Multi-Scale Structural Taxonomy with HyperStructure and SuperHyperStructure

Hyperstructures and their hierarchical extensions—SuperHyperStructures—provide a versatile algebraic language for modeling multi-level and interdependent systems 1,2. In materials and chemical sciences, structural descriptions naturally span a broad spectrum of characteristic length scales, commonly organized as Macrostructure, Mesostructure, Microstructure, Sub–microstructure, Nanostructure, Sub–nanostructure/˚A–scale structure, Atomic structure, Local structure/short–range order, and Nuclear structure. This paper revisits these scale notions through precise, measure-theoretic and set-theoretic definitions of geometric confinement, state fields, and correlation descriptors, and then elevates each scale to a hyperstructural setting by specifying appropriate base sets and multivalued operations that capture admissible unions, intersections, fusions, or pointwise selections. Building on these, we introduce superhyperstructural counterparts by means of iterated powersets and (𝑚, 𝑛)–SuperHyperOperations (including Krasner-type lifts, consensus maps, and budget-feasible bundling), establishing general closure and reduction theorems. Inter-level coarse–graining maps are shown to act as monotone hyper-homomorphisms, ensuring compatibility of operations across scales and recovering classical single-valued models via singleton embeddings. The resulting framework unifies deterministic and multivalued representations across all characteristic regimes, offering a mathematically consistent toolbox for scale-bridging analysis and design.