Z-Geometric Dynamics II: Geometric Entropy and High-Redshift Predictions

Building on the previous work "Z-Geometric Dynamics I: Galactic Scales and Dark Matter", this paper further introduces the geometric entropy axiom, reveals the emergent nature of the speed of light, and establishes the concept of the universal limiting radius R∞. By linking geometric entropy to the conservation of total information capacity of the universe, we define a geometric constant α that does not evolve with cosmic time. From α and the Planck length l_P, we derive an expression for the speed of light c and obtain an exact relation with the fine-structure constant α_EM = 32/31 α. Two independent paths—the dynamical path (based on the limiting background acceleration a∞ determined from galactic rotation curves in Z-Geometric Dynamics I) and the geometric information path (from the definition of the geometric entropy constant α)—yield a completely consistent universal limiting radius R∞ ≈ 4.8×10²⁶ m, with a deviation of about 0.13%, verifying the self-consistency of the theory. Starting from the definition of the universe as a closed system, we introduce the initial condition axiom R_i = 2GM/c² and, combined with general relativity and the Bekenstein-Hawking entropy formula from black hole thermodynamics, reveal the physical origin of the geometric entropy constant α—it is determined by the ratio of the initial cosmic scale to the Planck length. We also show that during cosmic expansion, the observable part satisfies the critical density condition H R_P = c in the radiation-dominated era, providing a geometric interpretation for the evolution of the background acceleration. These results unify macroscopic dynamics, microscopic information structure, and the fundamental definition of the universe within a self-consistent geometric framework, laying a solid foundation of constants for the subsequent unification of forces.