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

Abstract

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.

Disclosure

“for ordinary BH after global stopping. Characterizing useful joint sequential conditions under which a stopped BH procedure remains valid (and whether that it is too strong to be useful) is an open question. AI-assisted editing statement OpenAI’s GPT-5.5 Instant was used during development of the manuscript for language and ex- position, generating and refining illustrative examples, assistance with the presentation of proofs, literature searches, and checks of LaTeX cross-references. In p”

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2026-08-23T05:08:31.116351Z

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