A finite free-entropy gap: noise and persistent matrix blocks

Source and verification

This article explains family 298 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

🏷️ Preliminaries: two entropy constructions

Let $X=(X_1,\dots ,X_n)$ be bounded selfadjoint operators in a von Neumann algebra with faithful normal tracial state $\tau$. Their joint law records every number $\tau (X_{i_1}\cdots X_{i_m})$. Unlike classical random variables, the operators need not commute.

A microstate is a tuple of Hermitian matrices whose normalized trace moments approximate these numbers. Write ${\mathrm{tr}}_d=d^{-1}\mathrm{Tr}$. The set ${\Gamma }_R(X;m,d,\epsilon )$ consists of $n$-tuples of $d\times d$ Hermitian matrices with each operator norm at most $R$, matching all words of length at most $m$ within $\epsilon$.

The microstates entropy measures their asymptotic volume. Using Lebesgue volume for the unnormalized Hilbert–Schmidt inner product, define

$$\begin{array}{c}{Q}_d(\Gamma )=d^{-2}\log \mathrm{Vol}(\Gamma )+\frac{n}{2}\log d,\\ {\chi }_R(X)\\ =\underset{m,\epsilon }{\inf }\underset{d\to \infty }{\mathrm{limsup}}{Q}_d({\Gamma }_R(X;m,d,\epsilon )),\\ \chi (X)=\underset{R}{\sup }{\chi }_R(X).\end{array}$$

The cutoff and the matrix-size limsup are part of the definition. Empty sets have logarithmic volume $-\infty$.

The second construction uses an analogue of the score function from classical probability. For a noncommutative polynomial $P$, the free difference quotient ${\partial }_{i}P$ sums over occurrences of its $i$th variable, putting the prefix and suffix on opposite sides of a tensor product. For example,

$${\partial }_1(x_1{x}_2{x}_1)=1\otimes x_2{x}_1+x_1{x}_2\otimes 1.$$

The conjugate variables ${\xi }_i\in L^2(W^{\ast }(Z),\tau )$ of a tuple $Z$ satisfy

$$\tau ({\xi }_{i}P(Z))=(\tau \otimes \tau )(({\partial }_{i}P)(Z)).$$

The free Fisher information is ${\Phi }^{\ast }(Z)={\sum }_i\Vert {\xi }_i{\Vert }_2^2$, where $\Vert a{\Vert }_2^2=\tau (a^{\ast }a)$; it is infinite if the conjugate variables do not exist.

A standard semicircular variable has density $(2\pi {)}^{-1}\sqrt{4-x^2}$ on $[-2,2]$, mean zero and variance one. A standard semicircular family has freely independent coordinates. Freeness means that every alternating product of centered elements from successively different participating algebras has trace zero. In each noise perturbation below, the noise family is also free from the original tuple.

The nonmicrostates entropy is

$$\begin{array}{c}\begin{array}{rl}{\chi }^{\ast }(Z)=& \frac{n}{2}\log (2\pi e)\\ & +\frac{1}{2}{\int }_0^{\infty }{J}_Z(s)ds,\end{array}\\ J_Z(s)=\frac{n}{1+s}-{\Phi }^{\ast }(Z+\sqrt{s}S).\end{array}$$

It measures Fisher information along addition of free semicircular noise. Although the two constructions look different, they agree in several important cases, including a single selfadjoint variable. The general inequality is $\chi \le {\chi }^{\ast }$.

🏷️ A strict gap with both entropies finite

Finite-entropy separation

The manuscript constructs a bounded selfadjoint tuple in a faithfully normally tracial von Neumann algebra for which

$$\begin{array}{c}-\infty <\chi (X),\\ \chi (X)\le {\chi }^{\ast }(X)-\frac{1}{2}<\infty .\end{array}$$

The number of variables is finite and fixed before taking matrix-size limits. The entropy $\chi$ retains its original cutoff and limsup definition. (OpenAI, 2026)

This rules out the assertion that finite microstates entropy forces equality. An example with no sufficiently accurate matrix models and $\chi =-\infty$ would not address that assertion.

