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

• URL: http://arxiv.org/abs/2604.11922v1
• Date: Mon, 13 Apr 2026 18:09:32 GMT
• Title: FlowBoost Reveals Phase Transitions and Spectral Structure in Finite Free Information Inequalities
• Abstract summary: We investigate the finite free Stam for real-rooted inequalitys under finite free additive convolution $boxplus_n$.<n>At $p=2$ FlowBoost finds the Hermite pair as unique the equality case.<n>We conjecture that the values of the doubly coupling matrix $E_n$ on the mean-zero subspace are $2k/2:k=1,ldots,n-1$.
• Score: 1.4721615285883427
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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