FlowBoost Reveals Phase Transitions and Spectral Structure in Finite Free Information Inequalities

2026-04-13

AI Summary

This work investigates the extremal structure, phase transition behavior, and spectral properties of the finite free Stam inequality under ℓ^p generalization. Leveraging the FlowBoost closed-loop deep generative optimization framework together with finite free convolution theory and ℓ^p-Fisher information analysis, the study establishes that for p = 2, the Hermite pair is the unique extremizer, and the spectrum of the associated linearized convolution operator is precisely characterized. For p ≠ 2, a sharp critical exponent p* emerges: when p < 2, extremal configurations undergo bifurcation, exhibiting bimodal root distributions. The paper further formulates a universal conjecture on the singular values of the coupling matrix E_n, yielding dimension-independent local stability constants and convergence rates in the central limit theorem, all corroborated by numerical experiments confirming the critical nature of p*.

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

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.

Problem

• finite free convolution
• Stam inequality
• phase transition
• extremal structure
• spectral structure

Innovation

• FlowBoost
• finite free convolution
• Stam inequality
• phase transition
• spectral structure