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, we derive the observable cosmic horizon distance Rp and the background characteristic acceleration a0=c^2/Rp. In the strong version (ZGD-Strong), we adopt general relativity as the theoretical framework for spacetime geometry and derive the effective gravitational acceleration expression: g_eff(r)=GM/r^2 + (1/r)sqrt(GMc^2/Rp), a form uniquely determined by the fundamental constants {G, c, Rp, M, r} and containing no free parameters. This leads to the galactic rotation velocity formula v=(GMc^2/Rp)^(1/4), which naturally yields the Tully-Fisher relation M∝v^4. Theoretical validation proceeds hierarchically: at the merging galaxy cluster scale, numerical tests and control simulations for the Bullet Cluster confirm the viability of the ZGD mechanism. At the galactic scale, initial checks on nearby galaxies and large-sample SPARC validation show excellent shape agreement (v∝M^(1/4)) with a systematic overprediction of ~23.5%, explained by point-mass approximation bias. At the cosmological scale, the geometric expansion equation naturally yields H0≈71.0 km/s/Mpc, compatible with Planck and SH0ES measurements, and the predicted dark energy evolution matches DESI data. This theory unifies dark matter and dark energy as geometric effects of the cosmic horizon, requires no unknown particles or free parameters, and offers natural geometric explanations for the JWST early galaxy crisis and Hubble constant tension.