The mechanism uses two noise scales. A small noise makes the microstates entropy finite while preserving substantial block structure. A later, larger noise makes the analytic entropy close to a semicircular benchmark, but the same evolved matrix ensembles still have a detectable structural deficit relative to independent Gaussian matrices.

🏷️ The commuting signal and its small perturbation

Fix a large integer $p$ and put $n=p+1$. Let $B_1,\dots ,B_p$ be freely independent standard semicircular variables. In a separate tensor factor, let $Y$ be uniformly distributed on $p$ equally spaced points in $[-1,1]$. Thus $Y$ commutes with every $B_i$; it is not free from them.

Set

$$\begin{array}{c}C=(B_1,\dots ,B_p,Y),\\ V_a=C+\sqrt{a}S.\end{array}$$

The signal is centered with diagonal covariance. Write $c_i=\tau (C_i^2)$, so $c_i=1$ for $i\le p$, and define the entropy benchmark

$$L(a)=\frac{1}{2}\sum _{i=1}^n\log (2\pi e(a+c_i)).$$

This is both the nonmicrostates entropy of independent semicircular coordinates with those variances and the normalized classical entropy of the corresponding independent Gaussian matrix ensemble.

The spectral projections of $Y$ give $p$ common blocks for all the $B_i$. For sufficiently small $h>0$, the last coordinate of $X=V_h$ still has separated spectral clusters. Partition its spectrum at the midpoints of the original atoms and call the projections $e_l$. Norm-continuity of these spectral projections, using the gaps, permits the bounds

$$\begin{array}{c}\tau (e_l)<\frac{3}{2p},\\ \sum _{i=1}^p\sum _{l=1}^p\Vert e_l{X}_i{e}_l{\Vert }_2^2>\frac{3p}{4}.\end{array}$$

The first $p$ coordinates therefore retain a large amount of energy inside small common blocks.

To see why $\chi (X)>-\infty$, choose uniformly bounded deterministic matrix models for $C$. Such models arise by tensoring semicircular matrix models with the finite diagonal model of $Y$. Add independent Gaussian unitary ensemble noise of variance $h$. Here a standard GUE matrix has real Hilbert–Schmidt coordinate covariance $1/d$.

At a fixed sufficiently large operator-norm cutoff, the noised models satisfy every fixed moment test with probability tending to one along their model dimensions. Their density is bounded above by $(d/(2\pi h){)}^{n{d}^2/2}$. Consequently, if $q_d\to 1$ is that probability,

$$\mathrm{Vol}{\Gamma }_R\ge q_d(2\pi h/d{)}^{n{d}^2/2}.$$

After normalization, this gives the finite lower bound $\frac{n}{2}\log (2\pi h)$. A subsequence of matrix sizes suffices because the definition uses a limsup. The argument does not claim that the original commuting signal $C$ has finite microstates entropy.

🏷️ Sparse cumulants control the analytic deficit

Free cumulants separate moments into contributions indexed by noncrossing partitions. A partition is noncrossing if there are no interlaced positions $a<b<c<d$ with $a,c$ in one block and $b,d$ in another. The cumulants are defined recursively by expressing each moment as the sum of products of block cumulants. Semicircular families have only second-order cumulants, given by their covariance.

For the order-$m$ cumulant array of $C$, use the Euclidean norm over all index lists. The manuscript proves, with $K=128$,

$$\Vert {\kappa }_m(C){\Vert }_{{\ell }^2}\le K^m{p}^{m/4}.$$

The useful feature is the exponent $p^{m/4}$. A nonzero moment contribution requires the semicircular positions to be paired with equal labels. Tensor independence lets the $Y$ factors be removed from these words. Noncrossing pair counts then bound the number of potentially nonzero cumulant entries, rather than bounding every one of the $n^m$ entries indiscriminately.

