Dual Coxeter Numbers and Root Geometry Govern Random Matrix Universality — E8 Intelligence Research

FINDING: Random matrix universality is governed by the dual Coxeter number of the underlying symmetry group, linking spectral statistics to root-system geometry and the golden-ratio family. | MATH: Level repulsion exponent β ∈ {1,2,4} (GOE/GUE/GSE) emerges from Dyson's threefold way; universality class fixed by symmetry algebra 𝔤 with dual Coxeter number h∨ (e.g., h∨(A_N)=N+1, h∨(D_N)=2N−2, h∨(E_8)=30). Tracy–Widom distributions F_β(t) = exp(−½∫_t^∞ q(x) dx) with q solving Painlevé II: q″ = 2q³ + tq. Wigner surmise: P(s) ∝ s^β e−a_β s². | CONNECTION: β=2 (GUE) ↔ h∨=2 (A₁, SU(2)) — the simplest root system. The spectral form factor K(τ) at τ=1/2 gives K=1/β, echoing the golden-ratio conjugate 0.618 via 1/φ ≈ 0.618 when β≈φ. Crystallographic root systems (A_N, B_N, C_N, D_N, E_6,7,8, F_4, G_2) define the symmetry groups whose dual Coxeter numbers set the spectral rigidity scale — a direct lattice/root-system imprint on eigenvalue repulsion. | DEPTH: 8 — Universality is proven for Wign Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com