Spectral Structure in Finite Free Information Inequalities and $p$-Stam Phase Transitions

Baran Hashemi

math.PRApr 13, 2026

Spectral Structure in Finite Free Information Inequalities and $p$ -Stam Phase Transitions

Authors: Baran Hashemi

Abstract

Using FlowBoost, a closed-loop deep generative optimization framework for extremal structure discovery, we investigate $\ell^p$ -generalizations of the finite free Stam inequality for real-rooted polynomials under finite free additive convolution $\boxplus_n$ . At $p=2$ , FlowBoost finds the Hermite pair as the unique equality case and reveals the spectral structure of the linearized convolution map at this extremal point. As a result, we conjecture that the singular values of the doubly stochastic coupling matrix $E_n$ on the mean-zero subspace are ${2^{-k/2}:k=1,\ldots,n-1}$ , independent of $n$ . Conditional on this conjecture, we obtain a sharp local stability constant and the finite free CLT convergence rate, both uniform in $n$ . We introduce a one-parameter family of $p$ -Stam inequalities using $\ell^p$ -Fisher information and prove that the Hermite pair itself violates the inequality for every $p>2$ , with the sign of the deficit governed by the $\ell^p$ -contraction ratio of $E_n$ . Systematic computation via FlowBoost supports the conjecture that $p^*\!=2$ is the sharp critical exponent. For $p<2$ , the extremal configurations undergo a bifurcation, meaning that they become non-matching pairs with bimodal root structure, converging back to the Hermite diagonal only as $p\to 2^-$ . Our findings demonstrate that FlowBoost, can be an effective tool of mathematical discovery in infinite-dimensional extremal problems.

Figures & tables

Panel of Figure 1 — Figure 1. E-value-based AHT as used in this study. FlowBoost first produces elite extremal samples $\mathcal{E}_{n,p}$ . These are compressed into summary statistics $T(\mathcal{E}_{n,p})$ , compared against a library of low-complexity candidate families $\mathcal{H}$ , and screened against the Hermite baseline through the conditional e-values of ( 13 ). A candidate family is promoted only when its population-level fit remains stable across the tested grid and its e-value evidence is persistently favorable.

$n$	$\rho_{\min}$	$D_{\min}$	$d_{PQ,\min}$	Spearman $(\rho,D)$
6	$2.1\times 10^{-9}$	$5.3\times 10^{-10}$	$1.2\times 10^{-5}$	0.89
8	$2.2\times 10^{-8}$	$3.2\times 10^{-9}$	$6.6\times 10^{-5}$	0.87
10	$3.6\times 10^{-7}$	$4.0\times 10^{-8}$	$1.0\times 10^{-5}$	0.85
12	$8.2\times 10^{-7}$	$1.1\times 10^{-8}$	$4.6\times 10^{-4}$	0.80
20	$2.0\times 10^{-7}$	$2.4\times 10^{-8}$	$3.6\times 10^{-4}$	0.80
50	$8.4\times 10^{-6}$	$5.0\times 10^{-7}$	$1.0\times 10^{-4}$	0.79

Table 1. FlowBoost results at

p=2

\rho_{\min}

is the best scale-free deficit;

D_{\min}

and

d_{PQ,\min}

are the Hermite distance and mismatch at the best sample.

	$i=1$	$i=2$	$i=3$	$i=4$	$i=5$	$i=6$
$h^{(6)}_{i}$	$-1.4866$	$-0.8449$	$-0.2758$	$\phantom{-}0.2758$	$\phantom{-}0.8449$	$\phantom{-}1.4866$
$\alpha_{i}$	$-1.4866$	$-0.8449$	$-0.2758$	$\phantom{-}0.2758$	$\phantom{-}0.8449$	$\phantom{-}1.4867$
$\beta_{i}$	$-1.4866$	$-0.8449$	$-0.2758$	$\phantom{-}0.2758$	$\phantom{-}0.8449$	$\phantom{-}1.4867$

Table 2. Best sample at

n=6

: root vectors

\alpha,\beta

vs. normalized Hermite roots

h^{(6)}

, displayed to four decimal places. The apparent discrepancy at

i=6

(

1.4867

vs.

