Lean statement

Infinite finitely presented periodic group

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.SourceBurnside.thm_main

Lean statement file: PeriodicGroup.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 247 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 thm_main : MainStatement ∧ BurnsideStatement := 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.

structure OAI.SourceBurnside.Root line 9

structure Root (n : ℕ) where  row : Fin n  col : Fin n  ne : row ≠ col  deriving DecidableEq

abbreviation OAI.SourceBurnside.Generator line 15

abbrev Generator (n : ℕ) (R : Type u) := Root n × R

definition OAI.SourceBurnside.bracket line 17

def bracket {G : Type u} [Group G] (x y : G) : G := x * y * x⁻¹ * y⁻¹

definition OAI.SourceBurnside.letter line 19

def letter {n : ℕ} {R : Type u} (p : Root n) (a : R) :    FreeGroup (Generator n R) := FreeGroup.of (p, a)

inductive type OAI.SourceBurnside.SteinbergRel line 22

inductive SteinbergRel (n : ℕ) (R : Type u) [Ring R] :    FreeGroup (Generator n R) → Prop  | add (p : Root n) (a b : R) :      SteinbergRel n R (letter p (a + b) * (letter p a * letter p b)⁻¹)  | commute (p q : Root n) (h₁ : p.col ≠ q.row) (h₂ : p.row ≠ q.col) (a b : R) :      SteinbergRel n R (bracket (letter p a) (letter q b))  | mul (i j k : Fin n) (hij : i ≠ j) (hjk : j ≠ k) (hik : i ≠ k) (a b : R) :      SteinbergRel n R        (bracket (letter ⟨i, j, hij⟩ a) (letter ⟨j, k, hjk⟩ b) *          (letter ⟨i, k, hik⟩ (a * b))⁻¹)

definition OAI.SourceBurnside.Steinberg line 33

def Steinberg (n : ℕ) (R : Type u) [Ring R] :=  PresentedGroup {r | SteinbergRel n R r}  deriving Group

definition OAI.SourceBurnside.Periodic line 37

def Periodic (G : Type u) [Group G] : Prop := ∀ g : G, ∃ m : ℕ, 0 < m ∧ g ^ m = 1

definition OAI.SourceBurnside.MainStatement line 39

def MainStatement : Prop :=  ∃ (R : Type) (_ : Ring R) (_ : Algebra (ZMod 2) R),    Infinite (Steinberg 12 R) ∧ Group.IsFinitelyPresented (Steinberg 12 R) ∧      Periodic (Steinberg 12 R)

definition OAI.SourceBurnside.BurnsideStatement line 44

def BurnsideStatement : Prop :=  ∃ (G : Type) (_ : Group G), Infinite G ∧ Group.IsFinitelyPresented G ∧ Periodic G

Possible paper match, with no recorded review

No paper statement clearly matched the Lean wording. Statement Map, the program that collects Atlas’s source records, selected this possible match using words in OpenAI’s catalogue and Lean note. No review of whether the statements mean the same thing is recorded.

Shared with the catalogue’s wording or OpenAI’s note, none of them matched in the Lean text: algebra, finite, finitely, gives, group, infinite, periodic and presented.

Statements in An infinite finitely presented periodic group come first because the release lists only that paper for this Lean statement.

Theorem 1.1 of An infinite finitely presented periodic group

Theorem

There is an infinite periodic group with an ordinary finite presentation. More precisely, the construction below gives a unital associative \F_2-algebra R for which \St_{12}(R) is infinite, finitely presented, and periodic.

Section 1, Introduction Read it in the paper TeX source, line 88

Show the TeX

There is an infinite periodic group with an ordinary finite
presentation. More precisely, the construction below gives a unital
associative $\F_2$-algebra $R$ for which $\St_{12}(R)$ is infinite,
finitely presented, and periodic.

Other possible paper matches, closest wording first

Corollary 1.2 of An infinite finitely presented periodic group. Words of the Lean statement, its names split into words, that also appear in the theorem: thm, periodic, infinite, finitely, presented and group.
Theorem 4.1 of An infinite finitely presented periodic group. Words of the Lean statement, its names split into words, that also appear in the theorem: steinberg, generators, relations, letter, algebra, presentation and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: two, gives, order and finite.
Proposition 5.3 of An infinite finitely presented periodic group. Words of the Lean statement, its names split into words, that also appear in the theorem: generators, relations, presentation and group.
Corollary 3.8 of An infinite finitely presented periodic group. Words of the Lean statement, its names split into words, that also appear in the theorem: thm, integer, infinite, finitely, algebra and presented. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: radical and nil.
Proposition 3.6 of An infinite finitely presented periodic group. Words of the Lean statement, its names split into words, that also appear in the theorem: letter. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: characteristic, two and finite.
Theorem 1.1 of An infinite finitely presented residually finite 2-group. Words of the Lean statement, its names split into words, that also appear in the theorem: infinite, group and presentation. Found in the paper’s title, which its main theorem states: finitely. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: element, order and finite.

