Lean statement

Kadison’s similarity theorem and uniform hyperreflexivity

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.

Kadison's similarity theorem and uniform hyperreflexivity

Theorems selected for verification: OAI.KadisonSimilarity.universalCommutatorTheorem OAI.KadisonSimilarity.universalHyperreflexivity OAI.KadisonSimilarity.similarityTheorem

Lean statement file: KadisonSimilarity.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.

Read the cited theorem in the paper 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 universalCommutatorTheorem : UniversalCommutatorTheorem.{u} := by  sorry⋮theorem universalHyperreflexivity : UniversalHyperreflexivity.{u} universalConstant := by  sorry⋮theorem similarityTheorem : SimilarityTheorem.{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.

abbreviation OAI.KadisonSimilarity.BoundedUnitalHom line 16

abbrev BoundedUnitalHom :=  {π : A →ₐ[ℂ] (H →L[ℂ] H) // Continuous π}

definition OAI.KadisonSimilarity.SimilarToStar line 19

def SimilarToStar (π : BoundedUnitalHom A H) : Prop :=  ∃ S : (H →L[ℂ] H)ˣ, ∀ a : A,    (S : H →L[ℂ] H) * π.1 (star a) * (↑S⁻¹ : H →L[ℂ] H) =      star ((S : H →L[ℂ] H) * π.1 a * (↑S⁻¹ : H →L[ℂ] H))

definition OAI.KadisonSimilarity.commutator line 30
```
def commutator (S T : R) : R := S * T - T * S
```

abbreviation OAI.KadisonSimilarity.HilbertCopies line 42

abbrev HilbertCopies (H : Type u) (n : ℕ) := PiLp 2 (fun _ : Fin n => H)

definition OAI.KadisonSimilarity.matrixOperator line 44

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

definition OAI.KadisonSimilarity.amplify line 50

def amplify (Y : H →L[ℂ] H) (n : ℕ) :    HilbertCopies H n →L[ℂ] HilbertCopies H n :=  (PiLp.continuousLinearEquiv 2 ℂ (fun _ : Fin n => H)).symm.toContinuousLinearMap.comp    (ContinuousLinearMap.pi fun i => Y.comp (PiLp.proj 2 (fun _ : Fin n => H) i))

definition OAI.KadisonSimilarity.innerDerivation line 55

def innerDerivation (Y : H →L[ℂ] H) :    (H →L[ℂ] H) →L[ℂ] (H →L[ℂ] H) :=  ContinuousLinearMap.mul ℂ (H →L[ℂ] H) Y -    (ContinuousLinearMap.mul ℂ (H →L[ℂ] H)).flip Y

definition OAI.KadisonSimilarity.scalarCommutatorNorm line 62

def scalarCommutatorNorm (P : VonNeumannAlgebra H) (Y : H →L[ℂ] H) : ℝ := by  let : SeminormedAddCommGroup P.toStarSubalgebra.toSubalgebra.toSubmodule := inferInstance  let : NormedSpace ℂ P.toStarSubalgebra.toSubalgebra.toSubmodule := inferInstance  exact ‖(innerDerivation Y).comp P.toStarSubalgebra.toSubalgebra.toSubmodule.subtypeL‖

definition OAI.KadisonSimilarity.CommutatorEstimate line 67

def CommutatorEstimate (P : VonNeumannAlgebra H) (C : ℝ) : Prop :=  ∀ (Y : H →L[ℂ] H) (n : ℕ) (X : Matrix (Fin n) (Fin n) (H →L[ℂ] H)),    (∀ i j, X i j ∈ P) →    ‖commutator (amplify Y n) (matrixOperator X)‖ ≤      C * scalarCommutatorNorm P Y * ‖matrixOperator X‖

definition OAI.KadisonSimilarity.UniformCommutatorEstimate line 73

def UniformCommutatorEstimate (C : ℝ) : Prop :=  ∀ (K : Type u) [NormedAddCommGroup K] [InnerProductSpace ℂ K] [CompleteSpace K]    (P : VonNeumannAlgebra K), CommutatorEstimate P C

definition OAI.KadisonSimilarity.UniversalCommutatorTheorem line 77

def UniversalCommutatorTheorem : Prop := ∃ C : ℝ, 0 ≤ C ∧ UniformCommutatorEstimate.{u} C

definition OAI.KadisonSimilarity.SimilarityTheorem line 79

def SimilarityTheorem : Prop :=  ∀ (A : Type u) [CStarAlgebra A] (K : Type v) [NormedAddCommGroup K]    [InnerProductSpace ℂ K] [CompleteSpace K] (π : BoundedUnitalHom A K),    SimilarToStar A K π

abbreviation OAI.KadisonSimilarity.CommutantProjection line 84

abbrev CommutantProjection (M : VonNeumannAlgebra H) :=  {e : H →L[ℂ] H // e ∈ M.commutant ∧ star e = e ∧ e * e = e}

