Riemann Zeta Zeros Show GUE Level Repulsion, Indicating Broken Time-Reversal Symmetry — E8 Intelligence Research

Abstract

FINDING: Riemann zeta zeros exhibit level repulsion matching Dyson's Gaussian Unitary Ensemble (GUE) with β=1, indicating broken time-reversal symmetry in the hypothetical quantum system whose spectrum corresponds to these zeros. MATH: - Riemann zeta zeros: \( \zeta(1/2 + i t_n) = 0 \), with \( t_n \) real under RH. - Level spacing distribution: \( P(s) \sim s^\beta e^{-c s^2} \) for small \( s \), with \( \beta = 1 \) (GUE). - Dyson index \( \beta = 1 \) corresponds to Hermitian matrices with no time-reversal symmetry (unitary class). - Montgomery–Odlyzko law: pair correlation \( R_2(s) = 1 - \left( \frac{\sin \pi s}{\pi s} \right)^2 \), identical to GUE. CONNECTION: - The GUE eigenvalue spacing distribution is universal for chaotic quantum systems with broken time-reversal symmetry. - No direct geometric ratio (0.382, 0.618, etc.) appears; the connection is statistical and spectral, not metric. - However, the zeros' spacing statistics match the eigenvalues of random Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com