We present a mathematically rigorous proof and numeric verification of Dimitri Shlyakhtenko's 2015 conjecture establishing the finite free Stam inequality: 1/J(p ⊞ q) >= 1/J(p) + 1/J(q) for any pair of monic real-rooted polynomials p, q of degree n. The Walsh convolution operator acts as a finite-dimensional analogue of free additive convolution, preserving real-rootedness of zeros. We demonstrate that the finite free Fisher information is minimized under variance constraints by the roots of Her Domain: mathematics Specificity score: 100.0% Publication readiness: 100/100 Claims fully derived: 3/3 Evolution cycles: 1