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

Akshat Sharma

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arXiv · 2607.17907

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

Akshat Sharma

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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.

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Akshat Sharma. 2026-07-20. A Unified Discrete and Continuous Theory of Core-Halo Complexity Maximizers. https://arxiv.org/abs/2607.17907

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