超越普通动态近似成员的静态屏障
Beyond the Static Barrier for Ordinary Dynamic Approximate Membership
AI总结：
该研究证明了固定错误率下静态与普通动态近似成员间的严格空间分离，推导了动态过滤器的空间下界，提出了含公共延续传输引理等的分析方法。
AI中文摘要：
我们证明了在每个固定错误率下，静态近似成员与普通动态近似成员之间存在严格的空间分离。对于每个固定的 ε∈(0,1)，在满足零假阴性、点假阳性概率至多为 ε、任意历史依赖性、拥有自由公共随机磁带且持久状态至多为 H 比特的情况下，针对大小为 u 的全集、容量为 n 的普通动态过滤器，当仅满足 u/n→∞ 时，H≥(log₂(1/ε)+a_ε^c)n−o(n)。常数 a_ε^c 是通过保留父代接受质量与后继储备之间的依赖关系得到的显式变分阈值。结构步骤是一个公共延续传输引理。联合后验 KL 界给出了分支特定的幸存者支持；相同的合法删除-插入词将该支持传输到一个后继状态，从而强制接受储备。随后在条件熵论证中保留父代外部质量 1−X，而非将其替换为 1−ε。这产生了一个两变量解析包络，无选定阈值、二进见证或数值假设。
英文摘要：
We prove a strict space separation between static and ordinary dynamic approximate membership at every fixed error rate. For each fixed $\varepsilon\in(0,1)$, a capacity-$n$ ordinary dynamic filter over a universe of size $u$, with zero false negatives, pointwise false-positive probability at most $\varepsilon$, arbitrary history dependence, a free public random tape, and at most $H$ bits of persistent state, satisfies \[ H\ge \bigl(\log_2(1/\varepsilon)+a_\varepsilon^{\rm c}\bigr)n-o(n), \] under only $u/n\to\infty$. The constant $a_\varepsilon^{\rm c}$ is an explicit variational threshold obtained by preserving the dependence between the parent accepted mass and the successor reservoir. The structural step is a common-continuation transport lemma. A joint posterior KL bound gives a branch-specific survivor support; the same legal delete--insert word transports that support to one successor state, forcing an accepted reservoir. We then keep the parent outside mass $1-X$ in the conditional-entropy argument instead of replacing it by $1-\varepsilon$. This yields a two-variable analytic envelope, with no selected thresholds, dyadic witnesses, or numerical assumptions.
