Description (Zenodo): This paper introduces a self-adjoint operator K̃ on the prime number lattice, demonstrates GUE spectral statistics (KS p-value = 0.905) matching Riemann zeros, proves an exact rank-one decomposition with a closed-form secular equation, and provides numerical evidence for a non-vanishing spectral gap Δ∞ ≈ 0.961. Keywords: prime number lattice, Hilbert–Pólya conjecture, GUE statistics, random matrix theory, spectral gap, rank-one perturbation, secular equation, Riemann zeta function, quantum chaos, Wigner–Dyson, level repulsion, arithmetic operator