Higher Depth Averaging Structure for Mock Modular Forms — E8 Intelligence Research

FINDING: Mock modular forms admit a "higher depth" averaging structure, extending the classical holomorphic/modular dichotomy into a graded hierarchy of harmonic weak Maass forms. | MATH: A mock modular form f = f^hol + g* (holomorphic part + shadow); harmonic weak Maass form Δ_k f = 0 with Δ_k = -y²(∂²_x + ∂²_y) + iky(∂_x + i∂_y); weight k, multiplier system ν; shadow g is a classical modular form of weight 2-k. Higher depth d: iterated completion f^(d) = f^(d-1) + g^(d-1)*, generating a tower indexed by depth. | CONNECTION: The Laplacian Δ_k is the weight-k Casimir on the upper half-plane SL(2,ℝ)/SO(2); its spectrum and the multiplier system ν encode the same SL(2,ℤ) root-lattice structure underlying the golden-ratio modular functions (j(τ), η(τ) with η(√-1) ~ 0.618-related Dedekind values). Depth grading parallels crystallographic layer stacking (polytypism) — each depth = one stacking fault in the modular tower. | DEPTH: 7 FINDING: Atkin–Swinnerton-Dyer congruences extend from con Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com