Lean statement

Finite separation of microstates and nonmicrostates free entropy

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.FiniteEntropySeparation.main

Lean statement file: FiniteEntropySeparation.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 298 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 :    ∃ (H : Type) (_ : NormedAddCommGroup H) (_ : InnerProductSpace ℂ H)      (_ : CompleteSpace H) (M : VonNeumannAlgebra H) (τ : TracialVector M)      (n : ℕ) (X S : Fin n → H →L[ℂ] H),      2 ≤ n ∧      (∀ i, X i ∈ M ∧ IsSelfAdjoint (X i)) ∧      (∀ i, S i ∈ M ∧ IsSelfAdjoint (S i)) ∧      FreeStandardNoise τ X S ∧      (⊥ : EReal) < chi τ X ∧      chi τ X ≤ chiStar τ X S - (1 / 2 : EReal) ∧      chiStar τ X S - (1 / 2 : EReal) < (⊤ : EReal) := 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.

structure OAI.FiniteEntropySeparation.TracialVector line 14

structure TracialVector (M : VonNeumannAlgebra H) where  vacuum : H  norm_vacuum : ‖vacuum‖ = 1  tracial : ∀ A ∈ M, ∀ B ∈ M,    inner ℂ vacuum ((A * B) vacuum) = inner ℂ vacuum ((B * A) vacuum)  separating : ∀ A ∈ M, A vacuum = 0 → A = 0

definition OAI.FiniteEntropySeparation.trace line 21

def trace {M : VonNeumannAlgebra H} (τ : TracialVector M) (A : H →L[ℂ] H) : ℂ :=  inner ℂ τ.vacuum (A τ.vacuum)

definition OAI.FiniteEntropySeparation.wordEval line 24

def wordEval {n : ℕ} (X : Fin n → H →L[ℂ] H) (w : List (Fin n)) : H →L[ℂ] H :=  (w.map X).prod

definition OAI.FiniteEntropySeparation.wordMoment line 27

def wordMoment {M : VonNeumannAlgebra H} (τ : TracialVector M) {n : ℕ}    (X : Fin n → H →L[ℂ] H) (w : List (Fin n)) : ℂ := trace τ (wordEval X w)

definition OAI.FiniteEntropySeparation.FreeSubalgebras line 30

def FreeSubalgebras {ι : Type} {M : VonNeumannAlgebra H} (τ : TracialVector M)    (A : ι → Subalgebra ℂ (H →L[ℂ] H)) : Prop :=  ∀ (k : ℕ), 0 < k → ∀ (j : Fin k → ι) (a : Fin k → H →L[ℂ] H),    (∀ r, a r ∈ A (j r)) → (∀ r, trace τ (a r) = 0) →    (∀ (r : ℕ) (hr : r + 1 < k),      j ⟨r, Nat.lt_trans (Nat.lt_succ_self r) hr⟩ ≠ j ⟨r + 1, hr⟩) →    trace τ (List.ofFn a).prod = 0

definition OAI.FiniteEntropySeparation.FreeStandardNoise line 38

def FreeStandardNoise {M : VonNeumannAlgebra H} (τ : TracialVector M) {n : ℕ}    (X S : Fin n → H →L[ℂ] H) : Prop :=  (∀ i k, trace τ (S i ^ k) =    if Even k then (catalan (k / 2) : ℂ) else 0) ∧  FreeSubalgebras τ (fun j : Option (Fin n) => match j with    | none => Algebra.adjoin ℂ (Set.range X)    | some i => Algebra.adjoin ℂ {S i})

definition OAI.FiniteEntropySeparation.polynomialL2 line 46

def polynomialL2 {M : VonNeumannAlgebra H} (τ : TracialVector M) {n : ℕ}    (X : Fin n → H →L[ℂ] H) : Submodule ℂ H :=  (Submodule.span ℂ (Set.range (fun w : List (Fin n) => wordEval X w τ.vacuum))).topologicalClosure

definition OAI.FiniteEntropySeparation.derivativeMoment line 50

def derivativeMoment {M : VonNeumannAlgebra H} (τ : TracialVector M) {n : ℕ}    (X : Fin n → H →L[ℂ] H) (i : Fin n) (w : List (Fin n)) : ℂ :=  ∑ r : Fin w.length, if w[r.val] = i then    wordMoment τ X (w.take r.val) * wordMoment τ X (w.drop (r.val + 1)) else 0

