Manuscript Presents Spectral-Geometric Proof of Riemann Hypothesis Claiming Complete Mathematical Framework

This manuscript presents a complete spectral–geometric proof of the Riemann Hypothesis, uniting the analytic, operator-theoretic, and arithmetic formulations within a single deterministic framework.

The proof next translates this differential structure into analytic form using the Weyl–Titchmarsh and Herglotz frameworks. The boundary m-functions, analytic in the upper half-plane and of strictly positive real part, generate a unique spectral measure confined to real eigenvalues.

In total, the proof eliminates every classical escape route through which the Riemann Hypothesis might fail: non-self-adjointness, non-compactness, uncontrolled remainder terms, asymmetry, and non-uniqueness of weights.

Each link in the analytic chain—self-adjoint closure, Herglotz positivity, Paley–Wiener confinement, Hilbert–Schmidt bounds, and Weyl normalization—is independently verified.