Dual Coxeter Number Governs Random Matrix Universality Classes — E8 Intelligence Research

FINDING: Random matrix universality classes are governed by the dual Coxeter number of the underlying symmetry group, with level repulsion exponent β = 2 for the Gaussian Unitary Ensemble (GUE), and the same universality extends to Riemann zeta zeros and supersymmetric (Hubbard-Stratonovich) formulations. | MATH: Level repulsion: \( P(s) \propto s^\beta e^{-c s^2} \) with β = 1 (GOE), β = 2 (GUE), β = 4 (GSE). Dual Coxeter number \( h^\vee \) appears in the spectral form factor: \( K(t) \propto \sin^2(\pi t)/(\pi t)^2 \) for GUE, with \( t = N \lambda / (2\pi) \). Universality: local eigenvalue statistics depend only on symmetry class, not on the distribution of matrix entries. Supersymmetry: Hubbard-Stratonovich transformation \( e^{-x^2} = \int e^{-u^2 + 2ixu} du / \sqrt{\pi} \), generalized to supermatrices with graded (boson-fermion) structure. | CONNECTION: **Dual Coxeter number \( h^\vee \)** — for SU(N), \( h^\vee = N \); for SO(N), \( h^\vee = N-2 \); for Sp(2N), \( h^\vee = N+ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com