{"id":"bebda1ff-71de-46fe-9e92-a584ec3418ce","arxiv_id":"2608.09927","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Dynamic e-closure controls stopped and supremum FDR for growing hypothesis families with continuing evidence, and shows coherent admissible pointwise mergers require one global weight sequence.","lead":"A new statistical framework, dynamic e-closure, controls false discovery rates when hypotheses arrive over time and evidence keeps updating. It proves that the only admissible pointwise evidence-merging rules spanning all horizons need a single global weight budget, which forces symmetric rules to become trivial on open-ended universes.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The theory's forward guarantee depends on deterministic active sets and global-filtration validity; under random hypothesis arrivals, Lemma 3.4 has no fixed terminal intersection, so the stopped-FDR conclusion is unsupported in the motivating setting.","rationale":"The reader's weakest assumption points at exactly the same structural premise: deterministic finite active sets and global-filtration validity. I see no flaw in the closure algebra itself. The forward proof of Theorem 3.6 is a direct application of Lemma 3.4, and the maximal statement in Lemma 3.4 is sound: the T-to-infinity passage uses monotone convergence on the increasing variables M_T = sup_{t≤T} Z_t, so it does not require the terminal e-processes themselves to converge. The converse representation theorems and the cross-horizon rigidity theorem also check out; the coefficient comparison in Theorem 4.8 is valid. The only genuinely load-bearing limitation is that Lemma 3.4 breaks when active sets are random, because then no fixed terminal intersection N_T(P) exists to receive optional stopping. The paper states this limitation explicitly and does not claim a random-arrival theorem, so this is a scope condition rather than a hidden error. I therefore recommend leaving the verdict unchanged: ACCEPT with moderate confidence remains appropriate, with the caveat that the practical online guarantee is limited to deterministic arrival schedules unless a new localization argument is developed.","tokens_in":29120,"tokens_out":24708,"duration_ms":229042,"concrete_test":"Construct a two-horizon example with random active sets: let I_1 be {1} or {2} according to a fair coin, set I_2 = {1,2}, and define globally valid e-processes E^S for every finite S satisfying future-extension coherence pathwise at every realization. Define a global stopping time that certifies an all-null rejection set selected from the active intersection, and compute the stopped FDR. If the stopped FDR exceeds alpha, Lemma 3.4's deterministic-active-set premise is genuinely essential and the scope limitation is real. If it does not, attempt a replacement localization proof under random I_t with I_t independent of the data, which would show the limitation is avoidable.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central forward theorem (Theorem 3.6) rests on Lemma 3.4, which is the only place where hypothesis arrival interacts with optional stopping. For a bounded stopping time sigma_T = tau ∧ T, the lemma uses N_sigma_T(P) = N_T(P) ∩ I_sigma_T; because I_t is deterministic and finite, N_T(P) is a fixed finite set and E^{N_T(P)} is a fixed e-process to which optional stopping applies. This step fails when arrivals are random predictable: I_T, and hence N_T(P), is random, so there is no fixed terminal intersection available, and future-extension coherence alone does not supply a substitute. The paper acknowledges this in Section 8.1 and Section 2.1, but it remains the weakest premise of the headline stopped-FDR and SupFDR claims. The same assumption requires all coordinate and intersection processes to be valid in the single global filtration used for stopping; Section 4.4's lifting is potentially conservative. None of this is an internal inconsistency: Theorem 3.6 is explicitly conditional on deterministic active sets. But it means the advertised 'new hypotheses may arrive over time' setting is covered only for deterministic arrival schedules, and the motivating platform-trial application typically has random arm additions.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"Many modern testing problems are sequential along two axes: new hypotheses may arrive over time, and evidence for existing hypotheses continues to evolve and may be inspected at arbitrary global stopping times. The paper develops dynamic e-closure for this setting under explicit assumptions of deterministic finite active sets I_t (Assumption (2.3)) and validity of all coordinate and intersection processes in a single global filtration.","tokens_in":29350,"tokens_out":39524,"duration_ms":311565,"significance":"Within its stated scope, this is a substantial contribution to e-process multiple testing. I spot-checked the localization lemma, the necessity counterexample, the rigidity coefficient argument, and the stopped-BH example and found no internal errors; the appendix contains complete proofs of all main results. The necessary-and-sufficient representation up to containment cleanly extends fixed-time e-closure to the joint online/any-time setting, and Theorem 4.8 sharply delimits what admissible pointwise arbitrary-dependence mergers can achieve under horizon coherence. The paper is well documented for a theory paper: complete proofs, constructive counterexamples, and explicit limitation statements in Section 8.1. The main caveat on significance is that the headline stopped-FDR