Lean statement

Coefficient-group injectivity for unimodular one-relator quotients

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.Kervaire.coefficient_injective

Lean statement file: Kervaire.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 256 link it.

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.

theorem coefficient_injective (A : Type u) [Group A] (w : WordGroup A)    (hw : Unimodular w) : Function.Injective (coefficientMap w) := 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.

abbreviation OAI.Kervaire.WordGroup line 11

abbrev WordGroup (A : Type u) [Group A] := Monoid.Coprod A (Multiplicative ℤ)

definition OAI.Kervaire.exponentSum line 13

def exponentSum {A : Type u} [Group A] (w : WordGroup A) : ℤ :=  Multiplicative.toAdd (Monoid.Coprod.snd w)

definition OAI.Kervaire.Unimodular line 16

def Unimodular {A : Type u} [Group A] (w : WordGroup A) : Prop :=  exponentSum w = 1 ∨ exponentSum w = -1

definition OAI.Kervaire.relatorClosure line 19

def relatorClosure {A : Type u} [Group A] (w : WordGroup A) : Subgroup (WordGroup A) :=  Subgroup.normalClosure ({w} : Set (WordGroup A))

abbreviation OAI.Kervaire.RelatorQuotient line 25

abbrev RelatorQuotient {A : Type u} [Group A] (w : WordGroup A) :=  WordGroup A ⧸ relatorClosure w

definition OAI.Kervaire.coefficientMap line 28

def coefficientMap {A : Type u} [Group A] (w : WordGroup A) : A →* RelatorQuotient w :=  (QuotientGroup.mk' (relatorClosure w)).comp Monoid.Coprod.inl

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: unimodular, injective, word and group.

Statements in The Kervaire theorem for groups come first because the release lists only that paper for this Lean statement.

Theorem 1.1 of The Kervaire theorem for groups

Theorem

For every group A and every unimodular word w\in A*\langle t\rangle, the canonical homomorphism

A\longrightarrow (A*\langle t\rangle)/\langle\!\langle w\rangle\!\rangle_H

is injective.

Section 1, Introduction Read it in the paper TeX source, line 23

Show the TeX

For every group $A$ and every unimodular word $w\in A*\langle t\rangle$,
the canonical homomorphism
\[
 A\longrightarrow (A*\langle t\rangle)/\langle\!\langle w\rangle\!\rangle_H
\]
is injective.

Other possible paper matches, closest wording first

Theorem 1 of Nonsingular systems of equations over arbitrary groups. Words of the Lean statement, its names split into words, that also appear in the theorem: exponent, integers, injective and group.
Theorem 4.1 of The Kervaire theorem for groups. Words of the Lean statement, its names split into words, that also appear in the theorem: word. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: one.
Proposition 4.1 of Nonsingular systems of equations over arbitrary groups. Words of the Lean statement, its names split into words, that also appear in the theorem: exponent and word. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: one.
Corollary 1.2 of The Kervaire theorem for groups. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: conjecture.
Theorem 2 of Nonsingular systems of equations over arbitrary groups. Words of the Lean statement, its names split into words, that also appear in the theorem: integers and groups.

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

three independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet; Neal McBurnett’s openai-math-lean-tooling. 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

Neal McBurnett’s openai-math-lean-tooling

How they ran it, in their words

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; the run was still in progress at this snapshot (192 accepted, 224 pending).

Accepted Their term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 719 s; peak memory 6.6 GiB).
Reported 11 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 256’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: Only the Kervaire special case is stated: coefficient_injective (A) (w : WordGroup A) (hw : Unimodular w) : Function.Injective (coefficientMap w) with WordGroup A := Monoid.Coprod A (Multiplicative ℤ) and Unimodular w := exponentSum w = 1 ∨ exponentSum w = -1 (one equation in one unknown with exponent sum ±1; docs: 'the selected unimodular-relator result underlying the conjecture'). The summary's headline, Howie's conjecture for arbitrary finite systems whose exponent-sum matrix has full row rank over ℚ (simultaneous solution in an overgroup), is not stated; the Kervaire companion claim is covered (coefficient injectivity implies nontriviality).
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: Howie's theorem on finite nonsingular systems (the headline's lead clause, from the October 5 manuscript) is not stated, not even the general one-equation case (exponent sum != 0). The coefficient-embedding clause is stated only for one unimodular relator.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The linked formalization proves the stronger coefficient-injectivity statement used in the paper: for any group $A$ and any relator in $A*\mathbb Z$ whose exponent sum in the cyclic generator is $1$ or $-1$, the natural map from $A$ into the quotient by that relator is injective. This is the selected unimodular-relator result underlying the conjecture.”
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

