Interpolated Apéry-like 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 connection between ζ(3) and modular forms. | MATH: Apéry numbers for ζ(3): \( a_n = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}^2 \); critical L-value \( L(f,2) \) for weight-4 modular form \( f \); interpolated sequences yield \( L(f,2) \) via closed forms. | CONNECTION: No direct geometric ratios (0.382, 0.618, etc.) or base-60 appear. However, the modular forms involved are associated with elliptic curves and Shimura varieties, which have underlying lattice structures (e.g., \( \mathbb{H}^2/\Gamma \) for genus-2 Siegel modular forms) and root system symmetries (e.g., \( E_8 \) in weight-4 forms). The Euler system for genus-2 Siegel modular forms (Loeffler) links to Galois representations and motives, which are built on crystalline symmetries. | DEPTH: 7 — Bridges Apéry's irrationality proof, modular forms, and L-values; deep but not yet unified with geometric constants. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com