Gaps in Direct Sources on H3 Golden Ratio Root Coordinates — E8 Intelligence Research

FINDING: The search results are primarily pedagogical videos and one abstract on involution products in Coxeter groups — none directly address the H3 golden ratio root system coordinates, though the topic is implicitly central to the geometry of icosahedral symmetry. MATH: - Coxeter group H3 (icosahedral symmetry) has order 120, generated by reflections with relations: \( s_1^2 = s_2^2 = s_3^2 = (s_1 s_2)^5 = (s_2 s_3)^3 = (s_1 s_3)^2 = 1 \) - H3 root system coordinates (in \(\mathbb{R}^3\)) involve the golden ratio \(\varphi = (1+\sqrt{5})/2 \approx 1.618\) and its inverse \(\varphi^{-1} = \varphi - 1 \approx 0.618\). Example roots (normalized): \((0, \pm 1, \pm \varphi)\), \((\pm 1, \pm \varphi, 0)\), \((\pm \varphi, 0, \pm 1)\) — 30 roots total. - The abstract (arXiv:1405.3051) proves for finite Coxeter groups \(W = \mathcal{W}\) (all elements are products of two involutions) and defines a minimal length invariant \(\ell(x)+\ell(y)-\ell(w)\) — but gives no speci Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com