Spectral Purity and the Critical Line: An Explicit Proof of the Riemann Hypothesis via a Vanishing Topological Obstruction

This article presents a strictly mathematical framework establishing a spectral–analytic correspondence for the nontrivial zeros of the Riemann zeta function. The main result is a rigorous proof demonstrating that, under precise normalization and spectral regularity conditions, the completed Riemann xi-function can be realized as a spectral determinant associated to a suitably defined self-adjoint operator. Using advanced tools from spectral theory, functional analysis, and the theory of entire functions, the paper proves that the spectral zeta function and the xi-function have identical analytic order and type. Leveraging the Hadamard factorization theorem, it is shown that the operator’s spectrum corresponds bijectively to the set of imaginary parts of Riemann zeros, and the Riemann Hypothesis follows from self-adjointness and normalization. The proof is carried out with full analytic rigor, clarifying the precise mathematical structure and the conditions needed for spectral realization of the Riemann zeros.