Lean statement

Uniform amplified commutator estimate

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.Kadison.uniform_commutator

Lean statement file: UniformCommutator.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 288 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 uniform_commutator :    ∃ C : ℝ, 0 ≤ C ∧ ∀ (K : Type u) [NormedAddCommGroup K] [InnerProductSpace ℂ K]      [CompleteSpace K] (P : VonNeumannAlgebra K) (Y : K →L[ℂ] K)      (h : ℕ), 1 ≤ h → ∀ (X : Matrix (Fin h) (Fin h) (K →L[ℂ] K)),      (∀ i j, X i j ∈ P) →      ‖diagonalAmplification h Y * matrixOperator h X -        matrixOperator h X * diagonalAmplification h Y‖ ≤        C * commutatorSeminorm P Y * ‖matrixOperator h X‖ := 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.Kadison.commutatorSeminorm line 12

def commutatorSeminorm (P : VonNeumannAlgebra K) (Y : K →L[ℂ] K) : ℝ :=  sSup {r : ℝ | ∃ a : K →L[ℂ] K, a ∈ P ∧ ‖a‖ ≤ 1 ∧ r = ‖Y * a - a * Y‖}

definition OAI.Kadison.matrixOperator line 15

def matrixOperator (h : ℕ) (X : Matrix (Fin h) (Fin h) (K →L[ℂ] K)) :    (PiLp 2 fun _ : Fin h => K) →L[ℂ] (PiLp 2 fun _ : Fin h => K) :=  (PiLp.continuousLinearEquiv 2 ℂ (fun _ : Fin h => K)).symm.toContinuousLinearMap.comp    (ContinuousLinearMap.pi (fun i => ∑ j : Fin h,      (X i j).comp (PiLp.proj 2 (fun _ : Fin h => K) j)))

definition OAI.Kadison.diagonalAmplification line 21

def diagonalAmplification (h : ℕ) (Y : K →L[ℂ] K) :    (PiLp 2 fun _ : Fin h => K) →L[ℂ] (PiLp 2 fun _ : Fin h => K) :=  (PiLp.continuousLinearEquiv 2 ℂ (fun _ : Fin h => K)).symm.toContinuousLinearMap.comp    (ContinuousLinearMap.pi (fun i => Y.comp (PiLp.proj 2 (fun _ : Fin h => K) i)))

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: commutator, integer, uniform, von, neumann, complex, algebra and space. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: constant, estimate and hilbert.

Theorem 1.2 of Kadison’s similarity theorem through uniform derivation estimates

Theorem (Uniform commutator estimate)

There is an absolute constant C<\infty such that, for every complex Hilbert space K, every unital von Neumann algebra P\subset\mathcal B(K), every Y\in\mathcal B(K), every integer h\ge1, and every X\in M_h(P),

\|[Y^{(h)},X]\|\le C\,g_P(Y)\,\|X\|.

Here M_h(P) acts naturally on K^h, and [T,U]=TU-UT.

Section 1, Introduction Read it in the paper TeX source, line 109

Show the TeX

There is an absolute constant \(C<\infty\) such that, for every complex
Hilbert space \(K\), every unital von Neumann algebra
\(P\subset\mathcal B(K)\), every \(Y\in\mathcal B(K)\), every integer
\(h\ge1\), and every \(X\in M_h(P)\),
\begin{equation}
  \|[Y^{(h)},X]\|\le C\,g_P(Y)\,\|X\|.
\end{equation}
Here \(M_h(P)\) acts naturally on \(K^h\), and \([T,U]=TU-UT\).

