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 rarely
identifying because distinct dynamics can produce similar apparent tail indices. This paper does
not 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 of
stabilizers 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-based
multiplicative concentration theorem, a Banach fixed-point Wasserstein contraction theorem for
random affine maps, and a stationary-mean identity that yields a sharp falsification rule for Type
I stabilization. We validate the diagnostic bundle by simulation (showing separation even when
Hill tail indices are similar) and illustrate it on US place populations and scientific citations.