Sharp Poincaré Interpolation Along Wasserstein Geodesics

We prove a sharp interpolation inequality for the Poincar\'e constant along quadratic Wasserstein geodesics. Let $\mu_i$, $i=0,1$, be $\kappa_i$-strongly log-concave probability measures on $\mathbb R^n$, and let $(\mu_t)_{t\in[0,1]}$ be their optimal displacement interpolation. Then $$\sqrt{C_P(\mu_t)} \leq \frac{1-t}{\sqrt{\kappa_0}} + \frac{t}{\sqrt{\kappa_1}}.$$ This estimate is optimal for every $t,\kappa_0,\kappa_1$, holds for all test functions without symmetry assumptions, and remains valid for extended-valued potentials. We also characterize equality at an interior time: it holds if and only if the two endpoints split off curvature-saturating Gaussian factors in a common direction. The equality directions form the maximal subspace on which both endpoints have the corresponding Gaussian factors. As a special case, we resolve a question of Aishwarya and Rotem concerning odd functions along optimal interpolations between even strongly log-concave measures. The proof uses a two-endpoint Bochner method, which is also one of the most important contribution at the methodological level: it converts curvature information available only at the endpoints directly into a sharp spectral estimate along the connecting geodesic, bypassing the generally inaccessible curvature of the intermediate measures.