We present a mathematically rigorous dual-sector quantum cosmology in which the Hilbert space of the Universe factorises as H = Hm ⊗ Hv, with Hv carrying a Wave Function of the Void Ψv. Starting from the full ADM action on a compact, orientable three-manifold, we derive the void-extended Wheeler–DeWitt (WDW-V) constraint. The void sector is modelled as a standard (non-phantom) harmonic-oscillator field with positive kinetic term and positive-definite zero-point energy. Entanglement between the matter and void sectors, generated by a minimal bilinear coupling Hˆ int ∝ a3 φˆ2m φˆ2v , drives a monotonically increasing von Neumann entropy Sent(a), providing a first-principles derivation of the second law of thermody- namics from wave-function entanglement alone, with no low-entropy boundary condi- tion imposed by hand. A WKB analysis of the gravitational sector yields an emergent time parameter t via the Page–Wootters mechanism, and the arrow of time follows from the asymmetric spectra of the two sectors. We prove that normalisability of Ψv together with simple connectivity uniquely selects the three-sphere S3 as the spa- tial topology via the Poincar ́e–Perelman theorem, eliminating the initial-singularity problem. Crucially, the observed accelerated recession of galaxies is re-interpreted ge- ometrically: since the spatial volume of S3 scales as V (a) = 2π2a3, a linear change in the curvature radius produces a cubic change in volume and a proportional stretch- ing of geodesic distances d = aχ. No dark-energy field, phantom sector, or negative pressure is required. Observational signatures include characteristic oscillations in the CMB power spectrum and a purely geometric redshift–distance relation. Keywords: quantum cosmology; Wheeler–DeWitt equation; entanglement entropy; arrow of time; Page–Wootters mechanism; three-sphere topology; geometric expan- sion.
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