Emergence Metric and Universal Emergence Force in Many-body Systems

We propose a universal emergence metric defined as ε_emerg = S_r × I_k × D_m, which combines coherence (S_r), information rate (I_k), and mode diversity (D_m) in coherent many-body systems. Numerical studies across multiple platforms—including superconducting qubits, Bose-Einstein condensates, and Rydberg lattices—show that ε_emerg scales linearly with system size, while the per-particle emergence density (ε_emerg / N) remains constant within each platform. From basic postulates of stability and maximum control rate, a universal emergence force F_emerg = F_P × ε_emerg arises as a necessary effective force to maintain coherent stability. This framework is scale-invariant, platform-independent, and consistent with fundamental physical limits, providing a universal quantitative characterization of emergent phenomena in complex quantum systems.