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 247’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: “weaker-statement”.
The challenge statements do not imply the headline claim: they state a nontrivially weaker statement. Their record
Their note: thm_main : MainStatement ∧ BurnsideStatement only gives an infinite finitely presented PERIODIC group (Periodic G := ∀ g, ∃ m > 0, g^m = 1, namely Steinberg 12 R for some F₂-algebra R, and a general ∃ G version). The summary's group is a residually finite 2-group (every element of 2-power order) with accompanying nil-algebra consequences; neither 'residually finite' nor '2-power order' nor the algebra statements is stated (docs: 'no common exponent is asserted ... separate nil-algebra and radical-algebra conclusions are outside these selected statements'). The negative answer to the finitely presented Burnside problem itself is covered.
Definitions the group asks readers to compare with the paper: Steinberg n R := PresentedGroup {r | SteinbergRel n R r} with relations x_{ij}(a+b) = x_{ij}(a) x_{ij}(b), [x_{ij}(a), x_{kl}(b)] = 1 for p.col ≠ q.row ∧ p.row ≠ q.col, and [x_{ij}(a), x_{jk}(b)] = x_{ik}(ab): the standard Steinberg presentation, hand-written (the carrier ring R is existentially chosen).
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: Residual finiteness and the 2-power order of every element (the October 5 manuscript's contribution; undeclared), plus the nil algebra and algebraic-unitization clause (declared outside).

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization gives a negative answer by constructing an infinite finitely presented periodic group, including a witness realized as a Steinberg group over an algebra of characteristic two. Periodicity means that each element has some finite order; no common exponent is asserted. The paper's separate nil-algebra and radical-algebra conclusions are outside these selected statements.”
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

56 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 fileOther linked names open the current documentation for Mathlib, Lean’s mathematical library.

Links in the file: 27 to its own declarations, 21 to Mathlib’s documentation. 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 thm_main 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 9 other declarations of the file; with the theorem itself they span 31 lines. 2 of them are named in the statement itself.

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

structure Rootline 9
abbrev Generatorline 15
def bracketline 17
def letterline 19
inductive SteinbergRelline 22
def Steinbergline 33
def Periodicline 37
def MainStatementline 39in the statement
def BurnsideStatementline 44in the statement
theorem thm_mainline 51selected theorem

import Mathlibnamespace OAIuniverse unamespace SourceBurnsidestructure Root (n : ℕ) where  row : Fin n  col : Fin n  ne : row ≠ col  deriving DecidableEqabbrev Generator (n : ℕ) (R : Type u) := Root n × Rdef bracket {G : Type u} [Group G] (x y : G) : G := x * y * x⁻¹ * y⁻¹def letter {n : ℕ} {R : Type u} (p : Root n) (a : R) :    FreeGroup (Generator n R) := FreeGroup.of (p, a)inductive SteinbergRel (n : ℕ) (R : Type u) [Ring R] :    FreeGroup (Generator n R) → Prop  | add (p : Root n) (a b : R) :      SteinbergRel n R (letter p (a + b) * (letter p a * letter p b)⁻¹)  | commute (p q : Root n) (h₁ : p.col ≠ q.row) (h₂ : p.row ≠ q.col) (a b : R) :      SteinbergRel n R (bracket (letter p a) (letter q b))  | mul (i j k : Fin n) (hij : i ≠ j) (hjk : j ≠ k) (hik : i ≠ k) (a b : R) :      SteinbergRel n R        (bracket (letter ⟨i, j, hij⟩ a) (letter ⟨j, k, hjk⟩ b) *          (letter ⟨i, k, hik⟩ (a * b))⁻¹)def Steinberg (n : ℕ) (R : Type u) [Ring R] :=  PresentedGroup {r | SteinbergRel n R r}  deriving Groupdef Periodic (G : Type u) [Group G] : Prop := ∀ g : G, ∃ m : ℕ, 0 < m ∧ g ^ m = 1def MainStatement : Prop :=  ∃ (R : Type) (_ : Ring R) (_ : Algebra (ZMod 2) R),    Infinite (Steinberg 12 R) ∧ Group.IsFinitelyPresented (Steinberg 12 R) ∧      Periodic (Steinberg 12 R)def BurnsideStatement : Prop :=  ∃ (G : Type) (_ : Group G), Infinite G ∧ Group.IsFinitelyPresented G ∧ Periodic Gend SourceBurnsidenamespace SourceBurnsidetheorem thm_main : MainStatement ∧ BurnsideStatement := by  sorryend SourceBurnsideend 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.

An infinite finitely presented periodic group

Theorem

Section 1, Introduction Read it in the paper TeX source, line 88

Show the TeX

There is an infinite periodic group with an ordinary finite
presentation. More precisely, the construction below gives a unital
associative $\F_2$-algebra $R$ for which $\St_{12}(R)$ is infinite,
finitely presented, and periodic.

Corollary (A periodic Kazhdan group)

The group \St_{12}(R) of Theorem thm:main is infinite, finitely presented, and periodic, and has Kazhdan’s property (T). In particular, it is nonamenable.

Section 1, Introduction Read it in the paper TeX source, line 104

Show the TeX

The group \(\St_{12}(R)\) of Theorem~\ref{thm:main} is infinite,
finitely presented, and periodic, and has Kazhdan's property~\((T)\).
In particular, it is nonamenable.

Configuration and file facts

File: lean/ComparatorChallenges/PeriodicGroup.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 247 link it.
Solution module: OAI.GroupTheory.PeriodicGroups.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: 9 local vocabulary declarations
Lines of Lean, without comments or blank lines: 31
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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