Lean statement

Separable nonunitarizable witnesses for countable nonamenable groups

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.Dixmier.main_theorem_all_universes

Lean statement file: Dixmier.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 251 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 main_theorem_all_universes : MainTheorem.{u,v} := 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.Dixmier.leftTranslate line 12

def leftTranslate {G : Type u} [Group G] [TopologicalSpace G]    [DiscreteTopology G] (g : G) (f : G →ᵇ ℂ) : G →ᵇ ℂ :=  f.compContinuous ⟨fun x => g⁻¹ * x, continuous_of_discreteTopology⟩

definition OAI.Dixmier.Amenable line 17

def Amenable (G : Type u) [Group G] [TopologicalSpace G]    [DiscreteTopology G] : Prop :=  ∃ m : (G →ᵇ ℂ) →ₗ[ℂ] ℂ,    m (BoundedContinuousFunction.const G 1) = 1 ∧    (∀ f : G →ᵇ ℂ, (∀ x, 0 ≤ (f x).re ∧ (f x).im = 0) →      0 ≤ (m f).re ∧ (m f).im = 0) ∧    ∀ (g : G) (f : G →ᵇ ℂ), m (leftTranslate g f) = m f

definition OAI.Dixmier.SimilarToUnitary line 25

def SimilarToUnitary {G : Type u} [Group G]    {H : Type v} [NormedAddCommGroup H] [InnerProductSpace ℂ H]    (π : G →* (H →L[ℂ] H)) : Prop :=  ∃ S : H ≃L[ℂ] H, ∀ (g : G) (x : H), ‖S (π g (S.symm x))‖ = ‖x‖

definition OAI.Dixmier.NonamenableWitness line 32

def NonamenableWitness (G : Type u) [Group G] : Prop :=  ∃ (H : Type v) (_ : NormedAddCommGroup H) (_ : InnerProductSpace ℂ H)    (_ : CompleteSpace H) (_ : TopologicalSpace.SeparableSpace H)    (π : G →* (H →L[ℂ] H)),    (∀ g : G, ‖π g‖ ≤ 101) ∧ ¬ SimilarToUnitary π

definition OAI.Dixmier.MainTheorem line 39

def MainTheorem : Prop :=  ∀ (G : Type u) [Group G] [Countable G] [TopologicalSpace G]    [DiscreteTopology G], ¬ Amenable G → NonamenableWitness.{u, v} G

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: pi, amenable, hilbert, similar, complex, space, separable, discrete and five more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: chosen.

Statements in Unitarizability implies amenability for discrete groups come first because the release lists only that paper for this Lean statement.

Theorem 1.1 of Unitarizability implies amenability for discrete groups

Theorem

A discrete group is amenable if and only if it is unitarizable. For every discrete nonamenable group G and every \varepsilon>0, there are a complex Hilbert space \mathcal K and a representation \pi:G\to\mathrm{GL}(\mathcal K) with |\pi|\le1+\varepsilon that is not similar to a unitary representation. If G is countable, \mathcal K may be chosen separable.

Section 1, Introduction Read it in the paper TeX source, line 38

Show the TeX

A discrete group is amenable if and only if it is unitarizable.
For every discrete nonamenable group \(G\) and every \(\varepsilon>0\),
there are a complex Hilbert space \(\mathcal K\) and a representation
\(\pi:G\to\mathrm{GL}(\mathcal K)\) with \(|\pi|\le1+\varepsilon\)
that is not similar to a unitary representation. If \(G\) is countable,
\(\mathcal K\) may be chosen separable.

Other possible paper matches, closest wording first

Proposition 2.2 of Unitarizability implies amenability for discrete groups. Words of the Lean statement, its names split into words, that also appear in the theorem: positive, space, separable, pi, hilbert, similar and seven more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: uniform.
Proposition 4.4 of Unitarizability implies amenability for discrete groups. Words of the Lean statement, its names split into words, that also appear in the theorem: uniformly, unitary, bounds, representation and norm.
Theorem 1.1 of Strong Ulam Stability Characterizes Amenability. Words of the Lean statement, its names split into words, that also appear in the theorem: amenable, discrete, group and countable.
Proposition 3.2 of Unitarizability implies amenability for discrete groups. Words of the Lean statement, its names split into words, that also appear in the theorem: products, function, nonamenable and countable.
Proposition 2.2 of Strong Ulam Stability Characterizes Amenability. Words of the Lean statement, its names split into words, that also appear in the theorem: positive, discrete, nonamenable and group.
Corollary 4.3 of Unitarizability implies amenability for discrete groups. Words of the Lean statement, its names split into words, that also appear in the theorem: bounds and norm.

