Unified Cosmological History in the SV0 Model: From a Non-Singular Geometric Bounce to Late-Time Dark Energy

This manuscript is the ninth paper in a series establishing the Signed-Volume Effective-Fluid (SV0) model for dark energy. In Papers I-VIII, we introduced the SV0 model, derived its microphysics (non-minimal coupling and K-essence), formulated its covariant structure, analyzed its perturbations, established its dynamical stability, forecasted its observational signatures, and provided a string-theoretic embedding. In this ninth work, we unify the early-universe bounce cosmology with the late-time SV0 dark energy model. We demonstrate that the L³-sphere constraint V1 + V2 = V0 (previously introduced to resolve the Big Bang singularity) is mathematically equivalent to the SV0 constraint |V_A| + |V_B| = V0 under the identification V1 ≡ |V_A| and V2 ≡ |V_B|. By extending the SV0 exchange law with a symmetric Planck-scale regularization factor F(|V_A|,|V_B|), we show that the early universe undergoes a non-singular bounce, smoothly transitioning from contraction to expansion. Furthermore, we show that the same dynamical attractor that drives the phantom crossing at late times also resolves the trajectory consistency problem of the early bounce. This unified framework provides a single, continuous cosmological history: from a geometric Planck-scale bounce, through a radiation-dominated era, to the late-time dark energy attractor. Related Works:LSB (Geometric Bounce): https://github.com/uditha-sandaruwan/13-bouncePaper I: https://doi.org/10.5281/zenodo.24026844Paper II: https://doi.org/10.5281/zenodo.24032977Paper III: https://doi.org/10.5281/zenodo.24033838Paper IV: https://doi.org/10.5281/zenodo.24034927Paper V: https://doi.org/10.5281/zenodo.24035268Paper VI: https://doi.org/10.5281/zenodo.24035911Paper VII: https://doi.org/10.5281/zenodo.24038624Paper VIII: https://doi.org/10.5281/zenodo.24039486