Quantitative Prolate-Proxy Estimates for Zeta Spectral Models

Abstract

We isolate and prove the analytic part of a proxy-to-ground-state bridge in recent spectral models for the Riemann zeta function. On $I_\lambda=[\lambda^{-1},\lambda]$ we distinguish the semilocal Weil quadratic form from its associated self-adjoint operator, replace endpoint normalization of an $L^2$ eigenvector by an invariant inner-product normalization, and derive the exact $L^2$-to-weighted-$L^1$ loss needed for convergence on horizontal strips. For the localized prolate proxy $K_\lambda$ we supply the exterior Gaussian tail required in the comparison with the limiting function $k=\mathcal E(h)$ and prove $K_\lambda\to k$ in $L^2$ and in every weighted $L^1_b$ norm with $0<b<1/2$. In the stated Fourier convention, $\widehat{k}=\Xi/4$. We then prove an exact residual-gap estimate converting an operator residual into an $L^2$ eigenvector error and formulate a quantitative criterion that would imply local-uniform convergence to $\Xi$ and hence, together with an all-real-zero theorem, the Riemann hypothesis. The residual, spectral-gap and finite-to-continuum estimates required by that criterion are left explicit rather than assumed.