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 pₙ^ (1/3) and off-diagonal interaction kernel |cos (pi* (ln pₘ - ln pₙ) ) | / 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 pₘ/pₙ, 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.