Root Systems as Reflection Lattices: Axioms and Rank-2 Angle Restrictions — E8 Intelligence Research

FINDING: The search results are a scattered set of educational videos and one lattice QCD paper; the only substantive mathematical content is the formal definition of root systems (A1, rank-2) as reflection-generated lattices, with no direct competition-problem solution or Vieta jumping connection provided. MATH: Root system axioms: Φ spans Euclidean space V; for all α ∈ Φ, reflection s_α(β) = β − 2(β·α)/(α·α) α ∈ Φ; 2(β·α)/(α·α) ∈ ℤ. For rank 2, angles between roots are restricted: 2cos²θ ∈ {0,1,2,3,4} → θ ∈ {90°, 60°, 45°, 30°} with length ratios √2, √3, 2. A1: {±α}, single reflection group ℤ₂. The lattice QCD paper (arXiv:0710.4339) gives heavy-quark masses via Fermilab action — no explicit constants beyond standard QCD scales. CONNECTION: Root systems are crystallographic reflection groups (Weyl groups). The rank-2 angles yield ratios: 60° → cos60°=0.5, 30° → cos30°=√3/2≈0.866, 45° → √2/2≈0.707. The golden ratio φ=1.618 does NOT appear in any classical root system (only in no Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com