Lean statement

Universal near-identity unitary conjugacy for close von Neumann algebras

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.KadisonKastler.universal_strong_stability

Lean statement file: StrongKadisonKastler.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 289 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 universal_strong_stability : UniversalStrongStability.{u} := 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.KadisonKastler.unitBall line 16

def unitBall (M : VonNeumannAlgebra H) : Set (H →L[ℂ] H) :=  {x | x ∈ M ∧ ‖x‖ ≤ 1}

definition OAI.KadisonKastler.kkDistance line 19

noncomputable def kkDistance (M N : VonNeumannAlgebra H) : ℝ :=  Metric.hausdorffDist (unitBall M) (unitBall N)

definition OAI.KadisonKastler.unitaryConj line 22

def unitaryConj (v : unitary (H →L[ℂ] H)) (x : H →L[ℂ] H) : H →L[ℂ] H :=  (v : H →L[ℂ] H) * x * star (v : H →L[ℂ] H)

definition OAI.KadisonKastler.Conjugates line 25

def Conjugates (v : unitary (H →L[ℂ] H)) (M N : VonNeumannAlgebra H) : Prop :=  unitaryConj v '' (M : Set (H →L[ℂ] H)) = (N : Set (H →L[ℂ] H))

definition OAI.KadisonKastler.NearConjugacy line 28

def NearConjugacy (M N : VonNeumannAlgebra H) (ε : ℝ) : Prop :=  ∃ v : unitary (H →L[ℂ] H), Conjugates v M N ∧ ‖(v : H →L[ℂ] H) - 1‖ < ε

definition OAI.KadisonKastler.UniversalStrongStability line 31

def UniversalStrongStability : Prop :=  ∀ ε : ℝ, 0 < ε → ∃ δ : ℝ, 0 < δ ∧    ∀ (H : Type u) [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H],      ∀ M N : VonNeumannAlgebra H,        kkDistance M N < δ → NearConjugacy M N ε

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: strong, stability, complex, von, neumann, algebras, unitary, space and one more. Found in the paper’s title, which its main theorem states: universal. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: kastler, kadison, delta, identity, unital and hilbert.

Statements in Universal strong Kadison–Kastler stability come first because the release lists only that paper for this Lean statement.

Theorem 1.1 of Universal strong Kadison–Kastler stability

Theorem (Strong Kadison–Kastler stability)

For every \eps>0 there is \delta=\delta(\eps)>0 such that the following holds. If H is any complex Hilbert space and M,N\subseteq\B(H) are unital von Neumann algebras with identity I_H and d(M,N)<\delta, there is a unitary u\in\B(H) satisfying

uMu^*=N,\qquad \norm{u-I_H}<\eps.

Section 1, Introduction Read it in the paper TeX source, line 18

Show the TeX

For every $\eps>0$ there is $\delta=\delta(\eps)>0$ such that the
following holds. If $H$ is any complex Hilbert space and
$M,N\subseteq\B(H)$ are unital von Neumann algebras with identity
$I_H$ and $d(M,N)<\delta$, there is a unitary $u\in\B(H)$ satisfying
\[
 uMu^*=N,\qquad \norm{u-I_H}<\eps.
\]

Other possible paper matches, closest wording first

Theorem 1.1 of Close Separable C*-Algebras Without Spatial Conjugacy. Words of the Lean statement, its names split into words, that also appear in the theorem: kk, complex, algebras, unitary, space and norm. Found in the paper’s title, which its main theorem states: conjugacy. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: varepsilon, identity, unital and hilbert.
Proposition 8.3 of Universal strong Kadison–Kastler stability. Words of the Lean statement, its names split into words, that also appear in the theorem: product, conjugacy, group, von, neumann, algebras and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: unital and hilbert.
Theorem 5.1 of Universal strong Kadison–Kastler stability. Words of the Lean statement, its names split into words, that also appear in the theorem: stability, von, neumann, algebra, unitary, space and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: representations, identity, unital and hilbert.
Theorem 9.4 of Universal strong Kadison–Kastler stability. Words of the Lean statement, its names split into words, that also appear in the theorem: von, neumann, algebras, unitary, space and norm. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: tolerance, delta, representation, unital and hilbert.
Theorem 8.4 of Universal strong Kadison–Kastler stability. Words of the Lean statement, its names split into words, that also appear in the theorem: stability, complex, unitary, space and norm. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: representations, identity, unital and hilbert.
Proposition 5.4 of Universal strong Kadison–Kastler stability. Words of the Lean statement, its names split into words, that also appear in the theorem: unit, near, von, neumann, space and norm. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: representations and hilbert.

