A Majorization-Theoretic Framework for Comparing Scale-Free Mechanisms: Dynamic Majorization Equilibrium (DME)

Many mechanisms generate heavy-tailed and scale-free distributions: random multiplicative growth with reinjection (Kesten), stabilized geometric Brownian motion, preferential attachment, and various critical phenomena. This paper does \emph{not} claim a new generative theorem for power laws. Instead it develops a majorization-theoretic \emph{comparative framework}---based on the Lorenz order and convex order---for organizing and empirically distinguishing these mechanisms by how they move distributions in inequality order. Mathematically, we provide a self-contained toolkit for Lorenz/convex order reasoning on nonnegative size variables and prove that non-degenerate multiplicative updates increase Lorenz concentration (including a conditional/correlated extension). We clarify additive effects: mean-zero noise is convex-order dispersive, but Lorenz-order claims for size variables require nonnegativity conditions that are often violated unless the model includes truncation, reinjection, or reflecting boundaries. We also give a Wasserstein contraction theorem for random affine maps that is explicitly framed as a bulk-stability (Class~B) condition and does not replace Kesten--Goldie tail analysis. Empirically, we operationalize DME as an identification toolkit: Lorenz drift statistics, conditional increment estimation with intercept tests, entry/attachment diagnostics, log-increment saturation checks, and tail-index estimation (Hill). We demonstrate the methodology on three canonical datasets where mechanism identification is debated (US city sizes, World Bank country populations, and scientific citation counts) and on a simulation ground-truth benchmark (Kesten, preferential attachment, reflected GBM). The empirical results show that the diagnostics separate mechanisms even when estimated tail indices are similar, moving analysis beyond the ``tail-fitting trap.'' A companion reproducible Colab notebook is provided as supplementary material.