Symmetry and Geometry of Dissipative Qubit Dynamics — E8 Intelligence Research

FINDING: The Bloch sphere's symmetry group for spin-1/2 qubits is the binary polyhedral group (double cover of SO(3)), and its geometry is governed by the Lie algebra su(2) ≅ so(3), with dissipative dynamics extending to gl(3,R)⋉R³. | MATH: su(2) generators {σ_x, σ_y, σ_z} with commutation [σ_i, σ_j] = 2iε_ijk σ_k; Bloch vector r ∈ R³, |r| ≤ 1; pure states on S² (Riemann sphere); dissipative semigroup: gl(3,R) ⊕ R³ acting on Bloch ball; Wilson lines: holonomy W = P exp(i∮ A·dl) for Berry connection A = ⟨ψ|dψ⟩; Poincaré sphere = same SU(2)/U(1) coset structure as Bloch sphere. | CONNECTION: The binary tetrahedral group (order 24) and binary icosahedral group (order 120) are finite subgroups of SU(2) — the latter's orbit on S² gives the icosahedral symmetry with golden ratio φ = (1+√5)/2 ≈ 1.618, and its squared inverse φ⁻² ≈ 0.382 appears in the icosahedron's dihedral angles; the binary octahedral group (order 48) relates to the F4 root system (crystallographic, 24 roots + 24 weights). Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com