Two Spectral Regimes in a Prime-Based Operator: GUE Statistics, a Stable Phase Boundary, and Connection to Tate's Thesis

We study a symmetric operator H on the sequence of prime numbers with diagonal elements p_n^(1/3) and off-diagonal interaction kernel |cos(pi*(ln p_m - ln p_n))| / sqrt(|m-n|). Numerical experiments for N up to 3000 reveal two distinct spectral regimes: the lower ~75% of the spectrum exhibits near-perfect GUE level repulsion (small gap fraction ~0.00-0.03), while the upper ~25% shows classical diffusion (small gap fraction ~0.06-0.09). The boundary between regimes is stable at 0.747 +/- 0.046 across all tested sizes and is specific to prime numbers: it disappears when primes are replaced by the smooth sequence n*ln(n). The interaction kernel is shown to depend only on the ratio p_m/p_n, suggesting a natural interpretation as a discretization of Tate's adelic zeta integral on orbits of Q*. No claim of proof of the Riemann Hypothesis is made.