Entropy Reimagined: Theoretical Foundations of HyperEntropy and SuperHyperEntropy with Real-World and Engineering Applications

Classical mathematical structures can be systematically extended into hyperstructures and superhyperstructures through the powerset and its n-fold iterations. These enriched frameworks are particularly well suited for modeling hierarchical or multi‐layered relationships across diverse domains. Meanwhile, entropy quantifies the average information‐theoretic uncertainty of a discrete probability distribution. In this paper, we introduce two new constructs: HyperEntropy, which extends Shannon entropy to distributions over subsets of a base set, and SuperHyperEntropy, which further generalizes this concept across iterated powerset levels. We develop their fundamental properties, present illustrative examples, and outline potential applications, thereby laying the groundwork for future advances in entropy theory. Additionally, we present a preliminary engineering application.