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): \( A_n = ∑ₖ₌₀^n {n}{k}^2 {n+k}{k}^2 \); interpolated version \( A_n(t) \) 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 \( C_g \)). 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