From Born Projection to Bell Nonseparability: A Hopf-Bundle Formulation of Retained Complex Topology in Deductive Entropy

Abstract

This paper is the second in a Deductive Entropy (DE) sequence examining whether established quantum geometry provides a natural cross-scale description of persistence, projection, and complexity. It follows Hidden Complex Topology, Born Quadratic Projection, and Dark Energy, which proposed that Born projection extracts observable probability without necessarily exhausting the complex geometric structure of the quantum state. The present paper develops that proposal using established quantum mechanics and quantum geometry, without introducing a new action, force, or microscopic dynamics. For a single qubit, retained structure is represented through the first Hopf fibration, its U(1) connection, curvature, geometric-phase holonomy, and characteristic class. For two qubits, the construction extends to the second quaternionic Hopf fibration, concurrence and tangle, the non-Abelian SU(2) connection, Wilson-loop holonomy, and second characteristic class. Clifford parallels provide a geometric representation of nonintersection with persistent global linkage, while concurrence supplies the graded measure of entanglement. The central DE interpretation is: Entanglement links what geometry separates, because the underlying quantum topology never factorised. Bell-inequality violations are therefore interpreted as geometric separation without topological separation. Bell factorisation fails through the nonfactorisability of the joint quantum state and its canonical bundle geometry, while parameter independence, Born probabilities, standard quantum correlations, and no-signalling remain unchanged. The formulation is deliberately parameter-free at this level. Rather than constructing a phenomenological scalar topology functional with adjustable coefficients, it uses canonical geometric quantities already present in quantum theory. The proposal is constrained by local-unitary covariance and must reproduce standard two-qubit concurrence, the singlet correlation E(a,b)=−a⋅b, and the Tsirelson bound 22. The paper positions DE not as a replacement for quantum mechanics, but as an extension of the implications of established quantum geometry into the study of persistence and complexity. It also establishes the geometric foundation for a subsequent paper examining consciousness, recursive memory, and the biological implications of the U(1) fibre.