Three-Dimensional or Multi-Dimensional Molecular Hyperstructure and Molecular HyperGraph

This paper develops a unified, geometry-aware framework for representing molecules and their higher-order organization. We formalize 𝑑-dimensional molecular structures, hyperstructures, and superhyperstructures by combining labeled (hyper)graphs with Euclidean embeddings and rank-aware operations on iterated power sets. Our definitions render geometric observables—bond lengths, bond angles, and dihedral torsions—intrinsic, and we specify configuration spaces modulo rigid motions. A canonical lifting scheme propagates coordinates from atoms to sets-of-sets, enabling consistent embeddings of hyperedges and superedges via convex carriers or simplex parametrizations. We prove reduction theorems: hyperstructures collapse to embedded molecular graphs when all hyperedges are binary, and superhyperstructures reduce to hyperstructures at rank zero. Worked chemical examples (staggered ethane in 3D, planar benzene in 2D, and a hydrogen-bonded water dimer as a hyper/superhyper instance) verify the axioms with explicit numerical coordinates. The framework supports coarse-grained modeling, constraint-based geometry, and machine-learning representations, while remaining compatible with classical cheminformatics notions of atoms, bonds, functional groups, and molecular assemblies.