Two notions of free entropy differ even when both are finite

What OpenAI claims

Constructs a bounded self-adjoint tuple in a tracial von Neumann algebra whose microstates and nonmicrostates free entropies satisfy -∞ < χ < χ^* < ∞. This answers Voiculescu’s finite-entropy equality question negatively: the matrix-approximation and free-Fisher-information definitions differ even when both are finite. (Lean)

Paper

1. A finite-entropy separation of microstates and nonmicrostates free entropy

   25 September 2026 · Preprint, not peer reviewed · PDF

   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 -∞ < χ(X) ≤ χ^*(X)-1/2 < ∞. 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.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.