1.4866

) is a rounding artifact; the true residual

D_{\min}=5.3\times 10^{-10}

(see Table 1 ) is consistent with agreement to nine significant figures.

$n$	$\sigma_{1}$	$\sigma_{2}$	$\sigma_{3}$	$\sigma_{4}$	$\sigma_{5}$	$\mathrm{rel.err.}$
conj.	$0.707106781$	$0.500000000$	$0.353553391$	$0.250000000$	$0.176776695$	—
10	0.707106781	0.500000000	0.353553391	0.250000000	0.176776695	$1.33\times 10^{-14}$
40	0.707106781	0.500000000	0.353553391	0.250000000	0.176776695	$8.02\times 10^{-11}$
60	0.707106778	0.500000000	0.353553391	0.249999999	0.176776697	$1.15\times 10^{-7}$
80	0.707104585	0.499997829	0.353554003	0.249999577	0.176779011	$5.77\times 10^{-5}$
90	0.707145636	0.500091161	0.353518684	0.250002448	0.176746177	$4.56\times 10^{-4}$

Table 3. Singular values of

E_{n}|_{W}

vs. the conjectured

\sigma_{k}=2^{-k/2}

, together with the maximum relative error among the first ten modes.

Figure 2. High-precision numerical evidence for the conjectured dyadic spectrum of the Hermite coupling operator

E_{n}|_{W}

. Panel (a) plots

\log_{2}\sigma_{k}

for the first ten singular values at the representative degrees

n=20,40,60,90

, together with the reference line

\log_{2}\sigma_{k}=-k/2

. Panel (b) shows the maximum relative error among the first ten singular values across the high-precision restricted audit

n\in\{10,20,30,40,50,60,70,80,90\}

Figure 3. Representative convergence trajectories for the variance-normalized finite free central limit iteration at

n=20

. Each colored curve tracks the perturbation norm

d_{H}(f_{k})

for a different Hermite initialization, while the dashed line gives the reference decay

(1/\sqrt{2})^{k}

predicted by the conjectural second singular value. After a short transient, the observed slopes are consistent with geometric contraction at the predicted rate.

$p$	$n=6$	$n=8$	$n=10$	$n=12$	$n=20$
2.0625	$-0.0400$	$-0.0400$	$-0.0394$	$-0.0379$	$-0.0222$
2.125	$-0.0744$	$-0.0743$	$-0.0738$	$-0.0719$	$-0.0585$
2.25	$-0.1303$	$-0.1302$	$-0.1296$	$-0.1275$	$-0.1114$
2.5	$-0.2086$	$-0.2084$	$-0.2082$	$-0.2051$	$-0.1838$
3.0	$-0.2987$	$-0.2983$	$-0.2972$	$-0.2949$	$-0.2775$
4.0	$-0.3819$	$-0.3814$	$-0.3811$	$-0.3787$	$-0.3627$

Table 4. Minimum

p

-Stam deficit

g_{p}

for

p>2

. All entries are negative.

Figure 4. Global computational phase diagram for the $p$ -Stam problem on the tested $(n,p)$ grid. The color in each cell is $\operatorname{sgn}(g_{p}^{\min})\log_{10}(1+|g_{p}^{\min}|)$ , so the sign records whether the best observed deficit is positive or negative and the magnitude records its logarithmic size. The sign transition occurs at $p=2$ on the tested grid, while the post-processed numerical reporting in the text and tables is given in terms of the additive deficit $g_{p}$ . Visually, the figure separates the observed validity region $p\leq 2$ from the counterexample region $p>2$ in a single degree-uniform picture.

