The Least Action Principle for Complex Systems Emergence

The unity of physics is rooted in the principle of least action. From Lagrange’s Mécanique analytique (1788) to Einstein’s general relativity field equations (1915), Dirac’s quantum electrodynamics (1928), and the Weinberg-Salam-Glashow Standard Model (1967–1979), the core equations of all modern physics can be derived from a single variational principle, δS = 0. This is the fundamental reason physics has become a unified science. In sharp contrast, the science of complex systems has long lacked a comparable unified variational foundation. Onsager’s principle of minimum entropy production (1931), Prigogine’s far-from-equilibrium dissipative structures (1947, 1955), Jaynes’s maximum entropy (1957), Ziegler’s maximum entropy production principle, MEPP (1963), Bak-Tang-Wiesenfeld self-organized criticality, SOC (1987), and Friston’s free energy principle, FEP (2005, 2010) each have their own variational statement, yet there is no unified foundation spanning hierarchical levels and time scales. These principles are not merely disunified—they appear contradictory (Onsager and MEPP give opposite predictions in the far-from-equilibrium regime). This paper proposes the Least Action Principle for Complex Systems Emergence: the evolution of any complex system minimizes a complexity action functional that spans hierarchical levels and time scales. We show that this principle derives the five core phenomena of complex systems science—the nonuniformity potential evolution equation, the intrinsic exponential relaxation law, the cross-level downward constitution mechanism, the information compression quantification of emergence strength, and the two-stage phase transition path of cross-minimum jumps—while unifying all existing complex systems variational principles (Friston FEP, Onsager, MEPP, SOC, Jaynes, Kahneman are all special cases of the present principle under particular limiting conditions). The principle is presented with complete mathematical rigor, including: information-theoretic non-dimensionalization (resolving cross-domain dimensional consistency); three classes of boundary conditions (open/closed/adiabatic); three classes of variational constraints (spontaneous/constrained/forced); the complete relaxation spectrum (exponential/power-law/stretched exponential/logarithmic/Kramers, five forms); the Kramers quantitative formula for phase transition thresholds; stochastic action with fluctuations (Onsager-Machlup form); a multidimensional emergence index; bidirectional hierarchical dynamics; and the variational distinction between purposeful and non-purposeful evolution. The principle provides variational answers to questions concerning the universality of complex emergence, the irreducibility of strong emergence, the physical basis of free will, and precise criteria for the emergence of artificial consciousness—all as necessary consequences of the least action path. A complete falsifiability framework with five concrete falsifiable predictions (cognitive transitions, social revolutions, AGI emergence, and others) is provided. The principle endows complex systems theory with a unified variational foundation at the same level as fundamental physics, opening the variational era of complex systems science.