HyperGraph- and superhypergraph-based molecular models: weighted, rough, neural, and multipolar frameworks

We present a unified, set–theoretic framework that extends molecular graphs to hypergraphs and superhypergraphs via iterated power sets. We define Molecular Graphs, Molecular HyperGraphs, and Molecular SuperHyperGraphs, and develop four complements over them: Weighted, Rough, Neural, and Multipolar frameworks. We prove concise inclusion results—most notably, that the Weighted Molecular SuperHyperGraph strictly contains both Weighted Molecular HyperGraphs and (unweighted) Molecular SuperHyperGraphs—while preserving alternating–path distances under canonical embeddings. Compact examples (e.g., methane, ethanol, acetic acid) illustrate how atoms, bonds, functional groups, and higher–order motifs appear as vertices, hyperedges, and superedges under rank constraints. We also provide implementation–agnostic message–passing rules for variable–arity interactions, enabling property prediction and hierarchical analysis in chemistry and chemical biology. This paper is devoted to theoretical analysis, and it is hoped that quantitative studies by domain experts will be developed in future work.