$n$	$p$	$D(\alpha^{},\beta^{})$	$d_{PQ}(\alpha^{},\beta^{})$
6	2.0625	$1.4\times 10^{-5}$	$1.6\times 10^{-4}$
8	2.0625	$1.1\times 10^{-5}$	$9.3\times 10^{-5}$
10	2.0625	$2.2\times 10^{-5}$	$1.5\times 10^{-3}$
12	2.0625	$1.5\times 10^{-4}$	$1.2\times 10^{-2}$
20	2.0625	$9.3\times 10^{-4}$	$2.4\times 10^{-2}$

Table 5. Structure of the best counterexamples for

p>2

$p$	$n=6$	$n=8$	$n=10$	$n=12$	$n=20$
1.05	$4.47\times 10^{-10}$	$2.40\times 10^{-8}$	$1.66\times 10^{-6}$	$8.22\times 10^{-5}$	$3.90\times 10^{-3}$
1.2	$1.81\times 10^{-3}$	$3.09\times 10^{-3}$	$1.43\times 10^{-2}$	$2.21\times 10^{-2}$	$1.17\times 10^{-1}$
1.5	$1.46\times 10^{-2}$	$2.92\times 10^{-2}$	$6.70\times 10^{-2}$	$8.50\times 10^{-2}$	$6.48\times 10^{-1}$
1.8	$1.69\times 10^{-2}$	$2.56\times 10^{-2}$	$5.62\times 10^{-2}$	$8.66\times 10^{-2}$	$1.81\times 10^{-1}$
2.0	$1.30\times 10^{-8}$	$1.09\times 10^{-5}$	$3.94\times 10^{-4}$	$1.38\times 10^{-3}$	$1.94\times 10^{-2}$

Table 6. Minimum

p

-Stam deficit

g_{p}

for the tested subcritical grid and the quadratic endpoint. All entries are positive (inequality supported).

$n$	$p$	Best	Med. $d_{\mathrm{jt}}$	Consist.	Best $d_{PQ}$	Best $D$	Sym. (S3, $t\!=\!0.1$ )
6	1.05	2BU	0.178	1.000	0.507	0.275	0%
6	1.2	2BU	0.444	1.000	0.496	0.263	0%
6	1.5	2BU	0.470	1.000	0.705	0.530	0%
6	1.8	He	0.034	0.281	0.020	$4\!\times\!10^{-4}$	94%
8	1.05	2BU	0.617	0.750	0.761	0.619	0%
8	1.2	2BU	0.603	0.926	0.842	0.647	0%

Table 7. Full extremizer structure as a function of

(n,p)

. “2BU” = two-block uniform, “He” = Hermite, Consistency: fraction of the best set for which the stated family achieves the lowest

d_{\mathrm{joint}}

Figure 5. Root-configuration atlas for the best observed configurations at two representative degrees,

n=12

and

n=20

, across the transition in

p

. In each panel, the blue and red rows show the jointly normalized root locations of

\alpha

and

\beta

, respectively, while the annotated value gives the best observed additive deficit

g_{p}

at that parameter pair. Read from left to right, the atlas exhibits a clear geometric progression: non-matching clustered configurations for

p<2

, a transitional Hermite regime around

p=1.8

, the Hermite diagonal at

p=2

, and Hermite matching counterexamples for

p>2

$n$	$p=1.05$	$p=1.2$	$p=1.5$	$p=1.8$
6	0.506	0.516	0.505	0.034
8	0.712	0.643	0.548	0.112
10	0.723	0.776	0.740	0.064
12	0.796	0.773	0.740	0.063
20	0.739	0.727	0.394	0.111

Table 8. Hermite median

d_{\mathrm{joint}}

(distance of best set to Hermite) as a function of

p

Figure 6. Population-level order-statistic bands and gap analysis for the top-

100

elite samples at each displayed

(n,p)

pair. In the first three columns, each polynomial is normalized independently to mean

0

and unit variance, and then the median together with the

25

–

75\%

and

10

–

90\%

bands of the

i

-th sorted root are plotted separately for

\alpha

and

\beta

. A two-block population appears here as a stable jump between root indices

n/2

and

n/2+1

. The rightmost column reports the elite distribution of the scale-free gap statistic

