Spectral Realization of the Riemann Zeta Function via Laplace Perturbation

This work presents a purely geometric and spectral construction of a self-adjoint operator on the modular surface whose completed spectral zeta function coincides identically with the completed Riemann zeta function ξ(s)\xi(s)ξ(s). The construction is based on a perturbation of the hyperbolic Laplacian using a compactly supported automorphic potential. We derive the spectral zeta function from heat kernel methods and Mellin transformation, prove its analytic properties, and show that it satisfies the same functional equation, zero structure, and order as ξ(s)\xi(s)ξ(s). The identification is made via Hadamard’s uniqueness theorem. As a consequence, the Riemann Hypothesis is reformulated — and derived — as a spectral statement about the reality of eigenvalues of a geometric operator. No arithmetic input is used in the construction; the analytic identity with ξ(s)\xi(s)ξ(s) arises strictly from spectral data. This approach provides a new framework for RH based entirely on classical spectral theory. SHA256 cbfa06e78e37859af723c04960c7609b2894860e910a41729799fefd21d5131e.