definition OAI.KadisonSimilarity.offDiagonalSeminorm line 87

def offDiagonalSeminorm (M : VonNeumannAlgebra H) (T : H →L[ℂ] H) : ℝ :=  sSup (Set.range fun e : CommutantProjection M =>    ‖(1 - (e : H →L[ℂ] H)) * T * (e : H →L[ℂ] H)‖)

definition OAI.KadisonSimilarity.UniversalHyperreflexivity line 91

def UniversalHyperreflexivity (C : ℝ) : Prop :=  ∀ (K : Type u) [NormedAddCommGroup K] [InnerProductSpace ℂ K] [CompleteSpace K]    (M : VonNeumannAlgebra K) (T : K →L[ℂ] K),    Metric.infDist T (M : Set (K →L[ℂ] K)) ≤ 2 * C * offDiagonalSeminorm M T

definition OAI.KadisonSimilarity.rowConstant line 100
```
def rowConstant : ℝ := 4 * Real.sqrt 2
```

definition OAI.KadisonSimilarity.cyclicConstant line 101

def cyclicConstant : ℝ := (1 + 4 * rowConstant) ^ 2

definition OAI.KadisonSimilarity.cornerConstant line 102

def cornerConstant : ℝ := 6 * Real.pi ^ 2 * (1 + 4 * cyclicConstant)

definition OAI.KadisonSimilarity.factorConstant line 103

def factorConstant : ℝ := max (max (3 * cornerConstant + 2) (1 + 2 * rowConstant)) 2

definition OAI.KadisonSimilarity.universalConstant line 104

def universalConstant : ℝ := 3 * factorConstant + 2

Paper theorem cited in the Lean source

A source comment for OAI.KadisonSimilarity.similarityTheorem names this theorem’s label in the paper’s TeX source. Read the source comment from line 14

/-- Theorem `thm:similarity`: every bounded unital complex-algebra
representation of a unital C-star algebra is similar to a star representation. -/

Theorem 1.1 of 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 \(a\in A\).

Other possible paper matches, closest wording first

Theorem 1.2 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: commutator, integer, uniform, absolute, von, constant and seven more.
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: universal, hyperreflexivity, dist, absolute, von, constant and seven more.
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, similar, similarity, linear, pi, derivation and seven more.
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, projection, corner, integer, pi and estimate.
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: derivation, bounded, representation, operator, norm, algebra and one more.
Proposition 6.3 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: finite, factor, constant and norm.

Reported Comparator runs

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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

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 formalization proves this for arbitrary Hilbert spaces. It also gives one universal hyperreflexivity constant for all unital von Neumann algebras.”
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

122 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: 26 to its own declarations, 62 to Mathlib’s documentation. 23 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 universalCommutatorTheorem 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 20 of the file’s 22 other declarations; with the theorem itself they span 58 lines. 4 of them are named in the statement itself.

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

abbrev BoundedUnitalHomline 16library
def SimilarToStarline 19library
def commutatorline 30library
abbrev HilbertCopiesline 42library
def matrixOperatorline 44library
def amplifyline 50library
def innerDerivationline 55library
def scalarCommutatorNormline 62library
def CommutatorEstimateline 67library
def UniformCommutatorEstimateline 73library
def UniversalCommutatorTheoremline 77in the statementlibrary
def SimilarityTheoremline 79in the statementlibrary
abbrev CommutantProjectionline 84library
def offDiagonalSeminormline 87library
def UniversalHyperreflexivityline 91in the statementlibrary
def rowConstantline 100library
def cyclicConstantline 101library
def cornerConstantline 102library
def factorConstantline 103library
def universalConstantline 104in the statementlibrary
theorem universalCommutatorTheoremline 107selected theoremlibrary
theorem universalHyperreflexivityline 111selected theoremlibrary
theorem similarityTheoremline 115selected theoremlibrary

Other Lean rules that may affect the statement 3

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.

run_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {} line 40
run_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {} line 98
attribute [-instance] NonUnitalCStarAlgebra.toNormedSpace line 38

