Xuan-Liang Unified Field Theory: From Multi-Velocity Xuan-Liang Construction to Cosmology and Astrophysical Tests

Modern physics rests on two pillars: general relativity and quantum field theory. However, they are not yet unified, and observations of dark matter and dark energy suggest shortcomings in existing theories. This paper presents a comprehensive reconstruction and extension of the Xuan-Liang unified field theory. Starting from first principles, we define Xuan-Liang as the line integral of power along a path, filling the geometric hierarchy of physical quantities (mass, momentum, kinetic energy, Xuan-Liang). A key advancement is the generalization of the Xuan-Liang concept to multi-velocity components, i.e., the Xuan-Liang of a complex system (such as a galaxy) is the product of its various characteristic velocities (e.g., rotation, revolution, bulk motion). This naturally leads to a modified Newtonian potential of the Yukawa form: Φ(r) = −GM r [1 + δ(1 − e−r/λ)], where the coupling strength δ and characteristic scale λ arise from multi-velocity coupling. Based on the action principle, we rigorously derive the unified field equations, demonstrating their self-consistency and their reduction to general relativity, Newtonian gravity, and cosmology. The theory’s explanatory power is demonstrated through applications: (i) it perfectly fits galaxy rotation curves from dwarf galaxies to the Milky Way and Andromeda, spanning a huge mass range, without requiring dark matter particles; (ii) it provides a dynamical dark energy model whose energy density smoothly transitions from matter-like behavior (w ≈ 0) at high densities to cosmological-constantlike behavior (w ≈ −1) at low densities, consistent with cosmic acceleration; (iii) it predicts testable modifications to black hole thermodynamics and strong-field gravity, including changes in black hole shadows and gravitational wave signals. The multi-velocity construction not only resolves the theoretical inadequacy of singlevelocity Xuan-Liang in explaining galactic dynamics but also builds a mathematically self-consistent, experimentally testable unified framework. Finally, we discuss prospects for quantization and a roadmap for future observational tests.