FB(S³)R: Bell Correlations as Abelianised SU(2) Holonomy. A Self-Contained Formulation and Proof with Yang-Mills Bridge

We provide a self-contained formulation and proof of the following geometric theo- rem: the quantum correlation function E(a, b) = − cos θab for a maximally entangled two-qubit Bell state is the real part of the U(1) Berry􏰀Pancharatnam holonomy of the Hopf principal bundle π: S3 → S2, where S3 ≃ SU(2) carries the non-abelian geometry of the qubit state space and S2 ≃ CP1 is the projective measurement space. The abelianisation is made precise as a reduction of structure group: the Hopf projection reduces the SU(2) principal bundle over the full state sphere to a U(1) principal bundle over the Bloch sphere, and the Bell correlation is the holonomy of the induced U(1) connection. We prove gauge invariance of this holonomy, indepen- dence from the choice of representative path (dependence only on solid angle), and express the result via the Fubini􏰀Study metric as E(a,b) = −cos(2dFS). The violation of the CHSH inequality is derived as a geometric bound on the sum of four solid angles, with the Tsirelson value 2 2 arising directly from the non-abelian commutator σ · a, σ · b = 2i σ · (a × b). We conclude by identifying a precise structural correspondence (not isomor- phism) between this Berry holonomy and the non-abelian Wilson loop of Yang􏰀Mills gauge theory on S3 × R.