Vacuum of Emergence and Spacetime Homogeneity from the Causal–Entropy Law

Order Mechanics (OM) , derived from the first principles of the Causal-Entropy Law (CEL) 【1-4】, aims to establish a unified formulation of gravitational and quantum phenomena. Within the OM framework, this paper develops a rigorous and operational definition of the order density ρ_O: ρ_O= R/(V_p∙r) where R is the causal curvature, r represents the entropy curvature, and V_P denotes the Planck volume. Based on this definition, this paper presents a geometric description of the vacuum ground-state order and demonstrates the emergence of the vacuum. By leveraging the CEL mechanism, we not only resolve the vacuum energy catastrophe but also derive the four-dimensional homogeneity of the vacuum order field. Additionally, we derive the modified Einstein field equations within the OM framework, demonstrating that the cosmological constant Λ is not an empirical constant but rather a dynamic geometric manifestation of the entropy curvature tensor r_μv. Further, this work conceptualizes microscopic particles as semi-aggregated state (SAS) solitons within the order density field. By introducing a complex local order field ψ_SAS= Ae^iθ to characterize their intrinsic structure, we directly deduce the Heisenberg uncertainty relation from the gradient dynamics of the complex field, employing the Fourier bandwidth-limiting theorem for continuous media. Our findings indicate that gravity and quantum mechanics—traditionally viewed as fragmented domains—are not mutually independent physical frameworks, but rather two asymptotic limits of a single geometric paradigm, Order Mechanics. This study establishes a preliminary logical closed loop and mathematical synthesis integrating OM with general relativity, quantum mechanics, and the second law of thermodynamics, thereby offering a promising new paradigm toward a paradox-free grand unified theory.