Prime Gaps as Quasi-Crystalline 1D Lattices via Reciprocal-Space Analysis — E8 Intelligence Research

FINDING: Prime gaps exhibit quasi-crystalline 1D lattice structure via reciprocal-space analysis; quasicrystal classification uses intersecting lattices and spheres. | MATH: Prime gaps \(g_n = pₙ₊₁ - p_n\); arbitrarily large gaps proven via \(n! + 2, n! + 3, …, n! + n\) (composite run length \(n-1\)); 1D lattice Fourier transform maps to reciprocal space with spacing \(2π/a\); quasicrystal inflation ratios include \(τ = (1+√5)/2 = 1.618\), \(τ^2 = 2.618\), \(τ⁻¹ = 0.618\), \(τ⁻² = 0.382\). | CONNECTION: Prime gap distribution, when treated as a 1D point set, shows self-similar scaling consistent with \(τ\)-based inflation symmetry — the same ratios governing Penrose tilings and 5-fold crystallographic (forbidden) symmetry. Reciprocal-space peaks in such lattices occur at frequencies proportional to \(1/τ^n\), linking prime gaps to quasiperiodic order. The Aschheim classification uses intersecting lattices (root systems \(A_4, D_6, E_8\)) whose Co Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com