Screw Functions, de Branges Spaces, and Zeta Spectral Triples: A Spectral-Geometric Approach to RH and GRH

This paper reorganizes the preceding determinant-convergence and Weil-positivity program around a sharper spectral-geometric synthesis of three inputs: Connes-Consani-Moscovici zeta spectral triples, Suzuki’s screw-function formulation of Weil’s quadratic form, and de Branges Hilbert spaces of entire functions. The immediate aim is not to announce an unconditional proof of the Riemann Hypothesis (RH) or the Grand Riemann Hypothesis (GRH). Rather, the paper gives a mathematically explicit framework in which the finite, unconditional mechanisms and the remaining infinite-limit estimates are separated. Four obstructions are treated in detail: the non-self-adjointness caveat in continuous localized Weil realizations, finite-cutoff negative spectral blocks and possible spectral pollution, semilocal Sonin/prolate leakage in adelic phase space, and non-abelian L-packet interference in a GRH extension. Two analytic targets are then isolated as the decisive closure points: weighted eigenvector convergence (WEC) in a de Branges-screw topology and trace-class resolvent convergence (TR) for a nonlocal self-adjoint realization. The paper proves several conditional implications: WEC implies compact-uniform Mellin convergence; TR implies compact-uniform Fredholm determinant convergence; compact-uniform convergence of real-zero determinant approximants to the completed zeta or L-function implies RH or GRH by Hurwitz’s theorem. It also retains the earlier unconditional prime-clock trace theorem, which extracts the Weil prime-power weights from a half-density damped operator trace. The result is a rigorous conditional spectral-geometric theorem and a concrete proof program, rather than an unsupported claim of a completed solution.