Dynamic e-closure for online hypotheses with any-time-valid evidence: closure principles and projective mergers

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.