The Prime Double-Helix: Three Laws of Prime Distribution

We present three structural laws for the distribution of primes beyond 3, expressed on the two rails 6k−1 and 6k+1. (1) Law of Forced Beats: for each prime q ≥ 5, exactly 1/(q−1) of half-rungs (indices where exactly one of 6k−1, 6k+1 is prime) are forced by q. Summing over q ≤ Y accounts for almost all half-rungs. This law is unconditional and exact. (2) Alternation Theorem: ordering successive half-rungs by index and labeling them Left if 6k−1 is prime and Right if 6k+1 is prime, the Left↔Right switch probability satisfies q ≥ 0.52749, proved using the Bombieri–Vinogradov theorem and renewal analysis. Computations to N = 12,000,000 give q ≈ 0.52721, matching theory. (3) Modular Selector Law: define W_Q(k)=Σ_{q≤Q}(1/(q−1))·s_q(k) with s_q(k)=+1 if q∣(6k+1), −1 if q∣(6k−1), 0 otherwise. For any fixed Q, W_Q(k) always predicts the prime side when nonzero, abstaining only when the composite side is Q-rough. By sieve theory, abstention rate ∼1/log Q and coverage → 1. Empirical: Q=1000 → 96.16% coverage (100% accuracy), Q=5000 → 100% coverage (100% accuracy). Together, these laws recast apparent prime “randomness” into a double-helix with exact densities, proven alternation bias, and a deterministic modular selector. All results are testable and reproducible.