Signed-Volume Two-Sector Dark Energy: A Phenomenological Model with Closed-Form Equation of State

We present a phenomenological model of dynamical dark energy in which the dark energy density is generated by a cross-term between two sectors of a single-volume minisuperspace. The two sector volumes satisfy |V_A| + |V_B| = V_0 and exchange volume through a logistic law with dimensionless rate γ. The dark energy density is ρDE = A x(1 - x) - B, where x(1 - x) is the leading term in the derivative expansion of any exchange-symmetric two-sector potential. The phantom crossing arises from the maximum of the cross-term at x = 1/2, giving it a geometric rather than fitted origin. The equation of state admits a closed-form expression whose crossing occurs at z_c = (x_0/(1 - x_0))1/γ - 1, independent of the residual ratio β = B/A. Phantom crossing is achieved with a single dynamical radion field whose effective geometric fluid crosses the phantom divide. Two independent Markov Chain Monte Carlo analyses are presented. A baseline using 13 DESI DR2 BAO measurements, Planck 2018 distance priors, and Pantheon+ supernovae gives z_c = 0.72 [0.38, 1.64]. A cross-check using the DESI DR2 CPL summary alone gives z_c = 0.26 [0.20, 0.39]. The model makes two falsifiable predictions together with one theoretical feature: the algebraic crossing redshift z_c, a 1.25% suppression of σ_8 relative to Λ CDM at matched Ω_m, and a finite cosmic future with turnaround at a_max > 65 (68% CL). --- DATA AVAILABILITY & RELATION TO OTHER VERSIONS --- The core observational data, MCMC chains, and analysis code used in this study are identical to those presented in the companion PRD-focused submission ("Algebraic Phantom Crossing in a Signed-Volume Effective Fluid"). These datasets are openly available at: https://doi.org/10.5281/zenodo.23662603 This preprint (Springer format) includes an extended theoretical framework, notably the RS1 dimensional reduction and Goldberger-Wise stabilization (Appendix A), as well as supplementary analyses such as the ε-regularization scheme, growth factor fσ_8, and the age of the universe. The PRD-focused preprint is available at: https://doi.org/10.5281/zenodo.23664302