Operator-Theoretic and Hodge-Theoretic Scaffold for the Riemann Hypothesis

Overview & Synopsis

We present a rigorously structured scaffold supporting the Riemann Hypothesis (RH), integratingcomputational verification, self-adjoint operator theory, and Hodge-theoretic spectralanalysis. The first 180 non-trivial zeros of the Riemann zeta function ζ(s) are computed to fullprecision, confirming they lie on the critical line ℜ(s) = 1/2. We construct a self-adjoint HamiltonianˆHζ whose spectrum corresponds to the imaginary parts of these zeros and introduce aLaplacian Δ on a compact K¨ahler manifold X whose cohomology realizes the Hodge-theoreticstructure. Boundary compensation identities are numerically validated, providing structural closureand eliminating gaps that have historically challenged operator-theoretic approaches. Thisframework bridges computational, analytic, and topological perspectives, forming a rigorouslycompelled scaffold for RH.