Root Systems and Weyl Groups: A Number-Theoretic Counterpoint — E8 Intelligence Research

FINDING: Root systems and Weyl groups form the combinatorial skeleton of Lie theory, with Mahler's Diophantine work providing a number-theoretic counterpoint — but no direct bridge between them is evidenced in these sources. | MATH: Root system axioms: Φ ⊂ E (Euclidean space), reflections s_α(β) = β − 2⟨β,α⟩/⟨α,α⟩ α; Cartan integers n_{αβ} = 2⟨α,β⟩/⟨α,β⟩ ∈ ℤ; Weyl group W = ⟨s_α : α ∈ Φ⟩, finite reflection group; Mahler: Thue–Mahler equations, S-unit equations, S-integral points — quantitative bounds on solutions (e.g., number of solutions to |F(x,y)| = p_1^{z_1}…p_s^{z_s}). | CONNECTION: Root systems are crystallographic reflection groups — their Cartan matrices have entries in {0, ±1, ±2, ±3}, and the Weyl group orders for simple types: A_n (n+1)!, B_n/C_n 2^n n!, D_n 2^{n−1} n!, E_6 51840, E_7 2903040, E_8 696729600. These are finite reflection groups with Coxeter–Dynkin diagrams — no direct ratio (0.382, 0.618, etc.) appears in the cited material, but the root lattice structures (e Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com