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 \emph{Dynamic Inequality Equilibrium} (DIE)
framework therefore shifts inference from tail shape to inequality \emph{dynamics}—specifically, how
distributions move under the Lorenz (equivalently convex) order.

This repaired/extended version strengthens the framework in two ways.

\textbf{(i) Stabilized regime operationalization (non-entropic repair).} In stabilized multiplicative environments,
the bulk is often well approximated by a log-normal distribution. Prior ``entropic'' intuitions (treating
Shannon differential entropy as an equality/heterogeneity proxy under fixed mean) are mathematically
invalid for the log-normal family: constrained differential entropy is \emph{non-monotone} in the Lorenz
parameter and therefore violates Schur-concavity on this family. We therefore replace entropy-based language
with \emph{Lorenz-consistent, scale-invariant} proxies derived from the log-normal Gini coefficient:
\[
R_{II}(\sigma)=G(\sigma)=\erf\!\Big(\frac{\sigma}{2}\Big),\qquad
R_I(\sigma)=1-G(\sigma)=\erfc\!\Big(\frac{\sigma}{2}\Big),
\]
which are strictly monotone in the Lorenz parameter $\sigma$.

\textbf{(ii) A continuous bridge to criticality.} We upgrade the discrete Type I/Type II stabilizer taxonomy by
linking it 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, we revise the diagnostic protocol: in near-equilibrium stabilized regimes, the observed Gini
coefficient can be mapped to a latent regime preference parameter (``$\alpha$-inference'') \emph{conditional}
on a Cobb--Douglas aggregation assumption; and boundary diagnostics are refined by an explicit drift-saturation
test in log-space.