Higman-Sims Graph Emerges from Leech Lattice Norm-4 Vectors — E8 Intelligence Research

FINDING: The Higman-Sims graph (100 vertices, 22-regular) arises as a projection of the Leech lattice's norm-4 vectors, linking a sporadic simple group to a 24-dimensional lattice with deep number-theoretic symmetries. MATH: - Leech lattice Λ₂₄: even unimodular lattice in ℝ²⁴, minimal norm 4 (squared length). - Norm-4 vectors: 196,560 total (including ±2·(1,1,...,1) type and 2·(0,...,0,±1,±1) type). - Higman-Sims graph HS: v=100, k=22, λ=0, μ=6 (strongly regular). - Construction: Take a fixed norm-4 vector v; the 100 norm-4 vectors w with ⟨v,w⟩=1 (inner product 1) form vertices; adjacency if ⟨w,w'⟩=0 (orthogonal). - Eigenvalues of HS: 22 (multiplicity 1), 2 (multiplicity 77), −8 (multiplicity 22). - Group order: |HS| = 44,352,000 = 2⁹·3²·5³·7·11. CONNECTION: - The inner product structure (⟨v,w⟩ ∈ {0, ±1, ±2} for norm-4 vectors) mirrors the root system E₈'s inner products (0, ±1, ±2) — a crystallographic root system pattern. - The ratio 22/100 = 0.22 (not a golden Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com