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Voiculescu's question: do microstates and nonmicrostates free entropy agree when finite?

For a bounded self-adjoint tuple XX in a tracial von Neumann algebra, Voiculescu defined the microstates free entropy χ(X)\chi(X) from volumes of matrix approximations and the nonmicrostates entropy χ∗(X)\chi^*(X) from free Fisher information along free semicircular noise. They agree for one variable, and Biane-Capitaine-Guionnet proved χ≤χ∗\chi\le\chi^* in general; equality is known under convexity hypotheses. Agreement was one of the questions of Voiculescu's unification problem, and Guionnet stated the finite-entropy form as a conjecture. Does χ(X)>−∞\chi(X)>-\infty imply χ(X)=χ∗(X)\chi(X)=\chi^*(X)?

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Construction
Field
Free probability; operator algebras
Posed by
Dan Voiculescu, Free entropy, Bull. London Math. Soc. 34 (2002), Sections 3.5 and 3.8 (unification problem); finite-entropy form stated by Alice Guionnet, Probab. Surv. 1 (2004), Conjecture 7.1
Year posed
2002
Years open
24y
Solved
2026-09-25
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
34 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Claims a negative answer: a bounded self-adjoint tuple with −∞<χ(X)≤χ∗(X)−12<∞-\infty<\chi(X)\le\chi^*(X)-\tfrac12<\infty, built as a small semicircular perturbation of free semicirculars tensored with a commuting discrete variable. It does NOT settle equality for every smaller number of variables (the example uses a large fixed nn), and uses the limsup version of χ\chi; equality under the known convexity conditions is unaffected.

What the AI did

The release README says all results were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result. Its named exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. The manuscripts are credited to OpenAI with no human author named. Single manuscript; it mentions the release's free group factor isomorphism paper only for contrast. The main theorem has a Lean comparator challenge in the release.

Verification

No independent mathematician has checked this yet. Theorem 1.1 was read against the question: a bounded self-adjoint nn-tuple (n≥2n\ge2, large but fixed) in a von Neumann algebra with faithful normal trace with −∞<χ(X)≤χ∗(X)−1/2<∞-\infty<\chi(X)\le\chi^*(X)-1/2<\infty, χ\chi taken with operator-norm cutoff and limsup over matrix sizes. The Lean challenge ComparatorChallenges/FiniteEntropySeparation.json (theorem OAI.FiniteEntropySeparation.main, solution module OAI.Analysis.FreeEntropy.Main, present at the pinned commit) is not in formalization.yaml; it was found through lean/docs/298.md. Its statement was read here: it defines χ\chi via microstate volumes with cutoff and limsup, χ∗\chi^* via conjugate systems along a free standard semicircular tuple in the same algebra, and asserts exactly the displayed inequalities. Permitted axioms propext, Quot.sound, Classical.choice. Not rebuilt here.

Sources

Changelog1 change

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