and SupFDR guarantees require deterministic finite arrival schedules and a single global filtration; the platform-trial motivation in Section 1.1 involves random arm additions, deferred to future work in Section 8.2. This is a disclosed scope restriction rather than an internal inconsistency, but the abstract should carry the restriction explicitly.","major_comments":[],"minor_comments":[{"comment":"The forward theorems are explicitly conditional on deterministic finite active sets (Assumption (2.3)) and on global-filtration validity, and Section 8.1 discloses this; nevertheless, the abstract's opening sentence and the platform-trial motivation in Section 1.1 invite the reader to expect coverage of random hypothesis arrivals, a setting in which Lemma 3.4's fixed-terminal-intersection comparison has no immediate substitute. The stress-test concern about random predictable arrivals therefore lands as a real but disclosed scope limitation; I recommend stating the deterministic-arrival restriction explicitly in the abstract and in Section 1.1.","section":"Abstract; §1.1; §2.1; §8.1"},{"comment":"The reduction claim that the online SupFDR e-closure theorem of Xu et al. [2026] is recovered as a one-axis special case of Theorem 3.7 is asserted without verifying that the reduced process (E-hat)^S_t := E^{S ∩ [t]} is an e-process in the global filtration when the index S ∩ [τ] is random at a stopping time τ. The claim is correct: for bounded τ ≤ T, increasingness of the online e-collection gives E^{S ∩ [τ]} ≤ E^S pointwise, so E[(E-hat)^S_{τ∧T}] ≤ 1, and Fatou extends this to unbounded almost surely finite τ; adding this one-line justification, or a pointer to the corresponding property in Xu et al. [2026], would make the remark self-contained.","section":"Remark 3.8"},{"comment":"In the stopped-BH counterexample, the step-up rule k* = max{k : p_(k) ≤ kα/m} is what makes BH reject both hypotheses at time three when (P_{1,3}, P_{2,3}) = (α, α), even though p_(1) = α > α/2; stating this rule explicitly would prevent readers from concluding that the individual threshold p_(1) ≤ α/2 must also hold.","section":"§6.4"},{"comment":"The abstract as rendered contains spacing and hyphenation artifacts ('Dynamice-closure', 'forpointwisemergers', 'ordinaryarbitrarydependencee-merging') that should be cleaned before submission; if these artifacts come from the extraction rather than the source, the authors should confirm that the source compiles cleanly.","section":"Abstract"},{"comment":"In the AI-assisted editing statement, 'the LLMh' appears to be a typo for 'the LLM'.","section":"AI-assisted editing statement"}],"recommendation":"minor_revision","confidential_remarks":"The paper is a good fit for a mathematical statistics journal: the contributions are definitional and structural, the proofs are complete and checkable, and the paper builds transparently on very recent work (Xu et al. 2025, 2026; Wang 2025; Choe and Ramdas 2026; Tavyrikov et al. 2026) with clear differentiation in Section 1.6. I have no concerns about citation practice, novelty disclosure, or the disclosed AI-assisted editing statement. The only substantive reservation is the deterministic-arrival scope, which is disclosed in the manuscript and would be better reflected in the abstract; I support publication after the minor revisions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe one thing to know: this paper is the real unification of fixed-time e-closure and online e-closure, and the cross-horizon rigidity theorem is the part I'd remember in six months. It proves forward and converse dynamic closure principles for stopped FDR and SupFDR, with localization handling the random active true-null intersection. The converse uses normalized loss certificates, which is the standard representation trick, not circular.\n\nWhat is genuinely new: the two-axis setup with a single global filtration and deterministic active sets, future-extension coherence, and the localization lemma that replaces the random intersection index at stopping time with a fixed terminal intersection. The rigidity theorem says that admissible pointwise arbitrary-dependence mergers across horizons force a single globally summable weight sequence, and on a countably infinite universe symmetric nontrivial mergers are impossible. That is a clean structural result, and it is not in the prior literature the paper cites. The proofs in the appendix are complete and I did not find internal errors.\n\nThe soft spot, in proportion: the forward theory only works when the active sets are deterministic and all processes are valid in the global filtration used for stopping. As the stress-test note says, under random predictable arrivals the terminal true-null set is random and Lemma 3.4 does not have a fixed intersection to stop, so the stopped-FDR conclusion is unsupported in that case. The paper does not hide this: Section 8.1 and Section 2.1 state it plainly, and the future-work section points to the need for a different localization argument. So it is a clear scope condition rather than a hidden flaw. The only caveat is presentation: the motivating platform-trial story often involves random arm additions, so the reader should not buy the full motivational packaging. But the theorems are correctly stated for deterministic schedules.\n\nThe merger results are also narrower than they first look: they apply to pointwise mergers valid under arbitrary dependence. The paper says so and notes that structured dependence or history-dependent mergers can get around the rigidity. Good.