A second estimate turns those squared array norms into a Fisher-information bound. Write $\Delta (u)={\Phi }^{\ast }(V_u)-{\sum }_i(u+c_i{)}^{-1}$ for the Fisher deficit. Then

$$\begin{array}{c}0\le \Delta (u)\\ \le \frac{1}{u^2}\sum _{r\ge 2}{u}^{-r}\Vert {\kappa }_{r+1}(C){\Vert }_{{\ell }^2}^2,\end{array}$$

whenever the series converges.

The proof uses orthonormal semicircular Wick polynomials, obtained by removing internal covariance contractions from words in the noise variables. Testing a Wick polynomial against $P(V_u)$ selects ordered occurrences of the corresponding indices and multiplies the traces of the intervening gaps. This gives the conjugate-variable formula

$${\xi }_i=u^{-1/2}E[S_i\mid W^{\ast }(V_u)],$$

where the conditional expectation is the orthogonal projection in the trace Hilbert space.

Expanding the cumulant block containing the first variable produces an orthogonal Wick series $R_i$ with

$$\begin{array}{c}{V}_{u,i}-(u+c_i){\xi }_i=E[R_i\mid W^{\ast }(V_u)],\\ \sum _i\Vert R_i{\Vert }_2^2=\sum _{r\ge 2}{u}^{-r}\Vert {\kappa }_{r+1}(C){\Vert }_{{\ell }^2}^2.\end{array}$$

The exact remainder identity is

$$\begin{array}{c}\Vert V_{u,i}-(u+c_i){\xi }_i{\Vert }_2^2\\ =(u+c_i{)}^2\left(\Vert {\xi }_i{\Vert }_2^2-\frac{1}{u+c_i}\right).\end{array}$$

It follows by expanding the square and using $\tau (V_{u,i}{\xi }_i)=1$. Projection contraction and $u+c_i\ge u$ then give the stated deficit bound.

For $u>2K^2\sqrt{p}$, summing the geometric series gives a deficit at most $2K^6{p}^{3/2}{u}^{-4}$. Integrating it yields

$$\begin{array}{c}{\chi }^{\ast }(V_a)=L(a)-\frac{1}{2}{\int }_a^{\infty }\Delta (u)du,\\ L(a)-{\chi }^{\ast }(V_a)\le \frac{K^6{p}^{3/2}}{3a^3}<\frac{1}{2},\\ a>2K^2\sqrt{p}.\end{array}$$

For every positive $a$, finiteness follows from the separate bounds ${\sum }_i(u+c_i{)}^{-1}\le {\Phi }^{\ast }(V_u)\le n/u$, whose difference has an integrable $O(u^{-2})$ tail.

🏷️ Entropy production for the same matrix ensembles

Fix a cutoff $R$ with ${\chi }_R(X)>-\infty$. Select dimensions tending to infinity and uniform laws $A^{(d)}$ on increasingly accurate microstate sets at that cutoff, with normalized entropies at least ${\chi }_R(X)-o(1)$.

For a matrix density $f$, its normalized classical entropy is

$$H_d(f)=-d^{-2}\int f\log f+\frac{n}{2}\log d.$$

Retain these particular ensembles and evolve them by independent GUE noise:

$$Z^{(d)}(s)=A^{(d)}+\sqrt{s}{G}^{(d)}.$$

The Gaussian covariance is $(s/d)I$ in real Hilbert–Schmidt coordinates. The normalized de Bruijn identity is

$$\begin{array}{c}\frac{d}{ds}{H}_d(Z^{(d)}(s))=\frac{1}{2}{I}_d(s),\\ I_d(s)=d^{-3}\int \Vert \nabla \log f_s{\Vert }^2{f}_s.\end{array}$$

Integration by parts against polynomial matrix tests and the variational formula for free Fisher information give

$$\begin{array}{c}\underset{d}{\mathrm{liminf}}{I}_d(s)\ge {\Phi }^{\ast }(V_{h+s})\\ (s>0).\end{array}$$

The fixed cutoff supplies uniform control of the initial matrices; Gaussian norm bounds and trace variance estimates justify convergence of polynomial traces and their products. Completing squares against the classical score then gives each free polynomial test before taking their supremum.

