OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Gives an alternative to Tanaka’s ZFC construction of a unital infinite-dimensional simple complex C∗-algebra with a faithful tracial state and exactly one nonzero irreducible representation up to unitary equivalence. Thus the unrestricted compact-operator characterization fails without additional set-theoretic assumptions; the counterexample is nonseparable. (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.
Both independent assessments find that the Lean statements express or imply the whole result described above.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.
There exists a unital infinite-dimensional simple complex C^*-algebra A with a faithful tracial state such that all its nonzero irreducible representations are unitarily equivalent. In particular, A is not isomorphic to K(H) for any complex Hilbert space H.
Reported proof verification
One Lean statement in the release; three independent groups report that Comparator accepted runs for this file at the version shown here.
We construct in ZFC a unital infinite-dimensional simple complex C∗-algebra with a faithful tracial state whose nonzero irreducible representations are all unitarily equivalent. This gives a negative answer to Naimark’s problem without additional set-theoretic assumptions.
One Lean statement in this family
Lean is a language for mathematical statements and proofs that a computer can verify. The Lean statement below is shown beside its possible paper counterpart, with the reason they were linked.
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.
There exists a unital infinite-dimensional simple complex C^*-algebra A with a faithful tracial state such that all its nonzero irreducible representations are unitarily equivalent. In particular, A is not isomorphic to K(H) for any complex Hilbert space H.
The words they share: equivalent, isomorphic, unitarily, complex, space, dimensional, irreducible, representations, faithful, simple, tracial, state, algebra, infinite, hilbert, unital.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
Naimark.lean, the release lists it as a supporting statement.
theoremmain : ∃ (A : Type u) (_ : CStarAlgebra A), IsSimpleCStar A ∧ ¬ FiniteDimensional ℂ A ∧ (∃ τ : A →L[ℂ] ℂ, IsFaithfulTracialState τ) ∧ HasUniqueIrreducibleRepresentation.{u, v, w} A ∧ ∀ (H : Type v) [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H], ¬ IsIsomorphicToCompacts A H := bysorry
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.
Seven definitions are compared by name and type only: IsSimpleCStar, IsState, IsFaithfulTracialState, IsIrreducible, UnitarilyEquivalent, HasUniqueIrreducibleRepresentation and 1 more.
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”.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
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 or imply the whole advertised result (their term: “full”).
Their note:main : ∃ (A : Type u) (_ : CStarAlgebra A), IsSimpleCStar A ∧ ¬ FiniteDimensional ℂ A ∧ (∃ τ, IsFaithfulTracialState τ) ∧ HasUniqueIrreducibleRepresentation A ∧ ∀ (H : Type v) [Hilbert], ¬ IsIsomorphicToCompacts A H (universe-polymorphic): a unital infinite-dimensional simple C*-algebra with a faithful tracial state all of whose irreducible representations (nonzero, no nontrivial closed invariant subspace) are unitarily equivalent, yet not isomorphic to the compact operators on any Hilbert space. Matches the summary (nonseparability is a consequence, not stated; the summary’s ‘alternative to Tanaka’ framing is irrelevant to the statement).
Definitions the group asks readers to compare with the paper:Hand-written C*-algebra representation theory: IsSimpleCStar, IsFaithfulTracialState, IsIrreducible π := π ≠ 0 ∧ no nontrivial closed π-invariant subspace, UnitarilyEquivalent, IsIsomorphicToCompacts A H := ∃ injective φ : A →⋆ₙₐ[ℂ] (H →L H), range = compact operators; standard, quantified over Hilbert spaces in explicit universes.
“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.
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.
Naimark’s problem asks whether a $C^*$-algebra whose nonzero irreducible representations are all unitarily equivalent must be an algebra of compact operators. The formalization gives a counterexample in ZFC: a unital infinite-dimensional simple complex $C^*$-algebra with a faithful tracial state has that uniqueness property for irreducible representations but is not isomorphic to the compact operators on any Hilbert space. No additional set-theoretic assumption is used.
