Selberg Trace Formula Links Hyperbolic Geodesics to GUE Random Matrix Theory — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: Selberg trace formula connects spectral statistics of Laplacians on hyperbolic surfaces to GUE random matrix theory, linking prime geodesic lengths to Riemann zeta zeros via trace-class operators. | MATH: Selberg trace formula: ∑_n h(r_n) = (area/4π)∫_{-∞}^{∞} r tanh(πr) h(r) dr + ∑_{γ} ∑_{k=1}^{∞} (Λ(γ)/sinh(kℓ(γ)/2)) g(kℓ(γ)). Riemann zeta zeros: ζ(1/2 + it_n) = 0, with GUE spacing distribution P(s) ≈ (πs/2)exp(-πs²/4). Montgomery-Odlyzko law: N(T) ~ (T/2π)log(T/2πe) + O(log T). | CONNECTION: Hyperbolic geodesic lengths ℓ(γ) correspond to log(p^m) for primes p; Selberg zeta function Z(s) = ∏_{γ} ∏_{k=0}^{∞} (1 - e^{-(s+k)ℓ(γ)}) mirrors Euler product ζ(s) = ∏_p (1 - p^{-s})^{-1}. The GUE eigenvalue spacing (0.618, 1.618) emerges in nearest-neighbor distribution of zeta zeros — a golden ratio signature in quantum chaos. | DEPTH: 9 — This is a profound unification of number theory (prime distribution), quantum physics (energy levels of chaotic systems), and geometry (hyperbolic Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7171839621
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