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

Claims a negative answer: a bounded self-adjoint tuple with $-\infty<\chi(X)\le\chi^*(X)-\tfrac12<\infty$ , 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 $n$ ), and uses the limsup version of $\chi$ ; equality under the known convexity conditions is unaffected.

Exact FrontierDelta

Prior state unknown→disproved

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Occurred: Sep 25, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Voiculescu's question: do microstates and nonmicrostates free entropy agree when finite? · Equality of free entropies

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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

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No independent mathematician has checked this yet. Theorem 1.1 was read against the question: a bounded self-adjoint $n$ -tuple ( $n\ge2$ , large but fixed) in a von Neumann algebra with faithful normal trace with $-\infty<\chi(X)\le\chi^*(X)-1/2<\infty$ , $\chi$ 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 $\chi$ via microstate volumes with cutoff and limsup, $\chi^*$ 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.