FB(S³)R: A Hopf-Bundle Holonomy Representation of Spin-Singlet Bell Correlations

We present a conservative geometric representation of the spin-singlet Bell correlation E(a,b) = −a · b as the real part of a closed U(1) holonomy on the standard Hopf bundle S¹ → S³ → S², associated with normalized spinors and Bloch-sphere rays. The construction uses the standard Berry connection and a closed loop on the Bloch sphere enclosing solid angle Ω = 2(π − θ), where θ is the angle between the measurement settings. The framework does not introduce local hidden variables, does not modify Born probabilities, and does not alter the no-signaling structure of the singlet state. Within this setting, we establish a restricted rigidity result: in the Hopf-bundle lune construction, an integer Chern charge reproduces the singlet kernel for all settings if and only if q = ±1, with the conventional orientation corresponding to q = +1. We further record the associated rotational symmetry, the reduced-state no-signaling property, and the standard CHSH/Tsirelson bound (2√2). The result provides a self-contained representation theorem linking: the Pauli-algebra formulation, the Hopf fibration S¹ → S³ → S², and Berry holonomy, without modifying the operational content of quantum mechanics. Version prior to submission to Physical Review A