import Mathlibnamespace OAInoncomputable sectionnamespace KadisonSimilarityuniverse u vsection Homomorphismsvariable (A H : Type*) [CStarAlgebra A]  [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]abbrev BoundedUnitalHom :=  {π : A →ₐ[ℂ] (H →L[ℂ] H) // Continuous π}def SimilarToStar (π : BoundedUnitalHom A H) : Prop :=  ∃ S : (H →L[ℂ] H)ˣ, ∀ a : A,    (S : H →L[ℂ] H) * π.1 (star a) * (↑S⁻¹ : H →L[ℂ] H) =      star ((S : H →L[ℂ] H) * π.1 a * (↑S⁻¹ : H →L[ℂ] H))end Homomorphismssection Commutatorsvariable {R : Type*} [NormedRing R]def commutator (S T : R) : R := S * T - T * Send Commutatorsvariable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H]sectionattribute [-instance] NonUnitalCStarAlgebra.toNormedSpacerun_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}abbrev HilbertCopies (H : Type u) (n : ℕ) := PiLp 2 (fun _ : Fin n => H)def matrixOperator {n : ℕ} (X : Matrix (Fin n) (Fin n) (H →L[ℂ] H)) :    HilbertCopies H n →L[ℂ] HilbertCopies H n :=  (PiLp.continuousLinearEquiv 2 ℂ (fun _ : Fin n => H)).symm.toContinuousLinearMap.comp    (ContinuousLinearMap.pi fun i =>      ∑ j : Fin n, (X i j).comp (PiLp.proj 2 (fun _ : Fin n => H) j))def amplify (Y : H →L[ℂ] H) (n : ℕ) :    HilbertCopies H n →L[ℂ] HilbertCopies H n :=  (PiLp.continuousLinearEquiv 2 ℂ (fun _ : Fin n => H)).symm.toContinuousLinearMap.comp    (ContinuousLinearMap.pi fun i => Y.comp (PiLp.proj 2 (fun _ : Fin n => H) i))def innerDerivation (Y : H →L[ℂ] H) :    (H →L[ℂ] H) →L[ℂ] (H →L[ℂ] H) :=  ContinuousLinearMap.mul ℂ (H →L[ℂ] H) Y -    (ContinuousLinearMap.mul ℂ (H →L[ℂ] H)).flip Yvariable [CompleteSpace H]def scalarCommutatorNorm (P : VonNeumannAlgebra H) (Y : H →L[ℂ] H) : ℝ := by  let : SeminormedAddCommGroup P.toStarSubalgebra.toSubalgebra.toSubmodule := inferInstance  let : NormedSpace ℂ P.toStarSubalgebra.toSubalgebra.toSubmodule := inferInstance  exact ‖(innerDerivation Y).comp P.toStarSubalgebra.toSubalgebra.toSubmodule.subtypeL‖def CommutatorEstimate (P : VonNeumannAlgebra H) (C : ℝ) : Prop :=  ∀ (Y : H →L[ℂ] H) (n : ℕ) (X : Matrix (Fin n) (Fin n) (H →L[ℂ] H)),    (∀ i j, X i j ∈ P) →    ‖commutator (amplify Y n) (matrixOperator X)‖ ≤      C * scalarCommutatorNorm P Y * ‖matrixOperator X‖def UniformCommutatorEstimate (C : ℝ) : Prop :=  ∀ (K : Type u) [NormedAddCommGroup K] [InnerProductSpace ℂ K] [CompleteSpace K]    (P : VonNeumannAlgebra K), CommutatorEstimate P Cdef UniversalCommutatorTheorem : Prop := ∃ C : ℝ, 0 ≤ C ∧ UniformCommutatorEstimate.{u} Cdef SimilarityTheorem : Prop :=  ∀ (A : Type u) [CStarAlgebra A] (K : Type v) [NormedAddCommGroup K]    [InnerProductSpace ℂ K] [CompleteSpace K] (π : BoundedUnitalHom A K),    SimilarToStar A K πabbrev CommutantProjection (M : VonNeumannAlgebra H) :=  {e : H →L[ℂ] H // e ∈ M.commutant ∧ star e = e ∧ e * e = e}def offDiagonalSeminorm (M : VonNeumannAlgebra H) (T : H →L[ℂ] H) : ℝ :=  sSup (Set.range fun e : CommutantProjection M =>    ‖(1 - (e : H →L[ℂ] H)) * T * (e : H →L[ℂ] H)‖)def UniversalHyperreflexivity (C : ℝ) : Prop :=  ∀ (K : Type u) [NormedAddCommGroup K] [InnerProductSpace ℂ K] [CompleteSpace K]    (M : VonNeumannAlgebra K) (T : K →L[ℂ] K),    Metric.infDist T (M : Set (K →L[ℂ] K)) ≤ 2 * C * offDiagonalSeminorm M Tendrun_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}def rowConstant : ℝ := 4 * Real.sqrt 2def cyclicConstant : ℝ := (1 + 4 * rowConstant) ^ 2def cornerConstant : ℝ := 6 * Real.pi ^ 2 * (1 + 4 * cyclicConstant)def factorConstant : ℝ := max (max (3 * cornerConstant + 2) (1 + 2 * rowConstant)) 2def universalConstant : ℝ := 3 * factorConstant + 2/-- One absolute constant controls every finite matrix commutator. -/theorem universalCommutatorTheorem : UniversalCommutatorTheorem.{u} := by  sorry/-- Every von Neumann algebra has hyperreflexivity constant at most `2 * universalConstant`. -/theorem universalHyperreflexivity : UniversalHyperreflexivity.{u} universalConstant := by  sorry/-- Every bounded unital representation is similar to a star representation. -/theorem similarityTheorem : SimilarityTheorem.{u,v} := by  sorryend KadisonSimilarityendend 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)

\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)

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 \([T,U]=TU-UT\).

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/KadisonSimilarity.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.KadisonSimilarity.UnconditionalSimilarity
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: 20 local vocabulary declarations
Lines of Lean, without comments or blank lines: 62
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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