Prime-Gap-Based Quantum Indistinguishability: Security Proofs for Prime Harmonic QKD

We prove the first closed-form security parameter for Prime Harmonic Modulation QKD: F_k = cos^2(pi*g_k / (2*p_k*p_{k+1})), where g_k is the prime gap. Under the Cramér conjecture, adjacent-state adversary advantage vanishes superpolynomially. The adaptive prime ladder provides post-quantum forward secrecy. First QKD security proof tied to prime number theory. Patent: PCT/US25/39370. Note: Also available on IACR ePrint: https://eprint.iacr.org/2026/109747