The Principle of Dynamic Inequality Equilibrium: A Lorenz-Order Framework with a Stabilized Regime and a Diffusion Bridge to Criticality

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).

We develop (i) a stabilized-regime module for multiplicative environments where the bulk is often well approximated by a log-normal family (used as a one-parameter Lorenz coordinate via $\sigma$), 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 inequality proxies derived from the log-normal Gini coefficient.

Crucially, we add a representation theorem for long-run viability in independent multiplicative systems: if the stabilized macro-state admits additive sufficient statistics under convolution and the viability evaluator is additive in log under independence, then the evaluator is necessarily multiplicatively separable in the exponentiated statistics. In the canonical log-normal case this yields $V(\mu,\sigma^2)=C\exp(a\mu+b\sigma^2)$, i.e.\ Cobb--Douglas form in interpretive (calligraphic) coordinates $\mathcal{X}=\exp(\mu)$ and $\mathcal{Y}=\exp(\sigma^2)$. Any trade-off weights (e.g.\ $a,b$ or a one-parameter reduction) are properties of the evaluator and are \emph{not} identified by Lorenz geometry alone.

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.