Verified-Height Conditioning in the Riemann Xi Moment Tower: Phase-Shear Confinement, Directional Christoffel Leverage, and Finite-Node Schur Certificates

This preprint develops unconditional finite-verification results for an RH-equivalent reciprocal-zero moment representation of the Riemann Xi function. Assuming the Riemann hypothesis has been verified through a finite height T, the unverified zero tail is shown to admit dimension-uniform low-rank confinement, while the associated Hausdorff/Bernstein difference triangle has explicit phase-safe regions and phase-resolved defect bounds. The conditioning analysis separates global monomial ill-conditioning from the directions actually reached by the unknown tail. The retained finite jet has a uniformly controlled negative phase-shear component of order S_{10}(T); finite verified zero subsets yield explicit upper Christoffel certificates; and a verified-metric residual leverage estimate gives a finite-node small-gain criterion implying positivity of the full Hankel matrix. At the verified height T_0=3\times10^{12}, the phase-shear floor is below 4.382\times10^{-113}. The revised release also analyzes discrete double scaling. Because the verified measure is finite and atomic, Green-function asymptotics for a regular interval measure do not describe its Christoffel growth. A discrete deletion-polynomial lower bound is proved instead. Combined with Riemann–von Mangoldt asymptotics, it shows that the particular uniform phase-shear-majorant certification route has an asymptotic ceiling of order T^{2/3}\log T, rather than T\log T. High-precision diagnostics using the first 300 zeta zeros independently reject the naive interval-Green extrapolation on the tested scaling rays. The paper does not claim a proof of the Riemann hypothesis, an arithmetic-to-polarization theorem, or an exact transform between Weil support and moment depth. Its contribution is an unconditional quantitative analysis of what finite zero verification controls, how that information is conditioned under different representations, and how finite-data Schur-positivity certificates can be constructed.