The First Principle of Complex Systems: The Axiom of Coupling Conservation and the Vitality Equation (V = N · C^2)

Traditional scientific paradigms—from classical mechanics to neoclassical economics—are deeply entrenched in "entity reductionism," treating macroscopic systemic properties as linear aggregations of isolated elements. This framework fundamentally fails to explain the highly non-linear phase transitions observed in complex adaptive systems. This study establishes the definitive foundational axiom of the "Coupling Ontology": The Axiom of Coupling Conservation. It posits that no entity exists in isolation; the essence of any system is the dynamic topological network of its relationships. By mathematically reducing the previously established System Stability Index (SSI) and integrating complex network topology, we derive the universal Vitality Equation: V = N · C^2. This equation rigorously demonstrates that a system's macroscopic vitality (V) is not determined by the linear accumulation of nodes (N), but by the non-linear, square-level emergence of "Coupling quality" (C). By demonstrating broad structural isomorphism across chemistry, biology, psychology, and macro-sociology, this study breaks the boundaries of traditional reductionism, offering a unified mathematical syntax for understanding the survival, collapse, and phase transitions of all complex systems.