Lean statement

Prescribed odd-characteristic counterexample

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.OddKaplansky.main_theorem

Lean statement file: OddKaplansky.leanShow on the mapFamily 197. OpenAI lists this as a main result in its Lean catalogue.

See the candidate theorem Who ran the proof-verification program Report a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

def MainClaim : Prop :=  sourcePrime.Prime ∧ Odd sourcePrime ∧  ∃ (K : Type) (_ : Field K) (_ : Fintype K) (_ : CharP K sourcePrime),    Fintype.card K = sourcePrime^4 ∧    ∃ (G : Type) (_ : Group G) (_ : Group.FG G),      (∃ g : G, g ≠ 1 ∧ IsOfFinOrder g) ∧      ∃ a b : MonoidAlgebra K G,        a * b = 1 ∧ b * a ≠ 1 ∧        Function.Injective (cellular b) ∧ ¬ Function.Surjective (cellular b)⋮theorem main_theorem : MainClaim := by  sorry

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.

Definitions needed to read it

Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.

definition OAI.OddKaplansky.sourceM line 7

@[irreducible] def sourceM : ℕ := Nat.choose 1200 600

definition OAI.OddKaplansky.sourcePrime line 8

@[irreducible] def sourcePrime : ℕ := Nat.minFac ((Nat.factorial sourceM)^2 + 1)noncomputable section

definition OAI.OddKaplansky.cellular line 12

def cellular {K G : Type} [Field K] [Group G]    (d : MonoidAlgebra K G) (x : G → K) (g : G) : K :=  d.coeff.sum fun u c => c * x (g * u)

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: min, prime, odd, order, integer, field, finite and group. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: binom, generated, torsion and finitely.

Statements in A Counterexample to Kaplansky’s Direct-Finiteness Conjecture in Odd Characteristic 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 Odd Characteristic

Theorem

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.

Section 1, Introduction Read it in the paper TeX source, line 13

Show the TeX

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 $ba\ne1$.

Other possible paper matches, closest wording first

Theorem 1.1 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, odd, order, field, finite and group. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: characteristic, element and finitely.
Proposition 8.1 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: cellular, injective, field, finite and group.
Corollary 7.3 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: surjective, cellular, injective, field and group.
Proposition 4.2 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: natural, algebras, injective and group.
Proposition 7.2 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: surjective, cellular, injective, field, finite and group.
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, odd, order, integer, field, finite and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: characteristic, finitely and element.

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.

Carlos Toledo’s cert-machine

How they ran it, in their words

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…

Accepted Their term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Reported 10 October 2026, at the release version shown here. Original verification report

MathVet

How they ran it, in their words

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.

Accepted Their term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Reported 8 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

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.

Allowed axioms: propext, Quot.sound, Classical.choice

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.

MathVet
Finds part of it stated, or a narrower or weaker statement Their 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).
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their 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.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“Let $p$ be the smallest prime divisor of $\bigl(\binom{1200}{600}!\bigr)^2+1$; this prime is odd. The formalized result constructs a field $K$ of order $p^4$, a finitely generated group $G$ with torsion, and elements $a,b$ violating this implication.”
Quoted from OpenAI’s Lean note for this family

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

33 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: 8 to its own declarations, 17 to Mathlib’s documentation. 5 declared names also link 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.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves main_theorem exactly as written here. To compare that statement with the paper, also read the definitions and lemmas listed below. Atlas found them by following names in the source text and lists each after the names it uses. For x.foo, Atlas includes every foo defined here for a type, even when the statement may use only one of them.

It reaches all 4 other declarations of the file; with the theorem itself they span 16 lines. 1 of them is named in the statement itself.

Names in the namespace OAI.OddKaplansky are listed without it.

def sourceMline 7library
def sourcePrimeline 8library
def cellularline 12library
def MainClaimline 16in the statementlibrary
theorem main_theoremline 26selected theoremlibrary

import Mathlibnamespace OAInamespace OddKaplansky@[irreducible] def sourceM : ℕ := Nat.choose 1200 600@[irreducible] def sourcePrime : ℕ := Nat.minFac ((Nat.factorial sourceM)^2 + 1)noncomputable sectiondef cellular {K G : Type} [Field K] [Group G]    (d : MonoidAlgebra K G) (x : G → K) (g : G) : K :=  d.coeff.sum fun u c => c * x (g * u)def MainClaim : Prop :=  sourcePrime.Prime ∧ Odd sourcePrime ∧  ∃ (K : Type) (_ : Field K) (_ : Fintype K) (_ : CharP K sourcePrime),    Fintype.card K = sourcePrime^4 ∧    ∃ (G : Type) (_ : Group G) (_ : Group.FG G),      (∃ g : G, g ≠ 1 ∧ IsOfFinOrder g) ∧      ∃ a b : MonoidAlgebra K G,        a * b = 1 ∧ b * a ≠ 1 ∧        Function.Injective (cellular b) ∧ ¬ Function.Surjective (cellular b)theorem main_theorem : MainClaim := by  sorryendend OddKaplanskyend OAI

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.

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

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.

Section 1, Introduction Read it in the paper TeX source, line 8

Show the TeX

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.

A Counterexample to the Group-Ring Determinant Conjecture

Theorem

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.

Section 1, Introduction and proof overview Read it in the paper TeX source, line 90

Show the TeX

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.

Theorem ([A Counterexample to Kaplansky’s…, Theorem 1.1])

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.

Section 1, Introduction and proof overview Read it in the paper TeX source, line 107

Show the TeX

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 $b_2a_2\ne1$.

A Counterexample to Kaplansky’s Direct-Finiteness Conjecture in Odd Characteristic

Theorem

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.

Section 1, Introduction Read it in the paper TeX source, line 13

Show the TeX

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 $ba\ne1$.

Configuration and file facts

File: lean/ComparatorChallenges/OddKaplansky.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Algebra/OddKaplansky/Main.lean
Allowed axioms: propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports: Mathlib
Imports from outside Mathlib: None
Local vocabulary: 4 local vocabulary declarations
Lines of Lean, without comments or blank lines: 16
Second proof-verification program (nanoda): Off in OpenAI's configuration.

A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.

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