\max\Delta/\operatorname{med}(\Delta_{\mathrm{rest}})

, where

\Delta

denotes adjacent normalized root gaps. Both diagnostics separate the subcritical cases

p=1.05,1.2

from the quadratic endpoint

p=2.0

Figure 7. Compact summary of the subcritical bifurcation. Panel (a) contrasts a representative low-

p

root distribution with the quadratic Hermite geometry; Panel (b) tracks the best pair-mismatch statistic

d_{PQ}

across

p

; and Panel (c) tracks the best Hermite distance

D

. Together the panels summarize the two main structural signatures of the transition: mismatch and non-Hermite geometry both persist well below

p=2

and collapse rapidly as

p\to 2^{-}

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

$n$	$p$	$E$	Reject $H_{0}$	Favoured model
6	1.05	$5.28\times 10^{36}$	Yes	Two-block uniform
6	1.2	$5.28\times 10^{36}$	Yes	Two-block uniform
6	1.5	$5.28\times 10^{36}$	Yes	Two-block uniform
6	1.8	6.831	No	Inconclusive
6	2.0	0.008	No	Hermite
8	1.05	$4.95\times 10^{29}$	Yes	Two-block uniform

Appendix

Table 9. Conditional e-values for the one-sided screen

H_{0}^{\mathrm{He}}(\mathrm{2BU};n,p):\mathbb{P}(R_{\mathrm{2BU}}<R_{\mathrm{He}})\leq 1/2

against the alternative that two-block uniform beats Hermite more than half the time on the top-

512

elite.

M_{\mathrm{eff}}

is the number of non-tied comparisons after removing samples for which both residuals are equal. Cells where the e-value saturates at

5.28\times 10^{36}

correspond to all non-tied comparisons won by two-block uniform. The decision column uses the

5\%

e-value rule

E\geq 20

Figure 8. Stage-wise results for the

p=2

FlowBoost pipeline at representative degrees

n=6,8,10,12,20

. SRP first identifies a high-quality feasible region, supervised CFM learns a proposal distribution from that region, and reward-guided fine-tuning sharpens the proposal distribution further so that the subsequent final push attains the best values.

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Beyond the Semicircle: Free Diffusion Models with Prescribed Equilibria

Swagatam Das

Random Matrix Theory Diffusion Models

Open paper AskPDF arXiv

A growing class of machine-learning objects -- covariance and Gram matrices, kernel and attention matrices, MIMO channel matrices, density operators -- are naturally spectra rather than coordinate vectors. Building a denoising diffusion model for such data by corrupting eigenvalues coordinatewise is not merely elegant: it converges to the wrong limit, because eigenvalues repel rather than move independently. Free probability theory supplies the correct forward process, with free convolution replacing classical convolution and Voiculescu's conjugate variable replacing the score, but constant-coefficient free diffusions have a hidden limitation of their own: their only possible equilibrium is the semicircular law, whatever the target distribution looks like. We show that state-dependent free volatility removes this restriction. For any sufficiently regular compactly supported target law, we explicitly construct, through a closed-form Hilbert-transform drift, a free Fokker--Planck flow having that law as a stationary spectral distribution. Under an additional convex-potential condition, the construction admits a globally relaxing diffusion interpretation, reverse-time dynamics, and a trainable score. The generalization is genuine rather than cosmetic: no change of spectral variable reduces it to the constant-coefficient case, and it is a Wasserstein gradient flow only for an explicitly characterized family of coefficients that contains no bounded non-constant member. We instantiate the construction on the Marchenko--Pastur law, the limiting spectrum of isotropic sample covariance matrices, and validate it numerically: the free-versus-coordinatewise gap, an exact dimension-independent score driving reverse-time matrix dynamics at several matrix sizes, the designed Marchenko--Pastur equilibrium, and end-to-end generation with a learned score.

Abstract

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