OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Assuming ZF is consistent, constructs a model in which every surjective image of a set injects into that set, yet the axiom of choice fails. Choice for ordinal-indexed families still holds. From any countable transitive model of ZFC, a separate construction gives a transitive symmetric extension with these properties and no new countable sequences of ground-model elements. (OpenAI’s Lean note)
Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.
The independent assessments find that the Lean statements express only part of the result described above, or a narrower statement.Read the assessments
Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
Let V be a countable transitive model of \ZFC. There are a set forcing \mathbb Q\in V, a V-generic filter G, and a transitive symmetric submodel W of V[G] such that
The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.
If \ZF is consistent, then so is \ZF+\PP+\ACwo+\neg\AC.
No Lean statement in the release is paired with this theorem.
The release does not say which paper theorem its catalogue describes. Both statements above are shown whole, the closer match to the description’s wording first. The atlas has not reviewed either choice.
Reported proof verification
Two Lean statements in the release; independent groups report that Comparator accepted runs for all these files at the version shown here.
We prove that the Partition Principle does not imply the Axiom of Choice: if ZF is consistent, then so is ZF with the Partition Principle, Choice for ordinal-indexed families, and the negation of the Axiom of Choice. Separately, over every countable transitive model of ZFC, we construct a transitive symmetric model of this theory with the same ordinals and no new countable sequences of ground elements.
Two Lean statements in this family
Choose one to see it beside the paper theorem it is paired with.
For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.
The symmetric forcing relation is definable formula by formula in V, and it forces every axiom instance of full pure-set \ZF. If V is externally transitive, the evaluated class W is a transitive model of that theory, with
The truth lemma holds for this evaluated model. Over an arbitrary countable choice ground, it holds for the symmetric quotient of internal names described in Section 8.
The words they share: formula, relation, axiom, transitive, ground, zf, choice, model, countable, internal, externally.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
Let V be a countable transitive model of \ZFC. There are a set forcing \mathbb Q\in V, a V-generic filter G, and a transitive symmetric submodel W of V[G] such that
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
Other mathematical statements in OpenAI’s Lean library
Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express part of the advertised result (their term: “partial”).
Their note:Stated: PartitionConsistency.main_consistency : Consistent ZF → Consistent (ZF ∪ {PP, ACwo, ¬AC}) (a hand-coded first-order Derives calculus, full ZF with arbitrary Separation/Replacement instances, PP = every internal surjection X ↠ Y yields an internal injection Y ↪ X, ACwo = choice for ordinal-indexed families), which is the summary’s first two claims. The third claim, a transitive symmetric extension of any countable transitive model of ZFC with no new countable sequences, appears only as exists_model_partitionPrinciple_without_choice and needs an extra hypothesis (an internal strongly inaccessible K in the ground model, docs: ‘containing an internal strongly inaccessible cardinal’) and its conclusion lacks ordinal-indexed choice and the no-new-sequences property (docs: ‘stronger transitive-model preservation assertions are outside’).
Definitions the group asks readers to compare with the paper:Hand-written proof theory and set theory: Formula, Derives (natural deduction), Consistent T := ∀ n, ¬ Derives (T.at n).bot, ZF := {p | IsZFAxiom p} with separation/replacement coded by variable-index arithmetic, PP, AC, ACwo. The theorem is an implication from Consistent ZF, so a coding error that made the encoded ZF inconsistent would make it vacuous; PartitionPrinciple uses its own SetFormula/Realize semantics on ZFSet. (axiom_files hit for LocalRecords.lean is a doc-comment false positive.)
“partial”, in their words: Only a special case is stated, or a co-equal headline claim has no challenge statement.
Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.
cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — DECLARED”).
The gap they name:In the countable-transitive-model clause, the Lean adds an internal strongly inaccessible cardinal to the hypotheses and drops from the conclusion ACwo-in-N, “symmetric extension” and “no new countable sequences of ground-model elements”. The relative-consistency clauses are fully stated.
