On the Entropy-Based Localization of Inequality in Probability Distributions.

We present a novel method for localizing inequality within probability distributions by applying a recursive Hahn decomposition to the degree of uniformity-a measure derived from the exponential of Shannon entropy. This approach partitions the probability space into disjoint regions exhibiting progressively sharper deviations from uniformity, enabling structural insights into how and where inequality is concentrated. To demonstrate its broad applicability, we apply the method to both standard and contextualized systems: the discrete binomial and continuous exponential distributions serve as canonical cases, while two hypothetical examples illustrate domain-specific applications. In the first, we analyze localized risk concentrations in disease contraction data, revealing targeted zones of epidemiological disparity. In the second, we uncover stress localization in a non-uniformly loaded beam, demonstrating the method's relevance to physical systems with spatial heterogeneity. This decomposition reveals aspects of structural disparity that are often obscured by scalar summaries. The resulting recursive tree offers a multi-scale representation of informational non-uniformity, capturing the emergence and localization of inequality across the distribution. The framework may have implications for understanding entropy localization, transitions in informational structure, and the dynamics of heterogeneous systems.