Hierarchic Substitution Rules and Cyclotomic Tilings Unify Ammann–Beenker and Penrose Quasicrystals — E8 Intelligence Research

FINDING: Quasicrystalline long-range order is governed by hierarchic (inflationary) substitution rules whose diffraction spectra are pure-point but non-periodic; the Ammann–Beenker (8-fold, silver mean) and Penrose (5-fold, golden mean) tilings are the canonical 2D cases, unified under cyclotomic substitution tilings (CAST). | MATH: Silver mean λ_s = 1+√2 ≈ 2.41421356 (root of x²−2x−1=0); golden mean φ = (1+√5)/2 ≈ 1.61803399 (root of x²−x−1=0); Ammann–Beenker vertex set lies in ℤ[√2] (2n=8 cyclotomic field ℚ(ζ₈), ζ₈=e^{iπ/4}); Penrose in ℤ[φ] ⊂ ℚ(ζ₅). Inflation multipliers: AB → λ_s² = 3+2√2 ≈ 5.828; Penrose → φ² = 2.618. Substitution matrix eigenvalues = inflation multiplier (Perron–Frobenius) + algebraic conjugates. Diffraction = countable set of Bragg peaks on ℤ-module of rank 4 (AB) / rank 4 (Penrose), with intensities scaling as |amplitude|² over a dense Fourier–Bohr spectrum. | CONNECTION: Silver mean 1+√2 and golden mean φ are the two quadratic Pisot–Vijayaraghavan units of sma Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com