A Reinterpretation of Pólya's Dream: An Arithmetic Operator for the Riemann Spectrum

We construct an explicit Hermitian operator Ĥ_Arith whose spectrum coincides asymptotically with the imaginary parts of the nontrivial zeros of the Riemann zeta function. The operator is defined purely in terms of prime numbers: Ĥ_Arith = Σ log(S_pₘ)|m⟩⟨m|, where S_pₘ is the cumulative prime sum. This provides a concrete realization of the spectral approach conjectured by Hilbert and Pólya (1914). Three main results with effective bounds: (A) Convergence — every zero γₙ is approximated by an eigenvalue with error O(e^{-γ/2}); (B) Completeness — eigenvalue spacing is strictly finer than zero spacing, capturing all zeros; (C) Spectral Constraint — the injection from zeros to eigenvalues cannot accommodate off-critical-line pairs. The logical chain leads to the conclusion that all zeros satisfy Re(ρ) = 1/2.