definition OAI.FiniteEntropySeparation.ConjugateSystem line 55

def ConjugateSystem {M : VonNeumannAlgebra H} (τ : TracialVector M) {n : ℕ}    (X : Fin n → H →L[ℂ] H) (ξ : Fin n → H) : Prop :=  ∀ i, ξ i ∈ polynomialL2 τ X ∧ ∀ w : List (Fin n),    inner ℂ (ξ i) (wordEval X w τ.vacuum) = derivativeMoment τ X i w

definition OAI.FiniteEntropySeparation.freeFisher line 60

def freeFisher {M : VonNeumannAlgebra H} (τ : TracialVector M) {n : ℕ}    (X : Fin n → H →L[ℂ] H) : ℝ≥0∞ :=  ⨅ (ξ : Fin n → H) (_ : ConjugateSystem τ X ξ), ENNReal.ofReal (∑ i, ‖ξ i‖ ^ 2)

definition OAI.FiniteEntropySeparation.heatTuple line 64

def heatTuple {n : ℕ} (X S : Fin n → H →L[ℂ] H) (s : ℝ) :    Fin n → H →L[ℂ] H := fun i => X i + (Real.sqrt s : ℂ) • S i

definition OAI.FiniteEntropySeparation.chiStar line 67

def chiStar {M : VonNeumannAlgebra H} (τ : TracialVector M) {n : ℕ}    (X S : Fin n → H →L[ℂ] H) : EReal :=  let b : ℝ → ℝ≥0∞ := fun s => ENNReal.ofReal ((n : ℝ) / (1 + s))  let f : ℝ → ℝ≥0∞ := fun s => freeFisher τ (heatTuple X S s)  ((n / 2 * Real.log (2 * Real.pi * Real.exp 1) : ℝ) : EReal) +    (1 / 2 : EReal) * (∫⁻ s in Set.Ioi (0 : ℝ), b s - f s).toEReal -    (1 / 2 : EReal) * (∫⁻ s in Set.Ioi (0 : ℝ), f s - b s).toEReal

abbreviation OAI.FiniteEntropySeparation.MatrixCoordinates line 75

abbrev MatrixCoordinates (n d : ℕ) := EuclideanSpace ℝ (Fin n × Fin d × Fin d)

definition OAI.FiniteEntropySeparation.hermitianMatrix line 77

def hermitianMatrix {n d : ℕ} (a : MatrixCoordinates n d) (i : Fin n) :    Matrix (Fin d) (Fin d) ℂ := fun j k =>  if j = k then (a (i, j, j) : ℂ)  else if j < k then ((a (i, j, k) : ℂ) + Complex.I * (a (i, k, j) : ℂ)) / Real.sqrt 2  else ((a (i, k, j) : ℂ) - Complex.I * (a (i, j, k) : ℂ)) / Real.sqrt 2

definition OAI.FiniteEntropySeparation.matrixOpNorm line 83

def matrixOpNorm {d : ℕ} (A : Matrix (Fin d) (Fin d) ℂ) : ℝ :=  ‖(Matrix.toEuclideanLin A).toContinuousLinearMap‖

definition OAI.FiniteEntropySeparation.matrixMoment line 86

def matrixMoment {n d : ℕ} (a : MatrixCoordinates n d) (w : List (Fin n)) : ℂ :=  (d : ℂ)⁻¹ * Matrix.trace (w.map (hermitianMatrix a)).prod

definition OAI.FiniteEntropySeparation.microstates line 89

def microstates {M : VonNeumannAlgebra H} (τ : TracialVector M) {n : ℕ}    (X : Fin n → H →L[ℂ] H) (R : ℝ) (m d : ℕ) (ε : ℝ) : Set (MatrixCoordinates n d) :=  {a | (∀ i, matrixOpNorm (hermitianMatrix a i) ≤ R) ∧    ∀ w : List (Fin n), w.length ≤ m → ‖matrixMoment a w - wordMoment τ X w‖ < ε}

definition OAI.FiniteEntropySeparation.chiCutoff line 94

def chiCutoff {M : VonNeumannAlgebra H} (τ : TracialVector M) {n : ℕ}    (X : Fin n → H →L[ℂ] H) (R : ℝ) : EReal :=  ⨅ m : ℕ, ⨅ ε : {ε : ℝ // 0 < ε},    limsup (fun d : ℕ =>      (((((d + 1 : ℕ) : ℝ) ^ 2)⁻¹ : ℝ) : EReal) *        (volume (microstates τ X R m (d + 1) ε.val)).log +      (((n : ℝ) / 2 * Real.log (d + 1) : ℝ) : EReal)) atTop

definition OAI.FiniteEntropySeparation.chi line 102

def chi {M : VonNeumannAlgebra H} (τ : TracialVector M) {n : ℕ}    (X : Fin n → H →L[ℂ] H) : EReal :=  ⨆ R : {R : ℝ // 0 < R}, chiCutoff τ X R.val

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: 1/2, microstates, von, algebra, tracial, chi, neumann, norm and six more. Found in the paper’s title, which its main theorem states: free. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: using, state, sizes, operator, normal, faithful, entropy and bounded.

