Covariant Formulation of the Signed-Volume Effective Fluid: Action, Thermodynamics, and Exchange Currents

This manuscript is the third paper in a series establishing the Signed-Volume Effective-Fluid (SV0) model for dark energy. In Paper I, we introduced the SV0 model and demonstrated a structurally guaranteed phantom crossing, fitting DESI DR2, Pantheon+, and Planck 2018 data. In Paper II, we derived the phenomenological exchange law from a two-field scalar theory with non-minimal coupling to gravity. In this third work, we provide the full 4D covariant completion of the framework. We construct a manifestly covariant action using the expansion scalar Θ = ∇_μ u^μ and derive the covariant exchange current Q^ν between the two sectors. We demonstrate that the background FLRW equations naturally emerge from this action without invoking a formal Lagrange multiplier. Furthermore, we analyze the thermodynamics of the exchange, showing that the background evolution is adiabatic. Finally, we perform a conformal transformation to the Einstein frame and show that while the non-minimal coupling modifies the effective potential and kinetic terms, the structural phantom crossing remains robust. This work solidifies the SV0 model as a mathematically rigorous, covariant field theory of dark energy. Related Works:Paper I (Preprint): https://doi.org/10.5281/zenodo.24026844Paper II (Preprint): https://doi.org/10.5281/zenodo.24032977