/\\#257 pair, as the totient-window discrepancy anti-concentration input needed\nto close \\#249's certificate-supply obligation (Part IV). The two open problems, in their\nLean-nearest forms, terminate at structurally the same missing analytic statement, applied to two\ndifferent arithmetic functions (a Mersenne binary-digit count versus Euler's totient).\n\n\\subsection{Summary}\n\nErd\\H{o}s \\#257 is open, in both the universal form (O1) and the single-target half-value form\n(O2). Every route surveyed above either (a) is proved equivalent to O2 and therefore offers no\nreduction, (b) is a genuine strict weakening (O3, O4, O5, O3-supply) whose own supply is\nunproved at cofinal scale, or (c) is certified only on a finite initial segment, however large a\nmargin that segment exhibits. Nothing in this paper closes any of these gaps, and no claim above\nshould be read as progress toward doing so.\n\n\\clearpage\n\\section*{Statements and declarations}\n\n\\paragraph{Artefact and data availability.} The linked Lean declarations,\nfixed toolchain, and library manifest are published in the companion\nrepository. This manuscript provides navigation and exposition rather than\nproof authority.\n\n\\paragraph{Declaration of generative AI use.} \\paperfullaideclaration\n\n\\paragraph{Authority boundary.} Kernel checking establishes that a proposition\nwas proved; it does not authorise the exposition, literature judgements, or any\nclaim that Problem 257 is solved.\n\n\\paragraph{Funding and competing interests.} This work received no external\nfunding. The author declares no competing interests.\n\n\\end{document}\n","dependencies":[],"external_verification_route":"https://raw.githubusercontent.com/wcook04/plectis-lean-erdos249-257/main/paper/erdos257-mersenne-reasoning-surface.tex"},{"record_type":"paper","paper_id":"erdos-269-three-prime-running-lcm","title":"The Three-Prime Running Least Common Multiple","source_repository":"plectis-lean-erdos249-257","source_path":"paper/erdos-269-three-prime-running-lcm.tex","hosted_pdf_path":"papers/erdos-269-three-prime-running-lcm.pdf","licence":"CC-BY-4.0","copyright":"2026 Will Cook","cite_as":"Plectis reader corpus, paper:erdos-269-three-prime-running-lcm","pdf_sha256":"sha256:7d4bf77200d6e87de77fd6a951de1e229642db0a44748b3c7bdc23619242fce9","pdf_byte_count":163018,"pdf_embedded":false,"pdf_omission_reason":"the exact LaTeX is embedded and retrieves better than extracted PDF text; the PDF is a presentation rendering and its hash is recorded for verification","source_embedded":true,"availability":"embedded_exact","embedded_copy_role":"native_manuscript","embedded_from_repository":"plectis-lean-erdos249-257","embedded_copy_path":"paper/erdos-269-three-prime-running-lcm.tex","media_type":"text/x-tex","body_format":"latex","sha256":"sha256:8b374d27bf539e7418d539175fa04b0f72a3a4959f62dbf4952189d97b14ca37","byte_count":68922,"body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% One of the Erd\u0151s Problem Notes: a short standalone treatment of a single\n% problem in the ErdosProblems expansion library.  Build with\n% `tectonic erdos-269-three-prime-running-lcm.tex` or `make`.\n\\documentclass[11pt]{article}\n\\input{problem-note-preamble}\n\\usepackage{longtable}\n\\usepackage{colortbl}\n% Problem-specific source pin: #269's local-window consumer advances\n% independently of the corpus default.\n\\renewcommand{\\commit}{d3adb95d6077a77081f98120599ed4b789b97bcf}\n\\renewcommand{\\commitshort}{d3adb95d6077}\n\\hypersetup{\n  pdftitle={Running least common multiples of smooth numbers: a two-prime transcendence theorem and exact three-prime obstructions},\n  pdfsubject={Erd\u0151s Problem Notes: Problem 269},\n  pdfkeywords={transcendence, least common multiple, smooth numbers, Sturmian words, Hecke--Mahler series, Lean 4},\n}\n\n% The three pure-power coordinates and the running height.\n\\newcommand{\\hgt}{\\operatorname{H}}\n\\newcommand{\\lo}{\\lambda}\n\\newcommand{\\hi}{\\eta}\n\\newcommand{\\Lprefix}{\\operatorname{L}}\n\\newcommand{\\Kernel}{\\operatorname{K}}\n\\newcommand{\\authnone}{\\textcolor{BrickRed!75!black}{\\sffamily\\bfseries No result here}}\n\\newcommand{\\authliterature}{\\textcolor{BurntOrange!85!black}{\\sffamily\\bfseries Literature}}\n\\newcommand{\\authlean}{\\textcolor{MidnightBlue!82!black}{\\sffamily\\bfseries Paper + Lean}}\n\\newcommand{\\authexternal}{\\textcolor{Violet!80!black}{\\sffamily\\bfseries Paper + external theorem}}\n\\newcommand{\\authcompute}{\\textcolor{Violet!80!black}{\\sffamily\\bfseries Direct computation}}\n\n\\runninghead{Erd\\H{o}s \\#269: smooth-number running lcms}\n\\title{Running Least Common Multiples of Smooth Numbers}\n\\subtitle{A two-prime transcendence theorem and exact three-prime obstructions}\n\\author{Will Cook}\n\\date{July 2026}\n\n\\begin{document}\n\\maketitle\n\\noteprovenance\n\n\\begin{abstract}\n\\noindent\nFor distinct primes $p$ and $q$, retain one reciprocal for each distinct value\nof the running least common multiple of the $\\{p,q\\}$-smooth numbers.  We prove\nthat the resulting series is transcendental for every pair $p\\ne q$.  The\nordered jumps form a shift of a lower mechanical Sturmian word.  This gives an\nexact affine expression for the series in terms of a Hecke--Mahler value, to\nwhich a theorem of Bugeaud and Laurent applies.  The result settles the\ntwo-prime de-duplicated series, but not the original series with repetitions.\n\nFor three primes we establish exact structure rather than irrationality.  The\nrunning least common multiple below $x$ is the product of the three largest pure\nprime powers below $x$.  In the case $\\{2,3,5\\}$, compressing its jumps between\nsuccessive powers of two gives precisely the radix alphabet\n$\\{2,6,10,30\\}$.  The smallest nontrivial restriction of the reciprocal kernel\nhas determinant $-1/15$, ruling out a single product of three one-variable\nfactors.  We also give an exact finite height-fibre normal form and a quadratic\nshell-multiplicity bound.  These claims are linked to Lean declarations; the\ntwo-prime transcendence argument uses an external analytic theorem and is not\nformalised here.\n\nThe remaining three-prime route is conditional.  Once a bounded integral carry\nwith the required smooth-factor divisibility is available, a\ndenominator-dependent local-window residue escape forces a contradiction.  We\ncheck the algebraic cancellation and the conditional consumer, and an\ninteger-only scan finds such a window for $106{,}666$ tested denominator/start\npairs.  The scan is finite evidence.  The rationality-to-carry bridge, including\nthe needed divisibility, and an unbounded escape producer are not proved.\nConsequently Erd\\H{o}s Problem~\\#269 remains open.\n\\end{abstract}\n\n\\section{Introduction}\\label{sec:problem}\n\nLet $P$ be a finite set of primes with $|P|\\ge2$, and let\n$a_1<a_2<\\cdots$ enumerate the positive integers all of whose prime factors lie\nin $P$.  Erd\\H{o}s Problem~\\#269 asks whether\n\\[\n \\sum_{n\\ge1}\\frac{1}{[a_1,\\ldots,a_n]}\n\\]\nis irrational, where $[a_1,\\ldots,a_n]$ is the least common\nmultiple~\\cite[p.~65]{erdosgraham1980}\\cite[p.~106]{erdos1988}.  Numbering and\ncurrent reported status follow Bloom's catalogue, whose current record\nreproduces the displayed problem and explicitly warns that its OPEN label is\nthe website owner's present assessment and may omit relevant literature\n\\cite{erdosproblems}.  The original publications, not the catalogue, carry\nthe mathematical claims.  The problem is open for every finite $P$ with\n$|P|\\ge2$.  In the primary 1988 source, Erd\\H{o}s states\nthe infinite-$P$ assertion as a simple exercise and presents persistence for\na finite number of\nprimes greater than one as a probable extension, not as a theorem; the current\nopen status is supplied by the catalogue rather than inferred from that\nconjectural wording.\n\nFor one prime, $a_n=p^{n-1}$ and the series is the rational number $p/(p-1)$.\nFor infinitely many primes, Erd\\H{o}s calls irrationality a simple exercise\n\\cite[p.~106]{erdos1988}.  In a letter dated 1 January 1973 he also stated that\nirrationality follows after repeated running-LCM values are deleted\n\\cite[p.~335]{erdos1974letter}, but the one-page letter gives no proof.\n\nWhen $|P|=2$, that de-duplicated series is transcendental for every pair of\ndistinct primes.  Section~\\ref{sec:two-prime} reads the interlacing pure-power\nchannels as a shifted lower mechanical Sturmian word and reduces the series to\nthe Hecke--Mahler value covered by Bugeaud and Laurent's Theorem~1.1\n\\cite[p.~61, Theorem~1.1]{bugeaudlaurent2023}.  This is an independent argument,\nnot a reconstruction of Erd\\H{o}s's unpublished proof.  It does not cover the\noriginal series, where a running value is repeated for every smooth integer in\nthe corresponding cell, and it does not extend to three primes.\n\nFor the three-prime analysis, fix pairwise distinct primes $p,q,r$.  Call $n$\n\\emph{smooth} when\n$n=p^{i}q^{j}r^{k}$ for some $i,j,k\\ge0$; this is the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{31}{smooth3Val}{smooth lattice\nvalue}.  For $x\\ge1$ put\n\\[\n \\Lprefix(x)=\\lcm\\{\\,n\\le x:\\ n\\ \\text{smooth}\\,\\},\n \\qquad\n \\hgt(x)=p^{\\lfloor\\log_p x\\rfloor}q^{\\lfloor\\log_q x\\rfloor}\n r^{\\lfloor\\log_r x\\rfloor}.\n\\]\nThese are the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{53}{smoothPrefixLcm}{running least\ncommon multiple} and the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{36}{threePrimeHeight}{pure-power\nheight}.  Here $\\lfloor\\log_b x\\rfloor$ is the largest $e$ for which\n$b^e\\le x$.  Since $a_1,\\ldots,a_n$ are exactly the smooth numbers up to\n$a_n$, the original summand is $1/\\Lprefix(a_n)$.  At a smooth point the\nreciprocal kernel is\n\\[\n \\Kernel(i,j,k)=\\frac{1}{\\hgt(p^{i}q^{j}r^{k})},\n\\]\nthe\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{40}{threePrimeKernelQ}{lattice\nkernel}.  We call the positive powers of a base its \\emph{channel}.\n\nAll later formal statements use the elementary identity\n$\\Lprefix(x)=\\hgt(x)$.  It is the finite-$P$ restriction of the classical\nprime-power decomposition of $\\lcm(1,\\ldots,N)$, so no novelty is claimed for\nthe identity itself\n\\cite[Ex.~1.21(a), p.~22]{apostol1976}\\cite[p.~183,\nExercise~6.2.7]{montgomeryvaughan2007}.  The running value is constant on\nlogarithmic cells and changes by one prime at each pure-power jump.  For\n$\\{2,3,5\\}$, grouping those jumps between\nsuccessive powers of two gives the exact block alphabet $\\{2,6,10,30\\}$.\nSeparability fails at the first nontrivial rectangle: the corresponding\n$2\\times2$ minor of the reciprocal kernel is $-1/15$.  We also derive the\nfinite height-fibre normal form and its quadratic shell bound.  These are\nunconditional structural statements; none is an irrationality theorem for a\nthree-prime series.\n\nFor the residue argument, suppose rationality produced a bounded integral carry\ndivisible by the $\\{2,3,5\\}$-smooth part of its denominator.  That factor could\nthen be cancelled, leaving a denominator $B$ coprime to $30$.  A suitable\nconsecutive window would force the reduced carry outside its bound.  Lean checks the\nabsorption-and-cancellation algebra and the implication from the stated\nlocal-window escape condition to nonexistence of the reduced carry.  It does\nnot derive the actual carry or its divisibility from the series, and it does not\nproduce escaping windows cofinally.  The finite scan reported below probes the\nsecond gap only; it cannot fill either one.\n\nThe statement of Problem~\\#269 also appears as an unproved conjecture in the\n\\emph{Formal Conjectures} collection~\\cite{formalconjectures269}.  Its\nNat-indexed normalisation includes the empty-prefix term, and its principal\nassertions end in \\texttt{sorry}.  The declarations linked in this note prove\nauxiliary propositions about the same objects; they do not solve the\nconjecture.  Kova\\v{c} and Tao~\\cite{kovactao2024} treat other Erd\\H{o}s unit\nfraction problems by elementary means, but their arguments are not used here.\nWe give no numerical estimate for the value of the original series.\n\n\\section{The de-duplicated two-prime sum is transcendental}\n\\label{sec:two-prime}\n\nLet $P=\\{p,q\\}$ with $p<q$, and write\n\\[\n L_{p,q}(t)=p^{\\lfloor\\log_p t\\rfloor}q^{\\lfloor\\log_q t\\rfloor}.\n\\]\nThe elementary least-common-multiple argument used later in\nTheorem~\\ref{res:lcm}, with one coordinate omitted, identifies $L_{p,q}(t)$ as\nthe running least common multiple of the $\\{p,q\\}$-smooth numbers up to $t$.\nDefine\n\\[\n \\mathcal D_{p,q}\n =1+\\sum_{t\\in\\{p,p^2,\\ldots\\}\\cup\\{q,q^2,\\ldots\\}}\n       \\frac1{L_{p,q}(t)} .\n\\]\nThus $\\mathcal D_{p,q}$ contains the initial value and exactly one reciprocal\nfor every later distinct running-LCM value.\n\n\\begin{theorem}[two-prime de-duplicated transcendence]\n\\label{res:two-prime-transcendence}\nFor every pair of distinct primes $p,q$, the number $\\mathcal D_{p,q}$ is\ntranscendental.\n\\end{theorem}\n\n\\begin{proof}\nInterchanging the primes if necessary, assume $p<q$, and set\n\\[\n \\theta=\\frac{\\log p}{\\log q},\\qquad x=\\frac1p,\\qquad y=\\frac1q,\\qquad\n m_n=\\lfloor n\\theta\\rfloor,\\qquad\n \\delta_n=m_{n+1}-m_n .\n\\]\nHere $0<\\theta<1$, and $\\theta$ is irrational: a rational relation would give\n$p^b=q^a$ for positive integers $a,b$.  Consequently\n$\\delta_n\\in\\{0,1\\}$.  The word $(\\delta_n)_{n\\ge0}$ is a shift of the lower\nmechanical Sturmian word of slope $\\theta$; the floor-increment formula is all\nthat the proof needs.\n\nThe initial value together with the $p$-channel contributes\n\\[\n A=\\sum_{n\\ge0}x^ny^{m_n}.\n\\]\nThere is a $q$-power strictly between $p^n$ and $p^{n+1}$ exactly when\n$\\delta_n=1$; it is then $q^{m_n+1}$, and its post-jump reciprocal is\n$x^ny^{m_n+1}$.  Hence the $q$-channel contributes\n\\[\n B=\\sum_{n\\ge0}\\delta_nx^ny^{m_n+1},\n \\qquad \\mathcal D_{p,q}=A+B.\n\\]\nAll series here converge absolutely.  Since\n$y^{m_{n+1}}-y^{m_n}=\\delta_n y^{m_n}(y-1)$, shifting the sum for $A$ gives\n\\[\n A-1-xA\n =x(y-1)\\sum_{n\\ge0}\\delta_nx^ny^{m_n}.\n\\]\nIt follows that\n\\begin{equation}\\label{eq:two-prime-affine}\n \\mathcal D_{p,q}\n =\\frac{y-x}{x(y-1)}A-\\frac{y}{x(y-1)}.\n\\end{equation}\n\nIn the notation\n\\[\n F_\\theta(x,y)\n =\\sum_{n\\ge1}\\sum_{k=1}^{\\lfloor n\\theta\\rfloor}x^ny^k\n\\]\nfor the Hecke--Mahler series, a finite geometric sum gives\n\\begin{equation}\\label{eq:hecke-mahler-boundary}\n A=\\frac1{1-x}-\\frac{1-y}{y}F_\\theta(x,y).\n\\end{equation}\nBugeaud and Laurent's Theorem~1.1 states, in particular, that\n$F_\\theta(\\beta,\\alpha)$ is transcendental when $\\theta\\in(0,1)$ is\nirrational, $\\alpha,\\beta$ are nonzero algebraic numbers,\n$|\\beta\\alpha^\\theta|<1$, and $|\\beta|<1$\n\\cite[p.~61, Theorem~1.1]{bugeaudlaurent2023}.  We may take\n$(\\beta,\\alpha)=(x,y)$, because\n\\[\n |xy^\\theta|=\\frac1p\\left(\\frac1q\\right)^{\\log p/\\log q}\n =\\frac1{p^2}<1.\n\\]\nThus $F_\\theta(x,y)$ is transcendental, and\n\\eqref{eq:hecke-mahler-boundary} makes $A$ transcendental.  Finally the\ncoefficient of $A$ in~\\eqref{eq:two-prime-affine} is a nonzero algebraic\nnumber, since $x\\ne y$; therefore $\\mathcal D_{p,q}$ is transcendental.\n\\end{proof}\n\n\\noindent\nThis proves more than the irrationality claimed for the de-duplicated sum in\nErd\\H{o}s's letter, but only when $|P|=2$.  The repeated sum assigns a\nmultiplicity to each running value, and a third prime replaces the single\nSturmian boundary by a genuinely two-dimensional ordering problem; neither is\ncovered by Theorem~\\ref{res:two-prime-transcendence}.\n\n\\statusboundary\n\nThe table records two independent facts.  The \\emph{authority} column says\nwhether the statement is supported by an argument in this paper together with\na linked Lean declaration, by a direct computation, or by no result in this\npaper.  The last column states its logical reach and the nearest boundary.\nThus a conditional implication may be checked in Lean while its hypothesis\nremains open, and the dyadic-window scan is a direct computation over\n$106{,}666$ instances rather than an unbounded theorem.\n\n{\\footnotesize\n\\setlength{\\tabcolsep}{4.5pt}\n\\renewcommand{\\arraystretch}{1.06}\n\\arrayrulecolor{gray!65}\n\\begin{longtable}{@{}\n  >{\\raggedright\\arraybackslash}p{0.33\\linewidth}\n  >{\\raggedright\\arraybackslash}p{0.19\\linewidth}\n  >{\\raggedright\\arraybackslash}p{0.40\\linewidth}@{}}\n\\caption{Authority and logical reach of the statements discussed in this note.}\\\\\n\\toprule\n\\textbf{Statement} & \\textbf{Authority} & \\textbf{Logical reach and nearest boundary} \\\\\n\\midrule\n\\endfirsthead\n\\toprule\n\\textbf{Statement} & \\textbf{Authority} & \\textbf{Logical reach and nearest boundary} \\\\\n\\midrule\n\\endhead\n\\midrule\n\\multicolumn{3}{r}{\\small\\itshape Continued on the next page.}\\\\\n\\endfoot\n\\bottomrule\n\\endlastfoot\n\\rowcolor{BrickRed!6}\n\\multicolumn{3}{@{}l}{\\sffamily\\bfseries\\textcolor{BrickRed!75!black}{A. Open target and prior literature}}\\\\\nIrrationality in any three-prime case & \\authnone & Open. \\\\\nIrrationality after deleting duplicate running-LCM values &\n\\authliterature\\ (Erd\\H{o}s 1974 letter) & Asserted on p.~335, but no proof is supplied there and\nthe letter's argument is not recovered here. \\\\\nTranscendence after deleting duplicate values when $|P|=2$ &\n\\authexternal\\ (Bugeaud--Laurent Theorem~1.1) & Unconditional for every pair of\ndistinct primes; Theorem~\\ref{res:two-prime-transcendence}.  It does not apply\nto repetitions or to three primes. \\\\\n\\addlinespace[2pt]\n\\rowcolor{MidnightBlue!6}\n\\multicolumn{3}{@{}l}{\\sffamily\\bfseries\\textcolor{MidnightBlue!82!black}{B. Unconditional finite structure}}\\\\\n$\\Lprefix(x)=\\hgt(x)$ & \\authlean & Unconditional for distinct primes; Theorem~\\ref{res:lcm}.  The unrestricted identity is classical. \\\\\n$\\hgt(x)\\le x^{3}$ & \\authlean & Unconditional; Proposition~\\ref{res:cube}. \\\\\nConstancy on logarithmic cells & \\authlean & Unconditional; Theorem~\\ref{res:cell}. \\\\\nSingle-coordinate jump ratios & \\authlean & Unconditional; Theorem~\\ref{res:jump}. \\\\\nExactly $3n+1$ jump points & \\authlean & Unconditional; Theorem~\\ref{res:count}. \\\\\nHeight-fibre normal form & \\authlean & Exact finite identity; Theorem~\\ref{res:fibre}. \\\\\nInfinite limit of that expansion & \\authnone & Not proved here; Section~\\ref{sec:fibre}. \\\\\nShell multiplicity $9\\,\\#\\mathcal S\\le(j+3)^{2}$ & \\authlean & Unconditional; Theorem~\\ref{res:shell}. \\\\\n$2\\times2$ kernel restriction has rank two at $\\{2,3,5\\}$ & \\authlean & Unconditional local obstruction; determinant $-1/15$ in Theorem~\\ref{res:rank}. \\\\\nDyadic block radix lies in $\\{2,6,10,30\\}$ & \\authlean & Unconditional; Theorem~\\ref{res:dyadic-alphabet}. \\\\\nCanonical least-positive-residue arithmetic & \\authlean & Unconditional finite lemma; Theorem~\\ref{res:consumer}. \\\\\n\\addlinespace[2pt]\n\\rowcolor{BurntOrange!7}\n\\multicolumn{3}{@{}l}{\\sffamily\\bfseries\\textcolor{BurntOrange!85!black}{C. Conditional bridge}}\\\\\nCarry contradiction assuming one window per reduced denominator & \\authlean & Conditional implication only; Theorem~\\ref{res:windowconsumer}. \\\\\nSmooth-factor absorption and carry cancellation & \\authlean & Conditional core; Theorem~\\ref{res:denominator-reduction}.  Divisibility of the actual carry remains unproved. \\\\\n\\addlinespace[2pt]\n\\rowcolor{Violet!6}\n\\multicolumn{3}{@{}l}{\\sffamily\\bfseries\\textcolor{Violet!80!black}{D. Bounded computation and remaining frontier}}\\\\\nDyadic-window scan for $B\\le1000$ & \\authcompute & $106{,}666$ pairs with no failures; bounded evidence, not a theorem. \\\\\nBridge from the actual summands to $(b_n,m_n,K)$ & \\authnone & Not proved here; Section~\\ref{sec:open}. \\\\\nLocal-window escape for the actual $\\{2,3,5\\}$ word & \\authnone & Open; Problem~\\ref{prob:producer}. \\\\\n\\end{longtable}\n}\n\n\\subsection*{Structure}\nSection~\\ref{sec:two-prime} proves the unconditional transcendence theorem.\nSections~\\ref{sec:lcm} and~\\ref{sec:cells} then identify the three-prime running\nvalue and its jumps.  Section~\\ref{sec:fibre} gives the finite three-prime\nnormal form, Section~\\ref{sec:shell} bounds fibre multiplicity, and\nSection~\\ref{sec:rank} records the obstruction to separating the kernel.\nSection~\\ref{sec:escape} determines the block alphabet, proves the conditional\ncontradiction, and states the remaining unproved hypothesis.\nSection~\\ref{sec:open} collects the questions that remain.  Linked phrases open\nthe corresponding Lean declaration at the pinned source\nrevision~\\commitshort; Theorem~\\ref{res:two-prime-transcendence} is an analytic\nargument using the cited external theorem and is not Lean-formalised here.\n\n\\medskip\n\\noindent\\textbf{Keywords.} irrationality; least common multiple; smooth\nnumbers; lattice sums; Lean~4.\n\\qquad\n\\textbf{MSC 2020.} 11J72 (primary); 11A05, 11N25, 68V20 (secondary).\n\n\\section{The running least common multiple as a product of pure powers}\\label{sec:lcm}\n\nThe smooth numbers up to $x$ are indexed by the exponent triples $(i,j,k)$ with\n$i\\le\\lfloor\\log_p x\\rfloor$, $j\\le\\lfloor\\log_q x\\rfloor$,\n$k\\le\\lfloor\\log_r x\\rfloor$ and $p^{i}q^{j}r^{k}\\le x$: the coordinate bounds\nmake the index set finite, and the last condition is the actual constraint.\nThis is the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{46}{smoothPrefixExponents}{smooth\nprefix index set}.  Both conditions matter: the coordinate box is strictly\nlarger than the prefix, since a product of three large pure powers can exceed\n$x$ while each factor does not.\n\nThe identification below is routine, and its unrestricted analogue is classical,\nas recalled in the introduction; the short proof is given because every later\nstatement is derived from it in the formal development.\n\n\\begin{theorem}[the running least common multiple]\\label{res:lcm}\nLet $p,q,r$ be pairwise distinct primes and $x\\ge1$.  Then\n$\\Lprefix(x)=\\hgt(x)$.\n\\end{theorem}\n\n\\begin{proof}\nFor divisibility in one direction, every smooth $n\\le x$ has exponents bounded\nby the corresponding integer logarithms, so $n\\mid\\hgt(x)$, and hence\n$\\Lprefix(x)\\mid\\hgt(x)$.  For the other, the three pure powers\n$p^{\\lfloor\\log_p x\\rfloor}$, $q^{\\lfloor\\log_q x\\rfloor}$ and\n$r^{\\lfloor\\log_r x\\rfloor}$ are themselves smooth numbers not exceeding $x$,\nso each divides $\\Lprefix(x)$.  Distinct primes have coprime powers, so their\nproduct divides $\\Lprefix(x)$ as well.  The two divisibilities give equality.\n\\end{proof}\n\n\\noindent Formalised as the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{123}{smoothPrefixLcm_eq_threePrimeHeight}{running-lcm\nidentity}, from the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{77}{smoothPrefixLcm_dvd_threePrimeHeight}{divisibility\ninto the height} and the three membership statements for the pure powers,\nnamely the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{84}{pureFirst_mem_smoothPrefixExponents}{first},\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{97}{pureSecond_mem_smoothPrefixExponents}{second},\nand\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{110}{pureThird_mem_smoothPrefixExponents}{third}\npure-power memberships.\n\nThe hypothesis that the primes are pairwise distinct is used exactly once, in\nthe coprimality step, and it is not decorative: without it the three pure\npowers need not have coprime orders and their product need not divide the least\ncommon multiple.  The identity is what makes every later statement about\n$\\Lprefix$ computable from three integer logarithms.\n\nAt $(p,q,r)=(2,3,5)$ it reads\n$\\Lprefix(x)=2^{\\lfloor\\log_2 x\\rfloor}3^{\\lfloor\\log_3 x\\rfloor}\n5^{\\lfloor\\log_5 x\\rfloor}$, whose first ten values are\n\\[\n\\begin{array}{c|cccccccccc}\nx&1&2&3&4&5&6&7&8&9&10\\\\ \\hline\n\\Lprefix(x)&1&2&6&12&60&60&60&120&360&360\n\\end{array}\n\\]\nSo $\\Lprefix(10)=8\\cdot9\\cdot5=360$, which is indeed the least common multiple\nof the smooth numbers $1,2,3,4,5,6,8,9,10$, and $\\Lprefix(6)=4\\cdot3\\cdot5=60$\nrather than $6$: the running value at a smooth cutoff already contains powers of\nthe other two primes that the cutoff itself does not.  These ten values also\nillustrate two of the statements of Section~\\ref{sec:cells}, since the value is\nconstant on $\\{5,6,7\\}$ and on $\\{9,10\\}$, and each change multiplies by a\nsingle prime, by $2$ at $x=2,4,8$, by $3$ at $x=3,9$, and by $5$ at $x=5$.\n\n\\begin{proposition}\\label{res:cube}\nFor every $x\\ge1$, $\\hgt(x)\\le x^{3}$.\n\\end{proposition}\n\n\\begin{proof}\nEach of the three factors is a power of its base not exceeding $x$.\n\\end{proof}\n\n\\noindent Formalised as the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{430}{threePrimeHeight_le_cube}{cubic\nmajorant}, which needs no primality.  The exponent $3$ is the number of\ngenerating primes, and the bound is the reason the reciprocal kernel is\ncomparable to $x^{-3}$ rather than to $x^{-1}$.\n\n\\section{Constancy on logarithmic cells and the jump points}\\label{sec:cells}\n\nBy Theorem~\\ref{res:lcm} the running value depends on $x$ only through the\nthree integer logarithms.  It is therefore constant wherever none of them\nchanges and moves only where one of them does, and the next definition names\nthe sets on which they are all constant, so that the next two theorems can say\nwhere the value stands still and by what factor it moves.\n\nSay that $x$ and $y$ lie in the same \\emph{logarithmic cell} when\n$\\lfloor\\log_b x\\rfloor=\\lfloor\\log_b y\\rfloor$ for each of $b=p,q,r$: the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{163}{SameThreePrimeLogCell}{cell\nrelation}.\n\n\\begin{theorem}[constancy on cells]\\label{res:cell}\nIf $x,y\\ge1$ lie in the same logarithmic cell, then\n$\\Lprefix(x)=\\Lprefix(y)$.  The same holds for the kernel at two smooth points\nwhose values lie in one cell.\n\\end{theorem}\n\n\\begin{proof}\nImmediate from Theorem~\\ref{res:lcm}, since $\\hgt$ depends on $x$ only\nthrough the three integer logarithms.\n\\end{proof}\n\n\\noindent Formalised as the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{179}{smoothPrefixLcm_eq_of_sameLogCell}{cell\nconstancy of the running value}, the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{169}{threePrimeHeight_eq_of_sameLogCell}{cell\nconstancy of the height}, and the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{191}{threePrimeKernelQ_eq_of_sameLogCell}{cell\nconstancy of the kernel}.  The running value therefore changes only when one of\nthe three logarithms changes, and the next theorem says by exactly how much.\n\n\\begin{theorem}[single-coordinate jump ratios]\\label{res:jump}\nLet $x,y\\ge1$.  If $\\lfloor\\log_p y\\rfloor=\\lfloor\\log_p x\\rfloor+1$ while the\nother two logarithms agree, then $\\Lprefix(y)=p\\,\\Lprefix(x)$; and similarly\nwith $q$ or $r$ in place of $p$.\n\\end{theorem}\n\n\\begin{proof}\nBy Theorem~\\ref{res:lcm} both sides are heights, and one factor of the height\ngains one in its exponent while the others are unchanged.\n\\end{proof}\n\n\\noindent Formalised as the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{326}{smoothPrefixLcm_firstLogStep}{first},\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{339}{smoothPrefixLcm_secondLogStep}{second},\nand\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{352}{smoothPrefixLcm_thirdLogStep}{third}\ncoordinate steps, over the corresponding statements for the height alone, which\nneed no primality: the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{294}{threePrimeHeight_firstLogStep}{first\nheight step}, the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{305}{threePrimeHeight_secondLogStep}{second},\nand the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{316}{threePrimeHeight_thirdLogStep}{third}.\n\nThe jump points are therefore the pure prime powers, and they do not collide\nacross channels.\n\n\\begin{theorem}[jump count]\\label{res:count}\nLet $n\\ge0$.  The set of the first $n$ positive powers of $p$, of $q$, and of\n$r$ has exactly $3n$ elements, and adjoining the common origin $1$ gives\nexactly $3n+1$.\n\\end{theorem}\n\n\\begin{proof}\nThe count is routine.  Within one channel the powers $b,b^{2},\\ldots,b^{n}$ are distinct because\n$b\\ge2$.  Across two channels a common value would be a positive power of two\ndistinct primes, which unique factorisation forbids.  Finally $1$ is not a\npositive power of any prime.\n\\end{proof}\n\n\\noindent Formalised as the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{249}{threePrimePositiveJumpSet_card}{positive\njump count} and the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{278}{threePrimeJumpSetWithOrigin_card}{jump\ncount with the origin}, over the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{208}{positivePrimePowers_card}{channel\ncardinality}, the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{229}{positivePrimePowers_disjoint}{channel\ndisjointness}, and the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{219}{one_not_mem_positivePrimePowers}{exclusion\nof the origin}; the channels themselves are the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{204}{positivePrimePowers}{positive\npower sets}.  The exponent $0$ is omitted from each channel because it is the\nshared initial value, which is why the origin is counted once rather than three\ntimes.\n\nTwo channels never meet, so at each positive pure power exactly one of the three\nlogarithms advances, and by Theorem~\\ref{res:jump} the running value is\nmultiplied there by the prime of that channel.  Reading those multipliers in\nincreasing order of the pure powers gives the \\emph{jump word} of $\\Lprefix$:\none letter from $\\{p,q,r\\}$ for each positive pure power, recording which prime\nthe value is multiplied by at that point.  At $\\{2,3,5\\}$ the pure powers in\nincreasing order are $2,3,4,5,8,9,16,25,27,32,\\ldots$, so the jump word begins\n\\[\n 2,\\;3,2,\\;5,2,\\;3,2,\\;5,3,2,\\;\\ldots\n\\]\nwhere the grouping is the one used in Section~\\ref{sec:escape}: each group ends\nat a power of two.\n\n\\section{The height-fibre normal form}\\label{sec:fibre}\n\nGrouping a lattice sum by the value of the running least common multiple turns\nit into a sum over heights whose coefficients are multiplicities.  We prove\nthis exactly, on a finite rectangular box.\n\nWrite $\\mathcal B(h_p,h_q,h_r)$ for the box of exponent triples with $i\\le h_p$,\n$j\\le h_q$, $k\\le h_r$, the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{368}{smoothExponentBox}{exponent\nbox}, and, for a height $H$, write $F(H)$ for the set of points of the box whose\nrunning height $\\hgt(p^{i}q^{j}r^{k})$ equals $H$, the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{377}{smoothHeightFiber}{height\nfibre}.  Distinct lattice points can carry the same height, and the coefficients\nof the normal form are exactly the sizes of these fibres.\n\n\\begin{theorem}[finite normal form]\\label{res:fibre}\nFor every box $\\mathcal B=\\mathcal B(h_p,h_q,h_r)$,\n\\[\n \\sum_{(i,j,k)\\in\\mathcal B}\\Kernel(i,j,k)\n =\\sum_{H}\\frac{\\#F(H)}{H},\n\\]\nthe outer sum ranging over the heights actually attained on $\\mathcal B$.\n\\end{theorem}\n\n\\begin{proof}\nThe identity is a regrouping.  Partition $\\mathcal B$ into the fibres of the\nheight map.  On $F(H)$ every summand is\n$1/H$ by definition of the kernel, so the fibre contributes $\\#F(H)/H$.\n\\end{proof}\n\n\\noindent Formalised as the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{406}{finiteSmoothKernelSum_groupedByHeight}{height-fibre\nnormal form}, over the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{384}{smoothHeightFiber_kernel_sum}{fibre\nsum}, with the height of a lattice point given by the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{373}{smoothPointHeight}{point\nheight}.\n\nFor a worked instance take $(p,q,r)=(2,3,5)$ and the box $\\mathcal B(1,1,1)$,\nwhose eight points carry the smooth values $1,2,3,5,6,10,15,30$ and the heights\n\\[\n\\begin{array}{c|cccccccc}\np^{i}q^{j}r^{k}&1&2&3&5&6&10&15&30\\\\ \\hline\n\\hgt&1&2&6&60&60&360&360&10800\n\\end{array}\n\\]\nSix heights occur, two of them twice: the points $5$ and $6$ share the height\n$60$, and $10$ and $15$ share the height $360$.  Theorem~\\ref{res:fibre} here\nreads\n\\[\n 1+\\tfrac12+\\tfrac16+\\tfrac1{60}+\\tfrac1{60}+\\tfrac1{360}+\\tfrac1{360}\n +\\tfrac1{10800}\n =1+\\tfrac12+\\tfrac16+\\tfrac2{60}+\\tfrac2{360}+\\tfrac1{10800}\n =\\tfrac{18421}{10800}.\n\\]\nThe two coefficients $2$ carry the whole content of the regrouping on this box;\nwhere the heights are pairwise distinct the identity is a relabelling and\nnothing more.\n\nTogether with Theorem~\\ref{res:cell} this is the finite core of the ordered\nprime-power jump expansion: the value is constant on cells, the cells are\nindexed by heights, and the coefficient of a height is the number of lattice\npoints it collects.  The passage to the infinite sum, and the explicit ordering\nof the pure powers that would make the expansion a series in the jumps, are not\nproved here.  Theorem~\\ref{res:fibre} is an identity between two finite sums,\nand it is stated over the full box rather than the smooth prefix, so it is not\na statement about $\\Lprefix$ at a cutoff.\n\nThe same module records a one-step map $\\tau(b,d,s)=b(s-d)$, the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{487}{tailStateStep}{variable-base\ntail step}, with the rewriting\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{490}{tailStateStep_eq}{that names its\nexpanded form}.  No orbit of this map is analysed.\n\n\\section{A quadratic bound for smooth exponent shells}\\label{sec:shell}\n\nA tail estimate needs to know how many lattice points can share a short\nmultiplicative interval.  Fix a box $\\mathcal B(h_p,h_q,h_r)$ and an interval\n$[\\lo,\\hi)$, and write $\\mathcal S$ for the set of exponent triples of that box\nwhose smooth value lies in that interval, the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{500}{smoothExponentShell}{smooth\nexponent shell}.  Every bound below assumes the interval short in the\nmultiplicative sense, meaning that $\\hi$ is at most a stated multiple of $\\lo$.\n\nInformally, the next lemma says that an interval whose right endpoint is at\nmost $b$ times its left endpoint contains at most one of the numbers $b^{a}w$.\n\n\\begin{lemma}[uniqueness in a short interval]\\label{res:short}\nLet $b\\ge1$ and suppose $\\hi\\le b\\,\\lo$.  If $b^{a}w$ and $b^{a'}w$ both lie in\n$[\\lo,\\hi)$, then $a=a'$.\n\\end{lemma}\n\n\\begin{proof}\nIf $a<a'$ then $\\hi\\le b\\,\\lo\\le b\\cdot b^{a}w=b^{a+1}w\\le b^{a'}w<\\hi$, which\nis impossible; the case $a>a'$ is symmetric.\n\\end{proof}\n\n\\noindent Formalised as the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{509}{exponent_unique_in_short_interval}{short-interval\nuniqueness}.  The hypothesis is that the interval has multiplicative width at\nmost $b$.\n\n\\begin{proposition}\\label{res:drop}\nLet $\\mathcal S$ be the shell of the box $\\mathcal B(h_p,h_q,h_r)$ in\n$[\\lo,\\hi)$.  If $\\hi\\le r\\,\\lo$ then $\\#\\mathcal S\\le(h_p+1)(h_q+1)$; if\n$\\hi\\le p\\,\\lo$ then $\\#\\mathcal S\\le(h_q+1)(h_r+1)$.\n\\end{proposition}\n\n\\begin{proof}\nSuppose $\\hi\\le r\\,\\lo$.  If two triples of $\\mathcal S$ agree in their first\ntwo coordinates, Lemma~\\ref{res:short} applied with $b=r$ and $w=p^{i}q^{j}$\nforces their third coordinates to agree as well.  So the projection forgetting\nthe third coordinate is injective on $\\mathcal S$, and its image lies in a\nrectangle with $(h_p+1)(h_q+1)$ points.  The case $\\hi\\le p\\,\\lo$ is the same\nwith the first coordinate projected away.\n\\end{proof}\n\n\\noindent Formalised as the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{585}{smoothExponentShell_card_le_dropThird}{third-coordinate\nprojection} and the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{543}{smoothExponentShell_card_le_dropFirst}{first-coordinate\nprojection}.\n\n\\begin{theorem}[quadratic multiplicity bound]\\label{res:shell}\nLet $\\mathcal S$ be the shell of the box $\\mathcal B(h_p,h_q,h_r)$ in\n$[\\lo,\\hi)$.  Suppose $\\hi\\le r\\,\\lo$ and $h_p\\le h_q\\le h_r$ with\n$h_p+h_q+h_r=j$.  Then\n\\[\n 9\\,\\#\\mathcal S\\le(j+3)^{2}.\n\\]\n\\end{theorem}\n\n\\begin{proof}\nBy Proposition~\\ref{res:drop}, $\\#\\mathcal S\\le(h_p+1)(h_q+1)$; under the\nsorting hypothesis the two surviving coordinates are the two smallest.  It\ntherefore\nsuffices to prove that $a\\le b\\le c$ with $a+b+c=j$ gives\n$9(a+1)(b+1)\\le(j+3)^{2}$.  From $a\\le b\\le c$ we get $a+2b\\le j$, so it is\nenough that $9(a+1)(b+1)\\le(a+2b+3)^{2}$; writing $b=a+d$ with $d\\ge0$, the\ndifference of the two sides is $d(3a+4d+3)\\ge0$.\n\\end{proof}\n\n\\noindent Formalised as the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{646}{smoothExponentShell_card_quadratic}{quadratic\nshell bound}, over the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{629}{sorted_pair_quadratic}{sorted\nquadratic estimate}.  The constant $9$ is the square of the number of\ngenerating primes and appears because the bound is the arithmetic--geometric\ncomparison for a sum of three sorted coordinates; sorting is a hypothesis, not\na normalisation, since the shell itself is not symmetric in the three bases.\nThe last inequality of the proof is an equality when $h_p=h_q=h_r$, both sides\nthen being $9(h_p+1)^{2}$, so no constant larger than $9$ survives that step.\n\nThe bound is uniform in $\\lo$ and $\\hi$ subject to the width condition, and it\nis stated for the actual filtered shell rather than for a lattice model of it.\nIt is an input to a tail estimate and is not itself one: no series is bounded\nhere.  The estimate is elementary and uses no analytic input on the\ndistribution of smooth numbers, only the projection of\nProposition~\\ref{res:drop}.\n\n\\section{Non-separability of the three-prime kernel}\\label{sec:rank}\n\nOne might hope to write the three-prime kernel as $f(i)g(j)h(k)$ and so reduce\nthe problem to one-dimensional criteria.  Such a factorisation would force the\nvalue at $(1,1,0)$ to be determined by the values at $(0,0,0)$, $(1,0,0)$ and\n$(0,1,0)$, since the four values would then satisfy\n$\\Kernel(0,0,0)\\,\\Kernel(1,1,0)=\\Kernel(1,0,0)\\,\\Kernel(0,1,0)$.\nTheorem~\\ref{res:rank} computes those four values at $\\{2,3,5\\}$ and finds that\nthey do not: the failure is exact, and it occurs on the smallest rectangle on\nwhich it could occur.\n\n\\begin{theorem}[non-separability at $\\{2,3,5\\}$]\\label{res:rank}\nWith $(p,q,r)=(2,3,5)$,\n\\[\n \\Kernel(0,0,0)\\,\\Kernel(1,1,0)\n -\\Kernel(1,0,0)\\,\\Kernel(0,1,0)\n =-\\tfrac1{15}\\ne0.\n\\]\nConsequently this two-by-two kernel restriction has rank two over\n$\\mathbb Q$.\n\\end{theorem}\n\n\\begin{proof}\nThe four values are $\\Kernel(0,0,0)=1$, $\\Kernel(1,0,0)=1/2$,\n$\\Kernel(0,1,0)=1/6$ and $\\Kernel(1,1,0)=1/60$, computed from\n$\\hgt(1)=1$, $\\hgt(2)=2$, $\\hgt(3)=2\\cdot3=6$ and\n$\\hgt(6)=4\\cdot3\\cdot5=60$.  Hence the determinant is\n$1/60-1/12=-1/15$, and a two-by-two matrix with nonzero determinant has rank\ntwo.\n\\end{proof}\n\n\\noindent Formalised as the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{479}{kernel_235_not_rankOne}{non-separation\nwitness}, over the four exact values, the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{443}{kernel_235_origin}{origin},\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{450}{kernel_235_two}{value at two},\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{458}{kernel_235_three}{value at\nthree}, and\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{467}{kernel_235_six}{value at six}.\nThe exact determinant calculation is the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{721}{kernel_235_minor_eq_neg_one_fifteen}{rank-two\ncertificate}.\nThe height at $6$ is $60$ rather than $6$ because the maximal pure powers below\n$6$ are $4$, $3$ and $5$: the running least common multiple at a smooth cutoff\nsees powers of the other primes that the cutoff itself does not contain.  That\nis the mechanism behind the non-separation.\n\nA nonzero two-by-two minor rules out writing the kernel as\n$f(i)g(j)h(k)$ on this box, so no argument may assume that single product form\nhere.  It rules out nothing further: a rank-two matrix is itself a sum of two\nrank-one matrices, so sums of separable terms, higher-rank decompositions,\nseparable majorants, changes of variable, and one-dimensional estimates applied\nafter a decomposition all remain available.  It gives no\nrank lower bound beyond two, and it is not an independence or irrationality\nstatement.\n\n\\section{Dyadic blocks and a conditional carry contradiction}\\label{sec:escape}\n\nThe section makes three moves and leaves one hypothesis standing.  We first\ncompress the jump word of Section~\\ref{sec:cells} into the blocks cut out by\nconsecutive powers of two, and show that the multiplier of a block takes only\nfour values (Theorem~\\ref{res:dyadic-alphabet}).  We then record what can be\ncancelled from a hypothetical denominator, and exactly where that cancellation\nis still conditional (Theorem~\\ref{res:denominator-reduction}).  Finally we\nleave the smooth numbers behind and argue with integer sequences alone: for a\nmultiplier coprime to $30$, no positive sequence obeying the cleared recurrence\ncan stay inside its bound once one window has the required residue escape\n(Theorem~\\ref{res:windowconsumer}).  The stronger cofinal condition introduced\nbelow is a natural way to supply such a window.  Neither producer is proved\nhere, and the closing subsection reports a finite computation rather than an\nunbounded theorem.\n\n\\subsection{The four-element block alphabet}\n\nTake $(p,q,r)=(2,3,5)$ and compress the jump word of Section~\\ref{sec:cells}\nbetween consecutive powers of two: a block starts just\nafter $2^a$, includes every pure $3$- or $5$-power strictly between $2^a$ and\n$2^{a+1}$, and ends with the jump at $2^{a+1}$.  A channel cannot occur twice\ninside one block.  Indeed, if\n\\[\n  2^a<b^e,b^f<2^{a+1}\\qquad(b\\ge2),\n\\]\nthen the ratio between the interval endpoints is $2\\le b$, so strict\nmonotonicity of the powers forces $e=f$.  This is the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{668}{dyadicInternalPower_exponent_unique}{checked\ninternal-power uniqueness lemma}.  Each block therefore contributes at most one\nletter from each of the $3$- and $5$-channels, and exactly one letter $2$ at its\nright end.\n\nLet $\\beta_a$, the \\emph{block radix}, be the product of the terminal dyadic\nfactor $2$, a factor $3$\nwhen the block contains an internal $3$-power, and a factor $5$ when it\ncontains an internal $5$-power; equivalently, $\\beta_a$ is the product of the\nletters of the jump word lying in block $a$.\n\n\\begin{theorem}[the dyadic block alphabet]\\label{res:dyadic-alphabet}\nWith $(p,q,r)=(2,3,5)$ and $\\beta_a$ as above, for every $a$,\n\\[\n  \\beta_a\\in\\{2,6,10,30\\};\n  \\qquad\\text{in particular}\\qquad 2\\le\\beta_a\\le30.\n\\]\n\\end{theorem}\n\n\\begin{proof}\nInternal-power uniqueness leaves two independent yes/no choices, one for the\n$3$-channel and one for the $5$-channel.  Multiplying the terminal factor $2$\nby the selected channel factors gives exactly the displayed four cases.\n\\end{proof}\n\nAll four letters already occur among the first five of the six blocks\ntabulated below:\n\\[\n\\begin{array}{c|c|c|c|c}\na & (2^{a},2^{a+1}) & \\text{internal }3\\text{-power} & \\text{internal }5\\text{-power} & \\beta_a\\\\ \\hline\n0 & (1,2)   & \\text{none} & \\text{none} & 2\\\\\n1 & (2,4)   & 3           & \\text{none} & 6\\\\\n2 & (4,8)   & \\text{none} & 5           & 10\\\\\n3 & (8,16)  & 9           & \\text{none} & 6\\\\\n4 & (16,32) & 27          & 25          & 30\\\\\n5 & (32,64) & \\text{none} & \\text{none} & 2\n\\end{array}\n\\]\nBlock $4$ is the only one of these six carrying an internal power in both\nchannels, and block $5$ carries neither, so its radix falls back to the\nterminal factor alone.  Multiplying the radices along a run of blocks gives the\nproduct of the jump-word letters over that run: for instance\n$\\beta_1\\beta_2=60$ is the product of the four multipliers at\n$3,4,5,8$.\n\nThe definition is the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{688}{dyadicBlockBase235}{dyadic\nblock base}; Lean checks both the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{699}{dyadicBlockBase235_cases}{exact\nfour-case alphabet} and the\n\\lword{Erdos269/ThreePrimeRunningLcm.lean}{711}{dyadicBlockBase235_mem_interval}{bounded-radix\nconsequence}.  The radix word is therefore constrained to four values, and no\ngrowth hypothesis on it is needed.  What is not yet checked in Lean is the\nidentification of the corresponding \\emph{block digit} $m_a$, the integer that\nthe forcing word attaches to block $a$, with the multiplicity forcing in the\noriginal series; that identification is the first obligation of\nSection~\\ref{sec:open}, and the block digits are not defined in this note.\n\n\\subsection{The denominator reduction and its boundary}\n\nLet $D$ be the reduced denominator of a hypothetical rational value, and write\n\\[\n  D=D_{\\mathrm{sm}}B,\\qquad\n  D_{\\mathrm{sm}}=2^{u}3^{v}5^{w},\\qquad \\gcd(B,30)=1.\n\\]\nThe coprimality condition below is therefore intended as the endpoint of a\nreduction from an arbitrary $D$, not as a restriction on which rational values\nare being considered.\n\nThe argument runs on a sequence of integers $c_n$, one for each step, called\nthe \\emph{carry states}; they satisfy a recurrence\n$c_{n+1}=b_nc_n-Dm_n$ of the shape displayed below, driven by a radix word\n$b_n$ and a forcing word $m_n$.  The name is meant to suggest the integer left\nafter clearing $D$ from the $n$-th tail of the series.  Supplying that reading\nfor the actual series, and with it the divisibility used in the next paragraph,\nis the unproved identification of Section~\\ref{sec:open}; nothing in this\nsubsection or the next depends on the reading, only on the recurrence.\n\nEvery fixed $\\{2,3,5\\}$-smooth factor divides the\nrunning height once the cutoff reaches that factor\n(\\lword{Erdos269/RestrictedFloorSum.lean}{513}{smooth3Val_dvd_threePrimeHeight_of_le}{checked\nabsorption}).  If the denominator-cleared carry states $c_n$ share the absorbed\nfactor, so that $c_n=D_{\\mathrm{sm}}d_n$, Lean cancels it from\n\\[\n  c_{n+1}=b_nc_n-D_{\\mathrm{sm}}B\\,m_n\n\\]\nand obtains\n\\[\n  d_{n+1}=b_nd_n-Bm_n\n\\]\n(\\lword{Erdos269/RestrictedFloorSum.lean}{546}{integralCarry_cancel_commonFactor}{checked\ncancellation}).  Positivity and the sharp denominator-dependent upper bound\ndescend through the same positive factor\n(\\lword{Erdos269/RestrictedFloorSum.lean}{566}{reducedCarry_pos_le_of_commonFactor}{checked\nbound transfer}), and the reduced carry inherits the exact window identity\n(\\lword{Erdos269/RestrictedFloorSum.lean}{577}{reducedIntegralCarry_window}{checked\nwindow transfer}).\n\nInformally: provided the smooth part $D_{\\mathrm{sm}}$ divides every carry\nstate, it can be divided out of the whole system, leaving the same four\nstatements with a multiplier coprime to $30$ in place of $D$.  That proviso is\nthe hypothesis of the theorem, and it is not proved here.\n\n\\begin{theorem}[conditional denominator reduction]\\label{res:denominator-reduction}\nIf the actual denominator-cleared carry for $D=D_{\\mathrm{sm}}B$ has the\ncommon-factor form $c_n=D_{\\mathrm{sm}}d_n$ with $D_{\\mathrm{sm}}>0$, then its\nrecurrence, positivity, bound, and window identity reduce to the same statements\nwith multiplier $B$, where $\\gcd(B,30)=1$.\n\\end{theorem}\n\n\\noindent The theorem is conditional at exactly one point: height absorption\ndoes not by itself prove that the \\emph{carry state} is divisible by\n$D_{\\mathrm{sm}}$.  That divisibility must come from the still-unproved\nrationality-to-carry identification for the actual series.  The phrase\n``$B$ coprime to $30$'' below is valid only downstream of this bridge.\n\n\\subsection{The window recurrence and the residue contradiction}\n\nThe statements of this subsection are about integer sequences: they do not\nrefer to $\\Lprefix$ or to the smooth numbers, and the identification of those\nsequences with the block data of the preceding subsection is the obligation\njust recorded.\n\nCall a pair $(\\ell,h)$ with $h>0$ a \\emph{window}: the stretch of $h$\nconsecutive steps beginning at index $\\ell$.  To locate a carry at the end of a\nwindow without performing any division, one needs the multiplier and the\nforcing accumulated across the window, and those are the two sequences defined\nnext.  For an integer radix word $b_n$ and forcing word $m_n$, define\n\\[\n\\begin{aligned}\n W_{\\ell,0}&=1,\n&W_{\\ell,h+1}&=b_{\\ell+h}W_{\\ell,h},\\\\\n F_{\\ell,0}&=0,\n&F_{\\ell,h+1}&=b_{\\ell+h}F_{\\ell,h}+m_{\\ell+h}.\n\\end{aligned}\n\\]\nThese are the\n\\lword{Erdos269/RestrictedFloorSum.lean}{417}{windowBase}{window base} and\n\\lword{Erdos269/RestrictedFloorSum.lean}{422}{windowForcing}{window forcing}.\nIf an integral carry satisfies\n\\[\n d_{n+1}=b_n d_n-Bm_n ,\n\\]\nthen induction gives the exact division-free identity\n\\[\n d_{\\ell+h}=W_{\\ell,h}d_\\ell-BF_{\\ell,h};\n\\]\nthis is the\n\\lword{Erdos269/RestrictedFloorSum.lean}{480}{integralCarry_window}{checked\nwindow identity}.\n\nFor $C>0$, let $\\operatorname{lpr}_C(x)\\in\\{1,\\ldots,C\\}$ be the least\npositive representative of $x\\bmod C$, with a zero residue represented by\n$C$.  This is the\n\\lword{Erdos269/ResidueEscape.lean}{19}{leastPositiveResidue}{canonical\nrepresentative}.  Lean checks both its\n\\lword{Erdos269/ResidueEscape.lean}{24}{leastPositiveResidue_pos_le}{positive\nrange} and its\n\\lword{Erdos269/ResidueEscape.lean}{45}{leastPositiveResidue_modEq}{congruence\nto the source integer}.  This convention matters: replacing\n$\\operatorname{lpr}_C(x)$ by $|x|$ would not be a modular statement.\n\nLet $K(B,n)$ be a bound on the reduced carry at denominator $B$ and step $n$.\nIt enters as a parameter: the statements below hold for whichever function $K$\nis supplied, and deriving the correct one for the actual series is part of the\nidentification of Section~\\ref{sec:open}.  Say that \\emph{local-window escape}\nholds when, for every $B>0$ coprime to $30$, there are $\\ell$ and $h>0$ such\nthat\n\\[\n C_{\\ell,h}:=|W_{\\ell,h}|>0\n \\quad\\text{and}\\quad\n K(B,\\ell+h)<\n \\operatorname{lpr}_{C_{\\ell,h}}(-BF_{\\ell,h}).\n\\tag{E}\\label{eq:escape}\n\\]\nBoth the quantifier over $B$ and the dependence of $K$ on $B$ are part of the\nstatement.  The exact minimal predicate is the\n\\lword{Erdos269/RestrictedFloorSum.lean}{652}{LocalWindowEscape}{local-window\nescape condition}.  The stronger \\emph{cofinal} version asks for such a window\nafter every lower bound $\\ell_0$; this is the\n\\lword{Erdos269/RestrictedFloorSum.lean}{594}{CofinalLocalWindowEscape}{cofinal\nlocal-window escape condition}.  Lean checks that the cofinal version implies\nthe minimal one by taking $\\ell_0=0$.\n\nThe next statement is the finite core of the argument.  Informally, it says\nthat a positive integer of size at most $K$ cannot be congruent modulo $C$ to a\nnumber whose canonical positive residue exceeds $K$.  The key point is the\nconvention just fixed: because $\\operatorname{lpr}_C$ represents a vanishing\nresidue by $C$ and not by $0$, the proof has to treat that case separately, and\nthe two branches conclude for different reasons.\n\n\\begin{theorem}[the finite residue contradiction]\\label{res:consumer}\nLet $C>0$, $c>0$, and $|c|\\le K$.  If\n$c\\equiv x\\pmod C$ and $K<\\operatorname{lpr}_C(x)$, then the hypotheses are\ncontradictory.\n\\end{theorem}\n\n\\begin{proof}\nThe canonical representative lies in $\\{1,\\ldots,C\\}$.  The inequalities put\n$c$ strictly between $0$ and $C$.  If $x\\equiv0\\pmod C$, its positive\nrepresentative is $C$ while $c\\bmod C=c\\ne0$.  Otherwise both $c$ and\n$\\operatorname{lpr}_C(x)$ are their own residues and congruence makes them\nequal, contradicting $|c|\\le K<\\operatorname{lpr}_C(x)$.\n\\end{proof}\n\n\\noindent The natural-state version is the\n\\lword{Erdos269/ResidueEscape.lean}{71}{no_bounded_positive_state_of_residue_escape}{finite\ncontradiction}; the integer carry version is the\n\\lword{Erdos269/ResidueEscape.lean}{104}{no_bounded_positive_int_state_of_leastPositiveResidue}{integer\nform of the contradiction}.\n\n\\begin{theorem}[a conditional contradiction for bounded carries]\\label{res:windowconsumer}\nLet $(b_n)$, $(m_n)$ and $K$ be the radix word, the forcing word and the bound\nof this subsection, and assume local-window escape.  Fix $B>0$ coprime to\n$30$.  There is no integral sequence $d_n$ satisfying simultaneously\n\\[\n d_{n+1}=b_nd_n-Bm_n,\\qquad d_n>0,\\qquad\n |d_n|\\le K(B,n)\\quad(n\\ge0).\n\\]\n\\end{theorem}\n\n\\begin{proof}\nChoose the window supplied for $B$.  The checked window identity\ngives\n\\[\n d_{\\ell+h}\\equiv -BF_{\\ell,h}\\pmod{|W_{\\ell,h}|}.\n\\]\nThe endpoint state is positive and at most $K(B,\\ell+h)$, whereas the\ncanonical positive residue of the right-hand side is larger than this bound.\nTheorem~\\ref{res:consumer} is the contradiction.\n\\end{proof}\n\n\\noindent This is formalised as the\n\\lword{Erdos269/RestrictedFloorSum.lean}{677}{no_positive_reducedCarry_of_localWindowEscape}{minimal\nreduced-carry extinction theorem}.  The\n\\lword{Erdos269/RestrictedFloorSum.lean}{666}{localWindowEscape_of_cofinalLocalWindowEscape}{cofinal-to-local\nimplication} recovers the earlier cofinal formulation.  Coprimality with $30$\nis used to select a window; once a window has been chosen, the finite\ncontradiction does not use it.  The formalisation carries the edge cases:\n$|W_{\\ell,h}|=0$ is excluded, a zero residue is represented by the full\nmodulus, and positivity prevents the endpoint carry from being zero.\n\n\\subsection{A finite check of the escape condition}\n\nA \\emph{certificate} here is a finite tuple of integers recording one instance\nof the inequality in~\\eqref{eq:escape} for the data the checker constructs.  It\nexists so that a reader can recheck the instance in a line, without running the\nchecker and without touching the infinite part of the argument.  The\ninteger-only\n\\href{\\sourceurl/scripts/check_erdos269_dyadic_windows.py}{dyadic-window\nchecker} constructs the ordered pure-power jumps, the block bases, and the\nblock digits from exact multiplicity counts.  It reproduces the following\ncertificates; the columns are denominator $B$, dyadic start $a$, window length\n$h$, endpoint jump index $n$, window base $W$, forcing $F$, least positive\nresidue $R=\\operatorname{lpr}_{W}(-BF)$, and short bound $K$.\n\\[\n\\begin{array}{c|c|c|c|r|r|r|r}\nB&a&h&n&W&F&R&K\\\\ \\hline\n1&1&2&4&60&47&13&9\\\\\n7&1&3&6&360&289&137&95\\\\\n16&1&4&9&10800&8735&640&352\n\\end{array}\n\\]\nThe first row reads as follows.  The window starts at $a=1$ and has length $2$,\nso its base is the product of the two block radices,\n$W=\\beta_1\\beta_2=6\\cdot10=60$; the forcing accumulated over the window is\n$F=47$; and $\\operatorname{lpr}_{60}(-47)=13$, since $-47+60=13$, which exceeds\nthe bound $K=9$.  The other two rows are read the same way, with\n$W=\\beta_1\\beta_2\\beta_3=360$ and $W=\\beta_1\\beta_2\\beta_3\\beta_4=10800$.\nThe third row lies outside the domain of the escape condition, since\n$\\gcd(16,30)=2$ while both~\\eqref{eq:escape} and\nTheorem~\\ref{res:windowconsumer} quantify only over $B>0$ coprime to $30$.  It\nis displayed to illustrate the window arithmetic at greater depth, not as an\ninstance of the escape condition.\n\nA fresh scan over every $B\\le1000$ coprime to $30$ and every\n$100\\le a\\le500$ tested $106{,}666$ pairs.  In every case a window of length\nat most $18$ made both $W>K$ and\n$\\operatorname{lpr}_{W}(-BF)>K$; the largest first successful length was\n$14$.  The computation uses integers only and is reproducible from the pinned\nchecker.  Neither the scan nor the three displayed certificates proves escape\nfor unbounded $B$ or for cofinally many starts.\n\n\\section{Complements and further questions}\\label{sec:open}\n\nTheorem~\\ref{res:two-prime-transcendence} closes the de-duplicated question\nfor two primes, and does so at the stronger level of transcendence.  The\noriginal series with repetitions remains open even there, while the\nthree-prime de-duplicated series is outside the one-dimensional Hecke--Mahler\nreduction used in that theorem.\n\nTwo statements would be needed to turn Section~\\ref{sec:escape} into an\nirrationality proof at $\\{2,3,5\\}$, and they are of different kinds.  The first\nis the complete problem-specific carry identification.  Starting with an\narbitrary reduced denominator $D=D_{\\mathrm{sm}}B$, it must identify the\nordered jump word and its block digits $m_a$, derive the bound $K(B,a)$, and\nprove that the denominator-cleared carry is divisible by\n$D_{\\mathrm{sm}}$.  Theorem~\\ref{res:denominator-reduction} then produces\n$d_{n+1}=b_nd_n-Bm_n$ with $\\gcd(B,30)=1$ and the radix\n$b_a=\\beta_a$ of Theorem~\\ref{res:dyadic-alphabet}.  The second obligation is\nthe Diophantine existence statement of Problem~\\ref{prob:producer}.  No Lean\ntheorem yet supplies that problem-specific carry identification or\nproves~\\eqref{eq:escape}.\n\n\\begin{problem}[cofinal local-window escape]\\label{prob:producer}\nLet $b_a=\\beta_a\\in\\{2,6,10,30\\}$ be the block radix of\nTheorem~\\ref{res:dyadic-alphabet}.  Fix an integer forcing word $(m_a)$\nattached block by block to the ordered jump word, and a function $K(B,n)$\nbounding the reduced carry at denominator $B$ and step $n$.  For that pair\n$(m,K)$, prove~\\eqref{eq:escape}: for every $B>0$ coprime to $30$\nand every starting index there are a later $\\ell$ and an $h>0$ with\n$|W_{\\ell,h}|>0$ and\n\\[\n \\operatorname{lpr}_{|W_{\\ell,h}|}(-BF_{\\ell,h})>K(B,\\ell+h).\n\\]\n\\end{problem}\n\n\\noindent The problem is well posed for each pair $(m,K)$ meeting that\ninterface, independently of whether the identification above is carried out.\nThe case of interest is the pair that identification would supply.\n\nProblem~\\ref{prob:producer} implies the local hypothesis of\nTheorem~\\ref{res:windowconsumer}, so it rules out the corresponding positive\nreduced carry.  Together with the identification above it would supply the\nmissing contradiction for the three-prime case.  Until both pieces are present,\nProblem~\\#269 remains open.\n\nFor each $B$, Problem~\\ref{prob:producer} asks for a cofinal family of windows\nsatisfying~\\eqref{eq:escape}, that is, a family containing windows that start\narbitrarily far out; it does not ask for every window to escape.  The radix word is\nconstrained to $\\{2,6,10,30\\}$, so the remaining arithmetic is in the block\ndigit and its carry into the residue class, and one seeks blocks for which the\nmodulus $|W_{\\ell,h}|$ grows faster than the carry window and the weighted\nforcing $-BF_{\\ell,h}$ does not remain trapped in the two short arcs adjacent\nto $0$ modulo that modulus.  No equidistribution of all windows is required.\nThe formal consumer uses exactly one escaping window for each reduced\ndenominator $B$, because the carry recurrence, positivity and bound hold at\nevery index.  Cofinal escape is retained as the natural producer shape and is\nnow connected to that minimal consumer by a separate Lean theorem.\n\nNone of the following weaker statements suffices, and each removes a false\nversion of~\\eqref{eq:escape}.  Merely proving $2\\le b_a\\le30$ loses the exact\nchannel alphabet.  A fixed finite list of denominator certificates does not\naddress the universal $B$-quantifier.  A bound independent of $B$ is not the\nbound delivered by denominator clearing.  Large absolute forcing is irrelevant\nunless its \\emph{least positive residue} is large.  In the other direction, a\nconstruction of infinitely many $(B,\\ell_0)$ for which every such window family\nfails would show that no argument of this shape can succeed.\n\n\\begin{problem}[direct nonintegral tails]\nAs an alternative to local windows, prove directly that for every positive\n$B$ coprime to $30$ the denominator-cleared tail of the actual $\\{2,3,5\\}$\nexpansion is nonintegral cofinally.  Such a theorem would\nbypass~\\eqref{eq:escape}, but it would still have to be joined to the\ndenominator-clearing step in the same way.\n\\end{problem}\n\nAn analytic treatment of the three logarithmic coordinates cannot begin by\nassuming that the kernel factors: by Theorem~\\ref{res:rank}, a single product\nof three one-variable functions is unavailable even at $\\{2,3,5\\}$.  A\ndecomposition into finitely many separable terms is not ruled out.\n\n\\pdfbookmark[1]{Statements and declarations}{statements-declarations}\n\\subsection*{Statements and declarations}\n\n\\paragraph{Artefact and data availability.} The\n\\href{\\sourceurl}{pinned formal-source revision} contains the Lean sources, the\nfixed toolchain, the library manifest, and the exact dyadic-window checker used\nin the finite experiment.  This manuscript provides navigation rather than\nproof authority.\n\n\\paragraph{Declaration of generative AI use.} \\paperfullaideclaration\n\nLean checks each proof term against the fixed library\nversion, and the sources linked here contain no proof placeholders and no\nproject-defined axioms; Lean does not authorise the exposition, the citation\nchoices, or the interpretation, for which the author remains responsible.\n\n\\paragraph{Funding and competing interests.} This work received no external\nfunding.  The author declares no competing interests.\n\n\\paragraph{Acknowledgements.} The problem numbering and status follow the\nErd\\H{o}s Problems catalogue maintained by Thomas Bloom~\\cite{erdosproblems}.\n\n\\appendix\n\\section{Guide to the formal sources}\\label{app:index}\n\nEach linked phrase opens its Lean declaration at the pinned source\nrevision~\\commitshort.  The running-LCM structure, residue arithmetic, and\nlocal-window bridge occupy three separate modules.  Four distinctions are\nworth carrying into the source.  The height statements hold for arbitrary\nbases, while the statements about $\\Lprefix$ need the three primes to be\ndistinct.  The normal form of Section~\\ref{sec:fibre} is a finite identity over\na rectangular box, not a convergence theorem.  The smooth part of a\nhypothetical denominator can be cancelled only after divisibility of the actual\ncarry states by that factor is proved.  The local-window predicate is an\nunproved hypothesis of Theorem~\\ref{res:windowconsumer}; the stronger cofinal\npredicate is a sufficient producer, and neither is asserted for the actual\n$\\{2,3,5\\}$ word.\n\n% The exhaustive source manifest is retained in the authored TeX for automated\n% validation, and kept outside the printed note.