Interpolated Apéry Sequences Link Sporadic Modular L-Values to ζ(3) — E8 Intelligence Research

FINDING: Interpolated Apéry sequences express critical L-values of modular forms, extending Zagier's result for ζ(3) to six sporadic sequences. | MATH: Apéry numbers for ζ(3): \( a_n = ∑ₖ₌₀^n {n}{k}^2 {n+k}{k}^2 \); interpolated version yields critical L-value of weight-4 modular form; generalization to six sporadic sequences (Zagier's list) gives L-values of modular forms of varying weights. | CONNECTION: No explicit geometric ratios (0.382, 0.618, etc.) or base-60 appear. However, modular forms are deeply tied to lattice symmetries (SL(2,Z) action on upper half-plane) and elliptic curves, which are tori (genus-1 curves) with lattice parameter τ. The critical L-values relate to periods and quasi-periods of elliptic curves, linking to root system E8 in Monstrous Moonshine indirectly. | DEPTH: 8 — Bridges Apéry's irrationality proof, modular forms, and L-functions; fundamental to arithmetic geometry and Langlands program. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com