The Impossibility of Cohesion Without Fragmentation

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.