# Two notions of free entropy differ even when both are finite

> Proves two standard notions of free entropy can differ by at least 1/2 even when both are finite. Plain-English explainer, Lean status and sources.

Canonical: https://closedform.nishachay.tech/e/openai-math-2026-298/
Last updated: 2026-10-14

**Proves two standard notions of free entropy can differ by at least 1/2 even when both are finite.**

**Trust:** Formal proof listed · **Papers:** 1 · **First paper:** September 25, 2026 · **Field:** Operator algebras · **Entry:** 298

*Explained by AI.* Drafted by one AI model and checked sentence by sentence against the paper by a different one. Spot a mistake? Report an error.

Free entropy measures how much room a family of noncommuting variables has, much as ordinary entropy does for random numbers. Voiculescu gave two definitions: one counts nearby matrix models, the other tracks added noise. They agree for one variable, and a known question asked if they agree whenever the first is finite. This paper builds a family where both are finite but differ by at least one half, and Lean checks it. The family uses a large number of variables.

## At a glance

- **First step:** 1994 (Voiculescu, who defined microstates free entropy (1994) and later nonmicrostates free entropy (1998))
- **Significance:** Major. It answers negatively the finite-entropy form of a question in Voiculescu's free-entropy unification problem, with a Lean check.
- **Trust:** Formal proof listed
- **Field:** Operator algebras (1 paper)

## The explanation

Start with random matrices. Fill a large d-by-d Hermitian matrix with Gaussian entries of variance 1, the GUE ensemble. In the paper's normalization, its entropy per entry is ½·log(2πe), about 1.419. Take a second, independent GUE coordinate of variance 4. Its value is ½·log(8πe), about 2.112. Independent coordinates add, so the pair has entropy about 3.531[1]. Entropy here measures spread: more variance, more room.

Free probability studies variables that do not commute, like large matrices. Free entropy measures their room. Voiculescu gave two definitions. The microstates entropy χ counts the volume of matrix tuples whose trace moments match the variables. The nonmicrostates entropy χ* instead adds free semicircular noise, the noncommutative bell curve, and integrates an information measure along the way[2]. The two agree for a single variable. Whether they agree in general was part of Voiculescu's unification problem[3]. Biane, Capitaine and Guionnet proved χ ≤ χ* in general[4]. Equality was known in regular convex settings. The finite-entropy question asked: if χ is finite, must χ equal χ*?

The paper answers no[5]. It builds a bounded tuple X with χ finite and χ at most χ* − ½[6]. The gap of ½ is about a third of the one-variable value 1.419 above. The construction mixes free semicircular variables with one coordinate Y that is spread evenly over p points in [−1, 1]. With p = 3, those points are −1, 0 and 1. Y commutes with the others and splits their energy into blocks. A small dose of free noise makes χ finite while keeping that block structure[7].

The obstacle is to show the two entropies see the block structure differently. After more noise, χ* becomes close to that of plain semicircular variables. Matrix models keep detectable blocks, which costs them volume. The noise time t = p^(3/4) balances the two estimates as p grows[8]. For p = 16, that gives t = 8. The example needs a large but fixed number of variables. It says nothing about smaller numbers[9].

How sure is this? Lean, a computer proof checker, has a formal statement of the main theorem: a bounded tuple with χ finite and χ ≤ χ* − ½. So the paper proves this, as far as the Lean statement matches the paper. No specialist review is published.

**What it doesn't show.** This does not settle the question for small numbers of variables, and it does not address cases where the microstates entropy is minus infinity.

## Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

## Hype check

A clean negative answer to a named question, with a Lean check. It needs many variables and leaves small numbers of variables open. No specialist review is published.

## Words used

- **Free probability:** A theory of noncommuting random variables, such as large random matrices, with freeness in place of independence.
- **Self-adjoint tuple:** A list of operators each equal to its own adjoint, the analogue of real random variables.
- **Microstates free entropy χ:** A measure of the volume of matrix tuples whose trace moments approximate those of the variables.
- **Nonmicrostates free entropy χ*:** A free entropy defined by adding free semicircular noise and integrating free Fisher information.
- **Semicircular variable:** The free-probability analogue of a Gaussian; its eigenvalue density is a semicircle.
- **GUE:** Random Hermitian matrices with independent Gaussian entries, whose eigenvalues follow the semicircle law.
- **Tracial state:** A positive normalized linear functional τ with τ(xy) = τ(yx), playing the role of an expectation.

## The original paper

- [A finite-entropy separation of microstates and nonmicrostates free entropy](https://github.com/openai/math/blob/main/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/paper.pdf) - September 25, 2026

- [Lean notes](https://github.com/openai/math/blob/main/lean/docs/298.md)

## Quoted lines from the paper (9)

[1] From [A finite-entropy separation of microstates and nonmicrostates free entropy](https://github.com/openai/math/blob/main/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/paper.pdf):

> Thus independent GUE coordinates of variances $w_i>0$ have normalized entropy $\frac12\sum_i\log(2\pi e w_i)$.

[2] From [A finite-entropy separation of microstates and nonmicrostates free entropy](https://github.com/openai/math/blob/main/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/paper.pdf):

> His nonmicrostates free entropy $\chi^*(X)$ instead uses conjugate variables, the free analogues of classical scores, and integrates free Fisher information along addition of freely independent semicircular noise.

[3] From [A finite-entropy separation of microstates and nonmicrostates free entropy](https://github.com/openai/math/blob/main/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/paper.pdf):

> The two quantities agree for a single self-adjoint variable, and their agreement for general tuples was one of the questions in Voiculescu's unification problem.

[4] From [A finite-entropy separation of microstates and nonmicrostates free entropy](https://github.com/openai/math/blob/main/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/paper.pdf):

> Biane, Capitaine, and Guionnet proved the general comparison $\chi\leq\chi^*$ through large-deviation bounds for matrix Brownian motion.

[5] From [A finite-entropy separation of microstates and nonmicrostates free entropy](https://github.com/openai/math/blob/main/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/paper.pdf):

> We answer the finite-entropy equality question for microstates and nonmicrostates free entropy negatively.

[6] From [A finite-entropy separation of microstates and nonmicrostates free entropy](https://github.com/openai/math/blob/main/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/paper.pdf):

> We construct a bounded self-adjoint tuple $X$ in a von Neumann algebra with faithful normal tracial state such that $-\infty<\chi(X)\leq\chi^*(X)-\tfrac12<\infty$.

[7] From [A finite-entropy separation of microstates and nonmicrostates free entropy](https://github.com/openai/math/blob/main/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/paper.pdf):

> A small perturbation $X=C+\sqrt h S$ has finite microstates entropy, while its last coordinate still determines spectral blocks containing much of the energy of the first $p$ coordinates.

[8] From [A finite-entropy separation of microstates and nonmicrostates free entropy](https://github.com/openai/math/blob/main/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/paper.pdf):

> The choice $t=p^{3/4}$ satisfies both conditions as $p\to\infty$.

[9] From [A finite-entropy separation of microstates and nonmicrostates free entropy](https://github.com/openai/math/blob/main/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/paper.pdf):

> It does not address equality for every smaller prescribed number of variables.

## Other sources

- [openai/math, Lean scope note for family 298](https://github.com/openai/math/blob/main/lean/docs/298.md): “The formalization disproves equality of microstates and nonmicrostates free entropy even when both quantities are finite.”
- [openai/math release notes (2026): model provenance and review status](https://github.com/openai/math/blob/main/README.md): “manuscripts and supporting proof artifacts produced by an internal OpenAI model”

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