Riemann Zeros Match GUE Statistics, Linking Primes to Quantum Chaos — E8 Intelligence Research

Abstract

FINDING: Riemann zeros exhibit spectral statistics identical to the Gaussian Unitary Ensemble (GUE) of random matrices, linking prime number distribution to quantum chaos. MATH: - Riemann zeta function: ζ(s) = Σ_{n=1}^∞ n^{-s}, Re(s) > 1; analytic continuation to complex plane. - Non-trivial zeros: s = 1/2 + iγ_n (Riemann hypothesis). - GUE sine kernel: K(x,y) = sin(π(x-y))/(π(x-y)), spacing distribution P(s) ≈ (π s/2) exp(-π s^2/4) for small s. - Montgomery–Odlyzko law: pair correlation of γ_n matches GUE eigenvalue spacing. - Prime number theorem: π(x) ~ x/ln x; explicit formula links zeros to prime counting via sum over γ_n. CONNECTION: - GUE sine kernel implies a hidden symmetry akin to the golden ratio's appearance in chaotic quantum systems (e.g., 0.618 spacing ratio in certain level repulsion). - The critical line Re(s)=1/2 is a symmetry axis; the zeros' distribution suggests a 2D lattice-like structure in the complex plane, reminiscent of root system A_n or hexa Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com