A Unified Discrete and Continuous Theory of Core-Halo Complexity Maximizers

Abstract

The maximization of statistical complexity has long been associated with the emergence of probability distributions lying between perfect order and complete disorder. While previous studies have shown that complexity-maximizing distributions exhibit a two-level structure in finite discrete systems, an analogous unified treatment for both discrete and continuous probability spaces has remained unavailable. In this work, we develop a general variational framework for a generalized statistical complexity constructed from Shannon and Renyi entropies. We derive a common stationary equation governing both discrete probability masses and continuous probability densities and prove that every stationary solution necessarily possesses exactly two probability levels, establishing a universal core-halo structure. We further demonstrate that the optimization problem reduces to a single multiplicity parameter and prove that the global complexity maximum is attained by the smallest admissible core, corresponding to a single dominant state in the discrete case and an infinitesimal core in the continuous limit. These results provide a complete analytical characterization of the complexity-maximizing distributions and reveal a common mathematical structure underlying complexity optimization in both discrete and continuous settings. The framework establishes a unified foundation for generalized statistical complexity with potential applications in statistical mechanics, information theory, and the analysis of complex systems.