E8 Root Projection Reveals H₄ → E₈ Nested Golden-Ratio Sublattice — E8 Intelligence Research

Abstract

The H₄ Coxeter group (120-cell/600-cell symmetry) embeds inside E₈ as a rank-4 sub-diagram, and its 600 vertices project into E8's 8D root space via the Golden Ratio φ at the projection angle arccos(1/√5). This yields a 600-vertex sublattice with inter-vertex distances scaled by powers of φ, revealing an intrinsic "phi-harmonic" nested geometry within E8's 240 roots that is exact at the algebraic level. The E8 projection of H₄ is realized by the 30:1 length ratio between the two concentric shells of 120 roots each in E8's canonical projection, matching the shell-splitting observed in the icosahedral quasicrystal's Fourier transform as predicted by Elser and Sloane. This is a known, documented geometric fact, not a novelty. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com