Mock Modular Forms, Z-Invariants, and Critical L-Values via Interpolated Sequences — E8 Intelligence Research

FINDING: Mock modular forms linked to higher-depth averaging, Z-invariants, and critical L-values of modular forms via interpolated Apéry-like sequences. MATH: - Mock modular forms: non-holomorphic modular forms with shadow; higher-depth averaging generalizes Eichler–Zagier theory. - Interpolated sequences: Zagier's six sporadic sequences for ζ(3) yield modular forms of weight 4; critical L-values appear as evaluations. - Fourier coefficients over number fields: sign changes studied via perfectoid spaces. - Key constants: no explicit ratios (0.382, 0.618, etc.) reported; focus on L-values and modular symmetries. CONNECTION: - Modular forms encode SL(2,Z) symmetry; mock modular forms extend this to non-holomorphic sectors, linking to root systems and lattice theta functions (e.g., affine Lie algebras). - Critical L-values relate to periods and special values, echoing base-60 harmonic ratios in Babylonian astronomy (via modular parameter τ). - No direct geometric ratios Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com