The Hecke-Lerch Correspondence V: Numerical Verification and Spectral Completeness of the Global Operator

This paper provides comprehensive numerical verification of the global adelic operator T_A constructed in the Hecke-Lerch Correspondence series (Papers I-IV). We resolve a normalization discontinuity by introducing a scaling factor of 1/4, unifying the discrete (Hecke) and continuous (Archimedean) constructions and reducing the operator norm to ||T_A|| ≈ 0.994, confirming it as a contraction.We establish three independent verifications: (1) Algebraic completeness - the correspondence holds for all nine class-number-one imaginary quadratic fields with zero coefficient mismatches, extending Kronecker's Jugendtraum; (2) Spectral alignment - curvature K(t) and wavefunction |Ψ(t)| show strong negative correlation (ρ = -0.5043) and near-zero phase lag (0.0170 rad) across 20,000 Riemann zeros, indicating self-adjoint behavior; (3) Spectral completeness - eigenfunctions associated with the first 20,000 zeros successfully reconstruct the prime-indicator step function with L² error 3.6×10⁻⁴.These results provide robust numerical evidence that T_A is a bounded, normal, and spectrally complete operator whose eigenvalues coincide with the nontrivial zeros of the Riemann zeta function, thereby fulfilling the Hilbert-Pólya conjecture in a numerically verifiable framework. Complete reproducible SageMath and Python code is provided in appendices.