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 compara-
tive, falsifiable framework—Dynamic Inequality Equilibrium —that uses inequality dynamics
(Lorenz/convex order) to constrain broad mechanism classes.
Conceptually, Class A is the inequality-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). The paper leverages classical foundations from majorization
theory and iterated random functions to derive disciplined diagnostic implications, including a
stationary-mean identity that yields a sharp, testable constraint 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 a firm-size-bin control series.