Quasi-Bragg Law with Hierarchical ℤ[φ] Metric Governs Quasicrystal Diffraction — E8 Intelligence Research

FINDING: Quasicrystal diffraction is governed by a quasi-Bragg law with a hierarchical metric, indexed by a ℤ[φ] module, revealing non-periodic order. | MATH: Quasi-Bragg law: \( \mathbf{q} \cdot \mathbf{R} = 2\pi n \) generalized to \( \mathbf{q} \cdot \mathbf{R}_i = 2\pi n_i \) for incommensurate vectors; diffraction peaks at positions \( \mathbf{k} = \sum_{i=1}^6 n_i \mathbf{a}_i^* \) with \( \mathbf{a}_i^* \) in a 6D reciprocal lattice projected to 3D; module ℤ[φ] where φ = (1+√5)/2 ≈ 1.618; peak intensities scale with powers of φ⁻¹. | CONNECTION: Icosahedral group (order 60) symmetry; golden ratio φ appears in peak positions and scaling; ratios 0.618 (φ⁻¹), 1.618 (φ), 2.618 (φ²) define self-similarity; base-60 implicit in icosahedral angles (e.g., 60°, 120°); crystallographic root system H₃ (non-crystallographic) with Coxeter number 10. | DEPTH: 9 — Directly links quasicrystal order to φ-based algebra, extending periodic Bragg diffraction to aperiodic systems, revealing a hidden h Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com