AI 中文总结
该研究解决四州席位分配的边界问题，证明不存在确定性方案同时满足配额与人口单调性公理，采用有限逻辑装置构造矛盾，为配额约束格取整提供全局兼容性障碍。
AI 中文摘要
席位分配将分数额转换为整数席位分配。配额要求每个州获得其标准配额的向下取整值或向上取整值，而人口单调性禁止人口弱增长的州失去席位给人口弱减少的州。Gölz、Peters和Procaccia近期通过构造五个州的案例，移除了经典不兼容性结果中使用的序保持假设；结合三个州的Webster方法可能性结果，这使得四个州成为未解决的边界情况。我们证明，在允许比较的议院规模不同的定义下，不存在确定性的四州席位分配方案同时满足这两个公理。该证明是一个有限逻辑装置：以中心配置处的一个配额选择为条件，十二个辅助配置编码三位信息并形成矛盾循环。该论证仅使用人口不变的州之间的席位转移，且不假设匿名性、中立性、序保持性、同质性或其他正则条件，也适用于相对人口单调性。从几何角度看，该结果是配额约束格取整的全局兼容性障碍，而非平均失真边界。
英文摘要
Apportionment converts fractional entitlements into integer seat allocations. Quota requires each state to receive the floor or ceiling of its standard quota, whereas population monotonicity prohibits a state whose population weakly increases from losing a seat to a state whose population weakly decreases. Gölz, Peters, and Procaccia recently removed the order-preservation assumptions used in classical incompatibility results by giving a five-state construction; together with the three-state Webster possibility result, this left four states as the unresolved boundary. We prove that no deterministic four-state apportionment solution satisfies both axioms under their definition, which permits the compared house sizes to differ. The proof is a finite logical gadget. Conditional on one quota choice at a central profile, twelve auxiliary profiles encode three bits and force a frustrated cycle. The argument uses only transfers between states whose populations are unchanged and assumes neither anonymity, neutrality, order preservation, homogeneity, nor other regularity conditions. It also applies to relative population monotonicity. Geometrically, the result is a global compatibility obstruction for quota-constrained lattice rounding, rather than an average-distortion bound.
