Symbolic Constraints in Polyhedral Enclosure and Tetrahedral Decomposition in Genus-0 Polyhedra
I present an exploratory, coordinate-free framework for organizing two related problems in polyhedral combinatorics: screening polygonal face multisets for genus-zero enclosure and examining tetrahedral decompositions that preserve the original boundary vertex set. The boundary is summarized using a face-degree excess measure that records how far the surface departs from full triangulation, while standard Euler and incidence relations provide exact constraints and parity-dependent bounds on the external structure. For tetrahedral decompositions, standard incidence relations for a triangulated 3-ball are separated from the additional patterns generated by a restricted boundary-driven decomposition procedure. Within the manually constructed examples, bipyramidal configurations provide an extremal reference class, and departures from bipyramidal structure are associated with changes in attainable tetrahedral states. The examples also suggest a contraction of the realized internal state space as boundary face-degree excess increases, motivating the notions of S-flexibility and incidence-gated transitions. These patterns are presented as empirical structural observations and conjectural directions rather than general classification theorems. The resulting framework provides a symbolic pre-screening workflow that can reject some impossible polygonal face multisets before explicit graph construction or geometric realization, and is implemented in the open-source R package polyenclose.