The 2-Sphere of Imaginary Units: Quaternions, Pauli Matrices, and Rolling SU(2) — E8 Intelligence Research

FINDING: Quaternions, Pauli matrices, and SU(2) form a unified algebraic structure where the square root of −1 is not unique but a 2-sphere of solutions, and Pfaffian systems on SU(2) encode rolling distributions with constant curvature. MATH: - Quaternion square roots of −1: any unit pure quaternion \( q = ai + bj + ck \), \( a^2+b^2+c^2=1 \), satisfies \( q^2 = -1 \). This is the 2-sphere \( S^2 ⊂ H \). - Pauli matrices \( σ_1, σ_2, σ_3 \) satisfy \( σ_i σ_j = δᵢⱼ I + i εᵢⱼₖ σ_k \), generating \( su(2) \). The map \( i ↦ iσ_1, j ↦ iσ_2, k ↦ iσ_3 \) gives an isomorphism \( H R ⊕ su(2) \). - Pfaffian systems: For a \((2,3,5)\)-distribution, the Cartan–Nurowski conformal structure is encoded by a Pfaffian form \( θ \) with \( dθ \) restricted to the distribution. The SU(2) Pfaffian system arises from rolling two surfaces of Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com