Theorem 1.1 of A finite-entropy separation of microstates and nonmicrostates free entropy

Theorem

There exist an integer n\geq2 and a bounded self-adjoint tuple X=(X_1,\ldots,X_n) in a von Neumann algebra with faithful normal tracial state such that

-\infty<\chi(X)\leq\chi^*(X)-\tfrac12<\infty.

Here \chi is the original microstates entropy using an operator-norm cutoff and a limsup over matrix sizes.

Section 1, Introduction Read it in the paper TeX source, line 42

Show the TeX

There exist an integer $n\geq2$ and a bounded self-adjoint tuple
$X=(X_1,\ldots,X_n)$ in a von Neumann algebra with faithful normal
tracial state such that
\[
 -\infty<\chi(X)\leq\chi^*(X)-\tfrac12<\infty.
\]
Here $\chi$ is the original microstates entropy using an operator-norm
cutoff and a limsup over matrix sizes.

Other possible paper matches, closest wording first

Proposition 3.2 of A finite-entropy separation of microstates and nonmicrostates free entropy. Words of the Lean statement, its names split into words, that also appear in the theorem: sqrt, fisher, standard, adjoint, tuple, self and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: semicircular and bound.

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 298’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 : ∃ H M τ n X S, 2 ≤ n ∧ X_i, S_i ∈ M self-adjoint ∧ FreeStandardNoise τ X S ∧ ⊥ < chi τ X ∧ chi τ X ≤ chiStar τ X S - 1/2 ∧ chiStar τ X S - 1/2 < ⊤: a bounded self-adjoint tuple in a von Neumann algebra with faithful normal trace (TracialVector), with S a freely independent semicircular family free from X, and chi (microstates with operator-norm cutoff R, limsup over matrix sizes, inf over moment orders m and tolerances ε, sup over R) strictly below the nonmicrostates chiStar (free Fisher information integral along X + √s S) by at least 1/2, both finite. Matches the summary (n >= 2 variables, finite and unequal).
Definitions the group asks readers to compare with the paper: Custom 'free entropy': chiCutoff τ X R := ⨅ m ε, limsup_d ((d+1)^{-2} (volume (microstates τ X R m (d+1) ε)).log + (n/2) log(d+1)) in EReal, chi := ⨆ R, chiCutoff, freeFisher := ⨅ conjugate systems ξ, ‖ξ‖² (∞ if none), chiStar := (n/2) log(2πe) + 1/2 ∫⁻ (n/(1+s) - Φ*(X + √s S))⁺ - 1/2 ∫⁻ (Φ* - n/(1+s))⁺ via ConjugateSystem (polynomial moments) and FreeSubalgebras; standard Voiculescu definitions written out by hand (EReal ⊤ - ⊤ conventions are not an issue because the claim bounds chiStar - 1/2 below ⊤).
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 disproves equality of microstates and nonmicrostates free entropy even when both quantities are finite. It constructs a bounded self-adjoint tuple $X$ in a von Neumann algebra with a faithful normal tracial state such that $-\infty<\chi(X)\le\chi^*(X)-1/2<\infty$. Here $\chi$ is the microstates entropy with an operator-norm cutoff and a limsup over matrix sizes, and $\chi^*$ is the nonmicrostates entropy defined using free semicircular noise. The example has a fixed finite number of variables.”
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

125 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: 43 to its own declarations, 117 to Mathlib’s documentation. 20 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.

It reaches all 19 other declarations of the file; with the theorem itself they span 85 lines. 4 of them are named in the statement itself.

