面向具有任意时刻有效证据的在线假设的动态e-闭包：闭包原理与投影合并
Dynamic $e$-closure for online hypotheses with any-time-valid evidence: closure principles and projective mergers
AI总结：
本文针对在线假设检验场景提出动态e-闭包，证明其可控制stopped-FDR与SupFDR，还扩展理论至有界损失并给出相关构造与反例。
AI中文摘要：
许多现代检验问题在两个维度上是序贯的：新假设可能随时间到达，而正在考虑的假设的证据会持续演变，并可在任意停止时刻被检查。我们针对该场景提出动态e-闭包。在全局停止时刻，活跃的真零交集是随机的。未来扩展一致性允许将其证书与固定终端交集的证书进行比较，从而实现同时的停止错误发现率（stopped-FDR）控制。若证书还具有时间单调性，则所得闭包可控制同时的SupFDR，且具有集合持续性。反之，满足任一准则的所有过程都包含于由规范归一化损失过程生成的动态闭包中。对于逐点合并，固定维容许性等价于普通的任意依赖e-合并。跨时间范围的一致性则要求单一全局可求和权重序列，且在e-值1填充下具有精确中立性；在可数无限假设空间上，这排除了容许逐点类中的非平凡对称合并。该理论从错误发现率（FDP）扩展到在可能的真零配置中单调、在报告动作中局部的有界损失。我们还给出一个一致性反例、持续构造以及全局有效的共享控制模型。
英文摘要：
Many modern testing problems are sequential along two axes: new hypotheses may arrive over time, while evidence for hypotheses already under consideration continues to evolve and may be inspected at arbitrary stopping times. We develop dynamic $e$-closure for this setting. At a global stopping time the active true-null intersection is random. Future-extension coherence allows its certificate to be compared with that of a fixed terminal intersection, yielding simultaneous stopped-FDR control. If the certificates are also time-monotone, the resulting closure controls simultaneous SupFDR and is setwise persistent. Conversely, every procedure satisfying either criterion is contained in a dynamic closure generated by canonical normalized-loss processes. For pointwise mergers, fixed-dimensional admissibility is equivalent to ordinary arbitrary-dependence $e$-merging. Coherence across horizons then forces a single globally summable weight sequence and exact neutrality under padding by the $e$-value one; on a countably infinite hypothesis universe, this rules out nontrivial symmetric mergers in the admissible pointwise class. The theory extends from FDP to bounded losses that are monotone in the possible true-null configuration and local in the reported action. We also give a coherence counterexample, persistent constructions, and a globally valid shared-control model.