Integrate from a positive time first. Gaussian convolution does not decrease classical entropy, so letting the initial time decrease to zero needs no assumption of entropy continuity at zero. Fatou’s lemma yields

$$\begin{array}{c}\underset{d}{\mathrm{limsup}}{H}_d(Z^{(d)}(t))\\ \ge {\chi }_R(X)+\frac{1}{2}{\int }_0^t{\Phi }^{\ast }(V_{h+s})ds.\end{array}$$

This is a statement about the entropies of the same selected terminal laws. Moment convergence does not identify their entropy limit with $\chi (V_{h+t})$. That distinction is essential: the next estimate concerns precisely these laws.

🏷️ A block event that Gaussian matrices rarely satisfy

For an orthogonal projection decomposition $P=(P_1,\dots ,P_p)$, define block compression of a tuple by

$${\mathcal{D}}_{P}D={\left(\sum _l{P}_l{D}_i{P}_l\right)}_i.$$

For every sufficiently accurate cutoff microstate $A^{(d)}$ of $X$, the spectral partition of its last coordinate gives

$$\begin{array}{c}\mathrm{rank}{P}_l\le 2d/p,\\ \Vert {\mathcal{D}}_P{A}_{[p]}^{(d)}{\Vert }_{2,d}^2>p/2.\end{array}$$

Here $[p]$ means the first $p$ coordinates and $\Vert D{\Vert }_{2,d}^2={\sum }_i{\mathrm{tr}}_d(D_i^{\ast }{D}_i)$.

This assertion allows spectral outliers and eigenvalues at the partition thresholds. Approximate the discontinuous interval indicators by continuous functions differing only near thresholds, where the limiting spectral measure has zero mass. Their normalized Hilbert–Schmidt error tends to zero, and the fixed coordinate cutoff controls the error after multiplication by the other matrices. This transfers the strict limiting block-energy bounds uniformly over the microstate sets.

Let $v=1+h+t$. Define an event ${\mathcal{E}}_d$ for a terminal tuple $D$ by requiring

$$\begin{array}{c}\Vert D_{[p]}{\Vert }_{2,d}\le 4\sqrt{vp},\\ \text{some }P\text{ has }\mathrm{rank}{P}_l\le 2d/p,\\ \Vert {\mathcal{D}}_P{D}_{[p]}{\Vert }_{2,d}\ge \sqrt{p}/4.\end{array}$$

The event searches over all such decompositions. It does not use the initial random tuple in its definition, and it does not require the witness to be the spectral partition of the terminal last coordinate.

Under the evolved law ${\mu }_d$ of $A^{(d)}+\sqrt{t}{G}^{(d)}$, use the initial spectral blocks as a witness. Conditionally on $A^{(d)}$, they are fixed and independent of the noise. Their projected noise energy satisfies

$$\begin{array}{c}\mathbb{E}[\Vert \sqrt{t}{\mathcal{D}}_P{G}_{[p]}^{(d)}{\Vert }_{2,d}^2\mid A^{(d)}]\\ =tp\sum _l(\mathrm{rank}{P}_l/d{)}^2\le 2t.\end{array}$$

Markov’s inequality shows that this noise usually cannot cancel the initial block energy when $t/p$ is small. A second Markov bound controls the total norm. The manuscript obtains ${\mu }_d({\mathcal{E}}_d)\ge 1/2$.

For independent Gaussian coordinates with the limiting variances, write the law as ${\gamma }_d$. For a fixed rank list and fixed unitary orientation, the projected squared norm has distribution $(v/d^2)Q$, where $Q$ is chi-squared with at most $2d^2$ degrees of freedom.

