A K-essence Realization of the SV0 Dark Energy Model: An Alternative UV Completion without Non-Minimal Coupling

The Signed-Volume (SV0) dark energy model provides a structurally guaranteed phantom crossing. In Paper II of this series, the phenomenological exchange law was derived from a scalar-tensor theory with non-minimal coupling to the Ricci scalar. In this paper, we present an alternative first-principles derivation using a purely K-essence framework. By introducing a specific kinetic coupling Z(x)X, where X = -1/2 (∂x)², we show that the required H² scaling emerges naturally from the tracker condition X ∼ H². We demonstrate that the logistic exchange law dx/dt = γ H x(1-x) arises as a stable dynamical attractor in the slow-roll regime. Furthermore, the rest-frame sound speed c_s² is derived rather than assumed, and we explicitly show that c_s² > 0 across the phantom divide, ensuring the absence of gradient instabilities. This work establishes that the SV0 background is theoretically robust and can be realized through distinct microphysical mechanisms. 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.24038382 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[4] M. U. Sandaruwan, "Gauge-Invariant Cosmological Perturbations in the SV0 Dark Energy Model..." (2026). Zenodo: https://doi.org/10.5281/zenodo.24034927[5] M. U. Sandaruwan, "Dynamical Systems Analysis of the Signed-Volume Dark Energy Model..." (2026). Zenodo: https://doi.org/10.5281/zenodo.24035268