\n\\iffalse\n\n\\begin{tabular}{@{}ll@{}}\ninformal statement & linked Lean source\\\\\nsmooth lattice value & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{31}{smooth3Val}\\\\\npure-power height & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{36}{threePrimeHeight}\\\\\nlattice kernel & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{40}{threePrimeKernelQ}\\\\\nsmooth prefix index set & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{46}{smoothPrefixExponents}\\\\\nrunning least common multiple & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{53}{smoothPrefixLcm}\\\\\nprefix value divides the height & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{59}{smooth3Val_dvd_threePrimeHeight_of_mem}\\\\\ndivisibility into the height & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{77}{smoothPrefixLcm_dvd_threePrimeHeight}\\\\\nfirst pure-power membership & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{84}{pureFirst_mem_smoothPrefixExponents}\\\\\nsecond pure-power membership & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{97}{pureSecond_mem_smoothPrefixExponents}\\\\\nthird pure-power membership & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{110}{pureThird_mem_smoothPrefixExponents}\\\\\nrunning-lcm identity & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{123}{smoothPrefixLcm_eq_threePrimeHeight}\\\\\ncell relation & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{163}{SameThreePrimeLogCell}\\\\\ncell constancy of the height & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{169}{threePrimeHeight_eq_of_sameLogCell}\\\\\ncell constancy of the running value & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{179}{smoothPrefixLcm_eq_of_sameLogCell}\\\\\ncell constancy of the kernel & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{191}{threePrimeKernelQ_eq_of_sameLogCell}\\\\\npositive power sets & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{204}{positivePrimePowers}\\\\\nchannel cardinality & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{208}{positivePrimePowers_card}\\\\\nexclusion of the origin & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{219}{one_not_mem_positivePrimePowers}\\\\\nchannel disjointness & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{229}{positivePrimePowers_disjoint}\\\\\npositive jump channels & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{243}{threePrimePositiveJumpSet}\\\\\npositive jump count & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{249}{threePrimePositiveJumpSet_card}\\\\\njump set with the origin & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{273}{threePrimeJumpSetWithOrigin}\\\\\njump count with the origin & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{278}{threePrimeJumpSetWithOrigin_card}\\\\\nfirst height step & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{294}{threePrimeHeight_firstLogStep}\\\\\nsecond height step & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{305}{threePrimeHeight_secondLogStep}\\\\\nthird height step & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{316}{threePrimeHeight_thirdLogStep}\\\\\nfirst coordinate step & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{326}{smoothPrefixLcm_firstLogStep}\\\\\nsecond coordinate step & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{339}{smoothPrefixLcm_secondLogStep}\\\\\nthird coordinate step & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{352}{smoothPrefixLcm_thirdLogStep}\\\\\nexponent box & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{368}{smoothExponentBox}\\\\\npoint height & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{373}{smoothPointHeight}\\\\\nheight fibre & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{377}{smoothHeightFiber}\\\\\nfibre sum & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{384}{smoothHeightFiber_kernel_sum}\\\\\nheight-fibre normal form & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{406}{finiteSmoothKernelSum_groupedByHeight}\\\\\ncubic majorant & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{430}{threePrimeHeight_le_cube}\\\\\norigin value & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{443}{kernel_235_origin}\\\\\nvalue at two & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{450}{kernel_235_two}\\\\\nvalue at three & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{458}{kernel_235_three}\\\\\nvalue at six & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{467}{kernel_235_six}\\\\\nnon-separation witness & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{479}{kernel_235_not_rankOne}\\\\\nvariable-base tail step & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{487}{tailStateStep}\\\\\nexpanded tail step & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{490}{tailStateStep_eq}\\\\\nsmooth exponent shell & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{500}{smoothExponentShell}\\\\\nshort-interval uniqueness & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{509}{exponent_unique_in_short_interval}\\\\\nfirst-coordinate projection & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{543}{smoothExponentShell_card_le_dropFirst}\\\\\nthird-coordinate projection & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{585}{smoothExponentShell_card_le_dropThird}\\\\\nsorted quadratic estimate & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{629}{sorted_pair_quadratic}\\\\\nquadratic shell bound & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{646}{smoothExponentShell_card_quadratic}\\\\\ndyadic internal power & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{662}{DyadicInternalPower}\\\\\ninternal-power uniqueness & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{668}{dyadicInternalPower_exponent_unique}\\\\\ndyadic block base & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{688}{dyadicBlockBase235}\\\\\nexact dyadic radix alphabet & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{699}{dyadicBlockBase235_cases}\\\\\nbounded-radix consequence & \\lrefx{Erdos269/ThreePrimeRunningLcm.lean}{711}{dyadicBlockBase235_mem_interval}\\\\\nleast positive residue & \\lrefx{Erdos269/ResidueEscape.lean}{19}{leastPositiveResidue}\\\\\npositive representative range & \\lrefx{Erdos269/ResidueEscape.lean}{24}{leastPositiveResidue_pos_le}\\\\\nrepresentative congruence & \\lrefx{Erdos269/ResidueEscape.lean}{45}{leastPositiveResidue_modEq}\\\\\nescape condition & \\lrefx{Erdos269/ResidueEscape.lean}{65}{ResidueEscapesWindow}\\\\\nfinite natural-state contradiction & \\lrefx{Erdos269/ResidueEscape.lean}{71}{no_bounded_positive_state_of_residue_escape}\\\\\ncontrapositive form & \\lrefx{Erdos269/ResidueEscape.lean}{92}{residue_le_bound_of_bounded_positive_state}\\\\\ninteger least-positive-residue consumer & \\lrefx{Erdos269/ResidueEscape.lean}{104}{no_bounded_positive_int_state_of_leastPositiveResidue}\\\\\nwindow base & \\lrefx{Erdos269/RestrictedFloorSum.lean}{417}{windowBase}\\\\\nwindow forcing & \\lrefx{Erdos269/RestrictedFloorSum.lean}{422}{windowForcing}\\\\\naffine window identity & \\lrefx{Erdos269/RestrictedFloorSum.lean}{455}{affineRecurrence_window}\\\\\nscaled forcing identity & \\lrefx{Erdos269/RestrictedFloorSum.lean}{469}{windowForcing_const_mul}\\\\\nintegral carry window & \\lrefx{Erdos269/RestrictedFloorSum.lean}{480}{integralCarry_window}\\\\\nabsorbed smooth factor & \\lrefx{Erdos269/RestrictedFloorSum.lean}{513}{smooth3Val_dvd_threePrimeHeight_of_le}\\\\\ncommon-factor cancellation & \\lrefx{Erdos269/RestrictedFloorSum.lean}{546}{integralCarry_cancel_commonFactor}\\\\\nreduced bound transfer & \\lrefx{Erdos269/RestrictedFloorSum.lean}{566}{reducedCarry_pos_le_of_commonFactor}\\\\\nreduced window transfer & \\lrefx{Erdos269/RestrictedFloorSum.lean}{577}{reducedIntegralCarry_window}\\\\\ncofinal window producer & \\lrefx{Erdos269/RestrictedFloorSum.lean}{594}{CofinalLocalWindowEscape}\\\\\nminimal window producer & \\lrefx{Erdos269/RestrictedFloorSum.lean}{652}{LocalWindowEscape}\\\\\ncofinal-to-minimal implication & \\lrefx{Erdos269/RestrictedFloorSum.lean}{666}{localWindowEscape_of_cofinalLocalWindowEscape}\\\\\nreduced-carry extinction & \\lrefx{Erdos269/RestrictedFloorSum.lean}{677}{no_positive_reducedCarry_of_localWindowEscape}\\\\\n\\end{tabular}\n\\fi\n\n\\begin{thebibliography}{10}\n\\small\n\\raggedright\n\\setlength{\\itemsep}{0pt}\n\\bibitem{erdosgraham1980} P.~Erd\\H{o}s and R.~L. Graham,\n  \\href{https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf}{\\emph{Old\n  and New Problems and Results in Combinatorial Number Theory}},\n  Monogr. Enseign. Math. 28, Geneva, 1980, p.~65.  For a possibly infinite\n  prime set $Q$, the page states the infinite-$Q$ irrationality and asks what\n  happens for finite $Q$ with more than one element.\n\\bibitem{erdos1988} P.~Erd\\H{o}s, \\emph{On the irrationality of certain series:\n  problems and results}, in A.~Baker (ed.), \\emph{New Advances in Transcendence\n  Theory}, Cambridge UP, 1988, pp.~102--109,\n  doi:\\href{https://doi.org/10.1017/CBO9780511897184.009}{10.1017/CBO9780511897184.009}.\n\\bibitem{erdos1974letter} P.~Erd\\H{o}s,\n  \\href{https://www.fq.math.ca/Scanned/12-4/letter.pdf}{\\emph{Letter to the\n  Editor}} (written 1~January 1973), Fibonacci Quart. \\textbf{12} (1974),\n  no.~4, p.~335.  The letter poses the full series as a conjecture and asserts\n  irrationality after retaining only the distinct running-LCM values.\n\\bibitem{erdosproblems} T.~F. Bloom,\n  \\href{https://www.erdosproblems.com/269}{\\emph{Erd\\H{o}s Problem \\#269}},\n  \\texttt{erdosproblems.com/269}, accessed 28 July 2026 (page displays\n  ``last edited 28 December 2025'').  The current record labels the\n  finite-support problem open, cites \\texttt{[ErGr80, p.~65]} and\n  \\texttt{[Er88c, p.~106]}, routes to the 1974 letter on p.~335, records the\n  infinite-prime and de-duplicated variants, and explicitly describes its\n  status as the website owner's present assessment rather than a\n  literature-completeness guarantee.\n\\bibitem{formalconjectures269} The Formal Conjectures Authors,\n  \\href{https://github.com/google-deepmind/formal-conjectures/blob/f776d2f2039351b00737ffcafb9d7d7666e1d9af/FormalConjectures/ErdosProblems/269.lean}\n  {\\emph{FormalConjectures.ErdosProblems.\\texttt{269}}},\n  Lean source at commit \\texttt{f776d2f}, 2025, accessed 28 July 2026.\n\\bibitem{apostol1976} T.~M. Apostol,\n  \\href{https://doi.org/10.1007/978-1-4757-5579-4}\n  {\\emph{Introduction to Analytic Number Theory}},\n  Springer, New York, 1976.\n\\bibitem{montgomeryvaughan2007} H.~L. Montgomery and R.~C. Vaughan,\n  \\emph{The Prime Number Theorem}, in \\emph{Multiplicative Number Theory I:\n  Classical Theory}, Cambridge Studies in Advanced Mathematics 97,\n  Cambridge University Press, 2007, pp.~168--198,\n  doi:\\href{https://doi.org/10.1017/CBO9780511618314.008}\n  {10.1017/CBO9780511618314.008}.\n\\bibitem{kovactao2024} V.~Kova\\v{c} and T.~Tao,\n  \\href{https://doi.org/10.1007/s10474-025-01528-0}\n  {\\emph{On several irrationality problems for Ahmes series}},\n  Acta Math. Hungar. \\textbf{175} (2025), 572--608,\n  doi:\\href{https://doi.org/10.1007/s10474-025-01528-0}\n  {10.1007/s10474-025-01528-0};\n  \\href{https://arxiv.org/abs/2406.17593}{arXiv:2406.17593}, 2024.\n\\bibitem{bugeaudlaurent2023} Y.~Bugeaud and M.~Laurent,\n  \\emph{Transcendence and continued fraction expansion of values of\n  Hecke--Mahler series}, Acta Arith. \\textbf{209} (2023), 59--90,\n  doi:\\href{https://doi.org/10.4064/aa220323-18-1}\n  {10.4064/aa220323-18-1};\n  \\href{https://irma.math.unistra.fr/~bugeaud/travaux/BuMLAA.pdf}\n  {authors' publisher-layout PDF},\n  \\href{https://arxiv.org/abs/2203.12901}{arXiv:2203.12901v1}.\n  Theorem~1.1 is on journal p.~61 (publisher-layout PDF p.~3); the authors\n  note there that its $\\rho=0$ case was already obtained by Loxton and van\n  der Poorten.\n\\end{thebibliography}\n\n\\end{document}\n","dependencies":[{"record_type":"paper_source_dependency","tex_reference":"problem-note-preamble","source_path":"paper/problem-note-preamble.tex","media_type":"text/x-tex","sha256":"sha256:e7378fe7784916681e7faf96c1ea7a8bb941a416cc651b97512e849bcaeb63f7","byte_count":4990,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Shared preamble for the Erd\u0151s Problem Notes: the short, standalone,\n% problem-owned manuscripts covering the expansion library `ErdosProblems`.\n%\n% One note owns one Erd\u0151s problem.  Each note repeats whatever context its own\n% reader needs, so that a reader who arrives at a single problem never has to\n% assemble it from the gateway paper.  Deliberate repetition across notes is the\n% design, not an oversight.\n%\n% Source links resolve against one pinned formal-source commit, declared once\n% below.  `scripts/check_problem_note_sources.py` verifies that every authored\n% (file, line, declaration) triple names that exact declaration in the pinned\n% snapshot, so a link cannot silently rot as later work moves lines.\n\n\\usepackage{paper-house-style}\n\\usepackage{array}\n\\usepackage{booktabs}\n\\usepackage{enumitem}\n\\setlist{topsep=0.35em,itemsep=0.2em}\n\\usepackage[colorlinks=true,linkcolor=linkinternal,urlcolor=linkexternal,\n  citecolor=linkinternal]{hyperref}\n\\hypersetup{bookmarksnumbered=true,bookmarksopen=true,bookmarksopenlevel=1,\n  pdfauthor={Will Cook},pdflang={en-GB}}\n\n% ---------------------------------------------------------------------------\n% Pinned formal source.  \\commit names the checkpoint every link resolves\n% against; the checker reads that snapshot rather than the working tree, so the\n% printed coordinates stay correct however later waves move declarations.\n% ---------------------------------------------------------------------------\n\\newcommand{\\commit}{d6d344d9557faee19276b01a1c1f20d7c08f03d5}\n\\newcommand{\\commitshort}{d6d344d9557f}\n\\newcommand{\\repobase}{https://github.com/wcook04/plectis-lean-erdos249-257/blob/\\commit}\n\\newcommand{\\sourceurl}{https://github.com/wcook04/plectis-lean-erdos249-257/tree/\\commit}\n\\newcommand{\\PX}{ErdosProblems}\n\n% Declaration names stay inside link targets.  The printed label is either a\n% spaced, lower-case reading of the declaration name (\\lref) or a short authored\n% phrase (\\lword); neither prints a module path or a raw Lean identifier.\n\\ExplSyntaxOn\n\\cs_new_protected:Npn \\noteleanlabel #1\n  {\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_to_str:n {#1} }\n    \\regex_replace_all:nnN { _+ } { \\c{space} } \\l_tmpa_tl\n    \\regex_replace_all:nnN\n      { ([a-z0-9])([A-Z]) }\n      { \\1\\c{space}\\2 }\n      \\l_tmpa_tl\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_lower_case:n { \\tl_use:N \\l_tmpa_tl } }\n    \\tl_use:N \\l_tmpa_tl\n  }\n\\ExplSyntaxOff\n\\newcommand{\\leanlabel}[1]{\\noteleanlabel{#1}}\n\\newcommand{\\lref}[3]{\\href{\\repobase/\\PX/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\lword}[4]{\\href{\\repobase/\\PX/#1\\#L#2}{#4}}\n\\newcommand{\\lloc}[2]{\\href{\\repobase/\\PX/#1\\#L#2}{Lean source}}\n\n% Links into a sibling library, named repository-relative.  The expansion\n% library is implicit in \\lref/\\lword, because that is where problem-owned\n% work lands.  Several problems have their headline theorem in the older\n% reviewed #249/#257 corpus, and a note that could not name that library\n% would have to paraphrase a checked statement instead of linking it.\n\\newcommand{\\mref}[3]{\\href{\\repobase/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\mword}[4]{\\href{\\repobase/#1\\#L#2}{#4}}\n\\newcommand{\\mloc}[2]{\\href{\\repobase/#1\\#L#2}{Lean source}}\n\\emergencystretch=2em\n\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\theoremstyle{definition}\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{example}[theorem]{Example}\n\\newtheorem{problem}[theorem]{Problem}\n\\theoremstyle{remark}\n\\newtheorem*{remark}{Remark}\n\n\\newcommand{\\N}{\\mathbb{N}}\n\\newcommand{\\Npos}{\\mathbb{N}_{>0}}\n\\newcommand{\\Z}{\\mathbb{Z}}\n\\newcommand{\\Q}{\\mathbb{Q}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\lcm}{\\operatorname{lcm}}\n\\newcommand{\\ph}{\\varphi}\n\n% The series line printed under every note's title block.\n\\newcommand{\\seriesline}{Erd\\H{o}s Problem Notes}\n\n% Problem-note compatibility name for the corpus-owned page-one disclosure in\n% paper-house-style.sty.\n\\newcommand{\\noteprovenance}{\\paperprovenance}\n\n% The status paragraph every note carries, in its own words, before any result.\n% Kernel checking establishes the exact Lean proposition; it does not establish\n% that the proposition is the interesting one, that it is new, or that the\n% Erd\u0151s problem is closed.\n\\newcommand{\\statusboundary}{%\n  \\emph{Status.}  The problem treated here is open, and this note does not\n  close it.  Every statement below marked as checked is a proposition that the\n  pinned Lean kernel accepts from the sources this note links to, with no\n  \\texttt{sorry}, no added axiom, and no unchecked evaluation.  That is a claim\n  about the formal statement, not about its mathematical interest, its\n  novelty, or the original problem.  The unresolved obligations are named\n  exactly, in their own section, and none of the finite computations,\n  reductions, or no-go results here removes one of them.}\n"}],"external_verification_route":"https://raw.githubusercontent.com/wcook04/plectis-lean-erdos249-257/main/paper/erdos-269-three-prime-running-lcm.tex"},{"record_type":"paper","paper_id":"erdos-1041-lemniscate-newton-flow","title":"Newton Flow and Critical-Value Ray Separation","source_repository":"plectis-lean-erdos249-257","source_path":"paper/erdos-1041-lemniscate-newton-flow.tex","hosted_pdf_path":"papers/erdos-1041-lemniscate-newton-flow.pdf","licence":"CC-BY-4.0","copyright":"2026 Will Cook","cite_as":"Plectis reader corpus, paper:erdos-1041-lemniscate-newton-flow","pdf_sha256":"sha256:7575a8c893089057c268d6f3d497cbd6ce90b21ac4bdee56f4e656ffee329e4d","pdf_byte_count":79137,"pdf_embedded":false,"pdf_omission_reason":"the exact LaTeX is embedded and retrieves better than extracted PDF text; the PDF is a presentation rendering and its hash is recorded for verification","source_embedded":false,"availability":"unavailable_at_build","reason":"resolved checkout for registry repository 'plectis-lean-erdos249-257' did not carry the paper source at build time; the source could not be embedded","external_verification_route":"https://raw.githubusercontent.com/wcook04/plectis-lean-erdos249-257/main/paper/erdos-1041-lemniscate-newton-flow.tex"},{"record_type":"paper","paper_id":"erdos-1049-rational-base-lambert","title":"Multiplicative Obstructions at Base 3/2","source_repository":"plectis-lean-erdos249-257","source_path":"paper/erdos-1049-rational-base-lambert.tex","hosted_pdf_path":"papers/erdos-1049-rational-base-lambert.pdf","licence":"CC-BY-4.0","copyright":"2026 Will Cook","cite_as":"Plectis reader corpus, paper:erdos-1049-rational-base-lambert","pdf_sha256":"sha256:9b25be837a8cd9b56117ccdd3f2e62e48666ee851653739db7eaf3a2741d78a4","pdf_byte_count":153097,"pdf_embedded":false,"pdf_omission_reason":"the exact LaTeX is embedded and retrieves better than extracted PDF text; the PDF is a presentation rendering and its hash is recorded for verification","source_embedded":true,"availability":"embedded_exact","embedded_copy_role":"native_manuscript","embedded_from_repository":"plectis-lean-erdos249-257","embedded_copy_path":"paper/erdos-1049-rational-base-lambert.tex","media_type":"text/x-tex","body_format":"latex","sha256":"sha256:a33758fe3c7f4e27ec8941aced881a22f4362e07adbcb2bb09c7daf6a8b8ab3a","byte_count":52419,"body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% One of the Erd\u0151s Problem Notes: a short standalone treatment of a single\n% problem in the ErdosProblems expansion library.  Build with\n% `tectonic erdos-1049-rational-base-lambert.tex` or `make`.\n\\documentclass[11pt]{article}\n\\input{problem-note-preamble}\n% Problem-specific source pin: #1049's primitive endpoint-jet obstruction advances\n% independently of the corpus default.\n\\renewcommand{\\commit}{d6d344d9557faee19276b01a1c1f20d7c08f03d5}\n\\renewcommand{\\commitshort}{d6d344d9557f}\n\\hypersetup{\n  pdftitle={Arithmetic obstructions at the rational base 3/2},\n  pdfsubject={Erd\u0151s Problem Notes: Problem 1049},\n  pdfkeywords={irrationality, Lambert series, rational base, Pad\u00e9 approximation, Lean 4},\n}\n\n\\runninghead{Erd\\H{o}s \\#1049: obstructions at base 3/2}\n\\title{Arithmetic Obstructions at the Rational Base $3/2$}\n\\subtitle{Two precise limits of integer-base methods for Erd\\H{o}s\nProblem~\\#1049}\n\\author{Will Cook}\n\\date{August 2026}\n\n\\begin{document}\n\\maketitle\n\\noteprovenance\n\n\\begin{abstract}\n\\noindent\nLet $F(t)=\\sum_{n\\ge1}(t^n-1)^{-1}$.  The irrationality of $F(3/2)$ remains\nopen.  We isolate two exact arithmetic obstructions that explain why familiar\ninteger-base mechanisms do not transfer automatically to this rational base.\n\nFirst, a cut-and-clear argument at $r/s$ leaves the denominator contribution\n$s^{N+1}$ inside its forcing term.  For a precisely stated coordinatewise\ncorridor, we prove\n\\[\n s^{N+K+1}<r(N+K).\n\\]\nAt $3/2$ this inequality is impossible for every nonempty shift and cleared\nwindow.  The associated cleared-tail recurrence\n$U_{N+1}=rU_N-Bc(N+1)s^{N+1}$ makes the integer-base boundary explicit: the\ngeometric factor disappears when $s=1$ and survives when $s\\ge2$.\n\nSecond, we study integer polynomial rows specialised homogeneously at $(3,2)$.\nDividing rows by their scalar contents is exactly neutral in the\nlocal-to-Archimedean determinant balance.  Unit endpoint coefficients also\nexclude both $2$ and $3$ from a common divisor of the two specialised\nevaluations.  Hence neither device supplies the required local gain.  An\nadditive alternative remains: with $M\\ge4R+2S$ coefficient pairs and $R>0$, a\nnonzero $\\{-1,0,1\\}$ relation cancels all four endpoint jets.  This counting\ntheorem does not show that the resulting polynomial pair or analytic remainder\nis nonzero.\n\nLean checks these finite algebraic statements.  We separately verify the\nelementary height inequality used by the published $7/2$ criterion and record\ndenominator-exponent identities for a homogenised Pad\\'e construction; the\nanalytic irrationality theorems remain external.  None of these results\ndecides the arithmetic nature of $F(3/2)$.\n\\end{abstract}\n\n\\section{Introduction}\\label{sec:problem}\n\nLet $t>1$ be a rational number and let $\\tau(n)$ count the divisors of $n$.\nErd\\H{o}s Problem~\\#1049 asks whether\n\\[\n F(t)=\\sum_{n\\ge1}\\frac{1}{t^{n}-1}=\\sum_{n\\ge1}\\frac{\\tau(n)}{t^{n}}\n\\]\nis irrational~\\cite[p.~102]{erdos1988}.  The two forms agree by expanding\n$(t^{n}-1)^{-1}=\\sum_{k\\ge1}t^{-nk}$ and collecting the terms with the same\nexponent, the coefficient of $t^{-m}$ being the number of divisors of $m$.\nThe question is a conjecture of Chowla; Erd\\H{o}s proved it for every integer\n$t\\ge2$~\\cite{erdos1948}.  Bloom's catalogue currently records the rational-base\nproblem as open~\\cite{erdosproblems}; we use it for numbering and reported\nstatus, and the original papers for the mathematical claims.  Some individual\nnon-integral bases are known, including $7/2$ below.\n\nWrite $t=r/s$ in lowest terms with $r>s\\ge1$, so that $s=1$ is exactly the\ninteger case Erd\\H{o}s settled.  The resistant explicit base of least naive\nheight $H(r/s)=\\max(r,s)$ is $t=3/2$.  A published height criterion of Bundschuh and\nV\\\"a\\\"an\\\"anen~\\cite[Thm.~2, p.~177; hypotheses pp.~175--176]{bv1994}\nsettles a family of rational bases restricted by a\nheight condition; that family contains $7/2$ and does not contain $3/2$.\nBetween the two lies the question this note is about: what exactly stops the\ninteger-base argument from running at $3/2$?\n\n\\paragraph{Relation to prior work.}\nErd\\H{o}s proved irrationality at every positive integer base in\n1948~\\cite{erdos1948}.  Pad\\'e methods later gave quantitative refinements at\ninteger bases: Borwein treated shifted Lambert series~\\cite{borwein1991}, Van\nAssche used little $q$-Legendre approximants~\\cite{vanassche2001}, and Zudilin\nobtained a uniform irrationality-measure bound through Heine's transform and a\npermutation group~\\cite{zudilin2004}.  All retain an integer base.  For\nnon-integral rational bases, the criterion of Bundschuh and V\\\"a\\\"an\\\"anen\ncontains $7/2$ but not $3/2$~\\cite{bv1994}.\n\nRivin's nonrationality theorem for the underlying Lambert function is a\nfunctional statement and does not decide a value at a particular rational\npoint~\\cite{rivin2026}.  The \\emph{Formal Conjectures} source checks the\nLambert-series identity but not the irrationality statements\n~\\cite{formalconjectures1049}.  Likewise, no theorem here derives the endpoint\nlemmas from the coefficient family of Zudilin or Rhin--Viola\n~\\cite{rhinviola1996}; that application theorem is genuinely missing.\n\n\\paragraph{Erd\\H{o}s's integer-base argument, in outline.}\nFor an integer $b\\ge2$, Erd\\H{o}s rewrites\n$F(b)=\\sum_{n\\ge1}\\tau(n)b^{-n}$.  A Chinese-remainder construction forces\narbitrarily long zero blocks in the base-$b$ expansion; tail bounds and\npositivity show that the expansion does not terminate.  Hence it has\narbitrarily long zero blocks without being eventually zero and is irrational\n~\\cite[pp.~63--66]{erdos1948}.\n\n\\paragraph{The direct cut-and-clear attempt studied here.}\nA more naive route is to suppose $F(b)=p/q$, cut at $N$, and multiply the\nremaining identity by $qb^N$.  This does not by itself trap a positive integer\nbelow $1$: the first uncleared tail contribution is\n$q\\tau(N+1)/b$.  We isolate additional corridor hypotheses under which a\nbounded-window version would work and show that they fail at $3/2$.\n\nWrite $\\beta=r/s$ for the base and $c(n)$ for the coefficient of $\\beta^{-n}$,\nso that $c=\\tau$ in the case at hand.  The term $c(n)\\beta^{-n}$ is\n$c(n)s^{n}/r^{n}$.  Clearing the\npower of $r$ leaves the numerator factor $s^{n}$ in place.  That factor is\ninvisible when $s=1$ and grows geometrically when $s\\ge2$.  At $3/2$ it is\n$2^{n}$.\n\n\\paragraph{Two theorem packages.}\nThe first concerns a precisely defined cut-and-clear corridor.  Its exact\nrecurrence retains a forcing term of size at least $2^{N+1}$ at every\nnon-integral rational base, and the corridor inequality is impossible at\n$3/2$ for all $N,K\\ge1$ (Theorems~\\ref{res:forcing} and\n\\ref{res:nocorridor}).  This excludes that predicate, not every argument one\nmight informally call ``clearing denominators''.\n\nThe second concerns polynomial coefficient rows.  Row content is neutral in\nthe determinant balance, and unit endpoints prevent a common divisor of two\nspecialised evaluations from carrying either local prime\n(Theorem~\\ref{res:content}, Proposition~\\ref{res:commonmult}).  The surviving\nalternative is additive: Theorem~\\ref{res:jetkernel} gives a nonzero\n$\\{-1,0,1\\}$ relation cancelling four endpoint jets once $M\\ge4R+2S$ and\n$R>0$.  It does not prove that the resulting polynomial pair or analytic\nremainder is nonzero.  Problem~\\ref{prob:kernel} records that application gate.\n\n\\paragraph{Sharpness.}\nThe corridor bound is exponential on the left and linear on the right, so for\nfixed $r$ and $s\\ge2$ it can hold only for bounded $N+K$.  At $3/2$ the crossing\nhas already occurred at the smallest admissible window.  The published\n$7/2$ result rests on a different height criterion~\\cite{bv1994}; nothing here\ntransfers that theorem to $3/2$.\n\n\\statusboundary\n\nThe Lean development proves finite algebra: determinant identities, endpoint\ncongruences, collision counts, the corridor exclusion, and the cleared-tail\nrecurrence.  The $7/2$ irrationality theorem is cited from\nBundschuh--V\\\"a\\\"an\\\"anen; only its elementary height inequality is checked\nhere.  The Pad\\'e section proves exponent arithmetic but supplies no remainder\nestimate.  No theorem in this note decides $F(3/2)$.\n\n\\paragraph{Structure.}\nSections~\\ref{sec:primitive}--\\ref{sec:endpoints} treat polynomial rows and\ntheir endpoint jets.  Sections~\\ref{sec:corridor}--\\ref{sec:tail} treat the\ncut-and-clear corridor.  The final sections separate the external $7/2$\ncriterion, Pad\\'e bookkeeping, and the open additive gate.  Linked phrases open\nthe corresponding Lean declaration at revision~\\commitshort.\n\n\\medskip\n\\noindent\\textbf{Keywords.} irrationality; Lambert series; rational base;\nPad\\'e approximation; Lean~4.\n\\qquad\n\\textbf{MSC 2020.} 11J72 (primary); 11J82, 68V20 (secondary).\n\n\\section{Row content does not change the local-to-Archimedean\nbalance}\\label{sec:primitive}\n\nAn irrationality argument by linear forms replaces the clearing scheme by\nexplicit rational approximation: one constructs integer linear forms in $1$ and\n$F(\\beta)$ whose analytic decay outruns the height of their common denominator.\nFor an integer coefficient pair \\((U,V)\\) and a real target \\(S\\), put\n\\[\n L_S(U,V)=US-V,\n \\qquad\n \\Delta\\bigl((U_n,V_n),(U_m,V_m)\\bigr)=U_nV_m-U_mV_n.\n\\]\nWe call \\(L_S(U,V)\\) the \\emph{error} of the row at \\(S\\); a good approximation\nis one that makes it small.  The second expression is the exterior determinant\nof the two rows, and it eliminates \\(S\\) exactly:\n\\[\n \\Delta=U_mL_S(U_n,V_n)-U_nL_S(U_m,V_m).\n\\]\nThis identity is what makes the determinant useful.  Since \\(\\Delta\\) is an\ninteger, if it does not vanish then\n\\[\n 1\\le|\\Delta|\n  \\le|U_m|\\,\\bigl|L_S(U_n,V_n)\\bigr|+|U_n|\\,\\bigl|L_S(U_m,V_m)\\bigr|,\n\\]\nand the two errors cannot both be smaller than\n\\(1/(|U_n|+|U_m|)\\).  The determinant therefore carries two competing\nquantities at once, an integer that divides it and its own absolute value, and\nthe theorem below is about how a rescaling moves them.\n\n\\begin{theorem}[row-content no-go]\\label{res:content}\nLet \\(S\\) be real, let \\((U_n,V_n)\\) and \\((U_m,V_m)\\) be pairs of integers, and\nlet \\(c_n,c_m\\) be integers.  Then\n\\[\n L_S(c_nU_n,c_nV_n)=c_nL_S(U_n,V_n),\n\\]\n\\[\n \\Delta\\bigl(c_n(U_n,V_n),c_m(U_m,V_m)\\bigr)\n =c_nc_m\\Delta\\bigl((U_n,V_n),(U_m,V_m)\\bigr),\n\\]\nand consequently\n\\[\n \\left|\\Delta\\bigl(c_n(U_n,V_n),c_m(U_m,V_m)\\bigr)\\right|\n =|c_n|\\,|c_m|\\,\n   \\left|\\Delta\\bigl((U_n,V_n),(U_m,V_m)\\bigr)\\right|.\n\\]\nIn particular \\(c_nc_m\\) divides the scaled determinant.  Hence a local\ndivisor supplied only by the two row contents is paid for by exactly the same\nfactor in the Archimedean determinant height.\n\\end{theorem}\n\n\\begin{proof}\nAll three identities are routine expansions in \\(\\Z\\) or \\(\\R\\); the\ndivisibility statement uses the primitive determinant as its witness.\n\\end{proof}\n\nInformally, Theorem~\\ref{res:content} says that a common factor introduced into\nthe rows appears twice over: once as a divisor of the exterior determinant, and\nonce as the same factor in the absolute value of that determinant.  It moves\nthe local gain and the Archimedean cost by exactly the same amount.\n\n\\begin{example}\\label{ex:content}\nTake \\((U_n,V_n)=(1,2)\\) and \\((U_m,V_m)=(3,5)\\), so that\n\\(\\Delta=1\\cdot5-3\\cdot2=-1\\).  Multiplying the first row by \\(c_n=6\\) and the\nsecond by \\(c_m=10\\) gives the rows \\((6,12)\\) and \\((30,50)\\), whose\ndeterminant is \\(6\\cdot50-30\\cdot12=-60\\).  That determinant is now divisible\nby \\(60\\), which looks like a local gain of \\(60\\); and its absolute value has\nrisen from \\(1\\) to \\(60\\), which is a cost of exactly the same size.\n\\end{example}\n\n\\noindent Lean checks the error identity in\n\\lword{Erdos1049/RationalPadeArithmetic.lean}{84}{rationalPadeError_mul}{error\nscaling}, the determinant identity in\n\\lword{Erdos1049/RationalPadeArithmetic.lean}{94}{rationalPadeExteriorDet_mul_contents}{content\nfactorisation}, the exact absolute-height identity in\n\\lword{Erdos1049/RationalPadeArithmetic.lean}{104}{natAbs_rationalPadeExteriorDet_mul_contents}{absolute\ndeterminant scaling}, and the divisor statement in\n\\lword{Erdos1049/RationalPadeArithmetic.lean}{116}{contentProduct_dvd_rationalPadeExteriorDet}{content-product\ndivisibility}.  The elimination identity is the checked\n\\lword{Erdos1049/RationalPadeArithmetic.lean}{124}{rationalPadeExteriorDet_cast_eq}{exterior\ndeterminant identity}.\n\nThe key point is that the two scalings are the same scaling.  Each identity on\nits own is a one-line expansion, and the interest of the theorem is not in any\none of them.  Taken together they say that the factor \\(c_nc_m\\) which a\nrescaling introduces into the determinant reappears undiminished, as\n\\(|c_n|\\,|c_m|\\), in the absolute value of that determinant.  The third\nidentity is displayed with absolute values for exactly that reason: it is what\nmakes the statement one about the Archimedean height and not about divisibility\nalone.  Whatever \\(c_n\\) and \\(c_m\\) are, a rescaling therefore leaves the\nbalance between the local divisor and that height where it was.\n\nThe theorem does not construct primitive Pad\\'e rows, estimate their\nremainders, or prove that their exterior determinant is nonzero.  It removes\none source of apparent gain: multiplying a useful row by a large common\ninteger cannot improve the local-to-Archimedean balance.  Every candidate\nfamily must first be divided by its rowwise common contents; the required\n\\(2\\)-adic and \\(3\\)-adic gain must remain afterwards.  The theorem quantifies\nover arbitrary integer coefficient pairs, so it applies to the rows of any\nconstruction of this shape, whatever produced them.\n\n\\section{Endpoint residues at $(3,2)$ and the four-jet\nkernel}\\label{sec:endpoints}\n\nZudilin's treatment of \\(q\\)-harmonic series~\\cite{zudilin2004} builds linear\nforms of the shape used in Section~\\ref{sec:primitive} out of Heine's basic\ntransform.  What matters here is one feature of that construction: its\ncoefficients are produced as polynomials with integer coefficients, and a\nnumerical base is substituted only at the end.  We call these the\nHeine--Zudilin forms and ask what the substitution at \\(3/2\\) does to the two\nprimes \\(2\\) and \\(3\\).  The endpoint and jet arguments below use no property\nof the construction beyond integrality of those coefficient polynomials, and\neach of them quantifies over arbitrary elements of \\(\\Z[X]\\).  The exception is\nTheorem~\\ref{res:scalar} at the end of the section, which concerns the scalar\nparameters of Zudilin's cone rather than the coefficient polynomials.\n\nSubstituting \\(X=3/2\\) into an integer polynomial produces a rational number,\nand multiplying by a power of \\(2\\) clears its denominator.  The following\nevaluation records that cleared numerator, so that all the arithmetic below\nstays inside \\(\\Z\\).  For \\(P(X)=\\sum_i p_iX^i\\in\\Z[X]\\) and a declared width\n\\(W\\ge0\\), put\n\\[\n H_W(P)=\\sum_{i=0}^{W}p_i\\,3^i2^{W-i}.\n\\]\nThis is the\n\\lword{Erdos1049/ZudilinConeArithmetic.lean}{96}{homEvalThreeTwo}{homogeneous\nendpoint evaluation}.  It is homogeneous in the sense that \\(X^{i}\\) is\nreplaced by \\(3^{i}2^{W-i}\\), so the numerator and the denominator of the base\nare carried symmetrically.  When \\(W\\ge\\deg P\\) it is exactly the cleared\nnumerator, since\n\\[\n \\sum_{i=0}^{W}p_i\\,3^i2^{W-i}=2^{W}\\sum_{i=0}^{W}p_i\\left(\\tfrac32\\right)^{i}\n =2^{W}P\\!\\left(\\tfrac32\\right).\n\\]\n\n\\begin{theorem}[endpoint residues]\\label{res:endpoints}\nLet \\(P=\\sum_ip_iX^i\\in\\Z[X]\\) and let \\(W\\ge0\\).  Then\n\\[\n H_W(P)\\equiv p_0\\,2^W\\pmod 3,\\qquad\n H_W(P)\\equiv p_W\\,3^W\\pmod 2.\n\\]\nConsequently a unit constant coefficient prevents divisibility by \\(3\\), and\na unit coefficient at the declared width \\(W\\) prevents divisibility by \\(2\\).\n\\end{theorem}\n\n\\begin{proof}\nModulo \\(3\\), every summand with \\(i>0\\) vanishes; modulo \\(2\\), every summand\nwith \\(i<W\\) vanishes.  The remaining powers are units in the corresponding\nresidue fields.\n\\end{proof}\n\nSince \\(2^{W}\\) is invertible modulo \\(3\\) and \\(3^{W}\\) is invertible modulo\n\\(2\\), the two congruences say more than the stated consequence: divisibility\nof \\(H_W(P)\\) by \\(3\\) is decided by \\(p_0\\) alone, and divisibility by \\(2\\) by\n\\(p_W\\) alone.  The rest of the coefficient vector is invisible to both primes.\nThe coefficient \\(p_W\\) is the top coefficient of \\(P\\) exactly when\n\\(\\deg P=W\\), and is zero when \\(\\deg P<W\\); the identity\n\\(H_W(P)=2^{W}P(3/2)\\) likewise holds only for \\(\\deg P\\le W\\).\n\n\\begin{example}\\label{ex:endpoints}\nTake \\(W=2\\).  The three polynomials below differ only at an endpoint.\n\\begin{center}\n\\begin{tabular}{@{}llcc@{}}\n\\toprule\n\\(P\\) & \\(H_2(P)\\) & \\(3\\mid H_2(P)\\) & \\(2\\mid H_2(P)\\)\\\\\n\\midrule\n\\(X^2+1\\)  & \\(1\\cdot4+0\\cdot6+1\\cdot9=13\\) & no  & no \\\\\n\\(X^2+3\\)  & \\(3\\cdot4+0\\cdot6+1\\cdot9=21\\) & yes & no \\\\\n\\(2X^2+1\\) & \\(1\\cdot4+0\\cdot6+2\\cdot9=22\\) & no  & yes \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\nThe first has both endpoints equal to \\(1\\) and its evaluation, \\(13\\), is\ndivisible by neither prime; as a check, \\(2^{2}\\bigl((3/2)^2+1\\bigr)=13\\).  The\nsecond and third show that each hypothesis is used: spoiling the constant\nendpoint admits \\(3\\), and spoiling the top endpoint admits \\(2\\).\n\\end{example}\n\n\\noindent The congruences are the\n\\lword{Erdos1049/ZudilinConeArithmetic.lean}{201}{homEvalThreeTwo_mod_three}{bottom-endpoint\nidentity} and\n\\lword{Erdos1049/ZudilinConeArithmetic.lean}{215}{homEvalThreeTwo_mod_two}{top-endpoint\nidentity}; the unit consequences are the\n\\lword{Erdos1049/ZudilinConeArithmetic.lean}{233}{three_not_dvd_homEvalThreeTwo_of_const_unit}{constant-endpoint\nobstruction} and\n\\lword{Erdos1049/ZudilinConeArithmetic.lean}{251}{two_not_dvd_homEvalThreeTwo_of_top_unit}{top-endpoint\nobstruction}.\n\nThe proof is two lines, but the shape of the statement is not incidental.  The\ndifficulty lies in the fact that the specialisation sees a different endpoint at\neach of the two primes: modulo \\(3\\) only the constant coefficient survives, and\nmodulo \\(2\\) only the top one does.  The two exclusions are therefore conditions\nat opposite ends of the coefficient vector, and the statement below imposes one\nat each end, on the two entries of a single coefficient pair.\n\n\\begin{proposition}[common divisor]\\label{res:commonmult}\nLet \\(U,V\\in\\Z[X]\\) and let \\(W\\ge0\\).  If \\(U\\) has unit top endpoint, \\(V\\)\nhas unit constant endpoint, and an integer\n\\(c\\) divides both \\(H_W(U)\\) and \\(H_W(V)\\), then\n\\[\n 2\\nmid c\\qquad\\text{and}\\qquad 3\\nmid c.\n\\]\n\\end{proposition}\n\n\\noindent This is the checked\n\\lword{Erdos1049/ZudilinConeArithmetic.lean}{293}{commonMultiplier_not_two_not_three_of_endpoint_units}{common-divisor\nexclusion}.  The proposition does not say that the two evaluations are coprime;\nit says that whatever they share misses both of the primes that matter at\n\\(3/2\\).\n\n\\begin{example}\\label{ex:commonmult}\nTake \\(W=2\\), \\(U=X^2+3\\) and \\(V=5X^2+1\\).  The top endpoint of \\(U\\) and the\nconstant endpoint of \\(V\\) are both \\(1\\), and\n\\[\n H_2(U)=3\\cdot4+1\\cdot9=21,\\qquad H_2(V)=1\\cdot4+5\\cdot9=49 .\n\\]\nHere \\(\\gcd(21,49)=7\\), so a common divisor does exist and is not small; it\nis simply coprime to \\(6\\).\n\\end{example}\n\n\\begin{corollary}[no gain from row content or a common divisor at\n\\(3/2\\)]\\label{res:nomult}\nUnder the endpoint hypotheses of Proposition~\\ref{res:commonmult}, neither\nrowwise scalar content nor a common divisor of the two specialised\nevaluations \\(H_W(U)\\) and \\(H_W(V)\\) can manufacture the two local primes\nrequired at \\(3/2\\).\n\\end{corollary}\n\n\\begin{proof}\nBy Theorem~\\ref{res:content} rowwise content multiplies the analytic error and\nthe exterior determinant, including the absolute determinant height, by exactly\nthe factors it introduces, so a divisor obtained that way is paid for by the\nsame factor in that height.  By Proposition~\\ref{res:commonmult} an integer\ndividing both specialised evaluations is divisible by neither \\(2\\) nor \\(3\\).\n\\end{proof}\n\n\\noindent Both ingredients are Lean-checked; the combination is an ordinary\ndeduction and is not separately formalised.  The corollary excludes two ways of\nproducing the required gain.  It does not show that a gain of that kind is\nnecessary for a proof by linear forms at \\(3/2\\).\n\nCorollary~\\ref{res:nomult} excludes the two multiplicative devices above, and\nwhat the rest of this section pursues in their place is additive: rather than multiplying one\nrow by a scalar, take an integer combination of several rows and ask that the\ncombination be divisible where the individual rows are not.  The endpoint\ncongruences are the first case of a divisibility condition that can be imposed\nto any depth, and it is that condition, read additively, which is counted\nbelow.\n\nWe first raise the two congruences to prime powers.  Fix depths \\(R,S\\ge0\\).\nFor \\(P\\in\\Z[X]\\) the \\emph{bottom jet} \\(J_{3,R}(P)\\) is the residue of\n\\(H_W(P)\\) modulo \\(3^R\\), and the \\emph{top jet} \\(J_{2,S}(P)\\) is its residue\nmodulo \\(2^S\\); Theorem~\\ref{res:endpoints} computes them at \\(R=S=1\\).  Their\nvanishing is exactly the requested divisibility:\n\\[\n J_{3,R}(P)=0\\iff 3^R\\mid H_W(P),\\qquad\n J_{2,S}(P)=0\\iff 2^S\\mid H_W(P).\n\\]\nThese are the checked\n\\lword{Erdos1049/ZudilinConeArithmetic.lean}{189}{bottomJet3_eq_zero_iff_dvd}{bottom-jet\ndivisibility criterion} and\n\\lword{Erdos1049/ZudilinConeArithmetic.lean}{195}{topJet2_eq_zero_iff_dvd}{top-jet\ndivisibility criterion}.  The \\emph{four-jet signature} of a coefficient pair\n\\((U,V)\\) is then the quadruple\n\\[\n \\bigl(J_{3,R}(U),J_{3,R}(V),J_{2,S}(U),J_{2,S}(V)\\bigr)\n \\in(\\Z/3^R\\Z)^2\\times(\\Z/2^S\\Z)^2 ,\n\\]\ntwo residues for each of the two primes, one from each entry of the pair.  By\nthe displayed criteria it vanishes exactly when \\(3^R\\) divides both specialised\nentries and \\(2^S\\) divides both.\n\nThis turns the missing local divisor into an additive\ncongruence-kernel problem: what is sought is no longer a common divisor of\nthe two evaluations, but a vector of small integer coefficients on which four\nresidues vanish at once.  Since \\(H_W\\) is linear in the coefficients of \\(P\\),\nthe four-jet signature of a combination is the corresponding combination of\nsignatures, which is what makes the following count possible.\n\n\\begin{theorem}[binary four-jet collision]\\label{res:jetkernel}\nFix a width \\(W\\) and depths \\(R,S\\), and let \\((U_j,V_j)_{j<M}\\) be any \\(M\\)\npairs of integral polynomials.  Call a subset of \\(\\{0,\\dots,M-1\\}\\), equivalently\na vector of \\(\\{0,1\\}^M\\), a \\emph{binary selector}.  If the\n\\(2^M\\) binary selectors outnumber the finite four-jet target\n\\[\n (\\Z/3^R\\Z)^2\\times(\\Z/2^S\\Z)^2,\n\\]\nthen two distinct subsets have the same four-jet sum.  Subtracting their\nindicator vectors gives a nonzero coefficient vector in\n\\(\\{-1,0,1\\}^M\\) cancelling all four jets.  The target has exact cardinality\n\\[\n (3^R)^2(2^S)^2.\n\\]\nIn particular, if \\(R>0\\) and \\(4R+2S\\le M\\), such a collision exists.\n\\end{theorem}\n\n\\begin{proof}\nSend each binary selector to the sum of the four-jet signatures it selects.\nThe claimed cardinal inequality and the pigeonhole principle give two\ndistinct selectors in the same fibre.  The cardinality formula is the product\nof the four cyclic-modulus cardinalities.  For \\(R>0\\),\n\\[\n (3^R)^2(2^S)^2<(4^R)^2(2^S)^2=2^{4R+2S}\\le2^M,\n\\]\nwhich proves the stated sufficient threshold.\n\\end{proof}\n\n\\noindent The target count is the checked\n\\lword{Erdos1049/ZudilinConeArithmetic.lean}{129}{fourJetSignature_card}{four-jet\ntarget cardinality}; the abstract collision is the checked\n\\lword{Erdos1049/ZudilinConeArithmetic.lean}{140}{exists_distinct_binary_selectors_same_fourJet}{four-jet\npigeonhole kernel}, and the linear sufficient condition is the checked\n\\lword{Erdos1049/ZudilinConeArithmetic.lean}{159}{exists_distinct_binary_selectors_same_fourJet_of_rank}{rank--depth\ncollision threshold}.  The count is a routine pigeonhole; the reformulation\nabove is what makes it relevant.  Pigeonhole cancellation itself requires no\nindependence.  Independence or a non-collapsed deformation is needed only to\nensure that the resulting nonzero selector difference has a nonzero\npolynomial pair and analytic remainder.  None of the statements proved here\nproduces a sufficiently large primitive non-collapsed family, or proves that a\ncollision has nonzero polynomial pair or nonzero remainder.\n\n\\begin{example}\\label{ex:jetcount}\nAt depths \\(R=S=1\\) the four-jet target is\n\\((\\Z/3\\Z)^2\\times(\\Z/2\\Z)^2\\), of cardinality \\(9\\cdot4=36\\), and the threshold\nreads \\(M\\ge4\\cdot1+2\\cdot1=6\\).  With six pairs there are \\(2^{6}=64\\) binary\nselectors against \\(36\\) targets, so two of them collide and their difference\nis a vector in \\(\\{-1,0,1\\}^{6}\\), not identically zero, killing all four jets.\n\\end{example}\n\nInformally, the theorem says only this: once there are at least \\(4R+2S\\) pairs\nand the bottom depth \\(R\\) is positive, some coefficient vector in\n\\(\\{-1,0,1\\}^M\\), not identically zero, kills all four jets of the\ncorresponding combination.  It does not say that the combination is nonzero as\na pair of polynomials, and it does not say that its remainder is nonzero.  Those\nare the two obligations Problem~\\ref{prob:kernel} carries.\n\nOne further exclusion is recorded here.  It concerns the scalar parameters of\nZudilin's cone rather than the coefficient polynomials.\n\n\\begin{theorem}[scalar margin no-go]\\label{res:scalar}\nLet \\(C_1>0\\).  If \\(C_0\\le0\\) or \\(2C_0\\le C_1\\), then\n\\[\n C_0\\log 3-C_1\\log 2<0.\n\\]\n\\end{theorem}\n\n\\noindent Written multiplicatively, the conclusion is \\(3^{C_0}<2^{C_1}\\).  The\ninequality is immediate from \\(\\log3<2\\log2\\) and \\(C_1>0\\), and\nis the checked\n\\lword{Erdos1049/ZudilinConeArithmetic.lean}{323}{three_two_scalar_margin_neg}{three-halves\nscalar margin}.  The interest is in the fence around it.  The primary Zudilin\ntheorem~\\cite[Theorem~1, p.~154; Sec.~5, pp.~161--162]{zudilin2004} supplies an integer-base\nirrationality-exponent estimate on its parameter cone, and the elementary\ninequality \\(\\mu\\ge2\\) then forces \\(2C_0\\le C_1\\) whenever \\(C_0>0\\); Lean\nchecks that implication separately.  The primary theorem assumes an integer\nbase \\(p=1/q\\).  It does not state a rational \\(p=3/2\\) theorem, so the\nall-scale coefficient construction and rational specialisation remain\nexternal to the checked result, which is the scalar parameter margin alone.\n\n\\section{Failure of coordinatewise clearing at $3/2$}\\label{sec:corridor}\n\nThis section and the next return to the elementary route and record what the\nresidue $s^{n}$ costs there.  We first isolate the arithmetic that the clearing\nscheme leaves behind, in a form that does not mention the series.  The point of\ndoing so is that the clearing argument asks for two things at once, that a\npower of the numerator divide what has been accumulated and that what survives\nbe small, and these are easier to play off against each other once the series\nhas been discarded and only six natural numbers remain.\n\n\\begin{definition}[coordinatewise corridor]\\label{def:corridor}\nLet $a,b,N,K,Q,D$ be natural numbers.  Say that $(a,b,N,K,Q,D)$ is a\n\\emph{coordinatewise corridor} when\n\\[\n a>0,\\qquad Q>0,\\qquad D>0,\\qquad D\\le N+K,\\qquad\n a^{K}\\mid QD,\\qquad Q\\,b^{\\,N+K+1}<a^{\\,K+1}.\n\\]\n\\end{definition}\n\n\\noindent The reading is: $a$ and $b$ are the numerator and denominator of the\nbase, playing the roles of $r$ and $s$ in the introduction, so that\n$(a,b)=(3,2)$ is the case of interest;\n$N$ is the shift, $K$ is the width of the cleared window, $Q$ is the\naccumulated clearing factor, and $D$ is the final coefficient being cleared.\nThe bound $D\\le N+K$ is the only property of the coefficient used; for the\ndivisor-counting coefficient it holds because $\\tau(n)\\le n$.  The divisibility\n$a^{K}\\mid QD$ is the requirement that clearing succeeded coordinatewise, and\nthe last inequality is the tail estimate that makes the trapped integer smaller\nthan~$1$.  The name records the shape of the constraint: the divisibility\nbounds $a^{K}$ from above by $Q(N+K)$, the tail estimate bounds $a^{K+1}$ from\nbelow by $Q\\,b^{\\,N+K+1}$, and admissible parameters must fit in the band\nbetween them.  The definition is the\n\\lword{Erdos1049/RationalBaseLambert.lean}{106}{CoordinatewiseCorridor}{corridor\npredicate}.\n\n\\begin{theorem}[corridor bound]\\label{res:corridorbound}\nIf $(a,b,N,K,Q,D)$ is a coordinatewise corridor, then\n\\[\n b^{\\,N+K+1}<a\\,(N+K).\n\\]\n\\end{theorem}\n\n\\begin{proof}\nA routine divisibility computation.  Since $QD>0$ and $a^{K}\\mid QD$, we have\n$a^{K}\\le QD$, and $D\\le N+K$ gives\n$a^{K}\\le Q(N+K)$.  Hence\n\\[\n Q\\,b^{\\,N+K+1}<a^{\\,K+1}=a^{K}\\cdot a\\le Q(N+K)\\cdot a,\n\\]\nand cancelling the positive factor $Q$ gives the claim.\n\\end{proof}\n\n\\noindent Formalised as the\n\\lword{Erdos1049/RationalBaseLambert.lean}{114}{coordinatewiseCorridor_implies_pow_lt_linear}{power-versus-linear\nconsequence}.\n\nThe inequality of Theorem~\\ref{res:corridorbound} is where the integer and\nrational cases part.  At $b=1$ it reads $1<a(N+K)$, which holds for every\n$a\\ge2$ and every nonempty window; the corridor imposes no obstruction at all.\nAt $b\\ge2$ the left side is exponential in $N+K$ and the right side is linear,\nso the corridor can survive only for small $N+K$.  At $b=2$ the crossing has\nalready happened at the smallest admissible window.\n\n\\begin{example}\\label{ex:corridor}\nTake the smallest window, $N=K=1$, and numerator $a=3$.  At $b=1$ the tuple\n$(3,1,1,1,3,1)$ is a corridor: $D=1\\le2$, the divisibility reads $3\\mid3$, and\nthe tail estimate reads $3\\cdot1^{3}=3<3^{2}=9$.  At $b=2$ no choice works.  The\ntail estimate becomes $Q\\cdot2^{3}<3^{2}$, which forces $Q=1$; the divisibility\nthen reads $3\\mid D$, and the only candidates are $D=1$ and $D=2$, neither\ndivisible by $3$.  This is the proof of\nTheorem~\\ref{res:corridorbound} in miniature: it derives $a^{K}\\le Q(N+K)$,\nhere $3\\le2$.\n\\end{example}\n\n\\begin{proposition}\\label{res:exp}\nFor every natural number $x\\ge2$ we have $3x<2^{\\,x+1}$.\n\\end{proposition}\n\n\\begin{proof}\nA routine induction from $x=2$, where $6<8$.  For the step, $2^{x+1}\\ge2^{2}>3$ when\n$x\\ge1$, so $3(x+1)=3x+3<2^{x+1}+2^{x+1}=2^{x+2}$.\n\\end{proof}\n\n\\noindent Formalised as the\n\\lword{Erdos1049/RationalBaseLambert.lean}{135}{three_mul_lt_two_pow_succ}{exponential\ncomparison}.\n\n\\begin{theorem}[no corridor at base $3/2$]\\label{res:nocorridor}\nFor all $N\\ge1$ and $K\\ge1$ and all natural $Q,D$, the tuple\n$(3,2,N,K,Q,D)$ is not a coordinatewise corridor.\n\\end{theorem}\n\n\\begin{proof}\nA corridor would give $2^{\\,N+K+1}<3(N+K)$ by\nTheorem~\\ref{res:corridorbound}, contradicting Proposition~\\ref{res:exp}\napplied to $x=N+K\\ge2$.\n\\end{proof}\n\n\\noindent Formalised as the\n\\lword{Erdos1049/RationalBaseLambert.lean}{148}{threeHalves_no_coordinatewiseCorridor}{corridor\nexclusion at three halves}.\n\nTheorem~\\ref{res:nocorridor} excludes the coordinatewise clearing scheme at\n$3/2$, and nothing else: it does not bound the denominator of $F(3/2)$, it does\nnot show that $F(3/2)$ is irrational, and it does not show that $F(3/2)$ is\nrational.  It also does not cover a clearing scheme of a different shape, since\nthe corridor fixes one divisibility pattern and one tail inequality.\n\n\\section{The cleared-tail recurrence and the size of the forcing\nterm}\\label{sec:tail}\n\nTheorem~\\ref{res:nocorridor} says that one scheme fails.  This section\nidentifies the quantity responsible, as an exact recurrence.\n\nLet $r,s,B,F$ be rationals with $r\\ne0$ and let $c:\\N\\to\\Q$ be arbitrary.\nDefine the prefix and the cleared tail state by\n\\[\n P_N=\\sum_{m=0}^{N-1}\\frac{c(m+1)\\,s^{\\,m+1}}{r^{\\,m+1}},\n \\qquad\n U_N=B\\,r^{N}\\bigl(F-P_N\\bigr).\n\\tag{$\\ast$}\\label{eq:tailstate}\n\\]\nThus $P_N$ is the partial sum of $\\sum_{n\\ge1}c(n)(s/r)^{n}$ through level $N$,\nand $U_N$ is the tail of a putative value $F$ after that level, scaled by\n$Br^{N}$.  These are the\n\\lword{Erdos1049/RationalBaseLambert.lean}{161}{rationalBasePrefixQ}{rational-base\nprefix} and the\n\\lword{Erdos1049/RationalBaseLambert.lean}{174}{rationalBaseClearedTailQ}{cleared\ntail state}.\n\n\\begin{theorem}[cleared-tail recurrence]\\label{res:tailrec}\nLet $r,s,B,F\\in\\Q$ with $r\\ne0$, let $c:\\N\\to\\Q$, and let $P_N$ and $U_N$ be as\nin~\\eqref{eq:tailstate}.  Then for every $N$,\n\\[\n U_{N+1}=r\\,U_N-B\\,c(N+1)\\,s^{\\,N+1}.\n\\]\n\\end{theorem}\n\n\\begin{proof}\nAn immediate computation.  Expanding\n$P_{N+1}=P_N+c(N+1)s^{N+1}/r^{N+1}$ and $r^{N+1}=r^{N}\\cdot r$ in the\ndefinition of $U_{N+1}$ and clearing the denominator $r^{N+1}$, which is\nnonzero, gives the identity.\n\\end{proof}\n\n\\noindent Formalised as the\n\\lword{Erdos1049/RationalBaseLambert.lean}{180}{rationalBaseClearedTailQ_succ}{cleared-tail\nrecurrence}.\n\nThe recurrence has a linear part $rU_N$ and a forcing term $Bc(N+1)s^{N+1}$,\nthe inhomogeneous term the state receives at each step.\nThe direct-clearing route studied here works only if the state remains in a\nbounded window, and the forcing term is what that window must absorb.  Its\nsize is what separates the two denominator regimes.\n\n\\begin{theorem}[the forcing term]\\label{res:forcing}\nLet $s,B$ be natural numbers and $c:\\N\\to\\N$, and put\n$G_N=B\\,c(N+1)\\,s^{\\,N+1}$.\n\\begin{enumerate}\n\\item If $s\\ge2$, $B\\ge1$ and $c(N+1)\\ge1$, then $2^{\\,N+1}\\le G_N$.\n\\item If $s=1$, then $G_N=B\\,c(N+1)$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nBoth parts are routine.  For the first,\n$2^{N+1}\\le s^{N+1}=1\\cdot s^{N+1}\\le Bc(N+1)s^{N+1}$,\nusing $B\\,c(N+1)\\ge1$.  The second part is the definition with $s=1$.\n\\end{proof}\n\n\\noindent Formalised as the\n\\lword{Erdos1049/RationalBaseLambert.lean}{197}{twoPow_le_rationalBaseForcingNat}{exponential\nlower bound} and the\n\\lword{Erdos1049/RationalBaseLambert.lean}{211}{rationalBaseForcingNat_one}{integer-base\ncollapse}.\n\nAt $s=1$ the forcing term is $Bc(N+1)$, so it grows only as fast as the\ncoefficient; for the divisor function that is $O(N^{\\varepsilon})$ for every\n$\\varepsilon>0$, and a bounded-state argument has room.  At $s\\ge2$ the same\nterm is at least $2^{N+1}$ whenever the coefficient is nonzero.\n\n\\begin{example}\\label{ex:forcing}\nTake $B=1$, $c=\\tau$ and $N=9$, so that the coefficient is $\\tau(10)=4$.  At\n$s=1$ the forcing term is $4$.  At $s=2$ it is $4\\cdot2^{10}=4096$, and part~(1)\nof Theorem~\\ref{res:forcing} already guarantees at least $2^{10}=1024$ without\nknowing the coefficient at all.\n\\end{example}\n\nThis is an\nexact lower bound on one quantity, and it is all that is proved.  It is not a\nproof that no bounded-state argument exists at $s\\ge2$; the theorem that one\nparticular scheme fails is Theorem~\\ref{res:nocorridor}, and the bound here\nrecords the size of the term that scheme would have to absorb.\n\n\\section{The height criterion at $7/2$}\\label{sec:sevenhalves}\n\nIn 1994 Bundschuh and V\\\"a\\\"an\\\"anen proved an irrationality criterion for a\nfamily of rational bases cut out by a height condition\n\\cite[Theorem~2, p.~177; hypotheses pp.~175--176]{bv1994}.  We keep their\nnotation: $q$ is the base, and $\\alpha$\nand $\\lambda$ are the parameters of the criterion.  In its special case\n$\\alpha=-1$, the printed hypothesis is\n\\[\n \\lambda<\\left(\\frac12+\\frac1{\\pi^2}\\right)^{-1}.\n\\]\nAt $q=7/2$ the Archimedean parameter is\n$\\lambda=\\log 7/\\log(7/2)$.  The criterion therefore applies once the following\nstrict inequality is checked.\n\n\\begin{theorem}[the $7/2$ height condition]\\label{res:sevenhalves}\n\\[\n \\frac{\\log 7}{\\log(7/2)}\n <\n \\left(\\frac12+\\frac1{\\pi^2}\\right)^{-1}.\n\\]\n\\end{theorem}\n\n\\noindent Numerically the two sides are $1.5533\\ldots$ and $1.6630\\ldots$, so\nthe condition holds with a margin of about $0.11$.  The Lean proof factors the\nestimate into the\n\\lword{Erdos1049/RationalBaseLambert.lean}{32}{sevenHalves_power_certificate}{integer\ncertificate} $2^{18}<7^7$, the resulting\n\\lword{Erdos1049/RationalBaseLambert.lean}{41}{sevenHalves_log_ratio_lt}{logarithmic\nratio bound} $\\log2/\\log7<7/18$, the\n\\lword{Erdos1049/RationalBaseLambert.lean}{57}{one_div_pi_sq_lt_one_nine}{$\\pi$-bound}\n$1/\\pi^2<1/9$, and the\n\\lword{Erdos1049/RationalBaseLambert.lean}{66}{sevenHalves_bundschuhVaananen_margin}{strict\nmargin}\n\\[\n \\frac{\\log2}{\\log7}<\\frac12-\\frac1{\\pi^2}.\n\\]\nThe\n\\lword{Erdos1049/RationalBaseLambert.lean}{76}{sevenHalves_archimedean_height_condition}{final\nheight inequality} then rewrites\n$\\log(7/2)=\\log7-\\log2$ and closes the displayed condition.\n\nThis formalises the complete elementary parameter check at $q=7/2$; it does\nnot formalise Bundschuh and V\\\"a\\\"an\\\"anen's analytic irrationality theorem,\nwhose proof occupies pp.~189--193 of the source.  The conclusion that\n$F(7/2)$ is irrational is consequently cited from that theorem, not claimed as\na Lean theorem here.\n\nThe criterion does not cover $3/2$, and the results of\nSections~\\ref{sec:corridor} and~\\ref{sec:tail} give no evidence either way\nabout whether $F(3/2)$ is irrational.\n\n\\section{Denominator exponents for a homogenised Pad\\'e\nconstruction}\\label{sec:pade}\n\nA second external route builds the linear forms of Section~\\ref{sec:primitive}\nby Pad\\'e approximation.  Such a construction produces its coefficients as sums\nof rational summands, and to obtain integer rows one multiplies through by a\nsingle common denominator.  The bookkeeping obligation is then to check that no\nsummand needs a larger denominator than the one proposed, which is a comparison\nbetween the exponents alone.  This section does that comparison, and only that.\n\nNo source construction is named here for the exponent expressions below, and\nthey are checked as displayed polynomial identities rather than derived.\nGiving them Pad\\'e-theoretic force would additionally require deriving the\nexpressions from a stated homogenised coefficient formula, proving integrality\nof the coefficients after multiplication by the proposed common denominator,\nand establishing nonvanishing and decay of the associated remainder.  None of\nthose three is claimed here.\n\nFor a\nhomogenised construction over integer parameters the proposed common\ndenominator exponent is $E_n=(3n^{2}-n)/2$.  Since only doubled exponents\noccur below, every statement lives over $\\Z$ and no parity bookkeeping is\nneeded; we write $\\widetilde{E}_n=2E_n=3n^{2}-n$.\n\n\\begin{proposition}[summand bound and exact gap]\\label{res:pade}\nLet $\\widetilde{E}_n=3n^{2}-n$ and put\n\\[\n \\widetilde{P}(n,k)=2\\bigl(k(n-k)+nk\\bigr)+k(k-1),\n\\]\n\\[\n \\widetilde{Q}(n,m)=2(n^{2}-n)+j^{2}+2jm+j-m^{2}+3m,\n \\qquad j=n-m-1 .\n\\]\nThen, for integers $n,k,m$:\n\\begin{enumerate}\n\\item if $0\\le k\\le n$, then $\\widetilde{P}(n,k)\\le\\widetilde{E}_n$, and the\ngap factors as\n$\\widetilde{E}_n-\\widetilde{P}(n,k)=(n-k)(3n-k-1)$;\n\\item $\\widetilde{E}_n-\\widetilde{Q}(n,m)=2\\bigl(n+m(m-1)\\bigr)$ identically;\n\\item if $n\\ge0$ and $m\\ge1$, then $\\widetilde{Q}(n,m)\\le\\widetilde{E}_n$.\n\\end{enumerate}\n\\end{proposition}\n\n\\noindent Part~(1) is the\n\\lword{Erdos1049/RationalPadeArithmetic.lean}{30}{rationalPadePSummandDenExpTwice_le}{summand\nexponent bound}, part~(2) the\n\\lword{Erdos1049/RationalPadeArithmetic.lean}{52}{rationalPadeQMaxDenExpTwice_gap}{exact\ngap identity}, and part~(3) the\n\\lword{Erdos1049/RationalPadeArithmetic.lean}{60}{rationalPadeQMaxDenExpTwice_le}{maximal\nexponent bound}.  The gap in~(2) is an identity in $\\Z[n,m]$, so~(3) follows\nfrom $m(m-1)\\ge0$ and $n\\ge0$; the hypothesis $m\\ge1$ names the range in which\nthe corresponding summand does not vanish, and it is not the sharpest\nhypothesis under which the inequality holds.\n\n\\begin{example}\\label{ex:pade}\nAt $n=2$ the proposed doubled exponent is $\\widetilde{E}_2=10$, and\n$\\widetilde{P}(2,k)$ takes the values $0,6,10$ at $k=0,1,2$.  The three gaps are\n$10,4,0$, matching the factorisation $(2-k)(5-k)$ of part~(1); the summand at\n$k=2$ is the one that saturates the proposed denominator.  For part~(2), at\n$m=1$ we have $j=0$ and $\\widetilde{Q}(2,1)=6$, with gap\n$4=2\\bigl(2+1\\cdot0\\bigr)$.\n\\end{example}\n\nThese are routine inequalities between polynomials in the exponents.  They\nestablish that the proposed exponent $\\widetilde{E}_n$ dominates the two\ndisplayed summand exponent expressions, and nothing further.  Positivity of the remainder, its rate of\ndecay, and the comparison of that rate against the denominator height are the\nanalytic obligations, and none of them is treated here, so nothing in this\nsection is an irrationality measure.\n\n\\section{Complements and further questions}\\label{sec:open}\n\nProblem~\\#1049 is open.  The following is the exact statement that a\nconstruction at \\(3/2\\) would have to supply.  Theorem~\\ref{res:jetkernel}\nsupplies the combinatorial step: at depths \\(R,S\\) with \\(R>0\\), once at least\n\\(4R+2S\\) pairs are in hand, a coefficient vector on which all four jets vanish\nexists.  What the problem adds is that those pairs be primitive-normalised\nHeine--Zudilin pairs drawn from a family that does not degenerate, that the\ndepths grow quadratically, and that the resulting remainder be nonzero with a\nlocal gain beating the Archimedean height.\n\n\\begin{problem}[growing-rank simultaneous endpoint-jet kernel]\\label{prob:kernel}\nFor positive quadratic target depths \\(R_n,S_n\\), exhibit at least\n\\(4R_n+2S_n\\) \\emph{primitive-normalised} Heine--Zudilin coefficient pairs\nfrom a genuinely non-collapsed deformation, and prove that at least one\nresulting checked collision satisfies\n\\[\n J_{3,R_n}\\!\\left(\\sum_j\\lambda_jU_j\\right)=\n J_{3,R_n}\\!\\left(\\sum_j\\lambda_jV_j\\right)=0,\n\\qquad\n J_{2,S_n}\\!\\left(\\sum_j\\lambda_jU_j\\right)=\n J_{2,S_n}\\!\\left(\\sum_j\\lambda_jV_j\\right)=0,\n\\]\nwhere \\(R_n,S_n\\) have the required quadratic order, while the resulting\nremainder is nonzero and its local gain beats the Archimedean height.  The\nchecked collision already has coefficients in \\(\\{-1,0,1\\}\\).  A\nrank-saturated contiguous-shift family or a genuinely independent deformation\nwould be a first candidate.\n\\end{problem}\n\nRescaling a row along a ray, letting the row contents grow, or computing a\nfurther common divisor of the two specialised evaluations does not address the\nobstruction that survives Corollary~\\ref{res:nomult}; each is one of the two\ndevices excluded there, and what Problem~\\ref{prob:kernel} asks for is\nadditive.  Devices outside that corollary are not addressed either way:\ninserting a multiplicative cyclotomic factor changes the coefficient\npolynomials rather than scaling a row, and cross-row common factors,\ndeterminant-specific arithmetic, additive combinations, and a different\narchitecture all lie outside the two devices it excludes.  A theorem showing that every\navailable family is too small or too degenerate to meet the requirements of\nProblem~\\ref{prob:kernel} would show that no argument of this shape can\nsucceed.\n\nThe unrestricted question, which rational bases give an irrational value,\nremains open, and is not reduced to the problem above.\n\n\\pdfbookmark[1]{Statements and declarations}{statements-declarations}\n\\subsection*{Statements and declarations}\n\n\\paragraph{Artefact and data availability.} The\n\\href{\\sourceurl}{pinned formal-source revision} contains the Lean sources, the\nfixed toolchain, and the library manifest used in the verification.  This\nmanuscript provides navigation rather than proof authority.\n\n\\paragraph{Declaration of generative AI use.} \\paperfullaideclaration\n\nLean checks each proof term against the fixed library\nversion, and the sources linked here contain no proof placeholders and no\nproject-defined axioms; Lean does not authorise the exposition, the citation\nchoices, or the interpretation, for which the author remains responsible.\n\n\\paragraph{Funding and competing interests.} This work received no external\nfunding.  The author declares no competing interests.\n\n\\paragraph{Acknowledgements.} The problem numbering and status follow the\nErd\\H{o}s Problems catalogue maintained by Thomas Bloom~\\cite{erdosproblems}.\n\n\\appendix\n\\section{Guide to the formal sources}\\label{app:index}\n\nEach linked phrase opens its Lean declaration at the pinned source\nrevision~\\commitshort.  The declarations of this note live in three modules:\nthe corridor, the cleared-tail recurrence, and the two arithmetic certificates\nin the first; the Pad\\'e exponent arithmetic in the second; and the\nHeine--Zudilin endpoint arithmetic in the third.  The link\ncoordinates are validated against that pinned revision, so they remain correct\nas later work moves lines in the working tree.\n\n% The exhaustive source manifest is retained in the authored TeX for automated\n% validation, and kept outside the printed note.\n\\iffalse\n\n\\begin{tabular}{@{}ll@{}}\ninformal statement & linked Lean source\\\\\ncorridor predicate & \\lrefx{Erdos1049/RationalBaseLambert.lean}{106}{CoordinatewiseCorridor}\\\\\ncorridor bound & \\lrefx{Erdos1049/RationalBaseLambert.lean}{114}{coordinatewiseCorridor_implies_pow_lt_linear}\\\\\nexponential comparison & \\lrefx{Erdos1049/RationalBaseLambert.lean}{135}{three_mul_lt_two_pow_succ}\\\\\nno corridor at $3/2$ & \\lrefx{Erdos1049/RationalBaseLambert.lean}{148}{threeHalves_no_coordinatewiseCorridor}\\\\\nrational-base prefix & \\lrefx{Erdos1049/RationalBaseLambert.lean}{161}{rationalBasePrefixQ}\\\\\nprefix step & \\lrefx{Erdos1049/RationalBaseLambert.lean}{166}{rationalBasePrefixQ_succ}\\\\\ncleared tail state & \\lrefx{Erdos1049/RationalBaseLambert.lean}{174}{rationalBaseClearedTailQ}\\\\\ncleared-tail recurrence & \\lrefx{Erdos1049/RationalBaseLambert.lean}{180}{rationalBaseClearedTailQ_succ}\\\\\nforcing magnitude & \\lrefx{Erdos1049/RationalBaseLambert.lean}{191}{rationalBaseForcingNat}\\\\\nexponential lower bound & \\lrefx{Erdos1049/RationalBaseLambert.lean}{197}{twoPow_le_rationalBaseForcingNat}\\\\\ninteger-base collapse & \\lrefx{Erdos1049/RationalBaseLambert.lean}{211}{rationalBaseForcingNat_one}\\\\\ninteger power comparison & \\lrefx{Erdos1049/RationalBaseLambert.lean}{32}{sevenHalves_power_certificate}\\\\\nexact rational boundary & \\lrefx{Erdos1049/RationalBaseLambert.lean}{36}{sevenHalves_rational_margin}\\\\\nlogarithmic ratio bound & \\lrefx{Erdos1049/RationalBaseLambert.lean}{41}{sevenHalves_log_ratio_lt}\\\\\n$\\pi$ reciprocal-square bound & \\lrefx{Erdos1049/RationalBaseLambert.lean}{57}{one_div_pi_sq_lt_one_nine}\\\\\nstrict source margin & \\lrefx{Erdos1049/RationalBaseLambert.lean}{66}{sevenHalves_bundschuhVaananen_margin}\\\\\ncomplete $7/2$ height condition & \\lrefx{Erdos1049/RationalBaseLambert.lean}{76}{sevenHalves_archimedean_height_condition}\\\\\ndoubled common exponent & \\lrefx{Erdos1049/RationalPadeArithmetic.lean}{19}{rationalPadeDenExpTwice}\\\\\ndoubled summand exponent & \\lrefx{Erdos1049/RationalPadeArithmetic.lean}{24}{rationalPadePSummandDenExpTwice}\\\\\nsummand exponent bound & \\lrefx{Erdos1049/RationalPadeArithmetic.lean}{30}{rationalPadePSummandDenExpTwice_le}\\\\\ndoubled maximal exponent & \\lrefx{Erdos1049/RationalPadeArithmetic.lean}{46}{rationalPadeQMaxDenExpTwice}\\\\\nexact gap identity & \\lrefx{Erdos1049/RationalPadeArithmetic.lean}{52}{rationalPadeQMaxDenExpTwice_gap}\\\\\nmaximal exponent bound & \\lrefx{Erdos1049/RationalPadeArithmetic.lean}{60}{rationalPadeQMaxDenExpTwice_le}\\\\\nPad\\'e error scaling & \\lrefx{Erdos1049/RationalPadeArithmetic.lean}{84}{rationalPadeError_mul}\\\\\nrow-content determinant factorisation & \\lrefx{Erdos1049/RationalPadeArithmetic.lean}{94}{rationalPadeExteriorDet_mul_contents}\\\\\nabsolute determinant scaling & \\lrefx{Erdos1049/RationalPadeArithmetic.lean}{104}{natAbs_rationalPadeExteriorDet_mul_contents}\\\\\ncontent-product divisibility & \\lrefx{Erdos1049/RationalPadeArithmetic.lean}{116}{contentProduct_dvd_rationalPadeExteriorDet}\\\\\nexterior determinant identity & \\lrefx{Erdos1049/RationalPadeArithmetic.lean}{124}{rationalPadeExteriorDet_cast_eq}\\\\\nhomogeneous endpoint evaluation & \\lrefx{Erdos1049/ZudilinConeArithmetic.lean}{96}{homEvalThreeTwo}\\\\\nfour-jet target cardinality & \\lrefx{Erdos1049/ZudilinConeArithmetic.lean}{129}{fourJetSignature_card}\\\\\nfour-jet pigeonhole kernel & \\lrefx{Erdos1049/ZudilinConeArithmetic.lean}{140}{exists_distinct_binary_selectors_same_fourJet}\\\\\nrank--depth collision threshold & \\lrefx{Erdos1049/ZudilinConeArithmetic.lean}{159}{exists_distinct_binary_selectors_same_fourJet_of_rank}\\\\\nbottom-jet divisibility & \\lrefx{Erdos1049/ZudilinConeArithmetic.lean}{189}{bottomJet3_eq_zero_iff_dvd}\\\\\ntop-jet divisibility & \\lrefx{Erdos1049/ZudilinConeArithmetic.lean}{195}{topJet2_eq_zero_iff_dvd}\\\\\nbottom endpoint modulo three & \\lrefx{Erdos1049/ZudilinConeArithmetic.lean}{201}{homEvalThreeTwo_mod_three}\\\\\ntop endpoint modulo two & \\lrefx{Erdos1049/ZudilinConeArithmetic.lean}{215}{homEvalThreeTwo_mod_two}\\\\\nunit constant endpoint obstruction & \\lrefx{Erdos1049/ZudilinConeArithmetic.lean}{233}{three_not_dvd_homEvalThreeTwo_of_const_unit}\\\\\nunit top endpoint obstruction & \\lrefx{Erdos1049/ZudilinConeArithmetic.lean}{251}{two_not_dvd_homEvalThreeTwo_of_top_unit}\\\\\ncommon-multiplier exclusion & \\lrefx{Erdos1049/ZudilinConeArithmetic.lean}{293}{commonMultiplier_not_two_not_three_of_endpoint_units}\\\\\nirrationality-exponent inequality & \\lrefx{Erdos1049/ZudilinConeArithmetic.lean}{311}{twice_le_of_irrationalityExponent_bounds}\\\\\nthree-halves scalar margin & \\lrefx{Erdos1049/ZudilinConeArithmetic.lean}{323}{three_two_scalar_margin_neg}\\\\\n\\end{tabular}\n\\fi\n\n\\begin{thebibliography}{99}\n\\small\n\\raggedright\n\\setlength{\\itemsep}{0pt}\n\\bibitem{erdos1948} P.~Erd\\H{o}s,\n  \\href{https://users.renyi.hu/~p_erdos/1948-04.pdf}{\\emph{On arithmetical\n  properties of Lambert series}},\n  J. Indian Math. Soc. (N.S.) \\textbf{12} (1948), 63--66.\n\\bibitem{erdos1988} P.~Erd\\H{o}s, \\emph{On the irrationality of certain series:\n  problems and results}, in A.~Baker (ed.), \\emph{New Advances in Transcendence\n  Theory}, Cambridge UP, 1988, pp.~102--109,\n  doi:\\href{https://doi.org/10.1017/CBO9780511897184.009}{10.1017/CBO9780511897184.009}.\n\\bibitem{borwein1991} P.~B. Borwein, \\emph{On the irrationality of\n  $\\sum1/(q^{n}+r)$},\n  J. Number Theory \\textbf{37} (1991), no.~3, 253--259,\n  doi:\\href{https://doi.org/10.1016/S0022-314X(05)80041-1}{10.1016/S0022-314X(05)80041-1}.\n\\bibitem{bv1994} P.~Bundschuh and K.~V\\\"a\\\"an\\\"anen,\n  \\href{https://numdam.org/item/CM_1994__91_2_175_0.pdf}{\\emph{Arithmetical\n  investigations of a certain infinite product}},\n  Compositio Math. \\textbf{91} (1994), no.