I present a coordinate-free, symbolic framework for determining 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 \emph{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 for internally decomposed polyhedra with the minimal number of added internal edges. 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.

翻译：本文提出一种坐标无关的符号化框架，用于判定给定多边形面集能否构成闭合的亏格为零的多面体表面，并预测此类表面如何分解为内部四面体。该方法仅使用离散关联变量，如内部四面体数量$T$、内部粘合三角形数量$N_i$和内部三角剖分段数量$S_i$，并在尝试几何嵌入前进行组合可行性检验。针对\emph{正规形式}的多面体，本文建立了关联$V,E,F$与平坦度参数$S:=\sum_f(\tmop{deg} f-3)$的精确恒等式，并识别了$E$、$F$和$S$中的奇偶敏感效应。外部恒等式与奇偶敏感边界对亏格为零的多面体图具有普适性。对于内部参量，本文证明了精确关系式$N_i=2T-V+2$和$T-N_i+S_i=1$（其中$S_i$视为内部边数量），并获得了具有最少添加内部边的内部分解多面体的受限线性范围。基于此，本文提出了一种符号化工作流程，可对结构不可行性进行快速预检验，从而减少计算几何学、图形学与自动化建模中昂贵的几何验证需求。