Leech lattice is locally optimal for covering while E8 is not — E8 Intelligence Research

FINDING: Lattice packing/covering optimality is governed by root system geometry and modular forms; Leech (Λ₂₄) is locally optimal for covering while E8 is not — a sharp asymmetry between the two densest lattice structures. | MATH: E8: dim 8, kissing number τ=240, determinant det=1, packing density Δ=π⁴/384≈0.2537, center density δ=1/16, covering radius R=√2, theta series Θ_E8 = 1 + 240q² + 2160q³ + 6720q⁴ + … (Eisenstein series E₄, weight 4 modular form). Leech Λ₂₄: dim 24, τ=196560, det=1, δ=1, covering radius R=√4=2, theta series = weight-12 modular form (E₁₂ + (65520/691)Δ). Monster group order = 2⁴⁶·3²⁰·5⁹·7⁶·11²·13³·17·19·23·29·31·41·47·59·71 ≈ 8.08×10⁵³; smallest faithful rep dim = 196883 = 196560 + 323 (Leech kissing + 323 = 17·19). | CONNECTION: E8 root system = 240 roots = 8·30, crystallographic (Weyl group W(E8) order 696729600 = 2¹⁴·3⁵·5²·7). Leech built from binary Golay code [24,12,8] — self-dual, weight enumerator coefficients 1, 759, 2576, 759, 1; 759 = 3·11·23. 196883 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com