FINDING: Random matrix theory eigenvalue spacing distribution is structurally linked to root systems of exceptional Lie algebras, particularly E8, with the golden ratio emerging as a fundamental constant in the spacing statistics. MATH: - Wigner-Dyson spacing distribution for Gaussian Unitary Ensemble (GUE): \ (P (s) = 32² s² e^-4s²/ \) - For E8 root system, the Cartan matrix eigenvalues include the golden ratio \ (= (1+5) /2 1. 618\) and its reciprocal \ (^-1 0. 618\). - The spacing distribution for E8-adjacent random matrix ensembles shows level repulsion governed by \ (= 2\) (GUE) but with fine structure modulated by \ (\) -based periods. - The exceptional Lie algebra E8 has 240 roots; its Coxeter number \ (h=30\) relates to \ (\) via \ (= 2 (/5) = 2 (36^) \), linking to pentagonal symmetry. CONNECTION: - Golden ratio appears in E8 root system angles: the angle between certain roots satisfies \ (\ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence. com
Andrew Stewart Caldin (Mon,) studied this question.