Interpolated Apéry-like Sequences Connect Six Sporadic Cases to Modular L-Values — E8 Intelligence Research

FINDING: Interpolated Apéry-like sequences express critical L-values of modular forms, extending Zagier's connection between ζ(3) and weight-4 modular forms to six sporadic sequences. | MATH: Apéry numbers for ζ(3) satisfy recurrence (n+1)^3 u_{n+1} = (2n+1)(17n^2+17n+5) u_n - n^3 u_{n-1}. Interpolated version yields critical L-value of weight-4 modular form. Generalization: six sporadic sequences (Zagier's list) interpolate to critical L-values of modular forms of weights 2, 3, 4. Key constants: ζ(3) ≈ 1.2020569, L-values at s = 1, 2, 3. | CONNECTION: No direct geometric ratio (0.382, 0.618, 1.618) or base-60 appears. However, modular forms are deeply tied to elliptic curves and lattice symmetries (SL(2,Z) action on upper half-plane). Critical L-values encode arithmetic of elliptic curves (Birch–Swinnerton-Dyer conjecture). The six sporadic sequences correspond to root systems of type A, D, E (ADE classification) — crystallographic symmetry groups. | DEPTH: 8 — Bridges number theory ( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com