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

We present a rigorously structured scaffold supporting the Riemann Hypothesis (RH), integrating
computational verification, self-adjoint operator theory, and Hodge-theoretic spectral
analysis. The first 180 non-trivial zeros of the Riemann zeta function ζ(s) are computed to full
precision, confirming they lie on the critical line ℜ(s) = 1/2. We construct a self-adjoint Hamiltonian
ˆH
ζ whose spectrum corresponds to the imaginary parts of these zeros and introduce a
Laplacian Δ on a compact K¨ahler manifold X whose cohomology realizes the Hodge-theoretic
structure. Boundary compensation identities are numerically validated, providing structural closure
and eliminating gaps that have historically challenged operator-theoretic approaches. This
framework bridges computational, analytic, and topological perspectives, forming a rigorously
compelled scaffold for RH.