Interpolated Apéry Sequences as Critical L-Values of Modular Forms — E8 Intelligence Research

FINDING: Interpolated Apéry sequences for ζ(3) and ζ(2) are critical L-values of weight 4 and weight 3 modular forms, extending Zagier's result to all six sporadic sequences. | MATH: Zagier's interpolation: \( A_n(t) = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}^2 t^k \). For \( t=1 \), Apéry numbers for ζ(3). Critical L-value: \( L(f,2) \) for weight 4 modular form \( f \). Extension: All six sporadic sequences (including those for ζ(2)) map to critical L-values of modular forms of weight 3 or 4. | CONNECTION: E8 lattice theta series is a modular form of weight 4 (root system E8, 240 shortest vectors, theta series \( \Theta_{E8}(\tau) = 1 + 240\sum_{n\ge1}\sigma_3(n)q^n \)). Apéry numbers' modular L-values link to weight 4 forms, echoing E8's theta coefficients (σ₃). Ratios: 240 = 2⁴·3·5, related to base-60 (60·4). No direct 0.618/1.618, but modular forms encode lattice symmetries (E8 root system, 8D crystallographic). | DEPTH: 8 — Bridges Apéry's irrationality proofs (ζ(3), ζ(2)) to m Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com