AI-drafted comparisons, with no recorded human review
Counterexample to Naimark’s problem in ZFCNaimark
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem in ZFC: a unital infinite-dimensional simple C*-algebra with a faithful tracial state, one irreducible representation up to equivalence, and not isomorphic to compact operators. Seven definitions are Comparator holes; their library bodies match the challenge file.
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.
Counterexample to Naimark’s problem in ZFCNaimark
Three reports, all 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.
Other people’s accounts of running proof-verification software, with links to their reports.
Jack McCarthy, Re-running Comparator with the definition holes pinned, 9 October 2026. Source
With the definition holes pinned so the solver’s definitions must match the challenge, Comparator reports the solution is okay for this challenge, so the hole was not being abused by OpenAI’s own solution.
Jack McCarthy, OpenAI math: Comparator fixes (mirror of openai/math at fd4aeeb), 9 October 2026. Source
With all 7 definition holes removed from the configuration, so that Comparator compares the definition bodies too, the release’s solution still passes; run with the Lean kernel only, on macOS without Comparator’s sandbox.
From github.com/BenKnill/openai-math-hol-cards, 8 October 2026. Source
Strict Comparator re-check with the definition holes removed, so hole bodies are compared too, accepted the solution with the three standard axioms. All 7 of its hole bodies matched the challenge. Run at adc7f124; the challenge file is unchanged at fd4aeeb2.
Robert Joseph George, Do the Lean statements in openai/math match their papers?, 7 October 2026. Source
The review reports that Comparator accepted OpenAI’s solution for this challenge, run end to end on Lean 4.34.1 with the default kernel and only the three standard axioms.
Problems reported by others
Reports of errors and corrections from outside readers, as they state them. The atlas has not judged them.
Alex Meiburg, Several of the comparator challenges in the OpenAI release are broken and hackable, 8 October 2026. Source
Reported as broken in the same way: its operator-algebra definitions are listed as holes, so a trivial replacement makes the checked statement vacuous.
Lean community, OpenAI releases solutions to hundreds of problems (117 messages, 6-10 Oct 2026), 6 October 2026. Source
Naimark’s Comparator file only proves existence under Lean’s axioms, leaving open reliance on inaccessible-cardinal axioms, which undercuts the ZFC point of the result.
Explanations by others
Explanations of this family written elsewhere, by the people named. Each is their own reading of the mathematics.
Timothy Chow relays Alex Meiburg’s Lean Zulip finding that some Comparator challenges let definitions be overridden and that the Naimark Lean statement proves provability in Lean, not in ZFC.
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 “A strong professor’s paper of the year”. The page says that nothing on it is verified.
Alex Meiburg, Several of the comparator challenges in the OpenAI release are broken and hackable, 9 October 2026. Source
The same author adds that the Naimark challenge shows only a construction under standard axioms, leaving open that it relies on inaccessible cardinal axioms, whereas the point of the result was to avoid them.
tparker, Philip Weiss, Will Sawin, Nik Weaver, Pace Nielsen, mathworker21, Joshua P. Swanson, Timothy Chow, Daniel Soltesz, MathOverflow: how many of the released results went against the conventional wisdom?, 8 October 2026. Source
An operator algebraist says the field expected a positive solution to Naimark’s problem to be consistent with ZFC, so a ZFC counterexample runs against their expectation.
tparker with nine answerers, How many of the results released by OpenAI went against the conventional mathematical wisdom?, 8 October 2026. Source
Weaver says a ZFC-provable counterexample to Naimark’s problem was not what they expected, given consistency results since 2004.
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
Read the definitions the solution supplies
OpenAI’s proof file supplies some definitions. Atlas records no comparison confirming that they mean the same objects as the paper’s definitions.IsSimpleCStar, IsState, IsFaithfulTracialState, IsIrreducible, UnitarilyEquivalent, HasUniqueIrreducibleRepresentation, IsIsomorphicToCompactsOpen the form
Identify the theorem the catalogue describes
Family 297The 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
The challenge statements imply the headline claim after unpacking definitions or one routine step.
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 the headline as OpenAI wrote it.
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 400 s; peak memory 6.5 GiB).
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.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.