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

Using FlowBoost, a closed-loop deep generative optimization framework for extremal structure discovery, we investigate ℓp-generalizations of the finite free Stam inequality for real-rooted polynomials under finite free additive convolution boxplusn. 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 En 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 ℓp-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 ℓp-contraction ratio of En. 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→ 2^-. Our findings demonstrate that FlowBoost, can be an effective tool of mathematical discovery in infinite-dimensional extremal problems.