AI 中文总结
该研究证明对于具有D个同等权利叶子的完全二叉梳状结构，不存在同时满足下限配额、上限配额和房屋单调性的规则，其最优最坏情况偏差为Θ(log D)，揭示了静态与动态配额可行分配间的差距。
AI 中文摘要
多级分配通过一组层级结构分配整数席位。Schmidt-Kraepelin、Suksompong和Wijaya证明，在每个固定的总席位规模下，可同时满足下限配额和上限配额；他们还构造了分别满足其中任一配额的房屋单调规则。他们留下一个未解决的问题：是否存在一个规则能同时满足下限配额、上限配额和房屋单调性，即使仅要求相对于根节点的配额。我们给出否定答案。对于具有D个同等权利叶子的完全二叉梳状结构，每个房屋单调分配序列会诱导出一个席位接收者序列。嵌套梳状组的配额要求会使得每个网格对齐前缀的偏差低于1。随后的中点嵌入将整个区间的偏差限制为该值加1/2，这与Schmidt的对数下界矛盾。相反，二元van der Corput席位调度的梳状前缀误差最多为log D/(3log 2)+1。因此，梳状结构上的最优最坏情况误差为Θ(log D)，且对于足够大的有限D，不存在房屋单调配额规则。该证明揭示了静态-动态差距：每个总席位规模都存在一个配额可行的分配，但可行分配无法嵌入到一条单调路径中。用量化语言表述，该结果刻画了梳状结构上渐进one-hot舍入的嵌入量化惩罚的阶数。
英文摘要
Multi-level apportionment allocates integer seats through a hierarchy of groups. Schmidt-Kraepelin, Suksompong, and Wijaya proved that, at every fixed house size, lower and upper quota can be satisfied simultaneously; they also constructed house-monotone rules satisfying either quota separately. They left open whether one rule can satisfy lower quota, upper quota, and house monotonicity together, even when quota is required only relative to the root. We give a negative answer. For a full binary comb with $D$ equally entitled leaves, every house-monotone allocation sequence induces a sequence of seat recipients. Quota for the nested comb groups would force every grid-aligned prefix discrepancy to be below one. A midpoint embedding then bounds the full interval discrepancy by this quantity plus $1/2$, contradicting Schmidt's logarithmic lower bound. Conversely, a binary van der Corput seat schedule has comb-prefix error at most $\log D/(3\log 2)+1$. Thus the optimal worst-case error on the comb is $Θ(\log D)$, and for sufficiently large finite $D$ no house-monotone quota rule exists. The proof isolates a static--dynamic gap: each house size admits a quota-feasible allocation, but the feasible allocations cannot be embedded into one monotone path. In quantization language, the result characterizes the order of the embedded-quantization penalty for progressive one-hot rounding on the comb.
