Emergence Metric and Universal Emergence Force in Many-body Systems

We propose a universal emergence metric defined as εₑmerg = Sᵣ × Iₖ × Dₘ, which combines coherence (Sᵣ), information rate (Iₖ), and mode diversity (Dₘ) in coherent many-body systems. Numerical studies across multiple platforms—including superconducting qubits, Bose-Einstein condensates, and Rydberg lattices—show that εₑmerg scales linearly with system size, while the per-particle emergence density (εₑmerg / N) remains constant within each platform. From basic postulates of stability and maximum control rate, a universal emergence force Fₑmerg = FP × εₑmerg 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.