Spectral-Geometric Approaches to RH and GRH: The Sobolev-Suzuki L² Framework for WEC and TR

This paper develops a non-circular Sobolev-Suzuki framework for the weighted eigenvector convergence (WEC) and trace-class resolvent convergence (TR) problems that arise in spectral-geometric approaches to the Riemann Hypothesis (RH) and the Grand Riemann Hypothesis (GRH). The main logical point is that a proof of RH cannot use, as a primary analytic input, the positive de Branges Hilbert-space realization of the completed Weil form attached to the Riemann -function, because that positivity is intertwined with Weil positivity and hence with RH itself. The proof domain is therefore shifted to the unconditional Hilbert space L²(-a,a) and the Sobolev scale H^s(-a,a). In this setting, one can formulate nonlocal self-adjoint realizations of the first-order generator, compact Suzuki screw-kernel operators, Galerkin truncations, Schatten-class resolvent criteria, Fredholm determinant continuity, and the Hurwitz passage without assuming RH. The paper proves several non-circular functional-analytic results: self-adjointness of the nonlocal operator , Hilbert-Schmidt compactness of continuous screw-kernel operators, closed lower-semibounded form realization for , a Sobolev-L² criterion for WEC, a Schatten-1 criterion for TR, determinant continuity under trace-norm convergence, and the real-zero Hurwitz implication. It also incorporates three concrete analytic strategies for attacking the remaining quantitative estimates: adelic Sonin-Prolate leakage bounds for WEC, Babuška-Osborn spectral approximation for TR, and trace-class decay of finite negative spectral energy. The resulting theorem is an exact non-circular reduction: if the explicitly stated WEC/TR estimates and normalization-identification conditions are proved in the Sobolev-Suzuki setting, then the normalized determinant approximants converge compact-uniformly to and RH follows by Hurwitz’s theorem. Automorphic analogues would imply GRH for the corresponding class. The final sections isolate the remaining analytic inequalities in a form suitable for direct verification and future completion.