Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.
Theorem selected for verification:OAI.KaplanskyCounterexample.finitelyPresented_counterexample
A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.
Possible paper match, with no recorded review
Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.
Words of the Lean statement, its names split into words, that also appear in the theorem: prime, presented, characteristic, odd, order, two, finitely, field and two more. Found in the paper’s title, which its main theorem states: direct, counterexample and finiteness. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: element.
Statements in A Counterexample to Kaplansky’s Direct-Finiteness Conjecture in Characteristic Two come first because the release’s name for this Lean statement shares words with that paper’s title.
Theorem 1.1 of A Counterexample to Kaplansky’s Direct-Finiteness Conjecture in Characteristic Two
Theorem
There exist a finite field K of characteristic two, a finitely presented group G containing an element of odd prime order, and elements a_{\mathrm{out}},b_{\mathrm{out}}\in K[G] such that
There exist a finite field $K$ of characteristic two, a finitely presented group $G$ containing an element of odd prime order, and elements $a_{\mathrm{out}},b_{\mathrm{out}}\in K[G]$ such that
\[
a_{\mathrm{out}}b_{\mathrm{out}}=1,
\qquad b_{\mathrm{out}}a_{\mathrm{out}}\ne1.
\]
The field, group, and finite sums are specified by a terminating prescription involving finite sets and finite-field arithmetic.
Other possible paper matches, closest wording first
Theorem 1.1 of A Counterexample to Kaplansky’s Direct-Finiteness Conjecture in Odd Characteristic. Words of the Lean statement, its names split into words, that also appear in the theorem: torsion, prime, odd, order, integer, finitely and three more. Found in the paper’s title, which its main theorem states: direct, counterexample, finiteness and characteristic. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: generated.
Theorem 3.1 of A Counterexample to Kaplansky’s Direct-Finiteness Conjecture in Characteristic Two. Words of the Lean statement, its names split into words, that also appear in the theorem: prime, presented, characteristic, odd, order, two and five more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: constructions and element.
Theorem 1.1 of A Torsion-Free Group Algebra That Is Not Directly Finite. Words of the Lean statement, its names split into words, that also appear in the theorem: torsion, presented, two, finitely, finite and group. Found in the paper’s title, which its main theorem states: algebra. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: elements.
Corollary 7.4 of A Counterexample to Kaplansky’s Direct-Finiteness Conjecture in Characteristic Two. Words of the Lean statement, its names split into words, that also appear in the theorem: characteristic, two, field, finite and group. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: elements.
Proposition 2.3 of A Counterexample to Kaplansky’s Direct-Finiteness Conjecture in Characteristic Two. Words of the Lean statement, its names split into words, that also appear in the theorem: prime, odd, order, two, field and finite.
Theorem 1.2 of A Counterexample to the Group-Ring Determinant Conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: characteristic, two, finitely, field, finite and group. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: generated.
How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.
Reported Comparator runs
two independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.
For a configured theorem, Comparator tests whether OpenAI’s proof proves the exact Lean statement. Some files instead configure definitions for comparison by name and type. Acceptance does not establish that the statement means the same thing as the paper’s theorem.
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 (.github/workflows/openai-math-kernel.yml), one j…
AcceptedTheir term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
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.
AcceptedTheir term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.
This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.
Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.
Do family 197’s Lean statements express the advertised result?
These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.
Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
Finds part of it stated, or a narrower or weaker statementTheir term: “partial”.
Only a special case is stated, or a co-equal headline claim has no challenge statement. Their record
Their note: Determinant counterexample is stated (GroupRingDeterminant.main: f.g. G, n >= 1, integral matrix A invertible over ℚ[G], bounded left-regular T invertible with 0 < fkDet T < 1, fkDet T = exp(Re tr(CFC.log(T*T))/2)) and a Gottschalk counterexample is stated only for odd characteristic (OddKaplansky: K of order p^4, f.g. G with torsion, ab = 1 ∧ ba ≠ 1, cellular b injective and not surjective). The summary's headline for Kaplansky is a finitely presented TORSION-FREE NONSOFIC group over F_2: the char-2 statements (KaplanskyDirectFiniteness, KaplanskyFinitelyPresented) give a f.g./f.p. group that has an element of odd prime order, and nonsofic/torsion-free are not stated (docs: 'The further conclusion that the group is nonsofic is outside these statements').
Definitions the group asks readers to compare with the paper: @[irreducible] def sourcePrime : ℕ := Nat.minFac ((Nat.factorial sourceM)^2 + 1) with sourceM := Nat.choose 1200 600 in OddKaplansky: the characteristic is a specific astronomically large, non-computable prime; existence of a counterexample is claimed only for this prime (not suspicious in logic, but the statement is for one prescribed p).
Finds part of it stated, or a narrower or weaker statementTheir term: “NARROWER — UNLINKED”.
