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 Aₙ (t) = Σ₊=₀ⁿ (C (n, k) ) ² C (n+k, k) tᵏ; 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
Andrew Stewart Caldin (Wed,) studied this question.