~2, 175--199.\n\\bibitem{zudilin2004} W.~Zudilin,\n  \\href{https://www.impan.pl/shop/en/publication/transaction/download/product/82435}{\\emph{Heine's\n  basic transform and a permutation group for \\(q\\)-harmonic series}},\n  Acta Arith. \\textbf{111} (2004), no.~2, 153--164,\n  doi:\\href{https://doi.org/10.4064/aa111-2-4}{10.4064/aa111-2-4}.\n\\bibitem{rhinviola1996} G.~Rhin and C.~Viola,\n  \\emph{On a permutation group related to $\\zeta(2)$},\n  Acta Arith. \\textbf{77} (1996), no.~1, 23--56,\n  doi:\\href{https://doi.org/10.4064/aa-77-1-23-56}{10.4064/aa-77-1-23-56}.\n\\bibitem{vanassche2001} W.~Van Assche,\n  \\href{https://arxiv.org/abs/math/0101187}{\\emph{Little \\(q\\)-Legendre\n  polynomials and irrationality of certain Lambert series}},\n  Ramanujan J. \\textbf{5} (2001), no.~3, 295--310,\n  doi:\\href{https://doi.org/10.1023/A:1012930828917}{10.1023/A:1012930828917}.\n\\bibitem{vandehey2013} J.~Vandehey,\n  \\href{https://arxiv.org/abs/1206.0340}{\\emph{On an incomplete argument of\n  Erd\\H{o}s on the irrationality of Lambert series}},\n  Integers \\textbf{13} (2013), Paper A58.\n\\bibitem{rivin2026} I.~Rivin,\n\\href{https://arxiv.org/abs/2604.25151}{\\emph{Zero Coefficients of Rational\n  Power Series and Rational Lambert Series}}, arXiv:2604.25151v1, 2026.\n  Theorem~1.1 is on p.~2 and proved on pp.~6--7; the periodic-coefficient\n  Corollary~6.4 is on p.~9.\n\\bibitem{kovactao2024} V.~Kova\\v{c} and T.~Tao,\n  \\href{https://arxiv.org/abs/2406.17593}{\\emph{On several irrationality\n  problems for Ahmes series}}, Acta Math. Hungar. \\textbf{175} (2025),\n  572--608; arXiv:2406.17593, 2024.\n\\bibitem{erdosproblems} T.~F. Bloom,\n  \\href{https://www.erdosproblems.com/1049}{\\emph{Erd\\H{o}s Problem \\#1049}},\n  \\texttt{erdosproblems.com/1049}, accessed 28 July 2026\n  (page displays ``last edited 28 September 2025'').  The current record\n  labels the problem open, cites \\textup{[Er88c, p.~102]} and \\textup{[Er48]},\n  and explicitly describes its status as the website owner's present\n  assessment rather than a literature-completeness guarantee.\n\\bibitem{formalconjectures1049} The Formal Conjectures Authors,\n  \\href{https://github.com/google-deepmind/formal-conjectures/blob/f776d2f2039351b00737ffcafb9d7d7666e1d9af/FormalConjectures/ErdosProblems/1049.lean}\n  {\\emph{FormalConjectures.ErdosProblems.\\texttt{1049}}},\n  Lean source at commit \\texttt{f776d2f}, 2026, accessed 28 July 2026.  The\n  irrationality declarations are unproved; the Lambert-series identity is\n  proved.\n\\end{thebibliography}\n\n\\end{document}\n","dependencies":[{"record_type":"paper_source_dependency","tex_reference":"problem-note-preamble","source_path":"paper/problem-note-preamble.tex","media_type":"text/x-tex","sha256":"sha256:e7378fe7784916681e7faf96c1ea7a8bb941a416cc651b97512e849bcaeb63f7","byte_count":4990,"licence":"CC-BY-4.0","body_format":"latex","body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n%\n% Shared preamble for the Erd\u0151s Problem Notes: the short, standalone,\n% problem-owned manuscripts covering the expansion library `ErdosProblems`.\n%\n% One note owns one Erd\u0151s problem.  Each note repeats whatever context its own\n% reader needs, so that a reader who arrives at a single problem never has to\n% assemble it from the gateway paper.  Deliberate repetition across notes is the\n% design, not an oversight.\n%\n% Source links resolve against one pinned formal-source commit, declared once\n% below.  `scripts/check_problem_note_sources.py` verifies that every authored\n% (file, line, declaration) triple names that exact declaration in the pinned\n% snapshot, so a link cannot silently rot as later work moves lines.\n\n\\usepackage{paper-house-style}\n\\usepackage{array}\n\\usepackage{booktabs}\n\\usepackage{enumitem}\n\\setlist{topsep=0.35em,itemsep=0.2em}\n\\usepackage[colorlinks=true,linkcolor=linkinternal,urlcolor=linkexternal,\n  citecolor=linkinternal]{hyperref}\n\\hypersetup{bookmarksnumbered=true,bookmarksopen=true,bookmarksopenlevel=1,\n  pdfauthor={Will Cook},pdflang={en-GB}}\n\n% ---------------------------------------------------------------------------\n% Pinned formal source.  \\commit names the checkpoint every link resolves\n% against; the checker reads that snapshot rather than the working tree, so the\n% printed coordinates stay correct however later waves move declarations.\n% ---------------------------------------------------------------------------\n\\newcommand{\\commit}{d6d344d9557faee19276b01a1c1f20d7c08f03d5}\n\\newcommand{\\commitshort}{d6d344d9557f}\n\\newcommand{\\repobase}{https://github.com/wcook04/plectis-lean-erdos249-257/blob/\\commit}\n\\newcommand{\\sourceurl}{https://github.com/wcook04/plectis-lean-erdos249-257/tree/\\commit}\n\\newcommand{\\PX}{ErdosProblems}\n\n% Declaration names stay inside link targets.  The printed label is either a\n% spaced, lower-case reading of the declaration name (\\lref) or a short authored\n% phrase (\\lword); neither prints a module path or a raw Lean identifier.\n\\ExplSyntaxOn\n\\cs_new_protected:Npn \\noteleanlabel #1\n  {\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_to_str:n {#1} }\n    \\regex_replace_all:nnN { _+ } { \\c{space} } \\l_tmpa_tl\n    \\regex_replace_all:nnN\n      { ([a-z0-9])([A-Z]) }\n      { \\1\\c{space}\\2 }\n      \\l_tmpa_tl\n    \\tl_set:Nx \\l_tmpa_tl { \\tl_lower_case:n { \\tl_use:N \\l_tmpa_tl } }\n    \\tl_use:N \\l_tmpa_tl\n  }\n\\ExplSyntaxOff\n\\newcommand{\\leanlabel}[1]{\\noteleanlabel{#1}}\n\\newcommand{\\lref}[3]{\\href{\\repobase/\\PX/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\lword}[4]{\\href{\\repobase/\\PX/#1\\#L#2}{#4}}\n\\newcommand{\\lloc}[2]{\\href{\\repobase/\\PX/#1\\#L#2}{Lean source}}\n\n% Links into a sibling library, named repository-relative.  The expansion\n% library is implicit in \\lref/\\lword, because that is where problem-owned\n% work lands.  Several problems have their headline theorem in the older\n% reviewed #249/#257 corpus, and a note that could not name that library\n% would have to paraphrase a checked statement instead of linking it.\n\\newcommand{\\mref}[3]{\\href{\\repobase/#1\\#L#2}{\\leanlabel{#3}}}\n\\newcommand{\\mword}[4]{\\href{\\repobase/#1\\#L#2}{#4}}\n\\newcommand{\\mloc}[2]{\\href{\\repobase/#1\\#L#2}{Lean source}}\n\\emergencystretch=2em\n\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\theoremstyle{definition}\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{example}[theorem]{Example}\n\\newtheorem{problem}[theorem]{Problem}\n\\theoremstyle{remark}\n\\newtheorem*{remark}{Remark}\n\n\\newcommand{\\N}{\\mathbb{N}}\n\\newcommand{\\Npos}{\\mathbb{N}_{>0}}\n\\newcommand{\\Z}{\\mathbb{Z}}\n\\newcommand{\\Q}{\\mathbb{Q}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\lcm}{\\operatorname{lcm}}\n\\newcommand{\\ph}{\\varphi}\n\n% The series line printed under every note's title block.\n\\newcommand{\\seriesline}{Erd\\H{o}s Problem Notes}\n\n% Problem-note compatibility name for the corpus-owned page-one disclosure in\n% paper-house-style.sty.\n\\newcommand{\\noteprovenance}{\\paperprovenance}\n\n% The status paragraph every note carries, in its own words, before any result.\n% Kernel checking establishes the exact Lean proposition; it does not establish\n% that the proposition is the interesting one, that it is new, or that the\n% Erd\u0151s problem is closed.\n\\newcommand{\\statusboundary}{%\n  \\emph{Status.}  The problem treated here is open, and this note does not\n  close it.  Every statement below marked as checked is a proposition that the\n  pinned Lean kernel accepts from the sources this note links to, with no\n  \\texttt{sorry}, no added axiom, and no unchecked evaluation.  That is a claim\n  about the formal statement, not about its mathematical interest, its\n  novelty, or the original problem.  The unresolved obligations are named\n  exactly, in their own section, and none of the finite computations,\n  reductions, or no-go results here removes one of them.}\n"}],"external_verification_route":"https://raw.githubusercontent.com/wcook04/plectis-lean-erdos249-257/main/paper/erdos-1049-rational-base-lambert.tex"},{"record_type":"paper","paper_id":"claim-faithful-publication-systems","title":"From Lean Proofs to Public Claims","source_repository":"plectis-lean-erdos249-257","source_path":"paper/claim-faithful-publication-systems-paper.tex","hosted_pdf_path":"papers/claim-faithful-publication-systems-paper.pdf","licence":"CC-BY-4.0","copyright":"2026 Will Cook","cite_as":"Plectis reader corpus, paper:claim-faithful-publication-systems","pdf_sha256":"sha256:5155ed424603a43a5e8148b0a61d9403379c0985fa56aaba000e9e0d5d0a11bb","pdf_byte_count":117372,"pdf_embedded":false,"pdf_omission_reason":"the exact LaTeX is embedded and retrieves better than extracted PDF text; the PDF is a presentation rendering and its hash is recorded for verification","source_embedded":true,"availability":"embedded_exact","embedded_copy_role":"native_manuscript","embedded_from_repository":"plectis-lean-erdos249-257","embedded_copy_path":"paper/claim-faithful-publication-systems-paper.tex","media_type":"text/x-tex","body_format":"latex","sha256":"sha256:df2b0666d39e9a41b83332b3189cee43e3ee6d76abae4e04740975715648028c","byte_count":53138,"body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n% Companion paper on release checks; build with tectonic.\n\\documentclass[11pt]{article}\n\\usepackage{iftex}\n\\ifpdftex\n  \\usepackage[T1]{fontenc}\n  \\usepackage[utf8]{inputenc}\n\\fi\n\\usepackage[a4paper,margin=1in]{geometry}\n\\usepackage{amsmath,amssymb,booktabs,array,float,placeins}\n\\usepackage[dvipsnames]{xcolor}\n\\usepackage{tikz}\n\\usetikzlibrary{arrows.meta,positioning,calc,shapes.geometric}\n\\usepackage[colorlinks=true,linkcolor=black,urlcolor=RoyalBlue,citecolor=black]{hyperref}\n\\hypersetup{\n  pdftitle={From Lean Proofs to Public Claims},\n  pdfauthor={Will Cook},\n  pdfsubject={Release checks and trust boundaries in one formal mathematics repository},\n}\n\\newcommand{\\repobase}{https://github.com/wcook04/plectis-lean-erdos249-257}\n\\newcommand{\\provenancenote}[1]{%\n  \\begingroup\n  \\renewcommand{\\thefootnote}{}%\n  \\footnotetext{#1}%\n  \\endgroup}\n\\newcommand{\\paperprovenance}{\\provenancenote{\\emph{Authorship and generative-system\nuse.} Every word of this manuscript was generated by agents based on large\nlanguage models operating within Will Cook's private research system for\nartificial intelligence. Cook set the objectives and acceptance criteria,\nreviewed and authorised the manuscript, and is responsible for its contents.\nThe agents are production tools, not authors.}}\n\\newcommand{\\paperfullaideclaration}{Every word of this manuscript was\ngenerated by agents based on large language models operating within Will\nCook's private research system for artificial intelligence. The formal proofs\nand repository software were likewise drafted and revised by the agents\nthrough that system under Cook's direction.\nCook set the objectives and acceptance criteria, selected and reviewed the\npublic claims, and approved the published version. Cook assumes responsibility\nfor the accuracy, interpretation, and presentation of the work. Generative\nsystems are production tools, not authors, and supply no independent authority.}\n\\emergencystretch=1.5em\n\\title{From Lean Proofs to Public Claims\\\\[2pt]\n{\\large Release checks and trust boundaries in one formal mathematics repository}}\n\\author{Will Cook}\n\\date{July 2026}\n\n\\begin{document}\n\\maketitle\n\\paperprovenance\n\n\\begin{abstract}\nLean is a proof assistant, software for writing and checking formal proofs.\nIts proof-checking core is called the kernel.  Lean verifies that a proof\nestablishes the formal statement written in the source.  It does not verify\nthat a repository's public front page (the README) or a paper describes that\nstatement faithfully.\nThat leaves a decision Lean cannot make: whether public wording has the formal\ntheorem's meaning, assumptions, and limits.  In the repository studied here, a\nmaintainer-reviewed record pairs each selected public claim with named Lean\ntheorems, the finite cases they cover, and the stronger conclusion that remains\nopen.  Before release, Lean rechecks the proofs; a second program tests whether\nthe recorded names, wording, and limits still agree after edits.  That program\ncan check only relationships a person selected and reviewed.\n\nThe worked example concerns a finite result around Erd\\H{o}s Problem 249.\nLean checks a finite calculation, called a certificate, at each of 28 named\nparameter values ending at \\(t=64\\).  The open statement requires certificates beyond\nevery fixed cutoff, and the development proves that this unbounded supply\nexists exactly when the totient constant in Problem 249 is irrational, that is,\nnot a ratio of integers.  Claiming that these\ncertificates continue beyond every cutoff would announce a solution while\nchanging no formal proof.  An earlier evaluation applied ten false edits.  One\nchanged the finite result into this stronger open claim and passed the release\nchecker because the finite/open distinction was not yet recorded.  After the\nclaim record and checker were repaired to include that distinction, the\nchecker rejected a test copy with the false wording.  This\ncovers one known failure; it gives no detection rate or general measure of\nreliability.  The repository also studies Erd\\H{o}s Problem 257.  Both\nmathematical problems remain open.\n\\end{abstract}\n\n\\section{The publication gap}\\label{sec:gap}\n\nThe \\href{\\repobase}{repository studied here}\nis a public development in the Lean 4 proof assistant~\\cite{lean4} around two\nunsolved problems in number theory,\nErd\\H{o}s Problems 249 and 257.  Both problems remain open.  The project\nproves intermediate results, exact reformulations, and finite certificates\naround them; it does not claim a solution to either problem.\n\nOne theorem illustrates the problem this paper is about.  Lean has checked a\nfinite certificate at each of 28 listed scales, the largest labelled\n$t = 64$.  Here a scale is a listed value of $t$.  A certificate is a\nfinite check proving that a particular value is not an integer, although\nrationality would require it to be one.  The corresponding open requirement asks for\ncertificates beyond every fixed cutoff.  Whatever the largest checked scale\nis, a cutoff lies beyond it, and the finite list says nothing there.  The\nfinite list therefore does not settle the open problem.\n\nA later README edit can nevertheless overstate the result.  By an equivalence\nproved in the development\n(Section~\\ref{sec:example}),\ncertificates beyond every fixed cutoff are not merely better evidence.  Their\nexistence is equivalent to answering Problem 249 affirmatively: the totient\nconstant is irrational.  Three objects recur throughout this paper: the \\emph{finite\ntheorem}, certificates at the 28 listed scales; the \\emph{open\nrequirement}, certificates beyond every cutoff; and the \\emph{equivalence}\nbetween the open requirement and the irrationality.  Lean has checked the\nfirst and the third; no Lean theorem carries the first to the second, and\nproving that implication would settle the problem.  An edit that presents the finite\ncases as completing the open requirement therefore asserts, in English,\nexactly the bridge the development does not contain; in substance it\nannounces a solution to an open problem.  Every Lean proof remains valid under\nthat edit, because the README is not part of any proof.  No program in the\nrepository decides whether a line of English has the same meaning as a formal\nstatement.  Lean acceptance therefore does not verify the public description.\n\nThe design has five parts: Lean source, the maintainer-reviewed claim\nrecord, authored public documents, generated\nindexes and summaries, and the release program and continuous-integration\nworkflow (CI) that runs both checks on each proposed change.\nThe repository keeps a \\emph{maintainer-reviewed claim record} in\n\\texttt{docs/claims.json}.  For each selected result the record\nstates the public wording, its status, the named Lean theorems or definitions\nthat support it (Lean calls such named items \\emph{declarations}), the bounded\ndomain it covers, and the stronger conclusion the record marks as open.\nA release checker,\n\\texttt{scripts/\\allowbreak check\\_\\allowbreak release.py}, then verifies that the\nrecorded relationships still hold across the Lean source, the record itself,\nthe authored public pages, and the generated files.  The record covers only\n\\emph{registered claims}, meaning claims entered into it, not every line of prose in the repository;\nsoftware cannot preserve a relationship it was never pointed at, and\nSection~\\ref{sec:failure} shows this limit operating in practice.  The\nshortest accurate summary of the division of labour is:\n\n\\begin{quote}\nLean checks the formal proofs.  A maintainer reviews what those proofs mean\nand how they may be described.  The release machinery checks that the recorded\nrelationships remain intact before a release.\n\\end{quote}\n\nReaders focused on the release design may skip\nSections~\\ref{sec:mathrole}--\\ref{sec:formal-evidence}; Section~\\ref{sec:record}\nresumes with the reviewed record.  Section~\\ref{sec:picture} shows the workflow;\nSections~\\ref{sec:trust}--\\ref{sec:limits} give its guarantees, observed failure,\nand limits.\n\n\\section{The release workflow}\\label{sec:picture}\n\n\\begin{figure}[!t]\n\\centering\n\\begin{tikzpicture}[\n  x=1cm, y=1cm,\n  artefact/.style={draw, semithick, rounded corners=1.5pt, align=center,\n    fill=white, inner xsep=5pt, inner ysep=5pt, text width=3.25cm, font=\\small},\n  humanbox/.style={draw=BurntOrange!80!black, densely dashed, rounded corners=3pt, align=center,\n    inner xsep=5pt, inner ysep=4pt, text width=2.75cm, font=\\small},\n  checkbox/.style={draw, thick, align=center, fill=white,\n    inner xsep=5pt, inner ysep=5pt, text width=4.35cm, font=\\small},\n  gate/.style={draw, thick, diamond, aspect=2.4, align=center, font=\\small,\n    fill=white, inner sep=1.2pt},\n  outcome/.style={draw, semithick, align=center, fill=white,\n    inner xsep=5pt, inner ysep=4pt, text width=3.4cm, font=\\small},\n  flow/.style={-{Stealth[length=2.4mm]}, semithick},\n  human/.style={-{Stealth[length=2.4mm]}, semithick, densely dashed,\n    draw=BurntOrange!80!black},\n  lab/.style={font=\\footnotesize, fill=white, inner sep=1.5pt}\n]\n\\node[artefact, text width=3.6cm] (lean) at (1.8, 0) {\\textbf{Lean source}\\\\[1pt]\n  {\\footnotesize\\texttt{Erdos249257.lean}\\\\ \\texttt{Erdos249257/*.lean}}\\\\[1pt]\n  {\\footnotesize\\itshape proves the finite theorem\\\\ and the equivalence}};\n\\node[artefact, text width=3.6cm] (claim) at (7.0, 0) {\\textbf{claim record}\\\\[1pt]\n  {\\footnotesize\\texttt{docs/claims.json}}\\\\[1pt]\n  {\\footnotesize\\itshape finite theorem: proved;\\\\ open requirement: open}};\n\\node[artefact, text width=3.6cm] (public) at (12.2, 0) {\\textbf{public documents}\\\\[1pt]\n  {\\footnotesize\\texttt{README.md}, \\texttt{SCOPE.md},\\\\ \\texttt{paper/*.tex}}\\\\[1pt]\n  {\\footnotesize\\itshape state the claim and\\\\ keep its limits visible}};\n\\node[humanbox] (hone) at (4.4, 2.00) {\\textbf{statement review}\\\\\n  {\\footnotesize intended meaning}};\n\\node[humanbox] (htwo) at (9.6, 2.00) {\\textbf{wording review}\\\\\n  {\\footnotesize claim strength and limits}};\n\\draw[human] (lean.north) |- (hone.west);\n\\draw[human] (hone.east) -| ([xshift=-7mm]claim.north);\n\\draw[human] ([xshift=7mm]claim.north) |- (htwo.west);\n\\draw[human] (htwo.east) -| (public.north);\n\\node[artefact, text width=4.3cm] (gen) at (9.6, -2.35) {\\textbf{generated views}\\\\[1pt]\n  {\\footnotesize reading map, code index,\\\\\n  machine-readable summary}};\n\\draw[flow] (claim.south) |- (gen.west) node[lab, pos=0.45, anchor=east, xshift=-0.5mm] {build};\n\\draw[semithick] (1.8,-3.35) -- (12.2,-3.35);\n\\draw[semithick] (lean.south) -- (1.8,-3.35);\n\\draw[semithick] (claim.south) -- (7.0,-3.35);\n\\draw[semithick] (gen.south) -- (9.6,-3.35);\n\\draw[semithick] (public.south) -- (12.2,-3.35);\n\\draw[rounded corners=4pt, fill=gray!8, draw=gray!55]\n  (-0.1,-6.32) rectangle (14.1,-3.90);\n\\node[font=\\scriptsize] at (6.0,-4.15)\n  {\\textsc{configured CI workflow} (fresh copy per job)};\n\\node[checkbox, anchor=north, draw=MidnightBlue!85, fill=MidnightBlue!5] (build) at (2.4, -4.40) {\\textbf{formal proof job}\\\\[1pt]\n  {\\footnotesize\\texttt{lake build}}};\n\\node[checkbox, text width=5.4cm, anchor=north, draw=Violet!75!black, fill=Violet!5] (release) at (9.6, -4.40)\n  {\\textbf{release-checking job}\\\\[1pt]\n   {\\footnotesize\\texttt{python3 scripts/check\\_release.py}\\\\\n   exact Lean files, claim fields,\\\\ generated files, required wording}};\n\\node[lab, anchor=east, text=MidnightBlue] at (2.15,-3.62) {proof sources};\n\\node[lab, anchor=east, text=Violet!75!black] at (9.35,-3.62) {all four artefacts reach the release job};\n\\draw[flow, draw=MidnightBlue!85] (2.4,-3.35) -- (build.north);\n\\draw[flow, draw=Violet!75!black] (9.6,-3.35) -- (release.north);\n\\draw[semithick] (build.south) |- (7.0,-6.55);\n\\draw[semithick] (release.south) |- (7.0,-6.55);\n\\node[gate] (gate) at (7.0, -7.35) {\\footnotesize both jobs pass?};\n\\draw[flow] (7.0,-6.55) -- (gate.north);\n\\node[outcome, draw=BrickRed!70!black, fill=BrickRed!8] (stop) at (1.9, -7.35) {\\textbf{workflow fails}\\\\\n  {\\footnotesize the failed job records the broken check}};\n\\node[outcome, draw=ForestGreen!65!black, fill=ForestGreen!10] (ship) at (12.0, -7.35) {\\textbf{workflow passes}\\\\\n  {\\footnotesize publication remains a human action}};\n\\draw[flow] (gate.west) -- (stop.east) node[lab, pos=0.45] {no};\n\\draw[flow] (gate.east) -- (ship.west) node[lab, pos=0.45] {yes};\n\\end{tikzpicture}\n\\caption{The configured checks for one saved change.  Dashed elements are human:\none maintainer performed both review steps here.  Solid elements are files,\nprograms, and automated checks.  Lean checks the formal proofs.  Human review\ncompares their intended meaning with the recorded public wording and limits.\nThe release job checks that recorded names, wording, Lean files, and generated\nviews still agree.  Continuous integration reruns the Lean and release jobs; neither\ninterprets unrestricted prose.  In the worked example of\nSection~\\ref{sec:example}, the release job protects the recorded distinction\nbetween the finite theorem and the open requirement.  Blue marks the Lean job,\namber review, and violet the release job.}\n\\label{fig:architecture}\n\\end{figure}\n\nRead the upper half of Figure~\\ref{fig:architecture} from left to right: Lean\nsource, claim record, and public documents, with generated views built\nfrom the record.  The dashed arrows show human review.  The lower half is\nautomated: each job runs on its own fresh copy.  The decision is: do both\njobs pass?\n\nSomeone must compare a formal theorem with its public wording and judge whether\nthe wording is faithful within stated limits.  Once that judgement is recorded,\na program can test whether later changes satisfy the recorded rules\nabout names, files, and wording.  It does not decide whether the original\njudgement was correct.\n\nThe workflow does not technically force a second independent mathematician.  A\nsingle maintainer can edit a Lean statement, its claim record, and the public\nwording together, and every check can pass while the shared interpretation is wrong;\nSection~\\ref{sec:limits} applies this limit to the worked claim.  The checks\ndetect departures from reviewed decisions.  They do not replace review or add\nauthority of their own.\n\nThe two jobs answer different questions.  \\texttt{lake build} asks whether\nLean accepts the imported formal statements and proofs.  It knows nothing about\nthe README, and it would still accept a change whose README falsely states that\nthe open requirement has been met, because no public document is among its inputs.  The\nsecond job, \\texttt{python3 scripts/check\\_release.py}, asks whether the\nrecorded relationships among proofs, record, public pages, and generated\nfiles still hold; it checks that the Lean source matches the recorded version\nbut does not run Lean.  The workflow in\n\\texttt{.github/workflows/lean.yml} runs them as two separate jobs whenever a\nsaved change is sent to GitHub, either directly or for review.\n\nEach job runs on a new GitHub-hosted virtual machine (a \\emph{runner}) and\ndownloads the selected repository version (a \\emph{revision})~\\cite{githubrunners}.\nA checker error stops the program; the workflow\ntreats that exit as a failing job.  Repository settings decide whether that\nblocks release.  Failure can stop the configured release process; a passing\nrelease job proves no mathematics.\n\nThe generated views select and reorganise recorded information (a reading\nmap, a code index, a\nmachine-readable summary) so that a reader or a program can enter the\nrepository without opening every file.  They create no new mathematics.  Each\nview records the program that rebuilds it from source, and stale output is\nregenerated rather than edited by hand.  The complete file map, ownership table, and\nmaintenance commands live in\n\\href{https://github.com/wcook04/plectis-lean-erdos249-257/blob/main/ARCHITECTURE.md}{\\texttt{ARCHITECTURE.md}};\nthis paper gives only what this argument needs.\n\n\\FloatBarrier\n\\section{One claim from Lean theorem to public page}\\label{sec:example}\n\nWe now follow one result from Lean source to public page.  We begin with its\npublic meaning, not its internal name:\n\n\\begin{quote}\nLean has checked a finite certificate at each of 28 listed scales through\n$t = 64$.  The registered open requirement asks for certificates beyond every\nfixed cutoff.  The finite list does not settle it.\n\\end{quote}\n\n\\subsection{What the certificates are for}\\label{sec:mathrole}\n\nErd\\H{o}s Problem 249 asks whether the totient constant\n\\[\nS=\\sum_{n\\ge 1}\\frac{\\varphi(n)}{2^{n}}\n\\]\nis \\emph{irrational}~\\cite{erdosgraham}, meaning that it is not $a/b$ for any integers $a$ and\n$b$ with $b\\ne 0$.  Here $\\varphi$ is Euler's totient function: $\\varphi(n)$\ncounts the integers from $1$ to $n$ sharing no factor greater than $1$ with\n$n$.  The following exact equivalence explains why the certificates matter.\nMultiplying $S$ by $2^N$ makes its first $N$ terms integral; write $R_N$ for\nthe remaining rescaled tail.  Thus $R_N$ differs from $2^N S$ by an integer.\nPut $D_{N,h}=R_{N+h}-R_N$.  It differs from $2^N(2^h-1)S$ by an integer.\nFor $h>0$, the factor $2^N(2^h-1)$ is a nonzero integer, so an integral\n$D_{N,h}$ would force $S$ to be rational.  Conversely, every rational number\nhas an eventually repeating binary expansion, the base-2 analogue of a\nrepeating decimal.  Thus some $h>0$ makes\n$D_{N,h}$ integral for every sufficiently large $N$.  Lean checks the exact\nequivalence:\n\\[\nS\\text{ is irrational}\\quad\\Longleftrightarrow\\quad\nD_{N,h}\\notin\\mathbb Z\\ \\text{ for every $h>0$ and every $N$}.\n\\]\nFor fixed $h$ and $N$, Lean proves $D_{N,h}\\notin\\mathbb Z$ exactly when a\ncertificate exists at a finite depth $L$.  Its integer calculation approximates\n$2^L D_{N,h}$ with error at most $r=N+h+L+2$.  The certificate checks that\nthe residue modulo $2^L$ lies strictly between $r$ and $2^L-r$, outside both\nbands compatible with an integral value.  The omitted infinite tail is thus\naccounted for by an explicit bound.  The claim record calls this a\n\\emph{certified kill}.  We use the shorter word \\emph{certificate}: it proves only that\nthis particular tail difference is not an integer.\n\nThe scales of the finite theorem compress this two-parameter requirement to\none.  The diagonal $H_t=\\operatorname{lcm}(1,\\dots,t)$ is the least common\nmultiple of $1,\\dots,t$: the smallest positive integer divisible by every\ninteger in that list.  It is therefore divisible by every candidate period\nup to $t$.  If $S$ were rational, its binary digits would have a period\n$h_0$ after a position $N_0$.  For $t\\geq\\max(h_0,N_0)$, one has\n$h_0\\mid H_t$ and $H_t\\geq N_0$, so rationality would make\n$D_{H_t,H_t}$ integral; a diagonal certificate rules this out.  The\ndevelopment proves the reduction exact: certificates along the diagonal\nbeyond every fixed cutoff\nare equivalent to the full two-parameter requirement, and therefore, through\nthe chain\nabove, to the irrationality itself.  Certificates on a finite list, however\nlong, are not.\n\nWriting $\\mathrm{Cert}(t)$ for ``a certificate exists at scale $t$'', the\ntwo statements then read:\n\\begin{align*}\n\\text{checked:}&\\ \\ \\mathrm{Cert}(t)\\ \\text{for every $t$ in a fixed list of\n28 scales;}\\\\\n\\text{open:}&\\ \\ \\text{for every cutoff }T,\\ \\mathrm{Cert}(t)\\ \\text{for\nsome }t>T.\n\\end{align*}\nThe boundary between them is a matter of logical form, not of scale.\nA list of 28 scales, of 280, or of 28{,}000 stands\nin the same relation to the open requirement: each is a fixed finite list\nwith a cutoff beyond it.  No\nfinite list, however long or expensive to verify, implies the second\nstatement without a further theorem that produces new witnesses.  Thus\n$t=64$ is the endpoint of this verified computation, not a frontier that\nLean has proved will continue.  This is the boundary an edit can\nsilently delete.\n\n\\subsection{The formal evidence}\\label{sec:formal-evidence}\n\nThe Lean source file (or \\emph{module})\n\\path{Erdos249257/DiagonalPincerCertificatesT64.lean}\nlists the 28 scales through $t=64$: 1 and the prime powers at which $H_t$\nchanges.  Intermediate scales add no new case.  Imported modules prove each\nlisted case.  The theorem\n\\path{certifiedKill_diagonal_all_imported_through_t64} combines them: for every\nlisted $t$, it gives a finite checking depth $d_t$ and proves a certificate\nwhose period and position are both $H_t$.  Lean evaluates the remaining finite\narithmetic at depth $d_t$.  Lean's \\emph{kernel} is the small trusted program\nthat checks every proof Lean accepts.  The certificate proofs use \\texttt{decide}:\nit evaluates the relevant finite yes-or-no proposition and, when the answer is\nyes, supplies a proof for the kernel to check~\\cite{leanapi}.  The equivalence is also\na proved Lean theorem, not an assumption of this paper.  The module\n\\path{Erdos249257/LcmConeFlatness.lean} proves the chain of\nSection~\\ref{sec:mathrole}: the tail-difference and pointwise certificate\nequivalences, followed by the diagonal characterisation\n\\path{irrational_totient_series_iff_lcm_diagonal_certificate_supply}.\nThe root \\texttt{Erdos249257.lean} imports both modules, so\n\\texttt{lake build} checks them.\n\n\\subsection{The reviewed record}\\label{sec:record}\n\nThe quoted record entry contains four names specific to this repository.  A\n\\emph{certified kill} is the certificate defined in\nSection~\\ref{sec:mathrole}.  The entry abbreviates least common multiple as\n``lcm'', so its \\emph{lcm-diagonal scales} are the values $H_t$.  The\n\\emph{small periods} are the periods 1 through 8, each handled by one explicit\ncertificate.  A \\emph{certificate shard} is the same calculation at a fixed\nperiod, position, and depth, isolated in another Lean file and checked by\n\\texttt{decide}.\n\nTable~\\ref{tab:record} shows the substance of the corresponding entry in\n\\texttt{docs/claims.json}, ordered as a reader meets it: what is claimed,\nwhere it stops, what remains open, then the supporting machinery.  The entry\nalso records an exact source line for each declaration; those coordinates are\nomitted from the table.\n\n\\begin{table}[H]\n\\centering\\footnotesize\n\\begin{tabular}{@{}>{\\raggedright\\arraybackslash}p{0.185\\textwidth}\n                    >{\\raggedright\\arraybackslash}p{0.745\\textwidth}@{}}\n\\toprule\nField & Content \\\\\n\\midrule\nPublic statement & ``Kernel-checked certified kills for small periods and at\n  28 explicit lcm-diagonal scales through $t = 64$, plus finite certificate\n  shards, each discharged by \\texttt{decide}.'' \\\\\n\\addlinespace\nBounded domain & The listed small periods, the 28 scales above, and the fixed\n  period, position, and depth of each separate certificate. \\\\\n\\addlinespace\nRemaining open & A link to the registered open requirement \\emph{unbounded\n  certificate supply}: produce a certified witness beyond every fixed cutoff. \\\\\n\\addlinespace\nStatus & \\emph{verified finite instance}: Lean checked the stated finite\n  inputs. \\\\\n\\addlinespace\nSupporting declarations & Five named Lean theorems with recorded modules\n  and source lines, ending in\n  \\texttt{certifiedKill\\_diagonal\\_all\\_imported\\_through\\_t64}. \\\\\n\\addlinespace\nClaim identifier & \\texttt{certified\\_kill\\_instances} \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{The reviewed claim entry for the worked example, exact source\ncoordinates omitted.  Together, the positive statement, bounded domain, and\nremaining-open link distinguish the finite result from the open problem.  In\nthe failure of Section~\\ref{sec:failure}, overstatement came from deleting\nthis boundary, not from inventing a theorem.}\n\\label{tab:record}\n\\end{table}\n\nThe table records the claim rather than explaining it.  The remaining-open\nfield points to the open requirement of Section~\\ref{sec:mathrole}, while the\nstatus \\emph{verified finite instance}\nclassifies the finite theorem.  Under the equivalence the remaining-open field\nis part of the claim: without it, the entry no longer separates the finite\ntheorem from an answer to Problem 249.  The record gives explanatory prose,\nthe release checker, and generated views one stable reference.  Prose must\nstill bridge the gap between an exact claim and a reader's understanding.\n\nThe fields differ in kind, and the difference matters for what\n``reviewed'' means.  The public statement, bounded domain, remaining-open\nlink, and status are fields about meaning.  They record a human judgement, and\nonly a person can meaningfully change them.  The record fixes the allowed\nstatus terms, and the checker rejects any other status.  The claim identifier\nlets software find the claim; it appears last, not in\nplace of the wording, and the per-declaration\nsource lines are location data that a program may refresh when code moves;\nupdating them does not repeat the judgement about meaning.  No field can by\nitself establish that the English is faithful to the Lean: the maintainer judged that the public\nstatement above describes the five declarations correctly, and the record\nstores the outcome of that judgement, not a justification of it.  Review and\nauthorisation are distinct: review asks whether the wording is faithful;\nauthorisation decides which wording the project will publish.  One\nmaintainer performed both here.  The review itself is concrete: read the final theorem, check that the\nwording claims the listed scales and no more, confirm that the open\nrequirement stays named as open.  It is also an event, not a permanent\nproperty: the review covered particular declarations and wording.  A later\nchange to either reopens it; later checks can only preserve what was recorded.\nThat decision must survive ordinary rewrites months later.\n\n\\subsection{What the checker verifies here}\n\nFor this claim, \\texttt{scripts/check\\_release.py} verifies a finite, listed set of\nrelationships.  Each declaration name must appear in the Lean\nsource within three lines of its recorded coordinate.  This checks that the\nname remains near its recorded position, not that the declarations entail the\npublic wording.  The checked Lean source itself must match the saved Git revision\nnamed in the claim record.  The checker compares the top-level Lean file and the\nentire library directory byte for byte with that recorded version.  It rejects\nany difference or extra Lean file absent from that version, so the verified names\ncannot refer to an older source than the one\nbeing shipped.  The\nrequired limitation wording must remain present on the registered public\npages; for this claim that wording is the visible trace of the boundary of\nSection~\\ref{sec:mathrole}, the clause keeping the open requirement open.\nThe generated views must equal a fresh rebuild from their sources.\nPresence, identity, and proximity are all the checker can test.  Lean checks\nthe theorem; the checker checks the recorded description around it; a person\nremains responsible for the claim that the two say the same thing.\n\n\\section{What the checks establish}\\label{sec:trust}\n\nThe word ``verification'' is easy to overread here.  Four separate questions\nare involved.  Did Lean accept the formal statements and proofs?  Did a\nmaintainer record a judgement that selected public wording describes them\nwithin stated limits?  Does the release checker find the recorded\nrelationships intact at this revision?  Did continuous integration run the\nconfigured jobs on that revision and report success?  Each question needs\ndifferent evidence.  A yes to one does not settle the others.\n\nWhen both jobs pass, Lean has accepted the proofs and the release checker has\nfound every recorded relationship intact.  This does not repeat the\nmaintainer's review, establish that the review was correct, or show that every\nconsequential claim was registered.\nTable~\\ref{tab:trust} states what each layer can and cannot establish.  Keeping\nthe layers separate prevents the single word ``verified'' from making their\ncombined guarantee sound stronger than it is.\n\n\\begin{table}[t]\n\\centering\\small\n\\begin{tabular}{@{}>{\\raggedright\\arraybackslash}p{0.16\\textwidth}\n                    >{\\raggedright\\arraybackslash}p{0.385\\textwidth}\n                    >{\\raggedright\\arraybackslash}p{0.365\\textwidth}@{}}\n\\toprule\nLayer & What it can establish & What it cannot establish \\\\\n\\midrule\nLean build and kernel\n& The written formal statements are proved from their imported definitions\n  and assumptions, using the recorded Lean version and dependencies.\n& That a formal statement is the one the author intended, or that any English\n  description of it is faithful. \\\\\n\\addlinespace\nMaintainer review\n& A recorded judgement that a named formal statement has the intended meaning\n  and selected public wording describes it within stated limits.\n& That Lean accepts the proofs; that every important claim was registered; or\n  that the review was independent or correct. \\\\\n\\addlinespace\nRelease checker\n& That recorded names, source lines, fields, links, required wording, Lean\n  files, and generated files satisfy the declared rules; that prohibited proof\n  shortcuts are absent; and that deliberately wrong examples still fail.\n& What unregistered prose means; whether the record is complete; or whether\n  the human judgements it preserves are correct. \\\\\n\\addlinespace\nContinuous integration\n& That each job ran on its own fresh copy of the uploaded revision and exited\n  successfully.\n& Independent mathematical approval; anything beyond what the two jobs\n  themselves establish. \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{What each layer establishes and leaves open.}\n\\label{tab:trust}\n\\end{table}\n\nInstantiated on the worked example, the development contains the finite theorem and the\nequivalence, but no theorem deriving the open requirement from the finite list.\nA maintainer recorded the judgement that the public wording\nasserts the finite theorem, confines it to the listed scales, and keeps the\nopen requirement named as open.  The release checker finds the recorded traces\nof that judgement intact: declaration names near their coordinates, agreement\nbetween the Lean files and saved revision, the limitation clause on the public pages, and freshly\nrebuilt views.  Continuous integration records that each job ran on its own\nfresh copy of the revision and passed.\n\nThe limits follow the same lines.  Lean cannot notice English that quietly\nimplies the open requirement met.  The recorded review does not show that every page\nrepeating the claim was found.  The checker cannot reject a paraphrase it was\nnever given.  Passing both jobs cannot make a mistaken reading of the equivalence\ncorrect.\n\nOne part of the release checker scans the repository's Lean source itself.\nAmong other patterns, it rejects proof placeholders (\\texttt{sorry} and\n\\texttt{admit}), project-defined \\texttt{axiom} declarations, and native\nevaluation in each form covered by the pinned Lean version:\n\\texttt{native\\_decide}, \\texttt{decide +native}, or \\texttt{decide} with its\nnative configuration enabled.  An axiom assumes a statement without a proof.\nThe ordinary \\texttt{decide} used for the certificates reduces a finite\nyes-or-no computation to a proof term that the kernel checks.  Native\nevaluation instead compiles and runs that computation, then adds a dedicated\naxiom recording the result.  The kernel can track the axiom but cannot replay\nthe compiled computation~\\cite{leanapi,leanref}.  Such a theorem therefore also trusts the\ncompiler and the code it ran.  The scan covers project source only; imported\nlibrary axioms remain part of the separate trust in the pinned dependencies.\nThe checker also compares\nall public Lean files byte for byte\nagainst the saved formal source revision and rejects extra Lean files.  It\nrequires generated views to match a fresh rebuild, verifies the source and PDF\nfile hashes of the shipped papers, and runs a set of deliberately wrong\nexamples that must keep failing.  For each covered example, this detects if a\nlater checker edit stops rejecting that known fault.  It does not show that\nevery rule has such an example or that an unseen fault will be rejected.\nNone of these comparisons determines meaning.  A file hash is a short fingerprint\nof a file's contents.  Different hashes prove files differ; matching hashes strongly\nsuggest identity.  No hash says which file expresses the right mathematics.\n\nWhen a mathematical file changes, local review begins with \\texttt{lake build}.\nA person then asks whether any statement, assumption, or intended meaning has\nchanged and updates the record and public pages when it has, following\n\\texttt{docs/methodology.json}.  The named builders regenerate the generated views,\nand \\texttt{python3 scripts/check\\_release.py} compares the resulting files.\nUploading any revision runs both jobs on a fresh copy.  During local\nreview, a prose-only edit need not repeat the proof build because no Lean file\nchanged, but the release checker and human review still matter.  If the words\nmake a stronger or different mathematical claim, the record and the review\nmust change with them.  The escaped edit of Section~\\ref{sec:failure} changed\nthe claim without either change.  The repository guide carries the full\ncommand list.\n\n\\pagebreak\n\\section{A boundary the checklist missed}\\label{sec:failure}\n\nThe evidence comes from three different times.  The historical exercise, the\npresent post-repair test, and the later executable reconstruction answer\ndifferent questions and must not be merged.  Appendix~\\ref{app:repro} identifies\ntheir files and commands.\n\n\\paragraph{Historical exercise.}\nThe structured report in \\texttt{docs/publication\\_evidence.json} identifies a\nsaved Git revision.  It says that ten deliberate false edits were applied to a\nseparate copy one at a time, restoring it between runs.  The release checker\nran after each edit; the Lean proof job did not.  The edits covered seven kinds\nof recorded relationship: allowed proof commands, record structure, source\ncoordinates, required paper links, boundary wording, generated-file freshness,\nand size budgets.  Examples included\nupgrading a conditional status to ``proved here'', deleting an open-boundary\nclause, moving a recorded source line, introducing \\texttt{native\\_decide} into\na proof, and changing a generated count.  The original run logs were not retained,\nso the file is a report of the runs rather than their original output.\n\nAccording to that record, nine of the ten edits were rejected.  One escaped:\nthe README clause saying that the finite cases \\emph{do not supply} the open\nrequirement was changed to say that they \\emph{complete} it.  Under the\nequivalence of Section~\\ref{sec:mathrole}, the altered page claimed in effect\nthat Problem 249 was settled.  It supplied in prose exactly the missing bridge\nfrom the finite theorem to the open requirement.  The Lean proofs were\nuntouched, and the release checker passed because this clause was absent from\nits checklist.  The escape therefore shows that a passing release checker does\nnot certify all public prose.  It is not a statistical result\nand gives no detection rate.\n\n\\paragraph{Repair and present witness.}\nThe repair recorded the clause as an \\emph{anchor}, meaning wording that must\nremain on the public page.  It also derived an opposing copy in which that\nwording is weakened or removed.  The current README passes and the false copy\nfails.  The repository calls this failing copy a \\emph{boundary witness}.  The\ntest shows that the repaired checker distinguishes the approved wording from\none known false neighbour; it measures no broader effectiveness.  The other nine edits\nwere not rerun against the extended checklist, so there is no\npost-repair aggregate result.\n\n\\paragraph{Later reconstruction.}\nThe reconstruction file in Appendix~\\ref{app:repro} specifies ten edits that can be run\nagainst the saved evaluation version.  Three scripted edits preserve\ntheir exact original targets; the other seven use fixed replacement targets\nbecause the original targets were not registered in the retained record.\nThis reconstruction is a new experiment, not the missing runs.  It is evidence\nabout its documented replacement edits, not about the original outputs or timings.\n\nThe evidence marks a coverage boundary, not a reliability score.  The edits were authored\nby the checker's author, so the human review gate of\nFigure~\\ref{fig:architecture} was deliberately omitted and the exercise tests\nthe checklist alone.  Five of the seven relationship kinds were probed by one\nedit.  The exercise covers one repository, one maintainer, and one working\nstyle, with no comparison against ordinary continuous integration or careful\nmanual review.  Reader error and transfer to other repositories were not\nmeasured.  The nine reported rejections rest on the historical report alone.\n\n\\begin{samepage}\n\\noindent The retained evidence supports one design lesson:\n\n\\begin{quote}\nThe program can preserve a relationship only after a person has identified\nand recorded that relationship.\n\\end{quote}\n\\end{samepage}\n\nRequiring a clause to be present does not detect a contradictory stronger\nclaim elsewhere on the page.  Coverage grows only when a person notices a\nrelationship, records it, and supplies an example that must fail.  Coverage can\nalso shrink: deleting a registered relationship leaves no rule requiring it,\nso every remaining check passes.\nRetiring a commitment therefore deserves the same review as creating one.\n\nThe shape of this limit is familiar from Section~\\ref{sec:mathrole}.  The\nrecorded checklist stands to unrestricted prose roughly as the finite\ncertificate list stands to the open requirement: every registered\nrelationship is real and checked, and no finite list of relationships\nexhausts an open-ended domain.  The comparison is structural, not exact: prose\nhas no equivalence theorem and no diagonal compresses it.  It shows why the\nchecker's silence outside its records is a continuing limit, not a defect one\nrepair can remove.\n\n\\section{Scope, reuse, and limits}\\label{sec:limits}\n\n\\paragraph{The failures the design addresses.}\nThe checks address accidental disagreement after a sound review.  A declaration\nmay move while its recorded line number stays fixed, and a generated view may\nbe stale or hand-edited.  A rewrite may remove a required limitation, a status\nmay be upgraded, or the shipped Lean files may cease to be the reviewed ones.\nA check may also decay until it can no longer fail.  In each case, one\nrecorded item disagrees with the others, which a mechanical comparison can\ndetect.\n\n\\paragraph{The failures it does not address.}\nThree failure modes remain outside the design.  \\emph{Unregistered wording:}\na paraphrase can strengthen a claim without touching a registered anchor, and\na contradiction, misleading emphasis, or new public document can escape for\nthe same reason.  Requiring one clause to be present does not require the page\nto agree with it.  \\emph{Coordinated but wrong change:} a maintainer can alter\nsource, record, and prose together, so every comparison agrees.  If the\ncertificate definition were weakened while the record and prose were updated\nto match, all checks could pass although the public reading of the equivalence\nwas wrong.  \\emph{Mistaken review:} if the original judgement was wrong, the\nmachinery preserves the mistake.  It checks persistence, not the quality of\nthe judgement.\n\n\\paragraph{Limits of the design and this implementation.}\nSome limits follow from the design.  Claims remain in unrestricted prose beside\nan external record, so human interpretation cannot be eliminated.  Only listed\nrelationships are checked.  The repository calls this set its \\emph{assurance\nperimeter}: recorded commitments inside it trigger automatic checks; outside it,\nthe checker is silent.  Choosing what to record remains a human judgement.\nThis repository gives priority to headline results and to wording whose\naccidental strengthening would change the project's public status, as the\ndeleted boundary clause did under the equivalence.  If the same claim appears\non several pages, the checker sees only the appearances named in the record.\nOther limits are implementation choices: one maintainer performs both review\nand authorisation; the checker accepts a theorem name within three lines of its\nrecorded location; anchors require exact wording; the exercise used one edit\nper relationship type; and the original logs were not retained.\n\n\\paragraph{Reuse.}\nAnother formal project could reuse the pattern without copying any file\nformat.\\par\\pagebreak\\noindent\nIt applies when bounded formal evidence sits beside a stronger public\ntarget.  Examples are finite cases beside a universal statement;\na conditional theorem beside its unresolved hypothesis; one direction of an\nimplication beside a claimed equivalence; or a result under named assumptions beside an unconditional\nheadline.  What transfers is the reviewed boundary between what is established\nand the adjacent stronger statement that is not.  The minimum obligations are concrete.\nEach public claim needs its own record entry, named formal evidence, and the\nexact source version containing it.  Its scope, assumptions, and the stronger\nconclusions it does not establish must be explicit.\nHuman review is distinct from automated checking.  Each generated\npublic view needs a named builder.  Proof checking and publication\nchecking must be separate and able to fail.  Every important recorded\nrelationship needs an example that deliberately violates it, and the record\nmust present its coverage as selective rather than complete.\n\nStronger variants impose more structure.  A project could generate public\nwording from the record instead of anchoring authored prose; extract formal\nstatements and assumptions instead of recording source lines; require an\nindependent reviewer for designated claims; constrain critical claims to a\ncontrolled vocabulary; or use tools that interpret prose to propose discrepancies for\nhuman review.  This repository instead keeps unrestricted prose and an\nexternal record.\n\n\\paragraph{What is not established.}\nThe architecture does not\nestablish a solution to either Erd\\H{o}s problem; that a formal statement is\nthe statement the author intended; that software understood any unregistered\nprose; that the record is complete; that one maintainer's review is\nindependent or adequate; or that the design transfers to other projects with\nits behaviour intact.  A large number of passing comparisons measure none of\nthose things.  What a passing workflow establishes is narrower: the\nconfigured jobs ran on the named revision, Lean accepted the formal proofs,\nand the release checker found every recorded relationship intact.  This paper\nis an architecture note with one bounded case study,\nnot an empirical evaluation of the design.  A credible evaluation would need\nseveral stages, each with its own limit.  Records naming the reviewer, revision,\nand time could establish who checked which revision, when, and how often, but\nnot effectiveness.  Deliberately wrong edits by people other than the checker's\nauthor could measure the response to unseen faults, but not\nbehaviour in ordinary use.  Only after those stages would comparison with ordinary review\non the same changes become meaningful.  None of those stages exists yet, and\nthis paper claims none of them.\n\n\\section{Related systems}\\label{sec:related}\n\nA \\emph{proof blueprint} pairs an informal outline with Lean declarations and\nrecords which results depend on which earlier results.  \\texttt{leanblueprint}\nstores that outline in \\TeX{} and checks that every named declaration\nexists~\\cite{leanblueprint}.  LeanArchitect stores blueprint data in Lean,\ninfers those dependencies, tracks formalisation progress, and exports\nsynchronised \\TeX{}~\\cite{leanarchitect}.  Both support a formalisation in\nprogress.  The claim record instead asks which public wording was reviewed\nafter Lean accepted a proof, and which stronger conclusion remains open.\n\nLean Atlas and EconCSLib ask whether a formalisation expresses its intended\nsource mathematics.  Lean Atlas leaves semantic verification to people.  Given\nchosen theorem statements, its Lean Compass selects the project declarations\nwhose meaning can affect them, narrowing what a person must inspect; it assumes\nLean's standard library and the mathematical library Mathlib are semantically\ncorrect rather than inspecting them~\\cite{leanatlas}.  EconCSLib uses a\ndifferent division: an AI language model writes Lean; Lean checks the formal\nstatements and proofs; a dashboard and separate language-model reviewers help\npeople assess whether a source paper was translated correctly.  Its paper\nreports that human review remained incomplete~\\cite{econcs}.  This paper\naddresses a later boundary: it starts from accepted Lean proofs and asks how\nsubsequent public wording can remain within a reviewed interpretation of them.\n\nIsabelle/DOF, a document system built on the Isabelle proof assistant, places\nformal and informal material in one checked document.\nAuthors define an \\emph{ontology}: document classes with typed fields and rules.\nThey label passages with those classes, and Isabelle's editor reports rule\nviolations as they edit~\\cite{isadof}.  The bounded domain and open\nconclusion in Table~\\ref{tab:record} could be typed fields in such an ontology.\nThis repository keeps prose unrestricted and checks a separate record; its\nchecker cannot inspect prose that the record does not name.\n\nCASCADE derives tests and a second implementation from the same documentation\nusing language models.  It reports a likely inconsistency only when a generated\ntest fails on existing code but passes on code generated from that\ndocumentation.  A person must confirm it~\\cite{cascade}.  Unlike this checker,\nit can inspect unrecorded documentation.\n\nMutation testing asks whether tests detect deliberately introduced\nfaults~\\cite{demillo,jiaharman}.  The deliberately wrong copies in\nSection~\\ref{sec:failure} serve that narrower purpose: a known wrong input\nmust continue to fail.  The ten-edit exercise is a small, hand-authored\nmutation run.  Since nine edits were not rerun after repair, it gives no\npost-repair detection rate.\n\nRequirements traceability follows a requirement from its origin through\ndevelopment, deployment, use, and revision~\\cite{gotel}.  An assurance case\nsets out a system claim, the argument for it, and supporting evidence.  The Goal\nStructuring Notation (GSN) is a graphical notation for that argument\nstructure~\\cite{gsn}.  The claim record resembles both, but it does not contain\nan argument that its Lean declarations justify the public wording.  It records\nthe approved wording, supporting declarations, scope, and limits; the justification remains\na human judgement.\n\n\\section{Conclusion}\\label{sec:conclusion}\n\nThe mathematical boundary is that Lean has checked\ncertificates at 28 listed scales through \\(t=64\\), while the registered open\nrequirement asks for certificates beyond every fixed cutoff.  The development\nproves that this unbounded supply is equivalent to answering Problem 249\naffirmatively; it does not prove the supply.  Both Erd\\H{o}s Problems 249 and 257\nremain open.\n\nThe evidence boundary is different.  The historical report says\nthat one of ten deliberate edits escaped the then-current checker.  That\nescape shows that a passing release checker does not certify all public prose;\nit gives no detection rate and no comparison with ordinary review.\nThe repaired checker accepts the intact wording and rejects the reconstructed\nfalse edit.\n\nWhether these checks reduce inaccurate public claims at acceptable cost remains\nuntested.  Until the evaluation described in Section~\\ref{sec:limits} exists,\na passing workflow means only that the configured jobs ran on the named revision,\nLean accepted the proofs, and the release checker found every recorded\nrelationship intact.\n\n\\appendix\n\\section{Reproducibility}\\label{app:repro}\n\nThe reviewed claim record, \\texttt{docs/claims.json}, names the saved Git\nrevision of the Lean source, which the release checker requires to match\nexactly.  This paper omits the changing commit identifier so that the claim\nrecord is the only place that states it.\n\n\\paragraph{Declaration of generative AI use.} \\paperfullaideclaration\n\nAI output is not proof authority: Lean checks each formal proof against the\nfixed library version, while Cook remains responsible for connecting those\nproofs to public wording. The release checker tests only recorded relationships\na person selected; passing it does not make generated wording faithful.\n\nThe paper inventory, \\path{docs/publication_contract.json}, records source\nand PDF cryptographic hashes and validation commands.  The evidence file for\nSection~\\ref{sec:failure}, \\path{docs/publication_evidence.json}, records the\nprotocol, outcomes, and limitations.  The reconstruction file,\n\\path{experiments/publication_mutations.json}, specifies the ten edits.\nThe script \\path{scripts/run_publication_mutations.py} checks that each edit\napplies once (\\texttt{--verify-operators}) and, by default, runs all ten in a\ncopy of the saved evaluation version (\\texttt{--all}).  For an edit whose exact\noriginal target was not preserved, the file names a fixed replacement.  The\noriginal raw outputs were not retained; the reconstruction does not claim to\nbe them.\n\nThe papers build with Tectonic or standard \\LaTeX{} via\n\\texttt{make -C paper}.  The tracked root PDF is the shipped copy, and the\ninventory checks its file hash.  These identities name artefacts; they do not\ninterpret them.\n\n{\\footnotesize\n\\begin{thebibliography}{15}\n\\setlength{\\itemsep}{0pt}\n\\setlength{\\parsep}{0pt}\n\\bibitem{lean4} L.~de Moura and S.~Ullrich, \\emph{The Lean 4 Theorem Prover\nand Programming Language}, in \\emph{Automated Deduction---CADE 28}, Lecture\nNotes in Computer Science 12699, 2021,\npp.~625--635,\n\\href{https://doi.org/10.1007/978-3-030-79876-5_37}{DOI}.\n\\bibitem{githubrunners} GitHub, \\emph{GitHub-hosted runners},\n\\href{https://docs.github.com/actions/concepts/runners/github-hosted-runners}{documentation},\naccessed 18 July 2026.\n\\bibitem{erdosgraham} P.~Erd\\H{o}s and R.~L.~Graham, \\emph{Old and New\nProblems and Results in Combinatorial Number Theory}, Monographies de\nL'Enseignement Math\\'ematique 28, L'Enseignement Math\\'ematique, Universit\\'e\nde Gen\\`eve, 1980, p.~61,\n\\href{https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf}{scan}.\n\\bibitem{leanapi} Lean project, \\emph{The Lean Language Reference: Validating\na Lean Proof},\n\\href{https://lean-lang.org/doc/reference/latest/ValidatingProofs/}{documentation},\naccessed 18 July 2026.\n\\bibitem{leanref} Lean project, \\emph{The Lean Language Reference: Axioms},\n\\href{https://lean-lang.org/doc/reference/latest/Axioms/}{documentation},\naccessed 18 July 2026.\n\\bibitem{leanblueprint} P.~Massot, \\emph{leanblueprint}, plasTeX plugin for\nLean formalisation blueprints, 2020,\n\\href{https://github.com/PatrickMassot/leanblueprint}{software repository},\naccessed 18 July 2026.\n\\bibitem{leanarchitect} T.~Zhu, P.~Monticone, S.~Welleck, and J.~Avigad,\n\\emph{LeanArchitect: Automating Blueprint Generation for Humans and AI},\nin \\emph{17th International Conference on Interactive Theorem Proving},\nLIPIcs 382, 2026, pp.~25:1--25:16,\n\\href{https://doi.org/10.4230/LIPIcs.ITP.2026.25}{DOI}.\n\\bibitem{leanatlas} B.~Yanahama and A.~Sannai, \\emph{Lean Atlas: An\nIntegrated Proof Environment for Scalable Human--AI Collaborative\nFormalization}, 2026,\n\\href{https://doi.org/10.48550/arXiv.2604.16347}{arXiv:2604.16347}.\n\\bibitem{econcs} N.~Garg, \\emph{EconCSLib: AI-Assisted Lean Formalization for\nEconomics \\& Computation research}, 2026,\n\\href{https://doi.org/10.48550/arXiv.2606.13306}{arXiv:2606.13306}.\n\\bibitem{isadof} A.~D.~Brucker and B.~Wolff, \\emph{Isabelle/DOF: Design and\nImplementation}, in \\emph{Software Engineering and Formal Methods}, Lecture\nNotes in Computer Science 11724, 2019, pp.~275--292,\n\\href{https://doi.org/10.1007/978-3-030-30446-1_15}{DOI}.\n\\bibitem{cascade} T.~Kiecker, J.~A.~Sparka, M.~Reuter, A.~Ziegler, and\nL.~Grunske, \\emph{CASCADE: Detecting\nInconsistencies between Code and Documentation with Automatic Test\nGeneration}, \\emph{Proceedings of the ACM on Software Engineering} 3 (FSE),\nArticle FSE168, July 2026, pp.~3816--3838,\n\\href{https://doi.org/10.1145/3808175}{DOI}.\n\\bibitem{demillo} R.~A.~DeMillo, R.~J.~Lipton, and F.~G.~Sayward, \\emph{Hints\non test data selection: Help for the practicing programmer}, IEEE Computer\n11(4), 1978, pp.~34--41,\n\\href{https://doi.org/10.1109/C-M.1978.218136}{DOI}.\n\\bibitem{jiaharman} Y.~Jia and M.~Harman, \\emph{An analysis and survey of the\ndevelopment of mutation testing}, IEEE Transactions on Software Engineering\n37(5), 2011, pp.~649--678,\n\\href{https://doi.org/10.1109/TSE.2010.62}{DOI}.\n\\bibitem{gotel} O.~C.~Z.~Gotel and A.~C.~W.~Finkelstein, \\emph{An analysis\nof the requirements traceability problem}, in \\emph{Proc.\\ First IEEE\nInternational Conference on Requirements Engineering}, 1994, pp.~94--101,\n\\href{https://doi.org/10.1109/ICRE.1994.292398}{DOI}.\n\\bibitem{gsn} Assurance Case Working Group, \\emph{Goal Structuring Notation\nCommunity Standard (Version 3)}, Safety-Critical Systems Club, 4 May 2021,\n\\href{https://scsc.uk/r1386.pdf}{standard}.\n\\end{thebibliography}\n}\n\\end{document}\n","dependencies":[],"external_verification_route":"https://raw.githubusercontent.com/wcook04/plectis-lean-erdos249-257/main/paper/claim-faithful-publication-systems-paper.tex"},{"record_type":"paper","paper_id":"cold-clone-to-proof-receipt","title":"From a Cold Clone to a Proof Receipt","source_repository":"plectis-lean-erdos249-257","source_path":"paper/cold-clone-to-proof-receipt.tex","hosted_pdf_path":"papers/cold-clone-to-proof-receipt.pdf","licence":"CC-BY-4.0","copyright":"2026 Will Cook","cite_as":"Plectis reader corpus, paper:cold-clone-to-proof-receipt","pdf_sha256":"sha256:c55bc8ec280f632a604bad8b2ccf35d38c32c375eb416ff66f7a6a8d974cba14","pdf_byte_count":75265,"pdf_embedded":false,"pdf_omission_reason":"the exact LaTeX is embedded and retrieves better than extracted PDF text; the PDF is a presentation rendering and its hash is recorded for verification","source_embedded":true,"availability":"embedded_exact","embedded_copy_role":"native_manuscript","embedded_from_repository":"plectis-lean-erdos249-257","embedded_copy_path":"paper/cold-clone-to-proof-receipt.tex","media_type":"text/x-tex","body_format":"latex","sha256":"sha256:285d110ab0ffefb53fde035bbce1038d5bddf41795dc84571ec1dc1ab0115335","byte_count":31734,"body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n% Companion paper on agent-native navigation and proof receipts.\n\\documentclass[11pt]{article}\n\\usepackage{paper-house-style}\n\\usepackage{booktabs,array,float,placeins,multicol,fancyvrb}\n\\usepackage{tikz}\n\\usetikzlibrary{arrows.meta,positioning,calc,fit,shapes.geometric}\n\\usepackage[colorlinks=true,linkcolor=linkinternal,urlcolor=linkexternal,citecolor=linkinternal]{hyperref}\n\\hypersetup{\n  bookmarksnumbered=true,bookmarksopen=true,bookmarksopenlevel=1,\n  pdftitle={From a Cold Clone to a Proof Receipt},\n  pdfauthor={Will Cook},\n  pdfsubject={Agent-native navigation and incremental validation for a large Lean development},\n  pdflang={en-GB},\n  pdfkeywords={Lean 4, formal mathematics, AI agents, navigation, proof receipts, incremental compilation, reproducibility},\n}\n\\newcommand{\\repobase}{https://github.com/wcook04/plectis-lean-erdos249-257}\n\\newcommand{\\auditedref}{\\href{\\repobase/tree/d6d344d9557faee19276b01a1c1f20d7c08f03d5}{\\texttt{d6d344d9557faee1}}}\n\\DefineVerbatimEnvironment{routeblock}{Verbatim}{\n  fontsize=\\scriptsize,\n  frame=single,\n  framerule=0.45pt,\n  framesep=5pt,\n  rulecolor=\\color{MidnightBlue!35},\n  xleftmargin=0.02\\linewidth,\n  xrightmargin=0.02\\linewidth\n}\n\\emergencystretch=1.6em\n\n\\runninghead{From a cold clone to a proof receipt}\n\\title{From a Cold Clone to a Proof Receipt}\n\\subtitle{Agent-native navigation and incremental validation for a large Lean development}\n\\author{Will Cook}\n\\date{July 2026}\n\n\\begin{document}\n\\maketitle\n\\paperprovenance\n\n\\begin{abstract}\nA large Lean repository may contain kernel-checked proofs while giving a new\nreader no reliable account of what exists or what remains open. We describe a\ncold-clone entry architecture evaluated at commit \\auditedref. Without\ncompiling Lean, its bounded overview exposes 665 source modules, 13,446\ndeclarations, ten mathematical programmes, 99 reviewed public claims, and four\nexact open propositions. A separate elaborated index contains 7,024\nsource-resolved constants and 32,902 direct type/value-reference pairs;\nanother 5,247 public pairs have an unresolved endpoint and are counted but not\nretained by identity.