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 74 s; peak memory 3.1 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 251’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: Stated: Dixmier's equivalence for all discrete groups (CurrentMainTheorem : ∀ G discrete, (Amenable G ↔ Unitarizable G) ∧ ∀ ε > 0, ¬ Amenable G → ∃ nonunitarizable π with ‖π g‖ ≤ 1 + ε on a complete Hilbert space, separable if G is countable) and the countable-group separable witness with bound 101 (Dixmier.main_theorem_all_universes). Not stated: the summary's second claim, equivalence of amenability with strong Ulam stability for countable discrete groups (it is not mentioned in the docs scope either, although the family title names Ulam stability).
Definitions the group asks readers to compare with the paper: Custom 'amenable': def Amenable (G) [DiscreteTopology G] : Prop := ∃ m : (G →ᵇ ℂ) →ₗ[ℂ] ℂ, m 1 = 1 ∧ (∀ f, (∀ x, 0 ≤ (f x).re ∧ (f x).im = 0) → 0 ≤ (m f).re ∧ (m f).im = 0) ∧ ∀ g f, m (leftTranslate g f) = m f, the standard positive left-invariant mean on ℓ^∞(G, ℂ); SimilarToUnitary π := ∃ S : H ≃L[ℂ] H, ∀ g x, ‖S (π g (S.symm x))‖ = ‖x‖ and Unitarizable quantify over Hilbert spaces in one universe (the theorem is universe polymorphic).
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 strong-Ulam-stability characterization of amenability for countable groups

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The Hilbert space can be chosen separable for countable groups; a separate linked statement records a separable witness with uniform bound $101$ in that case.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states a weaker form of a paper’s theorem”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

51 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: 5 to its own declarations, 20 to Mathlib’s documentation. 6 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_all_universes 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 5 other declarations of the file; with the theorem itself they span 24 lines. 1 of them is named in the statement itself.

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

def leftTranslateline 12library
def Amenableline 17library
def SimilarToUnitaryline 25library
def NonamenableWitnessline 32library
def MainTheoremline 39in the statementlibrary
theorem main_theorem_all_universesline 44selected theoremlibrary

import Mathlib.Analysis.InnerProductSpace.LinearMapimport Mathlib.Topology.ContinuousMap.Bounded.Normedimport Mathlib.Topology.Instances.Discretenamespace OAInoncomputable sectionuniverse u vnamespace Dixmieropen scoped BoundedContinuousFunctiondef leftTranslate {G : Type u} [Group G] [TopologicalSpace G]    [DiscreteTopology G] (g : G) (f : G →ᵇ ℂ) : G →ᵇ ℂ :=  f.compContinuous ⟨fun x => g⁻¹ * x, continuous_of_discreteTopology⟩/-- A positive normalized left-invariant complex mean. -/def Amenable (G : Type u) [Group G] [TopologicalSpace G]    [DiscreteTopology G] : Prop :=  ∃ m : (G →ᵇ ℂ) →ₗ[ℂ] ℂ,    m (BoundedContinuousFunction.const G 1) = 1 ∧    (∀ f : G →ᵇ ℂ, (∀ x, 0 ≤ (f x).re ∧ (f x).im = 0) →      0 ≤ (m f).re ∧ (m f).im = 0) ∧    ∀ (g : G) (f : G →ᵇ ℂ), m (leftTranslate g f) = m fdef SimilarToUnitary {G : Type u} [Group G]    {H : Type v} [NormedAddCommGroup H] [InnerProductSpace ℂ H]    (π : G →* (H →L[ℂ] H)) : Prop :=  ∃ S : H ≃L[ℂ] H, ∀ (g : G) (x : H), ‖S (π g (S.symm x))‖ = ‖x‖/-- A uniformly 101-bounded nonunitarizable representation on a separablecomplete complex Hilbert space, with its universe explicit. -/def NonamenableWitness (G : Type u) [Group G] : Prop :=  ∃ (H : Type v) (_ : NormedAddCommGroup H) (_ : InnerProductSpace ℂ H)    (_ : CompleteSpace H) (_ : TopologicalSpace.SeparableSpace H)    (π : G →* (H →L[ℂ] H)),    (∀ g : G, ‖π g‖ ≤ 101) ∧ ¬ SimilarToUnitary π/-- Every countable discrete nonamenable group has a bounded nonunitarizable witness. -/def MainTheorem : Prop :=  ∀ (G : Type u) [Group G] [Countable G] [TopologicalSpace G]    [DiscreteTopology G], ¬ Amenable G → NonamenableWitness.{u, v} G/-- The main theorem with independent group and Hilbert-space universes. -/theorem main_theorem_all_universes : MainTheorem.{u,v} := by  sorryend Dixmierendend 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.

Unitarizability implies amenability for discrete groups

Theorem

Section 1, Introduction Read it in the paper TeX source, line 38

Show the TeX

A discrete group is amenable if and only if it is unitarizable.
For every discrete nonamenable group \(G\) and every \(\varepsilon>0\),
there are a complex Hilbert space \(\mathcal K\) and a representation
\(\pi:G\to\mathrm{GL}(\mathcal K)\) with \(|\pi|\le1+\varepsilon\)
that is not similar to a unitary representation. If \(G\) is countable,
\(\mathcal K\) may be chosen separable.

Configuration and file facts

File: lean/ComparatorChallenges/Dixmier.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 251 link it.
Solution module: OAI.Analysis.Unitarizability.UniverseTransport
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.Analysis.InnerProductSpace.LinearMap, Mathlib.Topology.ContinuousMap.Bounded.Normed, Mathlib.Topology.Instances.Discrete
Imports from outside Mathlib: None
Local vocabulary: 5 local vocabulary declarations
Lines of Lean, without comments or blank lines: 24
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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