Most models in game theory and network formation implicitly assume that relations between agents are feasible whenever incentives are aligned or interaction opportunities exist. Under this premise analytical attention is directed toward equilibrium efficiency or probabilistic link formation while the possibility that a relation may be structurally infeasible is rarely examined. This paper develops a static axiomatic framework in which relation maintenance is treated as a problem of structural compatibility rather than strategic choice or stochastic realization. Agents occupy positions in an abstract space and relations are subject to minimum conditions defined over these positions. A bifurcation event such as a vote declaration or institutional assignment fixes agents positions and thereby determines which relations are compatible. We identify position dependent gain axes as the key source of structural selectivity and prove an impossibility result under any non degenerate positional constraint no bifurcation event can preserve all relations. Instead the post event network necessarily exhibits either the simultaneous emergence of fragmentation and cohesion or a degenerate trivial case in which constraints are position independent. The result is purely structural and does not rely on preferences beliefs incentives or dynamic adjustment. It establishes a fundamental limit on universally cohesive outcomes and reframes division not as a failure of design or coordination but as a logical consequence of positional constraints.

翻译：博弈论与网络形成模型大多隐含假定：当激励一致或互动机会存在时，智能体间的关系即可实现。在此前提下，分析焦点常集中于均衡效率或概率性连接形成，而关系可能存在的结构不可行性却鲜被探讨。本文构建了一个静态公理化框架，将关系维系视为结构兼容性问题而非策略选择或随机实现问题。智能体占据抽象空间中的位置，其关系需满足基于这些位置定义的最低条件。诸如投票声明或制度指派等分岔事件会固定智能体的位置，从而决定哪些关系具有兼容性。我们识别出位置依赖的收益轴作为结构选择性的关键来源，并证明了一个不可能性结果：在任何非退化的位置约束下，不存在能保全所有关系的分岔事件。取而代之的是，事件后的网络必然呈现以下两种情形：要么同时出现碎片化与凝聚现象，要么退化为约束与位置无关的平凡案例。该结果纯粹基于结构分析，不依赖于偏好、信念、激励或动态调整。它确立了实现普遍凝聚结果的基本限度，并将分化重新诠释为位置约束的逻辑必然结果，而非设计或协调的失败。