\n\nThe contribution is a composition of bounded intent routing, typed status\nrecords, exact source handles, stored Lean probes, focused builds, and release\nchecks. The proof workbench records free-text claims that cite an accepted\nprobe, but it does not verify that the prose follows from that probe, audit\ntransitive axiom dependencies, or protect the ledger from a writer with\nrepository access. Replay reruns stored input bytes and compares the recorded\nprocess verdict; it does not compare complete environments or dirty-file\ncontents. The\narchitecture therefore separates navigation, Lean process acceptance, and\nauthored interpretation. It is not a theorem prover, a trusted proof-receipt\nservice, a reconstruction of the original dirty environment, a portability\nstudy, or evidence that an agent's reasoning was optimal or externally novel.\n\\end{abstract}\n\n\\section{The cold-clone problem}\\label{sec:problem}\n\nThe first interaction with a large Lean repository is usually a filesystem.\nThe reader sees source files, generated files, scripts, papers, and build\nconfiguration, but not the shape of the mathematics. A language model can\nsearch filenames and text, yet those operations do not answer the questions\nthat matter: Which programmes exist? Which propositions are established,\nconditional, finite, or open? Where does a declaration sit in the formal\ndependency graph? Which surface is exhaustive, and which is a human selection?\nWhat may be trusted without compiling anything?\n\nThis is not only a convenience problem. When an agent cannot see a library's\noption surface, it tends to rediscover existing lemmas, confuse a finite result\nwith its open unbounded neighbour, or build a new local index because the\nexisting one was not discoverable. Conversely, loading every declaration,\npaper, and proof edge into one prompt destroys the distinctions the extra\ncontext was meant to reveal. A useful first-contact surface must therefore be\nsmall, expandable, and honest about what it omits.\n\nThe \\href{\\repobase}{development studied here} indexes six open\nErd\\H{o}s problems. Problems 249 and 257 are the two principal reviewed\nprogrammes; Problems 243, 251, 269, and 1049 are problem-owned expansion lanes\nwith their own notes and explicit nonclaims. The corpus is unusually deep for\na navigation experiment: it contains\nmany landed reductions, finite mechanisms, obstruction theorems, and\ncountermodels around a small number of hard frontiers. The design goal is not\nto replace reasoning with a fixed pipeline. It is to let a capable reader see\nenough structure to reason well, while reserving mathematical authority for\nthe checked Lean source at the audited revision.\n\n\\section{A layered mathematical option surface}\\label{sec:layers}\n\nNo single graph can serve all readers without hiding an important distinction.\nThe repository therefore commits four related layers (Figure~\\ref{fig:layers}).\nThey are projections with different coverage contracts, not successive claims\nof understanding.\n\n\\begin{figure}[H]\n\\centering\n\\begin{tikzpicture}[\n  wide/.style={draw=gray!55, rounded corners=3pt, align=left, inner sep=7pt,\n    text width=10.65cm, fill=white},\n  branch/.style={wide, text width=4.82cm, minimum height=3.35cm},\n  arr/.style={-{Stealth[length=2.2mm]}, semithick, draw=gray!70!black},\n  note/.style={font=\\scriptsize\\sffamily, text=gray!72!black, fill=white,\n    inner sep=1.5pt},\n  kicker/.style={font=\\scriptsize\\sffamily\\bfseries, text=gray!68!black}\n]\n\\node[wide, fill=MidnightBlue!5] (atlas) at (0,0)\n  {{\\scriptsize\\sffamily\\bfseries\\color{gray!68!black} SOURCE INVENTORY}\\\\[2pt]\n   \\textbf{Exhaustive declaration atlas}\\hfill\n   \\textcolor{MidnightBlue!85}{\\small\\textbf{665 modules \\(\\cdot\\) 13,446 declarations}}\\\\[2pt]\n   \\footnotesize Source coordinates, declaration kinds, module roles, and a route\n   for every live declaration.};\n\\node[branch, fill=TealBlue!5, below left=12mm and -5.34cm of atlas.south] (dep)\n  {{\\scriptsize\\sffamily\\bfseries\\color{gray!68!black} FORMAL STRUCTURE}\\\\[2pt]\n   \\textbf{Loaded-root dependency index}\\\\[2pt]\n   \\textcolor{TealBlue!75!black}{\\textbf{7,024 nodes \\(\\cdot\\) 32,902 resolved pairs}}\\\\[3pt]\n   \\footnotesize Direct type/value references joined at both endpoints to\n   source. Another 5,247 public pairs have an unresolved endpoint and are\n   retained only as a count.};\n\\node[branch, fill=BurntOrange!6, below right=12mm and -5.34cm of atlas.south] (semantic)\n  {{\\scriptsize\\sffamily\\bfseries\\color{gray!68!black} AUTHORED MEANING}\\\\[2pt]\n   \\textbf{Reviewed argument map}\\\\[2pt]\n   \\textcolor{BurntOrange!80!black}{\\textbf{99 claim nodes \\(\\cdot\\) 93 typed edges}}\\\\[3pt]\n   \\footnotesize Maintainer-authored relations such as \\emph{builds on},\n   \\emph{instantiates}, and \\emph{advances}. These edges organise exposition;\n   they are not additional proof objects.};\n\\node[wide, fill=ForestGreen!6, below=12mm of $(dep.south)!0.5!(semantic.south)$] (claims)\n  {{\\scriptsize\\sffamily\\bfseries\\color{gray!68!black} PUBLICATION SURFACE}\\\\[2pt]\n   \\textbf{Bounded query layer}\\hfill\n   \\textcolor{ForestGreen!70!black}{\\footnotesize\\textbf{17 routes; 10 programmes; 4 open propositions}}\\\\[2pt]\n   \\footnotesize Status vocabulary, exact expansion handles, reading routes,\n   and explicit stronger adjacent statements.};\n\\coordinate (fork) at ($(atlas.south)+(0,-4mm)$);\n\\draw[arr] (atlas.south) -- (fork) -| node[note,pos=.72] {formal slice} (dep.north);\n\\draw[arr] (fork) -| node[note,pos=.72] {authored selection} (semantic.north);\n\\draw[arr] (semantic.south) |- node[note,pos=.42] {reviewed selection} (claims.north);\n\\end{tikzpicture}\n\\caption{The navigation layers at commit \\auditedref. The dependency index is\nan elaborated partial view of the exhaustive atlas; the argument map and query\nlayer are reviewed interpretations, not proof authority.}\n\\label{fig:layers}\n\\end{figure}\n\n\\paragraph{Exhaustive inventory.}\nThe declaration atlas is a source-derived address book. It records every live\ndeclaration that its explicit source roots contain. Exhaustiveness here means\nthat a reader can locate a declaration, not that the system understands its\nmathematical role.\n\n\\paragraph{Exact formal neighbourhoods.}\nThe dependency index loads the two supported compact roots, extracts direct\nconstant references from elaborated types and values, and joins those\nconstants to atlas coordinates. The published edge array is exact only for the\nsource-resolved induced subgraph: 5,247 public reference pairs with at least\none unresolved endpoint contribute to a count, but their identities are not\nretained in the index. The separate unresolved-atlas table does retain 814\ndeclaration identities. Neither kind of absence licenses an independence\ninference, and direct references are not a transitive proof explanation.\n\n\\paragraph{Authored mathematical interpretation.}\nThe reviewed claim registry contains 99 public claim nodes and 93 typed\nargument edges. An edge says how the exposition positions two claims---for\nexample, that one builds on, instantiates, or advances another. It is not\ninferred from Lean and does not establish semantic entailment. Each claim\ninstead carries explicit status, wording, evidence handles, a claim ceiling,\nand a stronger adjacent statement. Exact declaration and source handles let a\nreader cross from that interpretation into formal evidence without conflating\nthe two.\n\n\\paragraph{Reviewed public claims.}\nThe claim map selects 99 results for public exposition. It records 36 as\nproved here, 7 as formalised here, 6 as unconditional progress, 39 as\nconditional reductions, 7 as verified finite instances, 2 as cited only, and\n2 as open. Four explicit frontier propositions describe the stronger\nobligations that survive. These labels are maintainer-reviewed public meaning,\nnot outputs inferred by the proof kernel.\n\n\\section{A bounded tour over an unbounded drilldown}\\label{sec:tour}\n\nThe overview command returns a bounded card rather than a database dump. At\nthe audited revision it reports corpus scale, ten mathematical programmes,\nseven claim-status classes, four exact open propositions, 21 publication\nfamilies, and 17 reading routes. Every row carries a typed handle that can be\nexpanded through the same command. Thus first contact is bounded without\nmaking the drilldown bounded.\n\nFive intent lenses cover the main transitions:\n\\begin{enumerate}\n\\item understand the mathematics through a selected programme and its claims;\n\\item locate any formal object through ordinary-language search followed by\n      exhaustive inventory;\n\\item inspect exact formal dependencies through proof cones and paths;\n\\item begin a checked change through the proof workbench and focused builder;\n\\item audit the agent and release system through the publication architecture\n      and release gate.\n\\end{enumerate}\nThe route logic names no specimen theorem or module. Repository-specific\nmathematics enters through data: programme rows, claims, declaration records,\nand frontier propositions. This distinction matters. The mechanism is generic\nover the repository's projection schemas, while the current projection content\nis intentionally specific to the mathematics it describes.\n\nOrdinary-language entry is likewise not an undocumented trigger list. The\n\\texttt{--vocabulary} packet is an executable Rosetta surface: it publishes\nquestion operators, aliases, transparent expansions, typed route hints, an\nall-problem registry projected from \\texttt{docs/problems.json}, and a\nroute-discovery lexicon projected from\n\\texttt{docs/claims.json::machine\\_readable\\_paper.entrypoints}. A unique exact\nauthored term may take a constant-time fast path; ambiguous or unseen language\nfalls through to visible vocabulary interpretation and ranked corpus search.\nEvery search response records that interpretation, so an agent can inspect how\nits wording was translated instead of depending on phrase luck.\n\nThe query layer also answers four different cold readers. A research mathematician\ngets programmes, claim statuses, and surviving open propositions. A\nformalisation engineer gets declaration coordinates, exact value references,\nand a route into a checked change. An AI-lab researcher gets coverage and\nauthority distinctions. An independent contributor gets the no-build\norientation, focused build route, and public handoff check. Each reader\nsees the same facts through a different question contract rather than a\nseparate hard-coded example.\n\n\\section{Crossing from navigation to authority}\\label{sec:authority}\n\nThe architecture treats navigation and proof as separate capabilities\n(Figure~\\ref{fig:sequence}). A static query may nominate a theorem or dependency\npath. It cannot establish that a proposed application typechecks, that a proof\ncloses a goal, or that an English interpretation is mathematically faithful.\n\n\\begin{figure}[H]\n\\centering\n\\begin{tikzpicture}[\n  stagebox/.style={draw, rounded corners=3pt, align=center, text width=1.86cm,\n    minimum height=1.36cm, fill=white, inner sep=5pt},\n  exact/.style={stagebox, draw=ForestGreen!70!black, fill=ForestGreen!6},\n  nav/.style={stagebox, draw=MidnightBlue!75, fill=MidnightBlue!5},\n  arr/.style={-{Stealth[length=2.2mm]}, semithick, draw=gray!78!black},\n  step/.style={font=\\tiny\\sffamily\\bfseries, text=gray!70!black},\n  phase/.style={draw, rounded corners=2pt, inner sep=3pt,\n    font=\\scriptsize\\sffamily\\bfseries}\n]\n\\node[nav] (clone)\n  {{\\tiny\\sffamily\\bfseries\\color{gray!70!black} 1 \u00b7 CLONE}\\\\[3pt]\n   \\small cold clone\\\\[-1pt]\\footnotesize committed projections};\n\\node[nav, right=3.5mm of clone] (tour)\n  {{\\tiny\\sffamily\\bfseries\\color{gray!70!black} 2 \u00b7 ROUTE}\\\\[3pt]\n   \\small bounded tour\\\\[-1pt]\\footnotesize intent route};\n\\node[nav, right=3.5mm of tour] (cone)\n  {{\\tiny\\sffamily\\bfseries\\color{gray!70!black} 3 \u00b7 FOCUS}\\\\[3pt]\n   \\small declaration\\\\[-1pt]\\footnotesize proof cone};\n\\node[exact, right=3.5mm of cone] (probe)\n  {{\\tiny\\sffamily\\bfseries\\color{gray!70!black} 4 \u00b7 VERIFY}\\\\[3pt]\n   \\small stored Lean probe\\\\[-1pt]\\footnotesize process verdict};\n\\node[exact, right=3.5mm of probe] (claim)\n  {{\\tiny\\sffamily\\bfseries\\color{gray!70!black} 5 \u00b7 BIND}\\\\[3pt]\n   \\small claim record\\\\[-1pt]\\footnotesize structural citation};\n\\draw[arr] (clone) -- (tour);\n\\draw[arr] (tour) -- (cone);\n\\draw[arr] (cone) -- (probe);\n\\draw[arr] (probe) -- (claim);\n\\draw[densely dashed, gray!70] ($(cone.east)!0.5!(probe.west)+(0,-11mm)$) --\n  ($(cone.east)!0.5!(probe.west)+(0,13mm)$);\n\\node[font=\\scriptsize\\sffamily\\bfseries, text=gray!70!black, fill=white,\n  inner sep=1pt, above=15mm of $(cone.east)!0.5!(probe.west)$]\n  {LEAN PROCESS BOUNDARY};\n\\node[phase, draw=MidnightBlue!35, fill=MidnightBlue!4,\n  below=5mm of $(clone.south)!0.5!(cone.south)$]\n  {NAVIGATION \\(\\cdot\\) fast, bounded, advisory};\n\\node[phase, draw=ForestGreen!35, fill=ForestGreen!4,\n  below=5mm of $(probe.south)!0.5!(claim.south)$]\n  {VALIDATION \\(\\cdot\\) exact input bytes, process-derived verdict};\n\\end{tikzpicture}\n\\caption{The shortest path from first contact to a stored Lean-process\nreceipt. The workbench checks the probe; the final claim remains authored\ninterpretation and requires separate comparison with the checked statement.}\n\\label{fig:sequence}\n\\end{figure}\n\nThe proof workbench records a typed move grammar through an append operation:\nobservations, conjectures, plans, interpretations, abandonments, probes,\nclaims, and closure. Conjectures may declare a falsifier. Dead ends remain in\nthe record instead of being rewritten into a linear success story. This is a\ncooperative workflow convention over ordinary writable files, not a\ntamper-evident ledger.\n\nA probe is different from a note. The workbench stores the exact Lean input\nbytes, runs \\texttt{lake env lean --stdin --json} in the current checkout, and\ncomputes one of three verdicts from the exit code and emitted error and\n\\texttt{hasSorry} messages: accepted, accepted with an admitted placeholder,\nor rejected. A command-line caller cannot supply that verdict. However,\n\\texttt{kernel\\_accepted} is a Lean-process result, not a proof of the\nfree-text claim: the runner neither rejects new \\texttt{axiom} declarations\nnor audits transitive axiom dependencies. The \\texttt{claim} command checks\nonly that the record cites an accepted probe; it does not establish that the\nclaim text follows from the stored input.\n\nReplay verifies the stored input hash, reruns the bytes in the current\ncheckout, and compares only the coarse verdict. The environment fingerprint\nstores the Git HEAD, dirty-path count, and toolchain string. It does not store\ndirty file identities or contents, and replay does not require the current\nfingerprint to match the opening fingerprint. Replay therefore answers a\nnarrow question---whether the current Lean process assigns the same verdict\nto the same stored bytes---rather than reconstructing or comparing complete\nenvironments.\n\nThis design deliberately leaves search policy open. LeanDojo couples an open\nprogrammatic Lean environment with extracted proof data, premise annotations,\nretrieval, and a theorem-proving benchmark~\\cite{leandojo}. Pantograph exposes\ntactic execution, proof states, metavariables, and data extraction through\nmachine interfaces~\\cite{pantograph}. Those systems make machine-to-Lean\ninteraction richer. The workbench here addresses a complementary question: after\nan agent chooses its own policy, which parts of the resulting reasoning record\nare advisory, and which probes have a reproducible Lean-process result?\n\n\\section{Compilation after comprehension}\\label{sec:incremental}\n\nA committed navigation projection is portable text; an \\texttt{.olean} file is\na toolchain- and platform-specific build product. A cold clone can therefore\norient immediately but must compile the selected formal target before relying\non a changed proof. Existing compatible Lake artifacts may shorten that build,\nbut this paper does not specify or authenticate their transport.\n\nThe focused build wrapper accepts exact module or source-file targets, or Lean\nfiles changed since a Git reference. It parses the reachable local import\ncone, orders stale or missing modules into dependency waves, and invokes Lake\nin batches bounded by the requested job count. It then performs an ordinary\nLake build of the requested target or targets; with no target it builds the\npublic root. Staleness is an operational optimisation based on source,\ndependency-output, and project-configuration timestamps. The final Lake run,\nnot the planner's timestamp test, is the validation step.\n\nThe exhaustive dependency-index validator is expensive for a different\nreason: it traverses the elaborated environment and exports direct\nreferences. The current \\texttt{--check} path always runs\n\\texttt{lake build Erdos249257}, exports the full supported environment, joins\nthe result to the declaration atlas, and compares the newly encoded packet\nwith the committed JSON. There is no dependency-index cache receipt or\n\\texttt{--full-check} bypass in this revision. Lake may still reuse current\nbuild artifacts, but the exporter scans the supported environment again.\n\n\\section{Dogfood receipt}\\label{sec:dogfood}\n\nThe committed prospective session is a naturalistic use of the workbench, not\na synthetic benchmark. Its ledger contains six reasoning notes, one accepted\nprobe, and two free-text claim records citing that probe. The session records\nthe development of a binary carry-pivot normal form for numeral-adjacent\nboundary words and an exact divisor-incidence identity sharpening a one-sided\ncoefficient bound to equality. The mathematical authority for those results\nis the checked declarations in\n\\texttt{Erdos249257/SuffixCylinderCarryPivot.lean}, not the ledger prose.\n\nThe session opened at Git HEAD \\texttt{7307d007d8306647} with 36 dirty paths.\nThe fingerprint did not preserve their names or contents. At the audited HEAD,\nreplay reran the stored bytes in the current dirty checkout and returned the\nsame accepted verdict. This supports only the claim that the same bytes\nreceive the same coarse Lean-process verdict now. It neither reconstructs the\nopening environment nor checks the two claim texts against the probed theorem\nstatements. The session also does not establish that the agent's route was\noptimal, that every candidate was searched, or that the mathematics is new\noutside the repository history.\n\nThe navigation surface was dogfooded separately as four audiences. The\nresearch-mathematics route reaches the programme map and explicit frontier;\nthe formalisation route reaches a source coordinate and exact proof cone; the\nAI-lab route exposes exhaustive-versus-selective coverage and receipt\nauthority; the contributor route reaches a focused build and release check.\nAdversarial tests remove or distort each protected transition and require the\ncold-clone validator to reject the mutation. This evaluates the declared\nfirst-contact contract, not subjective comprehension.\n\n\\section{Relation to prior work}\\label{sec:related}\n\nProof blueprints pair informal plans with named Lean declarations and\nauthor-supplied dependency links.  \\texttt{leanblueprint} stores a blueprint\nin \\TeX{} and its checker only confirms that each named declaration\nexists~\\cite{leanblueprint}; it neither infers those links nor checks that the\ninformal and formal statements agree.  LeanArchitect infers formal\ndependencies and proof-hole status and exports synchronised blueprint\nmaterial~\\cite{leanarchitect}. The Carleson project shows a blueprint used as\na large collaboration surface. Its public blueprint split the formalisation\ninto about 180 claimable tasks, most corresponding to one blueprint lemma;\nformalisation feedback produced localized corrections, modifications, and\nextensions to the mathematical guide, including a few changes to the general\nsetup and main theorems~\\cite{carleson}. The present tour has a different first-contact role. It\nbegins after the repository already contains a large corpus, distinguishes\nexhaustive inventory from authored interpretation, and routes a reader into\nexact dependency queries and stored Lean-process probes.\n\nLeanDojo and Pantograph provide stronger interaction substrates for learned or\nscripted theorem proving~\\cite{leandojo,pantograph}. This work does not propose\na new proof-search policy. It makes the policy slot explicit and records enough\nof an agent-chosen trajectory to separate notes, nominations, rejected probes,\nand accepted claims.\n\nNor is its declaration graph or retrieval route novel in isolation.\nLeanGraph extracts typed elaborator-level edges across Lean projects, while a\nseparate network study analyses Mathlib's multilayer declaration\ngraph~\\cite{theoremgraph,mathlibnetwork}. LeanExplore combines semantic,\nlexical, and graph ranking behind Python and MCP interfaces, and LeanSearch v2\ntargets global premise sets through iterative\nsketch--retrieve--reflect~\\cite{leanexplore,leansearchv2}. The narrower claim\nhere is that one cold-clone contract composes source addresses, selective\nsemantics, dependency cones, incremental validation, and typed process records\nwithout requiring a hosted service or an initial Lean build.\n\nLarge-library maintenance supplies the relevant compilation lesson.\n\\emph{Growing Mathlib} describes performance-aware library design,\ndeprecation, semantic linters, benchmarks, review tooling, and explicit\ntechnical-debt management~\\cite{growingmathlib}. The changed-cone planner and\nfull dependency-index validator apply a smaller-scale version of that\nmaintenance posture. The\nlocal measurements do not transfer Mathlib's scale or social evidence.\n\nLean Atlas narrows the declarations whose meaning can affect selected theorem\nstatements and leaves semantic verification to people~\\cite{leanatlas}. That\nis close to the authority distinction here: formal dependency can focus human\ninspection without deciding intended meaning. The cold-clone tour adds a\nfront-door contract across exhaustive inventory, selective semantics, public\nclaims, and typed agent-session records.\n\n\\section{Limits and transfer conditions}\\label{sec:limits}\n\nThis is a case study of one repository at one audited revision. Its projection\nschemas and routing engine are reusable mechanisms, but another project must\nsupply its own source roots, programme map, claim vocabulary, frontier\nstatements, and authority owners. The paper gives no evidence that a new\nproject can adopt the architecture cheaply or that agents using it outperform\nagents with an ordinary README.\n\nCounts describe artifact coverage, not mathematical understanding. Direct\ndependency edges are not proof explanations. Authored argument edges may be\nwrong or incomplete. Maintainer-reviewed claims may also be wrong; automation\npreserves recorded relationships but does not judge unrestricted prose.\n\nRecorded reasoning is inspectable rather than necessarily good. A model may\nchoose unproductive probes, omit a relevant source, or rationalise a dead end.\nReplay detects a change in the coarse process verdict for stored bytes; it\ndoes not compare full environments, certify claim entailment, or detect every\nflaw in the surrounding interpretation. Context-blind historical experiments\nalso cannot establish training-data blindness.\n\nFinally, the first full project build and the first elaborated graph export\nremain material costs. The architecture moves them after comprehension and\npermits focused validation while editing. The dependency-index checker still\nperforms a root build and full supported-environment export on every run; it\ndoes not eliminate that cost.\n\n\\section{Authority-aware agent entry}\\label{sec:conclusion}\n\nAn agent-native formal repository should not merely contain many proofs. It\nshould reveal, from a cold clone, what mathematics exists, what each navigation\nlayer knows, what remains open, and which next action crosses into proof\nauthority. The architecture here realises that sequence with committed\nprojections, a bounded intent tour, exact loaded-root dependency data, a\ntyped proof-workbench ledger, focused builds, and a full-export validator.\n\nThe central design choice is separation. Comprehension comes before\ncompilation; nomination comes before application; reasoning notes remain\ndistinct from Lean-process verdicts; exhaustive routing remains distinct from\nselective interpretation; and a public claim remains distinct from the formal\nstatement it describes. Those boundaries let a capable agent move quickly\nwithout making speed, replay, or a structural citation look like mathematical\nauthority.\n\n\\section*{Statements and declarations}\n\n\\paragraph{Declaration of generative AI use.} \\paperfullaideclaration\n\n\\paragraph{Authority boundary.} Navigation does not receive proof authority.\nOnly the pinned Lean process can accept a proof term; the projections, routes,\nand session records described here remain navigation or evidence about a run.\n\n\\appendix\n\\section{Reproduction routes}\\label{app:repro}\n\nThe public repository commits the tour, route, projections, workbench session,\nand validators used in this paper. The shortest reproduction route has three\ntasks. Run them from the repository root at commit \\auditedref; a dirty\ncheckout is not a reconstruction of the audited source state.\n\n\\paragraph{1. Orient without elaborating Lean.}\nThese commands inspect committed projections only:\n\\begin{routeblock}\npython3 scripts/query_corpus.py --overview --format card\npython3 scripts/query_corpus.py --route instant_orientation --format card\npython3 scripts/query_corpus.py --search \"<ordinary-language-query>\" --format card\npython3 scripts/query_corpus.py --declaration <Lean-qualified-name> --format card\npython3 scripts/query_corpus.py --proof-cone <Lean-qualified-name> --depth 2 --format card\npython3 scripts/check_cold_clone_comprehension.py --quick\n\\end{routeblock}\nA declaration name returned by search can be expanded into a source record,\ndirect neighbourhood, or bounded proof cone.\n\n\\paragraph{2. Inspect and replay the committed session.}\nThe first two commands are read-only; replay executes the stored Lean probe in\nthe current checkout. The final command validates the landed target module:\n\\begin{routeblock}\npython3 scripts/proof_workbench.py show --session carry_pivot_2026_07_27\npython3 scripts/proof_workbench.py replay --session carry_pivot_2026_07_27\npython3 scripts/lean_fast_build.py --jobs 2 Erdos249257/SuffixCylinderCarryPivot.lean\n\\end{routeblock}\n\n\\paragraph{3. Rebuild and compare the dependency index.}\nThis check performs a root build and full supported-environment export before\ncomparing with the committed index:\n\\begin{routeblock}\npython3 scripts/build_lean_dependency_index.py --check\n\\end{routeblock}\n\n{\\scriptsize\n\\begin{multicols}{2}\n\\begin{thebibliography}{10}\n\\setlength{\\itemsep}{0pt}\n\\setlength{\\parsep}{0pt}\n\\bibitem{leanblueprint} P.~Massot, \\emph{leanblueprint}, plasTeX plugin for\nLean formalisation blueprints, 2020,\n\\href{https://github.com/PatrickMassot/leanblueprint}{software repository},\naccessed 28 July 2026.\n\\bibitem{leanarchitect} T.~Zhu, P.~Monticone, S.~Welleck, and J.~Avigad,\n\\emph{LeanArchitect: Automating Blueprint Generation for Humans and AI},\nin \\emph{17th International Conference on Interactive Theorem Proving},\nLIPIcs 382, 2026, pp.~25:1--25:16,\n\\href{https://doi.org/10.4230/LIPIcs.ITP.2026.25}{DOI}.\n\\bibitem{carleson} L.~Becker et al., \\emph{A Blueprint for the Formalization\nof Carleson's Theorem on Convergence of Fourier Series}, 2025,\n\\href{https://doi.org/10.48550/arXiv.2405.06423}{arXiv:2405.06423}.\n\\bibitem{leandojo} K.~Yang, A.~M.~Swope, A.~Gu, R.~Chalamala, P.~Song,\nS.~Yu, S.~Godil, R.~Prenger, and A.~Anandkumar, \\emph{LeanDojo: Theorem\nProving with Retrieval-Augmented Language Models}, in \\emph{Advances in\nNeural Information Processing Systems 36}, 2023,\n\\href{https://doi.org/10.48550/arXiv.2306.15626}{arXiv:2306.15626}.\n\\bibitem{pantograph} L.~Aniva, C.~Sun, B.~Miranda, C.~Barrett, and S.~Koyejo,\n\\emph{Pantograph: A Machine-to-Machine Interaction Interface for Advanced\nTheorem Proving, High Level Reasoning, and Data Extraction in Lean 4}, in\n\\emph{Tools and Algorithms for the Construction and Analysis of Systems},\n2025, pp.~116--137,\n\\href{https://doi.org/10.1007/978-3-031-90643-5_6}{DOI}.\n\\bibitem{theoremgraph} S.~Kurgan et al., \\emph{TheoremGraph: Bridging Formal\nand Informal Mathematics}, 2026,\n\\href{https://doi.org/10.48550/arXiv.2606.25363}{arXiv:2606.25363}.\n\\bibitem{mathlibnetwork} X.~Li, N.~Peng, S.~Severini, and P.~Shafto,\n\\emph{The Network Structure of Mathlib}, 2026,\n\\href{https://doi.org/10.48550/arXiv.2604.24797}{arXiv:2604.24797}.\n\\bibitem{leanexplore} J.~Asher, \\emph{LeanExplore: A Search Engine for Lean 4\nDeclarations}, 2025,\n\\href{https://doi.org/10.48550/arXiv.2506.11085}{arXiv:2506.11085}.\n\\bibitem{leansearchv2} G.~Gao et al., \\emph{LeanSearch v2: Global Premise\nRetrieval for Lean 4 Theorem Proving}, 2026,\n\\href{https://doi.org/10.48550/arXiv.2605.13137}{arXiv:2605.13137}.\n\\bibitem{growingmathlib} A.~Baanen, M.~R.~Ballard, J.~Commelin,\nB.~Gin-ge Chen, M.~Rothgang, and D.~Testa, \\emph{Growing Mathlib:\nMaintenance of a Large Scale Mathematical Library}, in \\emph{Intelligent\nComputer Mathematics}, 2025,\n\\href{https://doi.org/10.48550/arXiv.2508.21593}{arXiv:2508.21593}.\n\\bibitem{leanatlas} B.~Yanahama and A.~Sannai, \\emph{Lean Atlas: An\nIntegrated Proof Environment for Scalable Human--AI Collaborative\nFormalization}, 2026,\n\\href{https://doi.org/10.48550/arXiv.2604.16347}{arXiv:2604.16347}.\n\\end{thebibliography}\n\\end{multicols}\n}\n\n\\end{document}\n","dependencies":[],"external_verification_route":"https://raw.githubusercontent.com/wcook04/plectis-lean-erdos249-257/main/paper/cold-clone-to-proof-receipt.tex"},{"record_type":"paper","paper_id":"plectis-public-system","title":"Plectis: What a Stranger Can Check","source_repository":"plectis","source_path":"paper/plectis-public-system.tex","hosted_pdf_path":"papers/plectis-public-system.pdf","licence":"CC-BY-4.0","copyright":"2026 Will Cook","cite_as":"Plectis reader corpus, paper:plectis-public-system","pdf_sha256":"sha256:e673cd0cb30f8329684c15ee12c758064b9e6843f52fb719a064b68849d2b0bd","pdf_byte_count":125110,"pdf_embedded":false,"pdf_omission_reason":"the exact LaTeX is embedded and retrieves better than extracted PDF text; the PDF is a presentation rendering and its hash is recorded for verification","source_embedded":true,"availability":"embedded_exact","embedded_copy_role":"native_manuscript","embedded_from_repository":"plectis","embedded_copy_path":"paper/plectis-public-system.tex","media_type":"text/x-tex","body_format":"latex","sha256":"sha256:a0e18f742b4c8521234ec160448ac3aaed6fbfa71668b52359a7ff9957921f11","byte_count":73542,"body":"% SPDX-FileCopyrightText: 2026 Will Cook\n% SPDX-License-Identifier: CC-BY-4.0\n\\documentclass[11pt]{article}\n\\usepackage{paper-house-style}\n\\usepackage{booktabs,array,enumitem}\n\\usepackage[colorlinks=true,linkcolor=linkinternal,urlcolor=linkexternal,\n  citecolor=linkinternal]{hyperref}\n\\usepackage{fancyvrb}\n\\usepackage{tikz}\n\\usetikzlibrary{positioning,arrows.meta,calc}\n\\usepackage{float}\n\\setlist{nosep,leftmargin=*}\n\\emergencystretch=2em\n\\newcommand{\\code}[1]{\\texttt{#1}}\n\\newcommand{\\component}[1]{\\nolinkurl{#1}}\n\\newcommand{\\repo}{\\href{https://github.com/wcook04/plectis}{\\nolinkurl{github.com/wcook04/plectis}}}\n\n% Shared figure styles: black mechanism, dashed unavailable/unestablished\n% material, and two accents used with one meaning each across all figures:\n% blue for what a stranger can do, amber for words and choices the project\n% itself supplies.\n\\colorlet{strangerblue}{RoyalBlue}\n\\colorlet{authoramber}{BurntOrange!85!black}\n\\tikzset{\n  part/.style={draw=black!75, line width=0.6pt, rounded corners=2pt,\n    align=center, inner sep=4pt, font=\\footnotesize, fill=gray!8},\n  words/.style={part, fill=white, draw=authoramber!75},\n  ghost/.style={draw=black!45, dashed, line width=0.6pt, rounded corners=2pt,\n    align=center, inner sep=4pt, font=\\footnotesize, text=black!60, fill=none},\n  act/.style={font=\\scriptsize\\itshape, text=strangerblue, align=center},\n  gnote/.style={font=\\scriptsize, text=black!60, align=center},\n  flow/.style={-{Stealth[length=2.2mm]}, line width=0.6pt, draw=black!75},\n  softflow/.style={-{Stealth[length=2.0mm]}, line width=0.6pt, draw=black!50, dashed},\n  authlink/.style={softflow, draw=authoramber!70},\n  actflow/.style={-{Stealth[length=2.0mm]}, line width=0.6pt, draw=strangerblue},\n  flabel/.style={font=\\scriptsize, inner sep=2pt},\n  gaplabel/.style={font=\\scriptsize, fill=white, inner sep=2.2pt, align=center,\n    text=authoramber},\n}\n\n% These values are checked against the public registries by\n% scripts/check_public_system_paper.py.  Do not update them from memory.\n\\newcommand{\\snapshotcommit}{57ffb7f5830cc446a2cbd8083d5d80bcab2af563}\n\\newcommand{\\snapshotshort}{57ffb7f5830c}\n\\newcommand{\\componentcount}{88}\n\\newcommand{\\familycount}{7}\n\\newcommand{\\entrycount}{2}\n\\newcommand{\\mappingcount}{12}\n\\newcommand{\\mathcount}{20}\n\\newcommand{\\safetycount}{20}\n\\newcommand{\\researchcount}{9}\n\\newcommand{\\publiccopycount}{20}\n\\newcommand{\\recoverycount}{5}\n\\newcommand{\\leanfilecount}{58}\n\\newcommand{\\copiedsourcecount}{21}\n\\newcommand{\\refactorcount}{27}\n\\newcommand{\\runtimecount}{22}\n\\newcommand{\\validatorcount}{18}\n\n\\hypersetup{\n  pdftitle={Plectis: What a Stranger Can Check},\n  pdfauthor={Will Cook},\n  pdfsubject={Public evidence from a private system: what a stranger can check, what the author can only report, and what remains unestablished},\n}\n\\runninghead{Plectis: what a stranger can check}\n\\title{Plectis: What a Stranger Can Check}\n\\subtitle{What public evidence can and cannot show about a private system}\n\\author{Will Cook}\n\\date{July 2026}\n\n\\begin{document}\n\\maketitle\n\n\\paperprovenance\n\n\\begin{abstract}\nThis paper treats one named version of Plectis as a single case in a bounded\ndescriptive analysis of testing public claims about software that cannot itself\nbe published.  The author releases selected code, fixed inputs, expected\nresults, decision rules, and run records while the larger system remains\nprivate.  What can a stranger conclude after rerunning those tests?  The answer\nis narrow.  The repository lets a reader inspect and rerun its published\nprocedures.  A matching result shows repeatability for a stated public case; it\ndoes not establish where the code came from, whether the expected answer is\ncorrect, whether the published selection resembles the private whole, or\nwhether the private system works reliably.\n\nThe contribution is a method for making one selected disagreement local: the\nregistered components (separately testable parts) place a prose claim beside a\nfixed test case, runnable procedure, expected result, decision rule, run\nrecord, and explicit inference limit.  The repository calls this package a\nclaim--evidence--limit contract.  It is an executable-disclosure case study,\nnot an evaluation of reviewer performance.  A worked component calculates an\naudio level from five small sample arrays.  A reader can repeat the calculation and,\nfor one row, derive the expected number independently of the implementation.\nEven there, the run cannot show whether the pass rule tests the claim as\nwritten, how the code behaves on untested inputs, or whether the chosen formula\nis the right account of audio level.  I chose the published fragments, cases,\nanswers, rules, and limits.  The paper therefore treats the repository as an\nauthor-curated test collection, not as independent validation of the private\nsystem.  I report that its published material was copied or adapted from that\nsystem and that I developed it by directing coding tools based on artificial\nintelligence (AI).  The argument does not depend on that account or on the\nclaimed origin of the published material.\n\\end{abstract}\n\n\\section{The problem}\n\\label{sec:problem}\n\nI have a larger private software system that I cannot responsibly publish:\nit contains personal records, credentials, live browser and account\ninformation, my working notes, and unfinished tools. I would still like\nstrangers to be able to test some claims about it instead of accepting my\nsay-so. I use \\emph{answerable} in a deliberately local sense: a claim is\nanswerable when a stranger can run a public test that bears on it, see how the\nresult is judged, and locate a disagreement in a named input, rule, or output.\nThis is not a new synonym for truth, reproducibility, verification, or\nindependent review. For the rest of the paper I use the plainer phrase\n\\emph{publicly testable claim}. The question is what conclusions such a test\ncan support when the larger system and the unpublished pool from which the\ntest was selected remain invisible.\n\nPlectis is my working answer: a public software repository, stored at\n\\repo. A \\emph{repository} is an organised project folder, usually kept\nwith its revision history; to \\emph{clone} a repository is to copy that\nfolder and history to another computer. Everything this paper says a\nstranger can check is in that repository. This paper describes how its\nclaims are exposed and where the evidence stops.\n\nThe main result is a boundary, not a score. Public code, inputs, rules, and\nrun records can make a selected claim easier to challenge. They cannot by\nthemselves establish three missing premises: that the public code came from\nthe private system, that the project's expected answers and pass rules are\ncorrect, or that the selected cases represent the unseen whole. The audio\nexample in Section~\\ref{sec:run} shows all three limits in one elementary\ncalculation. The later inventory records what was published; it is not\nevidence of breadth, quality, or independence.\nRuneson and H\\\"ost likewise treat configuration-managed project documents as\narchival data while warning that records created for another purpose may omit\nneeded information and be difficult to assess without complementary sources\n\\cite[\\S4.4, p.~148]{runeson2009}.\nThe repository therefore lets a reader test claims about the public objects it\nactually exposes, within the limits stated for each test.\n\n\\paragraph{Who made the decisions.}\nI built the private system by directing AI coding agents: I set the objectives\nand review criteria, accepted or rejected their output, and remained\nresponsible for the cases, expected answers, decision rules, and final public\nwording.  No other person worked on the system.  This account explains the\nconcentration of control, but it remains testimony about origin: Plectis\ncannot independently prove how the private work was produced, and the public\nrepository does not prove the story.  The same evidential limit would hold if\nevery line had been written by hand.  One process can give an implementation\nand its check the same mistake; this paper compares no development methods and\nestimates no error rates.\n\nThe boundary of the evidence can be stated before any command is run:\n\n\\begin{center}\n\\small\n\\begin{tabular}{>{\\raggedright\\arraybackslash}p{0.24\\linewidth}>{\\raggedright\\arraybackslash}p{0.67\\linewidth}}\n\\toprule\n\\textbf{Status} & \\textbf{What belongs in it} \\\\\n\\midrule\nPublicly checkable & The contents of a named public version, the commands listed for its components, the files those commands write, and the results of its public tests. \\\\\nReported here & The private system's history, the role of AI agents in producing it, and my claim that the published material came from the private system. \\\\\nNot established here & Whether the public selection represents the private whole, whether the published code works beyond its fixed inputs, whether the project's own pass rules are correct or even test the stated claims, and whether the private system works reliably at all. \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\n\\paragraph{What I examined.} The paper examines one author-curated software\ncollection. At the named repository version, I read the list\nand plain descriptions of all \\componentcount{} components, counted them by\nproject group and by the kind of public evidence offered, traced the worked\ncomponent from input to limit, and reran the appendix's public commands. I did\nnot compare every pass rule with its prose claim or derive fresh expected\nanswers; Section~\\ref{sec:stronger} reserves those checks for an outside\nevaluator. The paper defines the component contract, then follows\none ordinary component from input to limit. Five distinctions separate what a\npassing run shows from conclusions it may appear to support. The final\nsections identify the choices an evaluator would need to make independently\nof me.\n\nThis is a descriptive analysis of one public repository, not a statistical\nstudy. Runeson and H\\\"ost define the case as the object of study and its units\nof analysis as items contained within it\n\\cite[\\S2.5, p.~138]{runeson2009}. They also note that this boundary depends\non the study context and research goals \\cite[\\S3.1, p.~139]{runeson2009}. I\nborrow only those two terms: the named repository version is the single case,\nand each component is a \\emph{unit of analysis} (one item examined inside that\ncase). This paper does not otherwise claim a full case-study design. No subset\nof the components is used to estimate performance or represent the private\nsystem.\n\nReading this paper requires no programming. Reproducing its runs requires\nbasic command-line use, a terminal (a text window for commands), Git (for\ncopying and versioning repositories), Python 3.11 or later, and \\code{pip}\n(for installing Python software). The appendix gives the exact commands,\nenvironment details, and installation notes. It is a dated record of one\nverification pass at the version named there.\n\n\\section{One public test before the general design}\n\\label{sec:early-example}\n\nConsider four decimal audio samples:\n\\[\n0,\\quad 0.05,\\quad -0.05,\\quad 0.1.\n\\]\nThe published procedure squares them, averages the squares, takes the square\nroot, and multiplies by eight.  Their squared average is \\(0.00375\\), so the\nstated rule gives \\(8\\sqrt{0.00375}=0.489897\\ldots\\).  The public program\nproduces \\(0.489897949\\).  Anyone with a calculator can reproduce the expected\nnumber without trusting the program.\n\nThat agreement establishes something real and small: this public program\nagrees with the stated formula on this test case.  It does not establish that\nthe formula is the right account of audio level, that the public code came\nfrom the private system, that other inputs behave similarly, or that important\nrisks were selected for testing.  Those conclusions require additional\npremises that the observed match does not contain.\n\nThe paper asks the same six questions of every published item:\n\\begin{enumerate}\n\\item What sentence is offered for belief?\n\\item What public object or procedure was checked?\n\\item Who supplied the interpretation or expected result?\n\\item What does the observed match establish?\n\\item What remains unestablished?\n\\item Which independently controlled evidence would strengthen it?\n\\end{enumerate}\nSection~\\ref{sec:run} gives the full command, all five audio cases, and the\nsaved run record.  We now generalise from this example to the project's common\nrecord format.\n\n\\section{The component contract}\n\\label{sec:contract}\n\nA \\emph{component} in Plectis is one registered, testable part of the\nproject. Its \\emph{fixture} is a fixed sample input: in ordinary testing\nlanguage, a test case. A \\emph{receipt} is the saved record of a run. The\nrepository supplies one from an earlier run; rerunning can write another.\nA \\emph{validator} is the program that applies a component's pass and refusal\nrules. In testing terminology, the rule and expected result together act as\nthe test oracle: they say what should count as correct for that case. A\n\\emph{commit} is a named, saved version of the repository. These are Plectis's\ninternal names for familiar objects---test case, test oracle, run record, and\nsaved software version---and the paper uses the familiar terms whenever the\ninternal filename is irrelevant.\n\nHere \\emph{evidence} means the public input, procedure, output, and judgement\nrule offered in support of a claim. Calling them evidence does not establish\ntheir adequacy. \\emph{Contract} means only that the claim, those materials,\nand a limit are stated together so readers can check their fit. Nothing in\nthese labels grades the importance of what a component does.\n\nAt commit \\code{\\snapshotshort} the registry, the component list read by the\nprograms, contains \\componentcount{} entries. Every entry is required to\nconnect the same things: a claim stated in prose; a fixture; a command; the\noutput files the command writes; a stored receipt from a prior run; an\nexplicit rule for what counts as a pass and what must be refused; and a\nstated limit on what a pass establishes. Figure~\\ref{fig:contract} lays the\nrequired parts out, together with the fact about them that the rest of this\npaper keeps returning to: every part is supplied by me, and every part is\nexposed to a stranger.\n\n\\begin{figure}[t]\n\\centering\n\\begin{tikzpicture}[x=1cm,y=1cm]\n% words row\n\\node[words] (claim) at (3.45,2.05) {The claim, stated in prose};\n\\node[words] (limit) at (13.0,2.05) {The stated limit};\n% mechanism row\n\\node[part] (fix) at (0,0) {Fixed input\\\\(the fixture)};\n\\node[part] (code) at (3.45,0) {Public code};\n\\node[part] (out) at (6.55,0) {Fresh output\\\\files};\n\\node[part, text width=2.9cm] (val) at (9.55,0) {Comparison, pass rule, refusal rule\\\\(the validator)};\n\\node[part] (verdict) at (13.0,0) {Pass, or\\\\declared refusal};\n\\node[part] (rec) at (9.55,-2.0) {Fresh receipt\\\\(saved run record)};\n% flows\n\\draw[flow] (fix) -- node[flabel, above=1pt]{command} (code);\n\\draw[flow] (code) -- node[flabel, above=1pt]{writes} (out);\n\\draw[flow] (out) -- (val);\n\\draw[flow] (val) -- (verdict);\n\\draw[flow] (val) -- node[flabel, right]{records} (rec);\n\\draw[authlink] (code) -- node[flabel, right, align=left]{said to\\\\do this} (claim);\n\\draw[authlink] (verdict) -- node[flabel, right, align=left]{read under\\\\this ceiling} (limit);\n% stranger actions\n% top-anchored so the action row reads as one aligned row despite\n% differing line counts\n\\node[act, anchor=north] (a1) at (0,-2.95) {change the input};\n\\node[act, anchor=north] (a2) at (3.45,-2.95) {read it;\\\\rerun it};\n\\node[act, anchor=north] (a3) at (6.55,-2.95) {regenerate\\\\them};\n\\node[act, anchor=north] (a4) at (9.55,-2.95) {compare fresh\\\\with stored;\\\\dispute the rule};\n\\node[act, anchor=north] (a5) at (13.0,-2.95) {hold every\\\\conclusion\\\\to the limit};\n\\draw[RoyalBlue!45, dotted, line width=0.6pt] (a1) -- (fix);\n\\draw[RoyalBlue!45, dotted, line width=0.6pt] (a2) -- (code);\n\\draw[RoyalBlue!45, dotted, line width=0.6pt] (a3) -- (out);\n\\draw[RoyalBlue!45, dotted, line width=0.6pt] (a4) -- (rec);\n\\draw[RoyalBlue!45, dotted, line width=0.6pt] (a5) -- (verdict);\n\\end{tikzpicture}\n\\caption{How to read the component contract, at commit\n\\code{\\snapshotshort}. Read the middle row from left to right: a command applies\npublic code to a fixed input and the code writes output. The validator judges it\nunder the published pass and refusal rules, then writes a fresh receipt recording\nthe run. The stored receipt from the earlier run is available for comparison; it\ndoes not determine the new verdict. The two\nboxes above are words rather than mechanisms: the project states the claim and\nthe limit on what a pass may support. Every part is public, and the project\nsupplied every part. The italic blue row names what a stranger can do without\nasking anyone; amber marks the project's words and choices. Across the figures,\na dashed outline marks something outside what the public run establishes. This\nfigure has no dashed boxes because none of its parts is hidden. In\nFigures~\\ref{fig:boundary} and~\\ref{fig:gaps}, dashes mark private material or\nconclusions not established by the observed run. Section~\\ref{sec:run} follows\none component through this chain.}\n\\label{fig:contract}\n\\end{figure}\n\nThe repository cannot by itself support my assertions about the unseen private\nsystem. A component does something narrower: it gives a related public claim a\nprocedure that can fail in front of a stranger. The procedure\ndoes not make the claim true or establish the published fragment's origin. It\nchanges the form disagreement can take: a reader who doubts a component's\nclaim can point at a particular input, a particular line of the validator, or a\nparticular number in the output, instead of arguing with my description of\nsoftware they cannot see. A component that fails is still answerable in\nthis sense, and a component can be answerable and trivial; answerability\nconcerns the form a claim takes, not its truth and not its importance.\n\nA component may also end in a declared refusal rather than an answer. The\nworked example below refuses an input format it does not support. Components\nthat depend on an external tool, such as the Lean proof checker (a program\nthat verifies mathematical proofs), must record a refusal with a stated limit\nwhen the tool is absent, rather than reporting success.\n\\emph{Runnable} therefore means that the operation and its declared failure\nboundary are present in public form; it does not mean that every optional\ndependency exists on every machine. A refusal is informative only when it is\ndeclared in advance and distinguishable from a breakdown: a predeclared\nrefusal marks the edge of what the component claims, while an unexpected\ncrash is a defect. The two must not be read as the same kind of event. One\nhonesty condition on refusals cannot be discharged from inside the\nrepository at all: the public record shows each boundary as declared at the\nnamed commit, and it cannot show whether the boundary was drawn before or\nafter an awkward case was first met. A boundary drawn afterwards turns a\nfailure into a designed refusal in the telling. Saving the version before\nan outside evaluation begins, as Section~\\ref{sec:stronger} proposes, is\nwhat would make that ordering checkable.\n\nThe idea does not depend on software. The contract needs a claim in words; a\ndeterminate public case; a procedure a stranger can\ncause to run; an observable result; a rule connecting result to claim; a\ndeclared boundary of application; and a stated ceiling on inference. That\nthe case here is a file of numbers, the procedure a command, and the record a\ntext file is a convenience of this example. The same contract could be\nsatisfied in another field by a laboratory protocol with declared rejection\nconditions, or by a sealed evaluation service with published scoring rules.\nI make no claim about those settings; the point is only that the contract\nis a shape for exposing claims to strangers, and this repository is one\nexample. Read simply as software, it is a test suite with a registry and\nunusually complete paperwork. That is a fair description. The paper asks\nwhat such a test suite lets a stranger challenge, and which conclusions\nstill lie beyond it.\n\nThis local notion overlaps with reproducibility and transparency but is not\na synonym for either. A run can repeat perfectly with no stated claim attached,\nand a claim can be answerable while still awaiting a good test. The design\nalso begins from partial disclosure: it offers one limited point of contact,\nnot a view of the hidden system. A component exposes the published fragment\nto challenge. It does not make the private whole answerable.\n\nAnswerability is not the same property as computational reproducibility,\nalthough the component contract supplies materials needed to test the\nlatter. Under the convention used by the National Academies, computational\nreproducibility means obtaining consistent results using the same input\ndata, computational steps, methods, code, and conditions of analysis\n\\cite[conclusion 3-1]{nasem2019}. The report separately cautions that exact\nreproducibility does not guarantee computational correctness: undetected\nerroneous code can reproduce the same erroneous result\n\\cite[p.~9]{nasem2019}.\nIn software testing, a \\emph{test oracle} is a procedure used to distinguish\ncorrect from incorrect behaviour. The challenge of making that distinction is\ncalled the test oracle problem \\cite[p.~507]{barr2015}. Barr et al.\\ distinguish\na partial test oracle \\(D\\) from a conceptual ground-truth oracle that always\ngives the right answer, modelled as a total oracle \\(G\\), while observing that\nsuch ground truth is unknowable except in trivial cases\n\\cite[Defs.~2.4 and 2.6--2.8, pp.~509--510]{barr2015}. Soundness requires\n\\(D(a)\\Rightarrow G(a)\\), and completeness requires\n\\(G(a)\\Rightarrow D(a)\\). A passing validator therefore applies its published\nrule; it does not by itself establish that the rule is sound. The risk\nthat a curated collection disproportionately exposes favourable cases is\nanalogous to selective publication and the file-drawer problem\n\\cite[p.~638]{rosenthal1979}.\nAn \\emph{assurance case} links system claims to arguments and evidence. The\nObject Management Group's Structured Assurance Case Metamodel (SACM), version\n2.3, defines such a case as a collection of auditable claims, arguments, and\nevidence. It separately represents an artifact citation and the\n\\emph{AssertedEvidence} relationship said to connect that artifact to a claim;\nthe relationship is itself an assertion by the case's author. Representing an\nargument is therefore not the same as establishing its validity\n\\cite[secs. 1.1, 4, 7.3, 11.9--11.15]{omgsacm2023}. The GSN Community Standard\nstates the boundary more directly: GSN documents an asserted chain of\nreasoning and its evidential references, but the notation itself establishes\nneither the truth of the argument nor the sufficiency of its support\n\\cite[\\S0:3.2, p.~11; \\S\\S0:4.11--0:4.13, p.~15]{gsn}. Plectis is not an\nimplementation of that standard or of GSN, and makes no claim that the private\nsystem is safe. Its validator applies a published rule and records a result;\nit neither models nor independently justifies the evidential link from that\nresult to the prose claim.\nThe Association for Computing Machinery (ACM) calls digital research materials\nsuch as code, scripts, and data\n\\emph{artifacts}. Its \\emph{Artifacts Evaluated---Functional} badge means\nthose materials passed an independent audit for documentation, consistency,\ncompleteness, ability to run, and evidence of verification and validation.\nThe NISO recommended practice separates object review, reproduction with the\nauthor's objects, and replication by a new study\n\\cite[secs.~2.2--2.4]{nisobadging2021}. ACM's harmonised current names are\n\\emph{Results Reproduced} and \\emph{Results Replicated}; its badges are\nindependent, and results validation requires a peer-reviewed report\n\\cite{acmartifact}. Plectis claims neither status: its receipts are\nauthor-generated, not independent review or validation. These standards are\ncomparison points, not statuses awarded here. This paper claims no new theory of\nassurance, testing, or reproducibility; it reads each public procedure against\nits claim and stated limit.\n\n\\section{One run, examined}\n\\label{sec:run}\n\nThe component \\component{batch8_audio_level_rms_port} computes a loudness\nlevel from short arrays of audio samples. I chose it because it is ordinary.\nIt needs only a square root and no knowledge of the private system; its claim\nis small enough that every part of the evidence can be seen at once.\n\nThe reported origin is this. The private system contains a small macOS\nrecording application, and one of that application's functions reduces a\nshort stretch of audio samples held in memory to a level between zero and\none. The public\ncomponent re-expresses that calculation in Python. The component's code and\nits stored receipts name the private file it claims to re-express\n(\\component{AudioLevelMonitor.swift}); a stranger cannot open that file, so\nthe name is a report, not evidence. What the stranger can check is\neverything else.\n\nThe declared calculation begins with root-mean-square (the \\code{rms} in\nthe component's name): square each sample, average the squares, and take\nthe square root. It then applies the project's gain of eight and caps the\nresult at one. Signed sixteen-bit integer samples are first divided by\n32{,}767, the largest positive value in that representation, to bring them\nonto the same scale. An empty input must produce zero. An unrecognised\nformat name must be refused rather than guessed at.\n\nThe component's public command runs it against its fixture and writes fresh\noutput outside the repository (the backslashes split one long command across\nlines):\n\n\\begin{Verbatim}[fontsize=\\small]\nPYTHONPATH=src python3 -m plectis \\\n  batch8-audio-level-rms-port run \\\n  --input fixtures/first_wave/batch8_audio_level_rms_port/input \\\n  --out /tmp/plectis-audio-rms \\\n  --acceptance-out /tmp/plectis-audio-rms-acceptance.json\n\\end{Verbatim}\n\nAt commit \\code{\\snapshotshort}, the run produced:\n\n\\begin{center}\n\\small\n\\begin{tabular}{>{\\raggedright\\arraybackslash}p{0.21\\linewidth}>{\\raggedright\\arraybackslash}p{0.26\\linewidth}>{\\raggedleft\\arraybackslash}p{0.15\\linewidth}>{\\raggedleft\\arraybackslash}p{0.15\\linewidth}>{\\raggedright\\arraybackslash}p{0.09\\linewidth}}\n\\toprule\n\\textbf{Case} & \\textbf{Input} & \\textbf{Expected} & \\textbf{Observed} & \\textbf{Result} \\\\\n\\midrule\nDecimal samples & 0.0,\\ 0.05,\\ $-0.05$,\\ 0.1 & 0.489897949 & 0.489897949 & pass \\\\\nSixteen-bit integer samples & 0,\\ 1638,\\ $-1638$,\\ 3277 & 0.489892970 & 0.489892970 & pass \\\\\nValues above the cap & 1.0,\\ $-1.0$,\\ 0.5 & 1.000000000 & 1.000000000 & pass \\\\\nEmpty input & (none) & 0 & 0 & pass \\\\\nUnknown format & \\code{pcm24} & refusal & refusal & pass \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\nHere \\emph{pass} means ``refused as expected''; \\code{pcm24} was not accepted.\nBecause the inputs and the rule are both printed above, the first row can be\nchecked without the repository or a computer: the squares of the four\nsamples average to 0.00375, whose square root is $0.061237\\ldots$, and eight\ntimes that is $0.489897\\ldots$. A reader with a calculator can recompute this\nexpected value from the stated formula alone. For this one case the reader\ncan supply the test oracle: elementary arithmetic gives an expected number\nderived independently of the program's output, but not of the project's\nformula.\n\nAlmost nothing else in the collection is like that, which is why the example\nmatters. Elsewhere the project usually supplies the implementation, the\ninput, the expected values, and the validator through the same\nauthor-controlled process. A test\nassembled that way can pass for four different reasons: the implementation\nand the expectation are both correct; the two share a mistake; the chosen\ncases happen to sit where the implementation behaves; or the validator\nchecks a weaker property than the prose claim suggests. A matching result\ntherefore establishes repeatability under that public test. Independent\nevidence bearing on correctness needs an expected result justified apart from\nthe implementation. Arithmetic supplies that for the first row only after\naccepting the project's formula. Fresh inputs alone do not supply one.\n\nThe example suggests a question for any software test: where did the expected\nvalue come from, and who supplied it? The answer changes what a match can\nsupport:\n\n\\begin{center}\n\\small\n\\begin{tabular}{>{\\raggedright\\arraybackslash}p{0.44\\linewidth}>{\\raggedright\\arraybackslash}p{0.47\\linewidth}}\n\\toprule\n\\textbf{Who or what supplies the expected answer} & \\textbf{What a matching result then supports} \\\\\n\\midrule\nThe project itself: implementation, cases, expectations, and rule from one author-controlled process. & Repeatability under the project's own test; the four readings above all stay open. \\\\\n\\addlinespace\nThe reader derives a result from stated premises, as in the decimal-samples row. & Agreement with the stated formula on that case, not proof that the formula is right. \\\\\n\\addlinespace\nAn established external tool or reference. & Whatever the reference itself warrants, on the cases run, within the reference's own limits. \\\\\n\\addlinespace\nAn evaluator who chooses inputs and independently justifies expected results after the version is saved. & Independent evidence bearing on correctness for those cases, within the oracle's own justification and limits. \\\\\n\\addlinespace\nThe world: an outcome observed apart from any program. & Evidence about a real-world outcome, where one exists. \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\nMost components in the collection sit in the first position, except where\na reader promotes a case to the second by recomputing it, as the worked\nexample's decimal-samples row invites. Components that invoke a named\nexternal tool occupy a hybrid of the first and third positions: the project\nstill chooses the case and the rule, while the tool supplies a result that\ninherits its own limits. The evaluations in Section~\\ref{sec:stronger} describe\nways to move consequential choices out of the project's control.\n\n\\component{batch8_audio_level_rms_port_validation_receipt.json} is the stored\nreference for the table. It sits under\n\\component{receipts/first_wave/batch8_audio_level_rms_port/}, so a fresh run is\ncompared against a fixed record rather than against memory. A stored record by\nitself does not prove that the stated command produced it; its use is that\nanyone can generate a fresh one and compare the two.\n\nThe limit is part of the component. This run checks one public Python\ncalculation over synthetic sample arrays (arrays composed for the test, not\ncaptured from a device). It does not run the original application, start an\naudio session, request microphone permission, or capture sound, and it\ncannot establish that the named private file is where the calculation came\nfrom. Nor does the table show that the task was difficult (it was not),\nthat the component is useful elsewhere, or that the remaining entries\nresemble it. What it shows is that one assertion now has a form in which a\nstranger can rerun it, dispute its rule, perturb its input, or replace my\nexpected value with a better-derived one.\n\nThe calculator check derives one expected number without the implementation.\nIt shows agreement with the project-supplied formula on that case, not that\nthe formula is the right audio rule. It does not establish origin, reach, or\nrisks beyond the declared scans. Its correctness evidence is limited to that\ncalculation. The next section treats these five gaps separately.\n\n\\section{Five distinctions}\n\\label{sec:distinctions}\n\nFive labels organise the gaps in Figure~\\ref{fig:gaps}: \\emph{origin}, where\nthe material came from; \\emph{correctness}, whether an answer is right;\n\\emph{meaning}, whether a rule tests its claim; \\emph{reach}, whether shown\ncases stand for unshown ones; and \\emph{risk}, what the declared checks may\nmiss. These short labels correspond to familiar questions about provenance,\ntest-oracle validity, claim--test fidelity, representativeness, and residual\nrisk. They are not five new kinds of evidence. Each subsection separates what\nthe public run directly shows from the additional premise needed for a stronger\nconclusion.\n\n\\subsection*{Public execution versus where the code came from}\n\nA reader can establish what a named public commit contains and what its\ncommands do. No public run establishes where the material came from.\nPlectis records that boundary. Each copied file has a claimed source and a\nSHA-256 message digest, used here as a fingerprint: a 256-bit value calculated\nfrom its contents and used to detect change\n\\cite[abstract, p.~iii; \\S1 and Fig.~1, p.~3; \\S6.2, p.~21]{nistfips1804}.\nTwo different files can in principle have the same digest: FIPS 180-4 treats\nfinding such a pair as computationally infeasible, not mathematically\nimpossible \\cite[p.~iv]{nistfips1804}. For this comparison, the relevant question is\nwhether one can start with a specified file and find a different file with the\nsame fingerprint. The National Institute of Standards and Technology (NIST)\ncalls this second-preimage resistance and expects an approved hash function to\nmake such a search computationally infeasible. That differs from collision\nresistance, which asks only for any pair of different inputs that match\n\\cite[security-strength table]{nisthashfunctions}. A match therefore supports a\nnarrow claim: the public file's contents agree with the version represented in\nits public record. But I control both sides, so that agreement is evidence of\ninternal consistency, not of origin. The same holds for the\naudio component's named source file: the name is an entry I wrote. Origin\nclaims could only be strengthened by something a public repository cannot\nsupply about itself, such as commitments published before the fact, or a\ntrusted person permitted to examine both sides. The term \\emph{provenance}\nmeans an object's origin and history. Here, that origin remains reported\nrather than publicly established. FIPS 180-4 itself similarly cautions that\nconformance to the standard does not assure that a particular implementation\nis secure \\cite[qualification 12, p.~v]{nistfips1804}.\n\n\\subsection*{Repeatability versus correctness}\n\nA run that yields the same checked values and verdict as the stored receipt\nshows that the public code, on the supplied input, in a sufficiently similar\nenvironment, produces the same result again. Correctness is a different\nproperty: it needs a standard for\nwhat the result should have been, and a standard is only worth something if\nit does not merely restate the implementation. The worked example makes the\ndifference concrete. The first row's expected number comes from arithmetic\napplied to the project's formula, not from the implementation. That supports\nagreement with the formula on one case; it does not validate the formula.\nMost components have only the project's own expected values and validator.\nPassing tests anywhere in Plectis should be read at that strength and no more.