Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 298-01. A finite-entropy separation of microstates and nonmicrostates free entropy
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OpenAI@2026-10-07#rem:oai-p-298-01
Authors: OpenAI. Family 298.
Title: A finite-entropy separation of microstates and nonmicrostates free entropy.
Abstract: 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.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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