FlowBoost Reveals Phase Transitions and Spectral Structure in Finite Free Information Inequalities
This work provides new conjectures and insights into the structure of finite free convolution and extremal inequalities, which are foundational for free probability theory and random matrix theory.
The authors used FlowBoost, a deep generative optimization framework, to investigate finite free Stam inequalities for real-rooted polynomials. They conjectured that the singular values of the coupling matrix E_n are independent of n, and found a phase transition at p=2 where extremal configurations bifurcate from the Hermite pair to non-matching pairs for p<2.
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.