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Math-2824-A1 · revision r1 · on Math-2824

OpenAI claim summary for Math-2824

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

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.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Equality of microstates and nonmicrostates free entropy
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

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