A Walsh-Convolution Proof of Two Conjectures of Lachaume

For complex polynomials P and Q of degree at most p, this note identifies the derivative sum sum_{k=0}^p P^(k)(x)Q^(p-k)(x) exactly with p! times the Walsh symmetric additive convolution evaluated at 2x. The classical Walsh root-location theorem then places every output zero in the convex hull of all input zeros and proves Lachaume's Conjectures 4 and 5, together with the corresponding semi-symmetric hyperbolicity statement. The Walsh theory itself is classical, and Lachaume's 2018 Lovelock-gravity paper already uses Walsh's theorem for a related concavity implication; the contribution claimed here is limited to the exact identity, its application to Conjectures 4 and 5, and the complex convex-hull deduction. Status: Public Beta; internally verified short note; external mathematical review pending.