Result 298, Operator algebras

Two notions of free entropy differ even when both are finite

Constructs a bounded self-adjoint tuple in a tracial von Neumann algebra whose microstates and nonmicrostates free entropies satisfy −∞<χ<χ∗<∞-\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 formalization Disproof or counterexample

The bigger picture

Why it matters

Two ways of measuring free entropy, a notion of disorder for variables whose multiplication need not commute, can give different finite answers. This challenges the idea that they are interchangeable descriptions of the same quantity.

What changes?

The manuscript reports a bounded tuple of self-adjoint operators, the operator analogue of real-valued variables, in a von Neumann algebra with a faithful normal tracial state, which supplies an averaging rule. Its microstates entropy, defined using matrix approximations with an operator-norm cutoff and a limit superior over matrix sizes, is finite and at least one-half below its also-finite nonmicrostates entropy, defined through free Fisher information. The construction uses a large but fixed number of variables.

What does that help mathematicians do?

The reported example rules out equality based solely on both entropies being finite, answering Voiculescu's finite-entropy equality question negatively. The positive gap shows that the discrepancy is not just an artifact of infinite values. Researchers therefore cannot transfer an exact entropy value between the two definitions without further justification. The construction does not establish separation for every tuple or for small numbers of variables.

Are there practical applications?

The immediate value is foundational in operator algebras. Matrix approximation and free Fisher information provide distinct approaches to entropy, and this example identifies a limit on treating them as equivalent. It clarifies which definition a theorem must use and motivates identifying additional assumptions under which the two quantities might agree.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

A finite-entropy separation of microstates and nonmicrostates free entropy

September 25, 2026 15 pages

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)−12<∞-\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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026,
  author = {{OpenAI}},
  title = {{A finite-entropy separation of microstates and nonmicrostates free entropy}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/paper.pdf}{OAI:A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/298.md.

Two notions of free entropy differ even when both are finite

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization disproves equality of microstates and nonmicrostates free entropy even when both quantities are finite. It constructs a bounded self-adjoint tuple XX in a von Neumann algebra with a faithful normal tracial state such that −∞<χ(X)≤χ∗(X)−1/2<∞-\infty<\chi(X)\le\chi^*(X)-1/2<\infty. Here χ\chi is the microstates entropy with an operator-norm cutoff and a limsup over matrix sizes, and χ∗\chi^* is the nonmicrostates entropy defined using free semicircular noise. The example has a fixed finite number of variables.

Comparator links

Result Comparator statement
Finite separation of microstates and nonmicrostates free entropy FiniteEntropySeparation.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.