\n\n\\subsection*{A validator's rule versus the stated claim}\n\nA pass is a verdict from the validator, and a validator checks exactly the\nproperty written into it, which can be narrower than the claim as worded. A\ncomponent whose prose speaks of handling a class of inputs may in fact\nverify that one output file exists and matches a stored shape; a rule can\nhold while the sentence it stands under goes untested. The two previous\ndistinctions do not cover this gap: a result can be repeatable, and correct\nas far as the checked property goes, while the checked property is beside\nthe stated point. Plectis makes the gap inspectable rather than closed.\nEvery validator is public, so a reader can put the rule beside the prose\nand object where the rule is weaker, and the review in\nSection~\\ref{sec:conclusion} makes exactly that comparison its central step.\nSince the same author-controlled project produced both the claims and the\nrules, agreement between them records one judgement twice, and no count of\npassing components can substitute for reading one rule against one sentence.\n\n\\subsection*{Selected cases versus general behaviour}\n\nSelection acts at two levels. Within a component, the fixture covers only a\nfew possible inputs. Across the collection, the published components are my\nselection from a private whole.\n\n\\begin{samepage}\nI chose which mechanisms\ncould be given public form, which inputs to include, which expected values to\nstore, and which pass rules to enforce. Nothing in the public record shows\nhow many candidates there were, what was excluded beyond the privacy rules,\nor whether the collection resembles ordinary work in the private system. A\ncollection can consist entirely of genuine items and still give an\nunrepresentative picture of what it was drawn from. This is analogous to\nselective publication and the file-drawer problem, not the same process\n\\cite[p.~638]{rosenthal1979}.\nFigure~\\ref{fig:boundary} draws the situation: public checking reaches\neverything to the right of the boundary, and the pool the selection was\ndrawn from stays on the left.\n\\end{samepage}\n\n\\begin{figure}[H]\n\\centering\n\\begin{tikzpicture}[x=1cm,y=1cm]\n% private region\n\\draw[black!55, dashed, line width=0.7pt, rounded corners=3pt] (0,-0.1) rectangle (5.8,3.7);\n\\node[font=\\footnotesize, align=center] at (2.9,3.3) {The private system\\\\[-1pt]{\\scriptsize (not published)}};\n% mechanism squares: 5 x 3 grid, four of them selected (darker)\n\\foreach \\x in {0.75,1.80,2.85,3.90,4.95}\n  \\foreach \\y in {0.55,1.35,2.15}\n    \\draw[black!35, fill=gray!25] (\\x-0.14,\\y-0.14) rectangle (\\x+0.14,\\y+0.14);\n\\foreach \\sx/\\sy in {1.80/0.55,2.85/1.35,0.75/2.15,4.95/1.35}\n  \\draw[authoramber!80!black, fill=authoramber!55] (\\sx-0.14,\\sy-0.14) rectangle (\\sx+0.14,\\sy+0.14);\n\\node[font=\\small, text=black!45] at (5.42,1.35) {$\\cdots$};\n\\node[font=\\small, text=black!45] at (2.85,0.22) {$\\cdots$};\n% boundary\n\\draw[black!70, dashed, line width=0.8pt] (7.15,-1.95) -- (7.15,4.15);\n\\node[font=\\scriptsize, text=black!70, above] at (7.15,4.2) {the publication boundary};\n% selection arrow; the label plaque interrupts the boundary line symmetrically\n\\draw[flow, draw=authoramber] (5.8,1.8) -- (8.55,1.8);\n\\node[flabel, fill=white, align=center, inner sep=2.5pt, text=authoramber] at (7.15,2.44) {I chose and\\\\adapted these};\n% public box\n\\draw[black!75, line width=0.7pt, rounded corners=3pt, fill=gray!6] (8.55,0.55) rectangle (13.35,3.05);\n\\node[font=\\footnotesize, align=center] at (10.95,1.8) {The public repository\\\\at commit \\code{\\snapshotshort}:\\\\\\componentcount{} components in \\familycount{} groups};\n% stranger\n\\node[act, text width=2.0cm] (str) at (14.75,1.8) {a stranger:\\\\clone it, run it,\\\\compare, dispute};\n\\draw[actflow] (str.west) -- (13.35,1.8);\n% notes\n\\node[gnote, text width=5.6cm, anchor=north] at (2.9,-0.40) {outside public evidence: how many candidates there were, what was excluded and why, whether these resemble ordinary private work};\n\\node[gnote, text width=4.6cm, anchor=north] at (10.95,-0.40) {inside public evidence: files, commands, records, and tests of the published fragments};\n\\end{tikzpicture}\n\\caption{The selection problem. Each square stands for a private\nmechanism; the darker ones were given public form. Each public component\nhas the parts shown in Figure~\\ref{fig:contract}. The grid trails off\nbecause its true extent is exactly what the public side cannot know. The\ncount of squares on the left, the\nrule that picked the darker ones, and the resemblance between the two sides\nare all invisible from the right. The collection supplies directly\ncheckable evidence about the published files and their behaviour under\nnamed tests, but no basis by itself for generalising to the hidden whole.}\n\\label{fig:boundary}\n\\end{figure}\n\nThe registry's count should be read in that light. A registry of\n\\componentcount{} entries defines exactly what is published. A later\nevaluation would need that list to draw a sample and report failure rates\nagainst a fixed total. The count is an inventory, not a score, and\neven the unit of counting is a choice I control, since one mechanism could\nhave been registered as several components, or several as one.\n\n\\subsection*{Risk reduction versus guarantee}\n\nThe publication process runs checks whose honest description is that they\nreduce specific risks. Automated scans search copied material for specified\npatterns of credentials, personal information, and private file paths, and\nrecord what was omitted or rejected; a scan establishes that the scanner\ndid not find the patterns it was designed to find, and nothing stronger.\nPages such as the component map are generated from the registry, and tests\ncompare page and registry, which prevents the particular failure of prose\ndrifting away from records; a generator and its records can still be wrong\ntogether. The documented \\code{tour} run takes the public checkout as its\nexplicit input and does not automatically discover a neighbouring private\nrepository. The release-export contamination check is an explicit\nexception: when a copy of the private project folder is available, it may\ncompare the contents of public source files with private files to detect\nleakage. None of this\namounts to a guarantee that no secret, no identifying detail, and no\ninconsistency survives. Publication remains a judgement about acceptable\nrisk, and I made it.\n\nOrigin evidence and privacy pull in opposite directions. Establishing\nprovenance would require someone to examine private\nmaterial, and the privacy constraint that motivates the whole design forbids\nexactly that examination in public. Some claims are therefore permanently in\nthe reported category, short of a confidential review. The honest treatment\nis to label them, not to dress them up as public proof through bookkeeping\nthat one maintainer controls.\n\n\\medskip\n\nThe same mistake lies behind all five: treating an observation as if it proved\nsomething never tested. A broader conclusion needs an added assumption. For\norigin, one must assume\nthat the public object came from the private one. For correctness, one must\nknow that the expected answer was right; for meaning, that the rule tested\nwhat the claim said. For reach, one must assume that the chosen cases stand\nfor the unchosen; for risk, that the scan covered the relevant risks. The\nobservation itself supplies none of these assumptions. The contract makes\neach one explicit. Once named, a gap can be disputed, assigned a cost to\naddress, or deliberately left open. An unnamed gap is easy to cross without\nnoticing. These five are not exhaustive. Two more gaps appeared earlier:\nobserving a refusal today does not show when it was drawn\n(Section~\\ref{sec:contract}), and observing a number at one commit does not\ndescribe the next (the appendix). These are simply the five gaps this\ncollection most needs a reader to hold.\n\n\\begin{figure}[t]\n\\centering\n\\begin{tikzpicture}[x=1cm,y=1cm]\n\\node[part, text width=3.2cm] (obs) at (1.8,0) {Observed: at one commit, in one environment, a fresh run matches the stored record};\n\\draw[black!55, dashed, line width=0.7pt] (4.55,-3.75) -- (4.55,3.75);\n\\node[font=\\scriptsize, text=black!60, rotate=90] at (4.3,2.55) {edge of the observation};\n\\node[ghost, text width=4.6cm] (g1) at (13.1,2.9)  {\\ldots so it came from the private system as described};\n\\node[ghost, text width=4.6cm] (g2) at (13.1,1.45) {\\ldots so the result is what it should have been};\n\\node[ghost, text width=4.6cm] (g3) at (13.1,0)    {\\ldots so the test supports the claim as written};\n\\node[ghost, text width=4.6cm] (g4) at (13.1,-1.45){\\ldots so it behaves like this on inputs not shown};\n\\node[ghost, text width=4.6cm] (g5) at (13.1,-2.9) {\\ldots so no risk of that kind remains};\n% fanned starts so the five tails do not pile on one point; all lines are\n% drawn first, then all label plaques, so no later line strikes a label\n\\draw[softflow] ([yshift=22pt]obs.east) -- (g1.west);\n\\draw[softflow] ([yshift=11pt]obs.east) -- (g2.west);\n\\draw[softflow] (obs.east) -- (g3.west);\n\\draw[softflow] ([yshift=-11pt]obs.east) -- (g4.west);\n\\draw[softflow] ([yshift=-22pt]obs.east) -- (g5.west);\n\\path ([yshift=22pt]obs.east) -- (g1.west) node[gaplabel, pos=0.50]{origin: evidence about where it came from};\n\\path ([yshift=11pt]obs.east) -- (g2.west) node[gaplabel, pos=0.60]{correctness: an answer derived separately};\n\\path (obs.east) -- (g3.west) node[gaplabel, pos=0.50]{meaning: a rule that tests the words};\n\\path ([yshift=-11pt]obs.east) -- (g4.west) node[gaplabel, pos=0.60]{reach: cases I did not choose};\n\\path ([yshift=-22pt]obs.east) -- (g5.west) node[gaplabel, pos=0.50]{risk: checks beyond\\\\the listed patterns};\n\\end{tikzpicture}\n\\caption{Five conclusions a matching run invites and does not license.\nEach dashed arrow needs a further assumption that the observation itself\ncannot supply. From top to bottom, the tags follow the order discussed above:\norigin (provenance), correctness, meaning, reach, and risk. The contract's\ncontribution is not to cross these gaps but to make them nameable, one\ncomponent at a time.}\n\\label{fig:gaps}\n\\end{figure}\n\n\\section{What the collection contains}\n\\label{sec:collection}\n\nThis section is a dated inventory, not a result. The categories are useful for\nfinding a test and for seeing which kind of limitation accompanies it. Their\nnames and counts do not measure importance, correctness, independence, or\ncoverage of the private system.\n\nAt commit \\code{\\snapshotshort} the \\componentcount{} components fall into\n\\familycount{} groups:\n\n\\begin{center}\n\\small\n\\begin{tabular}{>{\\raggedright\\arraybackslash}p{0.38\\linewidth}r>{\\raggedright\\arraybackslash}p{0.47\\linewidth}}\n\\toprule\n\\textbf{Group} & \\textbf{Count} & \\textbf{What its components do} \\\\\n\\midrule\nGetting started & \\entrycount & Show a new reader where to begin and guide a first inspection. \\\\\nProject mapping & \\mappingcount & Map files and written rules, find relevant material, and route a question to the part of the project that owns it. \\\\\nComputer-checked mathematics & \\mathcount & Check records and fixed examples from formal mathematics, including small Lean statements and one exact finite calculation. \\\\\nAI reliability and safety tests & \\safetycount & Replay fixed cases involving hostile instructions, software boundaries, stored information, tool use, and benchmark rules. \\\\\nResearch-method tests & \\researchcount & Exercise fixed cases involving replication, conflicting forecasts, explaining model behaviour, simulation, and laboratory safety. \\\\\nPublic-copy maintenance & \\publiccopycount & Check copied source, generated pages, publication boundaries, and disagreement between records and derived files. \\\\\nWork recovery & \\recoverycount & Record unfinished changes and test how work resumes after interruption. \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\nThe groups do not share a subject. Their only unity is the contract of\nSection~\\ref{sec:contract}: each item became a small public test. The collection\nis organised by how claims are checked, not by a common scientific question.\n\nThe registry also classifies what kind of evidence each component supplies.\nIt calls this classification a \\emph{route}. A route says whether the\npublic component preserves source text, re-creates behaviour on fixed cases,\nrecords a run, or checks a declared rule. The four routes below cover all\n\\componentcount{} entries. Each leaves a different question open:\n\n\\begin{center}\n\\small\n\\begin{tabular}{>{\\raggedright\\arraybackslash}p{0.20\\linewidth}r>{\\raggedright\\arraybackslash}p{0.30\\linewidth}>{\\raggedright\\arraybackslash}p{0.32\\linewidth}}\n\\toprule\n\\textbf{Route} & \\textbf{Count} & \\textbf{What it preserves} & \\textbf{What it cannot show} \\\\\n\\midrule\nRecord-checked copied source & \\copiedsourcecount & The exact text of non-secret source files, checked against a public import record. & That the recorded origin is real; record and file share one maintainer. \\\\\nPublic reimplementation & \\refactorcount & One calculation's behaviour on fixed cases. & That it came from the private system; the private text is absent. \\\\\nDirect or external-tool run & \\runtimecount & A local computation, or a named external tool's result, captured with its context. & Behaviour where the tool or environment differs; the result is as strong as the tool and the case. \\\\\nContract checker & \\validatorcount & Whether a public input satisfies a declared structural rule. & Whether the declared rule is the right rule to require. \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\nCopied source preserves text but demonstrates little behaviour; a\nreimplementation does the converse; an external-tool run inherits the tool's\nlimits; and a contract checker is worth exactly the rule its author chose.\nTo inspect one item, use the public component map,\n\\component{ORGANS.md} (an inherited filename). Follow it rather than guessing,\nbecause some components share a page. The audio row points to\n\\component{paper_modules/batch8_audio_level_rms_port.md}, which in turn names\nthe fixture, receipts, route, and limit.\n\nTwo cautions govern the tables. First, components are not independent\nwitnesses: they share code, fixture style, authorship, and judgement.\n\\componentcount{} entries are registered claims, not \\componentcount{}\nconfirmations. Pages, entries, and receipts generated from the same underlying\nrecords repeat one witness rather than supply several. Machine checks can catch\ntheir accidental disagreement. They cannot create an independent witness.\nThe four routes impose four different limits on what may be concluded; they\nare not four independent methods confirming one claim.\n\nSecond, names such as computer-checked mathematics and AI reliability and\nsafety tests borrow gravity from serious fields. Read each row as the weakest\nproperty its tests actually check---often that a fixed case replays or a record\nhas a stated shape. The names help find components; they do not weigh them.\n\n\\section{The author's hand}\n\\label{sec:hand}\n\nAll five distinctions return to control. At the commit this paper describes,\nI hold every consequential choice on both sides of every test.\nI chose which private mechanisms were candidates, which were selected, and\nhow finely each was divided into components, and so what the count counts.\nI chose which of the four routes each took, which inputs went into the fixtures, what\nthe expected values are, what each validator checks, and where each\nrefusal boundary sits. And I chose which group each entry is filed under,\nwhich failures this paper explains and in what tone, which single\ncomponent the worked example honours, and\nwhere the publication boundary lies. Somebody had to make each of these\nchoices, and the privacy constraint meant it was me. None of that is\nmisconduct.\n\nBut that concentration of choice permits a failure mode, and it involves\nno false statement anywhere. Select the mechanisms that show well. Divide\nthe favourable ones finely. Narrow each claim until its fixed cases satisfy\nit. Give awkward material the least demanding check. Declare\nas refusals whatever would otherwise fail. File\nsmall replays under weighty names. Then let the counts, the categories, and\nthe machinery of records accumulate into an impression of scale and\nmaturity that no individual sentence asserts. A document can be cautious\nsentence by sentence while its arrangement still overstates, because a\nreader's impression forms on what is counted, named, worked through, and\nleft out, as much as on what is claimed. This paper is inside its own\narrangement, so declaring the risk does not discharge it; nothing in the\ntext can establish that the text is not doing what this paragraph\ndescribes.\n\nMaking disagreement local can also displace the larger question. A reader\ncan settle an argument about one input or one rule while leaving untouched\nwhether the selection is fair, whether the whole is sound, or what any of it\nis for. An objection about selection is not answered by the successful\nexecution of any number of selected items. The registry hosts the small\narguments; it does not settle the large ones.\n\nTwo corrections need no cooperation from me. First, read any component at the\nweakest property its validator checks, as Section~\\ref{sec:collection}\nproposes. Second, treat every count as an inventory, and every agreement among\nmy records as one witness. A third point concerns what no reader can inspect:\na private mechanism is easiest to publish when it can be separated into a\nsmall task, gives the same output from the same input, and runs in isolation.\nA collection like this\ntherefore over-represents pure calculation, replay, and record-checking, and\nunder-represents stateful, entangled, long-running behaviour. The\npublished picture tilts towards what publishes well before any question of\nhonesty arises.\n\nThe National Aeronautics and Space Administration (NASA) defines independence\nfor independent verification and validation (IV\\&V), a formal software check,\nthrough three separations: technical (the evaluator did not develop the system);\nmanagerial (a separate organisation chooses scope, methods, and schedule); and\nfinancial (an independent group controls the budget without adverse financial\npressure) \\cite[\\S4.4.1.2, p.~48]{nasa8739}.\nThese are criteria for NASA IV\\&V, not a universal test. Plectis has undergone\nno IV\\&V. No other outside evaluation report is cited or included, and I know\nof none; an unreported private rerun would remain invisible. I use the NASA\ncriteria only for the paper's five gaps: evidence about a choice depends less\non me once the choice is no longer mine. Section~\\ref{sec:stronger} lists\nevaluations that address those dependencies.\n\n\\section{What stronger evidence would look like}\n\\label{sec:stronger}\n\nA passing run shows that named public code produced a stated result from a\nsupplied input in a stated environment. It does not establish private\nprovenance, representativeness, correctness beyond self-supplied criteria,\nusefulness, security, originality, or the private system's reliability. The\ncomputer-checked mathematics label covers different checks: proof-work records,\nsmall public statements checked by Lean, and one exact finite calculation. The\nlabel itself supplies no theorem; each entry supports only its own public claim\nand stated limit. At the named commit, the repository's evidence still came\nfrom the project. It includes no outside evaluator's report; I know of none,\nalthough a private rerun could be unreported.\n\nFive forms of independent evaluation could add evidence that the public\nrepository cannot supply by itself. These are not the four publication routes\nin Section~\\ref{sec:collection}. A publication route classifies evidence already\nin the registry; the evaluations below describe new work by someone outside the\nproject. The order is only for exposition. It ranks nothing, and no evaluation\nis a prerequisite for another. Figure~\\ref{fig:ladder} shows which choice each\nevaluation transfers and which claim it still leaves open. The common question\nis who makes the choice that matters for that claim.\n\n\\begin{figure}[H]\n\\centering\n\\begin{tikzpicture}[x=1cm,y=1cm]\n% uniform box heights and bottom/top-anchored labels keep every dotted\n% leader the same length\n% a whisper of stranger-blue on the evaluations: each is evidence that lives\n% outside the author's hands\n\\node[part, text width=2.15cm, minimum height=1.52cm, fill=strangerblue!8] (s1) at (1.2,0) {Another person repeats the run};\n\\node[part, text width=2.15cm, minimum height=1.52cm, fill=strangerblue!8] (s2) at (4.3,0) {Evaluator chooses the cases};\n\\node[part, text width=2.15cm, minimum height=1.52cm, fill=strangerblue!8] (s3) at (7.4,0) {New inputs, answers worked out separately};\n\\node[part, text width=2.15cm, minimum height=1.52cm, fill=strangerblue!8] (s4) at (10.5,0) {Outside task, ordinary alternative};\n\\node[part, text width=2.15cm, minimum height=1.52cm, fill=strangerblue!8] (s5) at (13.6,0) {Confidential private-source review};\n\\node[act, text width=2.25cm, anchor=south] (b1) at (1.2,1.12) {who runs it};\n\\node[act, text width=2.25cm, anchor=south] (b2) at (4.3,1.12) {which cases count};\n\\node[act, text width=2.25cm, anchor=south] (b3) at (7.4,1.12) {what the answer should have been};\n\\node[act, text width=2.25cm, anchor=south] (b4) at (10.5,1.12){what counts as useful};\n\\node[act, text width=2.25cm, anchor=south] (b5) at (13.6,1.12){whether the origin account holds};\n\\node[gnote, text width=2.25cm, anchor=north] (n1) at (1.2,-1.12){does not by itself establish correctness};\n\\node[gnote, text width=2.25cm, anchor=north] (n2) at (4.3,-1.12){cannot reach the private pool};\n\\node[gnote, text width=2.25cm, anchor=north] (n3) at (7.4,-1.12){tested cases only, not generality};\n\\node[gnote, text width=2.25cm, anchor=north] (n4) at (10.5,-1.12){advantage only where compared};\n\\node[gnote, text width=2.25cm, anchor=north] (n5) at (13.6,-1.12){moves trust to the reviewer};\n\\foreach \\i in {1,...,5}{\n  \\draw[RoyalBlue!45, dotted, line width=0.6pt] (b\\i.south) -- (s\\i.north);\n  \\draw[black!30, dotted, line width=0.6pt] (s\\i.south) -- (n\\i.north);}\n\\end{tikzpicture}\n\\caption{Five forms of independent evaluation. Above each evaluation, in italic\nblue, the choice that leaves my hands; below it, in grey, what that evaluation\nstill cannot show. The evaluations are not cumulative stages. At the commit\nthis paper describes, the repository records none, and I know of none.}\n\\label{fig:ladder}\n\\end{figure}\n\n\\emph{Independent repetition} means a person with no involvement in the\nproject reruns the supplied commands in their own environment and publishes\nwhat happened, including failures. That would reduce dependence on my\nmachine and test whether the published instructions suffice elsewhere. It\nwould not by itself establish correctness.\n\n\\emph{Evaluator-controlled selection} means the evaluator chooses\ncomponents from the full registry, without substitution when one proves\nawkward, and original failures stay on the record even if I later repair\nthem. The units themselves are open to the same challenge: an evaluator\nwho finds that two entries are one dependent mechanism, or that a\nboundary excludes the difficult part of a task, is disputing the registry,\nnot misusing it. That addresses curation within the public collection. It\ncannot address whether the collection represents the private system,\nbecause the private pool stays invisible.\n\n\\emph{Fresh inputs with independently derived expectations} can move\nrepeatability towards evidence bearing on correctness: inputs chosen after\nthe version is saved, and expected results worked out by a method that\ndoes not pass through my code, whether independent calculation, a second\nimplementation, or an external reference. The result would support claims\nabout those cases only to the extent that the derivation is credible and\nindependent and the cases are adequate; general correctness still would not\nfollow.\n\n\\emph{Comparison on outside tasks} is what a claim of advantage over\nordinary alternatives would need: someone bringing a task that was not\ndesigned around the project's own categories and comparing the component\nagainst an ordinary alternative, which for several components would be a\nshort conventional script. Nothing in this paper makes that comparison.\n\n\\emph{A provenance review}, among the five evaluations shown here, bears most\ndirectly on the origin story: a trusted person examining selected private\nmaterial against the public collection under confidentiality. Prior public\ncommitments could provide another kind of provenance evidence. Even\ncompleted, this review would\ntransfer trust rather than remove it: public readers would be relying on\nthe reviewer's access, competence, and report instead of on mine, and the\nreview itself could not be published. The privacy cost may reasonably\nnever be paid. If it is not, the origin story remains testimony, and this\npaper is written so that its argument does not depend on it.\n\n\\subsection*{A falsifiable comparison}\n\nThe next study should compare two presentations of the same frozen public\nitems.  The baseline would provide ordinary documentation, code, outputs, and\ntests.  The comparison condition would additionally provide the explicit\nclaim, test case, expected result, decision rule, run record, and inference\nlimit used here.  Evaluators who did not design the items would choose from the\nfrozen public list, inspect deliberately mismatched claim--rule pairs, use\nfresh inputs where an expected result can be justified independently, and\npublish failures without author substitution.\n\nThe useful observations would be whether readers detect a mismatch, how often\nthey wrongly accuse a matching item, how long they take to locate the point of\ndisagreement, whether they reproduce the run, and how well their confidence\ntracks those outcomes.  That study would measure reviewer behaviour on the\npublic items.  It would still not measure the private system, the unseen\nselection pool, or correctness beyond the independently justified cases.  No\nsuch comparison is reported in this paper.\n\n\\subsection*{How the record should age}\n\nThe evaluations above may expose failures. Repairs should add facts, not erase\nthem. The appendix keeps three failures at the pinned commit although a later\ncommit repairs one; that repair does not subtract the first fact. If criticism\nshows a validator too weak or a claim too broad, record the revision's date and\ncause instead of making the project appear always to have meant the narrower\nclaim. A test rewritten after its outcome is known is a different test and may\nbe better. The known case can show whether the same failure returns; fresh\nconfirmation requires new cases whose answers were not used to design the\nrepair. Disappointment is in scope: if an evaluation finds a component trivial\nor no better than a short ordinary script, that conclusion belongs on the\nrecord with the same standing as a pass. Any outside evaluator's report should\nremain under that evaluator's control, cited rather than absorbed. If the\nrepository later falls short of this paragraph, the shortfall is itself a\nfinding, and Section~\\ref{sec:conclusion} says where findings go.\n\nThese evaluations do not pair one-for-one with the distinctions. They address\nthe runner, public-case selection, tested-case correctness, usefulness, and origin.\nNone repairs a validator--claim mismatch, proves representativeness, or turns\nrisk reduction into a guarantee.\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nPlectis makes selected public claims testable; it does not verify the private\nsystem as a whole. Its fixed cases, procedures, outputs, rules, and stated\nlimits let a stranger repeat a result and identify exactly what they dispute.\nA pass is no stronger than the test oracle---the rule and expected answer---or\nthan the basis for believing that oracle.\n\nThe worked audio calculation demonstrates the hierarchy. Repeating the stored\nresult supports computational reproducibility for that public case. Deriving\none expected number by arithmetic adds evidence that the implementation agrees\nwith the stated formula on that case. Neither step establishes that the formula\nis the right audio rule, that the code came from the private system, or that the\nselected case represents it.\n\nMore author-controlled components enlarge the inventory without changing who\nchose the cases or rules. Moving a relevant choice out of the author's hands\ncan strengthen evidence about that choice; it does not repair every gap.\n\n\\appendix\n\\small\n\\setlength{\\parskip}{0pt}\n\n\\section{Dated reproduction record}\n\\label{sec:appendix}\n\nExcept where a paragraph gives a later date, this appendix records one run on\n17~July~2026 under macOS 15.6.1 on Apple silicon (arm64) and Python 3.13.7.\nThe repository version was commit \\code{\\snapshotcommit}, abbreviated\n\\code{\\snapshotshort} elsewhere. Later commits or machines may differ; the body\ndoes not depend on them.\n\n\\paragraph{A first review.} Use the appendix's no-install block and select the\nversion analysed here. The tour describes the checkout but does not choose a\ncomponent claim. Ask the repository to find the audio example with\n\\par\\noindent{\\scriptsize\\Verb|PYTHONPATH=src python3 -m plectis comprehend --first-action \"audio level calculation\" --format text|}\\par\nThe response identifies the audio component, but do not run its first suggested\ncommand: it writes to the repository's saved receipt paths. Run the paper's exact\nno-install command near the start of Section~\\ref{sec:run} instead; it writes only\nto \\code{/tmp} and leaves the checkout unchanged. Under\n\\component{/tmp/plectis-audio-rms/}, open\n\\component{batch8_audio_level_rms_port_result.json};\nJSON is a plain-text field-and-value format. Open \\code{exercise}. In each\n\\component{reference_cases} item, compare \\code{expected\\_level} with\n\\code{observed\\_level}. The top-level \\component{anti_claim} says what a pass\ndoes not establish; \\component{claim_ceiling} inside \\code{exercise} gives the\nproject's strongest conclusion. The project wrote both cautions; no independent\nreviewer checked their limits. To challenge rather than repeat the supplied case,\nmake a disposable copy of the fixture input:\n\\begin{Verbatim}[fontsize=\\scriptsize]\ncp -R fixtures/first_wave/batch8_audio_level_rms_port/input/. /tmp/plectis-audio-rms-input\n\\end{Verbatim}\nIn the copied \\component{batch8_audio_level_rms_port_probe_manifest.json},\nchange the first case's second sample from \\code{0.05} to \\code{0.06}, but\nleave \\code{expected\\_level} unchanged. Then run:\n\\begin{Verbatim}[fontsize=\\scriptsize]\nPYTHONPATH=src python3 -m plectis batch8-audio-level-rms-port run \\\n  --input /tmp/plectis-audio-rms-input --out /tmp/plectis-audio-rms-probe \\\n  --acceptance-out /tmp/plectis-audio-rms-check.json; echo $?\n\\end{Verbatim}\nThe final number printed should be \\code{1}; in the acceptance file,\n\\code{status} should be \\code{blocked} and \\code{accepted} should be\n\\code{false}. This means the validator caught the deliberate mismatch; the\nexercise did not crash. Report any different result in\n\\code{CONTRIBUTING.md}. Treat a pass as evidence only for the published rule.\n\n\\paragraph{Names.} Repository: \\repo. Formerly Microcosm, Plectis retains\npackage \\code{microcosm\\_core}, page \\code{ORGANS.md} (``organs'' means\ncomponents), and the \\code{microcosm} alias. This paper uses command\n\\code{plectis}.\n\n\\paragraph{Getting the code and running without installing.}\n\n\\begin{Verbatim}[fontsize=\\small]\ngit clone https://github.com/wcook04/plectis\ncd plectis\ngit checkout 57ffb7f5830c\nPYTHONPATH=src python3 -m plectis tour --format text .\n\\end{Verbatim}\n\nThe final full stop means ``the current folder''. \\code{git checkout} pins the\nexact state described here; omit it to run the newest version, whose dated\nnumbers may differ.\n\nThe clean-clone tour read 5{,}018 files and chose the README-first order\n(internally \\component{readme_onboarding_route}) from five reading orders. It\nwrote 21 generated files under \\code{.microcosm/} without changing the examined\nsource. Those files can be rebuilt; the README's different count is another\ndated run, not a stable project size.\n\n\\paragraph{Installing.}\n\n\\begin{Verbatim}[fontsize=\\small]\npython3 -m pip install .\nplectis tour --format text .\n\\end{Verbatim}\n\nNo runtime library is required; \\code{pip} may fetch \\code{setuptools} to\nbuild. No check proves the absence of network or model calls, so inspect the\nsource.\n\n\\paragraph{The worked example.} A fresh Section~\\ref{sec:run} rerun reproduced\nall displayed passes and values and overwrote \\component{/tmp/plectis-audio-rms}.\nA stale field reports a copied private module, while the saved import record\nreports none because public code replaced it; the discrepancy remains.\n\n\\paragraph{A second Python version.} On 18~July~2026, a clean macOS clone under\nPython 3.12.7 matched the tour and example. Its offline install used only the\nmachine's \\code{setuptools} 80.9.0 (minimum 77.0.3). Only versions 3.12.7 and\n3.13.7 were tested.\n\n\\paragraph{Project-wide checks.} The project-wide Lean trust scan checks\n\\leanfilecount{} files for six text patterns but does not run Lean or verify\nproofs: unfinished placeholders; custom axioms\n(unproved assumptions); compiled calculations whose acceptance\nrelies on more than Lean's small proof-checking core;\n\\code{partial} definitions (runnable but not unfoldable in proofs);\n\\code{unsafe} definitions (barred from theorems); and removed computation\nlimits. No pattern appeared. The scan does not show whether files compile,\nproofs are valid, or statements express their authors' intent\n\\cite{leanvalidation}.\nSeparate components run Lean on small test projects; their results apply only\nto those projects.\nRegistry and trust checks passed; the \\leanfilecount{} files are distinct from\nthe \\mathcount{} mathematics components.\nThe limited \\code{make test} run covered 32 public test files, not every test\nfile: 356 of 361 tests passed, two skipped, and three failed. The\nfailures were an \\code{AGENTS.md} line-break expectation; outdated line or byte\ncounts in copied-source manifests; and root-layout rules that did not yet allow\nthe \\code{paper} directory or root PDF; the last was later repaired without\naltering this pinned run. Exact failure names, environment, and counts are in\n\\component{paper/public-test-receipt.json}. \\code{make ci} would add smoke and\npackage-install checks, but was not green because \\code{make test} failed. These\nchecks cover only the public checkout; this is my run, not an independent\nrepetition.\nThe paper's own\n\\component{scripts/check_public_system_paper.py} fails if counts, pinned facts,\nthe example, cautions, or citations disagree. It checks historical facts against\nthe pinned commit separately from current prose; with one maintainer, that is\ninternal consistency, not external truth.\n\n\\section*{Statements and declarations}\n\n\\paragraph{Declaration of generative AI use.} \\paperfullaideclaration\n\n\\paragraph{Authority boundary.}\nThe account of private production is testimony about origin, not independent\nevidence about the system's correctness, representativeness, or reliability.\nThe public commands and records establish only the cases and limits stated in\nthis paper.\n\n\\begin{thebibliography}{12}\n\\fontsize{7.5pt}{8.3pt}\\selectfont\n\\raggedright\n\\setlength{\\itemsep}{0pt}\n\\setlength{\\parsep}{0pt}\n\\setlength{\\parskip}{0pt}\n\n\\bibitem{runeson2009}\nPer Runeson and Martin H\\\"ost. ``Guidelines for Conducting and Reporting Case\nStudy Research in Software Engineering.'' \\emph{Empirical Software\nEngineering}, 14(2):131--164, 2009.\n\\href{https://doi.org/10.1007/s10664-008-9102-8}{doi:10.1007/s10664-008-9102-8}.\n\n\\bibitem{nasem2019}\nNational Academies of Sciences, Engineering, and Medicine.\n\\emph{Reproducibility and Replicability in Science}. Washington, DC:\nThe National Academies Press, 2019.\n\\href{https://doi.org/10.17226/25303}{doi:10.17226/25303}.\n\n\\bibitem{barr2015}\nEarl T. Barr, Mark Harman, Phil McMinn, Muzammil Shahbaz, and Shin Yoo.\n``The Oracle Problem in Software Testing: A Survey.''\n\\emph{IEEE Transactions on Software Engineering}, 41(5):507--525, 2015.\n\\href{https://doi.org/10.1109/TSE.2014.2372785}{doi:10.1109/TSE.2014.2372785}.\n\n\\bibitem{rosenthal1979}\nRobert Rosenthal. ``The File Drawer Problem and Tolerance for Null Results.''\n\\emph{Psychological Bulletin}, 86(3):638--641, 1979.\n\\href{https://doi.org/10.1037/0033-2909.86.3.638}{doi:10.1037/0033-2909.86.3.638}.\n\n\\bibitem{omgsacm2023}\nObject Management Group.\n\\href{https://www.omg.org/spec/SACM/2.3/PDF}{\\emph{Structured Assurance Case Metamodel (SACM)}},\n2.3, formal/23-05-08, October 2023.\n\n\\bibitem{gsn}\nSCSC Assurance Case Working Group (ACWG).\n\\emph{Goal Structuring Notation Community Standard, Version 3}.\nSCSC-141C, May 2021.\n\\href{https://scsc.uk/scsc-141c}{SCSC-141C}.\n\n\\bibitem{nisobadging2021}\nNational Information Standards Organization.\n\\href{https://doi.org/10.3789/niso-rp-31-2021}{\\emph{Reproducibility Badging\nand Definitions}},\nNISO RP-31-2021, approved 15 Jan. 2021.\n\n\\bibitem{acmartifact}\nAssociation for Computing Machinery.\n\\href{https://www.acm.org/publications/policies/artifact-review-and-badging-current}{``Artifact Review and Badging---Current,''}\nv1.1, 24 Aug. 2020; accessed 18 July 2026.\n\n\\bibitem{nistfips1804}\nNational Institute of Standards and Technology. \\emph{Secure Hash Standard},\nFIPS PUB 180-4, 2015.\n\\href{https://doi.org/10.6028/NIST.FIPS.180-4}{NIST FIPS 180-4}.\n\n\\bibitem{nisthashfunctions}\nNational Institute of Standards and Technology.\n\\href{https://csrc.nist.gov/projects/hash-functions}{``Hash Functions.''}\nUpdated 9 September 2024. Accessed 18 July 2026.\n\n\\bibitem{nasa8739}\nNational Aeronautics and Space Administration.\n\\emph{Software Assurance and Software Safety Standard},\nNASA-STD-8739.8B, Section 4.4.1.2, p.~48, 2022.\n\\href{https://standards.nasa.gov/standard/NASA/NASA-STD-87398}{NASA-STD-8739.8B}.\n\n\\bibitem{leanvalidation}\nLean Project. \\emph{Lean Language Reference} sections\n\\href{https://lean-lang.org/doc/reference/latest/ValidatingProofs/}{``Validating a Lean Proof''} and\n\\href{https://lean-lang.org/doc/reference/latest/Definitions/Recursive-Definitions/#partial-and-unsafe-definitions}{``Partial and Unsafe Definitions.''}\nAccessed 18 July 2026.\n\n\\end{thebibliography}\n\n\\end{document}\n","dependencies":[],"external_verification_route":"https://raw.githubusercontent.com/wcook04/plectis/main/paper/plectis-public-system.tex"}],"formal_claim_control_plane":[{"record_type":"formal_claim_control_plane_record","record_id":"lean_claims_registry","source_repository":"plectis-lean-erdos249-257","source_path":"docs/claims.json","media_type":"application/json","licence":"Apache-2.0","copyright":"2026 Will Cook","why_an_offline_reader_needs_this":"Single owner of release identity, per-claim status, the paper-label map, and the declarations backing each public claim.","embedded":true,"availability":"embedded","sha256":"sha256:5dadd1af656bff848f04719a68dedee2f5f67a62cb2c7ed38ad49793012c3499","source_byte_count":473232,"body":{"schema":"erdos249257-claims/3","comment":"Single owner of release identity, mathematical claim status, typed remaining-open propositions, and the machine-readable paper map. 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open."}},{"from":"largest_skip_gap_propagation","relation":"builds_on","to":"half_membership_seam_classification"},{"from":"boundary_pulse_normalization","relation":"builds_on","to":"largest_skip_gap_propagation"},{"from":"largest_skip_induction_reduction","relation":"builds_on","to":"largest_skip_gap_propagation"},{"from":"fixed_tail_survival_socket","relation":"builds_on","to":"half_membership_seam_classification"},{"from":"producer_carry_future_tail_socket","relation":"builds_on","to":"fixed_tail_survival_socket"},{"from":"quarter_band_endpoint_cells","relation":"builds_on","to":"concrete_seam_adapter"},{"from":"reset_deficit_escape_geometry","relation":"builds_on","to":"largest_skip_gap_propagation"},{"from":"seam_producer_carry_alignment","relation":"builds_on","to":"producer_carry_future_tail_socket"},{"from":"selected_window_depth18_base","relation":"builds_on","to":"selected_half_carry_window_compactness"},{"from":"rewind_seam_operational_bridge","relation":"builds_on","to":"localized_protected_seam_compactness"},{"from":"selected_suffix_cylinder_propagation","relation":"builds_on","to":"selected_window_depth18_base"},{"from":"selected_suffix_pre_feedback_run","relation":"builds_on","to":"selected_suffix_cylinder_propagation"},{"from":"concrete_producer_carry_coordinates","relation":"builds_on","to":"seam_producer_carry_alignment"},{"from":"denominator_exclusion","relation":"advances_open_target","to":"erdos_249","remaining_open_effect":{"remaining_open_proposition_id":"remaining_open.erdos_249_irrationality","operation":"narrows","statement":"Excludes every rational representation up to the stated denominator bound without proving irrationality."}},{"from":"certificate_reduction","relation":"advances_open_target","to":"erdos_249","remaining_open_effect":{"remaining_open_proposition_id":"remaining_open.unbounded_certificate_supply","operation":"re_expresses","statement":"Re-expresses the open irrationality step as the unbounded supply of checked finite certificates."}},{"from":"certificate_completeness","relation":"makes_exact","to":"certificate_reduction"},{"from":"primitive_coordinate_index_obstruction","relation":"builds_on","to":"mersenne_lambert_ladder"},{"from":"certified_kill_instances","relation":"instantiates","to":"certificate_reduction"},{"from":"diagonal_pincer_reduction","relation":"advances_open_target","to":"erdos_249","remaining_open_effect":{"remaining_open_proposition_id":"remaining_open.unbounded_certificate_supply","operation":"re_expresses","statement":"Re-expresses diagonal non-integrality as avoidance of an exact full-target interval; an unbounded avoidance supply remains open."}},{"from":"diagonal_fresh_loss_reduction","relation":"advances_open_target","to":"erdos_249","remaining_open_effect":{"remaining_open_proposition_id":"remaining_open.unbounded_certificate_supply","operation":"re_expresses","statement":"Transfers the open supply obligation to exact fresh-loss projections and binary suffix gaps without asserting that such a supply exists."}},{"from":"diagonal_fresh_loss_reduction","relation":"builds_on","to":"diagonal_pincer_reduction"},{"from":"fresh_prime_deficit_decomposition","relation":"builds_on","to":"diagonal_fresh_loss_reduction"},{"from":"power_two_bit_lift","relation":"builds_on","to":"diagonal_fresh_loss_reduction"},{"from":"parity_socket_obstruction","relation":"builds_on","to":"power_two_bit_lift"},{"from":"canonical_jump_slack_reduction","relation":"builds_on","to":"diagonal_fresh_loss_reduction"},{"from":"canonical_jump_slack_reduction","relation":"advances_open_target","to":"erdos_249","remaining_open_effect":{"remaining_open_proposition_id":"remaining_open.unbounded_certificate_supply","operation":"re_expresses","statement":"Re-expresses the canonical strict-jump centrality supply as the sign of one exact integer slack; cofinal nonnegative slack remains open."}},{"from":"diagonal_pincer_reduction","relation":"builds_on","to":"squared_mersenne_enclosure"},{"from":"actual_foreign_residue_projection","relation":"builds_on","to":"diagonal_pincer_reduction"},{"from":"actual_foreign_residue_projection","relation":"builds_on","to":"squared_mersenne_enclosure"},{"from":"actual_foreign_residue_projection","relation":"advances_open_target","to":"erdos_249","remaining_open_effect":{"remaining_open_proposition_id":"remaining_open.unbounded_certificate_supply","operation":"re_expresses","statement":"Places the finite foreign-residue projection behind an explicit analytic-complement bridge and an unbounded separated-projection supply; neither is asserted."}},{"from":"diagonal_pincer_reduction","relation":"builds_on","to":"mersenne_shadow_denominator"},{"from":"boolean_mobius_carry","relation":"builds_on","to":"tail_orbit_rigidity"},{"from":"reciprocal_mass_bound","relation":"builds_on","to":"boolean_mobius_carry"},{"from":"unbounded_carry_states","relation":"builds_on","to":"boolean_mobius_carry"},{"from":"sublog_zero_windows","relation":"builds_on","to":"unbounded_carry_states"},{"from":"reciprocal_mass_bound","relation":"advances_open_target","to":"universal_257","remaining_open_effect":{"remaining_open_proposition_id":"remaining_open.universal_257_all_infinite_supports","operation":"constrains","statement":"Adds a necessary reciprocal-mass condition for a rational value over an infinite support."}},{"from":"sublog_zero_windows","relation":"advances_open_target","to":"universal_257","remaining_open_effect":{"remaining_open_proposition_id":"remaining_open.universal_257_all_infinite_supports","operation":"constrains","statement":"Adds a necessary sublogarithmic zero-window condition for a rational value over an infinite support."}},{"from":"greedy_achievement_geometry","relation":"advances_open_target","to":"universal_257","remaining_open_effect":{"remaining_open_proposition_id":"remaining_open.universal_257_all_infinite_supports","operation":"constrains","statement":"Provides exact global geometry and finite non-membership tests for the value set without proving the universal irrationality statement."}},{"from":"half_greedy_two_thirds_band","relation":"builds_on","to":"greedy_achievement_geometry"},{"from":"half_greedy_two_thirds_band","relation":"advances_open_target","to":"universal_257","remaining_open_effect":{"remaining_open_proposition_id":"remaining_open.half_value_membership","operation":"constrains","statement":"Localizes an unsafe isolated greedy skip to an exact open interval of width below 1/9; it neither proves that the actual half-greedy orbit avoids the interval nor addresses arbitrary infinite supports."}},{"from":"dyadic_prefix_block_safety_interfaces","relation":"builds_on","to":"greedy_achievement_geometry"},{"from":"dyadic_prefix_block_safety_interfaces","relation":"advances_open_target","to":"universal_257","remaining_open_effect":{"remaining_open_proposition_id":"remaining_open.universal_257_all_infinite_supports","operation":"re_expresses","statement":"Gives exact block-safety, tail-gap, and primitive-overshoot witnesses sufficient for half-membership, but proves no cofinal supply of those witnesses and does not address arbitrary infinite supports."}},{"from":"half_second_channel_reduction","relation":"builds_on","to":"greedy_achievement_geometry"},{"from":"half_second_channel_reduction","relation":"advances_open_target","to":"universal_257","remaining_open_effect":{"remaining_open_proposition_id":"remaining_open.half_value_membership","operation":"re_expresses","statement":"Reduces membership of the target one half to an exact rational separation statement from level seven onward; it neither proves that statement nor settles arbitrary infinite supports."}},{"from":"transport_curvature_reductions","relation":"builds_on","to":"certificate_reduction"},{"from":"transport_curvature_reductions","relation":"advances_open_target","to":"erdos_249","remaining_open_effect":{"remaining_open_proposition_id":"remaining_open.unbounded_certificate_supply","operation":"re_expresses","statement":"Re-expresses diagonal non-integrality through several exact transport and curvature observables while leaving every unbounded supply open."}},{"from":"affine_transport_annihilation","relation":"builds_on","to":"transport_curvature_reductions"},{"from":"transport_certificate_instances","relation":"instantiates","to":"transport_curvature_reductions"},{"from":"first_harmonic_certificate_interface","relation":"builds_on","to":"certificate_reduction"},{"from":"first_harmonic_certificate_interface","relation":"builds_on","to":"certificate_completeness"},{"from":"first_harmonic_certificate_interface","relation":"advances_open_target","to":"erdos_249","remaining_open_effect":{"remaining_open_proposition_id":"remaining_open.unbounded_certificate_supply","operation":"re_expresses","statement":"Turns a cofinal first-harmonic estimate into the existing finite certificate supply; no such estimate is proved."}},{"from":"first_harmonic_pivot_decomposition","relation":"builds_on","to":"first_harmonic_certificate_interface"},{"from":"first_harmonic_pivot_decomposition","relation":"advances_open_target","to":"erdos_249","remaining_open_effect":{"remaining_open_proposition_id":"remaining_open.unbounded_certificate_supply","operation":"re_expresses","statement":"Gives an exact four-term first-harmonic decomposition and a sufficient 9X/10 budget; no prime-distribution or decorrelation estimate supplies those budgets cofinally."}},{"from":"adjacent_phase_separation_reduction","relation":"builds_on","to":"first_harmonic_certificate_interface"},{"from":"adjacent_phase_separation_reduction","relation":"advances_open_target","to":"erdos_249","remaining_open_effect":{"remaining_open_proposition_id":"remaining_open.unbounded_certificate_supply","operation":"re_expresses","statement":"Re-expresses the missing supply as cofinal adjacent-pair phase separation; no such supply is proved."}},{"from":"pivot_anti_reconstruction_reduction","relation":"builds_on","to":"adjacent_phase_separation_reduction"},{"from":"pivot_anti_reconstruction_reduction","relation":"advances_open_target","to":"erdos_249","remaining_open_effect":{"remaining_open_proposition_id":"remaining_open.unbounded_certificate_supply","operation":"re_expresses","statement":"Re-expresses the missing supply as cofinal pivot anti-reconstruction or window-pair separation; no cofinal arithmetic producer is proved."}},{"from":"fixed_precision_transport_no_go","relation":"builds_on","to":"transport_curvature_reductions"},{"from":"lcm_factor_ideal_anchor_pulse_no_go","relation":"builds_on","to":"transport_curvature_reductions"},{"from":"sigma_transcendence","relation":"positions","to":"mersenne_lambert_ladder"},{"from":"totient_coprimality_probability","relation":"builds_on","to":"mersenne_lambert_ladder"},{"from":"squared_lambert_gcd_moments","relation":"builds_on","to":"totient_coprimality_probability"},{"from":"stern_brocot_cylinder_law","relation":"builds_on","to":"squared_lambert_gcd_moments"},{"from":"stern_brocot_run_fibonacci_stability","relation":"positions","to":"stern_brocot_cylinder_law"},{"from":"prime_support_irrationality","relation":"positions","to":"universal_257"}]}},"release":{"version":"0.6.0","tag":"v0.6.0","date":"2026-07-12","lean_toolchain":"leanprover/lean4:v4.29.1","repository":"https://github.com/wcook04/plectis-lean-erdos249-257","formal_source":{"ref":"d6d344d9557faee19276b01a1c1f20d7c08f03d5","ref_kind":"commit","publication_state":"committed_checkpoint_pending_remote_publication","relationship_to_last_tag":"post_tag_checkpoint"},"public_projection":{"posture":"self_contained_public_projection_from_a_larger_ongoing_research_workflow","meaning":"This checkout is a deliberately curated, independently verifiable public output of a larger ongoing formal-mathematics workflow. No public claim depends on private or unreleased work.","public_evidence":["pinned formal-source identity","claim and open-proposition registry","machine-readable argument and module graphs","generated declaration atlas","bounded cold-clone and adversarial checks"],"boundary":"The workflow provenance explains the release discipline; it does not imply hidden proof authority or equivalence with private work."}},"status_taxonomy":{"proved here":"Lean theorem in the committed formal-source checkpoint; no novelty assertion is implied","formalised here":"Known mathematics represented and checked in Lean","unconditional progress":"Genuine theorem that does not settle the open problem","conditional reduction":"Exact implication whose remaining hypothesis is open","verified finite instance":"Kernel-checked computation at a bounded range","cited only":"External theorem used for positioning, not formalised","open":"Not proved by this release"},"remaining_open_propositions":[{"id":"remaining_open.erdos_249_irrationality","open_target_claim":"erdos_249","statement":"Prove that S = \u2211 \u03c6(n)/2\u207f is irrational, with no bound on a possible rational denominator.","paper_anchor":{"source":"paper/erdos249-257-main-paper.tex","environment":"problem","title":"Erd\\H{o}s \\#249","line":3439}},{"id":"remaining_open.unbounded_certificate_supply","open_target_claim":"erdos_249","statement":"Produce certified non-integrality witnesses at unbounded parameters. A sufficient one-parameter form is to produce them along the lcm diagonal.","paper_anchor":{"source":"paper/erdos249-257-main-paper.tex","environment":"problem","title":"unbounded certificate supply","line":3450}},{"id":"remaining_open.half_value_membership","open_target_claim":"universal_257","statement":"Decide whether 1/2 lies in the Mersenne achievement set. Yes refutes universal #257; no closes only this route.","paper_anchor":{"source":"paper/erdos249-257-main-paper.tex","environment":"problem","title":"the half-value counterexample route","line":3471}},{"id":"remaining_open.universal_257_all_infinite_supports","open_target_claim":"universal_257","statement":"Prove irrationality for every infinite support A, not only the named support families formalised here.","paper_anchor":{"source":"paper/erdos249-257-main-paper.tex","environment":"problem","title":"universal Erd\\H{o}s \\#257","line":3462}}],"claims":[{"id":"eb_full_support","label":"Full-support Erd\u0151s\u2013Borwein irrationality, every integer base b \u2265 2","status":"formalised here","statement":"For every integer b \u2265 2, the series \u2211_{n\u22651} 1/(b\u207f \u2212 1) is irrational (Erd\u0151s, 1948).","paper_label":"res:full","readme_headline":true,"declarations":[{"name":"irrational_erdosSum_full_support","module":"Erdos249257/CertificateKernel.lean","line":8000}]},{"id":"eb_constant","label":"The Erd\u0151s\u2013Borwein constant E is irrational","status":"formalised here","statement":"E = \u2211_{n\u22651} 1/(2\u207f \u2212 1) is irrational, as the base-2 corollary of the full-support theorem.","paper_label":"res:full","readme_headline":true,"declarations":[{"name":"irrational_erdosBorwein_series","module":"Erdos249257/CertificateKernel.lean","line":8007}]},{"id":"support_families","label":"Named infinite-support irrationality families toward #257","status":"formalised here","statement":"For every base b \u2265 2, \u2211_{n\u2208A} 1/(b\u207f \u2212 1) is irrational for factorial support, power-of-two support, multiples, pairwise-coprime supports with summable reciprocals, eventually-periodic supports, residue classes, and the odd numbers. Proved families, not the universal statement.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:support","readme_headline":true,"declarations":[{"name":"erdos257_family_factorial_instance","module":"Erdos249257/CertificateKernel.lean","line":5754},{"name":"erdos257_family_two_pow_instance","module":"Erdos249257/CertificateKernel.lean","line":5762},{"name":"irrational_erdosSupportSeries_pairwise_coprime","module":"Erdos249257/CertificateKernel.lean","line":10448}]},{"id":"periodic_nonnegative_weight_irrationality","label":"Eventually-periodic nonnegative weighted Lambert irrationality","status":"formalised here","statement":"For every integer base b \u2265 2, an eventually-periodic nonnegative rational coefficient sequence with a positive value at a positive index in its periodic region has irrational series \u2211 \u03b3(a)/(b^a \u2212 1). This includes eventually-periodic natural-number weights and indicator supports. Luca\u2013Tachiya prove a broader periodic rational-coefficient theorem by different methods; no priority claim is made here.","paper_label":"res:periodicnonnegative","readme_headline":false,"declarations":[{"name":"irrational_weightedErdosSeries_periodic","module":"Erdos249257/CertificateKernel.lean","line":12370},{"name":"irrational_weightedErdosSeries_eventuallyPeriodic","module":"Erdos249257/CertificateKernel.lean","line":12397},{"name":"irrational_ratWeightSeries_eventuallyPeriodic","module":"Erdos249257/CertificateKernel.lean","line":12483}]},{"id":"periodic_signed_weight_dichotomy","label":"Periodic signed-weight irrational-or-terminating dichotomy","status":"proved here","statement":"For every integer base b \u2265 2 and every periodic integer weight w, the signed series \u2211 w(a)/(b^a \u2212 1) is irrational or some power of b times its value is an integer. Excluding that terminating alternative implies irrationality. The terminating alternative is not excluded in general, so the full mixed-sign periodic-coefficient theorem is not claimed.","paper_label":"res:periodicsigned","readme_headline":false,"declarations":[{"name":"irrational_or_bpow_mul_eq_intCast_intWeightedErdosSeries_periodic","module":"Erdos249257/CertificateKernel.lean","line":13847},{"name":"irrational_intWeightedErdosSeries_periodic_of_bpow_mul_ne_intCast","module":"Erdos249257/CertificateKernel.lean","line":13872}]},{"id":"period_four_signed_residue_obstruction","label":"Period-four signed residue-selection obstruction","status":"proved here","statement":"For the period-four signed weight 1,0,\u22121,0, the associated signed divisor coefficient vanishes for every n congruent to 3 modulo 4. A whole arithmetic progression can therefore carry zero coefficient, so residue-blind selection cannot force a protected nonzero residue for every periodic signed weight.","paper_label":"res:periodfourobstruction","readme_headline":false,"declarations":[{"name":"intWeightedCoeff_periodFourSignWeight_eq_zero_of_mod_four_eq_three","module":"Erdos249257/CertificateKernel.lean","line":13898}]},{"id":"orthogonal_petal_sunflower_reduction","label":"Conditional orthogonal-petal sunflower reduction","status":"conditional reduction","statement":"For a finite-core orthogonal-petal bouquet at base 2, the exact uniform tail-selection predicate implies a forced-carry certificate supply and hence irrationality of the associated support series. Lean formalises the carry conversion and an explicit infinite alternating-core, squared-prime-petal bouquet whose elements have three prime factors with multiplicity. The tail-selection predicate itself is not proved, so this does not add an unconditional #257 support family.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:sunflower","readme_headline":false,"declarations":[{"name":"sunflowerForcedCarrySupply_of_orthogonalPetalBouquet","module":"Erdos249257/SupportSunflowerDichotomy.lean","line":531},{"name":"irrational_erdosSupportSeries_of_orthogonalPetalBouquet","module":"Erdos249257/SupportSunflowerDichotomy.lean","line":540},{"name":"alternatingPrimeSquareSupport_cardFactors_eq_three","module":"Erdos249257/SupportSunflowerDichotomy.lean","line":343}]},{"id":"composite_dilation_defect_identity","label":"Composite dilation defect identity","status":"proved here","statement":"For every positive support element a and positive x, the divisor-count coefficient at a\u00b7x is the coefficient at x, plus the distinguished new divisor when a does not divide x, plus an exact finite composite-dilation defect. For a finite-core orthogonal-petal bouquet, the ray defect is bounded by the finite exceptional-frame cardinality plus the petal-support divisor count. This local identity supplies no correlation estimate, tail selector, or irrationality theorem.","paper_label":"res:compositedefect","readme_headline":false,"declarations":[{"name":"supportCoeff_mul_eq_add_defect","module":"Erdos249257/CompositeDilationDefect.lean","line":30},{"name":"compositeDilationDefect_le_exceptional_add_petalCoeff","module":"Erdos249257/CompositeDilationDefect.lean","line":151},{"name":"compositeDilationDefect_two_six_fixture","module":"Erdos249257/CompositeDilationDefect.lean","line":218}]},{"id":"mixed_prime_power_layer_identity","label":"Mixed prime-power support-layer identity","status":"proved here","statement":"Exact p-adic and q-adic coefficient-layer operators commute. For distinct primes and a residual argument coprime to their product, the two-layer operator on the support coefficient equals the iterated exact-prime-power pullback. The checked singleton {12} fixture has mixed (2\u00b2,3) layer value one at residual argument one. This is finite coefficient algebra only, with no decimation transport, bounded-\u03a9 endpoint, or irrationality theorem.","paper_label":"res:mixedlayer","readme_headline":false,"declarations":[{"name":"primePowerLayer_comm","module":"Erdos249257/MaximalOmegaLayer.lean","line":29},{"name":"mixedPrimePowerLayerTwo_supportCoeffInt","module":"Erdos249257/MaximalOmegaLayer.lean","line":39},{"name":"mixedPrimePowerLayerTwo_twelve_fixture","module":"Erdos249257/MaximalOmegaLayer.lean","line":65}]},{"id":"dyadic_prefix_block_safety_interfaces","label":"Dyadic-prefix, block-safety, and primitive-overshoot interfaces","status":"proved here","statement":"Inside a hypothetical dyadic-rational support value, a reduced finite positive prefix transports exactly to a residual rational support value and wrap-ratio certificate. For the half-greedy orbit, reduced and raw margins, three-depth take-run excess, primitive-lattice cap violation, support-coefficient tail gaps, and gcd-scaled predecessor overshoot are connected by exact identities and equivalences; cofinal supplies of the named witnesses imply that 1/2 belongs to the Mersenne achievement set. Checked fixtures show that safe brackets with numerator monotonicity and a factor-11 predecessor rule are not inductive. No cofinal witness supply is proved, so neither half-membership nor universal Erd\u0151s #257 follows.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"readme_headline":false,"declarations":[{"name":"finiteFragment_dyadicPrefixCompression","module":"Erdos249257/DyadicPrefixCompression.lean","line":1481},{"name":"halfGreedyThreeDepthTakeRunExcessBound_iff_skippedExcessBound","module":"Erdos249257/DyadicPrefixCompression.lean","line":2021},{"name":"greedyHalf_badSkipCapViolation_iff_primitiveLattice","module":"Erdos249257/DyadicPrefixCompression.lean","line":2033},{"name":"halfGreedyPrefixSupportTailGap_eq_excess_div_denominator","module":"Erdos249257/DyadicPrefixCompression.lean","line":2126},{"name":"greedyHalf_badSkipCapViolation_iff_supportCoeffTailGap","module":"Erdos249257/DyadicPrefixCompression.lean","line":2164},{"name":"rawSafe_but_not_gcdScaled_fixture","module":"Erdos249257/DyadicPrefixCompression.lean","line":3294},{"name":"factorEleven_predecessorOvershoot_not_inductive_fixture","module":"Erdos249257/DyadicPrefixCompression.lean","line":3316},{"name":"unsafeSkipGcdOvershootSupply_iff_primitive","module":"Erdos249257/DyadicPrefixCompression.lean","line":3463},{"name":"half_mem_mersenneAchievementSet_of_unsafeSkipPrimitiveOvershootSupply","module":"Erdos249257/DyadicPrefixCompression.lean","line":3504}]},{"id":"half_trapping_return_carry_interface","label":"Half-trapping reverse-carry interface","status":"conditional reduction","statement":"Scalar-relation-invariant rational-linear channels have no nontrivial square determinant at any rank. Separately, a common coefficient word with a 1/0 seam and a common following overlap forces the terminal reverse-carry difference to be a power of two times an odd integer; a common nonnegative real carry bound yields the corresponding dyadic spacing inequality. The module neither supplies reverse-carry presentations nor a predecessor overlap, so it is not a greedy separation or irrationality theorem.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:halftrapping","readme_headline":false,"declarations":[{"name":"relationInvariantLinearChannels_det_eq_zero","module":"Erdos249257/HalfTrappingReturnCarry.lean","line":42},{"name":"overlappingReverseCarryWords_carryDifference_eq_twoPow_mul_odd","module":"Erdos249257/HalfTrappingReturnCarry.lean","line":124},{"name":"overlappingReverseCarryWords_twoPow_le_realBound","module":"Erdos249257/HalfTrappingReturnCarry.lean","line":191}]},{"id":"half_carry_reachability_compactness","label":"Half-carry reachability and compactness","status":"conditional reduction","statement":"At an unresolved even seam, the scalar transition reaches exactly every integer terminal carry other than the one value 2*\u03b4-c. If canonical seam coordinates are supplied at every sufficiently large even depth, then admissible finite half-carry words occur cofinally; compactness yields an infinite support A with erdosSupportSeries 2 A = 1/2. The canonical seam-supply hypothesis is not proved.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:halfcarryreachability","readme_headline":false,"declarations":[{"name":"evenSeamReachable_iff","module":"Erdos249257/HalfCarryReachability.lean","line":235},{"name":"cofinalAdmissibility_of_canonicalEvenSeamSupply","module":"Erdos249257/HalfCarryReachability.lean","line":999},{"name":"exists_infinite_support_half_of_canonicalEvenSeamSupply","module":"Erdos249257/HalfCarryReachability.lean","line":1027}]},{"id":"cofinal_strip_return_closure","label":"Cofinal square-root-strip return closure","status":"conditional reduction","statement":"The binary-scaled uncentred greedy half carry is antitone, and the square-root strip envelope divided by 2^N tends to zero. Therefore cofinal returns of the canonical greedy carry to that strip force the scaled carry to tend to zero and yield membership of 1/2 in the Mersenne achievement set. The cofinal return hypothesis is not proved.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:cofinalstripreturn","readme_headline":false,"declarations":[{"name":"greedy_integerHalfCarry_scaled_tendsto_zero_of_cofinalStripReturn","module":"Erdos249257/CofinalStripReturn.lean","line":84},{"name":"greedy_half_infinite_of_cofinalStripReturn","module":"Erdos249257/CofinalStripReturn.lean","line":130},{"name":"half_mem_mersenneAchievementSet_of_cofinalStripReturn","module":"Erdos249257/CofinalStripReturn.lean","line":157}]},{"id":"terminal_only_cofinal_closure","label":"Terminal-only cofinal square-root-strip closure","status":"conditional reduction","statement":"A normalized finite word with terminal half carry in the square-root strip has support-series value within (4M+12)/2^M of 1/2. Thus cofinally many such terminal-only witnesses, with neither coherent histories nor earlier-row bounds, force 1/2 into the Mersenne achievement set and yield an infinite support with support-series value 1/2. The cofinal terminal-witness supply is not proved.