FB(S³)R: The Wave Function of the Void. Dual-Sector Quantum Cosmology on S³, Emergent Time, Entropy, and Spacetime Geometry

Version 2.0 presents a substantially expanded and mathematically refined formulation of the Wave Function of the Void framework, introducing a dual-sector structure of quantum cosmology in which the total Hilbert space of the Universe factorises into coupled matter and vacuum components. The model is constructed on a compact, simply connected spatial manifold S³, enabling global coherence of the quantum state and eliminating singular boundary conditions. The central proposal is that the quantum state of the Universe includes an explicit vacuum sector described by its own wave function Ψᵥ. In contrast to conventional approaches where vacuum is treated as a passive background, the present work treats the void as a dynamical partner of observable matter. The resulting extended Wheeler–DeWitt–Void (WDW-V) equation governs the joint evolution of geometry, matter, and vacuum degrees of freedom within a unified probabilistic framework. The mathematical construction demonstrates that key cosmological properties can arise from entanglement dynamics between the two sectors: • Entropy is derived as von Neumann entanglement entropy of the reduced density matrix of the matter sector, producing monotonic entropy growth without requiring special low-entropy initial conditions. • Time appears as an emergent relational parameter obtained via WKB factorisation and the Page–Wootters mechanism, with the arrow of time determined by the phase differential between matter and vacuum spectral components. • Spacetime geometry is linked to the amplitude structure of the global wave function, allowing curvature and effective Friedmann dynamics to be expressed as expectation values of quantum operators. • The spatial topology S³ follows from conditions of normalisability, compactness, and simple connectivity of the universal wave function, consistent with the Poincaré–Perelman theorem. • The vacuum sector contributes an effective dark-energy term arising from the zero-point energy of the void Hamiltonian, providing a natural scale for the cosmological constant without introducing additional fields or arbitrary parameters. Compared to earlier conceptual versions of the model, Version 2.0 introduces explicit operator structure, interaction Hamiltonians, and a formal derivation of entanglement-driven entropy growth. The framework integrates elements of canonical quantum gravity, Wheeler–DeWitt quantisation, and relational time approaches into a single internally consistent structure. The present work does not claim phenomenological completeness but establishes a coherent mathematical basis for further investigation of dual-sector quantum cosmology, including potential observational signatures in the CMB power spectrum, dark-energy equation of state, and quantum corrections to cosmological expansion. The Wave Function of the Void framework suggests that vacuum degrees of freedom may play an active structural role in the global quantum state of the Universe. Version 2.0 therefore represents a step toward a more complete wave-based description of cosmology in which entropy growth, temporal asymmetry, and effective spacetime geometry emerge from entanglement structure rather than external postulates. Keywords:quantum cosmology, Wheeler–DeWitt equation, entanglement entropy, emergent time, Page–Wootters mechanism, S³ topology, vacuum energy, dark energy, quantum gravity, cosmological topology