Algebraic Phantom Crossing in a Signed-Volume Effective Fluid: Cosmological Signatures and Stability

We present a signed-volume two-sector effective-fluid model for dark energy in which the phantom divide is crossed algebraically. The dark energy density arises from a geometric cross-term between two sectors subject to the constraint |V_A|+|V_B|=V_0, yielding a closed-form equation of state. We do not claim a UV completion; rather, we adopt the effective-field-theory viewpoint that the late-time background and linear perturbations are fully specified by an equation of state w(z) and a rest-frame sound speed c_s^2. The crossing redshift is algebraically fixed at z_c=(x_0/(1-x_0))^(1/γ)-1, independent of the residual ratio β=B/A. Using DESI DR2 BAO, Planck 2018, and Pantheon+, we find a baseline posterior with median z_c=0.66 (16th-84th quantiles [0.38,1.63]; 95% CL [0.26,4.02]) and a cross-check anchored on the DESI CPL summary with z_c=0.26 (16th-84th quantiles [0.20,0.38]; 95% CL [0.13,1.07]). The residual β is bounded as β < 0.004 at 95% CL (with a median of ~2×10^-6), consistent with the small-residual limit β→0. The model predicts a percent-level deviation of σ_8 from ΛCDM whose sign would track z_c if resolved; current data yield Δσ_8/σ_8 = -0.40% ± 0.94%, leaving the sign unresolved with present sensitivity. A Bayesian evidence comparison (DESI DR2 BAO-only) yields ln B = -0.60 ± 0.18 against ΛCDM, which is inconclusive. We identify a ~1.8σ tension with the DESI CPL slope w_a, and argue it is a falsifiable prediction of the algebraic-crossing structure: the model fixes the triple (w_0, w_a, z_c) in terms of two parameters (x_0, γ), so the DESI-preferred triple is generically over-constrained by the algebraic-crossing structure. Future joint constraints will either confirm the predicted constraint surface or exclude the model.