Apéry Numbers Unify Sporadic Sequences via Weight-4 Modular L-Values — E8 Intelligence Research

FINDING: Apéry-like numbers interpolate to critical L-values of weight-4 modular forms, unifying sporadic sequences via modularity. | MATH: Apéry numbers \(A_n = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}^2\) satisfy \(A_n \sim \frac{\zeta(3)}{6} n^{-2}\) (asymptotic). Zagier's interpolation: \(A_n(t) = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k} t^k\) — at \(t=1\) gives Apéry; at \(t=-1\) gives a critical value \(L(f,2)\) for a weight-4 modular form \(f\) (specifically, \(L(f,2) = \frac{4\pi^2}{3} A_\infty(-1)\) up to rational factor). The six sporadic sequences (Zagier's list: \(A, B, C, D, E, F\)) all admit such interpolations, each mapping to a distinct weight-4 modular form's \(L\)-value at \(s=2\) (the critical point). | CONNECTION: The weight-4 modular forms correspond to elliptic curves of conductor \(N \in \{11, 14, 15, 20, 24, 27\}\) — these are precisely the levels with CM or complex multiplication, whose \(L\)-functions have root numbers \(-1\). The critical value \(L(f,2)\) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com