Affine Weyl Subgroups, Limit Roots, and Harmonic Ratios in Rank-Two Systems — E8 Intelligence Research

FINDING: Affine Weyl group reflection subgroups and their root distributions connect Coxeter theory to limit-root geometry, with potential harmonic ratios emerging from rank-2 affine systems. | MATH: Affine Weyl groups \( {W} \) act on root systems \( Φ \) with dual Coxeter number \( h^ \). Critical level shift \( k^ = -h^ \) in affine Lie algebras. Reflection subgroup \( W' ⊂ W \) induces root subsystem \( Φ' ⊂ Φ \); limit roots \( limn→∞ α_n/|α_n| \) from infinite reflection sequences. For affine \( Ã_1 \): simple roots \( α_1, α_2 \) with Cartan matrix \( {pmatrix} 2 & -2 \\ -2 & 2 {pmatrix} \), dual Coxeter number \( h^ = 2 \). Golden ratio appears in \( Ã_2 \) via \( cos(π/h)/cos(π/h^) \) ratios in Kac-Moody Weyl character formulas. | CONNECTION: Direct link to crystallographic root systems (types A, B, C, D, E, F, G) — these are the same lattices under Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com