Dynamic Phase-Transition Cosmological Model Based on Path-Integral Definition of Xuan-Liang: Complete Theory and Observational Tests

This paper presents a complete cosmological model of Xuan-Liang, achieving a unified description of dark matter and dark energy. Starting from the classical concept of "work" and extending it to "spatial accumulation of power," we first-principally derive the expression of Xuan-Liang: $X = \frac{1}{3} m v^3$. By fieldifying the concept of Xuan-Liang, we construct a dynamic phase-transition model with exact symmetry, whose evolution equation is: \[ \left( \frac{\rho_X}{\rho_t} \right)^{\Delta/2} + \left( \frac{\rho_X}{\rho_t} \right)^{-\Delta/2} = \left( \frac{a}{a_t} \right)^{-3\Delta/2} + \left( \frac{a}{a_t} \right)^{3\Delta/2} \] This equation reveals a profound duality between the cosmic scale factor and the density of the Xuan-Liang field. We present the complete Friedmann equations including ordinary matter, radiation, and the Xuan-Liang field, and numerically solve the cosmic evolution history. Using the latest observational data (Planck 2018, Pantheon+ supernovae, BAO), we constrain the model parameters, showing high compatibility with observations, with a $\chi^2$ improvement of about 8\% compared to the $\Lambda$CDM model. The model predicts a specific equation-of-state evolution $w(z)$, a precise phase-transition redshift $z_t = 0.65 \pm 0.08$, and a weak early dark energy component ($\Omega_{Xe} \sim 10^{-5}$). Theoretical analysis suggests that the path-integral origin of Xuan-Liang may reflect the topological structure of spacetime, providing new perspectives for quantum gravity.