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:terminalonlycofinal","readme_headline":false,"declarations":[{"name":"dist_half_erdosSupportSeries_wordSupport_le_of_terminalOnlyWitness","module":"Erdos249257/TerminalOnlyCofinal.lean","line":68},{"name":"half_mem_mersenneAchievementSet_of_cofinalTerminalOnlyStrip","module":"Erdos249257/TerminalOnlyCofinal.lean","line":134},{"name":"exists_infinite_support_half_of_cofinalTerminalOnlyStrip","module":"Erdos249257/TerminalOnlyCofinal.lean","line":192}]},{"id":"terminal_scaled_vanishing_closure","label":"Scaled terminal-vanishing closure","status":"conditional reduction","statement":"A normalized sequence of finite terminal words whose depths tend to infinity and whose terminal carries divided by the corresponding powers of 2 tend to zero has support-series values converging to 1/2. Closedness of the Mersenne achievement set then gives membership of 1/2 and an infinite support with that value. The prior cofinal square-root-strip condition maps into this weaker socket, but no scaled-vanishing sequence is constructed.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:terminalscaledvanishing","readme_headline":false,"declarations":[{"name":"dist_half_erdosSupportSeries_wordSupport_le_carry_add_tail","module":"Erdos249257/TerminalOnlyScaledVanishing.lean","line":24},{"name":"half_mem_mersenneAchievementSet_of_terminalScaledVanishing","module":"Erdos249257/TerminalOnlyScaledVanishing.lean","line":165},{"name":"exists_infinite_support_half_of_terminalScaledVanishing","module":"Erdos249257/TerminalOnlyScaledVanishing.lean","line":221}]},{"id":"half_cylinder_finite_shadow","label":"Finite shadows of the half cylinder","status":"conditional reduction","statement":"For admissible finite half-carry words, the residual from 1/2 is exactly the terminal carry minus the future coefficient tail, scaled by a power of two. At a seam, avoiding the integer strip is equivalent to full scalar strip reachability; the rank-three greedy fixture is raw dyadically safe but has an in-strip seam hole. If the governed frozen-margin producer holds, then 1/2 belongs to the Mersenne achievement set. That producer is not proved.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:halfcylindershadow","readme_headline":false,"declarations":[{"name":"halfStripAdmissible_residual_eq","module":"Erdos249257/HalfCylinderFiniteShadow.lean","line":55},{"name":"evenSeam_fullIntegerStrip_iff_escape","module":"Erdos249257/HalfCylinderFiniteShadow.lean","line":807},{"name":"half_mem_mersenneAchievementSet_of_governedFrozenMarginProducer","module":"Erdos249257/HalfCylinderFiniteShadow.lean","line":1256}]},{"id":"fixed_coefficient_rewind_contraction","label":"Fixed-coefficient rewind contraction","status":"conditional reduction","statement":"A finite sequence of canonical fixed-coefficient inverse-parent maps has an exact dyadic closed form. An interval no wider than its denominator contracts either to one ancestor or to an adjacent seam pair; the singleton alternative is exactly the stated phase-fit inequality. This does not exclude the seam-pair alternative or prove half-membership.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:fixedcoeffrewind","readme_headline":false,"declarations":[{"name":"rewind_eq_closedForm","module":"Erdos249257/HalfCarryCeilParentContraction.lean","line":49},{"name":"rewind_endpoint_eq_iff_phase_fit","module":"Erdos249257/HalfCarryCeilParentContraction.lean","line":99},{"name":"rewind_singleton_or_isSeamPair","module":"Erdos249257/HalfCarryCeilParentContraction.lean","line":180}]},{"id":"selected_half_carry_window_compactness","label":"Selected half-carry windows","status":"conditional reduction","statement":"If selected protected half-carry windows exist at every sufficiently large level, then their carry-one representatives yield cofinal admissibility and an infinite support with erdosSupportSeries 2 A = 1/2. The unresolved object-level input is a coherent history at each step: a common next-row divisor cylinder or the canonical adjacent seam. The module does not supply that input.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:selectedhalfwindow","readme_headline":false,"declarations":[{"name":"SelectedHalfWindow.HistoryAlternative","module":"Erdos249257/HalfCarrySelectedWindow.lean","line":236},{"name":"cofinalAdmissibility_of_selectedHalfWindows","module":"Erdos249257/HalfCarrySelectedWindow.lean","line":831},{"name":"exists_infinite_support_half_of_selectedHalfWindows","module":"Erdos249257/HalfCarrySelectedWindow.lean","line":864}]},{"id":"realized_rewind_phase_identity","label":"Realized rewind phase identity","status":"conditional reduction","statement":"For a realized fixed-coefficient rewind history, the phase is exactly the denominator minus one minus the binary suffix numeral. Thus the singleton rewind criterion is equivalent to the suffix numeral covering the other terminal carries. The global coverage inequality is not proved.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:rewindphase","readme_headline":false,"declarations":[{"name":"phase_eq_denom_sub_one_sub_suffixNumeral","module":"Erdos249257/HalfCarryRewindPhase.lean","line":39},{"name":"rewind_interval_singleton_iff_suffixNumeral_covers","module":"Erdos249257/HalfCarryRewindPhase.lean","line":65},{"name":"rewind_supportCoeffHistory_singleton_iff_prefixCarry_covers","module":"Erdos249257/HalfCarryRewindPhase.lean","line":352}]},{"id":"localized_protected_seam_compactness","label":"Localized protected seam compactness","status":"conditional reduction","statement":"At each realized localized one-hole seam, at least one of protected carries 3 and 4 gives an admissible finite word. If such seam realizations occur cofinally, then cofinal admissibility and an infinite support A with erdosSupportSeries 2 A = 1/2 follow. The cofinal realization supply is not proved.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:protectedseam","readme_headline":false,"declarations":[{"name":"ProtectedEvenSeamRealizedAt","module":"Erdos249257/HalfCarryProtectedSeamConsumer.lean","line":18},{"name":"cofinalAdmissibility_of_cofinalProtectedEvenSeamRealization","module":"Erdos249257/HalfCarryProtectedSeamConsumer.lean","line":31},{"name":"exists_infinite_support_half_of_cofinalProtectedEvenSeamRealization","module":"Erdos249257/HalfCarryProtectedSeamConsumer.lean","line":53}]},{"id":"mixed_cofinal_half_carry_fan_in","label":"Mixed cofinal half-carry fan-in","status":"conditional reduction","statement":"At every requested scale, either a selected protected half-carry window or a localized realized one-hole seam supplies an admissible finite word. If this mixed producer is cofinal, then compactness yields an infinite support A with erdosSupportSeries 2 A = 1/2. The mixed producer is not proved.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:cofinalwindowseam","readme_headline":false,"declarations":[{"name":"HalfCarryCofinalSelectedWindowOrProtectedSeam","module":"Erdos249257/HalfCarryCofinalWindowOrSeamConsumer.lean","line":25},{"name":"cofinalAdmissibility_of_cofinalSelectedWindowOrProtectedSeam","module":"Erdos249257/HalfCarryCofinalWindowOrSeamConsumer.lean","line":32},{"name":"exists_infinite_support_half_of_cofinalSelectedWindowOrProtectedSeam","module":"Erdos249257/HalfCarryCofinalWindowOrSeamConsumer.lean","line":58}]},{"id":"integer_greedy_first_wrap_reduction","label":"Integer-greedy first-wrap reduction","status":"conditional reduction","statement":"At a first half-cylinder wrap, the truncated Mersenne weights are gap-dominating, and a small positive seam subset-sum defect exists exactly when the descending integer-greedy remainder is small and positive. The remaining unbounded lower bound for that deterministic remainder sequence is not proved.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:integergreedy","readme_headline":false,"declarations":[{"name":"seamWeights_gapDominates","module":"Erdos249257/HalfCylinderIntegerGreedy.lean","line":420},{"name":"seam_smallPositiveDefect_iff_integerGreedyRemainder","module":"Erdos249257/HalfCylinderIntegerGreedy.lean","line":654},{"name":"quarterBandEscape_iff_bound_lt_nextRemainder","module":"Erdos249257/HalfCylinderIntegerGreedy.lean","line":1528}]},{"id":"concrete_seam_adapter","label":"Concrete finite seam adapter","status":"conditional reduction","statement":"Concrete finite seam words realize the abstract adjacent-cut recurrence, and the next seam word agrees with the integer-greedy choice. This implements the finite recurrence but does not establish the unbounded remainder estimate.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:concreteseam","readme_headline":false,"declarations":[{"name":"seamAdjacentCut_newCapacity","module":"Erdos249257/HalfCylinderConcreteSeamAdapter.lean","line":451},{"name":"seamCutNextWord_eq_greedyWord_succ","module":"Erdos249257/HalfCylinderConcreteSeamAdapter.lean","line":667}]},{"id":"fatal_gap_right_tail_classification","label":"Fatal-gap and eventual-right seam classification","status":"proved here","statement":"For the concrete integer seam words, eventual right extension is equivalent to nonmembership of 1/2 in the Mersenne achievement set. It is also equivalent to the existence of a finite fatal half gap and to the existence of a final skipped exponent in the real greedy orbit. This exact classification does not decide either membership branch.","remaining_open_proposition_ids":["remaining_open.half_value_membership"],"paper_label":"res:fatalright","readme_headline":false,"declarations":[{"name":"seamGreedyEventuallyRight_iff_existsFatalHalfGap","module":"Erdos249257/HalfCylinderFatalGapRightTail.lean","line":781},{"name":"seamGreedyEventuallyRight_iff_half_not_mem","module":"Erdos249257/HalfCylinderFatalGapRightTail.lean","line":787},{"name":"seamGreedyEventuallyRight_iff_exists_isLastHalfGreedySkip","module":"Erdos249257/HalfCylinderFatalGapRightTail.lean","line":802}]},{"id":"final_middle_cell_escape","label":"Upper and first middle producers cannot feed an all-right tail","status":"proved here","statement":"At a putative last false rank D \u2265 13, the upper producer is impossible unconditionally. On the middle branch, producer-carry cell -3 is also impossible: the lazy cofinite tail is below 1/2 while the corresponding Mobius-centred carry is negative, contradicting analytic nonnegativity. This removes the upper branch and one middle cell, not the right-tail branch or the remaining middle cells -2 and -1.","remaining_open_proposition_ids":["remaining_open.half_value_membership"],"paper_label":"res:finalskipexclusions","readme_headline":false,"declarations":[{"name":"upperProducer_not_last","module":"Erdos249257/HalfCylinderLastProducerContradiction.lean","line":250},{"name":"middleProducer_neg_three_not_last","module":"Erdos249257/HalfCylinderLastProducerContradiction.lean","line":315},{"name":"finalMiddleCell_neg_three_not_last","module":"Erdos249257/HalfCylinderFinalMiddleCellEscape.lean","line":587}]},{"id":"final_middle_neg_two_phase_sieve","label":"Rank-26 phase sieve for the final middle -2 cell","status":"conditional reduction","statement":"For every seam row s \u2265 27, the greedy decisions through rank 26 select exactly ranks 2, 3, 6, 7, 14, 20, 21, and 26. Consequently, in a hypothetical final middle -2 cell at rank D \u2265 27 followed by an all-right suffix, D mod 21 lies in {11, 14, 17, 20}, D mod 10 is not 8, the mod-10 class 7 cannot pair with the mod-21 class 11, D mod 13 is not 11, and the mod-13 class 10 cannot pair with the mod-21 class 11. Exactly 412 of the 2730 joint residue classes survive. This phase sieve restricts but does not eliminate the -2 cell, and it does not decide the middle -1 or right-tail branches.","remaining_open_proposition_ids":["remaining_open.half_value_membership"],"paper_label":"res:finalskipexclusions","readme_headline":false,"declarations":[{"name":"seamGreedy_postTwentyOne_decisions_through_twentySix","module":"Erdos249257/HalfFinalMiddlePhaseSieve.lean","line":990},{"name":"seamGreedyBits_head2367_14_20_21_26","module":"Erdos249257/HalfFinalMiddlePhaseSieve.lean","line":1190},{"name":"finalMiddleCell_neg_two_forces_phase_refinement_through_twentySix","module":"Erdos249257/HalfFinalMiddlePhaseSieve.lean","line":1913},{"name":"finalMiddleTwentySixPhaseSurvivors_card","module":"Erdos249257/HalfFinalMiddlePhaseSieve.lean","line":2050}]},{"id":"last_producer_tail_escape_reduction","label":"Last-producer tail-escape reduction","status":"conditional reduction","statement":"If every genuine middle producer other than the locally discharged -3 cell has producer carry larger than its complete future divisor-incidence tail, then an eventual right seam is impossible and 1/2 belongs to the Mersenne achievement set. A stated square-root producer lower bound is a sufficient stronger socket. Neither socket is proved here.","remaining_open_proposition_ids":["remaining_open.half_value_membership"],"paper_label":"res:taildominance","readme_headline":false,"declarations":[{"name":"half_mem_mersenneAchievementSet_of_middleProducerTailEscapeExceptNegThree","module":"Erdos249257/HalfCylinderLastProducerContradiction.lean","line":387},{"name":"half_mem_mersenneAchievementSet_of_middleProducerSqrtEscape","module":"Erdos249257/HalfCylinderLastProducerContradiction.lean","line":409}]},{"id":"middle_producer_cardinality_reduction","label":"Finite-prefix middle-producer cardinality reduction","status":"conditional reduction","statement":"For a finite seam prefix, the future divisor-incidence tail is at most the prefix cardinality. Hence the explicit inequality card(below support) + belowPulse + 5 < 4 * remainder forces half-membership in the Mersenne achievement set; the still stronger hypothesis that every late middle remainder is at least its row also suffices. Neither arithmetic hypothesis is proved here.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:taildominance","readme_headline":false,"declarations":[{"name":"half_mem_mersenneAchievementSet_of_middleProducerCardEscape","module":"Erdos249257/HalfCylinderMiddleCarryLowerBound.lean","line":1503},{"name":"half_mem_mersenneAchievementSet_of_middleProducerRowEscape","module":"Erdos249257/HalfCylinderMiddleCarryLowerBound.lean","line":1510}]},{"id":"upper_reset_band_certificates_through_thirty","label":"Upper-reset dyadic-band certificates through row 30","status":"verified finite instance","bounded_domain":"Upper transitions from rows 13 through 30, with every relevant subsequent dyadic danger band checked for each row.","statement":"For every upper transition from rows 13 through 30, every relevant subsequent dyadic danger band is avoided. The check shares one exact successor remainder across all band indices for each row. It is bounded evidence and does not prove the cofinal upper-reset band-escape hypothesis.","paper_label":"res:finalskipexclusions","readme_headline":false,"declarations":[{"name":"seamUpperResetDyadicBandEscape_through_thirty","module":"Erdos249257/HalfCylinderUpperResetBandCertificates.lean","line":78}]},{"id":"half_membership_seam_classification","label":"Positive half-membership classification in seam coordinates","status":"proved here","statement":"Membership of 1/2 in the Mersenne achievement set is equivalent to false successor terminal bits beyond every bound, to a cofinal sequence of such bits, and to a cofinal integer-seam sequence whose skipped ranks tend to infinity. These equivalences do not establish that any positive producer exists.","remaining_open_proposition_ids":["remaining_open.half_value_membership"],"paper_label":"res:halfmembership","readme_headline":false,"declarations":[{"name":"half_mem_mersenneAchievementSet_iff_unboundedTerminalFalse","module":"Erdos249257/HalfCylinderHalfMembershipClassification.lean","line":126},{"name":"half_mem_mersenneAchievementSet_iff_cofinalTerminalFalse","module":"Erdos249257/HalfCylinderHalfMembershipClassification.lean","line":203},{"name":"half_mem_mersenneAchievementSet_iff_exists_unboundedSkippedRanksAlong","module":"Erdos249257/HalfCylinderHalfMembershipClassification.lean","line":213}]},{"id":"largest_skip_gap_propagation","label":"Largest skipped-rank gap and successor propagation","status":"conditional reduction","statement":"At a named largest skipped rank in the late range, the adjacent seam words satisfy an exact division-free weight-gap identity. Upper or middle successors create a terminal largest skipped rank, while right successors preserve one. The global first-crossing branch producer is not proved.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:largestskip","readme_headline":false,"declarations":[{"name":"three_mul_largestSkipLowerWeight_add_exactLateGap_eq_upperWeight","module":"Erdos249257/HalfCylinderLargestSkipGap.lean","line":259},{"name":"seamGreedyWord_succ_isLargestFalseRank_terminal_of_upperOrMiddle","module":"Erdos249257/HalfCylinderLargestSkipGap.lean","line":328},{"name":"IsLargestFalseRank.seamGreedyWord_succ_of_rightBranch","module":"Erdos249257/HalfCylinderLargestSkipGap.lean","line":339}]},{"id":"boundary_pulse_normalization","label":"Boundary-pulse normalization at the largest-skip crossing","status":"conditional reduction","statement":"For a late largest skipped rank, the filled strict suffix has zero row pulse. At the first two-thirds crossing, the boundary rank is one of two exact incidence cases and has row pulse two or one. The producer estimate that would exploit this normalization is not proved.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:boundarypulse","readme_headline":false,"declarations":[{"name":"exists_lowerPrefix_wordPulse_eq_of_largestFalse_late","module":"Erdos249257/HalfCylinderBoundaryPulse.lean","line":85},{"name":"rowPulse_boundary_of_late_firstCrossing","module":"Erdos249257/HalfCylinderBoundaryPulse.lean","line":150}]},{"id":"largest_skip_induction_reduction","label":"Largest-skip induction from a first-crossing producer socket","status":"conditional reduction","statement":"If every late largest skipped rank either remains late at the next row or forces an upper-or-middle successor branch, then the late invariant holds from row 14 onward, produces cofinal skipped ranks, and places 1/2 in the Mersenne achievement set. The first-crossing socket is not proved.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:largestskipinduction","readme_headline":false,"declarations":[{"name":"LargestSkipLateStepSocket.largestSkipLateAt_succ","module":"Erdos249257/HalfCylinderLargestSkipInduction.lean","line":90},{"name":"largestSkipLateAt_of_stepSocket","module":"Erdos249257/HalfCylinderLargestSkipInduction.lean","line":120},{"name":"half_mem_mersenneAchievementSet_of_largestSkipLateStepSocket","module":"Erdos249257/HalfCylinderLargestSkipInduction.lean","line":167}]},{"id":"fixed_tail_survival_socket","label":"Fixed-tail survival classification for half-cylinder skips","status":"conditional reduction","statement":"A skipped rank is final exactly when its post-decision remainder lies strictly above the complete remaining Mersenne tail. Hence membership of 1/2 is equivalent to the pointwise survival inequality at every actual skipped rank. That inequality is not proved.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:fixedtailsocket","readme_headline":false,"declarations":[{"name":"isLastHalfGreedySkip_iff_skip_and_fatal","module":"Erdos249257/HalfCylinderFixedTailSocket.lean","line":22},{"name":"half_mem_iff_every_actual_skip_survives","module":"Erdos249257/HalfCylinderFixedTailSocket.lean","line":73}]},{"id":"producer_carry_future_tail_socket","label":"Producer-carry and future-tail comparison socket","status":"conditional reduction","statement":"For a producer-aligned finite support, its residual from 1/2 is exactly the integer producer carry minus the complete future divisor-incidence tail, scaled by a power of two. A negative carry places the support above 1/2, and carry at most -8 gives a Mersenne-gap margin. No occurrence theorem for these carries is proved.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:producercarry","readme_headline":false,"declarations":[{"name":"producerCarry_residual_identity","module":"Erdos249257/HalfCylinderProducerCarrySocket.lean","line":27},{"name":"producerCarry_gt_tail_iff_supportSeries_lt_half","module":"Erdos249257/HalfCylinderProducerCarrySocket.lean","line":48},{"name":"producerCarry_le_neg_eight_forces_gap_margin","module":"Erdos249257/HalfCylinderProducerCarrySocket.lean","line":190}]},{"id":"quarter_band_endpoint_cells","label":"Quarter-band endpoint cells for adjacent half-cylinder cuts","status":"conditional reduction","statement":"On the middle and terminal-reset branches of the concrete adjacent seam recurrence, failure of the next-row upper bound is exactly characterized by explicit multiple-of-four pulse cells. The module does not prove that the concrete seam orbit avoids those cells.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:quarterband","readme_headline":false,"declarations":[{"name":"seamAdjacentCut_middleNextUpperFailure_iff_pulseCell","module":"Erdos249257/HalfCylinderQuarterBandEndpoints.lean","line":153},{"name":"seamAdjacentCut_rightNextUpperFailure_iff_pulseCell","module":"Erdos249257/HalfCylinderQuarterBandEndpoints.lean","line":175}]},{"id":"reset_deficit_escape_geometry","label":"Reset-deficit escape geometry for the largest-skip crossing","status":"conditional reduction","statement":"A right branch at a late largest skipped rank pins the integer-greedy remainder into an explicit interval and satisfies an exact affine recurrence. A doubled-scale deficit forces an upper-or-middle branch within the stated run length. The global anti-concentration needed to exclude the pinned interval is not proved.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:resetdeficit","readme_headline":false,"declarations":[{"name":"rightBranch_remainder_window","module":"Erdos249257/HalfCylinderResetDeficitEscape.lean","line":200},{"name":"rightBranch_remainder_succ_eq","module":"Erdos249257/HalfCylinderResetDeficitEscape.lean","line":218},{"name":"upperOrMiddle_within_of_two_pow_deficit","module":"Erdos249257/HalfCylinderResetDeficitEscape.lean","line":261}]},{"id":"seam_producer_carry_alignment","label":"Seam-to-producer carry alignment identity","status":"proved here","statement":"For the left boundary support at a first-wrap seam, the producer-aligned carry is exactly four times the shifted seam hole minus the paired incidence pulse at the next two rows. This is a coordinate identity and gives no sign or occurrence estimate.","paper_label":"res:seamproducer","readme_headline":false,"declarations":[{"name":"producerCarry_insert_eq_four_mul_shiftedHole_sub_pulse","module":"Erdos249257/HalfCylinderSeamProducerAlignment.lean","line":24}]},{"id":"selected_window_depth18_base","label":"Computed depth-18 selected half-carry window base","status":"verified finite instance","bounded_domain":"The 12 terminal carries in the half strip at depth 18, together with their common depth-13 restriction and the next-row divisor-agreement certificate.","statement":"A kernel-reflected Boolean table supplies a selected half-carry window at depth 18 for every terminal carry in the half strip [1,12]. The representatives share a common depth-13 restriction, yielding the first next-row divisor-agreement certificate. This is a finite base and does not construct selected windows at unbounded depths.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:selectedwindowbase","readme_headline":false,"declarations":[{"name":"depth18TableCheck_eq_true","module":"Erdos249257/HalfCarrySelectedWindowBase.lean","line":79},{"name":"depth18SelectedHalfWindow","module":"Erdos249257/HalfCarrySelectedWindowBase.lean","line":129},{"name":"depth18SelectedHalfWindow_nextRowDivisorAgreement","module":"Erdos249257/HalfCarrySelectedWindowBase.lean","line":172}]},{"id":"rewind_seam_operational_bridge","label":"Operational bridge from rewind seams to protected next-row seams","status":"conditional reduction","statement":"For a selected window with a rewind seam, a two-value coefficient profile on the adjacent base ancestors induces the live next-row profile. Under the stated strip and buffer inequalities, this realizes a protected one-hole seam at the next row. The base-ancestor coefficient drop is not supplied.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:rewindseambridge","readme_headline":false,"declarations":[{"name":"nextRow_profile_of_rewindSeam","module":"Erdos249257/RewindSeamOperationalBridge.lean","line":140},{"name":"protectedEvenSeamRealizedAt_succ_of_profile","module":"Erdos249257/RewindSeamOperationalBridge.lean","line":217}]},{"id":"selected_suffix_cylinder_propagation","label":"Conditional suffix-cylinder feedback and child-gap recurrence","status":"conditional reduction","statement":"For the depth-18 selected window, the common depth-13 prefix and suffix numeral satisfy \u03bd + k = 17. Under a common next coefficient C the endpoint obeys E' = 2E - C. At a feedback row the full cylinder either advances or leaves one carry gap [g_-,g_+]. If the lower and upper families have coefficients C_- and C_+ and satisfy the explicit positivity, ordering, strip, and buffer inequalities, the next gap is exactly [2g_- - C_- - 1, 2g_+ - C_+]. Cutoff promotion appends forced bits 0 and 1 and preserves consecutive prefixes. A swallowed two-row update forces the sharp quarter-band bound B <= 4(g_+ - g_- + 1) + 2. Cofinal full stages would produce an infinite support of value 1/2, but no such cofinal family or uniform coefficient inequalities are proved.","remaining_open_proposition_ids":["remaining_open.half_value_membership"],"paper_label":"res:selectsuffix","readme_headline":false,"declarations":[{"name":"depth18SelectedHalfWindow_hasSuffixCylinderAt_thirteen","module":"Erdos249257/SelectedSuffixCylinder.lean","line":88},{"name":"SelectedHalfWindow.hasSuffixCylinderAt_stepExplicit","module":"Erdos249257/SelectedSuffixCylinder.lean","line":136},{"name":"promotedSuffixCylinder_or_protectedEvenSeamRealizedAt","module":"Erdos249257/SuffixCylinderPromotion.lean","line":322},{"name":"CylinderStage.feedbackStep_or_protectedSeam","module":"Erdos249257/SuffixCylinderGlobalProducer.lean","line":174},{"name":"SelectedHalfWindow.word_stepTailExplicit","module":"Erdos249257/SuffixCylinderBelow.lean","line":117},{"name":"hasSuffixCylinderAt_take","module":"Erdos249257/SuffixCylinderTwoSheet.lean","line":26},{"name":"CylinderStage.feedbackAdvance_or_protectedSeam_inStrip","module":"Erdos249257/SuffixCylinderFeedbackFanIn.lean","line":62},{"name":"CylinderStage.feedbackAdvance_or_inStripTwoSheet","module":"Erdos249257/SuffixCylinderInStrip.lean","line":665},{"name":"InStripGapStage.advanceOfCommonNextCoeff","module":"Erdos249257/SuffixCylinderFiniteGap.lean","line":256},{"name":"ProfiledGapStage.advanceOfNextCoeffProfile","module":"Erdos249257/SuffixCylinderProfiledGap.lean","line":124},{"name":"selectedHalfIntervalPromoteHigh","module":"Erdos249257/SuffixCylinderProfiledPromotion.lean","line":107},{"name":"ProfiledGapStage.lowerNextCoeff_le_upperNextCoeff_add_one","module":"Erdos249257/SuffixCylinderProfiledAdjacency.lean","line":232},{"name":"ProfiledGapStage.promoteCutoff_hasAdjacentPrefixes","module":"Erdos249257/SuffixCylinderProfiledPromotionAdjacency.lean","line":65},{"name":"ProfiledGapStage.promoteThenAdvance_hasAdjacentPrefixes","module":"Erdos249257/SuffixCylinderProfiledAutoStep.lean","line":110},{"name":"nextBound_le_four_gapWidth_add_two_of_macro_swallow","module":"Erdos249257/SuffixCylinderProfiledSwallow.lean","line":254},{"name":"cofinalTerminalOnlyStrip_of_cofinalCylinderStages","module":"Erdos249257/SuffixCylinderTerminalOnlyBridge.lean","line":125},{"name":"exists_infinite_support_half_of_cofinalCylinderStages","module":"Erdos249257/SuffixCylinderTerminalOnlyBridge.lean","line":138},{"name":"exists_infinite_positive_support_half_of_cofinalCylinderStages","module":"Erdos249257/SuffixCylinderTerminalOnlyBridge.lean","line":148}]},{"id":"selected_suffix_pre_feedback_run","label":"Verified finite suffix-cylinder stages through depth 52","status":"verified finite instance","bounded_domain":"Every depth from 18 through 29, plus explicit full stages at depths 51 and 52; depth 29 has common suffix cylinders at cutoffs 14 through 25 but not 26.","statement":"A common depth-13 prefix supports full selected windows at every depth 18 through 27, and the threshold continuation reaches depth 29 while covering the protected strip. At depth 29, cutoffs 14 through 25 remain common and cutoff 26 does not. A separate exact recurrence reaches depth 51 with endpoint 51,327,745 beyond the next head threshold, and the promoted depth-52 successor supplies a terminal-strip witness. These are finite stages, not an unbounded induction.","remaining_open_proposition_ids":["remaining_open.half_value_membership"],"paper_label":"res:prefeedback","readme_headline":false,"declarations":[{"name":"nonempty_stage_of_eighteen_le_of_le_twentySeven","module":"Erdos249257/SelectedSuffixCylinderPreFeedback.lean","line":118},{"name":"depth27SelectedHalfWindow_hasSuffixCylinderAt_thirteen","module":"Erdos249257/SelectedSuffixCylinderPreFeedback.lean","line":141},{"name":"depth27SelectedHalfWindow_endpoint_covers_strip","module":"Erdos249257/SelectedSuffixCylinderPreFeedback.lean","line":147},{"name":"stage27_endpoint_eq","module":"Erdos249257/SuffixCylinderFirstFeedback.lean","line":144},{"name":"depth29SelectedHalfWindow_endpoint_covers_strip","module":"Erdos249257/SuffixCylinderThreshold.lean","line":700},{"name":"depth29SelectedHalfWindow_not_commonRestrictionAt_twentySix","module":"Erdos249257/SuffixCylinderThreshold.lean","line":806},{"name":"fullCylinderStage51_feedback_headThreshold_margin","module":"Erdos249257/SuffixCylinderStage51.lean","line":536},{"name":"fullCylinderStage52_halfTerminalOnlyStripWitness","module":"Erdos249257/SuffixCylinderTerminalOnlyBridge.lean","line":118}]},{"id":"half_divisor_unit_drop","label":"Half-divisor unit coefficient drop at the feedback row","status":"proved here","statement":"At the doubled feedback row, switching only the terminal half-divisor bit from false to true increases the support coefficient by exactly one. The same identity holds for any explicitly identified adjacent boundary pair, and the rewind adapter transfers such a pair to the next protected seam while retaining the unit-drop witness. This does not construct such pairs globally.","paper_label":"res:halfdivisordrop","readme_headline":false,"declarations":[{"name":"supportCoeff_extend_true_eq_false_add_one_at_double","module":"Erdos249257/HalfDivisorUnitDrop.lean","line":20},{"name":"supportCoeff_boundaryPair_unitDrop_at_double","module":"Erdos249257/HalfDivisorUnitDrop.lean","line":35},{"name":"rewindBaseUnitDropAt_of_halfDivisorBoundaryPair","module":"Erdos249257/RewindHalfDivisorAdapter.lean","line":86},{"name":"protectedEvenSeamRealizedAt_succ_of_fullStrip_rewindHalfDivisorBoundaryPair","module":"Erdos249257/RewindHalfDivisorAdapter.lean","line":147}]},{"id":"rewind_boundary_pair_obstruction","label":"Finite rewind boundary-pair obstruction","status":"proved here","statement":"A concrete depth-26 selected two-word window has a depth-27 explicit rewind step with history [1], a width-two seam pair, and the doubled-row identity, but no depth-13 half-divisor boundary-pair witness. Therefore scalar one-row seam data alone does not entail the protected boundary predicate required by the cofinal consumer.","paper_label":"res:rewindboundarycounterexample","readme_headline":false,"declarations":[{"name":"depth27_actualRewindSeam_not_boundaryPair","module":"Erdos249257/RewindBoundaryPairCounterexample.lean","line":169}]},{"id":"concrete_producer_carry_coordinates","label":"Concrete adjacent-cut producer-carry coordinates","status":"proved here","statement":"For every adjacent cut at rank s \u2265 5, the producer carry of the terminal-augmented below support is 4 times the middle remainder minus the below pulse minus 4, while that of the terminal-augmented above support is the negative of 4 times the overshoot plus the above pulse plus 4. These are exact coordinate identities; they supply neither the needed exponential lower bounds nor a universal escape argument.","paper_label":"res:producercarrycoordinates","readme_headline":false,"declarations":[{"name":"producerCarry_insert_seamWordSupport_eq","module":"Erdos249257/HalfCylinderProducerLowerBound.lean","line":105},{"name":"producerCarry_insert_seamBelowSupport_eq_middleCoordinate","module":"Erdos249257/HalfCylinderProducerLowerBound.lean","line":150},{"name":"producerCarry_insert_seamAboveSupport_eq_neg_upperCoordinate","module":"Erdos249257/HalfCylinderProducerLowerBound.lean","line":183}]},{"id":"mersenne_lambert_ladder","label":"The Mersenne\u2013Lambert ladder identities","status":"formalised here","statement":"L(\u03bc) = 1/2, L(\u03c6) = 2, L(\u03c6*\u03bc) = S, and the M\u00f6bius-square lens S = 1/2 + \u2211 \u03bc(d)/(2^d \u2212 1)\u00b2, placing the open constant one convolution from the machine-checked irrational E.","paper_label":"res:lens","readme_headline":false,"declarations":[{"name":"tsum_totient_div_two_pow_sub_one_eq_two","module":"Erdos249257/CertificateKernel.lean","line":18101},{"name":"totient_series_eq_half_add_moebius_mersenne_square","module":"Erdos249257/CertificateKernel.lean","line":18126}]},{"id":"totient_coprimality_probability","label":"Fair-coin coprimality form of the totient constant","status":"proved here","statement":"For independent fair-coin geometric waiting times X and Y on the positive integers, S = 1/2 + P(gcd(X,Y)=1), where the probability is the visible coprime-pair series. That probability has no rational representation with denominator at most 39819823323350687661677887437915526. This is an exact reformulation and quantitative corollary; irrationality remains open.","remaining_open_proposition_ids":["remaining_open.erdos_249_irrationality"],"paper_label":"res:coprime","readme_headline":false,"declarations":[{"name":"totient_series_eq_half_add_visible_coprime_pairs","module":"Erdos249257/CertificateKernel.lean","line":18229},{"name":"tsum_visible_coprime_pairs_ne_int_div_of_den_le_39819823323350687661677887437915526","module":"Erdos249257/CertificateKernel.lean","line":18244}]},{"id":"squared_lambert_gcd_moments","label":"Squared-Lambert identities for divisor count and the first gcd moment","status":"proved here","statement":"The squared-Mersenne Lambert series with weight 1 equals the base-1/2 series of sigma(n)-tau(n), and the series with weight phi equals the base-1/2 series of P(n)-n, where P is Pillai's gcd-sum function. These are the exact divisor-count and first-moment identities for the fair-coin gcd law; the separately cited irrationality of the q-zeta middle row is not claimed as formalised here.","paper_label":"res:gcdmoment","readme_headline":false,"declarations":[{"name":"tsum_one_div_mersenne_sq_eq_sigma_sub_tau_series","module":"Erdos249257/GcdMomentCalculus.lean","line":216},{"name":"tsum_totient_div_mersenne_sq_eq_gcd_moment_series","module":"Erdos249257/GcdMomentCalculus.lean","line":235}]},{"id":"stern_brocot_cylinder_law","label":"Reduced-direction normalization and Stern\u2013Brocot cylinder law","status":"proved here","statement":"The Mersenne weights of positive reduced slopes sum exactly to one. At every positive Stern\u2013Brocot node (a,b), the cylinder mass 1/((2^a-1)(2^b-1)) splits into its stop mass and two child cylinders; the depth-d unfolding has error between zero and (2/3)^d times the cylinder mass and converges to the closed form.","paper_label":"res:sternbrocotcylinders","readme_headline":false,"declarations":[{"name":"tsum_pos_coprime_inv_mersenne_eq_one","module":"Erdos249257/GcdMomentCalculus.lean","line":349},{"name":"cylinderMass_split","module":"Erdos249257/GcdMomentCalculus.lean","line":474},{"name":"sternBrocotDepthMass_error","module":"Erdos249257/GcdMomentCalculus.lean","line":525},{"name":"tendsto_sternBrocotDepthMass","module":"Erdos249257/GcdMomentCalculus.lean","line":560}]},{"id":"stern_brocot_run_fibonacci_stability","label":"Fibonacci and continuant stability of Stern\u2013Brocot runs","status":"formalised here","statement":"A Stern\u2013Brocot word with r nonempty alternating runs has height at least F_(r+3), with equality on the all-unit run spine. Writing each run length as 1+e_i gives an exact multiaffine continuant-defect expansion, the sharp global gain F_(r+1) sum e_i, and exact one-site coefficient F_(i+2)F_(r-i+1). The first induced run layer has mass 1/2, while the natural denominator exponent is exactly F_(r+3)-2. These are source-level combinatorial and arithmetic facts; no analytic denominator survival or irrationality conclusion is claimed.","paper_label":"res:sternbrocotruns","readme_headline":false,"declarations":[{"name":"sbPair_runWord","module":"Erdos249257/SternBrocotRunGeometry.lean","line":125},{"name":"runHeight_fib_lower","module":"Erdos249257/SternBrocotRunGeometry.lean","line":185},{"name":"runHeight_defectRunLengths","module":"Erdos249257/SternBrocotRunGeometry.lean","line":263},{"name":"runHeight_defect_fib_sum_lower","module":"Erdos249257/SternBrocotRunGeometry.lean","line":343},{"name":"runHeight_oneSiteDefect","module":"Erdos249257/SternBrocotRunGeometry.lean","line":413},{"name":"oneRunMass_eq_half","module":"Erdos249257/SternBrocotRunGeometry.lean","line":467},{"name":"naturalRunDenominatorExponent_add_two","module":"Erdos249257/SternBrocotRunGeometry.lean","line":491}]},{"id":"primitive_coordinate_index_obstruction","label":"Primitive Euler-coordinate finite-jet obstruction","status":"proved here","statement":"For the primitive-conductor weight A = \u03c6*\u03bc, every positive index clearing the normalized coordinates A(n)/n through horizon N is divisible by the two-tier primorial: 4 at the prime 2 once N \u2265 4, p\u00b2 for odd p with p\u00b2 \u2264 N, and p otherwise. Hence no fixed positive index clears all normalized coordinates. This is a finite Euler/Witt-coordinate obstruction only; it gives no irrationality criterion for S.","paper_label":"res:primitiveindex","readme_headline":false,"declarations":[{"name":"twoTierPrimorial_dvd_of_integral_jet","module":"Erdos249257/PrimitiveDeterminantLift.lean","line":148},{"name":"no_fixed_integral_primitive_euler_index","module":"Erdos249257/PrimitiveDeterminantLift.lean","line":169}]},{"id":"primitive_unit_gap_ceiling","label":"Prime-power reduced-denominator unit-gap ceiling","status":"proved here","statement":"For the reduced-denominator unit-gap predicate at an odd prime power p^e, the candidate interval is empty or has exactly one candidate divisible by p; in particular its count is at most one. The exact row (u,N,K) = (3,3,5) satisfies the unit-gap predicate while the corresponding ordinary empty-gap inequality fails. This is a finite certificate-level strengthening, not an unbounded supply theorem or an irrationality criterion for S.","paper_label":"res:unitgap","readme_headline":false,"declarations":[{"name":"reducedDenominatorUnitGapCert_prime_pow_iff","module":"Erdos249257/PrimitiveWeightCertificate.lean","line":54},{"name":"reducedDenominatorCandidateCount_prime_pow_le_one","module":"Erdos249257/PrimitiveWeightCertificate.lean","line":111},{"name":"reducedDenominatorUnitGapCert_three_window_3_5","module":"Erdos249257/PrimitiveWeightCertificate.lean","line":151},{"name":"not_ordinary_gap_certificate_three_window_3_5","module":"Erdos249257/PrimitiveWeightCertificate.lean","line":157}]},{"id":"mersenne_tail_atom_boundary","label":"Finite Mersenne tail-atom determinant boundary","status":"proved here","statement":"A pointed Mersenne tail atom has the exact doubling-and-wrap carry law. For any finite atom matrix, clearing the product of its column Mersenne denominators yields the integral numerator determinant; a nonzero cleared weighted minor therefore has absolute value at least one. This is a finite-denominator boundary for an atom-minor route, not an irrationality criterion for S.","paper_label":"res:mersennetailatoms","readme_headline":false,"declarations":[{"name":"mersenneTailAtom_step","module":"Erdos249257/MersenneTailAtoms.lean","line":28},{"name":"mersenneAtom_denominatorProduct_mul_det","module":"Erdos249257/MersenneTailAtoms.lean","line":122},{"name":"one_le_abs_weightedMersenneAtom_cleared_det","module":"Erdos249257/MersenneTailAtoms.lean","line":181}]},{"id":"mobius_sign_support_no_go","label":"M\u00f6bius-sign support one-half no-go","status":"proved here","statement":"The Boolean support selected by the negative M\u00f6bius values has sum 1/2 plus the positive M\u00f6bius tail and is therefore at least 1/2 + 1/63. This closes the sign-truncation route to the target value 1/2; it does not exclude another infinite Boolean support with that value.","paper_label":"res:mobiussignnogo","readme_headline":false,"declarations":[{"name":"tsum_negativeMobius_eq_half_add_positiveMobiusTail","module":"Erdos249257/MobiusSignSupportNoGo.lean","line":111},{"name":"half_add_one_div_sixty_three_le_tsum_negativeMobius","module":"Erdos249257/MobiusSignSupportNoGo.lean","line":150}]},{"id":"totient_carry_kernel_anti_compression","label":"Unconditional carry-kernel anti-compression","status":"unconditional progress","statement":"If the totient coefficient series were rational, one positive-multiplier tempered integral carry would have canonical carry-section span dimension at least 2^e \u2212 1 for every e. The all-level kernel-independence producer is supplied by the CRT\u2013Dirichlet minor theorem. No finite-rank bound for every rationality-supplied carry is proved, so this is an infinite anti-compression barrier rather than an irrationality criterion for S.","remaining_open_proposition_ids":["remaining_open.erdos_249_irrationality"],"paper_label":"res:carryrank","readme_headline":false,"declarations":[{"name":"finrank_canonicalCarryKernel_ge_of_linearIndependent","module":"Erdos249257/TotientCarryKernelRigidity.lean","line":211},{"name":"not_irrational_totientSeries_implies_unbounded_carryRank_unconditional","module":"Erdos249257/TotientCarryKernelRigidity.lean","line":284}]},{"id":"denominator_exclusion","label":"Unconditional denominator exclusion for the totient constant","status":"unconditional progress","statement":"S = \u2211 \u03c6(n)/2\u207f is not equal to p/q for any integer p and any 1 \u2264 q \u2264 79 639 646 646 701 375 323 355 774 875 831 053 (\u2248 7.96 \u00d7 10\u00b3\u2074). A lower bound on any rational representation, not a proof of irrationality.","remaining_open_proposition_ids":["remaining_open.erdos_249_irrationality"],"paper_label":"res:farey","readme_headline":true,"declarations":[{"name":"tsum_totient_div_pow_two_ne_ratCast_of_den_le_79639646646701375323355774875831053","module":"Erdos249257/CertificateKernel.lean","line":18056}]},{"id":"certificate_reduction","label":"Exact reduction of #249 to finite decidable certificates","status":"conditional reduction","statement":"S is irrational if and only if certified non-integrality witnesses occur at unbounded parameters. The pointwise completeness theorem and the diagonal supply theorem together show that an unbounded lcm-diagonal supply is equivalent as well. Producing that supply remains open.","remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"],"paper_label":"res:diag","readme_headline":true,"declarations":[{"name":"irrational_totient_series_of_certificate_supply","module":"Erdos249257/TotientTailPeriodKiller.lean","line":394},{"name":"irrational_totient_series_of_lcm_diagonal_certificate_supply","module":"Erdos249257/LcmDiagonalReduction.lean","line":112},{"name":"rational_totient_series_forces_lcm_cone_flatness","module":"Erdos249257/LcmConeFlatness.lean","line":90},{"name":"irrational_totient_series_iff_pointwise_certificates","module":"Erdos249257/LcmConeFlatness.lean","line":399},{"name":"irrational_totient_series_iff_certificate_supply","module":"Erdos249257/LcmConeFlatness.lean","line":412},{"name":"irrational_totient_series_iff_lcm_diagonal_certificate_supply","module":"Erdos249257/LcmConeFlatness.lean","line":426}]},{"id":"certificate_completeness","label":"Certificate completeness","status":"proved here","statement":"At fixed parameters, a certified kill exists exactly when the corresponding tail difference is non-integral. At every positive shift, integrality of one tail difference forces S to be rational; thus irrationality is equivalent to pointwise non-integrality, and to pointwise finite certificates.","paper_label":"res:pointwisecomplete","readme_headline":true,"declarations":[{"name":"exists_certifiedKill_iff_tail_diff_notMem_int","module":"Erdos249257/LcmConeFlatness.lean","line":316},{"name":"tail_diff_mem_int_iff_scaled_series_mem_int","module":"Erdos249257/LcmConeFlatness.lean","line":327},{"name":"irrational_totient_series_iff_all_tail_diffs_nonintegral","module":"Erdos249257/LcmConeFlatness.lean","line":386},{"name":"irrational_totient_series_iff_pointwise_certificates","module":"Erdos249257/LcmConeFlatness.lean","line":399}]},{"id":"certified_kill_instances","label":"Initial certified-kill segment and generated certificates","status":"verified finite instance","statement":"Kernel-checked certified kills for small periods and at 28 explicit lcm-diagonal scales through t = 64, plus finite certificate shards, each discharged by decide.","bounded_domain":"The explicitly listed small periods; the lcm-diagonal scales t in {1,2,3,4,5,7,8,9,11,13,16,17,19,23,25,27,29,31,32,37,41,43,47,49,53,59,61,64}; and the parameters encoded by the finite certificate shards.","remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"],"paper_label":"res:deposits","readme_headline":true,"declarations":[{"name":"certifiedKill_all_small","module":"Erdos249257/TotientTailPeriodKiller.lean","line":404},{"name":"certifiedKill_periodLcm_diagonal_upto_six","module":"Erdos249257/LcmDiagonalReduction.lean","line":237},{"name":"certifiedKill_periodLcm_diagonal_at_seven","module":"Erdos249257/LcmDiagonalReduction.lean","line":242},{"name":"certifiedKill_periodLcm_diagonal_at_eight","module":"Erdos249257/LcmDiagonalReduction.lean","line":246},{"name":"certifiedKill_diagonal_all_imported_through_t64","module":"Erdos249257/DiagonalPincerCertificatesT64.lean","line":1967}]},{"id":"diagonal_pincer_reduction","label":"Exact full-target reduction on the lcm diagonal","status":"conditional reduction","statement":"At each lcm-diagonal scale, integrality is equivalent to a foreign-defect term landing in an explicit full-target interval. Globally, S is irrational if and only if the full target is avoided at arbitrarily large diagonal scales; that cofinal avoidance is not proved.","remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"],"paper_label":"res:fulltargetavoidance","readme_headline":true,"declarations":[{"name":"diagonal_int_iff_foreignDiagonalDefect_hits_fullTarget","module":"Erdos249257/DiagonalPincerDecomposition.lean","line":215},{"name":"irrational_totientSeries_of_full_target_avoidance_supply","module":"Erdos249257/DiagonalPincerDecomposition.lean","line":290}]},{"id":"diagonal_fresh_loss_reduction","label":"Fresh-loss projection and suffix-supply reduction","status":"conditional reduction","statement":"The fresh-loss residue certificate is exactly an integer projection inequality, and such a projection forces full-target avoidance. An unbounded projection supply\u2014or the stronger adjacent-suffix gap supplies formalised here\u2014implies irrationality; no unbounded supply is asserted.","remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"],"paper_label":"res:freshloss","readme_headline":false,"declarations":[{"name":"diagonalFreshLossResidueCert_iff_projectionCert","module":"Erdos249257/DiagonalFreshLossBridge.lean","line":72},{"name":"irrational_totientSeries_of_diagonalFreshLossProjectionSupply","module":"Erdos249257/DiagonalFreshLossBridge.lean","line":202}]},{"id":"diagonal_pincer_t64_primality_closure","label":"T64 diagonal-pincer primality closure","status":"verified finite instance","bounded_domain":"The generated Lucas primality dependencies required by the diagonal-pincer certificate family through t = 64.","statement":"A generated proof-producing Lucas-certificate module supplies the primality dependencies for the t=47,49,53,59,61,64 diagonal-pincer certificates. This is finite certificate infrastructure; it neither supplies an unbounded certificate family nor proves irrationality.","paper_label":"res:t64closure","readme_headline":false,"declarations":[{"name":"prime_lucas_2364533768205644535022723273","module":"Erdos249257/DiagonalPincerPrimeCertificates/ClosureT64.lean","line":51759}]},{"id":"diagonal_pincer_t64_endpoint_certificate","label":"T64 diagonal-pincer endpoint certificate","status":"verified finite instance","bounded_domain":"The single lcm-diagonal endpoint t = 64.","statement":"Lean checks the diagonal-pincer certified kill at t = 64 using the committed T64 Lucas-primality closure. This is one finite certificate and does not assert an unbounded certificate supply or irrationality.","paper_label":"res:t64endpoint","readme_headline":false,"declarations":[{"name":"certifiedKill_diagonal_t64","module":"Erdos249257/DiagonalPincerCertificateT64Endpoint.lean","line":1928}]},{"id":"fresh_prime_deficit_decomposition","label":"Fresh-prime endpoint-deficit decomposition","status":"proved here","statement":"At each positive height, every old-prime endpoint mass dominates the actual totient. The diagonal increment and literal foreign channel split exactly into the old M\u00f6bius increment and lower-minus-upper endpoint fresh deficits. The second and five-point curvature operators retain this split, and the nonnegative deficits bound the exact adverse weighted loss from the old-prime curvature margins. No cofinal deficit bound or irrationality conclusion is proved.","remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"],"paper_label":"res:freshprimedeficit","readme_headline":false,"declarations":[{"name":"endpointFreshDeficit_nonneg","module":"Erdos249257/FreshPrimeDeficitDecomposition.lean","line":91},{"name":"finiteForeignChannelIncrement_eq_endpointFreshDeficit_sub","module":"Erdos249257/FreshPrimeDeficitDecomposition.lean","line":184},{"name":"oldFivePoint_sub_secondBranchAdverseDeficit_le_actual","module":"Erdos249257/FreshPrimeDeficitDecomposition.lean","line":269}]},{"id":"flexible_odd_window_reduction","label":"Flexible three-rank odd-window reduction","status":"conditional reduction","statement":"At an analytically admissible odd power-of-two rank, centrality in any one of the canonical odd rank or its next two base-four successors supplies the existing adjacent-suffix gap interface and hence irrationality. The remaining assertion is a cofinal three-rank band supply; no such supply is proved.","remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"],"paper_label":"res:flexibleoddwindow","readme_headline":false,"declarations":[{"name":"diagonalAdjacentSuffixGapSupply_of_powerTwoOddGuardThreeRankBand","module":"Erdos249257/DiagonalFlexibleOddWindowSupply.lean","line":93},{"name":"irrational_totientSeries_of_powerTwoOddGuardThreeRankBandSupply","module":"Erdos249257/DiagonalFlexibleOddWindowSupply.lean","line":136}]},{"id":"flexible_odd_window_affine_frontier","label":"Affine three-scale frontier for the flexible odd window","status":"conditional reduction","remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"],"statement":"Under the stated short-correction bounds, the three half-word central-band predicates are exactly a three-scale affine escape disjunction. A uniform correction envelope alone cannot force that escape, as a zero-start affine orbit remains inside every scaled edge tube. Thus arithmetic state-correction correlation remains required.","paper_label":"res:flexibleoddaffine","readme_headline":false,"declarations":[{"name":"zeroAffineOrbit_stays_in_scaled_edge","module":"Erdos249257/DiagonalFlexibleOddWindowAffine.lean","line":225},{"name":"threeRankBandAt_iff_threeScaleAffineEscapeAt_of_shortCorrections","module":"Erdos249257/DiagonalFlexibleOddWindowAffine.lean","line":236}]},{"id":"power_two_bit_lift","label":"Exact power-of-two one-bit lift","status":"proved here","statement":"Given agreement modulo 2^k, agreement modulo 2^(k+1) is exactly evenness of the signed quotient after division by 2^k; failure gives the unique offset class. In particular, a signed difference 16*c agrees modulo 32 exactly when c is even. Under the stated correction and margin budget, the centered signed-margin socket is exactly failure of this next lift. This provides a finite fifth-bit interface and does not supply cofactor parity at unbounded scales.","remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"],"paper_label":"res:powertwobitlift","readme_headline":false,"declarations":[{"name":"two_pow_succ_dvd_iff_quotient_even","module":"Erdos249257/PowerTwoBitLift.lean","line":34},{"name":"modEq_thirtyTwo_iff_even_difference_cofactor","module":"Erdos249257/PowerTwoBitLift.lean","line":111},{"name":"not_modEq_thirtyTwo_iff_offset_sixteen","module":"Erdos249257/PowerTwoBitLift.lean","line":129},{"name":"actualPenultimateSignedMargin_centeredLift_iff_not_twice_dvd","module":"Erdos249257/PowerTwoCenteredBitLift.lean","line":121},{"name":"actualPenultimateSignedMargin_centeredThirtyTwo_iff_not_thirtyTwo_dvd","module":"Erdos249257/PowerTwoCenteredBitLift.lean","line":159}]},{"id":"parity_socket_obstruction","label":"Parity obstruction for the reduced margin socket","status":"proved here","statement":"For every positive even modulus and every radius in the stated nontrivial range, each parity class contains both a reduced-margin-socket hit and a miss. The same holds for a positive power-of-two modulus. Thus quotient parity alone cannot decide the reduced signed-margin socket. No t = 827 numeral or signed-margin certificate is asserted.","remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"],"paper_label":"res:paritysocket","readme_headline":false,"declarations":[{"name":"reducedMarginSocket_not_decided_by_parity","module":"Erdos249257/Totient827ParitySocketCeiling.lean","line":73},{"name":"powerTwo_reducedMarginSocket_not_decided_by_parity","module":"Erdos249257/Totient827ParitySocketCeiling.lean","line":110}]},{"id":"canonical_jump_slack_reduction","label":"Canonical strict-jump slack reduction","status":"conditional reduction","statement":"The two canonical adjacent-suffix central-band inequalities are equivalent to nonnegativity of one explicit integer slack. Cofinal nonnegative slack at strict lcm jumps implies irrationality of the totient constant; that cofinal sign statement remains open.","remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"],"paper_label":"res:slackcriterion","readme_headline":false,"declarations":[{"name":"canonicalAdjacentSuffixCentral_iff_slack_nonneg","module":"Erdos249257/DiagonalFreshLossBridge.lean","line":2667},{"name":"canonicalAdjacentSuffixJumpCentralSupply_iff_slackSupply","module":"Erdos249257/DiagonalFreshLossBridge.lean","line":2757},{"name":"irrational_totientSeries_of_canonicalAdjacentSuffixJumpSlackSupply","module":"Erdos249257/DiagonalFreshLossBridge.lean","line":2929}]},{"id":"squared_mersenne_enclosure","label":"Sharp squared-Mersenne diagonal enclosure","status":"proved here","statement":"The scaled diagonal tail difference is written as an explicit Lambert-projected centre plus a signed squared-Mersenne tail, with a sharp symmetric bound and a checked separation criterion for missing the full target.","paper_label":"res:squareenclosure","readme_headline":false,"declarations":[{"name":"scaleDiagonalTailDifference_sub_lambertProjectedDiagonal","module":"Erdos249257/SquaredMersenneDiagonalEnclosure.lean","line":116},{"name":"scaleFullTarget_miss_of_lambert_projected_separation","module":"Erdos249257/SquaredMersenneDiagonalEnclosure.lean","line":237}]},{"id":"actual_foreign_residue_projection","label":"Finite foreign-residue projection reduction","status":"conditional reduction","statement":"The finite M\u00f6bius residue diagonal splits exactly into projected foreign and divisor channels, and every finite stable tail window has a closed geometric bound. If an analytic kernel/complement bridge controls the actual foreign defect by that bound and the projection is separated from every integer, then the full target is missed. The analytic bridge and any unbounded separated-projection supply are not proved.","remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"],"paper_label":"res:foreignresidue","readme_headline":false,"declarations":[{"name":"finiteResidueDiagonal_eq_projectedForeign_add_divisor","module":"Erdos249257/ActualForeignResidueProjection.lean","line":307},{"name":"abs_foreignTailWindow_le_foreignComplementBound","module":"Erdos249257/ActualForeignResidueProjection.lean","line":275},{"name":"scaleFullTarget_miss_of_projected_separation","module":"Erdos249257/ActualForeignResidueProjection.lean","line":348}]},{"id":"mersenne_shadow_denominator","label":"Exact Mersenne-shadow denominator and survivor bounds","status":"proved here","statement":"The reduced denominator of the lcm-height scaled M\u00f6bius shadow is computed exactly, and an explicit upper-half Mersenne product both divides it and supplies a quantitative lower bound.","paper_label":"res:shadowden","readme_headline":false,"declarations":[{"name":"exists_upperHalf_mersenne_channel_dvd_den_with_bounds","module":"Erdos249257/MersenneShadowDenominatorGrowth.lean","line":99},{"name":"lcmHeight_scaledMobiusShadow_den_exact","module":"Erdos249257/MersenneShadowDenominatorGrowth.lean","line":147}]},{"id":"adelic_height_obstruction","label":"Scalar-localisation height obstruction","status":"proved here","statement":"If H divides the reduced denominator of a rational x and the reduced denominator of c\u00b7x divides H for an integer c, then the complementary factor x.den/H divides |c|. Scalar localisation transfers, rather than discards, that denominator information; this local obstruction is not an irrationality criterion.","paper_label":"res:adelic","readme_headline":true,"declarations":[{"name":"scalarLocalization_complement_dvd","module":"Erdos249257/AdelicHeightObstruction.lean","line":23},{"name":"scalarLocalization_integer_eq_mul_num","module":"Erdos249257/AdelicHeightObstruction.lean","line":56}]},{"id":"square_crt_correction_suppression","label":"Square-CRT correction suppression","status":"proved here","statement":"For every finite family of distinct primes and prescribed anchors, a bounded simultaneous square-CRT base removes the prime-dilation correction at every admissible finite shift. Checked clean cubes can have vanishing coefficients and can also have a nonzero coefficient; this finite mechanism supplies neither an unbounded certificate family nor an irrationality criterion for the totient constant.","paper_label":"res:squarecrt","readme_headline":false,"declarations":[{"name":"exists_squareCRT_clean_totient_family","module":"Erdos249257/SquareCRTCube.lean","line":198},{"name":"exists_squareCRT_clean_horizon","module":"Erdos249257/SquareCRTCube.lean","line":228},{"name":"squareCRT_clean_block_can_vanish","module":"Erdos249257/SquareCRTCube.lean","line":285},{"name":"squareCRT_clean_block_can_be_nonzero","module":"Erdos249257/SquareCRTCube.lean","line":303}]},{"id":"signed_dyadic_moment_substrate","label":"Signed dyadic-moment algebra","status":"proved here","statement":"A rectangular determinant expansion over all maps and a unique-largest-dyadic-exponent parity lemma are formalised. The latter makes a cleared finite dyadic sum odd, hence nonzero, when its selected numerator is odd and its exponent is strictly largest. This is finite algebraic substrate only: it proves neither non-vanishing of the actual totient Hankel determinants nor an irrationality criterion.","paper_label":"res:signedmoment","readme_headline":false,"declarations":[{"name":"det_mul_rectangular","module":"Erdos249257/SignedQMomentObstruction.lean","line":29},{"name":"scaled_dyadic_sum_odd","module":"Erdos249257/SignedQMomentObstruction.lean","line":78},{"name":"scaled_dyadic_sum_ne_zero","module":"Erdos249257/SignedQMomentObstruction.lean","line":96}]},{"id":"dyadic_totient_certificate_interface","label":"Unconditional dyadic totient-kernel rank","status":"unconditional progress","statement":"For every e, a CRT\u2013Dirichlet construction produces a separated nonzero evaluation minor for the canonical dyadic totient-kernel family, which is linearly independent with exact span dimension 2^e + 1. Consequently the span of the full dyadic totient kernel is not finite-dimensional over Q. Even residues reduce to odd-core channels, and the compressed-adjoint boundary remains available. These coefficient-side theorems do not prove that S is irrational.","remaining_open_proposition_ids":["remaining_open.erdos_249_irrationality"],"paper_label":"res:dyadictotient","readme_headline":false,"declarations":[{"name":"exists_separatedMinorCertificate_totientAffineOddFamily","module":"Erdos249257/TotientMahlerDefect.lean","line":847},{"name":"linearIndependent_canonicalTotientKernelFamily","module":"Erdos249257/TotientMahlerDefect.lean","line":900},{"name":"finrank_canonicalTotientKernel_eq","module":"Erdos249257/TotientMahlerDefect.lean","line":954},{"name":"not_finiteDimensional_span_fullTotientKernel","module":"Erdos249257/TotientMahlerDefect.lean","line":1014},{"name":"totientKernel_even_residue_reduce","module":"Erdos249257/TotientMahlerDefect.lean","line":159},{"name":"false_of_compressedAdjointCertificate","module":"Erdos249257/TotientMahlerDefect.lean","line":1051}]},{"id":"residual_gauge_obstruction","label":"Residual-gauge obstruction for monomial minors","status":"proved here","statement":"For a residual-weighted monomial matrix, common nonzero residual weights act by a right diagonal gauge and preserve determinant nonvanishing. Under the locked inverse-phase gauge, an exponent-one row is identically 1 even when the determinant remains nonzero; allowing row-dependent residuals reconstructs a consecutive-power matrix with zeroth row identically 1. Thus residual-blind rank, determinant, and conditioning criteria alone cannot exclude the target. This does not rule out a criterion with an additional arithmetic coupling identity, and it gives no irrationality criterion.","paper_label":"res:residualgauge","readme_headline":false,"declarations":[{"name":"det_residualMonomialMatrix_ne_zero_iff","module":"Erdos249257/ResidualGaugeObstruction.lean","line":55},{"name":"locked_reconstruction_preserves_nonzero_minor","module":"Erdos249257/ResidualGaugeObstruction.lean","line":94},{"name":"rowReconstruction_zero_row_one","module":"Erdos249257/ResidualGaugeObstruction.lean","line":145}]},{"id":"tail_orbit_rigidity","label":"Tempered tail-orbit rigidity criterion","status":"proved here","statement":"For coefficients c(n) \u2264 n, the binary series \u2211 c(n)/2\u207f is rational exactly when a tempered integer carry orbit exists, and every tempered orbit equals the scaled tail.","paper_label":"res:rigidity","readme_headline":false,"declarations":[{"name":"binaryCoeffSeries_rational_iff_exists_temperedBinaryOrbit","module":"Erdos249257/GenericTailOrbitRigidity.lean","line":426}]},{"id":"boolean_mobius_carry","label":"Boolean\u2013M\u00f6bius carry certificate equivalence","status":"proved here","statement":"A nonempty support with series value p/q exists exactly when a positive, square-root-bounded integer orbit with Boolean M\u00f6bius quotient exists; the support is reconstructed from the certificate.","paper_label":"res:carrycert","readme_headline":false,"declarations":[{"name":"exists_normalized_support_fraction_iff_exists_booleanMobiusCarry","module":"Erdos249257/BooleanMobiusCarry.lean","line":948}]},{"id":"reciprocal_mass_bound","label":"Rationality-forced reciprocal-mass lower bound","status":"proved here","statement":"A rational support-series value with odd denominator part v forces reciprocal mass at least 1/ord_v(2), or divergence; an infinite support with dyadic-rational value has reciprocal mass divergent or greater than one.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:orderwrap","readme_headline":false,"declarations":[{"name":"one_div_oddOrder_le_reciprocalMass_of_support_fraction","module":"Erdos249257/RationalSupportCarrySkeleton.lean","line":1478}]},{"id":"unbounded_carry_states","label":"Carry states are unbounded over infinite rational-valued supports","status":"proved here","statement":"An infinite support with rational series value has an unbounded positive carry state, closing finite-state readings of the carry system.","paper_label":"res:unbounded","readme_headline":false,"declarations":[{"name":"exists_unbounded_shifted_odd_tail_nat_state_of_support_fraction","module":"Erdos249257/RationalSupportCarrySkeleton.lean","line":2377}]},{"id":"sublog_zero_windows","label":"Sublogarithmic divisor-coverage zero windows","status":"proved here","statement":"A rational support-series value forces every zero window in the divisor-count coefficient to have length at most \u03b5\u00b7log\u2082(N+1) + O_{\u03b5,c,v}(1), uniformly in the support. This constrains coverage; it does not solve universal #257.