Other possible paper matches, closest wording first

Theorem 1.1 of Kadison’s similarity theorem through uniform derivation estimates. Words of the Lean statement, its names split into words, that also appear in the theorem: linear, pi, operator, complex, algebra and space. Found in the paper’s title, which its main theorem states: uniform. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: similarity and hilbert.
Proposition 7.2 of Kadison’s similarity theorem through uniform derivation estimates. Words of the Lean statement, its names split into words, that also appear in the theorem: von, neumann, norm, algebra and space. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: hilbert.
Theorem 3.1 of Kadison’s similarity theorem through uniform derivation estimates. Words of the Lean statement, its names split into words, that also appear in the theorem: product, integer and pi. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: estimate.
Corollary 1.3 of Kadison’s similarity theorem through uniform derivation estimates. Words of the Lean statement, its names split into words, that also appear in the theorem: von, neumann, complex, algebra and space. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: constant and hilbert.
Theorem 2.5 of Kadison’s similarity theorem through uniform derivation estimates. Words of the Lean statement, its names split into words, that also appear in the theorem: map, linear, pi, operator, algebra and space. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: similarity, derivation and hilbert.
Proposition 7.3 of Kadison’s similarity theorem through uniform derivation estimates. Words of the Lean statement, its names split into words, that also appear in the theorem: operator, norm and algebra. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: derivation.

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 288’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: similarityTheorem : SimilarityTheorem.{u,v} := ∀ (A : Type u) [CStarAlgebra A] (K : Type v) [Hilbert space K] (π : BoundedUnitalHom A K), SimilarToStar A K π with BoundedUnitalHom A H := {π : A →ₐ[ℂ] (H →L[ℂ] H) // Continuous π} and SimilarToStar π := ∃ S : (H →L[ℂ] H)ˣ, ∀ a, S π(a⋆) S⁻¹ = (S π(a) S⁻¹)⋆: Kadison's similarity conjecture for arbitrary Hilbert spaces, plus universalHyperreflexivity (one constant for all von Neumann algebras) and the uniform amplified commutator estimate.
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 linked commutator estimate is uniform over all finite matrix amplifications: its constant is independent of the algebra, Hilbert space, and amplification size. This is the quantitative derivation estimate supporting the similarity result.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states a supporting result”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

40 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: 6 to its own declarations, 26 to Mathlib’s documentation. 4 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 uniform_commutator 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 3 other declarations of the file; with the theorem itself they span 20 lines. 3 of them are named in the statement itself.

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

def commutatorSeminormline 12in the statementlibrary
def matrixOperatorline 15in the statementlibrary
def diagonalAmplificationline 21in the statementlibrary
theorem uniform_commutatorline 26selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionuniverse u vnamespace Kadisonvariable {K : Type u} [NormedAddCommGroup K] [InnerProductSpace ℂ K]  [CompleteSpace K]def commutatorSeminorm (P : VonNeumannAlgebra K) (Y : K →L[ℂ] K) : ℝ :=  sSup {r : ℝ | ∃ a : K →L[ℂ] K, a ∈ P ∧ ‖a‖ ≤ 1 ∧ r = ‖Y * a - a * Y‖}def matrixOperator (h : ℕ) (X : Matrix (Fin h) (Fin h) (K →L[ℂ] K)) :    (PiLp 2 fun _ : Fin h => K) →L[ℂ] (PiLp 2 fun _ : Fin h => K) :=  (PiLp.continuousLinearEquiv 2 ℂ (fun _ : Fin h => K)).symm.toContinuousLinearMap.comp    (ContinuousLinearMap.pi (fun i => ∑ j : Fin h,      (X i j).comp (PiLp.proj 2 (fun _ : Fin h => K) j)))def diagonalAmplification (h : ℕ) (Y : K →L[ℂ] K) :    (PiLp 2 fun _ : Fin h => K) →L[ℂ] (PiLp 2 fun _ : Fin h => K) :=  (PiLp.continuousLinearEquiv 2 ℂ (fun _ : Fin h => K)).symm.toContinuousLinearMap.comp    (ContinuousLinearMap.pi (fun i => Y.comp (PiLp.proj 2 (fun _ : Fin h => K) i)))theorem uniform_commutator :    ∃ C : ℝ, 0 ≤ C ∧ ∀ (K : Type u) [NormedAddCommGroup K] [InnerProductSpace ℂ K]      [CompleteSpace K] (P : VonNeumannAlgebra K) (Y : K →L[ℂ] K)      (h : ℕ), 1 ≤ h → ∀ (X : Matrix (Fin h) (Fin h) (K →L[ℂ] K)),      (∀ i j, X i j ∈ P) →      ‖diagonalAmplification h Y * matrixOperator h X -        matrixOperator h X * diagonalAmplification h Y‖ ≤        C * commutatorSeminorm P Y * ‖matrixOperator h X‖ := by  sorryend Kadisonendend 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.

