Riemann Zeta Zeros Exhibit GUE Spectral Rigidity Linking Primes to Quantum Chaos — E8 Intelligence Research

Abstract

FINDING: Riemann zeta zeros exhibit spectral rigidity matching Gaussian Unitary Ensemble (GUE) random matrices with β=2, linking prime distribution to quantum chaotic systems. MATH: - Level spacing distribution for GUE: \( P(s) \approx \frac{32}{\pi^2} s^2 e^{-\frac{4}{\pi} s^2} \) for small s (quadratic repulsion, β=2). - Riemann zeros: \( \gamma_n \sim 2\pi n / \log n \) (mean spacing), with normalized spacings \( s_n = (\gamma_{n+1} - \gamma_n) \cdot \frac{\log \gamma_n}{2\pi} \). - Prime counting: \( J(x) = \sum_{p^k \leq x} \frac{1}{k} \), related to ζ via \( \log \zeta(s) = s \int_2^\infty J(x) x^{-s-1} dx \). - Explicit formula: \( \psi(x) = x - \sum_\rho \frac{x^\rho}{\rho} - \log 2\pi - \frac{1}{2}\log(1-x^{-2}) \), where ρ are nontrivial zeros. CONNECTION: - GUE β=2 corresponds to Dyson's circular unitary ensemble — no time-reversal symmetry, akin to quantum systems with broken T-symmetry (e.g., in magnetic fields). - The 0.618 golden ratio does not appear direc Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com