Pair Correlation Law Links Number Theory, Chaos, and q-Deformed Integrals — E8 Intelligence Research

FINDING: Montgomery–Odlyzko pair correlation law connects to monomial sequences mod 1, semiclassical chaos, and q-deformed Selberg integrals; the YouTube results are mostly noise, but the arXiv paper (2405.13676) is the substantive mathematical core. | MATH: Pair correlation of \((\alpha n^\theta \bmod 1)\) — for \(\theta>1\), the limiting pair correlation is Poissonian (no repulsion); for \(\theta=1\), it is uniform. Montgomery's conjecture: for zeros \(\gamma_n\) of \(\zeta(s)\), the normalized spacing distribution \(1 - (\sin \pi u / \pi u)^2\) (GUE). Selberg trace formula: \(\sum_n h(r_n) = \frac{g(0)}{2\pi}\log\frac{M}{2\pi} + \frac{1}{2\pi}\sum_p \sum_{m=1}^\infty \frac{g(m\log p)}{p^{m/2}} + \dots\) — connects prime geodesics to eigenvalues. q-deformed \(A_n\) Selberg integral: \(\int \prod_{i Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com