From Finite Certificates to Uniform Coercivity: Loewner–Schur Propagation for Localized Weil Comparisons

We develop a quantitative finite-certificate method for proving uniform coercivity of parameter-dependent Hermitian block families and apply it to a localized Weil eight-mode comparison on 5.01 <= X < 7. The main analytic result is a fixed-target Loewner-Schur propagation theorem: finite anchor inequalities and two-sided Loewner derivative bounds propagate relative coupling estimates over a continuous parameter interval without tracking generalized eigenvalues or eigenvectors. If m is the critical metric floor, mu the remote floor, and beta < mu the total relative coupling price, then $$\Gamma(m,\mu,\beta)=\frac{m+\mu-\sqrt{(\mu-m)^2+4m\beta}}{2}$$ is a uniform coercivity margin. This bound is sharp among universal bounds depending only on m, mu, and beta. Applied to the localized Weil family, the certified data give $$D_{X,8}-\frac{1}{5}I\succeq\frac{203}{625000}I,$$ and therefore $$D_{X,8}\succeq 0.2003I,\qquad5.01\le X<7.$$ The underlying numerical certificates are unchanged from the previously released verification package; the improved 0.2003 bound is an exact analytic consequence of those certified data. Version 2.0r1. Paper DOI: 10.5281/zenodo.22321620Verification supplement: 10.5281/zenodo.22127800