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 \ (Aₙ = ₊=₀ⁿ nk² n+kk² \) satisfy recurrence \ ( (n+1) ³ A₍+₁ = (34n³ + 51n² + 27n + 5) Aₙ - n³ A₍-₁ \). Interpolated variant \ (Aₙ (t) \) yields \ (Aₙ (1) = Aₙ \). Zagier showed \ (A₀ (t) = L (f, t) \) for weight-4 modular form \ (f \). This paper generalizes to all six sporadic sequences (e. g. , \ ( (3) \), \ ( (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