Toward a Formal Framework of Physical SuperHyperStructures: Multi-Layered Extensions of Hyperstructures in Physics

Abstract

A hyperstructure replaces ordinary single-valued operations by set valued ones: outcomes of interactions are subsets of a universe, naturally organized through the powerset construction. Pushing this idea further, a superhyperstructure works with iterated powersets (the n-fold powerset) so that one can represent genuinely hierarchical collections of possibilities across multiple levels. In this work we adapt these frameworks to physical modeling and present concrete motivating examples. A Physical HyperStructure takes a set of physical states as its base and equips it with interaction hyperoperations subject to physically meaningful constraints, such as conservation (possibly with bounded losses) and covariance under a prescribed symmetry group. A Physical SuperHyperStructure then lifts the same principles to iterated powersets, allowing operations on ensembles and higher-order families of ensembles while still enforcing conservation-type bounds and symmetry consistency across levels.