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 (particle horizon radius) and the background characteristic acceleration a0 = c²/Rp. Based on the physical picture of causal delay and characteristic gravitational potential velocity, we derive the effective gravitational acceleration expression: gₑff (r) = GM/r² + (1/r) ×√ (GMc²/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 = (GMc²/Rp) ^ (1/4), which naturally yields the Tully-Fisher relation M ∝ v⁴. After completing the theoretical construction, we perform independent calibration primarily using the Milky Way and M33, with deviations of –5. 1% and +4. 2% between predicted and observed values, respectively; the +14. 8% deviation for M31 suggests possible systematic errors in its mass estimation and is not taken as core evidence. Subsequently, we extend the validation to 175 galaxies from the SPARC database, finding that the theoretical predictions perfectly match the observed shape (v ∝ M^ (1/4) ), but are systematically higher by about 30%. This systematic deviation shows a significant numerical correlation with the mass-to-light ratio baseline value of 0. 7: from v ∝ M^ (1/4), the mass underestimation factor is (1. 30) ⁴ ≈ 2. 86, suggesting the true mass-to-light ratio should be approximately 0. 7 × 2. 86 ≈ 2. 0, close to the typical value of a heavy initial mass function (IMF). To test this conjecture, we further perform an independent verification using gravitational lensing data: under the standard dark matter framework, the theoretical deviation is large; when adopting a heavy IMF that permits no dark matter (e. g. , similar to the Salpeter IMF), the deviation decreases significantly, supporting the conjecture. Finally, using nearby galaxies for independent calibration of the theoretical formula yields Rp = (4. 14 ± 0. 72) ×10²⁶ m, consistent with the cosmological observed value within the error range, demonstrating the theory’s self-consistency. This theory, centered on the geometry of the cosmic horizon, provides a unified explanatory framework for dark matter and dark energy, requiring no introduction of unknown particles or free parameters, and offers a natural geometric explanation for cutting-edge cosmological puzzles such as the JWST early galaxy crisis and the Hubble constant tension.