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. (Lean)
Manuscript
A finite-entropy separation of microstates and nonmicrostates free entropy
September 25, 2026 · source · cite as \cite{OAI:A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026}
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.
Result 298 · Operator algebras. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md