39 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: 19 to its own declarations, 20 to Mathlib’s documentation. 7 declared names also link to the same declaration in OpenAI’s library. 1 identifier stays plain because it resolves to no single declaration: a variable Lean binds automatically, notation, or a name the scan of the pinned sources did not find. 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 coefficient_injective 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 6 of the file’s 7 other declarations; with the theorem itself they span 14 lines. 3 of them are named in the statement itself.

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

abbrev WordGroupline 11in the statementlibrary
def exponentSumline 13library
def Unimodularline 16in the statementlibrary
def relatorClosureline 19library
abbrev RelatorQuotientline 25library
def coefficientMapline 28in the statementlibrary
theorem coefficient_injectiveline 31selected theoremlibrary

Other Lean rules that may affect the statement 1

Lean can use instances, notation and attributes without naming them in the statement. These are rules for interpreting or finding mathematical objects. Atlas’s scan of names cannot follow all such uses; read the listed source as well.

instance {A : Type u} [Group A] (w : WordGroup A) : (relatorClosure w)… line 22

import Mathlibnamespace OAInoncomputable sectionnamespace Kervaireuniverse uabbrev WordGroup (A : Type u) [Group A] := Monoid.Coprod A (Multiplicative ℤ)def exponentSum {A : Type u} [Group A] (w : WordGroup A) : ℤ :=  Multiplicative.toAdd (Monoid.Coprod.snd w)def Unimodular {A : Type u} [Group A] (w : WordGroup A) : Prop :=  exponentSum w = 1 ∨ exponentSum w = -1def relatorClosure {A : Type u} [Group A] (w : WordGroup A) : Subgroup (WordGroup A) :=  Subgroup.normalClosure ({w} : Set (WordGroup A))instance {A : Type u} [Group A] (w : WordGroup A) : (relatorClosure w).Normal :=  inferInstanceAs (Subgroup.normalClosure ({w} : Set (WordGroup A))).Normalabbrev RelatorQuotient {A : Type u} [Group A] (w : WordGroup A) :=  WordGroup A ⧸ relatorClosure wdef coefficientMap {A : Type u} [Group A] (w : WordGroup A) : A →* RelatorQuotient w :=  (QuotientGroup.mk' (relatorClosure w)).comp Monoid.Coprod.inltheorem coefficient_injective (A : Type u) [Group A] (w : WordGroup A)    (hw : Unimodular w) : Function.Injective (coefficientMap w) := by  sorryend Kervaireendend 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.

The Kervaire theorem for groups

Theorem

For every group A and every unimodular word w\in A*\langle t\rangle, the canonical homomorphism

A\longrightarrow (A*\langle t\rangle)/\langle\!\langle w\rangle\!\rangle_H

is injective.

Section 1, Introduction Read it in the paper TeX source, line 23

Show the TeX

For every group $A$ and every unimodular word $w\in A*\langle t\rangle$,
the canonical homomorphism
\[
 A\longrightarrow (A*\langle t\rangle)/\langle\!\langle w\rangle\!\rangle_H
\]
is injective.

Corollary (The Kervaire conjecture)

If A\ne1, then (A*\langle t\rangle)/\langle\!\langle w\rangle\!\rangle_H\ne1 for every w\in A*\langle t\rangle.

Section 1, Introduction Read it in the paper TeX source, line 36

Show the TeX

If $A\ne1$, then
$(A*\langle t\rangle)/\langle\!\langle w\rangle\!\rangle_H\ne1$
for every $w\in A*\langle t\rangle$.

Configuration and file facts

File: lean/ComparatorChallenges/Kervaire.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 256 link it.
Solution module: OAI.GroupTheory.Kervaire.Main
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: 7 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.

Plectis Atlas