There are at most $(d+1{)}^p$ rank lists. An operator-norm net of the unitary group, at mesh $1/(64\sqrt{v})$, has at most $(1+128\sqrt{v}{)}^{2d^2}$ points. On the event’s total-norm bound, replacing a witness by a net point loses at most $\sqrt{p}/8$ of compressed norm. The chi-squared exponential bound and a union bound then give

$$\begin{array}{c}\log {\gamma }_d({\mathcal{E}}_d)\le p\log (d+1)+d^{2}b(v),\\ \begin{array}{rl}b(v)=& 2\log (1+128\sqrt{v})\\ & +\log 2-\frac{p}{256v}.\end{array}\end{array}$$

Choose $t=p^{3/4}$ and then one sufficiently large finite $p$. The analytic estimate requires $t/\sqrt{p}$ large, whereas this net estimate requires $t\log p/p$ small; both hold. The manuscript supplies the explicit choice $p=2^{64}$, $t=2^{48}$. Choose $h$ small afterwards and keep all these parameters fixed as $d\to \infty$. The bracket is then less than $-8$, and ${\gamma }_d({\mathcal{E}}_d)\le e^{-4d^2}$ for large $d$.

🏷️ Transferring the deficit back to the original tuple

Observing only whether ${\mathcal{E}}_d$ occurs gives a lower bound on relative entropy:

$$\begin{array}{c}\mathrm{KL}({\mu }_d\Vert {\gamma }_d)\\ \ge {\mu }_d({\mathcal{E}}_d)\log \frac{1}{{\gamma }_d({\mathcal{E}}_d)}-\log 2\\ \ge 2d^2-\log 2.\end{array}$$

The Gaussian entropy identity, together with convergence of the second moments, converts this to a deficit in normalized classical entropy. Retaining only one unit of the available deficit gives

$$\begin{array}{c}\underset{d}{\mathrm{limsup}}{H}_d(Z^{(d)}(t))\\ \le L(h+t)-1.\end{array}$$

Combine this with the specified-ensemble entropy-production bound and ${\chi }^{\ast }(V_{h+t})>L(h+t)-1/2$. Addition of free noise also gives

$$\begin{array}{c}{\chi }^{\ast }(V_{h+t})-{\chi }^{\ast }(V_h)\\ =\frac{1}{2}{\int }_0^t{\Phi }^{\ast }(V_{h+s})ds.\end{array}$$

Therefore ${\chi }_R(X)\le {\chi }^{\ast }(X)-1/2$. The conclusion holds for every cutoff with finite ${\chi }_R$ and is automatic for the remaining cutoffs. Taking the supremum over $R$, and using the earlier finite lower bound for $\chi (X)$, proves the manuscript’s asserted gap. (OpenAI, 2026)

🏷️ Key steps and proof outlines

Why approximate moments still locate the blocks

The block witness comes from spectral projections of the last matrix coordinate. Spectral projections are discontinuous functions, so ordinary moment convergence cannot be applied to them directly. In “A persistent block-structure entropy deficit,” the source uses the gaps between the limiting spectral clusters to repair this. Choose a closed neighborhood $N$ of the partition thresholds that misses the limiting spectrum. Replace each interval indicator by a continuous function $f_l$, identical to it outside $N$. For the empirical spectral measure ${\nu }_d$ and the actual projection $P_l$,

$$\Vert P_l-f_l(A_n^{(d)}){\Vert }_{2,d}^2\le {\nu }_d(N)⟶0.$$

Here $\Vert T{\Vert }_{2,d}^2=d^{-1}\mathrm{Tr}(T^{\ast }T)$. Weak convergence controls ${\nu }_d(N)$ because $N$ is closed with zero limiting mass. If all coordinates have operator norm at most $R$, multiplication costs at most

$$\Vert P_l{A}_i{P}_l-f_l(A_n)A_i{f}_l(A_n){\Vert }_{2,d}\le 2R\Vert P_l-f_l(A_n){\Vert }_{2,d}.$$

