Lean statement

Nonamenability of Thompson’s group $F$

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.

Nonamenability of Thompson's group $F$

Theorem selected for verification: OAI.ThompsonNonamenability.thompson_F_nonamenable_composition

Lean statement file: ThompsonNonamenability.leanShow on the mapFamily 248. 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.

theorem thompson_F_nonamenable_composition :    ∃ group : Group F, letI := group;

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.ThompsonNonamenability.UnitInterval line 16
```
abbrev UnitInterval := Set.Icc (0 : ℝ) 1
```

definition OAI.ThompsonNonamenability.unitIntervalZero line 18

def unitIntervalZero : UnitInterval := ⟨0, le_rfl, zero_le_one⟩

definition OAI.ThompsonNonamenability.unitIntervalOne line 19

def unitIntervalOne : UnitInterval := ⟨1, zero_le_one, le_rfl⟩

structure OAI.ThompsonNonamenability.IntervalHomeomorph line 21

structure IntervalHomeomorph where  toHomeomorph : UnitInterval ≃ₜ UnitInterval

definition OAI.ThompsonNonamenability.IsDyadic line 31

def IsDyadic (x : ℝ) : Prop :=  ∃ (k : ℤ) (n : ℕ), x = (k : ℝ) / (2 : ℝ) ^ n

structure OAI.ThompsonNonamenability.DyadicPLWitness line 34

structure DyadicPLWitness (f : IntervalHomeomorph) where

Show the remaining 14 lines

  pieceCount : ℕ  positive : 0 < pieceCount  knots : Fin (pieceCount + 1) → UnitInterval  strictMono_knots : StrictMono knots  first : knots 0 = unitIntervalZero  last : knots (Fin.last pieceCount) = unitIntervalOne  dyadic : ∀ i, IsDyadic (knots i : ℝ)  cover : ∀ x : UnitInterval, ∃ i : Fin pieceCount,    knots i.castSucc ≤ x ∧ x ≤ knots i.succ  exponent : Fin pieceCount → ℤ  affine : ∀ (i : Fin pieceCount) (x : UnitInterval),    knots i.castSucc ≤ x → x ≤ knots i.succ →    (f x : ℝ) = (f (knots i.castSucc) : ℝ) +      (2 : ℝ) ^ exponent i * ((x : ℝ) - (knots i.castSucc : ℝ))

definition OAI.ThompsonNonamenability.HasDyadicPLPieces line 50

def HasDyadicPLPieces (f : IntervalHomeomorph) : Prop := Nonempty (DyadicPLWitness f)

definition OAI.ThompsonNonamenability.F line 52

def F := {f : IntervalHomeomorph // StrictMono f ∧ HasDyadicPLPieces f}

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: group and thompsons. Found in the paper’s title, which its main theorem states: nonamenable.

Theorem 1.1 of Thompson’s group F is nonamenable

Theorem

Thompson’s group F is not amenable.

Section 1, Introduction Read it in the paper TeX source, line 37

Show the TeX

Thompson's group \(F\) is not amenable.

Other possible paper matches, closest wording first

Proposition 2.3 of Thompson’s group F is nonamenable. Words of the Lean statement, its names split into words, that also appear in the theorem: nonempty.
Corollary 3.2 of Thompson’s group F is nonamenable. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: generating.
Proposition A.1 of Thompson’s group F is nonamenable. Words of the Lean statement, its names split into words, that also appear in the theorem: real.

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 248’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.

MathVet and cert-machine agree on how much of the advertised result the Lean statements express.

MathVet
Finds the whole advertised result stated or implied Their term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
Their note: thompson_F_nonamenable_composition : ∃ group : Group F, (∀ h g x, (h * g) x = h (g x)) ∧ ¬ Nonempty (InvariantMean F) where F := {f : IntervalHomeomorph // StrictMono f ∧ HasDyadicPLPieces f} (dyadic breakpoints, slopes 2^k via DyadicPLWitness) with group law forced to be composition, and InvariantMean G = positive normalized left-invariant linear functional on lp (fun _ : G => ℝ) ∞. Exactly the nonamenability of Thompson's F.
Definitions the group asks readers to compare with the paper: Custom 'amenable': InvariantMean G (positive, toLinearMap 1 = 1, toLinearMap (leftPull h f) = toLinearMap f on ℓ^∞(G, ℝ)) is the standard invariant-mean definition but is not a Mathlib notion; F itself is hand-defined (Group structure is existentially provided inside the theorem).
Carlos Toledo’s cert-machine
Finds the whole advertised result stated or implied Their term: “MATCHES”.
The challenge statements state the headline as OpenAI wrote it. Their record

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalized result rules out such a mean for the standard group of dyadic piecewise-linear homeomorphisms of the interval, proving nonamenability. No explicit boundary constant or prescribed generating set is given.”
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

89 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: 29 to its own declarations, 24 to Mathlib’s documentation. 12 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 thompson_F_nonamenable_composition 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 8 of the file’s 12 other declarations; with the theorem itself they span 30 lines. 1 of them is named in the statement itself.

