Golden Ratio as Recurrence Operator in Coxeter Group Involutions — E8 Intelligence Research

FINDING: Golden ratio φ emerges as a stable self-application recurrence operator in Coxeter group theory, with finite Coxeter groups fully decomposable into involutions. MATH: φ = (1+√5)/2 ≈ 1.618; φ⁻¹ = φ-1 ≈ 0.618; recurrence: φ = 1 + 1/φ. For finite Coxeter group W, W = {xy | x²=y²=1} (every element is a product of two involutions). Excess e(w) = min{ℓ(x)+ℓ(y)-ℓ(w)} for w=xy, x²=y²=1. CONNECTION: The 120-cell (H₄ Coxeter group) has circumradius = φ² ≈ 2.618, linking φ to 4D regular polytope symmetry. The golden ratio appears as the fundamental ratio in H₄ root system (600-cell, 120-cell). Crystallographic split groups (affine Coxeter) show homogeneity, implying lattice symmetries may encode φ-based recurrence. DEPTH: 7 — The connection between Coxeter involution products and golden ratio recurrence is mathematically rigorous, but the explicit geometric proof for 120-cell circumradius = φ² is not provided in these abstracts; it is a known result from Coxeter's own work. The "st Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com