“NARROWER — DECLARED”, in their words: The challenge statements state less than the headline, and OpenAI’s Lean note says so.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
The Partition Principle says that every surjection admits an injection in the reverse direction. The formalization proves the relative-consistency implication: if ZF is consistent, then so is ZF with the Partition Principle, Choice for ordinal-indexed families, and failure of the Axiom of Choice.
The linked model-construction result also constructs a transitive model of the Partition Principle without Choice from an externally countable transitive ground model satisfying the stated axioms, including Choice, and containing an internal strongly inaccessible cardinal. The paper’s stronger transitive-model preservation assertions are outside that selected construction.
AI-drafted comparisons, with no recorded human review
Partition Principle without Choice from a ground modelPartitionPrinciple
Unreviewed AI draft: states a weaker form of a paper’s theorem. States a transitive model of the Partition Principle without Choice from a countable transitive ground model that also contains a strongly inaccessible cardinal, an assumption the paper’s transitive-model theorem does not make; the theorem’s preservation assertions are outside the statement.
Relative consistency of the Partition Principle with ordinal Choice and failure of ChoicePartitionConsistency
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main consistency theorem: if ZF is consistent, so is ZF with the Partition Principle, ordinal-indexed choice and the failure of the Axiom of Choice.
Who ran the proof-verification program?
These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.
These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.
Partition Principle without Choice from a ground modelPartitionPrinciple
One report, at the release’s recorded version.
Reported by
Their word
Their account
Date
Original report
cert-machine
“CERTIFIED”
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
The program that collects Atlas’s source records, Statement Map, could not obtain or interpret some source material. These notes describe those limits; they do not identify mathematical errors.
PartitionPrinciple: Tracked source lean/OAI/SetTheory/PartitionPrinciple/Main.lean unavailable: bytes were not captured at fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb
What these papers cite in the release, and what cites them
These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.
Papers elsewhere in the release that this family’s papers cite
This family’s papers cite no paper elsewhere in the release.
Papers elsewhere in the release that cite this family’s papers
No paper elsewhere in the release cites this family’s papers.
References outside the release
Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release
Explain a result, compare statements or propose a correction
Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.
Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.
Commentary on this family by others, dated and credited. It is commentary, not a finding.
An unsigned public tier list, the OpenAI Math Tier List, classes it as a counterexample and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.
Asaf Karagila, Why I don’t trust Lean code from AI, 10 October 2026. Source
The author says they did not read or run the Lean proof and do not plan to, but are willing to accept that the formal statement of this principle is easy enough to get right.
Asaf Karagila, OpenAI, the Partition Principle, and Mathematics, 8 October 2026. Source
The author, who works on this problem, skimmed the manuscript but not the Lean code, found its structure and terminology hard to follow, and made no judgement on whether the result is correct or new.
From mathstodon.xyz/@johncarlosbaez, 7 October 2026. Source
Quotes Karagila’s blog post on the Partition Principle paper to their followers.
Robert Joseph George, Do the Lean statements in openai/math match their papers?, 7 October 2026. Source
The Lean statement assumes an inaccessible cardinal in the ground model and concludes failure of choice without the retained choice for ordinal-indexed families; the documentation discloses the cardinal but not that omission.
Questions awaiting review
Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.
Read the Lean statement against the paper
An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded.Open the form
An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded.Open the form
Identify the theorem the catalogue describes
Family 244The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means.Open the form
Address a question raised in an independent assessment
Only a special case is stated, or a co-equal headline claim has no challenge statement.
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.
The challenge statements state less than the headline, and OpenAI’s Lean note says so.
Their method
Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.
Re-run through Comparator with Lean’s kernel: accepted (the check took 54 s).
Date
10 October 2026
Version
at the release’s recorded version
Their method
Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory.
Relative consistency of the Partition Principle with ordinal Choice and failure of Choice
Role
the release lists it as a supporting statement
Paired with
Theorem 1.2 (candidate pairing)
Who ran the proof-verification program
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.