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

structure TracialVectorline 14in the statementlibrary
def traceline 21library
def wordEvalline 24library
def wordMomentline 27library
def FreeSubalgebrasline 30library
def FreeStandardNoiseline 38in the statementlibrary
def polynomialL2line 46library
def derivativeMomentline 50library
def ConjugateSystemline 55library
def freeFisherline 60library
def heatTupleline 64library
def chiStarline 67in the statementlibrary
abbrev MatrixCoordinatesline 75library
def hermitianMatrixline 77library
def matrixOpNormline 83library
def matrixMomentline 86library
def microstatesline 89library
def chiCutoffline 94library
def chiline 102in the statementlibrary
theorem mainline 108selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionopen scoped ENNReal NNReal ComplexConjugateopen MeasureTheory Filternamespace FiniteEntropySeparationvariable {H : Type} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]structure TracialVector (M : VonNeumannAlgebra H) where  vacuum : H  norm_vacuum : ‖vacuum‖ = 1  tracial : ∀ A ∈ M, ∀ B ∈ M,    inner ℂ vacuum ((A * B) vacuum) = inner ℂ vacuum ((B * A) vacuum)  separating : ∀ A ∈ M, A vacuum = 0 → A = 0def trace {M : VonNeumannAlgebra H} (τ : TracialVector M) (A : H →L[ℂ] H) : ℂ :=  inner ℂ τ.vacuum (A τ.vacuum)def wordEval {n : ℕ} (X : Fin n → H →L[ℂ] H) (w : List (Fin n)) : H →L[ℂ] H :=  (w.map X).proddef wordMoment {M : VonNeumannAlgebra H} (τ : TracialVector M) {n : ℕ}    (X : Fin n → H →L[ℂ] H) (w : List (Fin n)) : ℂ := trace τ (wordEval X w)def FreeSubalgebras {ι : Type} {M : VonNeumannAlgebra H} (τ : TracialVector M)    (A : ι → Subalgebra ℂ (H →L[ℂ] H)) : Prop :=  ∀ (k : ℕ), 0 < k → ∀ (j : Fin k → ι) (a : Fin k → H →L[ℂ] H),    (∀ r, a r ∈ A (j r)) → (∀ r, trace τ (a r) = 0) →    (∀ (r : ℕ) (hr : r + 1 < k),      j ⟨r, Nat.lt_trans (Nat.lt_succ_self r) hr⟩ ≠ j ⟨r + 1, hr⟩) →    trace τ (List.ofFn a).prod = 0def FreeStandardNoise {M : VonNeumannAlgebra H} (τ : TracialVector M) {n : ℕ}    (X S : Fin n → H →L[ℂ] H) : Prop :=  (∀ i k, trace τ (S i ^ k) =    if Even k then (catalan (k / 2) : ℂ) else 0) ∧  FreeSubalgebras τ (fun j : Option (Fin n) => match j with    | none => Algebra.adjoin ℂ (Set.range X)    | some i => Algebra.adjoin ℂ {S i})def polynomialL2 {M : VonNeumannAlgebra H} (τ : TracialVector M) {n : ℕ}    (X : Fin n → H →L[ℂ] H) : Submodule ℂ H :=  (Submodule.span ℂ (Set.range (fun w : List (Fin n) => wordEval X w τ.vacuum))).topologicalClosuredef derivativeMoment {M : VonNeumannAlgebra H} (τ : TracialVector M) {n : ℕ}    (X : Fin n → H →L[ℂ] H) (i : Fin n) (w : List (Fin n)) : ℂ :=  ∑ r : Fin w.length, if w[r.val] = i then    wordMoment τ X (w.take r.val) * wordMoment τ X (w.drop (r.val + 1)) else 0def ConjugateSystem {M : VonNeumannAlgebra H} (τ : TracialVector M) {n : ℕ}    (X : Fin n → H →L[ℂ] H) (ξ : Fin n → H) : Prop :=  ∀ i, ξ i ∈ polynomialL2 τ X ∧ ∀ w : List (Fin n),    inner ℂ (ξ i) (wordEval X w τ.vacuum) = derivativeMoment τ X i wdef freeFisher {M : VonNeumannAlgebra H} (τ : TracialVector M) {n : ℕ}    (X : Fin n → H →L[ℂ] H) : ℝ≥0∞ :=  ⨅ (ξ : Fin n → H) (_ : ConjugateSystem τ X ξ), ENNReal.ofReal (∑ i, ‖ξ i‖ ^ 2)def heatTuple {n : ℕ} (X S : Fin n → H →L[ℂ] H) (s : ℝ) :    Fin n → H →L[ℂ] H := fun i => X i + (Real.sqrt s : ℂ) • S idef chiStar {M : VonNeumannAlgebra H} (τ : TracialVector M) {n : ℕ}    (X S : Fin n → H →L[ℂ] H) : EReal :=  let b : ℝ → ℝ≥0∞ := fun s => ENNReal.ofReal ((n : ℝ) / (1 + s))  let f : ℝ → ℝ≥0∞ := fun s => freeFisher τ (heatTuple X S s)  ((n / 2 * Real.log (2 * Real.pi * Real.exp 1) : ℝ) : EReal) +    (1 / 2 : EReal) * (∫⁻ s in Set.Ioi (0 : ℝ), b s - f s).toEReal -    (1 / 2 : EReal) * (∫⁻ s in Set.Ioi (0 : ℝ), f s - b s).toERealabbrev MatrixCoordinates (n d : ℕ) := EuclideanSpace ℝ (Fin n × Fin d × Fin d)def hermitianMatrix {n d : ℕ} (a : MatrixCoordinates n d) (i : Fin n) :    Matrix (Fin d) (Fin d) ℂ := fun j k =>  if j = k then (a (i, j, j) : ℂ)  else if j < k then ((a (i, j, k) : ℂ) + Complex.I * (a (i, k, j) : ℂ)) / Real.sqrt 2  else ((a (i, k, j) : ℂ) - Complex.I * (a (i, j, k) : ℂ)) / Real.sqrt 2def matrixOpNorm {d : ℕ} (A : Matrix (Fin d) (Fin d) ℂ) : ℝ :=  ‖(Matrix.toEuclideanLin A).toContinuousLinearMap‖def matrixMoment {n d : ℕ} (a : MatrixCoordinates n d) (w : List (Fin n)) : ℂ :=  (d : ℂ)⁻¹ * Matrix.trace (w.map (hermitianMatrix a)).proddef microstates {M : VonNeumannAlgebra H} (τ : TracialVector M) {n : ℕ}    (X : Fin n → H →L[ℂ] H) (R : ℝ) (m d : ℕ) (ε : ℝ) : Set (MatrixCoordinates n d) :=  {a | (∀ i, matrixOpNorm (hermitianMatrix a i) ≤ R) ∧    ∀ w : List (Fin n), w.length ≤ m → ‖matrixMoment a w - wordMoment τ X w‖ < ε}def chiCutoff {M : VonNeumannAlgebra H} (τ : TracialVector M) {n : ℕ}    (X : Fin n → H →L[ℂ] H) (R : ℝ) : EReal :=  ⨅ m : ℕ, ⨅ ε : {ε : ℝ // 0 < ε},    limsup (fun d : ℕ =>      (((((d + 1 : ℕ) : ℝ) ^ 2)⁻¹ : ℝ) : EReal) *        (volume (microstates τ X R m (d + 1) ε.val)).log +      (((n : ℝ) / 2 * Real.log (d + 1) : ℝ) : EReal)) atTopdef chi {M : VonNeumannAlgebra H} (τ : TracialVector M) {n : ℕ}    (X : Fin n → H →L[ℂ] H) : EReal :=  ⨆ R : {R : ℝ // 0 < R}, chiCutoff τ X R.valopen MeasureTheorytheorem main :    ∃ (H : Type) (_ : NormedAddCommGroup H) (_ : InnerProductSpace ℂ H)      (_ : CompleteSpace H) (M : VonNeumannAlgebra H) (τ : TracialVector M)      (n : ℕ) (X S : Fin n → H →L[ℂ] H),      2 ≤ n ∧      (∀ i, X i ∈ M ∧ IsSelfAdjoint (X i)) ∧      (∀ i, S i ∈ M ∧ IsSelfAdjoint (S i)) ∧      FreeStandardNoise τ X S ∧      (⊥ : EReal) < chi τ X ∧      chi τ X ≤ chiStar τ X S - (1 / 2 : EReal) ∧      chiStar τ X S - (1 / 2 : EReal) < (⊤ : EReal) := by  sorryend FiniteEntropySeparationendend 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 finite-entropy separation of microstates and nonmicrostates free entropy

Theorem

There exist an integer n\geq2 and a bounded self-adjoint tuple X=(X_1,\ldots,X_n) in a von Neumann algebra with faithful normal tracial state such that

-\infty<\chi(X)\leq\chi^*(X)-\tfrac12<\infty.

Here \chi is the original microstates entropy using an operator-norm cutoff and a limsup over matrix sizes.

Section 1, Introduction Read it in the paper TeX source, line 42

Show the TeX

There exist an integer $n\geq2$ and a bounded self-adjoint tuple
$X=(X_1,\ldots,X_n)$ in a von Neumann algebra with faithful normal
tracial state such that
\[
 -\infty<\chi(X)\leq\chi^*(X)-\tfrac12<\infty.
\]
Here $\chi$ is the original microstates entropy using an operator-norm
cutoff and a limsup over matrix sizes.

Configuration and file facts

File: lean/ComparatorChallenges/FiniteEntropySeparation.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 298 link it.
Solution module: OAI.Analysis.FreeEntropy.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: 19 local vocabulary declarations
Lines of Lean, without comments or blank lines: 85
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