Two notions of free entropy differ even when both are finite
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Constructs a bounded self-adjoint tuple in a tracial von Neumann algebra whose microstates and nonmicrostates free entropies satisfy \(-\infty\lt \chi\lt \chi^*\lt \infty\). This answers Voiculescu’s finite-entropy equality question negatively: the matrix-approximation and free-Fisher-information definitions differ even when both are finite. (OpenAI’s Lean note)
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Both independent assessments find that the Lean statements express or imply the whole result described above.Read the assessments
Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
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.
Reported proof verification
One Lean statement in the release; two independent groups report that Comparator accepted runs for this file at the version shown here.
We answer the finite-entropy equality question for microstates and nonmicrostates free entropy negatively. We construct a bounded self-adjoint tuple X in a von Neumann algebra with faithful normal tracial state such that \(-\infty\lt \chi(X)\leq\chi^*(X)-\tfrac12\lt \infty\). Here χ is the original microstates entropy with an operator-norm cutoff and a limsup over matrix sizes. The counterexample uses a large but fixed number of variables.
One Lean statement in this family
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For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s 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.
The words they share: 1/2, microstates, von, algebra, tracial, chi, neumann, norm, cutoff, integer, matrix, adjoint, tuple, self, free, using, state, sizes, operator, normal, faithful, entropy, bounded.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
FiniteEntropySeparation.lean, the release lists it as a supporting statement.
theoremmain : ∃ (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) := bysorry
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
Other mathematical statements in OpenAI’s Lean library
Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).
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 ⊤).
“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.
Their method: 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.
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.
AI-drafted comparisons, with no recorded human review
Finite separation of microstates and nonmicrostates free entropyFiniteEntropySeparation
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: a bounded self-adjoint tuple with −∞ < χ(X) ≤ χ*(X) − 1/2 < ∞, separating the two free entropies.
Who ran the proof-verification program?
These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.
These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.
Finite separation of microstates and nonmicrostates free entropyFiniteEntropySeparation
Two reports, all at the release’s recorded version.
Reported by
Their word
Their account
Date
Original report
cert-machine
“CERTIFIED”
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
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Papers elsewhere in the release that this family’s papers cite
Papers elsewhere in the release that cite this family’s papers
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Identify the theorem the catalogue describes
Family 298The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means.Open the form
Address a question raised in an independent assessment
The challenge statements imply the headline claim after unpacking definitions or one routine step.
Their method
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.
The challenge statements state the headline as OpenAI wrote it.
Their method
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…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
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.