A Unified, Deterministic Locator for Primes on the 6k±1 Rails

We present a deterministic procedure that, for each pair (6k−1,6k+1), correctly identifies the prime side. The method combines (i) a residue-based “Modular Selector” aggregating congruence information for primes p ≤ Q and (ii) a deterministic primality proof on the vanishingly small set of cases where the selector abstains. We prove an explicit sieve bound showing the abstention share is at most C2(log Q)−2(1+O(1/log Q)) with C2 = 3e^(−2γ)C_twin ≈ 0.625. We also express the global left/right alternation rate as a convergent renewal identity indexed by the least prime factor of the composite rail. Empirical runs up to N = 3·10^7 confirm 100% correctness and match the theory. This yields a unified, constructive law of prime location on the 6k±1 rails.