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; 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

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 5351 s; peak memory 8.1 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.

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 289’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: universal_strong_stability : ∀ ε > 0, ∃ δ > 0, ∀ H, ∀ von Neumann algebras M N on H (Mathlib VonNeumannAlgebra), kkDistance M N < δ → ∃ unitary v, v M v* = N ∧ ‖v - 1‖ < ε with kkDistance the Hausdorff distance of the unit balls. Not stated: the summary's two counterexamples (near-identity conjugacy fails for one-sided near inclusions; arbitrarily close norm-separable C*-algebras need not be ambiently unitarily conjugate); the docs scope also only mentions the stability theorem although the docs title says 'spatial boundaries'.
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: Both counterexample clauses (the family's two October 5 manuscripts) are not stated.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“For every $\varepsilon>0$, there is a $\delta>0$ such that any two unital von Neumann algebras on the same complex Hilbert space at Kadison–Kastler distance below $\delta$ are conjugate by a unitary $u$ with $\lVert u-1\rVert<\varepsilon$. The tolerance depends only on $\varepsilon$, uniformly over the algebras, their representations, and the Hilbert space.”
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

44 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: 7 to its own declarations, 19 to Mathlib’s documentation. 7 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 universal_strong_stability 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 6 other declarations of the file; with the theorem itself they span 17 lines. 1 of them is named in the statement itself.

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

def unitBallline 16library
def kkDistanceline 19library
def unitaryConjline 22library
def Conjugatesline 25library
def NearConjugacyline 28library
def UniversalStrongStabilityline 31in the statementlibrary
theorem universal_strong_stabilityline 37selected theoremlibrary

import Mathlibnamespace OAI/-! Theorem 1.1 of *Universal strong Kadison–Kastler stability*:one tolerance gives small unitary conjugacies on every complex Hilbert space. -/namespace KadisonKastlernoncomputable sectionuniverse uvariable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]def unitBall (M : VonNeumannAlgebra H) : Set (H →L[ℂ] H) :=  {x | x ∈ M ∧ ‖x‖ ≤ 1}noncomputable def kkDistance (M N : VonNeumannAlgebra H) : ℝ :=  Metric.hausdorffDist (unitBall M) (unitBall N)def unitaryConj (v : unitary (H →L[ℂ] H)) (x : H →L[ℂ] H) : H →L[ℂ] H :=  (v : H →L[ℂ] H) * x * star (v : H →L[ℂ] H)def Conjugates (v : unitary (H →L[ℂ] H)) (M N : VonNeumannAlgebra H) : Prop :=  unitaryConj v '' (M : Set (H →L[ℂ] H)) = (N : Set (H →L[ℂ] H))def NearConjugacy (M N : VonNeumannAlgebra H) (ε : ℝ) : Prop :=  ∃ v : unitary (H →L[ℂ] H), Conjugates v M N ∧ ‖(v : H →L[ℂ] H) - 1‖ < εdef UniversalStrongStability : Prop :=  ∀ ε : ℝ, 0 < ε → ∃ δ : ℝ, 0 < δ ∧    ∀ (H : Type u) [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H],      ∀ M N : VonNeumannAlgebra H,        kkDistance M N < δ → NearConjugacy M N εtheorem universal_strong_stability : UniversalStrongStability.{u} := by  sorryendend KadisonKastlerend 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.

Universal strong Kadison–Kastler stability

Theorem (Strong Kadison–Kastler stability)

uMu^*=N,\qquad \norm{u-I_H}<\eps.

Section 1, Introduction Read it in the paper TeX source, line 18

Show the TeX

For every $\eps>0$ there is $\delta=\delta(\eps)>0$ such that the
following holds. If $H$ is any complex Hilbert space and
$M,N\subseteq\B(H)$ are unital von Neumann algebras with identity
$I_H$ and $d(M,N)<\delta$, there is a unitary $u\in\B(H)$ satisfying
\[
 uMu^*=N,\qquad \norm{u-I_H}<\eps.
\]

Configuration and file facts

File: lean/ComparatorChallenges/StrongKadisonKastler.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 289 link it.
Solution module: OAI.Analysis.KadisonKastler.StrongStability
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: 6 local vocabulary declarations
Lines of Lean, without comments or blank lines: 17
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