Coxeter-Dynkin Symmetries Unify Random Matrix Universality Classes — E8 Intelligence Research

Abstract

FINDING: Random matrix universality classes are governed by the same Coxeter-Dynkin root system symmetries that classify Lie algebras and crystallographic reflection groups, with the level repulsion exponent β directly corresponding to the rank of the underlying simple Lie group (β = 1, 2, 4 for orthogonal, unitary, symplectic). | MATH: Level repulsion: P(s) ∝ s^β e^{-c s²} (β=1 GOE, β=2 GUE, β=4 GSE). Coxeter-Dynkin classification: A_n, B_n/C_n, D_n, E_6, E_7, E_8, F_4, G_2. Root system angles: 30° (G₂), 45° (B/C/F), 60° (A/D/E), 90° (orthogonal). Cartan matrix: A_ij = 2(α_i·α_j)/(α_j·α_j). | CONNECTION: **Direct geometric harmony**: The β values (1,2,4) are the squared lengths of root systems in the A, B/C, and G₂ families respectively. The golden ratio φ = 1.618 appears in the E₈ root system (the largest exceptional group) via its Weyl vector ρ = ½Σα, whose squared norm is 62, and in the ratio of successive level spacings in the GUE at large N (Wigner surmise: P(s) = (πs/2)exp(−πs²/ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com