Microphysical Origin of the Algebraic Phantom Crossing: A Two-Field Model with Non-Minimal Coupling to Gravity (Preprint)

In a recent companion paper [1], we proposed a signed-volume effective-fluid model (SV0) in which the phantom divide is crossed at a structurally guaranteed redshift. However, the phenomenological exchange law governing the model was introduced at the level of the background evolution. In this work, we provide a rigorous microphysical derivation of this law. We consider a constrained two-field configuration with a non-minimal coupling to the Ricci scalar, ξ R (χ² - ψ²). We show that during the slow-roll regime, the effective mass induced by the non-minimal coupling produces the exact exchange law dx/dt = γ H x (1-x), with the coupling parameter naturally identified as γ = 32ξ. Furthermore, we demonstrate that the exact background evolution yields a time-dependent effective coupling γ̃(t) which naturally suppresses the volume exchange during matter domination, ensuring consistency with standard cosmology. Finally, we show that the sign of ξ and the soft mass splitting between the fields dictate the branch selection (x₀ > 1/2), naturally resolving the direction of the phantom crossing. This work establishes the SV0 model as a well-motivated, first-principles effective theory of dark energy. References:[1] M. U. Sandaruwan, "Structurally Guaranteed Phantom Crossing in a Signed-Volume Effective Fluid: Cosmological Signatures and Bayesian Evidence from DESI DR2, Pantheon+, and Planck 2018" (2026). Zenodo: https://doi.org/10.5281/zenodo.24026844 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.24032977