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 \((α n^θ 1)\) — for \(θ>1\), the limiting pair correlation is Poissonian (no repulsion); for \(θ=1\), it is uniform. Montgomery's conjecture: for zeros \(γ_n\) of \(ζ(s)\), the normalized spacing distribution \(1 - (sin π u / π u)^2\) (GUE). Selberg trace formula: \(∑_n h(r_n) = g(0)/2π/2π + 1/2π∑_p ∑ₘ₌₁^∞ {g(mlog p)}{pm/2} + \) — connects prime geodesics to eigenvalues. q-deformed \(A_n\) Selberg integral: \(∫ ∏_{i Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com