Z-Geometric Dynamics: A Unified Theory of Dark Matter and Dark Energy Based on the Cosmic Horizon

This paper presents a geometric framework for cosmology---Z-Geometric Dynamics (ZGD). The theory is based on three fundamental axioms: (1) finite speed of light, (2) finite age of the Universe, and (3) the cosmological principle. From these axioms, the observable cosmic horizon distance R_p (particle horizon radius) and the background characteristic acceleration a_0 = c^2 / R_p are derived. In the strong version (ZGD-Strong), general relativity is adopted as the theoretical framework for spacetime geometry, and, based on the physical picture of causal delay and characteristic gravitational potential velocity, the effective gravitational acceleration expression is derived: g_eff(r) = GM/r^2 + (1/r) * (GMc^2/R_p)^{1/2} a form uniquely determined by the fundamental constants {G, c, R_p, M, r} and containing no free parameters. This leads to the galactic rotation velocity formula v = (G M c^2 / R_p)^{1/4}, which naturally yields the Tully-Fisher relation M ∝ v^4. The validation of the theory follows a hierarchically progressive logic. At the merging galaxy cluster scale, β-model numerical validation, pure-Newtonian control tests, and physical-origin verification for the Bullet Cluster establish the viability of the ZGD mechanism---the position of the lensing peak is rigidly determined by the subcluster's own mass, with the main cluster gas contribution being negligible. The resulting minimalist physical picture and the core inference that the point-mass approximation is the supremum of the geometric correction term strength provide the physical foundation for the galactic-scale analysis. At the galactic scale, initial validation with nearby galaxies (Milky Way, M31, M33) yields results broadly consistent with observations; SPARC large-sample validation confirms a perfect match in shape (v ∝ M^{1/4}) between theoretical predictions and observations, though the overall values are systematically higher by about 23.5%. This systematic offset finds a self-consistent explanation within the density-dependence framework of the point-mass approximation. SPARC bulge-fraction group statistics further confirm this trend---the strong-bulge group shows a deviation of only +8.1%, while the weak-bulge group shows a deviation of +26.4%. The effective mass-to-light ratio calibrated from the strong-bulge group is approximately 0.51, consistent with the normal range of the Chabrier IMF in the 3.6 μm band. The gravitational lensing systems converge with SPARC at the same mass-to-light ratio, though the point-mass approximation overestimation is more severe at lensing impact parameters, and the current lensing verification strength is limited. At the cosmological scale, the geometric expansion equation H(z) = 1/(T_now - τ(z)) naturally yields H_0 ≈ 71.0 km/s/Mpc, lying between the Planck and SH0ES measurements, with the bare geometric expansion equation together with a low-redshift passive deceleration correction providing a preliminary description of the cosmic expansion history. This theory, centered on the geometry of the cosmic horizon, provides a unified geometric explanation for dark matter and dark energy without introducing any unknown particles or free parameters, and offers a natural geometric perspective on cutting-edge cosmological puzzles such as the JWST early galaxy crisis and the Hubble constant tension.