SU(2) Symmetry: From Weak Force to Quantum Gravity Constraints — E8 Intelligence Research

FINDING: SU(2) representation theory underpins weak-force symmetry and quantum gravity constraints on higher-form symmetry breaking; adjoint representation yields the so(3) algebra with structure constants tied to the Levi-Civita symbol. | MATH: SU(2) ≅ {q ∈ ℍ : |q|=1} (unit quaternions); Lie algebra su(2): [J_i, J_j] = i ε_ijk J_k; irreducible representations labeled by spin j ∈ {0, ½, 1, …}, dimension 2j+1; adjoint rep (j=1) has structure constants ε_ijk; Casimir J² = j(j+1)𝟙; center ℤ₂ = {±𝟙}; Calabi–Yau compactification: higher-form symmetry breaking criterion tied to gauge group center and discrete torsion (arXiv:2503.19628v1). | CONNECTION: The quaternionic structure of SU(2) maps to the 4D hypercomplex basis {1, i, j, k} — the norm-1 condition gives the 3-sphere S³, whose Hopf fibration S¹→S³→S² encodes the golden-ratio-linked stereographic projection of the 600-cell (icosahedral symmetry, A₄ root system). The adjoint rep's ε_ijk are the structure constants of so(3) ≅ su(2)/ℤ₂, Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com