Interpolated Apéry Sequences Yield Critical L-Values of Weight-4 Modular Forms — E8 Intelligence Research

FINDING: Interpolated Apéry-like sequences express critical L-values of weight-4 modular forms, extending Zagier's ζ (3) connection. | MATH: Apéry numbers for ζ (3): \ (Aₙ = ₊=₀ⁿ nk² n+kk² \) ; interpolated version \ (Aₙ (t) \) yields \ (L (f, 4) \) for weight-4 modular form \ (f \) ; Eichler-Shimura correspondence links modular forms to Galois representations and L-functions. | CONNECTION: No explicit geometric ratios (0. 382, 0. 618, etc. ) or base-60 appear. However, the Eichler-Shimura decomposition for Siegel modular forms (Caraiani) and Euler systems for genus-2 forms (Loeffler) hint at higher-dimensional lattice symmetries (e. g. , symplectic groups Sp (2g) related to root systems like \ (Cg \) ). The weight-4 L-values may encode periods of K3 surfaces or Calabi-Yau threefolds, which often exhibit golden-ratio-like modular parameters in their mirror symmetry. | DEPTH: 7 — Bridges number theory (Apéry's irrationality proof), modular forms, and arithmetic geomet Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence. com