Lean statement

Counterexample to Naimark’s problem in ZFC

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.

Counterexample to Naimark's problem in ZFC

Theorem selected for verification: OAI.NaimarkZFC.main

Lean statement file: Naimark.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 297 link it.

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

Problems reported

Problem reported Reported as broken in the same way: its operator-algebra definitions are listed as holes, so a trivial replacement makes the checked statement vacuous. Alex Meiburg, Several of the comparator challenges in the OpenAI release are broken and hackable, 2026-10-08 Source

The Lean statement

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

theorem main :    ∃ (A : Type u) (_ : CStarAlgebra A),      IsSimpleCStar A ∧ ¬ FiniteDimensional ℂ A ∧      (∃ τ : A →L[ℂ] ℂ, IsFaithfulTracialState τ) ∧      HasUniqueIrreducibleRepresentation.{u, v, w} A ∧      ∀ (H : Type v) [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H],        ¬ IsIsomorphicToCompacts A H := 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.NaimarkZFC.IsSimpleCStar line 11

def IsSimpleCStar (A : Type u) [CStarAlgebra A] : Prop :=  Nontrivial A ∧ ∀ I : TwoSidedIdeal A, IsClosed (I : Set A) → I = ⊥ ∨ I = ⊤

definition OAI.NaimarkZFC.IsState line 14

def IsState {A : Type u} [CStarAlgebra A] (f : A →L[ℂ] ℂ) : Prop :=  ‖f‖ = 1 ∧ ∀ a : A, (f (star a * a)).im = 0 ∧ 0 ≤ (f (star a * a)).re

definition OAI.NaimarkZFC.IsFaithfulTracialState line 17

def IsFaithfulTracialState {A : Type u} [CStarAlgebra A] (f : A →L[ℂ] ℂ) : Prop :=  IsState f ∧ (∀ a b : A, f (a * b) = f (b * a)) ∧    ∀ a : A, a ≠ 0 → 0 < (f (star a * a)).re

definition OAI.NaimarkZFC.IsIrreducible line 21

def IsIrreducible {A : Type u} [CStarAlgebra A]    {H : Type v} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]    (π : A →⋆ₙₐ[ℂ] (H →L[ℂ] H)) : Prop :=  π ≠ 0 ∧ ∀ M : Submodule ℂ H, IsClosed (M : Set H) →    (∀ a : A, ∀ x ∈ M, π a x ∈ M) → M = ⊥ ∨ M = ⊤

definition OAI.NaimarkZFC.UnitarilyEquivalent line 27

def UnitarilyEquivalent {A : Type u} [CStarAlgebra A]    {H : Type v} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]    {K : Type w} [NormedAddCommGroup K] [InnerProductSpace ℂ K] [CompleteSpace K]    (π : A →⋆ₙₐ[ℂ] (H →L[ℂ] H)) (ρ : A →⋆ₙₐ[ℂ] (K →L[ℂ] K)) : Prop :=  ∃ U : H ≃ₗᵢ[ℂ] K, ∀ a : A, ∀ x : H, U (π a x) = ρ a (U x)

definition OAI.NaimarkZFC.HasUniqueIrreducibleRepresentation line 33

def HasUniqueIrreducibleRepresentation (A : Type u) [CStarAlgebra A] : Prop :=  ∀ (H : Type v) [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]    (K : Type w) [NormedAddCommGroup K] [InnerProductSpace ℂ K] [CompleteSpace K]    (π : A →⋆ₙₐ[ℂ] (H →L[ℂ] H)) (ρ : A →⋆ₙₐ[ℂ] (K →L[ℂ] K)),    IsIrreducible π → IsIrreducible ρ → UnitarilyEquivalent π ρ

definition OAI.NaimarkZFC.IsIsomorphicToCompacts line 39

def IsIsomorphicToCompacts (A : Type u) [CStarAlgebra A]    (H : Type v) [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] : Prop :=  ∃ φ : A →⋆ₙₐ[ℂ] (H →L[ℂ] H), Function.Injective φ ∧    ∀ T : H →L[ℂ] H, (∃ a : A, φ a = T) ↔ IsCompactOperator T

Definitions taken from OpenAI’s library 7

Comparator compares only the name and type of each definition listed here. When a theorem is configured, its proof uses the definition in OpenAI’s solution library. Read that definition as well as the one in this file when comparing the statement with the paper.

