The Principle of Dynamic Majorization Equilibrium (DME): A Unified Framework for Diagnosing the Emergence of Scale-Free Structures

Many mechanisms generate heavy-tailed and scale-free distributions: random multiplica-tive growth with reinjection (Kesten), stabilized geometric Brownian motion with boundaries,preferential attachment with entry, and various critical phenomena. Static tail fitting is rarelyidentifying because distinct dynamics can produce similar apparent tail indices. This paper doesnot propose a new generative theorem for power laws. Instead it develops a comparative, falsi-fiable framework—Dynamic Majorization Equilibrium (DME)—that uses inequality dynamics(Lorenz/convex order) to constrain broad mechanism classes.Conceptually, Class A is the Lorenz-concentrating drive and Class B is the super-category ofstabilizers that stabilize Class A. For rigor and diagnostics, Class B is subdivided into Type I(affine reinjection: a global baseline in levels) and Type II (boundary/entry stabilization: floors,reflection, resets, and entry flux). We prove a Lorenz–convex equivalence theorem, a Jensen-basedmultiplicative concentration theorem, a Banach fixed-point Wasserstein contraction theorem forrandom affine maps, and a stationary-mean identity that yields a sharp falsification rule for TypeI stabilization. We validate the diagnostic bundle by simulation (showing separation even whenHill tail indices are similar) and illustrate it on US place populations and scientific citations.