The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link. Their record
Their note: The headline's lead claim — a TORSION-FREE, nonsofic, finitely presented group with 𝔽2[G] not directly finite — is not stated. The finitely presented Lean example is required to have torsion (∃ g with orderOf g = an odd prime) and lives over an unspecified finite field of characteristic 2. Nonsoficity is declared outside; torsion-freeness and the field 𝔽2 are not mentioned by the scope note, which does not link the October 4 torsion-free manuscript.
OpenAI’s own note on what this Lean statement covers
“The formalized results construct a finite field of characteristic two and a group algebra violating this implication. One statement records the counterexample for a finitely generated group; the more detailed construction gives a finitely presented group with an element of odd prime order.”
An AI agent drafted an interpretation of that note, classifying this statement as “states a paper’s theorem as the paper does”. No human review of the draft is recorded.
Supporting material
The complete Lean statement file
14 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.
Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.
Links in the file: 0 to its own declarations, 8 to Mathlib’s documentation. 1 declared name also links to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.
import Mathlibnamespace OAI.KaplanskyCounterexample/-- A finite characteristic-two field and a finitely presented group with odd-primeorder torsion give a counterexample to direct finiteness (paper, Theorem 1.1). -/theoremfinitelyPresented_counterexample : ∃ (K : Type) (_ : Field K) (_ : Fintype K) (_ : CharP K 2), ∃ (G : Type) (_ : Group G) (_ : Group.IsFinitelyPresented G), (∃ (ℓ : ℕ) (_ : ℓ.Prime) (_ : Odd ℓ) (g : G), orderOf g = ℓ) ∧ ∃ a_out b_out : MonoidAlgebra K G, a_out * b_out = 1 ∧ b_out * a_out ≠ 1 := bysorryend OAI.KaplanskyCounterexample
Every extracted statement of the accompanying papers
The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.
There exist a finite field K of characteristic two, a finitely presented group G containing an element of odd prime order, and elements a_{\mathrm{out}},b_{\mathrm{out}}\in K[G] such that
There exist a finite field $K$ of characteristic two, a finitely presented group $G$ containing an element of odd prime order, and elements $a_{\mathrm{out}},b_{\mathrm{out}}\in K[G]$ such that
\[
a_{\mathrm{out}}b_{\mathrm{out}}=1,
\qquad b_{\mathrm{out}}a_{\mathrm{out}}\ne1.
\]
The field, group, and finite sums are specified by a terminating prescription involving finite sets and finite-field arithmetic.
There are a finitely generated discrete group G_D, an integer n\ge1, and A\in\Mat_n(\Z[G_D]) invertible over \Q[G_D] such that
0<\det_{\N(G_D)}(T_A)<1.
The integral in Equation (eq:conjecture) is finite and strictly negative. Thus the unrestricted group-ring Determinant Conjecture has a negative resolution.
There are a finitely generated discrete group $G_D$, an integer
$n\ge1$, and $A\in\Mat_n(\Z[G_D])$ invertible over $\Q[G_D]$ such
that
\[
0<\det_{\N(G_D)}(T_A)<1.
\]
The integral in Equation~\eqref{eq:conjecture} is finite and strictly
negative. Thus the unrestricted group-ring Determinant Conjecture
has a negative resolution.
There exist a finite field K_2 of characteristic two, a finitely generated group H_2, and finite sums a_2,b_2\in K_2[H_2] with a_2b_2=1 and b_2a_2\ne1.
There exist a finite field $K_2$ of characteristic two, a finitely
generated group $H_2$, and finite sums $a_2,b_2\in K_2[H_2]$ with
$a_2b_2=1$ and $b_2a_2\ne1$.
Then p is an odd prime. There are a specified field K of order p^4, a specified finitely generated group G containing torsion, and specified finite sums a,b\in K[G] satisfying ab=1 and ba\ne1.
Put
\[
m=\binom{1200}{600},\qquad
p=\min\{h\in\mathbb Z:h>1,\ h\mid(m!)^2+1\}.
\]
Then $p$ is an odd prime. There are a specified field $K$ of order $p^4$,
a specified finitely generated group $G$ containing torsion, and specified
finite sums $a,b\in K[G]$ satisfying $ab=1$ and $ba\ne1$.
Family 197: A torsion-free group algebra that is not directly finite
Candidate pairing
Theorem
There exist a finite field K of characteristic two, a finitely presented group G containing an element of odd prime order, and elements a_{\mathrm{out}},b_{\mathrm{out}}\in K[G] such that
A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.
Recorded comparisons with the paper
Atlas records no contributed comparison of this statement with the paper.
Contributed Comparator run reports
Atlas records no contributed report of running Comparator for this statement file.
More about this statement
Commentary The headline claims a finitely presented torsion-free nonsofic group over the field with two elements; the statements allow any finite field of characteristic two and one needs an element of odd prime order. Robert Joseph George, Do the Lean statements in openai/math match their papers?, 2026-10-07Source
Questions to resolve when comparing this statement with the paper
Both independent readers find this family's challenges state less than its headline, while the atlas's AI draft classes this file as stating a paper's theorem as the paper does; check which part of the headline it covers.