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:sublog","readme_headline":true,"declarations":[{"name":"supportCoeffZeroWindow_length_le_eps_logb_add","module":"Erdos249257/SublogDivisorCoverage.lean","line":392}]},{"id":"greedy_achievement_geometry","label":"Greedy geometry of the Mersenne achievement set","status":"formalised here","statement":"The Mersenne achievement set is compact, perfect, totally disconnected, nowhere dense, and has Lebesgue measure one. Membership is equivalent to greedy survival at every level; for the target 1/2 it is exactly equivalent to the canonical greedy orbit omitting infinitely many exponents. No finite normalized support has value 1/2, so positive membership would automatically produce an infinite rational-valued support and refute universal #257. Finite exact-rational death certificates soundly prove non-membership (3/4 is excluded at level one). The strict-tail Cantor geometry is recorded by Kova\u010d\u2013Tao; no novelty claim is made for the formalised refinements.","remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"],"paper_label":"res:greedy","readme_headline":true,"declarations":[{"name":"volume_mersenneAchievementSet","module":"Erdos249257/GreedyAchievementSet.lean","line":996},{"name":"perfect_mersenneAchievementSet","module":"Erdos249257/GreedyAchievementSet.lean","line":1620},{"name":"isTotallyDisconnected_mersenneAchievementSet","module":"Erdos249257/GreedyAchievementSet.lean","line":1636},{"name":"isNowhereDense_mersenneAchievementSet","module":"Erdos249257/GreedyAchievementSet.lean","line":1645},{"name":"mem_mersenneAchievementSet_iff_greedy_survival","module":"Erdos249257/GreedyAchievementSet.lean","line":1445},{"name":"three_fourths_not_mem_mersenneAchievementSet","module":"Erdos249257/GreedyAchievementSet.lean","line":1748},{"name":"half_mem_mersenneAchievementSet_iff_greedySkippedSupport_infinite","module":"Erdos249257/GreedyAchievementSet.lean","line":2514},{"name":"finite_boolSupport_ne_half","module":"Erdos249257/HalfCarryReachability.lean","line":589}]},{"id":"half_greedy_two_thirds_band","label":"Exact band localization of the final skip between two takes","status":"proved here","statement":"Suppose a take at rank b is followed by the next take at c > b+1, put q = 2^b \u2212 1 and m = 2^(c\u22121), and write the pre-take residual as 1/R with 0 < R < q. The final intervening skip is dyadically unsafe exactly when q(m\u22121)/(q+m\u22121) < R < qm/(q+m), an interval of exact width q^2/((q+m)(q+m\u22121)). For one intervening skip this specializes to the two-thirds band (q(2q+1)/(3q+1), 2q(q+1)/(3q+2)), whose width is below 1/9; integral reciprocals never lie there, and the corresponding positive odd numerator is at least 7. These are local constraints: no theorem here says that the actual greedy orbit for 1/2 avoids a band or identifies the surviving last-producer cells with unsafe skips.","remaining_open_proposition_ids":["remaining_open.half_value_membership"],"paper_label":"res:twothirdsband","readme_headline":false,"declarations":[{"name":"postTakeUnsafeAt_iff_band","module":"Erdos249257/HalfGreedyTwoThirdsBand.lean","line":88},{"name":"band_width_general","module":"Erdos249257/HalfGreedyTwoThirdsBand.lean","line":104},{"name":"singletonUnsafe_iff_twoThirdsBand","module":"Erdos249257/HalfGreedyTwoThirdsBand.lean","line":127},{"name":"band_width","module":"Erdos249257/HalfGreedyTwoThirdsBand.lean","line":154},{"name":"band_width_lt_ninth","module":"Erdos249257/HalfGreedyTwoThirdsBand.lean","line":163},{"name":"not_twoThirdsBand_of_int","module":"Erdos249257/HalfGreedyTwoThirdsBand.lean","line":185},{"name":"seven_le_of_intBand_odd","module":"Erdos249257/HalfGreedyTwoThirdsBand.lean","line":231}]},{"id":"half_second_channel_reduction","label":"Exact rational second-channel reduction for the target one half","status":"conditional reduction","statement":"A decidable rational separation predicate controls the greedy Mersenne orbit for the target 1/2. Lean verifies levels 1 through 6 exactly; proving the predicate for every n >= 7 would place 1/2 in the Mersenne achievement set, equivalently would force infinitely many canonical greedy skips. The universal tail statement remains open.","remaining_open_proposition_ids":["remaining_open.half_value_membership"],"paper_label":"res:secondcriterion","readme_headline":false,"declarations":[{"name":"HalfSecondChannelSeparatedRat","module":"Erdos249257/GreedyAchievementSet.lean","line":2695},{"name":"halfSecondChannelSeparatedRat_of_pos_le_six","module":"Erdos249257/GreedyAchievementSet.lean","line":2723},{"name":"half_mem_mersenneAchievementSet_of_secondChannelSeparationRat_from_seven","module":"Erdos249257/GreedyAchievementSet.lean","line":2901},{"name":"half_mem_mersenneAchievementSet_of_skipped_twoChannelCap","module":"Erdos249257/GreedyAchievementSet.lean","line":1412},{"name":"half_mem_mersenneAchievementSet_iff_greedySkippedSupport_infinite","module":"Erdos249257/GreedyAchievementSet.lean","line":2514}]},{"id":"campbell_shift_synchronization_boundary","label":"Campbell shift-synchronization strategy boundary","status":"unconditional progress","statement":"The quarter-exponent prime-progression parameters cannot be synchronized with the last-skip fatal window under the stated period-freezing bounds. Moreover, the proposed shifted-zero interface is exactly equivalent to infinitely many actual greedy skips and to membership of 1/2 in the Mersenne achievement set. This closes a synchronization strategy and an exact re-expression; it does not provide an independent producer or prove half-membership.","paper_label":"res:campbellboundary","readme_headline":false,"declarations":[{"name":"campbellQuarterExponent_periodFreeze_noSync","module":"Erdos249257/CampbellShiftSynchronization.lean","line":299},{"name":"campbellPhaseFreeze_parameters_impossible","module":"Erdos249257/CampbellShiftSynchronization.lean","line":320},{"name":"shiftWindowZero_iff_greedyMersenneSkippedSupport_infinite","module":"Erdos249257/CampbellShiftSynchronization.lean","line":471},{"name":"shiftWindowZero_iff_half_mem_mersenneAchievementSet","module":"Erdos249257/CampbellShiftSynchronization.lean","line":488}]},{"id":"sigma_transcendence","label":"Transcendence of \u2211 \u03c3(m)/2^m","status":"cited only","statement":"Nesterenko (1996); the ladder rung L(Id). Cited for positioning, not formalised here.","declarations":[]},{"id":"prime_support_irrationality","label":"Irrationality of the prime-support series \u2211_p 1/(2^p \u2212 1)","status":"cited only","statement":"Tao\u2013Ter\u00e4v\u00e4inen; settles the prime-support special case of #257. Only the identity bridge to \u2211 \u03c9(n)/2\u207f is formalised, not the irrationality.","declarations":[]},{"id":"transport_curvature_reductions","label":"Transport and curvature reductions for Erd\u0151s #249","status":"conditional reduction","statement":"Curvature, fixed-three exponent transport, anchored (3,5) transport, prime-jump, and balanced three-transport observables each have exact finite non-integrality consumers whose unbounded supply would imply irrationality of the totient constant. Every supply hypothesis remains open.","remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"],"readme_headline":false,"declarations":[{"name":"irrational_totientSeries_of_sharpCurvatureSupply","module":"Erdos249257/CurvatureCarry.lean","line":206},{"name":"irrational_totient_series_of_exponentOnlyThreeTransportSupply","module":"Erdos249257/ExponentOnlyTransport.lean","line":205},{"name":"irrational_totient_series_of_anchoredJoint35ConeSupply","module":"Erdos249257/JointExponentTransport.lean","line":217},{"name":"irrational_totient_series_of_primeJumpSharpKill_supply","module":"Erdos249257/PrimeJumpWindow.lean","line":193},{"name":"irrational_totientSeries_of_roughnessBoundaryLatticeSupply","module":"Erdos249257/ThreeTransportBoundary.lean","line":177}]},{"id":"affine_transport_annihilation","label":"Affine residue state and old-channel annihilation","status":"proved here","statement":"The exact M\u00f6bius residue kernel has a periodic affine slope-and-intercept state. First-order exponent transport and every coefficient system with vanishing zeroth and first moments annihilate all divisor channels already present at the base height; a prime jump additionally has an explicit migrated-channel correction.","readme_headline":false,"declarations":[{"name":"transportResidueKernel_eq_affineState","module":"Erdos249257/ExponentOnlyTransport.lean","line":141},{"name":"oldChannel_affine_moment_annihilation","module":"Erdos249257/JointExponentTransport.lean","line":36},{"name":"residueHeckeDefect_eq_postJump_add_migration","module":"Erdos249257/PrimeJumpMigration.lean","line":38}]},{"id":"transport_certificate_instances","label":"Verified transport-window certificates","status":"verified finite instance","bounded_domain":"Exactly the prime-jump parameter triple (H,p,L)=(12,5,15) and the balanced three-transport pair (H,L)=(60,12).","statement":"Lean verifies the sharp prime-jump certificate at (H,p,L)=(12,5,15) and the balanced three-transport certificate at (H,L)=(60,12). These bounded instances do not establish any unbounded supply.","readme_headline":false,"declarations":[{"name":"primeJumpSharpKill_twelve_five","module":"Erdos249257/PrimeJumpWindow.lean","line":186},{"name":"sharpThreeTransportCert_60_12","module":"Erdos249257/ThreeTransportBoundary.lean","line":203}]},{"id":"first_harmonic_certificate_interface","label":"First-harmonic gap implies a finite certificate","status":"conditional reduction","statement":"Under the explicit dyadic room inequality, a constant-saving first-harmonic bound on one block, or on any nonempty finite set of indices below its right endpoint, forces a certified non-integrality witness in that set. No cofinal harmonic estimate is asserted.","remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"],"readme_headline":false,"declarations":[{"name":"exists_certifiedKill_of_first_harmonic_gap_subset","module":"Erdos249257/FirstHarmonicGap.lean","line":104},{"name":"exists_certifiedKill_of_first_harmonic_gap","module":"Erdos249257/FirstHarmonicGap.lean","line":128}]},{"id":"first_harmonic_pivot_decomposition","label":"Complex first-harmonic pivot decomposition","status":"conditional reduction","statement":"Removing the pivot phase gives an exact decomposition of the block first-harmonic sum into a centred supplier-fibre correlation, a supplier-fibre mean term, a bad-supplier term, and a non-supplier term. At a positive scale satisfying the explicit dyadic-room inequality, if the real part of the centred term is at most 14X/25 and the other three norms are at most X/100, X/100, and 8X/25, then the real first-harmonic sum is at most 9X/10 and yields a finite certificate. A cofinal supply of these budgets implies irrationality of S. No prime-distribution or decorrelation estimate supplying the budgets is proved.","remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"],"readme_headline":false,"declarations":[{"name":"windowFirstExp_sum_eq_pivot_decomposition","module":"Erdos249257/FirstHarmonicPivot.lean","line":342},{"name":"PivotBudgetAt","module":"Erdos249257/FirstHarmonicPivot.lean","line":371},{"name":"first_harmonic_gap_of_pivotBudgetAt","module":"Erdos249257/FirstHarmonicPivot.lean","line":377},{"name":"DTWPivotResidualDecorrelation","module":"Erdos249257/FirstHarmonicPivot.lean","line":397},{"name":"irrational_totient_series_of_pivotResidualDecorrelation","module":"Erdos249257/FirstHarmonicPivot.lean","line":405}]},{"id":"adjacent_phase_separation_reduction","label":"Adjacent-pair phase separation reduction","status":"conditional reduction","statement":"At scales satisfying the explicit dyadic room inequality, a squared chordal separation of at least 19/25 between two adjacent first-harmonic phases in a dyadic block forces a certified non-integrality witness at one of the two indices. A cofinal supply of such adjacent pairs implies irrationality of the totient constant; no such supply is proved.","remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"],"readme_headline":false,"declarations":[{"name":"exists_certifiedKill_of_phasePairSeparation","module":"Erdos249257/AdjacentPhaseSeparation.lean","line":258},{"name":"irrational_totient_series_of_adjacentPhaseSeparation","module":"Erdos249257/AdjacentPhaseSeparation.lean","line":295}]},{"id":"pivot_anti_reconstruction_reduction","label":"Pivot anti-reconstruction and window-pair reduction","status":"conditional reduction","statement":"On a finite supplier fibre, sufficient counted phase-pair separation feeds a certified non-integrality witness through exact reconstruction-defect and variance identities. Cofinal pivot anti-reconstruction, reconstruction variance, or counted-pair producers imply irrationality of S. The fibre-free cofinal window-pair predicate is exactly equivalent to irrationality of S. No cofinal arithmetic producer is proved.","remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"],"readme_headline":false,"declarations":[{"name":"exists_certifiedKill_of_pivotFiberSeparatedPairs","module":"Erdos249257/PivotAntiReconstruction.lean","line":1105},{"name":"irrational_totient_series_of_pivotAntiReconstruction","module":"Erdos249257/PivotAntiReconstruction.lean","line":1502},{"name":"dtwWindowSeparatedPairs_iff_irrational_totient_series","module":"Erdos249257/PivotAntiReconstruction.lean","line":1744}]},{"id":"fixed_precision_transport_no_go","label":"Fixed-precision valuation-unit no-go theorem","status":"unconditional progress","statement":"At every fixed positive precision, each finite compatible valuation-unit word has a prefix-locked centred completion. 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The construction does not assert that its forcing letters are actual totient differences and does not rule out nonlinear use of fresh divisor channels.","readme_headline":false,"declarations":[{"name":"lcm_factorIdeal_finiteRank_shiftAlgebra_not_sufficient","module":"Erdos249257/LcmFactorIdealPulseObstruction.lean","line":798},{"name":"lcm_factorIdeal_sparseAnchor_not_sufficient","module":"Erdos249257/LcmFactorIdealPulseObstruction.lean","line":866},{"name":"lcmAnchorPulse_t3_letters","module":"Erdos249257/LcmFactorIdealPulseObstruction.lean","line":895}]},{"id":"erdos_249","label":"Erd\u0151s #249: irrationality of S = \u2211 \u03c6(n)/2\u207f","status":"open","statement":"Not proved by this release. The reductions convert it into the unboundedness of the certificate supply, which is untouched.","remaining_open_proposition_ids":["remaining_open.erdos_249_irrationality","remaining_open.unbounded_certificate_supply"],"declarations":[]},{"id":"universal_257","label":"Erd\u0151s #257, universal form","status":"open","statement":"Irrationality of \u2211_{n\u2208A} 1/(2\u207f \u2212 1) for every infinite A is not proved by this release; only named families are formalised.","remaining_open_proposition_ids":["remaining_open.half_value_membership","remaining_open.universal_257_all_infinite_supports"],"declarations":[]}],"non_claims":[{"id":"not_erdos_257_solution","meaning":"This release does not settle the universal Erd\u0151s #257 problem"},{"id":"not_erdos_249_solution","meaning":"This release does not settle Erd\u0151s #249"},{"id":"not_publication_authority","meaning":"This release does not itself constitute a peer-reviewed publication record"},{"id":"not_private_root_equivalence","meaning":"This release is not asserted to be equivalent to any private development"},{"id":"not_provider_proof_authority","meaning":"Provider or model output is not proof authority; only the Lean kernel check is"},{"id":"not_hidden_proof_body_authority","meaning":"No unpublished proof sketch or hidden artefact carries proof authority for this release"}]},"body_format":"json_parsed","byte_parity_note":"body is the parsed object; re-serialisation differs from source bytes. 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A sufficient one-parameter form is to produce them along the lcm diagonal.","paper_anchor":{"source":"paper/erdos249-257-main-paper.tex","environment":"problem","title":"unbounded certificate supply","line":3450}},{"id":"remaining_open.half_value_membership","open_target_claim":"universal_257","statement":"Decide whether 1/2 lies in the Mersenne achievement set. Yes refutes universal #257; no closes only this route.","paper_anchor":{"source":"paper/erdos249-257-main-paper.tex","environment":"problem","title":"the half-value counterexample route","line":3471}},{"id":"remaining_open.universal_257_all_infinite_supports","open_target_claim":"universal_257","statement":"Prove irrationality for every infinite support A, not only the named support families formalised here.","paper_anchor":{"source":"paper/erdos249-257-main-paper.tex","environment":"problem","title":"universal Erd\\H{o}s \\#257","line":3462}}],"non_claims":[{"id":"not_erdos_257_solution","meaning":"This release does not settle the universal Erd\u0151s #257 problem"},{"id":"not_erdos_249_solution","meaning":"This release does not settle Erd\u0151s #249"},{"id":"not_publication_authority","meaning":"This release does not itself constitute a peer-reviewed publication record"},{"id":"not_private_root_equivalence","meaning":"This release is not asserted to be equivalent to any private development"},{"id":"not_provider_proof_authority","meaning":"Provider or model output is not proof authority; only the Lean kernel check is"},{"id":"not_hidden_proof_body_authority","meaning":"No unpublished proof sketch or hidden artefact carries proof authority for this release"}],"principal_claims":[{"id":"eb_full_support","status":"formalised here","statement":"For every integer b \u2265 2, the series \u2211_{n\u22651} 1/(b\u207f \u2212 1) is irrational (Erd\u0151s, 1948).","paper_label":"res:full","declarations":[{"name":"irrational_erdosSum_full_support","module":"Erdos249257/CertificateKernel.lean","line":8000}]},{"id":"eb_constant","status":"formalised here","statement":"E = \u2211_{n\u22651} 1/(2\u207f \u2212 1) is irrational, as the base-2 corollary of the full-support theorem.","paper_label":"res:full","declarations":[{"name":"irrational_erdosBorwein_series","module":"Erdos249257/CertificateKernel.lean","line":8007}]},{"id":"support_families","status":"formalised here","statement":"For every base b \u2265 2, \u2211_{n\u2208A} 1/(b\u207f \u2212 1) is irrational for factorial support, power-of-two support, multiples, pairwise-coprime supports with summable reciprocals, eventually-periodic supports, residue classes, and the odd numbers. Proved families, not the universal statement.","paper_label":"res:support","declarations":[{"name":"erdos257_family_factorial_instance","module":"Erdos249257/CertificateKernel.lean","line":5754},{"name":"erdos257_family_two_pow_instance","module":"Erdos249257/CertificateKernel.lean","line":5762},{"name":"irrational_erdosSupportSeries_pairwise_coprime","module":"Erdos249257/CertificateKernel.lean","line":10448}],"remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"]},{"id":"denominator_exclusion","status":"unconditional progress","statement":"S = \u2211 \u03c6(n)/2\u207f is not equal to p/q for any integer p and any 1 \u2264 q \u2264 79 639 646 646 701 375 323 355 774 875 831 053 (\u2248 7.96 \u00d7 10\u00b3\u2074). A lower bound on any rational representation, not a proof of irrationality.","paper_label":"res:farey","declarations":[{"name":"tsum_totient_div_pow_two_ne_ratCast_of_den_le_79639646646701375323355774875831053","module":"Erdos249257/CertificateKernel.lean","line":18056}],"remaining_open_proposition_ids":["remaining_open.erdos_249_irrationality"]},{"id":"certificate_reduction","status":"conditional reduction","statement":"S is irrational if and only if certified non-integrality witnesses occur at unbounded parameters. The pointwise completeness theorem and the diagonal supply theorem together show that an unbounded lcm-diagonal supply is equivalent as well. Producing that supply remains open.","paper_label":"res:diag","declarations":[{"name":"irrational_totient_series_of_certificate_supply","module":"Erdos249257/TotientTailPeriodKiller.lean","line":394},{"name":"irrational_totient_series_of_lcm_diagonal_certificate_supply","module":"Erdos249257/LcmDiagonalReduction.lean","line":112},{"name":"rational_totient_series_forces_lcm_cone_flatness","module":"Erdos249257/LcmConeFlatness.lean","line":90},{"name":"irrational_totient_series_iff_pointwise_certificates","module":"Erdos249257/LcmConeFlatness.lean","line":399},{"name":"irrational_totient_series_iff_certificate_supply","module":"Erdos249257/LcmConeFlatness.lean","line":412},{"name":"irrational_totient_series_iff_lcm_diagonal_certificate_supply","module":"Erdos249257/LcmConeFlatness.lean","line":426}],"remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"]},{"id":"certificate_completeness","status":"proved here","statement":"At fixed parameters, a certified kill exists exactly when the corresponding tail difference is non-integral. At every positive shift, integrality of one tail difference forces S to be rational; thus irrationality is equivalent to pointwise non-integrality, and to pointwise finite certificates.","paper_label":"res:pointwisecomplete","declarations":[{"name":"exists_certifiedKill_iff_tail_diff_notMem_int","module":"Erdos249257/LcmConeFlatness.lean","line":316},{"name":"tail_diff_mem_int_iff_scaled_series_mem_int","module":"Erdos249257/LcmConeFlatness.lean","line":327},{"name":"irrational_totient_series_iff_all_tail_diffs_nonintegral","module":"Erdos249257/LcmConeFlatness.lean","line":386},{"name":"irrational_totient_series_iff_pointwise_certificates","module":"Erdos249257/LcmConeFlatness.lean","line":399}]},{"id":"certified_kill_instances","status":"verified finite instance","statement":"Kernel-checked certified kills for small periods and at 28 explicit lcm-diagonal scales through t = 64, plus finite certificate shards, each discharged by decide.","paper_label":"res:deposits","declarations":[{"name":"certifiedKill_all_small","module":"Erdos249257/TotientTailPeriodKiller.lean","line":404},{"name":"certifiedKill_periodLcm_diagonal_upto_six","module":"Erdos249257/LcmDiagonalReduction.lean","line":237},{"name":"certifiedKill_periodLcm_diagonal_at_seven","module":"Erdos249257/LcmDiagonalReduction.lean","line":242},{"name":"certifiedKill_periodLcm_diagonal_at_eight","module":"Erdos249257/LcmDiagonalReduction.lean","line":246},{"name":"certifiedKill_diagonal_all_imported_through_t64","module":"Erdos249257/DiagonalPincerCertificatesT64.lean","line":1967}],"remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"]},{"id":"diagonal_pincer_reduction","status":"conditional reduction","statement":"At each lcm-diagonal scale, integrality is equivalent to a foreign-defect term landing in an explicit full-target interval. Globally, S is irrational if and only if the full target is avoided at arbitrarily large diagonal scales; that cofinal avoidance is not proved.","paper_label":"res:fulltargetavoidance","declarations":[{"name":"diagonal_int_iff_foreignDiagonalDefect_hits_fullTarget","module":"Erdos249257/DiagonalPincerDecomposition.lean","line":215},{"name":"irrational_totientSeries_of_full_target_avoidance_supply","module":"Erdos249257/DiagonalPincerDecomposition.lean","line":290}],"remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"]},{"id":"adelic_height_obstruction","status":"proved here","statement":"If H divides the reduced denominator of a rational x and the reduced denominator of c\u00b7x divides H for an integer c, then the complementary factor x.den/H divides |c|. Scalar localisation transfers, rather than discards, that denominator information; this local obstruction is not an irrationality criterion.","paper_label":"res:adelic","declarations":[{"name":"scalarLocalization_complement_dvd","module":"Erdos249257/AdelicHeightObstruction.lean","line":23},{"name":"scalarLocalization_integer_eq_mul_num","module":"Erdos249257/AdelicHeightObstruction.lean","line":56}]},{"id":"sublog_zero_windows","status":"proved here","statement":"A rational support-series value forces every zero window in the divisor-count coefficient to have length at most \u03b5\u00b7log\u2082(N+1) + O_{\u03b5,c,v}(1), uniformly in the support. This constrains coverage; it does not solve universal #257.","paper_label":"res:sublog","declarations":[{"name":"supportCoeffZeroWindow_length_le_eps_logb_add","module":"Erdos249257/SublogDivisorCoverage.lean","line":392}],"remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"]},{"id":"greedy_achievement_geometry","status":"formalised here","statement":"The Mersenne achievement set is compact, perfect, totally disconnected, nowhere dense, and has Lebesgue measure one. Membership is equivalent to greedy survival at every level; for the target 1/2 it is exactly equivalent to the canonical greedy orbit omitting infinitely many exponents. No finite normalized support has value 1/2, so positive membership would automatically produce an infinite rational-valued support and refute universal #257. Finite exact-rational death certificates soundly prove non-membership (3/4 is excluded at level one). The strict-tail Cantor geometry is recorded by Kova\u010d\u2013Tao; no novelty claim is made for the formalised refinements.","paper_label":"res:greedy","declarations":[{"name":"volume_mersenneAchievementSet","module":"Erdos249257/GreedyAchievementSet.lean","line":996},{"name":"perfect_mersenneAchievementSet","module":"Erdos249257/GreedyAchievementSet.lean","line":1620},{"name":"isTotallyDisconnected_mersenneAchievementSet","module":"Erdos249257/GreedyAchievementSet.lean","line":1636},{"name":"isNowhereDense_mersenneAchievementSet","module":"Erdos249257/GreedyAchievementSet.lean","line":1645},{"name":"mem_mersenneAchievementSet_iff_greedy_survival","module":"Erdos249257/GreedyAchievementSet.lean","line":1445},{"name":"three_fourths_not_mem_mersenneAchievementSet","module":"Erdos249257/GreedyAchievementSet.lean","line":1748},{"name":"half_mem_mersenneAchievementSet_iff_greedySkippedSupport_infinite","module":"Erdos249257/GreedyAchievementSet.lean","line":2514},{"name":"finite_boolSupport_ne_half","module":"Erdos249257/HalfCarryReachability.lean","line":589}],"remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"]}],"mathematical_programmes":[{"id":"erdos257_half_story","title":"Achievement-set 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avoidance on the lcm diagonal, fresh-prime endpoint defects, binary suffix arithmetic, affine escape sockets, and finite residue projections.","claim_ceiling":"Each interface sharpens or re-expresses the same supply problem; no unbounded avoidance, slack, or separated-projection supply is proved.","core_claim_count":7,"representative_claim_ids":["diagonal_pincer_reduction","diagonal_fresh_loss_reduction"],"remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"]},{"id":"boolean_mobius_constraints","title":"Boolean M\u00f6bius carry and global rationality constraints","mathematical_focus":"Tempered binary carry orbits, Boolean M\u00f6bius reconstruction, reciprocal mass, unbounded carry states, and sublogarithmic divisor-coverage windows.","claim_ceiling":"These are exact characterisations and necessary consequences of rationality; they do not contradict every hypothetical rational value or settle universal Erd\u0151s #257.","core_claim_count":5,"representative_claim_ids":["tail_orbit_rigidity","boolean_mobius_carry"],"remaining_open_proposition_ids":["remaining_open.universal_257_all_infinite_supports"]},{"id":"transport_curvature_programme","title":"Transport, curvature, phase separation, and no-go results","mathematical_focus":"Finite differences of totient tails, exponent transport, affine moment annihilation, the exact first-harmonic pivot budget, harmonic phases, reconstruction defects, and countermodels to bounded local signatures.","claim_ceiling":"The consumers and countermodels are exact, but no cofinal curvature, transport, phase-separation, or anti-reconstruction producer is proved.","core_claim_count":8,"representative_claim_ids":["transport_curvature_reductions","affine_transport_annihilation"],"remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"]},{"id":"lambert_obstruction_interfaces","title":"Mersenne\u2013Lambert identities and obstruction interfaces","mathematical_focus":"Dirichlet convolution identities, the M\u00f6bius-square lens, primitive-coordinate finite jets, exact tail atoms, sign-support no-go results, and conditional carry-rank growth.","claim_ceiling":"These are exact identities, obstruction theorems, and one conditional rank consequence; they do not form an irrationality criterion for S.","core_claim_count":6,"representative_claim_ids":["mersenne_lambert_ladder","primitive_coordinate_index_obstruction"],"remaining_open_proposition_ids":["remaining_open.unbounded_certificate_supply"]},{"id":"probabilistic_gcd_geometry","title":"Coprimality probability, gcd moments, and Stern\u2013Brocot geometry","mathematical_focus":"Fair-coin gcd distributions, squared-Mersenne Lambert transforms, Pillai's gcd-sum function, Stern\u2013Brocot cylinder measures, and Fibonacci/continuant run stability.","claim_ceiling":"Exact identities, denominator transfer, convergence, and combinatorial stability only; no irrationality or novelty claim.","core_claim_count":4,"representative_claim_ids":["totient_coprimality_probability","squared_lambert_gcd_moments"],"remaining_open_proposition_ids":["remaining_open.erdos_249_irrationality"]},{"id":"half_carry_compactness_programme","title":"Half-carry compactness, windows, seams, and finite propagation","mathematical_focus":"Exact dyadic prefix coordinates, compactness from finite half-carry words, protected seams, inverse-parent contraction, suffix-cylinder propagation, and checked stages through depth 52.","claim_ceiling":"The closure implications, local propagation laws, and bounded stages are exact; no cofinal or coherent unbounded window family is proved, so neither half-membership nor universal Erd\u0151s #257 follows.","core_claim_count":13,"representative_claim_ids":["half_carry_reachability_compactness","cofinal_strip_return_closure"],"remaining_open_proposition_ids":["remaining_open.half_value_membership","remaining_open.universal_257_all_infinite_supports"]},{"id":"arithmetic_obstruction_interfaces","title":"Denominator, CRT, moment, and gauge obstructions","mathematical_focus":"Squared-Mersenne tail enclosures, exact reduced denominators, adelic divisibility, square-CRT correction removal, dyadic determinant substrate, certificate dimensions, and residual-gauge no-go results.","claim_ceiling":"These are exact arithmetic interfaces, finite mechanisms, and obstruction theorems; they do not supply an unbounded certificate family or an irrationality criterion for the totient 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Both are assumed by the final theorem and neither is derived, so that theorem constrains the state system rather than the original sequence.","Any exclusion of negative excursions whose magnitudes are unbounded along a cofinal set.","The criterion of Erdos and Straus and Duverney's conditional characterisation, which are cited rather than formalised and are both stronger than anything checked here."],"open_obligations":[{"id":"unbounded_negative_excursions","statement":"Exclude, or construct, an exact orbit whose centred state is negative infinitely often with magnitudes unbounded along that cofinal set. Every checked exclusion consumes some finiteness: a fixed set of prime divisors, a fixed period, or a fixed bound on the negative part."},{"id":"derive_the_two_analytic_hypotheses","statement":"Derive eventual strict centring and normalised vanishing from the growth hypothesis and rationality, or show that they do not follow."},{"id":"formalise_the_published_criteria","statement":"Kernel-check the Erdos-Straus weighted criterion and Duverney's conditional characterisation under explicit analytic hypotheses."}],"finite_search":"An exact search over initial states below five thousand produced forced prefixes of length seventeen and no longer. It is superseded: the constant-negative template is now excluded outright, for every seed and at every scale."},{"problem_id":"erdos_249","erdos_number":249,"short_title":"The binary totient series","status":"open","question":"Is the binary Lambert series sum phi(n)/2^n irrational?","library_root":"ErdosProblems.lean","claim_registry_status":"not_registered; the claim registry does not carry these declarations and kernel checking them does not promote them into reviewed public claims","directory":"ErdosProblems/Erdos249","modules":[{"module":"ErdosProblems.Erdos249.TotientStrictPrimeEscape","path":"ErdosProblems/Erdos249/TotientStrictPrimeEscape.lean","present":true,"line_count":72,"declaration_count":3,"theorem_count":2,"content_digest":"sha256:9361d904a02a52e115aebefcea9b5a329f079b6f3f417188ac74945484876fa7"},{"module":"ErdosProblems.Erdos249.FiniteEulerSieve","path":"ErdosProblems/Erdos249/FiniteEulerSieve.lean","present":true,"line_count":69,"declaration_count":6,"theorem_count":4,"content_digest":"sha256:a0172184d831cb4d6004781027464922460d762c4a93b17b04fefc950dd62e93"},{"module":"ErdosProblems.Erdos249.PrimeRayCyclotomicCurvature","path":"ErdosProblems/Erdos249/PrimeRayCyclotomicCurvature.lean","present":true,"line_count":184,"declaration_count":12,"theorem_count":7,"content_digest":"sha256:afa772249fb970ccb167cae7f4b73130f3d66f454f466ffd6e305d27f926418c"}],"note":{"artifact_id":"erdos_249_note","title":"Exact Finite-Level Ranks in the 2-Kernel of Euler's Totient","source_path":"paper/erdos-249-binary-totient-series.tex","rendered_path":"erdos-249-binary-totient-series.pdf","authority_posture":"authored_problem_owned_synthesis_of_reviewed_gateway_claims_and_unregistered_expansion_modules_not_Lean_proof_authority"},"what_is_checked":["Conditionally: a cofinal strict 9/10 gap for the exact natural prime-tail orbit supplies the existing finite pivot-point escape and hence the reviewed irrationality endpoint.","The local s=1 and s=2 squared M\u00f6bius Euler-factor identities and the first two prime-power divisor-sum differences.","That the two-by-two checkerboard kills separable backgrounds and is unique up to scale under the four row and column cancellations.","An abstract four-point divisor-layer identity, stated from its four explicit factorisation hypotheses.","An affine completion no-go: every residue class modulo an even modulus has a digit that recentres one carry step.","Exact producer and consumer interfaces for prime-ray layer supply, bounded-degree order witnesses, and finite-prime-support escape."],"what_is_not_checked":["The strict prime-tail orbit gap, which is the unproved producer consumed by the conditional endpoint.","The packet's sharp parabolic Stern-Brocot cusp theorem and its passage from a geometric x-parameter to the arithmetic totient series.","Eventual Hankel nonvanishing for the M\u00f6bius-Mersenne power ladder, or its denominator obstruction.","The natural-boundary and quasimodular-transcendence steps of the infinite Euler-sieve tower.","Polynomial resultant realisability, exact-order finite-field supply, Frobenius freezing, or the Archimedean growth needed by the prime-ray strategy.","Unconditional irrationality of the binary totient series."],"open_obligations":[{"id":"strict_prime_tail_orbit_gap","statement":"Prove the cofinal strict 9/10 real-part gap for the exact natural prime-tail orbit, or replace it with another producer for the reviewed pivot-escape endpoint."},{"id":"hankel_denominator_fan_in","statement":"Prove eventual nonvanishing or a usable denominator obstruction for the M\u00f6bius-Mersenne power ladder and connect it to the target value."},{"id":"euler_sieve_limit_theorem","statement":"Control the infinite Euler-sieve limit strongly enough that finite-stage quasimodularity yields a valid arithmetic obstruction."},{"id":"prime_ray_resultant_supply","statement":"Realise the layer system by cyclic resultants and prove prime-support escape with the growth estimates needed to contradict a rational carry orbit."},{"id":"stern_brocot_arithmetic_bridge","statement":"Prove the sharp cusp asymptotic and then supply a non-formal bridge from the geometric parameter regime to the exact totient value."}],"finite_search":"The returned strict-orbit packet reports an exact finite deposit at parameters (147,23), verified through harmonic height 5022. It is research evidence, not a Lean-checked unbounded certificate."},{"problem_id":"erdos_251","erdos_number":251,"short_title":"The prime-gap dyadic series","status":"open","question":"Is the dyadic series sum p_n/2^n over consecutive primes irrational? Equivalently, is the corresponding consecutive-prime-gap dyadic series irrational?","library_root":"ErdosProblems.lean","claim_registry_status":"not_registered; the claim registry does not carry these declarations and kernel checking them does not promote them into reviewed public claims","directory":"ErdosProblems/Erdos251","modules":[{"module":"ErdosProblems.Erdos251.PrimeGapDyadicTail","path":"ErdosProblems/Erdos251/PrimeGapDyadicTail.lean","present":true,"line_count":782,"declaration_count":64,"theorem_count":43,"content_digest":"sha256:56c54f523c84d9a2e69799d4be5a2be283758505c3658b25a26ae07e68bce4d6"}],"note":{"artifact_id":"erdos_251_note","title":"The Prime-Gap Equivalence and the Integral-Shift Frontier","source_path":"paper/erdos-251-prime-gap-dyadic-series.tex","rendered_path":"erdos-251-prime-gap-dyadic-series.pdf","authority_posture":"authored_mathematical_exposition_of_an_unregistered_expansion_module_not_Lean_proof_authority"},"what_is_checked":["The finite summation-by-parts identity relating the prime dyadic partial sums to the prime-gap dyadic partial sums, with an exact endpoint term and no convergence premise.","The factor-of-two normalisation between the two displayed indexings, at every finite horizon.","Conditional on summability of the prime dyadic series, the exact infinite prime-gap identity and the equivalence of irrationality of the prime and prime-gap series.","For an abstract integer-digit dyadic tail recurrence: the exact block identity, and that a shift of fixed length is integral exactly when an explicit scaled state is.","That one integral shift propagates to every later index.","That repeated doubling removes the power-of-two denominator, so every rational-valued dyadic recurrence has one fixed positive shift integral at all sufficiently late indices.","For a real-valued dyadic recurrence, cofinal non-integrality of every fixed positive shift implies irrationality of the initial state.","That the actual consecutive-prime-gap sequence is unbounded, by the factorial prime-free-interval construction, and therefore is not eventually periodic with any positive period.","For any integer-digit dyadic tail recurrence, two adjacent fixed-length shifts lying strictly between -1 and 1 cannot both be integral when the corresponding digits differ.","For the actual prime gaps, a cofinal supply of adjacent small-shift pairs with a mismatching gap excludes eventual integrality of that fixed shift.","The generic distance-to-integers core of the finite-truncation criterion: an approximation farther than its error bound from every integer certifies non-integrality."],"what_is_not_checked":["The target irrationality.","Summability of the concrete prime series and the definition and recurrence of its concrete prime-gap tail as Lean infinite sums; the note supplies these at paper level from the prime number theorem.","A theorem supplying cofinally many adjacent actual-prime-gap tail shifts in the open unit interval with the required gap mismatch.","Any theorem forcing periodicity or automaticity of the prime gaps from rationality; the free-carry identity disconfirms that route."],"open_obligations":[{"id":"prime_gap_cofinal_shift_escape","statement":"For every fixed h >= 1 and every N0, find N >= N0 for which sum_{j>=1} (g_{N+h+j}-g_{N+j})/2^j is not an integer."},{"id":"cofinal_adjacent_small_mismatch","statement":"For every fixed h >= 1 and every N0, find N >= N0 such that both adjacent full tail shifts have absolute value less than 1 and g_{N+h+1} != g_{N+1}. The checked local consumer then excludes eventual integrality of the h-shift."},{"id":"actual_prime_gap_tail_formal_bridge","statement":"Formalise summability of the actual prime and prime-gap dyadic series and identify the concrete infinite prime-gap tail with the checked real/rational dyadic recurrence, so the local consumer reaches the target series without a paper-only analytic bridge."}],"finite_search":null},{"problem_id":"erdos_257","erdos_number":257,"short_title":"Reciprocal sums over infinite exponent supports","status":"open","question":"Is the sum of 1/(2^n-1) over every infinite set of positive exponents irrational?","library_root":"ErdosProblems.lean","claim_registry_status":"not_registered; the claim registry does not carry these declarations and kernel checking them does not promote them into reviewed public claims","directory":"ErdosProblems/Erdos257","modules":[{"module":"ErdosProblems.Erdos257.MersenneSubseriesRigidity","path":"ErdosProblems/Erdos257/MersenneSubseriesRigidity.lean","present":true,"line_count":421,"declaration_count":28,"theorem_count":23,"content_digest":"sha256:7dd147a2bef89f7f23a168c02a8dfb4c7fee887314a7a5367d49336330eb1419"}],"note":{"artifact_id":"erdos_257_note","title":"Forced Structure in Mersenne Subseries","source_path":"paper/erdos-257-mersenne-support-subseries.tex","rendered_path":"erdos-257-mersenne-support-subseries.pdf","authority_posture":"authored_problem_owned_synthesis_of_reviewed_gateway_claims_and_unregistered_expansion_modules_not_Lean_proof_authority"},"what_is_checked":["That the strict Mersenne tail inequality is hereditary under deleting an arbitrary collection of future weights.","That binary digit strings restricted to any chosen support retain unique Mersenne coding."],"what_is_not_checked":["The universal irrationality problem, or irrationality for any new infinite support.","The packet's measure classification: cofinite supports have positive measure and infinite-complement supports have measure zero.","The fair-bit law, the singularity dichotomy, and the explicit dimensions for periodic stride supports.","Any bridge from geometric uniqueness or measure rigidity to arithmetic irrationality."],"open_obligations":[{"id":"arithmetic_rigidity_for_thin_supports","statement":"Convert hereditary unique coding into an arithmetic obstruction to rational values for every infinite support, including zero-measure Cantor supports."},{"id":"formalise_measure_and_stride_geometry","statement":"Kernel-check the exact measure dichotomy, Bernoulli-law classification, and periodic-stride dimension formulas without overstating what they imply arithmetically."}],"finite_search":null},{"problem_id":"erdos_269","erdos_number":269,"short_title":"Three-prime running least common multiples","status":"open","question":"For a finite set of at least two primes, is the sum of reciprocals of the running least common multiples of the smooth numbers irrational? This library treats the three-prime case.","library_root":"ErdosProblems.lean","claim_registry_status":"not_registered; the claim registry does not carry these declarations and kernel checking them does not promote them into reviewed public claims","directory":"ErdosProblems/Erdos269","modules":[{"module":"ErdosProblems.Erdos269.ThreePrimeRunningLcm","path":"ErdosProblems/Erdos269/ThreePrimeRunningLcm.lean","present":true,"line_count":729,"declaration_count":50,"theorem_count":34,"content_digest":"sha256:b3279ce084c98e4e7521e0bb94d3e366de010a30548fab0b326390da16ff6c3c"},{"module":"ErdosProblems.Erdos269.ResidueEscape","path":"ErdosProblems/Erdos269/ResidueEscape.lean","present":true,"line_count":130,"declaration_count":7,"theorem_count":5,"content_digest":"sha256:c401a6f6018d13117a798eed62b8faf0abadb87cb2c36d5e755aa58fe98f81d1"},{"module":"ErdosProblems.Erdos269.RestrictedFloorSum","path":"ErdosProblems/Erdos269/RestrictedFloorSum.lean","present":true,"line_count":648,"declaration_count":41,"theorem_count":28,"content_digest":"sha256:21ad5930e03718ed931b2d05d94fc17b97171b017f6fbef869ca062807f686c2"}],"note":{"artifact_id":"erdos_269_note","title":"The Three-Prime Running Least Common Multiple","source_path":"paper/erdos-269-three-prime-running-lcm.tex","rendered_path":"erdos-269-three-prime-running-lcm.pdf","authority_posture":"authored_mathematical_exposition_of_an_unregistered_expansion_module_not_Lean_proof_authority"},"what_is_checked":["That for three pairwise distinct primes the least common multiple of the smooth prefix equals the product of the three maximal pure prime powers below the cutoff.","That equal logarithmic coordinates give an equal prefix least common multiple, so the running value is constant on logarithmic cells.","That advancing exactly one logarithmic coordinate multiplies the running value by the corresponding prime.","That the positive power channels of distinct primes are disjoint, so the first powers in three channels give an exact jump count.","That a finite smooth lattice sum groups exactly by running height, with the height fibre cardinality as coefficient.","A quadratic bound on the cardinality of the actual filtered exponent shell.","That each dyadic block contains at most one internal pure power from each non-dyadic channel, so the compressed {2,3,5} block radix belongs to the exact alphabet {2,6,10,30} and in particular lies between 2 and 30.","That the smallest {2,3,5} kernel rectangle has determinant exactly -1/15, so its two-by-two matrix has rank two over the rationals and no rank-one separation exists.","The canonical least-positive-residue interval and congruence facts, including the zero-residue representative.","The exact local-window identity for an integral carry.","A denominator-dependent CofinalLocalWindowEscape proposition and the full theorem that it rules out every positive reduced integral carry satisfying the matching bound."],"what_is_not_checked":["Irrationality or transcendence in any three-prime case.","The exact formal identification of the block digit with the multiplicity forcing in the original series, and the derivation of the denominator-dependent short bound from a hypothetical rational value.","CofinalLocalWindowEscape for those actual words; it is an explicit hypothesis of the checked consumer, not a theorem.","The infinite limit of the jump expansion; only its finite core is checked.","Any unbounded denominator exclusion. The finite certificates exclude denominators below an explicit bound and no further."],"open_obligations":[{"id":"actual_local_window_residue_escape","statement":"For every positive B coprime to 30 and every starting index, produce a later actual {2,3,5} window whose least positive forcing residue exceeds the exact denominator-dependent carry bound."},{"id":"actual_rational_carry_instantiation","statement":"Identify the formal radix word with the checked four-symbol dyadic block base, formalize the exact block digit and its link to the original summand multiplicities, and derive the positive reduced carry recurrence and its bound from a hypothetical rational value."},{"id":"nonstationary_analytic_theorem","statement":"Prove a genuinely higher-dimensional analytic theorem for the phase cocycle or the associated contraction."},{"id":"unbounded_height_certificate","statement":"Replace one finite denominator exclusion by a family whose exclusion bound tends to infinity."}],"finite_search":"The integer-only dyadic-window checker reproduces three exact packet certificates and, for all 106,666 pairs with denominator B <= 1000 coprime to 30 and dyadic start 100 <= a <= 500, finds an escaping window of length at most 18; the largest first successful length is 14. This is finite proof-consumer evidence only and proves neither unbounded-denominator nor cofinal escape."},{"problem_id":"erdos_1049","erdos_number":1049,"short_title":"Lambert-type series at rational bases","status":"open","question":"For which rational bases is the corresponding series irrational? The smallest resistant explicit base is three halves.","library_root":"ErdosProblems.lean","claim_registry_status":"not_registered; the claim registry does not carry these declarations and kernel checking them does not promote them into reviewed public claims","directory":"ErdosProblems/Erdos1049","modules":[{"module":"ErdosProblems.Erdos1049.RationalBaseLambert","path":"ErdosProblems/Erdos1049/RationalBaseLambert.lean","present":true,"line_count":216,"declaration_count":15,"theorem_count":11,"content_digest":"sha256:fe46fedeec9b8c90f38af320855a5f03542ed8bebf636ac2f141b4e67f33aa56"},{"module":"ErdosProblems.Erdos1049.RationalPadeArithmetic","path":"ErdosProblems/Erdos1049/RationalPadeArithmetic.lean","present":true,"line_count":222,"declaration_count":19,"theorem_count":14,"content_digest":"sha256:c7b54971556577d73bbc563dcf14eaf62aef8d31e1450c4e7337028e46013690"},{"module":"ErdosProblems.Erdos1049.ZudilinConeArithmetic","path":"ErdosProblems/Erdos1049/ZudilinConeArithmetic.lean","present":true,"line_count":345,"declaration_count":29,"theorem_count":19,"content_digest":"sha256:7c5542b1d2d9951ce43f5f92f43dc0884f079620c9805ddc96b4dd856d5f36d3"}],"note":{"artifact_id":"erdos_1049_note","title":"Rational Bases and the Integer-Base Argument","source_path":"paper/erdos-1049-rational-base-lambert.tex","rendered_path":"erdos-1049-rational-base-lambert.pdf","authority_posture":"authored_mathematical_exposition_of_an_unregistered_expansion_module_not_Lean_proof_authority"},"what_is_checked":["That a literal coordinatewise transfer of the integer-base clearing argument forces a power-versus-linear inequality, and that this inequality is impossible at base three halves.","The exact denominator-cleared rational-base tail recurrence, including its forcing term.","That for every noninteger base with denominator at least two the forcing magnitude grows at least exponentially, while at denominator one it collapses to a bounded expression.","The complete elementary Archimedean height inequality used by the published Bundschuh-V\u00e4\u00e4n\u00e4nen criterion at base seven halves; the analytic irrationality theorem itself remains external.","Polynomial bounds on the doubled denominator exponents in the homogenised Pad\u00e9 construction, together with an exact gap identity.","The exact exterior-determinant elimination identity, and that rowwise contents scale the Pad\u00e9 errors, exterior determinant, and absolute determinant height by the same factors.","The exact Heine-Zudilin cone endpoint exponent arithmetic used to identify the bottom and top coefficient endpoints.","That homogeneous evaluation at (3,2) sees only the constant endpoint modulo three and the top endpoint modulo two, so unit endpoints obstruct common local factors.","That under unit top and constant endpoint hypotheses, a common multiplier of the two specialised coefficient channels is divisible by neither two nor three; the missing local gain must therefore survive primitive normalisation.","That vanishing of the higher bottom and top endpoint jets is exactly divisibility by the corresponding powers of three and two.","That the source integer-base ratio inequality implies a strictly negative scalar product-formula margin at three halves.","That the four-jet target has exact cardinality (3^R)^2(2^S)^2 and, for positive R, any family with n at least 4R+2S forces two distinct binary selectors to collide; their {-1,0,1} difference cancels all four endpoint jets."],"what_is_not_checked":["Irrationality at base three halves, or for any rational base.","The published analytic height theorem, which is cited and used as an external result and is not formalised here.","The all-scale Heine-Zudilin coefficient construction at rational base three halves; the primary source assumes an integer base.","A sufficiently large primitive-normalised, non-collapsed coefficient family meeting the checked n >= 4R+2S threshold at the required quadratic depths.","That one of the resulting bounded-coefficient collision vectors has a nonzero polynomial pair, lies outside the analytic remainder nullspace, and has sufficient local gain.","Positivity or decay of the elementary Pad\u00e9 remainder; only its exponent arithmetic is checked."],"open_obligations":[{"id":"three_halves_growing_rank_endpoint_jet_kernel","statement":"After dividing every coefficient pair by its rowwise common content, at positive quadratic target depths R_n,S_n exhibit at least 4R_n+2S_n primitive coefficient pairs from a genuinely non-collapsed deformation, prove that the required two-adic and three-adic gain survives primitive normalisation, and prove that at least one resulting checked {-1,0,1} collision has a nonzero polynomial pair and lies outside the analytic remainder nullspace; or prove an exact rank obstruction."}],"finite_search":null}],"reviewed_problems":[{"problem_id":"erdos_249","erdos_number":249,"short_title":"The totient constant","status":"open","library_root":"Erdos249257.lean","exposition":"paper/erdos249-257-main-paper.tex","problem_note":"paper/erdos-249-binary-totient-series.tex","route":"python3 scripts/query_corpus.py --route erdos249_diagonal_arithmetic","note":"The gateway paper and reviewed claim registry own status; this shorter theorem-led note synthesises unregistered expansion modules without promoting them."},{"problem_id":"erdos_257","erdos_number":257,"short_title":"Reciprocal sums over infinite exponent supports","status":"open","library_root":"Erdos249257.lean","exposition":"paper/erdos249-257-main-paper.tex","problem_note":"paper/erdos-257-mersenne-support-subseries.tex","route":"python3 scripts/query_corpus.py --route erdos257_half_story","note":"The gateway paper and reviewed claim registry own status; this shorter theorem-led note synthesises unregistered expansion modules without promoting them."}],"validation_commands":["python3 scripts/build_problem_index.py --check","python3 scripts/check_problem_note_sources.py --coverage","python3 scripts/check_publication_contract.py"]},"body_format":"json_parsed","byte_parity_note":"body is the parsed object; re-serialisation differs from source bytes. Verify against sha256 by hashing the upstream file."},{"record_type":"formal_claim_control_plane_record","record_id":"lean_methodology","source_repository":"plectis-lean-erdos249-257","source_path":"docs/methodology.json","media_type":"application/json","licence":"Apache-2.0","copyright":"2026 Will Cook","why_an_offline_reader_needs_this":"Distinguishes Lean kernel checking from repository checking and from human mathematical review; owns the standing rules for public claims.","embedded":true,"availability":"embedded","sha256":"sha256:d2dab3ef7bd853be6bd086ad2ff448d97996280c648878f6aa5066e1616a722e","source_byte_count":46084,"body":{"schema":"erdos249257-methodology/2","title":"Formal proofs, mathematical claims, and what remains open","artifact_role":"rules_for_mathematical_reasoning_and_public_claim_changes_not_proof_authority","scope":"This file defines how this repository distinguishes formal proof, intended mathematical meaning, public claims, and remaining open propositions. It does not prove any mathematical statement and does not replace human mathematical review.","human_preamble":["This page records the rules used to relate the Lean development to the mathematical claims made in this repository, and to keep partial results from being overstated. It is generated from `docs/methodology.json`; run `python3 scripts/build_methodology.py` after changing that source. Lean source checked by the pinned Lean kernel remains proof authority.","The machine source calls the six standing rules `method_axiom` records. They are repository rules, not Lean `axiom` declarations."],"imports_contracts":["docs/claims.json::machine_readable_paper.authority_order","docs/claims.json::status_taxonomy","docs/claims.json::remaining_open_propositions","docs/claims.json::non_claims"],"evidence_classes":{"lean_check":{"human_name":"Lean check","decides":"Whether the declaration elaborates and its proof term is accepted in the pinned environment. The associated axiom audit reports the transitive axioms the declaration depends on.","does_not_decide":"Whether the proposition expresses the intended mathematics, is relevant to an open problem, or is novel."},"repository_check":{"human_name":"Repository check","decides":"Whether identifiers, statuses, source coordinates, routes, projections, and generated files agree.","does_not_decide":"Whether an informal statement is mathematically faithful or significant."},"human_mathematical_review":{"human_name":"Human mathematical review","decides_label":"Records a reviewed judgement about","decides":"Whether a formal proposition matches the intended statement, how it relates to prior work, what public claim it supports, and what remains open.","does_not_decide":"Whether the Lean kernel accepts the proof term."}},"human_capsule":["The exact boundary between proved and open must remain explicit.","Lean checks the stated proposition; it does not check that the proposition expresses the intended mathematics.","A public claim is transferred between representations only through a proved implication or equivalence.","A reduction must state the hypothesis or quantifier that remains open.","A finite result remains finite unless a theorem supplies the unbounded or universal step.","Progress toward an open problem must state which outstanding obligation it proves, narrows, decomposes, re-expresses, or constrains."],"ontology":{"mathematical_objects":[{"id":"open_target","definition":"An unresolved mathematical proposition whose status matters to this programme; in the claim registry these are the claims with status open."},{"id":"representation","definition":"Coordinates, an encoding, a recurrence, a series identity, or another model in which a target is studied."},{"id":"necessary_consequence","definition":"A proposition proved from a hypothetical assumption such as rationality or the existence of a counterexample."},{"id":"proved_reduction","definition":"A checked implication or equivalence that transfers a mathematical obligation in a stated direction."},{"id":"finite_certificate","definition":"Finite data accepted by a proved checker as establishing a bounded proposition."},{"id":"remaining_open_proposition","definition":"The exact unresolved proposition left after the proved work."}],"repository_objects":[{"id":"lean_declaration","definition":"A named Lean object in the pinned checked environment."},{"id":"public_claim","definition":"A public mathematical statement with a status, intended meaning, and links to any supporting Lean declarations."},{"id":"projection","definition":"A human-facing or machine-facing view derived from formal source, public claims, or this methodology source."}]},"mathematical_research_cycle":["open_target","representation","necessary_consequence","proved_reduction","finite_certificate","lean_declaration","remaining_open_proposition"],"public_claim_cycle":["lean_declaration","checked_assumptions","reviewed_intended_meaning","public_claim","authored_exposition","generated_projection","public_checkout"],"method_axioms":[{"id":"method_axiom.proof_is_exact_checked_proposition","title":"Proof means the exact proposition checked by Lean","human_name":"A formal result is the exact proposition Lean checked","statement":"A result is proved here only as the exact elaborated proposition accepted by the pinned public Lean environment under its transitive assumptions.","evidence_classes":["lean_check"],"applies_to":["object:lean_declaration","change_type:proof_body_changed","change_type:lean_proposition_changed"],"checked_by":["release.methodology_schema_and_references","release.proof_trust"],"scope_limit":"Kernel acceptance does not establish intended meaning, relevance, novelty, or claim status.","local_instances":[{"relation":"illustrates","ref_kind":"public_claim","ref":"eb_full_support"},{"relation":"enforces","ref_kind":"validation_guard","ref":"release.proof_trust"}]},{"id":"method_axiom.formal_proof_does_not_verify_intended_meaning","title":"Formal proof does not verify intended meaning","human_name":"Formal validity and mathematical faithfulness are separate questions","statement":"A checked Lean declaration and the intended mathematical statement remain distinct until human mathematical review records their correspondence.","evidence_classes":["lean_check","human_mathematical_review"],"applies_to":["object:lean_declaration","object:public_claim","change_type:lean_proposition_changed"],"checked_by":["release.methodology_schema_and_references","release.claim_transition_requirements"],"scope_limit":"The repository can require and route review; it cannot compute mathematical faithfulness.","local_instances":[{"relation":"illustrates","ref_kind":"public_claim","ref":"support_families"},{"relation":"states_limit","ref_kind":"remaining_open","ref":"remaining_open.universal_257_all_infinite_supports"},{"relation":"enforces","ref_kind":"validation_guard","ref":"release.claim_transition_requirements"}]},{"id":"method_axiom.representation_changes_require_proved_transport","title":"Representation changes require a proved implication or equivalence","human_name":"A public claim crosses representations only through proved transport","statement":"A claim may move between coordinates, encodings, normalisations, or models only through a checked implication or equivalence whose direction and losses are explicit.","evidence_classes":["lean_check","human_mathematical_review"],"applies_to":["object:representation","change_type:representation_changed","argument_relation:makes_exact"],"checked_by":["release.methodology_schema_and_references","release.claim_transition_requirements"],"scope_limit":"A transport theorem establishes the stated relation; it does not establish that the new representation is useful.","local_instances":[{"relation":"illustrates","ref_kind":"public_claim","ref":"mersenne_lambert_ladder"},{"relation":"states_limit","ref_kind":"remaining_open","ref":"remaining_open.erdos_249_irrationality"},{"relation":"enforces","ref_kind":"validation_guard","ref":"release.claim_transition_requirements"}]},{"id":"method_axiom.reduction_states_remaining_open_hypothesis","title":"A reduction states the remaining open hypothesis","human_name":"A reduction keeps the remaining hypothesis visible","statement":"Every conditional theorem, criterion, or reduction names the exact hypothesis or quantifier that remains open.","evidence_classes":["lean_check","repository_check","human_mathematical_review"],"applies_to":["claim_status:conditional reduction","object:proved_reduction","argument_relation:makes_exact"],"checked_by":["release.methodology_schema_and_references","release.claim_transition_requirements"],"scope_limit":"An exact reduction can expose an open proposition without proving it.","local_instances":[{"relation":"illustrates","ref_kind":"public_claim","ref":"certificate_reduction"},{"relation":"states_limit","ref_kind":"remaining_open","ref":"remaining_open.unbounded_certificate_supply"},{"relation":"enforces","ref_kind":"validation_guard","ref":"release.claim_transition_requirements"},{"relation":"tests_rejection","ref_kind":"negative_fixture","ref":"conditional_reduction_without_remaining_open_proposition"}]},{"id":"method_axiom.finite_results_remain_finite_without_uniform_theorem","title":"A finite result remains finite without an unbounded or universal theorem","human_name":"Finite verification does not change quantifiers","statement":"Finite certificates, bounded searches, and verified initial segments remain finite evidence unless a checked theorem supplies the unbounded or universal step.","evidence_classes":["lean_check","repository_check","human_mathematical_review"],"applies_to":["claim_status:verified finite instance","object:finite_certificate","change_type:finite_certificate_range_changed"],"checked_by":["release.methodology_schema_and_references","release.claim_transition_requirements"],"scope_limit":"Increasing a checked bound does not change the quantifier of the public claim.","local_instances":[{"relation":"illustrates","ref_kind":"public_claim","ref":"certified_kill_instances"},{"relation":"states_limit","ref_kind":"remaining_open","ref":"remaining_open.unbounded_certificate_supply"},{"relation":"enforces","ref_kind":"validation_guard","ref":"release.claim_transition_requirements"},{"relation":"tests_rejection","ref_kind":"negative_fixture","ref":"finite_result_changed_to_open_status"}]},{"id":"method_axiom.open_problem_progress_names_changed_open_proposition","title":"Progress toward an open problem names the outstanding obligation it affects","human_name":"A claim of progress names both the theorem and the outstanding obligation","statement":"A result advances an open target only when it states which remaining open proposition it proves, narrows, decomposes, re-expresses, or constrains.","evidence_classes":["repository_check","human_mathematical_review"],"applies_to":["argument_relation:advances_open_target","object:remaining_open_proposition","change_type:open_problem_relation_changed"],"checked_by":["release.methodology_schema_and_references","release.claim_transition_requirements"],"scope_limit":"The repository checks that the claimed effect is explicit; human review decides whether the effect is mathematically significant.","local_instances":[{"relation":"illustrates","ref_kind":"public_claim","ref":"denominator_exclusion"},{"relation":"illustrates","ref_kind":"public_claim","ref":"sublog_zero_windows"},{"relation":"states_limit","ref_kind":"non_claim","ref":"not_erdos_249_solution"},{"relation":"enforces","ref_kind":"validation_guard","ref":"release.claim_transition_requirements"},{"relation":"tests_rejection","ref_kind":"negative_fixture","ref":"open_target_edge_without_stated_effect"}]}],"principles":[{"id":"principle.partial_result_states_remaining_open_proposition","title":"State a partial result and the remaining open proposition together","group":"publishing","statement":"Every frontier-facing partial result states both its positive conclusion and the exact proposition that survives it.","evidence_classes":["repository_check","human_mathematical_review"],"applies_to":["claim_status:conditional reduction","claim_status:unconditional progress","argument_relation:advances_open_target"],"checked_by":["release.methodology_schema_and_references","release.claim_transition_requirements"],"scope_limit":"Closed results that do not face an open target need not invent a remaining-open field."},{"id":"principle.derive_exact_consequences_from_hypotheses","title":"Derive exact consequences from a hypothetical assumption","group":"proving","statement":"Under a hypothetical rational value or counterexample, derive checked structural consequences instead of replacing proof with plausibility.","evidence_classes":["lean_check","human_mathematical_review"],"applies_to":["object:necessary_consequence","object:proved_reduction"],"checked_by":["release.methodology_schema_and_references"],"scope_limit":"This is a method for developing consequences, not a claim that every useful proof must use contradiction."},{"id":"principle.exact_finite_checker_needs_soundness_and_completeness","title":"Calling a finite checker exact requires soundness and completeness","group":"proving","statement":"A finite checker needs a soundness theorem; calling it exact additionally requires a completeness theorem at the stated parameters.","evidence_classes":["lean_check","human_mathematical_review"],"applies_to":["object:finite_certificate","argument_relation:makes_exact"],"checked_by":["release.methodology_schema_and_references","release.claim_transition_requirements"],"scope_limit":"Soundness and completeness at fixed parameters do not provide an unbounded supply of certificates."},{"id":"principle.record_what_representation_changes_preserve","title":"Record what a representation change preserves, reflects, or loses","group":"publishing","statement":"Each representation change identifies the checked theorem that transfers the claim and states the direction and any loss.","evidence_classes":["lean_check","human_mathematical_review"],"applies_to":["object:representation","change_type:representation_changed"],"checked_by":["release.methodology_schema_and_references","release.claim_transition_requirements"],"scope_limit":"Informal similarity between two representations is not a transfer theorem."},{"id":"principle.record_failed_routes_and_counterexamples","title":"Record failed routes and counterexamples when they constrain later work","group":"maintaining","statement":"A failed strategy, counterexample, or disproved finite-state interpretation is retained when it rules out a recurring mathematical route.","evidence_classes":["human_mathematical_review"],"applies_to":["object:necessary_consequence","change_type:authored_exposition_changed"],"checked_by":["release.methodology_schema_and_references"],"scope_limit":"Routine failed experiments and private working notes are not automatically public artefacts."