FB(S³)R: Compact Simply-Connected Cosmology and the Geometric Origin of Quantum Nonlocality. Bell Inequality Violations from S³ Topology and Fibonacci Geodesic Stratification

This work develops a geometric framework in which Bell inequality violations arise from the global topology and spectral structure of the Universe. The spatial geometry is modelled as a compact, simply-connected three-manifold uniquely identified with the three-sphere S³, derived from normalisability, orientability, and simple-connectivity conditions on the cosmological wave function. Within this setting, the Hopf fibration S³ → S² provides the natural geometric structure underlying qubit state space: the total space S³ ≅ SU (2) represents the full quantum state, while the Bloch sphere S² corresponds to the projective space of measurement outcomes. The Berry connection on the Hopf bundle generates the quantum correlation law: E (a, b) = − cos θₐb establishing (Theorem 3. 3: Berry Phase → Bell Correlations) that Bell correlations arise as a geometric holonomy of the U (1) bundle rather than as an independent postulate of quantum theory. The associated Clifford algebra of SU (2) yields the Tsirelson bound as a geometric norm theorem: Sₘax = 2√2 demonstrating that the maximal strength of quantum correlations reflects intrinsic curvature properties of state space (Hopf–Clifford geometry). A spectral decomposition of L² (S³) reveals a minimally redundant self-similar indexing governed by Fibonacci recursion. Using the Zeckendorf representation and Perron–Frobenius theory, the Fibonacci stratification is shown to be uniquely compatible with complete spectral coverage of the Laplacian on S³ (Theorem 4. 5: Uniqueness of Fibonacci Stratification). In this framework, Fibonacci structure appears as the minimal gapless additive hierarchy of spectral domains rather than as a fundamental physical constant. Non-factorisability of the cosmological wave function is strengthened using the Paley–Wiener theorem together with the unique continuation principle for elliptic operators (Theorem 6. 1: Global Non-Factorisability). Exact subsystem independence is incompatible with compact spectral support on S³: Ψ (x, y) ≠ ψ (x) ⊗ χ (y) Entanglement therefore appears as a structural property of global eigenfunctions rather than an additional physical assumption. The framework naturally connects to key principles of quantum information theory (Section 7: Quantum Information Bridge): • compatibility with the no-signalling condition• preservation of operator locality• emergence of monogamy of entanglement (CKW inequality) • geometric interpretation of classical vs quantum correlation bounds Observable implications include discrete curvature spectra in the CMB, topology-dependent constraints on matched-circle searches, and potential spectral signatures associated with hierarchical mode structure. Compared with earlier formulations of the FBS³R model, this Version reformulates the approach in terms of differential geometry, spectral theory, and operator algebra, clarifying that the appearance of Fibonacci structure reflects minimal self-similar spectral indexing of the Laplacian spectrum. The work proposes that quantum nonlocality may be interpreted as a geometric property of global state space: correlations exceeding classical bounds arise naturally when the full SU (2) structure of S³ is taken into account rather than its S² projection. The resulting framework is presented as an open geometric programme exploring possible connections between topology, quantum correlations, and cosmological structure.