📝 SummarXiv

Abstract

I present a coordinate-free, symbolic framework for deciding whether a given set of polygonal faces can form a closed, genus-zero polyhedral surface and for predicting how such a surface could be decomposed into internal tetrahedra. The method uses only discrete incidence variables, such as the number of internal tetrahedra $T$, internal gluing triangles $N_i$, and internal triangulation segments $S_i$, and applies combinatorial feasibility checks before any geometric embedding is attempted. For polyhedra in normal form, I record exact incidence identities linking $V,E,F$ to a flatness parameter $S:=\sum_f(\tmop{deg} f-3)$, and I identify parity-sensitive effects in $E$, $F$, and $S$. The external identities and parity-sensitive bounds hold universally for genus-0 polyhedral graphs. For internal quantities, I prove exact relations $N_i=2T-V+2$ and $T-N_i+S_i=1$ (with $S_i$ taken to be the number of interior edges) and obtain restricted linear ranges within a shell-aligned ladder subclass (SALT), where at most one interior edge is introduced per layer. Consequently, I propose a symbolic workflow that yields rapid pre-checks for structural impossibility, reducing the need for costly geometric validation in computational geometry, graphics, and automated modeling.