Spectral Gap Optimality, Weyl's Law, and Modular L-Values on Hyperbolic Surfaces — E8 Intelligence Research

Abstract

FINDING: Spectral gap optimality on typical hyperbolic surfaces; Weyl's law for Laplacian eigenvalues; modular forms linked to critical L-values and interpolated sequences. MATH: - Weyl's asymptotic law: \( N(\lambda) \sim \frac{\mathrm{vol}(M)}{4\pi} \lambda \) for surfaces (leading term). - Spectral gap \(\lambda_1\): optimal lower bound for typical hyperbolic surfaces (Monk). - Interpolated Apéry numbers \(a_n\) satisfy \( \sum a_n t^n \) expressed as critical L-value \(L(f, k)\) for modular form \(f\) of weight 4. - Vector-valued modular forms of half-integral weight: algebro-geometric sheaf theory. CONNECTION: - Hyperbolic surfaces have constant negative curvature \(-1\), linking to modular group \(\mathrm{PSL}(2,\mathbb{Z})\) and its fundamental domain — a geometric tessellation with angles \(\pi/3, \pi/3, \pi/3\) (equilateral triangle) or \(\pi/2, \pi/3, \pi/6\). - The optimal spectral gap \(\lambda_1 \geq 1/4\) for arithmetic hyperbolic surfaces (Selberg conjectur Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com