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

TL;DR

This paper uses FlowBoost, a deep generative optimization framework, to explore spectral structures and phase transitions in finite free information inequalities related to $ ext{ell}^p$-generalizations of the finite free Stam inequality, revealing new extremal configurations and stability properties.

Contribution

The paper introduces FlowBoost for discovering extremal structures in finite free inequalities, conjectures spectral properties of coupling matrices, and analyzes phase transitions at $p=2$ in $ ext{ell}^p$-generalized inequalities.

Findings

FlowBoost identifies the Hermite pair as the unique extremal at $p=2$.

Conjecture on the spectral values of the coupling matrix $E_n$.

Bifurcation of extremal configurations for $p<2$.

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.