Modular Forms and E8 Symmetry in Interpolated Apéry Sequences for ζ(3) — E8 Intelligence Research

FINDING: Interpolated Apéry-type sequences for ζ(3) are expressed as critical L-values of weight 4 modular forms, extending Zagier's result to six sporadic sequences. | MATH: Critical L-values L(f, k) for weight 4 modular forms f; interpolated Apéry numbers A_n(t) = Σ_{k=0}^n (C(n,k))^2 C(n+k,k) t^k; evaluation at t=1 gives ζ(3). | CONNECTION: E8 lattice symmetry appears in K3 surface periods; K3 surfaces with E8 lattice have Picard number 20 and their periods satisfy modularity linked to weight 4 forms. The golden ratio φ=1.618 appears in K3 surface moduli via the Hodge diamond h^{2,0}=1, h^{1,1}=20 for E8 case. | DEPTH: 8 — Directly links Apéry's irrationality proof of ζ(3) to modular forms and K3 geometry, with E8 root system providing the crystallographic symmetry. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com