Riemann Zeros Match Quantum Chaos: GUE Statistics Link Primes to Symmetric Spaces — E8 Intelligence Research

FINDING: Riemann zeta zero spacings follow GUE random matrix eigenvalue statistics, linking prime distribution to quantum chaos and symmetric space spectral theory. | MATH: Montgomery-Odlyzko law: normalized zero spacings \( \delta_n = (\gamma_{n+1}-\gamma_n) \frac{\log(\gamma_n/2\pi)}{2\pi} \) match GUE eigenvalue spacing distribution \( p(s) = \frac{32}{\pi^2} s^2 e^{-4s^2/\pi} \) (small-s cubic repulsion \( \propto s^2 \), not linear). Pair correlation conjecture: \( R_2(s) = 1 - \left(\frac{\sin \pi s}{\pi s}\right)^2 \). Zeros \( \rho = 1/2 + i\gamma \) — critical line symmetry. | CONNECTION: GUE is the large-N limit of unitary group \( U(N) \) — a symmetric space of type AIII. The \( s^2 \) repulsion is the universal eigenvalue repulsion of the Gaussian Unitary Ensemble, tied to the root system \( A_{N-1} \) (Weyl chamber volume). The sine kernel \( \frac{\sin \pi(x-y)}{\pi(x-y)} \) is the universal scaling limit of all symmetric spaces of compact type — a crystallographic lattic Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com