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

abbrev UnitIntervalline 16library
def unitIntervalZeroline 18library
def unitIntervalOneline 19library
structure IntervalHomeomorphline 21library
def IsDyadicline 31library
structure DyadicPLWitnessline 34library
def HasDyadicPLPiecesline 50library
def Fline 52in the statementlibrary
theorem thompson_F_nonamenable_compositionline 78selected 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 CoeFun IntervalHomeomorph (fun _ => UnitInterval → UnitInterval) line 26

3 declarations the statement does not reach by name

These definitions and lemmas were not found by following names in the statement. Lean may still use them through rules such as instances or macros.

abbrev BoundedReal line 54
def leftPull line 58
structure InvariantMean line 71

import Mathlib.Topology.Homeomorph.Lemmasimport Mathlib.Topology.Instances.Real.Lemmasimport Mathlib.Analysis.Normed.Lp.lpSpaceimport Mathlib.Analysis.Normed.Operator.Basicnamespace OAInoncomputable sectionopen scoped ENNRealuniverse uGnamespace ThompsonNonamenabilityabbrev UnitInterval := Set.Icc (0 : ℝ) 1def unitIntervalZero : UnitInterval := ⟨0, le_rfl, zero_le_one⟩def unitIntervalOne : UnitInterval := ⟨1, zero_le_one, le_rfl⟩structure IntervalHomeomorph where  toHomeomorph : UnitInterval ≃ₜ UnitIntervalnamespace IntervalHomeomorphinstance : CoeFun IntervalHomeomorph (fun _ => UnitInterval → UnitInterval) :=  ⟨fun f => f.toHomeomorph⟩end IntervalHomeomorphdef IsDyadic (x : ℝ) : Prop :=  ∃ (k : ℤ) (n : ℕ), x = (k : ℝ) / (2 : ℝ) ^ nstructure DyadicPLWitness (f : IntervalHomeomorph) where  pieceCount : ℕ  positive : 0 < pieceCount  knots : Fin (pieceCount + 1) → UnitInterval  strictMono_knots : StrictMono knots  first : knots 0 = unitIntervalZero  last : knots (Fin.last pieceCount) = unitIntervalOne  dyadic : ∀ i, IsDyadic (knots i : ℝ)  cover : ∀ x : UnitInterval, ∃ i : Fin pieceCount,    knots i.castSucc ≤ x ∧ x ≤ knots i.succ  exponent : Fin pieceCount → ℤ  affine : ∀ (i : Fin pieceCount) (x : UnitInterval),    knots i.castSucc ≤ x → x ≤ knots i.succ →    (f x : ℝ) = (f (knots i.castSucc) : ℝ) +      (2 : ℝ) ^ exponent i * ((x : ℝ) - (knots i.castSucc : ℝ))def HasDyadicPLPieces (f : IntervalHomeomorph) : Prop := Nonempty (DyadicPLWitness f)def F := {f : IntervalHomeomorph // StrictMono f ∧ HasDyadicPLPieces f}abbrev BoundedReal (G : Type uG) := lp (fun _ : G => ℝ) ∞variable {G : Type uG} [Group G]def leftPull (h : G) : BoundedReal G →ₗ[ℝ] BoundedReal G where  toFun f := ⟨fun g => f (h * g), memℓp_infty ⟨‖f‖, by    rintro y ⟨g, rfl⟩    exact lp.norm_apply_le_norm ENNReal.top_ne_zero f (h * g)⟩⟩  map_add' f g := by    apply lp.ext    funext x    rfl  map_smul' r f := by    apply lp.ext    funext x    rflstructure InvariantMean (G : Type uG) [Group G] where  toLinearMap : BoundedReal G →ₗ[ℝ] ℝ  positive : ∀ f : BoundedReal G, (∀ g, 0 ≤ f g) → 0 ≤ toLinearMap f  normalized : toLinearMap 1 = 1  left_invariant : ∀ (h : G) (f : BoundedReal G),    toLinearMap (leftPull h f) = toLinearMap ftheorem thompson_F_nonamenable_composition :    ∃ group : Group F, letI := group;      (∀ (h g : F) (x : UnitInterval),        (h * g).val.toHomeomorph x = h.val.toHomeomorph (g.val.toHomeomorph x)) ∧      ¬ Nonempty (InvariantMean F) := by  sorryend ThompsonNonamenabilityendend 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.

Thompson’s group F is nonamenable

Theorem

Thompson’s group F is not amenable.

Section 1, Introduction Read it in the paper TeX source, line 37

Show the TeX

Thompson's group \(F\) is not amenable.

Configuration and file facts

File: lean/ComparatorChallenges/ThompsonNonamenability.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/GroupTheory/Thompson/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.Topology.Homeomorph.Lemmas, Mathlib.Topology.Instances.Real.Lemmas, Mathlib.Analysis.Normed.Lp.lpSpace, Mathlib.Analysis.Normed.Operator.Basic
Imports from outside Mathlib: None
Local vocabulary: 12 local vocabulary declarations
Lines of Lean, without comments or blank lines: 51
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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