Xuan-Liang Theory: Mathematical Construction from Basic Formula to Unified Equation

This paper starts from the basic definition of Xuan-Liang \( X = \frac{1}{3}mv^3 \) and, through rigorous mathematical-physical derivation, constructs the unified equation of Xuan-Liang theory. We first establish the geometric hierarchy theory of Xuan-Liang, generalize it to relativistic form, and then extend the Xuan-Liang concept to high-dimensional manifolds using differential geometry and topological methods. The core innovation lies in: starting from a simple algebraic expression, through a series of natural mathematical generalizations, ultimately deriving a unified equation with profound geometric implications: \( \int_{\mathcal{M}} \left[ \text{Tr}(\mathbb{X} \wedge \star \mathbb{X}) + \langle \Psi_X, \mathcal{D} \Psi_X \rangle + \alpha \mathbb{X} \wedge \mathcal{R} \right] = \chi(\mathcal{M}) \rho_X^{\text{min}} + \beta \int_{\partial\mathcal{M}} \Phi_{\text{obs}} \)This equation achieves a unified description of mass, motion, and spacetime geometry, providing a new theoretical framework for addressing problems of dark matter, dark energy, and quantum gravity. Specifically, we prove that under appropriate limits, the unified equation can naturally reduce to Einstein's field equations of general relativity and Newton's gravitational potential equation, which provides a solid foundation for the physical plausibility of the theory.