Harmonic Architecture of Prime Pairs: Empirical Evidence for Localized Deterministic Constraints in Twin, Cousin, and Sexy Primes

This paper presents empirical evidence demonstrating that the local distribution of prime numbers is strictly governed by localized deterministic constraints, challenging purely probabilistic macro-level frameworks like Cramér's random model. Through a computational analysis of twin (gap 2), cousin (gap 4), and sexy (gap 6) prime pairs up to 2 million, the study introduces the Information Potential heuristic I=2p+(g−1). The findings reveal a staggering near-universal adherence to a harmonic lattice of step 6: approximately 99% of twin and cousin primes—and 49% of sexy primes, with the remaining half structurally excluded by modulo 3 constraints—land within a ±48 offset of another prime. This research highlights the intense, localized modular architecture that operates beneath the pseudo-random macro-distribution of primes.