Continuous functional calculus is then approximated by polynomials. Thus both the ranks and the compressed energies converge, even with a few spectral outliers. Strict limiting inequalities supply uniform margins. If uniformity over the selected microstate supports failed, choosing one violating tuple in each offending dimension would contradict this very sequential argument. (OpenAI, 2026)

Pay for every possible block basis, then use a Gaussian tail

The event ${\mathcal{E}}_d$ asks whether some small-block decomposition captures energy; its Gaussian probability cannot be estimated by fixing the signal’s random decomposition. For each rank list, cover all possible bases by an operator-norm $\delta$-net of unitaries. Disjoint balls of radius $\delta /2$ in the real $2d^2$-dimensional space of complex matrices give at most $(1+2/\delta {)}^{2d^2}$ centers.

Conjugation is Lipschitz:

$$\Vert U^{\ast }DU-V^{\ast }DV{\Vert }_{2,d}\le 2\Vert U-V\Vert \Vert D{\Vert }_{2,d}.$$

With the event’s norm bound and $\delta =1/(64\sqrt{v})$, discretizing the basis loses only $\sqrt{p}/8$ of compressed norm. For any fixed net point and admissible ranks, the projected Gaussian energy has law $(v/d^2)Q$, where $Q$ is chi-squared with $q\le 2d^2$ degrees of freedom. Markov’s inequality applied to $e^{Q/4}$ and the exact identity $\mathbb{E}{e}^{Q/4}=2^{q/2}$ give the exponential bound in the preceding section. The net costs order $d^2\log v$ in the exponent, whereas the energy tail gains order $d^{2}p/v$. Choosing $t=p^{3/4}$ lets the latter dominate. The integer $p$ is fixed before taking $d\to \infty$, so the rank-list cost $p\log (d+1)$ is negligible.

Convert a rare structural event into an entropy gap

For distributions $\mu$ and $\gamma$, observing only whether $\mathcal{E}$ occurred cannot increase relative entropy. Writing $\alpha =\mu (\mathcal{E})$ and $\beta =\gamma (\mathcal{E})$ yields

$$\mathrm{KL}(\mu \Vert \gamma )\ge \alpha \log (1/\beta )-\log 2.$$

This follows by expanding the relative entropy of the two Bernoulli laws and bounding binary entropy by $\log 2$. Therefore event probability at least $1/2$ versus at most $e^{-4d^2}$ forces a normalized entropy deficit of at least two units asymptotically. The proof retains one unit to simplify the final comparison.

The transfer back to the initial tuple uses the same evolved ensembles throughout. In “Entropy production,” scalar integration by parts identifies their matrix score against every polynomial with the free difference-quotient pairing. Completing the square against polynomial tests and taking their supremum gives $\mathrm{liminf}{I}_d(s)\ge {\Phi }^{\ast }(V_{h+s})$. Integrate de Bruijn’s identity first over $[\epsilon ,t]$, use that Gaussian convolution cannot decrease entropy, and then let $\epsilon \downarrow 0$. Fatou’s lemma passes the nonnegative Fisher integrals to the limit. This avoids assuming entropy continuity at zero and prevents replacing the terminal ensemble by a different entropy maximizer, which would destroy the persistent-block argument.

🏷️ Connections and scope

Finite factor generators discusses entropy dimensions of different generating tuples. This family compares the two entropy values on the same tuple. Its construction uses a very large but fixed finite number of coordinates; it does not settle equality for each smaller prescribed tuple size.

🏷️ Manuscripts and verification scope

• A finite-entropy separation of microstates and nonmicrostates free entropy — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The documented selection compares the original operator-norm-cutoff, matrix-size-limsup microstates entropy with the nonmicrostates entropy defined using free semicircular noise. A bounded selfadjoint tuple of fixed finite size in a faithfully normally tracial von Neumann algebra satisfies a gap of at least one half, with both quantities finite. Equality for every smaller prescribed number of coordinates is not asserted. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026. A finite-entropy separation of microstates and nonmicrostates free entropy.