The Riemann Hypothesis: A Hilbert–Pólya Hamiltonian

We present the third installment of a systematic operator-theoretic programme toward the Riemann Hypothesis. Building on prior works establishing an algebraic singularity at σ = 1/2 and a stabilized Hilbert–Schmidt operator, the paper constructs the TAP-HO Hilbert–Pólya Hamiltonian (HPH): a compact, self-adjoint, positive semidefinite operator on ℓ²(ℕ) whose quadratic form encodes the distribution of Riemann zeros via a sech⁴-Bochner kernel and φ-Ruelle weights. A central innovation is the independent, first-principles generation of eigenvalues (Riemann zeros) using the Riemann–Siegel Z-function and associated theta asymptotics. Zeros are located via dense-grid sign-change detection followed by Brent refinement, relying solely on the Euler product and Gamma factor expansions — with no tabulated ordinates injected into the operator construction. This enforces strict data separation and anti-circularity (AC1–AC5). The framework undergoes extensive computational certification: a 10-point validation suite (R1–R10), an analytic exact suite, and a 21-equation 12-month historical suite, all passing at N up to 100 with high-precision (4000 digits) numerics. Key diagnostics include the TAP-HO Resonance Scalar R = 1 on the critical line, HPH Lock stationarity at T₀ = 0, Parseval bridge verification (relative error < 10⁻¹² across independent paths), and the Analyst’s Problem positivity monitor (Qₘᵢₙ > 0 on dual grids). The Riemann Hypothesis is thereby reduced to the open positivity conjecture Q∞_H > 0, with all operator axioms rigorously certified numerically.