A Fourier–Lévy Normal Form for Localized Weil Positivity: Finite-Rank Boundary Terms, Scalar Resolvents, and a Certified Finite Block and Continuum Fourier Tail

Abstract

This manuscript develops a Fourier–Lévy operator framework for localized Weil positivity. It derives the normal form A_X = L_X - c_X I + M_X^* J M_X, isolates the finite-rank boundary contribution, and reduces the parity sectors to scalar Herglotz-type resolvents. For the certified parameter choice X = 4 and N = 32, the relevant finite Fourier blocks are verified using interval arithmetic, while the high-frequency continuum Fourier tail is shown to be strictly positive for all Fourier modes |n| > 160. The remaining finite low-frequency Feshbach core is unresolved. Accordingly, this manuscript does not claim a proof of the Riemann Hypothesis. It provides a structural reduction and computer-assisted positivity certificates for the components stated above.