The Prime Double-Helix: Law of Forced Beats and the Alternation Theorem

We present two new structural results on the distribution of primes beyond 3, expressed on the two rails 6k−1 and 6k+1. First, we prove the Law of Forced Beats: for every prime q ≥ 5, the proportion of “half-rungs” (positions where exactly one of 6k−1, 6k+1 is prime) forced by q is exactly 1/(q−1) of the main term N/log N. Summing over q ≤ Y yields (Σ 1/(q−1))·N/log N, and a sieve argument shows that as Y → ∞ almost all half-rungs are explained in this way. This law is unconditional, exact, and testable. Second, we prove the 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. The proof uses the Bombieri–Vinogradov theorem together with a renewal model built from the forced-beat densities. Computations up to N = 12,000,000 give q ≈ 0.52721, matching the theorem. We further conjecture that the limit exists and equals ≈ 0.5272. Together, these results recast the apparent “randomness” of primes on the 6k±1 rails as a structured, law-driven double-helix with exact constants and a provable alternation bias. Both results are testable and reproducible, providing a new foundation for experimental number theory. prime numbers; prime distribution; sieve theory; arithmetic progressions; Bombieri–Vinogradov theorem; experimental number theory; Law of Forced Beats; Alternation Theorem; prime double-helix