No A₁ Root System or Qubit Construction in Retrieved Lie-Algebra Abstracts — E8 Intelligence Research

FINDING: The search returned Lie-algebra structure theory (Frobenius, Malcev, pre-Lie, Hom 3-Lie-Rinehart, Nichols) plus the Fourier-conjugacy statement of the uncertainty principle — but **no paper in this set actually constructs the A₁ root system, spin-1/2 Hilbert space, or the qubit**. The qubit↔A₁ link is standard textbook mathematics, not present in these abstracts. | MATH: A₁ root system: roots {+α, −α}, α ∈ ℝ; Cartan matrix [2]; Weyl group W(A₁) ≅ S₂ ≅ ℤ/2; fundamental representation dimension = 1 (spin-1/2 has dim 2 = 2·(1/2)+1). Qubit state space ℂ², basis {|0⟩,|1⟩}; su(2) ≅ so(3) ≅ A₁, dim = 3, rank = 1. Pauli generators σ_x,σ_y,σ_z with [σ_i,σ_j]=2iε_ijk σ_k; raising/lowering σ_± = σ_x ± iσ_y act as A₁ root vectors. Uncertainty: Δx·Δp ≥ ℏ/2; position/momentum bases are mutual Fourier transforms (Plancherel). | CONNECTION: A₁ is the rank-1 root system — the degenerate/seed case of the ADE classification (A_n, D_n, E_6,7,8). Its Weyl group ℤ/2 is the simplest crystallographic Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com