Quantitative Prolate-Proxy Estimates for Zeta Spectral Models

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_λ=[λ⁻¹,λ] we distinguish the semilocal Weil quadratic form from its associated self-adjoint operator, replace endpoint normalization of an L² eigenvector by an invariant inner-product normalization, and derive the exact L²-to-weighted-L¹ loss needed for convergence on horizontal strips. For the localized prolate proxy K_λ we supply the exterior Gaussian tail required in the comparison with the limiting function k= E(h) and prove K_λ→ k in L² and in every weighted L¹b norm with $0<b<1/2$. In the stated Fourier convention, k=Ξ/4. We then prove an exact residual-gap estimate converting an operator residual into an L² eigenvector error and formulate a quantitative criterion that would imply local-uniform convergence to Ξ 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.