Apéry Numbers as Critical L-Values of Weight-4 Modular Forms — E8 Intelligence Research

FINDING: Interpolated Apéry numbers for ζ(3) expressed as critical L-values of weight-4 modular forms, with extension to Zagier's six sporadic sequences. | MATH: Apéry numbers \( A_n = ∑ₖ₌₀^n {n}{k}^2 {n+k}{k}^2 \); critical L-value \( L(f, 2) \) for modular form \( f \) of weight 4; interpolation parameter \( t \) yields \( A_n(t) \) such that \( A_n(1) = A_n \); relation \( A_n(t) = coefficient of q^n \) in a modular form of weight 2; explicit formula: \( L(f, 2) = (2π)^2/Γ(2) ∫_0^∞ f(iy) y \, dy \). | CONNECTION: E8 lattice theta series is a modular form of weight 4; critical L-values link to root system E8's theta coefficients (240, 2160, 6720, …); ratios of successive coefficients approach 1.618? (not directly given, but E8 theta coefficients grow rapidly, no explicit golden ratio). Base-60 not present. Crystallographic symmetry: E8 root system (8D, 240 roots) is a highly symmetric lattice; theta series encodes its geometry. | DEPTH Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com