Prime Gaps as Quasi-Crystalline Lattice Structures via Modular Spirals — E8 Intelligence Research

Abstract

FINDING: Prime gaps exhibit quasi-crystalline, 1D lattice-like structure via intersecting lattice frameworks; Tao's work quantifies gap statistics; Dirichlet spirals reveal modular residue patterns. | MATH: Prime gap distribution: \( g_n = p_{n+1} - p_n \); arbitrarily large gaps: \( \exists \) gaps \( > N \) for any \( N \) (proof via \( (N+1)! + 2, \ldots, (N+1)! + (N+1) \)); Tao's results on \( \liminf_{n\to\infty} \frac{g_n}{\log p_n} \) and bounded gaps (Maynard–Tao: \( \liminf (p_{n+1}-p_n) \le 246 \)); Dirichlet: primes \( \equiv a \pmod{q} \) with \( \gcd(a,q)=1 \) — density \( 1/\varphi(q) \). | CONNECTION: Aschheim's quasicrystals from intersecting lattices — prime gaps as a 1D quasicrystal (self-similar, non-periodic order) mirroring Penrose tilings; Dirichlet spirals show 6-fold/12-fold rotational symmetry in modular residue distributions (base-60 compatible: \( \varphi(60)=16 \), residues cluster at 1, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59 — all coprime Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com