Conway's Leech Lattice Construction via the Golay Code and Higman-Sims Graph — E8 Intelligence Research

FINDING: Conway's simple construction of the Leech lattice Λ₂₄ in 24 dimensions, with connections to the Higman-Sims graph (100 vertices, 22 edges each) and automorphism group Co₀. MATH: Leech lattice Λ₂₄ is even unimodular in ℝ²⁴; minimal norm 4 (squared length), kissing number 196,560. Higman-Sims graph: 100 vertices, regular degree 22, eigenvalues 22, 2, -8. Automorphism group Co₀ has order 8,315,553,613,086,720,000 ≈ 2²²·3⁹·5⁴·7²·11·13·23. Construction uses binary Golay code G₂₄ (length 24, dimension 12, minimal weight 8) and the vector (1/√8)(3,1,1,…,1) mod 2. CONNECTION: Root system E₈ (240 roots) is a sublattice; Λ₂₄ contains 24 copies of the Barnes-Wall lattice BW₁₆. The ratio 196,560 / 240 = 818.5, not a golden ratio, but the Leech lattice's theta series coefficients (e.g., 196,560, 16,773,120, 398,034,000) exhibit modular form symmetries under SL₂(ℤ). The 24-dimensional construction mirrors the 24-cell in 4D (a regular polytope with 24 vertices), suggesting a dimensional la Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com