Mapping inequality in power law and exponential distributions

Abstract We present a theoretically grounded, entropy-based method for localizing inequality in probability distributions using recursive Hahn decomposition of an entropy-derived signed measure, while illustrating its applicability through examples drawn from physics and stochastic phenomena. Unlike traditional metrics that collapse entire distributions into single numbers, this method partitions the domain into regions of provably higher or lower uniformity, preserving monotonicity across recursive splits. Applied to two canonical distributions power law and exponential we obtain closed-form formulas for cutoff points and uniformity ratios. For power laws, these formulas reveal how inequality concentrates along heavy tails, replacing heuristic thresholds with analytic ones. For exponential distributions, including the Boltzmann form, we show that the entropy-based cutoff equals the conditional mean, giving it a new interpretation as an entropic equilibrium point where energy minimization and entropy maximization balance. This hierarchical decomposition moves beyond scalar summaries such as Gini or Theil, constructing a multiscale map of disparity that is both interpretable and computationally tractable. The approach generalizes to any continuous or discrete distribution, providing a unified, information-theoretic lens for analyzing inequality in economics, reliability theory, and statistical physics.