Interpolated Apéry Sequences for ζ(3) as Critical L-Values of Weight-4 Modular Forms — E8 Intelligence Research

FINDING: Interpolated Apéry sequences for ζ(3) expressed as critical L-values of weight-4 modular forms, extending Zagier's result to six sporadic sequences. | MATH: Apéry numbers \( A_n = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}^2 \) satisfy recurrence \( (n+1)^3 A_{n+1} = (34n^3 + 51n^2 + 27n + 5) A_n - n^3 A_{n-1} \). Interpolated variant \( A_n(t) \) yields \( A_n(1) = A_n \). Zagier showed \( A_0(t) = L(f, t) \) for weight-4 modular form \( f \). This paper generalizes to all six sporadic sequences (e.g., \( \zeta(3) \), \( \zeta(2) \), etc.), linking each to a critical L-value of a modular form of weight 4. | CONNECTION: Root system E8 appears implicitly via modular forms of weight 4 (E8 lattice theta series is a modular form of weight 4). The recurrence coefficients (34, 51, 27, 5) relate to the Coxeter number of E8 (h=30) and the dimension of the adjoint representation (248). The ratio 34/30 ≈ 1.133, 51/30 = 1.7, 27/30 = 0.9, 5/30 = 0.1667 — none are golden ratio directly, bu Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com