Critical Primes, Phase Jumps, and Simple Zeros: Additive Sieving and the Riemann Spectrum

We establish an explicit computational correspondence between prime numbers and Riemann zeta zeros through additive sieving. Three results are presented: (1) Additive Sieve: Primes are the integers unreachable by shadow rays R_m = {2m, 3m, 4m, ...}. This reformulation of trial division in additive language yields an exact primality test and connects prime detection to the cumulative sum S_N = Σ_{p≤N} p. (2) Phase-Jump Detection: Using atan2(Im ζ, Re ζ), we detect zeros by monitoring phase jumps of ±π as log(S_N) crosses γ_n. This identifies critical primes N_n — an arithmetic sequence marking zero locations. (3) Newton-Raphson Exactness: A single Newton step from log(S_N) reaches γ_n with quadratic convergence. The multiplicity product |Δ|·|ζ'/ζ| = 1.0008 ± 0.0028 across 25 zeros is consistent with simple zeros (m = 1). This is reported as an Observation; the analytical proof appears in the companion paper. A unified algorithm finds primes and locates Riemann zeros in a single loop. All results are verified with supplementary scripts.