Many mechanisms generate heavy-tailed and scale-free distributions. Static tail fitting is rarely identifying: distinct dynamics can yield similar apparent tail indices. The Dynamic Inequality Equilibrium (DIE) framework therefore shifts inference from tail shape to inequality dynamics---specifically, how distributions move under the Lorenz order (equivalently, convex order under equal-mean constraints). This paper develops (i) an operational stabilized-regime module for multiplicative environments where the bulk is often well approximated by a log-normal family, and (ii) a continuous bridge to criticality via a log-space diffusion model. In the stabilized log-normal module, Shannon differential entropy under a fixed arithmetic mean is non-monotone in the Lorenz parameter and therefore fails as a Lorenz-consistent equality proxy; we replace entropy language with Lorenz-consistent, scale-invariant proxies derived from the log-normal Gini coefficient. Conditionally, a closed-form equilibrium mapping between inequality and a latent regime parameter follows from a geometric viability aggregation. In the diffusion bridge, drift saturation in log-space yields exponential log-tails and power-law tails in levels, providing a mechanistic signature of critical regimes. We present an implementation-ready diagnostic protocol: inequality velocity and drift-area statistics for Lorenz dynamics, a falsifiable Class A null test, bulk approximation checks for stabilized log-normality, boundary-intercept tests for Type I stabilization, drift-shape tests for log-space saturation, and mobility coordinates to separate macro-stationarity from micro-mobility. Empirical illustrations using U.S. incorporated places (2000--2020) and a firm-size distribution (U.S. BDS, NAICS 61, 2010--2020) are provided, along with simulation-based validation designs.
Kevin Fathi (2025) studied this question.