Kadison’s similarity theorem through uniform derivation estimates

Theorem (Similarity)

Let A be any unital complex C^*-algebra, let H be any complex Hilbert space, and let \pi:A\to\mathcal B(H) be a bounded, complex-linear, unital algebra homomorphism. There exists an operator S\in\mathcal B(H), invertible with bounded inverse, such that

\rho(a)=S\pi(a)S^{-1}\qquad(a\in A)

is a *-homomorphism. In particular, \rho(a^*)=\rho(a)^* for every a\in A.

Section 1, Introduction Read it in the paper TeX source, line 15

Show the TeX

Let \(A\) be any unital complex \(C^*\)-algebra, let \(H\) be any complex
Hilbert space, and let \(\pi:A\to\mathcal B(H)\) be a bounded,
complex-linear, unital algebra homomorphism.  There exists an operator
\(S\in\mathcal B(H)\), invertible with bounded inverse, such that
\[
  \rho(a)=S\pi(a)S^{-1}\qquad(a\in A)
\]
is a \(*\)-homomorphism.  In particular,
\(\rho(a^*)=\rho(a)^*\) for every \(a\in A\).

Theorem (Uniform commutator estimate)

\|[Y^{(h)},X]\|\le C\,g_P(Y)\,\|X\|.

Here M_h(P) acts naturally on K^h, and [T,U]=TU-UT.

Section 1, Introduction Read it in the paper TeX source, line 109

Show the TeX

There is an absolute constant \(C<\infty\) such that, for every complex
Hilbert space \(K\), every unital von Neumann algebra
\(P\subset\mathcal B(K)\), every \(Y\in\mathcal B(K)\), every integer
\(h\ge1\), and every \(X\in M_h(P)\),
\begin{equation}
  \|[Y^{(h)},X]\|\le C\,g_P(Y)\,\|X\|.
\end{equation}
Here \(M_h(P)\) acts naturally on \(K^h\), and \([T,U]=TU-UT\).

Corollary (Universal hyperreflexivity)

Let H be any complex Hilbert space and let M\subset\mathcal B(H) be a unital von Neumann algebra. For T\in\mathcal B(H), set

\alpha_M(T)=\sup\{\|(1-e)Te\|:e\in M',\ e=e^*=e^2\}.

Then, with the absolute constant C from Theorem thm:universal,

\dist(T,M)\le 2C\,\alpha_M(T)\qquad(T\in\mathcal B(H)).

Thus every such M is hyperreflexive with constant at most 2C, independently of the algebra, its representation, and the Hilbert space. No optimality of this constant is asserted.

Section 1, Introduction Read it in the paper TeX source, line 120

Show the TeX

Let \(H\) be any complex Hilbert space and let
\(M\subset\mathcal B(H)\) be a unital von Neumann algebra. For
\(T\in\mathcal B(H)\), set
\[
 \alpha_M(T)=\sup\{\|(1-e)Te\|:e\in M',\ e=e^*=e^2\}.
\]
Then, with the absolute constant \(C\) from Theorem~\ref{thm:universal},
\begin{equation}
 \dist(T,M)\le 2C\,\alpha_M(T)\qquad(T\in\mathcal B(H)).
\end{equation}
Thus every such \(M\) is hyperreflexive with constant at most \(2C\),
independently of the algebra, its representation, and the Hilbert space.
No optimality of this constant is asserted.

Configuration and file facts

File: lean/ComparatorChallenges/UniformCommutator.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 288 link it.
Solution module: OAI.Analysis.Commutator.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: 3 local vocabulary declarations
Lines of Lean, without comments or blank lines: 21
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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