Why do division and cohesion often intensify together? This paper develops a static structural theory of relation maintenance based on minimal positional constraints. Rather than relying on utility-based or probabilistic models, social relations are formalized as constraint satisfaction problems over an abstract position space. When a bifurcation event -- such as a vote or institutional assignment -- fixes agents' positions, relational viability is determined solely by positional compatibility. We establish three structural results. First, under any non-degenerate positional constraint, fragmentation (relational collapse) and cohesion (condition confirmation) necessarily coexist as complementary outputs of a single compatibility function. Second, we prove a structural asymmetry of veto power: relation maintenance requires logical conjunction, while collapse requires only logical disjunction, implying that fragmentation operates as a unilateral structural veto. This yields a purely logical foundation for behavioral premises such as pairwise stability. Finally, we establish a conditional impossibility theorem: under positional plurality, avoiding relational collapse is structurally impossible, leaving coercive homogenization as the only design-level guarantee for universal cohesion. The framework isolates minimal boundary conditions and provides a formal language for analyzing polarized relational structures.

翻译：为何分裂与凝聚常同步加剧？本文基于最小化位置约束构建了关系维持的静态结构理论。不同于依赖效用模型或概率模型，我们将社会关系形式化为抽象位置空间上的约束满足问题。当分化事件（如投票或制度性指派）固定了行动者的位置时，关系的存续性完全取决于位置兼容性。我们建立了三个结构性结论：首先，在任何非退化位置约束下，分裂（关系崩溃）与凝聚（条件确认）必然作为同一兼容性函数的互补输出共存。其次，我们证明了否决权的结构不对称性：关系维持需要逻辑合取，而崩溃仅需逻辑析取，这意味着分裂发挥着单边结构性否决权的作用。这为成对稳定性等行为假设提供了纯粹的逻辑基础。最后，我们建立了一个条件性不可能定理：在位置多元性条件下，避免关系崩溃在结构上是不可能的，这使得强制性同质化成为实现普遍凝聚的唯一设计层面保障。该框架分离了最小边界条件，并为分析极化关系结构提供了形式化语言。