Comparator checks the name and type of IsSimpleCStar; its definition is in OpenAI’s library, in the modules that OAI.Analysis.Naimark.Tracial imports.
Comparator checks the name and type of IsState; its definition is in OpenAI’s library, in the modules that OAI.Analysis.Naimark.Tracial imports.
Comparator checks the name and type of IsFaithfulTracialState; its definition is in OpenAI’s library, in the modules that OAI.Analysis.Naimark.Tracial imports.
Comparator checks the name and type of IsIrreducible; its definition is in OpenAI’s library, in the modules that OAI.Analysis.Naimark.Tracial imports.
Comparator checks the name and type of UnitarilyEquivalent; its definition is in OpenAI’s library, in the modules that OAI.Analysis.Naimark.Tracial imports.
Comparator checks the name and type of HasUniqueIrreducibleRepresentation; its definition is in OpenAI’s library, in the modules that OAI.Analysis.Naimark.Tracial imports.
Comparator checks the name and type of IsIsomorphicToCompacts; its definition is in OpenAI’s library, in the modules that OAI.Analysis.Naimark.Tracial imports.

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: equivalent, isomorphic, unitarily, complex, space, dimensional, irreducible, representations and five more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: infinite, hilbert and unital.

Theorem 1.1 of A counterexample to Naimark’s problem in ZFC

Theorem

There exists a unital infinite-dimensional simple complex C^*-algebra A with a faithful tracial state such that all its nonzero irreducible representations are unitarily equivalent. In particular, A is not isomorphic to K(H) for any complex Hilbert space H.

Section 1, Introduction Read it in the paper TeX source, line 10

Show the TeX

There exists a unital infinite-dimensional simple complex $C^*$-algebra
$A$ with a faithful tracial state such that all its nonzero irreducible
representations are unitarily equivalent. In particular, $A$ is not
isomorphic to $K(H)$ for any complex Hilbert space $H$.

Other possible paper matches, closest wording first

Theorem 4.1 of A counterexample to Naimark’s problem in ZFC. Words of the Lean statement, its names split into words, that also appear in the theorem: finite, unique, dimensional, irreducible, representations, simple and four more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: unital.
Theorem 6.1 of A counterexample to Naimark’s problem in ZFC. Words of the Lean statement, its names split into words, that also appear in the theorem: representations, simple, algebra and states.
Proposition 3.1 of A counterexample to Naimark’s problem in ZFC. Words of the Lean statement, its names split into words, that also appear in the theorem: faithful, tracial, state and algebra. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: unital.
Theorem 5.5 of A counterexample to Naimark’s problem in ZFC. Words of the Lean statement, its names split into words, that also appear in the theorem: closed. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: property.
Proposition 3.4 of A counterexample to Naimark’s problem in ZFC. Words of the Lean statement, its names split into words, that also appear in the theorem: tracial and algebra. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: unital.
Corollary 5.7 of A counterexample to Naimark’s problem in ZFC. Words of the Lean statement, its names split into words, that also appear in the theorem: states.

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 400 s; peak memory 6.5 GiB).
Reported 10 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.

Definitions compared by name and type only (7): IsSimpleCStar, IsState, IsFaithfulTracialState, IsIrreducible, UnitarilyEquivalent, HasUniqueIrreducibleRepresentation, IsIsomorphicToCompacts. Their bodies come from OpenAI’s solution library.

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 297’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: main : ∃ (A : Type u) (_ : CStarAlgebra A), IsSimpleCStar A ∧ ¬ FiniteDimensional ℂ A ∧ (∃ τ, IsFaithfulTracialState τ) ∧ HasUniqueIrreducibleRepresentation A ∧ ∀ (H : Type v) [Hilbert], ¬ IsIsomorphicToCompacts A H (universe-polymorphic): a unital infinite-dimensional simple C*-algebra with a faithful tracial state all of whose irreducible representations (nonzero, no nontrivial closed invariant subspace) are unitarily equivalent, yet not isomorphic to the compact operators on any Hilbert space. Matches the summary (nonseparability is a consequence, not stated; the summary's 'alternative to Tanaka' framing is irrelevant to the statement).
Definitions the group asks readers to compare with the paper: Hand-written C*-algebra representation theory: IsSimpleCStar, IsFaithfulTracialState, IsIrreducible π := π ≠ 0 ∧ no nontrivial closed π-invariant subspace, UnitarilyEquivalent, IsIsomorphicToCompacts A H := ∃ injective φ : A →⋆ₙₐ[ℂ] (H →L H), range = compact operators; standard, quantified over Hilbert spaces in explicit universes.
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 formalization gives a counterexample in ZFC: a unital infinite-dimensional simple complex $C^*$-algebra with a faithful tracial state has that uniqueness property for irreducible representations but is not isomorphic to the compact operators on any Hilbert space. No additional set-theoretic assumption is used.”
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

57 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, 42 to Mathlib’s documentation. 8 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 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. For a definition marked “name and type only”, Comparator does not compare the definition’s body with the one here. When a theorem is configured, it verifies the proof using the definition in OpenAI’s library. Follow the library link to compare the two definitions.

