Ordinal Majorization Entropy (r-OME): Regime-Conditional Ordinal Aggregation with Dynamic Inequality Equilibrium, a Non-Entropic Stabilized-Regime Module, and Diffusion Diagnostics

Ordinal Majorization Entropy (OME) resolves the numerology trap in multidimensional ordinal databy switching the object of analysis from arbitrary numeric codings to the ordinal profile distributionp n-1 and then applying the majorization and Lorenz order with Schur-concaveentropy indices. Recent advances in dynamic inequality equilibrium and in stabilized multiplicativeenvironments reveal a latent failure mode of purely entropic architectures: within the mean-constrainedlog-normal family, Shannon differential entropy is non-monotone in the Lorenz parameter and thus failsLorenz consistency as an equality and robustness proxy over the full stabilized domain. The r-OMEarchitecture repairs this failure by adding a stabilized-regime module that replaces entropy-basedequality narratives with Lorenz-consistent, scale-invariant Gini proxies R₈ and R₈₈, and byimporting conditional equilibrium logic through a Cobb--Douglas meta-objective that yields a closed-formequilibrium inequality condition enabling alpha inference in near-stationary stabilized regimes. Dynamicdiagnostics are upgraded by a diffusion bridge in log-space and a drift-saturation test that distinguisheslinear restoring stabilization (log-normal bulk) from saturated-tail stabilization (power-law risk). This document introduces and proves Repair IV: a copula-theoretic extension that resolves themultidimensional index number problem for ordinal systems. Baseline OME treats the joint ordinal profileas a single flattened categorical variable, which destroys information about dependence among attributes. Repair IV integrates copula entropy, leveraging the equivalence between mutual information and negativecopula entropy, to produce an ordinally invariant, scale-separated, nonparametric measure ofmultidimensional lock-in. In the stabilized regime we define a normalized copula proxy R₈ₕ (aCopula--Gini mapping) that is monotone in dependence strength and compatible with the R₈ andR₈₈ stabilized-regime coordinates. The resulting r-OME-IV framework provides a unified diagnosticstate vector that separates marginal dispersion from cross-dimensional stratification, completing thegeometry needed to diagnose systemic risk and inequality in multidimensional ordinal environments.