Weyl Groups and Root Systems: The Symmetry Behind Lie Algebras and the Windmill Parity Invariant — E8 Intelligence Research

FINDING: Root systems and Weyl groups provide the symmetry backbone for classifying simple Lie algebras and their associated lattices, with the windmill problem's parity invariant emerging as a combinatorial shadow of these reflection groups. | MATH: Root system Φ ⊂ E (Euclidean space), reflection s_α(β) = β − 2⟨β,α⟩/⟨α,α⟩ α; Cartan integers A_ij = 2⟨α_i,α_j⟩/⟨α_i,α_i⟩ ∈ ℤ; Weyl group W = ⟨s_α : α ∈ Φ⟩, a finite Coxeter group with presentation (s_i s_j)^(m_ij) = 1, m_ij ∈ {2,3,4,6}; parity invariant in windmill problem corresponds to sign homomorphism W → {±1} (det(s_α) = −1), with the invariant living in ℤ/2ℤ — the same group that classifies the two orientations of each root reflection. | CONNECTION: Root systems are crystallographic — all Cartan integers are integers, forcing angle ratios to be cos(θ) ∈ {0, ±1/2, ±√2/2, ±√3/2} → angles {90°, 60°, 45°, 30°} and their supplements. This yields the golden-ratio-adjacent values: for A_2 (hexagonal lattice), the ratio of long to short root Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com