{"id":"openai-math/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy","pdf":"https://github.com/openai/math/blob/main/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026","tags":["free-probability","operator-algebras","free-entropy"],"score":0,"title":"A finite-entropy separation of microstates and nonmicrostates free entropy","domain":"Operator algebras","family":"298","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.OA","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":1,"published":"2026-09-25","reactions":{},"discussion":[{"url":"https://arxiv.org/abs/2610.27243","date":"2026-10-19","kind":"web","site":"arXiv.org","text":"It has been a long-standing open problem to determine when two versions of Voiculescu's free entropy, $χ$ and $χ^*$, agree. OpenAI recently generated the first example with $-\\infty < χ< χ^*$. We give a detailed presentation of a similar example, and explicitly show the separation between free…","added":"2026-10-24","title":"On counterexamples to the unification of free entropy","author":"David Jekel","record":"arxiv/2610.27243"}],"formalized":false,"family_title":"Two notions of free entropy differ even when both are finite"}