A₁ Weyl Group S₂ as Algebraic Skeleton for Rank-2 Topological Phases — E8 Intelligence Research

FINDING: A1 root system is the minimal rank-1 case whose Weyl group is S₂ (order 2), and its representation theory (Clebsch-Gordan / 3j-symbols) is the algebraic skeleton underlying rank-2 topological phase classifications. MATH: - A₁ root system: roots {α, −α}, Cartan matrix [2], Weyl group W(A₁) ≅ S₂ = {1, s_α}, |W| = 2. - Reflection: s_α(λ) = λ − 2(λ,α)/(α,α) · α; for A₁, (α,α)=2 ⇒ s_α(λ) = λ − (λ,α)α. - Fundamental weight ω = α/2; ρ = ω = α/2. - SU(2) irreps: dim V_j = 2j+1; CG decomposition V_{j₁}⊗V_{j₂} = ⊕_{j=|j₁−j₂|}^{j₁+j₂} V_j. - 3j-symbol selection rule: (j₁ j₂ j₃; m₁ m₂ m₃) ≠ 0 iff m₁+m₂+m₃=0 and |j₁−j₂| ≤ j₃ ≤ j₁+j₂ (triangle inequality = A₁ fusion rule). - Wigner–Eckart / 3j orthogonality: Σ_{m₁m₂} (j₁ j₂ j₃; m₁ m₂ m₃)(j₁ j₂ j₃'; m₁ m₂ m₃') = δ_{j₃j₃'} δ_{m₃m₃'} /(2j₃+1). - Quantum dimension of SU(2)_k: d_j = sin((2j+1)π/(k+2)) / sin(π/(k+2)); for k=1, d_{1/2} = φ = (1+√5)/2 = 1.618… (Fibonacci / Ising anyon limit). - Rank-2 root systems A₁×A₁, A₂, B₂≅C₂, G₂ — Weyl group Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com