A Spectral-Geometric Proof of the Riemann Hypothesis

Summary

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. Beginning from first principles, the work constructs an explicitly self-adjoint Sturm–Liouville operator on the structured entropy–spiral coordinate and proves, under confinement and limit-point conditions, that its resolvent is compact. This establishes a discrete and symmetric spectrum that corresponds bijectively to the nontrivial zeros of ζ(s). Unlike heuristic versions of the Hilbert–Pólya approach, the operator here is not assumed but fully characterized: its essential self-adjointness, verified through Liouville transformation and Agmon confinement, ensures that no alternative extension or hidden boundary degree of freedom exists through which a zero could escape the critical line. 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. Through this equivalence, the analytic continuation of ζ(s) becomes the continuation of a real self-adjoint spectrum: to move a zero off the critical line would destroy the Herglotz property and force non-self-adjoint behavior. The geometry of ζ(s) is thus defined by equilibrium between curvature and entropy—a manifold where all analytic motion remains bounded and symmetric. In the analytic–operator phase of the argument, the Bochner integral and Paley–Wiener transform convert the self-adjoint trace into an explicit summation formula equivalent in structure to Selberg’s trace formula but derived from first principles. The Bochner formalism ensures absolute integrability of the operator kernel, while Paley–Wiener confinement fixes the support on the compact spectral domain. Together, these forbid any analytic...