Z-Geometric Dynamics II: Information Conservation and Cosmic Evolution

In the geometric dynamics framework established by Z-Geometric Dynamics (Paper I), the observed particle horizon radius R_p(0) is taken as the fundamental input, from which the background acceleration a_0 = c^2/R_p(0) is derived, thereby establishing a unified geometric framework for dark matter and dark energy. This paper further explores the origin of this horizon. Starting from quantum mechanical unitarity, we introduce the axiom of information conservation, and together with the initial condition axiom and the maximum entropy hypothesis, define the geometric entropy constant α = 1/[π (R_i/l_P)^2] = 1/ln(R_p/l_P), establishing the concept of the information universe as a non-evolving abstract entity. Through a rigorous proof of dimensional reduction equivalence, we establish the universal upper bound I_obs ≤ S_info on the observer's information reception, and through dimensional reduction equivalence establish the equivalence relation between the observer's information entropy I_obs = ln(R_p/l_P) and the total information entropy of the universe S_info. Building on this, the paper further derives the ratio conjecture α_EM/α ≈ 32/31 between the geometric entropy constant and the fine-structure constant, laying a numerical anchor for the microscopic unification of Paper III. More importantly, starting from the core conclusions of Paper I---the horizon surface gravity a_0 = c^2/R_p and the 1/r decay of the gravitational field---and combining them with the standard theory of gravitational compression of spacetime in general relativity, we derive the self-consistency of spatial compression: any observer in the universe, using their own gravitationally compressed ruler, measures the same particle horizon radius, and this value remains constant in their respective time scales. This mechanism naturally explains the physical origin of the failure of the simultaneous equations at the early singularity, while providing a solid geometric foundation for the information saturation paradigm.