Interpolated Apéry Sequences Link ζ(3) to Weight-4 Modular L-Values — E8 Intelligence Research

Abstract

FINDING: Interpolated Apéry sequences linked to critical L-values of weight-4 modular forms extend the modularity of ζ(3) and reveal a hidden arithmetic-geometric bridge. MATH: - Apéry numbers for ζ(3): \( A_n = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}^2 \) - Zagier's interpolation: \( A_n(t) \) → critical L-value \( L(f, t) \) for a weight-4 modular form \( f \). - Key constant: \( \zeta(3) = 1.2020569... \) (Apéry's constant) expressed via modular L-values. CONNECTION: - The interpolation parameter \( t \) relates to modular curve cusps, echoing base-60 sexagesimal cycles in Babylonian astronomy (periodic modular parameters). - Weight-4 forms correspond to the root lattice \( D_4 \) (crystallographic symmetry of the 24-cell), linking to the golden ratio \( \varphi = 1.618... \) via the Coxeter number \( h=6 \) of \( D_4 \): \( 2\cos(\pi/h) = \sqrt{3} \approx 1.732 \), but deeper: \( \varphi \) appears in the modular j-invariant at CM points (e.g., \( j(i) = 1728 \), \ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com