Random Matrix Theory Links Quantum Chaos, Nuclear Spectra, and Prime Zeros — E8 Intelligence Research

Abstract

FINDING: Random matrix theory (RMT) spectral statistics match the spacing distribution of nontrivial Riemann zeta zeros, suggesting a hidden universality class linking quantum chaos, nuclear spectra, and prime number structure. MATH: - Riemann zeta zeros: \( \zeta(1/2 + i t_n) = 0 \) with \( t_n \) distributed according to GUE (Gaussian Unitary Ensemble) spacing: \[ P(s) \sim \frac{32}{\pi^2} s^2 e^{-\frac{4}{\pi} s^2} \quad \text{(small } s\text{)} \] and asymptotic Wigner-Dyson form \( P(s) \sim \frac{\pi}{2} s e^{-\frac{\pi}{4} s^2} \). - Nearest-neighbor spacing ratio: \( \langle r \rangle \approx 0.5996 \) (GUE), matching zeros. - Number variance: \( \Sigma^2(L) \sim \frac{1}{\pi^2} \log L + \text{constant} \). CONNECTION: - The GUE spacing distribution is linked to the symmetry of unitary matrices \( U(N) \), which underlies root systems of type \( A_N \) (crystallographic). - The golden ratio \( \phi = 1.618... \) appears indirectly via the Tracy-Widom Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com