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 framework therefore shifts inference from tail shape to inequality dynamics---specifically, how distributions move under the Lorenz (equivalently convex) order.

This version strengthens the framework in two ways.
(i) \textbf{Stabilized regime operationalization.} In stabilized multiplicative environments, the bulk is often well approximated by a log-normal distribution. Treating Shannon differential entropy as an equality/heterogeneity proxy under fixed mean is mathematically invalid for the log-normal family: constrained differential entropy is non-monotone in the Lorenz parameter and therefore violates Schur-concavity on this family. We therefore use Lorenz-consistent, scale-invariant proxies derived from the log-normal Gini coefficient:
\[
R_{II}(\sigma)=G(\sigma)=\mathrm{erf}\!\left(\frac{\sigma}{2}\right), \qquad
R_{I}(\sigma)=1-G(\sigma)=\mathrm{erfc}\!\left(\frac{\sigma}{2}\right),
\]
which are strictly monotone in the Lorenz parameter $\sigma$.
(ii) \textbf{A continuous bridge to criticality.} We link the discrete Type I/Type II stabilizer taxonomy to a diffusion model in log-space. In one-dimensional diffusions for $Y_t=\ln X_t$, if the stabilizing drift saturates to a constant negative value in the far tail, the stationary log-density is asymptotically exponential, implying a power-law tail in $X$. This provides a continuous parameter space connecting stabilized log-normal bulks to critical power-law tails.

Finally, the diagnostic protocol is revised: in near-equilibrium stabilized regimes, the observed Gini coefficient can be mapped to a latent regime parameter (``$\alpha$-inference'') conditional on a geometric aggregation assumption; and boundary diagnostics are complemented by an explicit drift-saturation test in log-space.