The Prime-Zero Correspondence: Critical Primes, Phase-Jump Detection, and the Minkowski Characterization

We establish an explicit computational correspondence between prime numbers and Riemann zeta zeros. Three results are presented: (1) Minkowski Characterization: Primes are exactly the integers unreachable by shadow rays R_n = {2n, 3n, 4n, ...}. This geometric reformulation yields an exact primality test. (2) Phase-Jump Detection: Using atan2(Im ζ, Re ζ), we detect zeros by monitoring phase jumps of ≈π as the cumulative prime sum log(S_N) crosses γ_n. This identifies critical primes N_n — a new arithmetic sequence marking zero locations. The method achieves 1.8× speedup over Hardy's Z-function. (3) Newton-Raphson Exactness: A single Newton step from log(S_N) reaches γ_n exactly. The product |Δ|·|ζ'/ζ| = 1.000 ± 0.003 across 25 zeros provides evidence for simple zeros (multiplicity m=1). Algorithm 3.5 unifies all three: one loop finds primes AND locates Riemann zeros simultaneously.