Poisson Statistics of Prime Gaps and Twin Primes — E8 Intelligence Research

FINDING: Prime gaps exhibit exponential/Poisson-like statistics at large scales, with twin primes as a special case of gap-2 events; the Poisson process framework provides the probabilistic backbone for gap distributions. | MATH: For primes ≤ x, the average gap ~ log x (PNT). Gap distribution conjecturally follows an exponential law: P(gap > λ log x) ~ e^{-λ}. Twin prime density ~ C₂ x/(log x)², where C₂ = 2∏_{p>2}(1 − 1/(p−1)²) ≈ 1.32032 (twin prime constant). Poisson process rate parameter λ = 1/log x for gaps. The arXiv paper (1405.2490) constructs a "cluster number set" with formula (2.3) to count twin-prime columns, attempting a sieve-based proof of infinitude. | CONNECTION: The twin prime constant C₂ = 1.32032 is not a golden ratio, but its reciprocal 1/C₂ ≈ 0.7574 is near 0.786 (√φ − 1? No, 0.786 = √(0.618)). More directly: the exponential gap distribution's mean (log x) and variance (log x) mirror a Poisson process — a lattice-like randomness. The sieve structure of primes is a Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com