Interpolated Apéry-like sequences link sporadic sequences to modular L-values. — 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 L-functions. MATH: - Apéry numbers for ζ(3): \( A_n = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}^2 \) - Zagier's interpolation: \( A_n(t) = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}^2 t^k \) → at t=1 gives ζ(3) L-value. - Generalization: Interpolated sequences \( S_n(t) \) for Zagier's six sporadic sequences → critical L-values of modular forms of weight 4. - Hypergeometric form: \( {}_4F_3 \) or \( {}_3F_2 \) series with rational parameters. - Constants: No explicit irrational ratios (0.382, 0.618, etc.) appear; focus on integer sequences and rational t. CONNECTION: - No direct geometric harmony (golden ratio, base-60, crystallographic symmetries) in the evidence. - Modular forms are linked to elliptic curves and lattice symmetries (SL(2,Z) action), but the finding does not emphasize specific ratios. - Root systems Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com