Gauge-Invariant Cosmological Perturbations in the SV0 Dark Energy Model: Resolution of the Phantom Crossing Instability (Preprint)

In the previous papers of this series, we established the SV0 model as a well-motivated, covariant dark energy framework that structurally guarantees a phantom crossing. The empirical analysis in Paper I relied on the Parameterized Post-Friedmann (PPF) approximation to handle the divergences at the phantom divide (w = -1). In this fourth paper, we provide the exact gauge-invariant perturbation theory for the SV0 fluid. Using the Bardeen formalism, we derive the scalar perturbation equations for the two-sector system. We demonstrate that the internal exchange current Q^ν acts as a dynamical regulator, eliminating the unphysical divergences associated with the adiabatic sound speed (c_a²) at the crossing. Furthermore, we show that the exact evolution of matter overdensities resolves the mild tension with the S₈ parameter, bringing the SV0 predictions into better agreement with weak lensing data without altering the background expansion history. Data Availability:The Python (SymPy) code used for the analytical verification of the mathematical derivations is publicly available on Zenodo at: https://doi.org/10.5281/zenodo.24034612 References:[1] M. U. Sandaruwan, "Structurally Guaranteed Phantom Crossing in a Signed-Volume Effective Fluid..." (2026). Zenodo: https://doi.org/10.5281/zenodo.24026844[2] M. U. Sandaruwan, "Microphysical Origin of the Algebraic Phantom Crossing..." (2026). Zenodo: https://doi.org/10.5281/zenodo.24032977[3] M. U. Sandaruwan, "Covariant Formulation of the Signed-Volume Effective Fluid..." (2026). Zenodo: https://doi.org/10.5281/zenodo.24033838