It reaches all 7 other declarations of the file; with the theorem itself they span 34 lines. 4 of them are named in the statement itself.

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

def IsSimpleCStarline 11in the statementname and type onlylibrary
def IsStateline 14name and type onlylibrary
def IsFaithfulTracialStateline 17in the statementname and type onlylibrary
def IsIrreducibleline 21name and type onlylibrary
def UnitarilyEquivalentline 27name and type onlylibrary
def HasUniqueIrreducibleRepresentationline 33in the statementname and type onlylibrary
def IsIsomorphicToCompactsline 39in the statementname and type onlylibrary
theorem mainline 44selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionuniverse u v wnamespace NaimarkZFCdef IsSimpleCStar (A : Type u) [CStarAlgebra A] : Prop :=  Nontrivial A ∧ ∀ I : TwoSidedIdeal A, IsClosed (I : Set A) → I = ⊥ ∨ I = ⊤def IsState {A : Type u} [CStarAlgebra A] (f : A →L[ℂ] ℂ) : Prop :=  ‖f‖ = 1 ∧ ∀ a : A, (f (star a * a)).im = 0 ∧ 0 ≤ (f (star a * a)).redef IsFaithfulTracialState {A : Type u} [CStarAlgebra A] (f : A →L[ℂ] ℂ) : Prop :=  IsState f ∧ (∀ a b : A, f (a * b) = f (b * a)) ∧    ∀ a : A, a ≠ 0 → 0 < (f (star a * a)).redef IsIrreducible {A : Type u} [CStarAlgebra A]    {H : Type v} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]    (π : A →⋆ₙₐ[ℂ] (H →L[ℂ] H)) : Prop :=  π ≠ 0 ∧ ∀ M : Submodule ℂ H, IsClosed (M : Set H) →    (∀ a : A, ∀ x ∈ M, π a x ∈ M) → M = ⊥ ∨ M = ⊤def UnitarilyEquivalent {A : Type u} [CStarAlgebra A]    {H : Type v} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]    {K : Type w} [NormedAddCommGroup K] [InnerProductSpace ℂ K] [CompleteSpace K]    (π : A →⋆ₙₐ[ℂ] (H →L[ℂ] H)) (ρ : A →⋆ₙₐ[ℂ] (K →L[ℂ] K)) : Prop :=  ∃ U : H ≃ₗᵢ[ℂ] K, ∀ a : A, ∀ x : H, U (π a x) = ρ a (U x)def HasUniqueIrreducibleRepresentation (A : Type u) [CStarAlgebra A] : Prop :=  ∀ (H : Type v) [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]    (K : Type w) [NormedAddCommGroup K] [InnerProductSpace ℂ K] [CompleteSpace K]    (π : A →⋆ₙₐ[ℂ] (H →L[ℂ] H)) (ρ : A →⋆ₙₐ[ℂ] (K →L[ℂ] K)),    IsIrreducible π → IsIrreducible ρ → UnitarilyEquivalent π ρdef IsIsomorphicToCompacts (A : Type u) [CStarAlgebra A]    (H : Type v) [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] : Prop :=  ∃ φ : A →⋆ₙₐ[ℂ] (H →L[ℂ] H), Function.Injective φ ∧    ∀ T : H →L[ℂ] H, (∃ a : A, φ a = T) ↔ IsCompactOperator Ttheorem main :    ∃ (A : Type u) (_ : CStarAlgebra A),      IsSimpleCStar A ∧ ¬ FiniteDimensional ℂ A ∧      (∃ τ : A →L[ℂ] ℂ, IsFaithfulTracialState τ) ∧      HasUniqueIrreducibleRepresentation.{u, v, w} A ∧      ∀ (H : Type v) [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H],        ¬ IsIsomorphicToCompacts A H := by  sorryend NaimarkZFCendend 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 Naimark’s problem in ZFC

Theorem

Section 1, Introduction Read it in the paper TeX source, line 10

Show the TeX

There exists a unital infinite-dimensional simple complex $C^*$-algebra
$A$ with a faithful tracial state such that all its nonzero irreducible
representations are unitarily equivalent. In particular, $A$ is not
isomorphic to $K(H)$ for any complex Hilbert space $H$.

Configuration and file facts

File: lean/ComparatorChallenges/Naimark.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 297 link it.
Solution module: OAI.Analysis.Naimark.Tracial
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: 34
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