The Field-Constrained Vitality Equation: A Dual-Axis Heterogeneity Evolution Model of Topology in Large-Scale Complex Social Systems

Previous studies have established the universal axiom of "Coupling Ontology" and derived the Vitality Equation V = N · C^2, along with the System Stability Index (SSI) model. However, any complex adaptive system exists within a specific physical and historical phase space, and its topological evolution is constrained by "field constraints." As an advanced extension of previous theories, this study introduces the "Environmental Field Constraint" parameter (E) to analyze the spatial heterogeneity of large-scale, high-density human settlement systems. The research indicates that fundamental parameters such as climate and topography determine the system's energy acquisition modes, thereby shaping the behavioral characteristics and emotional states of nodes, and ultimately constructing distinct coupling mechanisms (C). This study proposes a "Dual-Axis Heterogeneity Topology Model": along the longitudinal gradient, the system exhibits a phase transition of "Strong Centralized Command Field—Dual Coupling Game Transitional Zone—Strong Decentralized Contract Field"; along the latitudinal gradient, the system exhibits an evolution of "Exogenous Input Type—Transitional Transit Type—Endogenous Self-Generating Type." Based on this, the study constructs a "Three-Dimensional Topological Phase Diagram" and a "Mode Migration Navigation Mechanism," proving that after introducing spatial heterogeneity parameters, the equation can precisely diagnose structural vulnerabilities in different subsystems. This not only corrects the "system rejection" caused by a single model but also provides a mathematical structural isomorphism tool for cross-regional empirical research and evolutionary intervention in complex social systems.