The Prime Double-Helix and the Law of Forced Beats

We describe a new structural law governing the distribution of prime numbers. All primes greater than 3 lie on the two rails 6k−1 and 6k+1. A "half-rung" occurs when exactly one of these is prime. We prove that for every prime q ≥ 5, exactly a fraction 1/(q−1) of all half-rungs are "forced" by q—that is, one side is divisible by q, leaving only the other side to possibly be prime. Summing over small primes q shows that almost all half-rungs are accounted for by these forced beats. We call this the Law of Forced Beats. This result is unconditional, exact, and testable. Additionally, we observe a slight but measurable Left/Right alternation bias among half-rungs (≈52–53% flips), which we call the Alternation Law. The Alternation Law is conjectural but testable and admits a weak form under Bombieri–Vinogradov. Together, these laws show that the apparent randomness of prime distribution is structured, rhythmic, and law-driven.