Dynamic Phase-Transition Cosmological Model Based on Path-Integral Definition of Xuan-Liang

This paper proposes and develops a novel physical quantity called "Xuan-Liang" (mystery quantity) aimed at providing a new theoretical framework for the unified description of dark matter and dark energy. Starting from the classical concept of "work", we extend it to "accumulation of power along a spatial path" and derive the expression for Xuan-Liang from first principles: X = 13 mv3, with dimension [M][L]3[T]−3. We treat the Xuan-Liang field as a continuum description of this quantity and argue that its equation-of-state parameter w should vary smoothly with energy density ρX. Based on the hyperbolic tangent function, we construct a specific dynamic phase-transition model w(ρX), allowing the Xuan-Liang field to behave as dark matter (w ≈ 0) in the early high-density universe and as dark energy (w ≈ −1) in the late low-density universe. We rigorously solve the Friedmann equations within this model, providing an analytic implicit solution for energy density evolution and performing numerical simulations. Results show that this model naturally reproduces the cosmic evolution from matter dominance to dark energy dominance, compatible with current observational data (e.g., Planck CMB data). The core prediction of this model is a smoothly evolving equation of state w(z) that can be precisely tested by next-generation cosmological surveys (e.g., Euclid, LSST). Furthermore, the path-integral origin of Xuan-Liang suggests new connections to topological properties and quantum gravity theories.