},{"id":"principle.classify_change_before_updating_public_claims","title":"Classify a change before updating public claims","group":"maintaining","statement":"Distinguish proof-body, proposition, assumption, representation, finite-range, claim-status, exposition, and projection changes before deciding what must be updated.","evidence_classes":["repository_check","human_mathematical_review"],"applies_to":["change_type:proof_body_changed","change_type:lean_proposition_changed","change_type:assumptions_changed","change_type:representation_changed","change_type:finite_certificate_range_changed","change_type:claim_support_linkage_changed","change_type:claim_status_changed","change_type:authored_exposition_changed","change_type:projection_changed"],"checked_by":["release.methodology_schema_and_references","release.methodology_routes_exist"],"scope_limit":"The classification routes review; it does not decide the review outcome."},{"id":"principle.separate_lean_human_and_agent_authority","title":"Separate Lean, human, and agent authority","group":"maintaining","statement":"Lean owns formal validity; humans own intended meaning, significance, and public framing; agents may search, propose, test, and expose consequences without inheriting either authority.","evidence_classes":["lean_check","human_mathematical_review"],"applies_to":["object:lean_declaration","object:public_claim","object:projection"],"checked_by":["release.methodology_schema_and_references","release.methodology_routes_exist","release.proof_trust"],"scope_limit":"Human review does not replace kernel checking, and kernel checking does not replace human review."},{"id":"principle.update_authoritative_sources_before_projections","title":"Update authoritative sources before projections","group":"maintaining","statement":"Update Lean source first, then assumption and meaning review, then claim status, then authored exposition, then generated projections.","evidence_classes":["repository_check","human_mathematical_review"],"applies_to":["change_type:lean_proposition_changed","change_type:claim_status_changed","change_type:authored_exposition_changed","change_type:projection_changed"],"checked_by":["release.methodology_schema_and_references","release.methodology_routes_exist"],"scope_limit":"A proof-body-only change does not automatically authorize stronger prose."},{"id":"principle.public_checkout_explains_claims_and_open_boundary","title":"A public checkout must explain the claims and open boundary","group":"publishing","statement":"A reader using only the public checkout must be able to find the main argument, proof routes, public claim statuses, and remaining open propositions.","evidence_classes":["repository_check","human_mathematical_review"],"applies_to":["object:projection","change_type:projection_changed"],"checked_by":["release.methodology_schema_and_references","release.methodology_projection_matches_source","release.methodology_routes_exist"],"scope_limit":"Path and projection checks establish availability, not reader comprehension."},{"id":"principle.claim_assurance_is_two_sided","title":"Claim assurance checks underclaiming as well as overclaiming","group":"publishing","statement":"Public claim review checks both ceilings and coverage: prose must not exceed the checked theorem chain, and a source-backed high-value mechanism, exact reformulation, quantitative reduction, or distinct endpoint must be mechanically discoverable when omitting it would materially understate the corpus.","evidence_classes":["lean_check","repository_check","human_mathematical_review"],"applies_to":["object:lean_declaration","object:public_claim","object:projection","change_type:claim_support_linkage_changed","change_type:authored_exposition_changed","change_type:projection_changed"],"checked_by":["release.methodology_schema_and_references","release.methodology_projection_matches_source","release.methodology_routes_exist"],"scope_limit":"Coverage does not turn support lemmas into separate headline claims: group plumbing under the smallest faithful contribution and require human review for significance."}],"anti_principles":[{"id":"anti_principle.do_not_equate_lean_acceptance_with_intended_meaning","title":"Do not equate Lean acceptance with intended meaning","statement":"Do not use a successful Lean check to bypass review of whether the proposition expresses the intended mathematics.","evidence_classes":["human_mathematical_review"],"applies_to":["object:lean_declaration","object:public_claim","change_type:lean_proposition_changed"],"checked_by":["release.methodology_schema_and_references","release.claim_transition_requirements"],"scope_limit":"The repository can require review but cannot automate the judgement."},{"id":"anti_principle.do_not_report_reduction_as_solution","title":"Do not report a reduction as a solution","statement":"Do not narrate a conditional implication, equivalence, or criterion as closure of the original open problem.","evidence_classes":["repository_check","human_mathematical_review"],"applies_to":["claim_status:conditional reduction","object:proved_reduction"],"checked_by":["release.methodology_schema_and_references","release.claim_transition_requirements"],"scope_limit":"An exact and useful reduction remains a genuine result when its remaining hypothesis is explicit."},{"id":"anti_principle.do_not_generalize_finite_result_without_theorem","title":"Do not generalize a finite result without a theorem","statement":"Do not treat a large checked range as evidence for an unbounded or universal assertion.","evidence_classes":["repository_check","human_mathematical_review"],"applies_to":["claim_status:verified finite instance","object:finite_certificate","change_type:finite_certificate_range_changed"],"checked_by":["release.methodology_schema_and_references","release.claim_transition_requirements"],"scope_limit":"Finite results may be valuable without changing the universal claim."},{"id":"anti_principle.do_not_identify_representations_without_transport","title":"Do not identify two representations without proved transport","statement":"Do not speak of a transformed representation as the original claim unless a checked implication or equivalence supplies that relation.","evidence_classes":["lean_check","human_mathematical_review"],"applies_to":["object:representation","change_type:representation_changed"],"checked_by":["release.methodology_schema_and_references","release.claim_transition_requirements"],"scope_limit":"This rule concerns claim transfer, not the informal use of notation."},{"id":"anti_principle.do_not_treat_graph_model_or_private_work_as_proof","title":"Do not treat a graph, model output, or private work as proof authority","statement":"A dependency graph, generated explanation, language-model output, or private proof sketch is not public proof authority.","evidence_classes":["repository_check"],"applies_to":["object:projection","change_type:projection_changed"],"checked_by":["release.methodology_schema_and_references","release.methodology_routes_exist","release.proof_trust"],"scope_limit":"These artefacts may help navigation or discovery without becoming proof."},{"id":"anti_principle.do_not_use_output_volume_as_mathematical_progress","title":"Do not use output volume as mathematical progress","statement":"Counts of declarations, proof lines, certificates, or agent activity do not by themselves establish progress on an open problem.","evidence_classes":["human_mathematical_review"],"applies_to":["argument_relation:advances_open_target","change_type:open_problem_relation_changed"],"checked_by":["release.methodology_schema_and_references","release.claim_transition_requirements"],"scope_limit":"Scale metrics may describe the artefact; they do not change a mathematical claim."},{"id":"anti_principle.do_not_edit_projection_as_source","title":"Do not edit a projection as though it were the source","statement":"README, generated Markdown, the corpus descriptor, and declaration maps must remain below their owning source files.","evidence_classes":["repository_check"],"applies_to":["object:projection","change_type:projection_changed"],"checked_by":["release.methodology_schema_and_references","release.methodology_projection_matches_source","release.methodology_routes_exist"],"scope_limit":"The paper is authored exposition and is checked for bounded agreement rather than generated from this file."},{"id":"anti_principle.do_not_keep_rule_without_owner_check_and_scope","title":"Do not keep a rule without an owner, a check or review route, and a scope limit","statement":"A methodology rule is removed if it cannot name the objects or changes it applies to, the evidence it requires, the route that evaluates it, and what it does not establish.","evidence_classes":["repository_check"],"applies_to":["object:projection","change_type:methodology_changed"],"checked_by":["release.methodology_schema_and_references"],"scope_limit":"A human-review route is valid when the judgement is not machine-decidable; it must still be explicit."}],"worked_examples":[{"id":"example.exact_reduction_without_closure","title":"An exact reduction that does not close #249","claim_ids":["certificate_reduction","certificate_completeness"],"remaining_open_ids":["remaining_open.unbounded_certificate_supply"],"route":["a hypothetical rational value of S = \u2211 \u03c6(n)/2\u207f","an integrality constraint on totient tail differences","a finite non-integrality certificate at fixed parameters (certifiedKill)","an exact conditional reduction (irrational_totient_series_of_lcm_diagonal_certificate_supply)","the unbounded supply of such certificates, which remains open"],"narrative":"If S were rational, its totient tails would satisfy an integrality constraint. The certifiedKill predicate refutes that constraint at fixed parameters, and the completeness theorem shows the finite computation loses no information at those parameters. The reduction theorem then proves: if certificates occur at unbounded parameters, S is irrational. Producing that unbounded supply is the remaining open proposition; nothing in this repository claims it."},{"id":"example.finite_verification_stays_finite","title":"Finite verification that stays finite","claim_ids":["certified_kill_instances"],"remaining_open_ids":["remaining_open.unbounded_certificate_supply"],"narrative":"certifiedKill_all_small and the lcm-diagonal deposits through t = 8 (certifiedKill_periodLcm_diagonal_upto_six and the t = 7 and t = 8 endpoints) are kernel-checked with decide at fixed bounds. Each names a specific bounded region the checker has cleared; together with the generated certificate shards they show the reduction is not vacuous. The claim keeps the status verified finite instance and states its bounded domain. Raising the checked bound changes the stated bound; it does not change the quantifier of any public claim."},{"id":"example.necessary_structure_without_contradiction","title":"Necessary structure that does not yet contradict","claim_ids":["reciprocal_mass_bound","unbounded_carry_states","sublog_zero_windows"],"remaining_open_ids":["remaining_open.universal_257_all_infinite_supports"],"narrative":"The carry and zero-window theorems state what a rational series value would force: reciprocal-mass lower bounds, unbounded carry states, and sublogarithmic zero windows in divisor coverage. These are necessary consequences of rationality. They constrain a hypothetical rational value; none of them rules it out, and the universal #257 statement remains open."}],"human_review_triggers":[{"id":"claim_text_changed","description":"The wording of a public claim changes."},{"id":"claim_status_changed","description":"The status of a public claim changes."},{"id":"intended_meaning_changed","description":"The reviewed correspondence between a proposition and its intended statement changes."},{"id":"paper_statement_changed","description":"A paper-facing mathematical statement changes."},{"id":"significance_claim_changed","description":"What the repository says a result means for an open problem changes."},{"id":"rule_semantics_changed","description":"The meaning of a methodology rule changes, not only its wording or projection."}],"forbidden_effect_vocabulary":[{"id":"change_open_target_status","description":"The change must not alter the status of an open target claim."},{"id":"infer_unbounded_result","description":"The change must not derive an unbounded or universal statement from bounded evidence."},{"id":"strengthen_public_claim_wording","description":"The change must not strengthen the wording of any public claim."},{"id":"identify_representations_without_transport","description":"The change must not present a transformed representation as the original claim without a checked transport theorem."},{"id":"bypass_projection_owner","description":"The change must not edit a generated projection except by regenerating it from its owner."}],"change_types":["proof_body_changed","lean_proposition_changed","assumptions_changed","representation_changed","finite_certificate_range_changed","claim_support_linkage_changed","open_problem_relation_changed","claim_status_changed","authored_exposition_changed","projection_changed","methodology_changed"],"change_classes":[{"id":"proof_body_changed","summary":"A proof body changes; the stated proposition does not","minimum_evidence":["lean_check","repository_check"],"human_review":"on_trigger","review_triggers":["intended_meaning_changed","claim_text_changed"],"forbidden_effects":["strengthen_public_claim_wording","change_open_target_status"],"public_consequence":"No claim or exposition change follows automatically. Compare the axiom audit before and after."},{"id":"lean_proposition_changed","summary":"A stated proposition or its hypotheses change","minimum_evidence":["lean_check","repository_check","human_mathematical_review"],"human_review":"required","forbidden_effects":["change_open_target_status"],"public_consequence":"Reconcile the claim wording, the remaining-open boundary, and the paper statement."},{"id":"assumptions_changed","summary":"The transitive assumptions of a declaration change","minimum_evidence":["lean_check","repository_check","human_mathematical_review"],"human_review":"required","forbidden_effects":["strengthen_public_claim_wording"],"public_consequence":"Reassess the proof posture even when the proposition is unchanged."},{"id":"representation_changed","summary":"A claim moves between coordinates, encodings, or models","minimum_evidence":["lean_check","human_mathematical_review"],"human_review":"required","forbidden_effects":["identify_representations_without_transport"],"public_consequence":"Record the transport theorem, its direction, and anything not preserved."},{"id":"finite_certificate_range_changed","summary":"A checked bounded range or certificate family is extended","minimum_evidence":["lean_check","repository_check"],"human_review":"on_trigger","review_triggers":["claim_text_changed","claim_status_changed","intended_meaning_changed","paper_statement_changed"],"forbidden_effects":["change_open_target_status","infer_unbounded_result"],"public_consequence":"Keep the finite status; update only the stated bound."},{"id":"claim_support_linkage_changed","summary":"A claim's declaration coordinates or links are corrected; the claim is otherwise unchanged","minimum_evidence":["lean_check","repository_check"],"human_review":"on_trigger","review_triggers":["claim_text_changed","claim_status_changed","intended_meaning_changed"],"forbidden_effects":["strengthen_public_claim_wording","change_open_target_status"],"public_consequence":"Link or correct the supporting declarations of an unchanged claim, then regenerate dependent projections. The mathematical claim does not change; a Lean check confirms the linked declaration exists and elaborates."},{"id":"open_problem_relation_changed","summary":"An advances-open-target relation is added or changed","minimum_evidence":["repository_check","human_mathematical_review"],"human_review":"required","forbidden_effects":["change_open_target_status"],"public_consequence":"Record the outstanding obligation affected and the exact relation."},{"id":"claim_status_changed","summary":"The status of a public claim changes","minimum_evidence":["repository_check","human_mathematical_review"],"human_review":"required","forbidden_effects":["infer_unbounded_result"],"public_consequence":"State the theorem or review that licenses the new status."},{"id":"authored_exposition_changed","summary":"Authored paper or documentation prose changes mathematically","minimum_evidence":["repository_check","human_mathematical_review"],"human_review":"required","forbidden_effects":["change_open_target_status"],"public_consequence":"Prose must not exceed the status recorded in the claim registry."},{"id":"projection_changed","summary":"A generated projection is rebuilt","minimum_evidence":["repository_check"],"human_review":"not_required","forbidden_effects":["bypass_projection_owner","strengthen_public_claim_wording"],"public_consequence":"Regenerate below the existing owners; no mathematical effect."},{"id":"methodology_changed","summary":"docs/methodology.json changes","minimum_evidence":["repository_check"],"human_review":"on_trigger","review_triggers":["rule_semantics_changed"],"forbidden_effects":["change_open_target_status"],"public_consequence":"Regenerate METHODOLOGY.md and the corpus descriptor. Rules route review; they do not decide mathematics."}],"transition_contracts":[{"id":"transition.lean_declaration_to_public_claim","title":"From a Lean declaration to a public claim","trigger":"A new declaration or a changed proposition is proposed as support for a public claim.","required_evidence_classes":["lean_check","repository_check","human_mathematical_review"],"requirements":["The declaration exists in the pinned public source and passes the Lean check.","The public claim links to the declaration and uses a declared claim status.","Human mathematical review confirms that the proposition expresses the intended paper-facing statement.","Any remaining open proposition is stated explicitly."],"checked_by":["release.methodology_schema_and_references","release.claim_transition_requirements"]},{"id":"transition.representation_change_to_public_claim","title":"From a changed representation to a public claim","trigger":"A claim is moved into different coordinates, an encoding, a recurrence, or another model.","required_evidence_classes":["lean_check","human_mathematical_review"],"requirements":["A checked implication or equivalence identifies the direction of transfer.","The claim states anything not preserved by the transfer.","Human mathematical review confirms that the transferred statement is the intended one."],"checked_by":["release.methodology_schema_and_references","release.claim_transition_requirements"]},{"id":"transition.conditional_reduction_to_public_claim","title":"From a conditional reduction to a public claim","trigger":"A checked implication or equivalence is published with status conditional reduction.","required_evidence_classes":["lean_check","repository_check","human_mathematical_review"],"requirements":["The supporting declaration is linked.","The claim names at least one remaining open proposition from docs/claims.json.","The public statement says what the reduction does not prove."],"checked_by":["release.methodology_schema_and_references","release.claim_transition_requirements"]},{"id":"transition.finite_certificate_to_public_claim","title":"From a finite certificate to a public claim","trigger":"A checked finite computation or bounded family is added as a new public claim, or its interpretation changes.","required_evidence_classes":["lean_check","repository_check","human_mathematical_review"],"requirements":["The claim status remains verified finite instance.","The checked finite domain is explicit.","No open target changes status without a separate unbounded or universal theorem.","A purely mechanical extension of an already reviewed bounded family follows the finite_certificate_range_changed change class instead: Lean and repository checks, with review re-entered only when claim text, claim status, intended meaning, or the paper statement changes."],"checked_by":["release.methodology_schema_and_references","release.claim_transition_requirements"]},{"id":"transition.public_claim_to_open_target_edge","title":"From a public claim to an advances-open-target edge","trigger":"The argument graph says that a public claim advances an open target.","required_evidence_classes":["repository_check","human_mathematical_review"],"requirements":["The target claim has status open.","The edge names a remaining open proposition owned by that target.","The edge states whether the result proves, narrows, decomposes, re-expresses, or constrains that proposition."],"checked_by":["release.methodology_schema_and_references","release.claim_transition_requirements"]},{"id":"transition.methodology_source_to_public_projections","title":"From methodology source to public projections","trigger":"docs/methodology.json changes.","required_evidence_classes":["repository_check"],"requirements":["METHODOLOGY.md is regenerated exactly from docs/methodology.json.","The corpus descriptor records the methodology path, schema, digest, and compact capsule.","AGENTS.md and README.md route to the methodology without duplicating its full content.","README.md carries no methodology machine identifiers beyond the source link."],"checked_by":["release.methodology_schema_and_references","release.methodology_projection_matches_source","release.methodology_routes_exist"]}],"validation_guards":[{"id":"release.methodology_schema_and_references","description":"Checks the methodology schema, literal object and change references, evidence classes, rule fields, local-instance references, change-class evidence and review requirements, guard references, and reciprocal guard-to-rule bindings.","serves":["method_axiom.proof_is_exact_checked_proposition","method_axiom.formal_proof_does_not_verify_intended_meaning","method_axiom.representation_changes_require_proved_transport","method_axiom.reduction_states_remaining_open_hypothesis","method_axiom.finite_results_remain_finite_without_uniform_theorem","method_axiom.open_problem_progress_names_changed_open_proposition","principle.partial_result_states_remaining_open_proposition","principle.derive_exact_consequences_from_hypotheses","principle.exact_finite_checker_needs_soundness_and_completeness","principle.record_what_representation_changes_preserve","principle.record_failed_routes_and_counterexamples","principle.classify_change_before_updating_public_claims","principle.separate_lean_human_and_agent_authority","principle.update_authoritative_sources_before_projections","principle.public_checkout_explains_claims_and_open_boundary","principle.claim_assurance_is_two_sided","anti_principle.do_not_equate_lean_acceptance_with_intended_meaning","anti_principle.do_not_report_reduction_as_solution","anti_principle.do_not_generalize_finite_result_without_theorem","anti_principle.do_not_identify_representations_without_transport","anti_principle.do_not_treat_graph_model_or_private_work_as_proof","anti_principle.do_not_use_output_volume_as_mathematical_progress","anti_principle.do_not_edit_projection_as_source","anti_principle.do_not_keep_rule_without_owner_check_and_scope","transition.lean_declaration_to_public_claim","transition.representation_change_to_public_claim","transition.conditional_reduction_to_public_claim","transition.finite_certificate_to_public_claim","transition.public_claim_to_open_target_edge","transition.methodology_source_to_public_projections"]},{"id":"release.claim_transition_requirements","description":"Checks typed remaining-open propositions, conditional reductions, finite claim domains, and advances-open-target edge effects.","serves":["method_axiom.formal_proof_does_not_verify_intended_meaning","method_axiom.representation_changes_require_proved_transport","method_axiom.reduction_states_remaining_open_hypothesis","method_axiom.finite_results_remain_finite_without_uniform_theorem","method_axiom.open_problem_progress_names_changed_open_proposition","principle.partial_result_states_remaining_open_proposition","principle.exact_finite_checker_needs_soundness_and_completeness","principle.record_what_representation_changes_preserve","anti_principle.do_not_equate_lean_acceptance_with_intended_meaning","anti_principle.do_not_report_reduction_as_solution","anti_principle.do_not_generalize_finite_result_without_theorem","anti_principle.do_not_identify_representations_without_transport","anti_principle.do_not_use_output_volume_as_mathematical_progress","transition.lean_declaration_to_public_claim","transition.representation_change_to_public_claim","transition.conditional_reduction_to_public_claim","transition.finite_certificate_to_public_claim","transition.public_claim_to_open_target_edge"]},{"id":"release.methodology_projection_matches_source","description":"Checks that METHODOLOGY.md is the exact deterministic projection of docs/methodology.json.","serves":["principle.public_checkout_explains_claims_and_open_boundary","principle.claim_assurance_is_two_sided","anti_principle.do_not_edit_projection_as_source","transition.methodology_source_to_public_projections"]},{"id":"release.methodology_routes_exist","description":"Checks that the claim-map entrypoint, corpus descriptor, AGENTS.md, and README.md route to the methodology source and human projection, and that README carries no methodology machine identifiers.","serves":["principle.classify_change_before_updating_public_claims","principle.separate_lean_human_and_agent_authority","principle.update_authoritative_sources_before_projections","principle.public_checkout_explains_claims_and_open_boundary","principle.claim_assurance_is_two_sided","anti_principle.do_not_treat_graph_model_or_private_work_as_proof","anti_principle.do_not_edit_projection_as_source","transition.methodology_source_to_public_projections"]},{"id":"release.proof_trust","description":"Checks that public Lean source contains no sorry, admit, native_decide, or added axiom declarations.","serves":["method_axiom.proof_is_exact_checked_proposition","principle.separate_lean_human_and_agent_authority","anti_principle.do_not_treat_graph_model_or_private_work_as_proof"]}]},"body_format":"json_parsed","byte_parity_note":"body is the parsed object; re-serialisation differs from source bytes. Verify against sha256 by hashing the upstream file."},{"record_type":"formal_claim_control_plane_record","record_id":"lean_scope","source_repository":"plectis-lean-erdos249-257","source_path":"SCOPE.md","media_type":"text/markdown","licence":"Apache-2.0","copyright":"2026 Will Cook","why_an_offline_reader_needs_this":"Names the exact propositions that remain open, and states that private work and generated prose are not public proof authority.","embedded":true,"availability":"embedded","sha256":"sha256:bf0e5ee9be59ac1b3cb8af88015ebf272443dfa0650d2beeab2ee14bcedf07ed","source_byte_count":2681,"body":"<!-- SPDX-FileCopyrightText: 2026 Will Cook -->\n<!-- SPDX-License-Identifier: Apache-2.0 -->\n\n# Scope of this release\n\nThis release does not prove the irrationality of the Erd\u0151s #249 totient\nconstant, and does not prove the universal form of Erd\u0151s #257. Its claims\napply only to the source and dependencies identified by the immutable\nformal-source checkpoint in [`docs/claims.json`](docs/claims.json). The last\nrelease tag remains the citation and release identity; it can differ from that\ncheckpoint.\n\nMathematical proof authority is Lean source at that declared checkpoint after\nchecking by the pinned Lean kernel. Unreleased work, private repositories,\ngenerated prose, provider output, and hidden proof sketches are not part of\nthe public proof artefact. Work later than the declared checkpoint is out of\nscope even if it checks locally; a later tagged release is a separate citation\nand release action, not a substitute for an exact proof-source identity.\n\n## Exact open propositions\n\nThe prose boundary above is backed by machine-resolvable open propositions:\n\n- `remaining_open.erdos_249_irrationality`:\n  `python3 scripts/query_corpus.py --open remaining_open.erdos_249_irrationality`\n- `remaining_open.unbounded_certificate_supply`:\n  `python3 scripts/query_corpus.py --open remaining_open.unbounded_certificate_supply`\n- `remaining_open.universal_257_all_infinite_supports`:\n  `python3 scripts/query_corpus.py --open remaining_open.universal_257_all_infinite_supports`\n\nFinite instances, conditional reductions, cited neighbours, and named\ninfinite-support families do not discharge these propositions. Only a future\nclaim transition recorded against the exact identifier, with the evidence\nrequired by `docs/methodology.json`, can change this boundary.\n\n## Machine identifiers\n\nThese identifiers are the machine-readable form of the scope statement above.\nThey are declared once in [`docs/claims.json`](docs/claims.json) and\ncross-checked against this file by `scripts/check_release.py`, which fails\nwhen any public surface drifts from them.\n\n| Identifier | Public meaning |\n|---|---|\n| `not_erdos_249_solution` | The release does not settle Erd\u0151s #249 |\n| `not_erdos_257_solution` | The release does not settle the universal Erd\u0151s #257 |\n| `not_publication_authority` | The release is not itself a peer-reviewed publication record |\n| `not_private_root_equivalence` | The release is not asserted to be equivalent to any private development |\n| `not_provider_proof_authority` | Provider or model output is not proof authority; the Lean kernel check is |\n| `not_hidden_proof_body_authority` | No unpublished proof sketch or hidden artefact carries proof authority |\n","body_format":"markdown"},{"record_type":"formal_claim_control_plane_record","record_id":"lean_source_map","source_repository":"plectis-lean-erdos249-257","source_path":"docs/SOURCE_MAP.md","media_type":"text/markdown","licence":"Apache-2.0","copyright":"2026 Will Cook","why_an_offline_reader_needs_this":"Bounded reading order over the Lean modules. Navigation, explicitly not proof authority.","embedded":true,"availability":"embedded","sha256":"sha256:d87989c0c9827e7152a5ee6e870e245fed7e56684c942b963e443d66f7dc8400","source_byte_count":12416,"body":"<!--\nSPDX-FileCopyrightText: 2026 Will Cook\nSPDX-License-Identifier: Apache-2.0\n-->\n\n# Source map\n\n## Bounded reader route\n\nStart from one mathematical question rather than decoding the complete import\ngraph:\n\n1. Read `docs/orientation.json`, or its human projection\n   `docs/ORIENTATION.md`, and select one programme, claim, or exact remaining\n   open proposition.\n2. Run one bounded lookup:\n   `python3 scripts/query_corpus.py --route <programme_id>`,\n   `python3 scripts/query_corpus.py --claim <claim_id>`, or\n   `python3 scripts/query_corpus.py --open <remaining_open.id>`.\n3. Follow the returned paper label into the authored exposition and its\n   declaration or source handles into the modules grouped below.\n4. Read `Erdos249257.lean` only when the package topology itself is the\n   question. Read `docs/WAVE_INDEX.md` for chronology, not as the recommended\n   mathematical reading order.\n\nLean source checked by the pinned Lean kernel is proof authority. This map is\nauthored navigation: it does not strengthen a claim, replace\n`docs/claims.json`, or close an exact remaining-open proposition. Erd\u0151s #249\nand the universal form of #257 remain open.\n\n## Build entrypoint\n\n- `Erdos249257.lean`\n- Lake package: `erdos249257`\n- Lean library / stable root import: `Erdos249257`\n\n## #257 / Erd\u0151s\u2013Borwein families\n\n- `Erdos249257.CertificateKernel`\n- Relevant declarations include: `irrational_erdosSum_full_support`,\n  `irrational_erdosBorwein_series`, `irrational_erdosSum_factorial_support`,\n  `irrational_erdosSum_two_pow_support`, and\n  `irrational_erdosSupportSeries_eventuallyPeriodic`.\n\n## Periodic weighted Lambert series\n\n- `Erdos249257.CertificateKernel`\n- Nonnegative weights:\n  `irrational_weightedErdosSeries_periodic`,\n  `irrational_weightedErdosSeries_eventuallyPeriodic`, and\n  `irrational_ratWeightSeries_eventuallyPeriodic`.\n- Signed weights:\n  `irrational_or_bpow_mul_eq_intCast_intWeightedErdosSeries_periodic` gives\n  the irrational-or-base-terminating dichotomy, while\n  `irrational_intWeightedErdosSeries_periodic_of_bpow_mul_ne_intCast`\n  consumes an explicit nontermination hypothesis.\n- `intWeightedCoeff_periodFourSignWeight_eq_zero_of_mod_four_eq_three` is the\n  residue-selection obstruction explaining why the signed endpoint is not\n  unconditional irrationality.\n\n## #249 / totient denominator exclusions\n\n- `Erdos249257.GapFareyBound`\n- `Erdos249257.CertificateKernel`\n\n## #249 / certificate reductions\n\n- `Erdos249257.TotientTailPeriodKiller`\n- `Erdos249257.CarrySurvivorExtinction`\n- `Erdos249257.LcmDiagonalReduction`\n- `Erdos249257.LcmConeFlatness`\n- `Erdos249257.LcmConeNonflat`\n\nSupporting ladder modules:\n\n- `Erdos249257.MersenneLambertLadder`\n- `Erdos249257.GeometricCoprimality`\n- `Erdos249257.GcdMomentCalculus`\n\n## #249 / diagonal pincer frontier\n\nRead this band after the certificate reductions above.  It sharpens the same\nopen obligation; it does not prove the required unbounded supply.\n\n- `Erdos249257.DiagonalPincerDecomposition` \u2014 exact full-target interval\n  characterisation and the target-avoidance-to-irrationality reduction.\n- `Erdos249257.SquaredMersenneDiagonalEnclosure` \u2014 Lambert-projected centre,\n  exact signed square tail, and a sharp separation criterion.\n- `Erdos249257.DiagonalFreshLossBridge` \u2014 exact fresh-loss projection and\n  stronger adjacent-suffix supply interfaces.\n- `Erdos249257.DiagonalPincerCertificates` and\n  `Erdos249257.DiagonalPincerCertificatesT*` \u2014 the finite diagonal segment,\n  currently assembled at 28 explicit scales through `t = 64`.\n- `Erdos249257.DiagonalPincerPrimeCertificates.*` \u2014 finite prime certificate\n  shards used by that assembled segment; read the aggregators first.\n\nAlgebraic support for the same band:\n\n- `MersenneShadowCyclotomicNoncollapse \u2192 MersenneShadowDenominatorGrowth`\n- `RadicalMobiusShadow \u2192 RepunitMobiusNumerator \u2192 CyclicTensorMobiusShadow`\n- `CyclotomicProjectionOfShadow \u2192 PrimePowerJumpDynamics`\n- `RationalDenominatorSurvival`, `LambertDiagonalEnclosure`,\n  and `FullTargetPrimeAdjunctionNoGo`\n\nExploratory exact interfaces with explicit conditional boundaries live in\n`PrimitiveRationalGapSupply`, `PrimitiveSupportBridge`,\n`ActualForeignResidueProjection`, `SternBrocotRunGeometry`, and\n`SupportDilationDifferences`.\n\n### Transport and curvature companion band\n\nRead `CurvatureCarry \u2192 ExponentOnlyTransport \u2192 JointExponentTransport` for the\nmain proof spine, then `PrimeJumpMigration \u2192 PrimeJumpWindow` and\n`ThreeTransportBoundary` for the new-prime and balanced finite consumers.\nFor the harmonic route, read\n`FirstHarmonicGap \u2192 FirstHarmonicPivot \u2192 AdjacentPhaseSeparation`.\n`FirstHarmonicGap` turns a constant-saving cosine bound into a direct\ncertificate. `FirstHarmonicPivot` decomposes the complex harmonic sum exactly\ninto centred supplier-fibre, fibre-mean, bad-supplier, and non-supplier terms;\nthe one-sided budgets `14X/25`, `X/100`, `X/100`, and `8X/25` give the required\n`9X/10` saving. `AdjacentPhaseSeparation` gives a pairwise alternative. No\ncofinal harmonic, prime-distribution, or decorrelation estimate is proved.\n`TropicalCurvatureCarry` is a proved negative result showing that bounded\nvaluation\u2013unit precision alone cannot exclude all centred completions.\n\n- `CurvatureCarry` \u2014 exact curvature window, sharp radius, recurrence, and\n  conditional unbounded-supply endpoints.\n- `ExponentOnlyTransport` \u2014 squarefree support stability and the exact\n  periodic affine slope/intercept state.\n- `JointExponentTransport` \u2014 universal two-moment old-channel annihilation and\n  the anchored `(3,5)` four-vertex consumer.\n- `PrimeJumpMigration` / `PrimeJumpWindow` \u2014 migrated-channel correction,\n  sharp direct radius, and the finite `(12,5,15)` certificate.\n- `ThreeTransportBoundary` \u2014 balanced `(1,2,3,6)` consumer and the finite\n  `(60,12)` certificate.\n- `FirstHarmonicGap` \u2014 deterministic conversion of a blockwise harmonic gap\n  into a direct certificate.\n- `FirstHarmonicPivot` \u2014 exact four-term complex pivot decomposition and a\n  conditional cofinal budget-to-irrationality socket.\n- `TropicalCurvatureCarry` \u2014 fixed-precision local-signature no-go theorem.\n\nThe technical exposition is `paper/erdos249-transport-curvature-companion-note.tex`.  This\nband is included as a separate companion because it has its own definitions\nand proof spine; folding it into the principal paper would obscure the latter's\nglobal result map.  Every irrationality endpoint in the band remains a\nconditional reduction.\n\n### Paper-curation boundary\n\nThe paper treats `GcdMomentCalculus` as supporting mathematical exposition: its\nsquared Lambert transform, gcd moments, reduced-direction law, and cylinder\nrecursion explain the representation containing the #249 M\u00f6bius-square atom.\n`FullTargetPrimeAdjunctionNoGo` is included as a proved negative result because\nit closes a recurring prime-adjunction strategy without changing the open\nclaim. `SternBrocotRunGeometry` is registered as a source-level supporting\nclaim: its continuant/Fibonacci stability theorems are stated in the Lambert\ncomplements, but no endpoint argument currently consumes them. It therefore\nremains outside the paper's principal result spine; the cylinder law is owned\nby `GcdMomentCalculus`, while `SternBrocotRunGeometry` owns the induced-run\nstability package.\n\n## Binary-carry trunk (post-wave carry layer)\n\nBuilds on `Erdos249257.CertificateKernel`; not part of the wave chronology.\nReading order:\n`GenericTailOrbitRigidity \u2192 GreedyAchievementSet \u2192 CampbellShiftSynchronization \u2192 BooleanMobiusCarry \u2192 RationalSupportCarrySkeleton \u2192 DyadicPrefixCompression \u2192 SublogDivisorCoverage`.\n\n- `Erdos249257.GenericTailOrbitRigidity`\n- Relevant declarations include: `temperedBinaryOrbit_eq_scaledTail`, `binaryCoeffSeries_rational_iff_exists_temperedBinaryOrbit`, `balancedPulse_fanout_unbounded`, `balancedPulse_no_autonomous_decoder`\n\n- `Erdos249257.GreedyAchievementSet`\n- Relevant declarations include: `volume_mersenneAchievementSet`, `perfect_mersenneAchievementSet`, `isNowhereDense_mersenneAchievementSet`, `mem_mersenneAchievementSet_iff_greedy_survival`, `certifiedGreedyMersenneDeath_not_mem`, `three_fourths_not_mem_mersenneAchievementSet`, `rational_member_support_bit_iff`, `half_mem_mersenneAchievementSet_of_prefixForcingChain` (conditional interface only)\n\n- `Erdos249257.CampbellShiftSynchronization`\n- Relevant declarations include:\n  `campbellQuarterExponent_periodFreeze_noSync`,\n  `campbellPhaseFreeze_parameters_impossible`,\n  `shiftWindowZero_iff_greedyMersenneSkippedSupport_infinite`, and\n  `shiftWindowZero_iff_half_mem_mersenneAchievementSet`. These are a\n  synchronization no-go and an exact endpoint re-expression, not an\n  independent half-membership producer.\n\n- `Erdos249257.BooleanMobiusCarry`\n- Relevant declarations include: `erdosSupportSeries_rational_iff_exists_temperedCarry`, `support_fraction_iff_exists_temperedCarry`, `exists_normalized_support_fraction_iff_exists_booleanMobiusCarry`, `mobius_carryQuotient_recovers_support`, `support_half_iff_affineBinaryOrbit_tempered`\n\n- `Erdos249257.RationalSupportCarrySkeleton`\n- Relevant declarations include: `sum_doublingResidue_eq_mul_wrapCount`, `one_div_oddOrder_le_reciprocalMass_of_support_fraction`, `reciprocalMass_eq_residueMean_add_excessMean`, `dyadic_support_fraction_reciprocalMass_diverges_or_gt_one`, `exists_unbounded_shifted_odd_tail_nat_state_of_support_fraction`\n\n- `Erdos249257.DyadicPrefixCompression`\n- This is #257 support, not part of the #249 diagonal band. It gives exact\n  finite-fragment dyadic compression inside a hypothetical dyadic-rational\n  support value, reduced/raw block-safety coordinates, primitive-lattice and\n  support-tail-gap equivalences, conditional half-membership consumers, and\n  checked failures of naive induction.\n  Start with `finiteFragment_dyadicPrefixCompression`,\n  `halfGreedyThreeDepthTakeRunExcessBound_iff_skippedExcessBound`,\n  `greedyHalf_badSkipCapViolation_iff_primitiveLattice`,\n  `halfGreedyPrefixSupportTailGap_eq_excess_div_denominator`, and\n  `unsafeSkipGcdOvershootSupply_iff_primitive`. No cofinal witness supply is\n  proved.\n\n- `Erdos249257.SublogDivisorCoverage`\n- Relevant declarations include: `supportCoeffZeroWindow_length_le_eps_logb_add`\n\n## Half-value finite-prefix and boundary analysis\n\nThis band studies the test value `1/2` in ordinary finite-prefix language.\nA greedy prefix is the first `n` include/skip decisions. A seam is a pair of\nfinite words whose values bracket `1/2`. A cylinder is the set of infinite\nextensions of one fixed prefix. A producer is an unproved hypothesis that\nsupplies the required seams or return configurations arbitrarily far out.\n\nRead the band in four layers:\n\n1. `GreedyAchievementSet \u2192 HalfCutLocator` defines the achievement set, greedy\n   coding, and the finite cut around `1/2`.\n2. `HalfCarryReachability \u2192 HalfCylinderFiniteShadow` translates finite\n   prefixes into integer carry and finite-shadow coordinates.\n3. `HalfCylinderIntegerGreedy \u2192 HalfCylinderConcreteSeamAdapter \u2192\n   HalfCylinderFullShellSeamBridge \u2192 HalfCylinderSkippedEndpointClassifier \u2192\n   HalfCylinderFloorErrorReset \u2192 HalfCylinderSeamLimit` develops the exact\n   boundary arithmetic and limiting seam interface.\n4. `HalfCylinderFatalGapRightTail \u2192 HalfCylinderHalfMembershipClassification \u2192\n   HalfCylinderLastProducerContradiction \u2192 HalfCylinderMiddleCarryLowerBound`\n   gives the endpoint classification, eliminates the upper and `-3` middle\n   cases, and isolates the remaining carry-tail inequality for `-2` and `-1`.\n\nThe floor-error and shell modules are supporting arithmetic layers, not\nseparate endpoint theorems. The last layer remains conditional: no producer\nsupplying the required configurations cofinally is proved.\n\n## Claim registry\n\n- `docs/claims.json` \u2014 the single owner of release identity, claim statuses, and declaration coordinates (file and line), cross-checked by `scripts/check_release.py`\n\n## Generated finite certificates\n\n- `Erdos249257.GeneratedCertificates`\n- `Erdos249257.GeneratedCertificates.*`\n\nThe complete import graph and every declaration are generated in\n`docs/claims.json::machine_readable_paper.module_graph` and\n`docs/declaration_atlas.json`; use those surfaces when a grouped route above is\nnot specific enough. Current corpus counts live in the generated\n`docs/ORIENTATION.md`, not in this authored route map.\n","body_format":"markdown"}],"repository_maps":[{"schema":"plectis_public_repository_map_v0","repository":"plectis-lean-erdos249-257","role":"Machine-checked mathematics: Lean source, the public claims registry, scope and methodology records, and the mathematical manuscripts.","availability":"available","inventory_source":"git_head_tree","revision":"18b64bd2d932ecc2f64ac8f94994f0bb268734f0","public_tree_url":"https://github.com/wcook04/plectis-lean-erdos249-257/tree/main","tracked_file_count":842,"tracked_path_inventory_sha256":"sha256:15a27ece251543226915f719fd9a978e99fe37e0ddd65b39e7f60cafb3da747f","entrypoint_paths":["README.md","AGENTS.md","SCOPE.md","METHODOLOGY.md","docs/claims.json","docs/orientation.json","docs/methodology.json","docs/SOURCE_MAP.md","Erdos249257.lean","lakefile.toml","lean-toolchain"],"topology":{"max_depth":2,"complete_path_inventory_embedded":false,"why_compacted":"The counted depth-two tree orients a reader without repeating thousands of path prefixes. 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PDFs","reason":"presentation rendering of embedded LaTeX; hashes recorded per paper","external_verification_optional":true},{"what":"the full Lean source tree and Mathlib","reason":"thousands of declarations would crowd retrieval for ordinary questions; claim-to-declaration coordinates (module and line) are carried in the claims registry instead. repository_maps carries a counted, hash-bound topology of both public repositories without embedding every source body","external_verification_optional":true},{"what":"the flat list of every tracked repository path","reason":"repository_maps compresses each complete tracked-path inventory to a counted depth-two topology and retains a SHA-256 over the complete path list; repeating thousands of prefixes would add retrieval cost without adding orientation","external_verification_optional":true}]},"challenge_manifest":{"schema":"plectis_challenge_manifest_v0","artifact_kind":"post_freeze_evaluator_controlled_challenge_manifest","execution_boundary":{"chat_assistant_mode":"read_and_cite_only","packet_is_inert":true,"rule":"A chat assistant reading this packet may inspect and quote the challenge records, but it does not run the probe and must not say it executed a check. Any execution belongs to a separate Python interpreter or coding-agent session and must be attributed to that environment."},"why_this_exists":"Everything else in this packet was chosen by the same author: the mechanism, the fixture, the expected value, the validator rule and the presentation. Agreement among those shows self-consistency, not an independent reason to think the rule is right. A challenge moves one consequential choice -- which cases run -- to you.","how_to_run":{"surface":"separate_python_interpreter_or_coding_agent","primary":"Do not run this from the chat assistant reading the packet. If a human wants an independent execution result, move self_contained_probe.program_source into a separate Python 3 interpreter or coding-agent session. It carries the implementation under test verbatim and needs no repository, network, or third-party dependencies.","seed":"Any string. Choose it yourself, after reading this. The author has not seen it and cannot have tuned anything to it.","reproducibility":"Cases derive from sha256('{seed}|{label}|{index}') read as big-endian integers. Reimplement that in any language to confirm the cases follow from your seed rather than from a curated set.","chat_assistant_route":"The implementation source and its declared spec are both embedded. Read one against the other and report divergence. That is the packet-reading chat's valid challenge route, with nothing executed.","note_on_repository_routes":"The project also runs these challenges from its own repository. That route is not reproduced here: it needs a checkout you do not have, and a command you cannot run is not an instruction. Everything you need is in self_contained_probe."},"how_to_return_a_result":{"problem":"The packet-reading chat cannot write to this project, and a separate evaluator's run does not persist anywhere the project can see.","route":"The probe ends by printing a RECEIPT block. Hand that block back to whoever gave you this packet -- in your reply, or as a file. That printed JSON is the entire return path, and it is how an outside review becomes durable here rather than evaporating.","receipt_schema":"plectis_external_challenge_receipt_v0","what_the_project_does_with_it":"The project validates the receipt, binds it to the implementation hash it was actually run against -- so a receipt for bytes that have since changed cannot pass as current assurance -- and files it as external review state. A receipt naming a defeater is the strongest artifact this system can receive. You do not need to do any of that; you only need to print the block."},"outcome_semantics":{"score_authority":"none","asymmetry":"A challenge can only confirm or defeat. It never raises a score, count or pass rate. Surviving a challenge you did not design is weak evidence, because not finding a defeater is not proof of correctness. Failing one is strong evidence against the claim.","why_outside_input_is_admissible_here":"The anti-gaming replay quarantines cases outside the author's registry, correctly, because there an unregistered case could inflate a benchmark score. A challenge touches no score, so it can admit exactly the input that rule must refuse."},"open_challenges":[{"challenge_id":"audio_rms_independent_oracle","component_id":"batch8_audio_level_rms_port","what_you_control":"the input buffers, via your seed","declared_spec":"sqrt(sum(sample^2) / count) * 8, clamped to [0, 1]; int16 samples divide by Int16.max/32767","oracle_method":"Expected values are computed from the declared spec above by an oracle that never imports the implementation, and that accumulates with math.fsum where the implementation uses a running total -- so agreement is not an artifact of copying.","admissible_domain":"float32 and int16 buffers, length 0-64, amplitudes spanning silence, quiet, loud and clipping","metamorphic_relations":["permutation_invariance","sign_invariance","silence_is_zero","halving_halves_level","range_is_unit_interval","duplication_invariance"],"what_a_defeat_would_look_like":"any seeded buffer where the implementation and the independent oracle disagree beyond floating-point tolerance, any violated relation, or an undeclared format accepted instead of refused"},{"challenge_id":"claim_strength_mutation","component_id":"batch12_release_claim_language_gate","what_you_control":"the order claims are walked, via your seed","declared_spec":"Real public claims are strengthened minimally along quantifier, domain, modality, scope and negation axes, and the live release gate is asked whether it notices.","oracle_method":"Positive controls insert phrases already on the gate's own risky list. They establish that a catch would register. Without a caught control, a survivor count means nothing.","what_a_defeat_would_look_like":"a strengthened claim the gate does not classify as an active claim blocker -- reported as a survivor"}],"known_open_defeat":{"finding_id":"negative_context_bleed","found_by":"claim_strength_mutation","status":"open_unrepaired","summary":"The release gate decides that a risky phrase is a boundary clause if the line contains any negative marker anywhere, without checking that the negation scopes over the phrase. An unrelated disclaimer elsewhere in the same line therefore suppresses a live overclaim blocker.","minimal_reproduction":{"caught":"This system is production-ready.","escapes":"This system is production-ready. The colour is not blue."},"why_it_matters":"Public ceiling rows are built from disclaimer lists, so the gate is weakest on exactly the sentences that carry the strongest claims.","disclosure_posture":"Published unrepaired. Recording a defeat the project's own checks did not find is the point of the mechanism."},"self_contained_probe":{"schema":"plectis_self_contained_challenge_probe_v0","runs_where":"a separate Python 3 interpreter or coding-agent session; never the chat assistant merely reading the packet","dependencies":"python standard library only","how_to_use":["Leave the packet-reading chat in read-and-cite-only mode.","If execution is wanted, a human moves program_source into a separate Python 3 interpreter or coding-agent session.","Change SEED to a string you choose. Do not reuse the example.","Run it in that separate environment.","Attribute the printed RECEIPT to that separate run when handing it back."],"what_you_are_testing":"Whether the verbatim implementation agrees with an oracle derived from its own published spec, on inputs its author never chose, and whether six invariants hold that any correct implementation must.","chat_assistant_route":"Read the embedded implementation and declared spec, compare them, and report any divergence with packet paths. Do not execute the program or claim an execution result.","no_execution_path":"The implementation source and the declared spec are both embedded. Reading one against the other is a valid challenge and needs nothing executed.","implementation_under_test":{"source_path":"microcosm-substrate/src/microcosm_core/organs/batch8_audio_level_rms_port.py","file_sha256":"c7ab93c5ee6ce7032ec791b88d5a603ea5694344f8ab81d2763dbfb91368e600","embedded_verbatim":true},"program_bytes":13984,"program_source":"\"\"\"Plectis self-contained challenge probe.\n\nFor a separate evaluator environment. Standard library only. No network,\nno repository, no files written. The chat assistant that supplied or read\nthe packet must not claim it executed this program.\n\n1. Change SEED below to anything you like. Choose it yourself.\n2. Run this file.\n3. Hand the printed RECEIPT block back to whoever gave you this packet.\n\nYou are testing whether the implementation copied verbatim below agrees\nwith an oracle derived from its own published specification, on inputs\nits author never chose.\n\"\"\"\n\nfrom __future__ import annotations\n\nimport hashlib\nimport json\nimport math\nfrom typing import Any, Callable, Mapping, Sequence\n\n# Pick your own. The author has not seen it and cannot have tuned to it.\nSEED = \"change-me\"\n\nDECLARED_SPEC = 'sqrt(sum(sample^2) / count) * 8, clamped to [0, 1]; int16 samples divide by Int16.max/32767'\nIMPLEMENTATION_SOURCE = 'microcosm-substrate/src/microcosm_core/organs/batch8_audio_level_rms_port.py'\nIMPLEMENTATION_SHA256 = 'c7ab93c5ee6ce7032ec791b88d5a603ea5694344f8ab81d2763dbfb91368e600'\n\n\n# ======================================================================\n# IMPLEMENTATION UNDER TEST\n# Copied verbatim from microcosm-substrate/src/microcosm_core/organs/batch8_audio_level_rms_port.py.\n# Compare it against DECLARED_SPEC yourself -- that reading is itself a\n# challenge, and it needs no execution.\n# ======================================================================\ndef normalized_level(samples: Sequence[float | int], sample_format: str) -> float:\n    \"\"\"\n    [ACTION]\n    - Teleology: Compute normalized level for `batch8_audio_level_rms_port` using caller inputs and\n      module constants.\n    - Preconditions: Caller supplies `samples`, `sample_format` in the shape required by the signature.\n    - Reads: `samples`, `sample_format`, module constants, and any files named by those inputs.\n    - Writes: Return value only; durable writes, if any, are performed by explicitly named helper calls.\n    - Guarantee: Returns `float` with body-free status, card, receipt, or finding data as appropriate\n      for the function.\n    - Fails: Propagates parser, filesystem, JSON, and validation errors from called helpers; rejected\n      cases are represented as status or finding rows.\n    - Couples: Calls `sqrt`, `min`, `ValueError`, `max`, `float`; review those helpers when changing\n      this callable.\n    \"\"\"\n    if sample_format not in {\"float32\", \"int16\"}:\n        raise ValueError(f\"unsupported audio sample format: {sample_format}\")\n    if not samples:\n        return 0.0\n\n    total = 0.0\n    count = 0\n    if sample_format == \"float32\":\n        for sample in samples:\n            value = float(sample)\n            total += value * value\n            count += 1\n    else:\n        for sample in samples:\n            value = float(int(sample)) / 32767.0\n            total += value * value\n            count += 1\n\n    if count == 0:\n        return 0.0\n    rms = math.sqrt(total / count)\n    return min(max(rms * 8.0, 0.0), 1.0)\n\n\n# ======================================================================\n# INDEPENDENT ORACLE\n# Written from DECLARED_SPEC, never importing the implementation, and\n# accumulating with math.fsum where the implementation uses a running\n# total -- so agreement is not an artifact of copying.\n# ======================================================================\ndef oracle_audio_rms_level(samples: Sequence[float], sample_format: str) -> float:\n    \"\"\"Compute the declared normalized level independently of the organ.\n\n    Written from AUDIO_RMS_SPEC. Uses `math.fsum` over squares, whereas the\n    implementation under test accumulates with a running `total +=`. Agreement\n    across the two accumulation orders is therefore evidence about the\n    arithmetic, not an artifact of copying it.\n    \"\"\"\n    if sample_format not in (\"float32\", \"int16\"):\n        raise ValueError(f\"unsupported sample format: {sample_format}\")\n    if not samples:\n        return 0.0\n    if sample_format == \"int16\":\n        scaled = [float(int(value)) / 32767.0 for value in samples]\n    else:\n        scaled = [float(value) for value in samples]\n    mean_square = math.fsum(value * value for value in scaled) / len(scaled)\n    return min(max(math.sqrt(mean_square) * 8.0, 0.0), 1.0)\n\n\n# ======================================================================\n# SEEDED CASE GENERATION\n# Cases derive from sha256('{seed}|{label}|{index}'). Reimplement this\n# in any language to confirm they follow from your seed.\n# ======================================================================\ndef stream_block(seed: str, label: str, index: int) -> bytes:\n    \"\"\"Return the deterministic 32-byte block for (seed, label, index).\"\"\"\n    payload = f\"{seed}|{label}|{index}\".encode(\"utf-8\")\n    return hashlib.sha256(payload).digest()\n\n\nclass ChallengeRng:\n    \"\"\"Deterministic, portable value source driven by a reviewer-chosen seed.\"\"\"\n\n    def __init__(self, seed: str, label: str) -> None:\n        self.seed = seed\n        self.label = label\n        self._index = 0\n        self._buffer = b\"\"\n\n    def _take(self, count: int) -> bytes:\n        while len(self._buffer) < count:\n            self._buffer += stream_block(self.seed, self.label, self._index)\n            self._index += 1\n        head, self._buffer = self._buffer[:count], self._buffer[count:]\n        return head\n\n    def unit(self) -> float:\n        \"\"\"Uniform float in [0, 1).\"\"\"\n        return int.from_bytes(self._take(8), \"big\") / float(1 << 64)\n\n    def uniform(self, low: float, high: float) -> float:\n        return low + (high - low) * self.unit()\n\n    def below(self, bound: int) -> int:\n        \"\"\"Uniform integer in [0, bound).\"\"\"\n        if bound <= 0:\n            raise ValueError(\"bound must be positive\")\n        return int.from_bytes(self._take(8), \"big\") % bound\n\n    def between(self, low: int, high: int) -> int:\n        \"\"\"Uniform integer in [low, high].\"\"\"\n        return low + self.below(high - low + 1)\n\n    def pick(self, options: Sequence[Any]) -> Any:\n        return options[self.below(len(options))]\n\n\ndef generate_audio_cases(seed: str, count: int) -> list[dict[str, Any]]:\n    \"\"\"Derive audio buffers from the reviewer's seed.\n\n    The domain deliberately spans the clamp boundary, the empty buffer, single\n    samples and both declared formats, because those are where a normalisation\n    or clamping error would live.\n    \"\"\"\n    rng = ChallengeRng(seed, \"audio_rms\")\n    cases: list[dict[str, Any]] = []\n    for index in range(count):\n        sample_format = rng.pick([\"float32\", \"int16\"])\n        shape = rng.pick([\"quiet\", \"loud\", \"clipping\", \"single\", \"empty\", \"mixed\"])\n        if shape == \"empty\":\n            samples: list[float] = []\n        else:\n            length = 1 if shape == \"single\" else rng.between(2, 64)\n            if shape == \"quiet\":\n                span = 0.02\n            elif shape == \"loud\":\n                span = 0.2\n            elif shape == \"clipping\":\n                span = 1.0\n            else:\n                span = rng.uniform(0.01, 0.6)\n            if sample_format == \"int16\":\n                bound = max(1, int(span * 32767))\n                samples = [float(rng.between(-bound, bound)) for _ in range(length)]\n            else:\n                samples = [rng.uniform(-span, span) for _ in range(length)]\n        cases.append(\n            {\n                \"case_id\": f\"seeded_{index:03d}_{shape}_{sample_format}\",\n                \"sample_format\": sample_format,\n                \"samples\": samples,\n                \"shape\": shape,\n            }\n        )\n    return cases\n\n\n# ======================================================================\n# METAMORPHIC RELATIONS\n# Properties any correct implementation of the spec must satisfy, with\n# no stored expected value at all.\n# ======================================================================\ndef audio_metamorphic_relations(\n    level: Callable[[Sequence[float], str], float],\n    case: dict[str, Any],\n    rng: ChallengeRng,\n) -> list[dict[str, Any]]:\n    samples = list(case[\"samples\"])\n    fmt = case[\"sample_format\"]\n    base = level(samples, fmt)\n    results: list[dict[str, Any]] = []\n\n    def record(relation: str, holds: bool, detail: str) -> None:\n        results.append({\"relation\": relation, \"holds\": bool(holds), \"detail\": detail})\n\n    # Order must not matter: the level is a function of the multiset of samples.\n    if len(samples) > 1:\n        shuffled = list(samples)\n        for position in range(len(shuffled) - 1, 0, -1):\n            swap = rng.below(position + 1)\n            shuffled[position], shuffled[swap] = shuffled[swap], shuffled[position]\n        got = level(shuffled, fmt)\n        record(\"permutation_invariance\", math.isclose(got, base, abs_tol=1e-12),\n               f\"base={base!r} permuted={got!r}\")\n\n    # Squaring destroys sign, so negation must be invisible.\n    if samples:\n        negated = [-value for value in samples]\n        got = level(negated, fmt)\n        record(\"sign_invariance\", math.isclose(got, base, abs_tol=1e-12),\n               f\"base={base!r} negated={got!r}\")\n\n    # Silence is exactly zero, never a small positive artifact.\n    zeros = [0.0 if fmt == \"float32\" else 0] * max(1, len(samples))\n    got = level(zeros, fmt)\n    record(\"silence_is_zero\", got == 0.0, f\"zeros -> {got!r}\")\n\n    # Below the clamp, the level is linear in amplitude.\n    if samples and base > 0.0 and base < 0.5:\n        halved = [value / 2 for value in samples] if fmt == \"float32\" else [\n            float(int(value // 2)) for value in samples\n        ]\n        got = level(halved, fmt)\n        tolerance = 1e-9 if fmt == \"float32\" else 2e-3\n        record(\"halving_halves_level\", math.isclose(got, base / 2, abs_tol=tolerance),\n               f\"base={base!r} halved={got!r}\")\n\n    # The declared range is closed: no input may escape [0, 1].\n    record(\"range_is_unit_interval\", 0.0 <= base <= 1.0, f\"level={base!r}\")\n\n    # Duplicating the buffer leaves the mean square unchanged.\n    if samples:\n        got = level(samples + samples, fmt)\n        record(\"duplication_invariance\", math.isclose(got, base, abs_tol=1e-12),\n               f\"base={base!r} doubled={got!r}\")\n\n    return results\n\n\n# ======================================================================\n# DRIVER\n# ======================================================================\ndef run(seed):\n    cases = generate_audio_cases(seed, 200)\n    rng = ChallengeRng(seed, \"audio_metamorphic\")\n    defeaters = []\n    oracle_checks = 0\n    relation_checks = 0\n\n    for case in cases:\n        samples, fmt = case[\"samples\"], case[\"sample_format\"]\n        expected = oracle_audio_rms_level(samples, fmt)\n        try:\n            observed = normalized_level(samples, fmt)\n        except Exception as error:\n            defeaters.append({\"case_id\": case[\"case_id\"],\n                              \"kind\": \"unexpected_exception\",\n                              \"detail\": \"%s: %s\" % (type(error).__name__, error)})\n            continue\n        oracle_checks += 1\n        if not math.isclose(observed, expected, rel_tol=1e-12, abs_tol=1e-12):\n            defeaters.append({\"case_id\": case[\"case_id\"],\n                              \"kind\": \"oracle_disagreement\",\n                              \"detail\": \"expected=%r observed=%r\" % (expected, observed),\n                              \"samples\": samples, \"sample_format\": fmt})\n        for row in audio_metamorphic_relations(normalized_level, case, rng):\n            relation_checks += 1\n            if not row[\"holds\"]:\n                defeaters.append({\"case_id\": case[\"case_id\"],\n                                  \"kind\": \"metamorphic_violation\",\n                                  \"detail\": \"%s: %s\" % (row[\"relation\"], row[\"detail\"]),\n                                  \"samples\": samples, \"sample_format\": fmt})\n\n    refusals = []\n    for bogus in (\"int24\", \"\", \"FLOAT32\", \"pcm\"):\n        try:\n            normalized_level([0.1, 0.2], bogus)\n        except Exception as error:\n            refusals.append({\"sample_format\": bogus, \"refused\": True})\n        else:\n            refusals.append({\"sample_format\": bogus, \"refused\": False})\n            defeaters.append({\"case_id\": \"refusal_\" + (bogus or \"empty\"),\n                              \"kind\": \"missing_refusal\",\n                              \"detail\": \"undeclared format %r accepted\" % bogus})\n\n    return {\n        \"schema\": \"plectis_external_challenge_receipt_v0\",\n        \"produced_by\": \"self_contained_probe\",\n        \"reviewer_seed\": seed,\n        \"implementation_source\": IMPLEMENTATION_SOURCE,\n        \"implementation_sha256\": IMPLEMENTATION_SHA256,\n        \"declared_spec\": DECLARED_SPEC,\n        \"oracle_comparison_count\": oracle_checks,\n        \"metamorphic_check_count\": relation_checks,\n        \"refusal_check_count\": len(refusals),\n        \"defeater_count\": len(defeaters),\n        \"defeaters\": defeaters,\n        \"verdict\": \"defeater_found\" if defeaters else \"no_defeater_found\",\n        \"score_authority\": \"none\",\n        \"outcome_semantics\": (\n            \"A challenge can only confirm or defeat. Not finding a defeater is \"\n            \"not proof of correctness; finding one is evidence against the claim.\"\n        ),\n    }\n\n\nif __name__ == \"__main__\":\n    if SEED == \"change-me\":\n        print(\"WARNING: SEED is still the example value. Change it, then rerun.\\n\")\n    receipt = run(SEED)\n    print(\"verdict            :\", receipt[\"verdict\"])\n    print(\"oracle comparisons :\", receipt[\"oracle_comparison_count\"])\n    print(\"metamorphic checks :\", receipt[\"metamorphic_check_count\"])\n    print(\"refusal checks     :\", receipt[\"refusal_check_count\"])\n    print(\"defeaters          :\", receipt[\"defeater_count\"])\n    for d in receipt[\"defeaters\"][:10]:\n        print(\"   -\", d[\"kind\"], \":\", d[\"detail\"])\n    print()\n    print(\"----- RECEIPT: hand this block back -----\")\n    print(json.dumps(receipt, indent=2, sort_keys=True))\n    print(\"----- END RECEIPT -----\")\n\n"},"authority_boundary":"Report-only. Grants no release, publication, scoring or correctness authority. A challenge covers the declared domain of one component and says nothing about the private system or deployed behaviour."},"evidence_profiles_ref":{"schema":"microcosm_public_complete_corpus_section_ref_v0","path":"content-graph.json","json_pointer":"/evidence_profiles","count":359,"note":"Evidence-profile records are not duplicated into this corpus; resolve evidence_profile_ref ids against content-graph.json."}}