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Math-2824 · mathematics · Analysis

Equality of microstates and nonmicrostates free entropy

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every bounded finite self-adjoint tuple in a tracial von Neumann algebra for which both entropies are finite, must classical microstates free entropy $\chi $, with its original operator-norm cutoff and limsup convention, equal nonmicrostates free entropy $\chi *$?

Short English statement

For every bounded finite self-adjoint tuple in a tracial von Neumann algebra for which both entropies are finite, must classical microstates free entropy $\chi $, with its original operator-norm cutoff and limsup convention, equal nonmicrostates free entropy $\chi *$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2824-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2824

Manuscript 1 (family 298, 2026-09-25): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed counterexample to that assertion in full.

Equality of microstates and nonmicrostates free entropy

OpenAI's claim

A bounded finite self-adjoint tuple is claimed to satisfy finite microstates and nonmicrostates free entropies separated by at least one half, disproving equality even under finite-entropy assumptions. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
For some $n\ge2$, OpenAI claims operators $X=(X_1,\ldots,X_n)$, each bounded and self-adjoint, belonging to a von Neumann algebra equipped with a normal faithful tracial state satisfying \[ -\infty<\chi(X)\le\chi^*(X)-\tfrac12<\infty. \] The microstates entropy $\chi$ uses the original operator-norm cutoff and limsup in matrix size; $\chi^*$ is nonmicrostates free entropy. Both are finite and separated by a fixed positive gap, so finiteness does not imply their equality.

Outline of the argument

Manuscript 1 (family 298, 2026-09-25): A commuting block signal is perturbed by a small semicircular noise to ensure finite microstates entropy. At a second noise scale, cumulant and Fisher estimates place nonmicrostates entropy near the corresponding Gaussian value, while matrix microstates retain a block event exponentially unlikely for Gaussian ensembles. Entropy data processing turns that discrepancy into a fixed gap with the variable count held finite.

Formal verification

For family 298, the scope page documents the following selections. The selected bounded tuple has both free entropies finite and a gap of at least one half between them, using the specified matrix-cutoff and semicircular-noise definitions. The documented statement covers the main claim of manuscript 1. Its selected declaration also covers the clause used for this problem. Comparator files: FiniteEntropySeparation.lean. Declarations: OAI.FiniteEntropySeparation.main.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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