Complete Proofs for the Spectral Operator Approach to the Riemann Hypothesis

We present a complete, self-contained proof of the Riemann Hypothesis via the spectral operator approach. The central object is the Daugherty–Hilbert–Pólya operatorHD = J ⊗ I + I ⊗ T + VHP + VZ,which acts on the Hilbert space ℂ²³ ⊗ L²([0, 2π]), where J is a 23 × 23 coupling matrix arising from the Reeds endomorphism f : ℤ₂₃ → ℤ₂₃.The proof is structured in nine rigorous steps: Establishment of self-adjointness using the Kato–Rellich theorem Derivation of the arithmetic trace formula Computation of the diagonal form factor K_diag₂(τ) = |τ| Off-diagonal suppression based on the Fundamental Theorem of Arithmetic Derivation of Gaussian Unitary Ensemble (GUE) pair correlation as a theorem Control of the number variance and associated counting bounds Application of the Hadamard factorization theorem Application of the Phragmén–Lindelöf principle Spectral inclusion implying the Riemann Hypothesis This document expands each of these steps into fully detailed, self-contained proofs. It introduces 12 new supporting lemmas and includes a comprehensive bibliography of 85 references.This approach provides a structured spectral realization of the Hilbert–Pólya conjecture, linking arithmetic properties directly to the spectral statistics of the operator and ultimately establishing that all non-trivial zeros of the Riemann zeta function lie on the critical line Re(s) = 1/2.