Finite free convolution gives a finite-dimensional analogue of free additive convolution, acting on real-rooted polynomials through their root configurations. I will discuss a finite free Stam inequality for the associated Fisher information and a $p$-deformation obtained by replacing the quadratic score norm with an $\ell^p$-score functional. The talk asks whether the equality structure of the finite free Stam inequality is a purely quadratic phenomenon. I will explain how extremal search with FlowBoost recovers the Hermite pair at p=2, and how the linearization of the finite free convolution root map at this point leads to a new spectral conjecture: on the mean-zero subspace, the singular values are \[ 2^{-k/2},\qquad k=1,\dots,n-1, \] independently of the degree. Conditional on this spectrum, one obtains a sharp local stability constant and a degree-uniform linearized finite free CLT rate. I will then describe the $p$-Stam phase transition. The Hermite pair itself violates the proposed inequality for every p>2, while computations support validity for $1'<p\le 2$. For $p'<2$, the extremizers bifurcate away from Hermite into non-matching bimodal root configurations. I will also explain the e-value-based abductive hypothesis testing used to distinguish genuine structural evidence from optimizer artifacts, and conclude with open problems suggested by the experiments.