\n\nMy verdict: solid, honest theory paper. The central claims hold under the stated assumptions, the limitations are explicit, and the open problems are precise. It deserves serious peer review. I'd cite it and bring it to reading group.","headline":"A genuine unification of fixed-time and online e-closure with a strong merger rigidity theorem, hampered only by the explicitly stated deterministic-arrival scope condition.","tokens_in":29852,"tokens_out":2195,"would_cite":true,"duration_ms":20513,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62L10","62J15","62F03"],"pacs":[],"model":"deepseek-v4-flash","headline":"A coherence condition called future-extension coherence lets dynamic e-closure certify rejection sets that remain valid at arbitrary global stopping times, even as hypotheses arrive and evidence evolves.","keywords":["dynamic e-closure","online hypotheses","any-time-valid evidence","stopped FDR","SupFDR","e-process mergers","future-extension coherence","neutral-padding projectivity"],"falsifier":"A concrete counterexample would be a future-extension coherent dynamic intersection e-process collection, with deterministic finite active sets and all processes valid in one global filtration, whose dynamic closure at an almost surely finite global stopping time has expected FDP greater than alpha; such an example would refute Theorem 3.6, while Proposition 3.2 already shows that without coherence the failure is real.","tokens_in":28929,"feed_emoji":"📈","tokens_out":7190,"duration_ms":59515,"temperature":0.7,"pith_summary":"This paper develops closure for multiple testing when both axes are sequential: new hypotheses arrive over time, and evidence for active hypotheses keeps updating and may be inspected at arbitrary stopping times. It proves that a single condition—future-extension coherence, meaning adding a dormant hypothesis never lowers the current intersection certificate—makes the closed family of certified rejection sets control the false discovery rate at every global stopping time, and additionally control the supremum over time when certificates are time-monotone. The result is sharp: every procedure that satisfies the error criterion is contained in a closure generated by its own normalized loss certificates, so closure is necessary and sufficient up to containment. The same logic extends from FDR to bounded losses that are monotone in the true-null configuration and local in the reported action.","feed_headline":"One coherence rule keeps FDR control as hypotheses arrive","feed_subtitle":"One coherence condition turns per-intersection e-processes into stopped-FDR and SupFDR guarantees.","key_machinery":"The engine is the random-intersection localization lemma. It compares the certificate of the random active true-null intersection at a stopping time with one fixed terminal-intersection e-process, using the fact that the active true-null set at a bounded stopping time is the terminal true-null set intersected with the active set; future-extension coherence then bounds the random-index certificate by the fixed-index process, to which optional stopping applies. The converse direction uses the canonical current-loss and running-loss processes, normalized by alpha, which are the pointwise smallest coherent certificates covering a given procedure.","core_discovery":"The central discovery is a dynamic closure principle for simultaneous stopped FDR and SupFDR. For a future-extension coherent dynamic intersection e-process collection, the candidate family of rejection sets defined by the closure constraints controls simultaneous stopped FDR at level alpha (Theorem 3.6); with time-monotone certificates it controls simultaneous SupFDR and the family is setwise persistent (Theorem 3.7). Conversely, any candidate-family process satisfying either guarantee is contained in a closure generated by canonical current-loss or running-loss e-processes (Theorems 3.10, 3.11). On the construction side, coherent pointwise mergers of arbitrary e-processes are exactly the affine rules with a single globally summable weight sequence, so arithmetic-mean mergers that renormalize per horizon break coherence (Theorem 4.8).","pith_inferences":["The localization premise (deterministic finite active sets, single global filtration) is where the theory is most exposed; random predictable arrivals likely require a random-terminal-set version of the localization lemma, which the paper explicitly leaves open.","A practical consequence the paper states but does not emphasize: designs should commit to a global weight budget before the hypothesis family is known, since per-horizon renormalization is incoherent; platform-trial protocols can test this by comparing pre-specified versus renormalized weights in simulation.","The rigidity result suggests that dependence structure between hypotheses is the natural route to richer coherent mergers; under arbitrary dependence the admissible class is exhausted, but under known correlation structure (like shared-control arms) non-affine coherent mergers may exist."],"forward_implications":["Any procedure that controls simultaneous stopped FDR can be represented, up to containment, as a dynamic e-closure from coherent intersection evidence, so the closure framework is complete for this guarantee.","If the intersection certificates are time-monotone, the same closure is setwise persistent and controls simultaneous SupFDR, making previously certified rejection sets remain certified.","On an open-ended hypothesis universe, admissible pointwise mergers under arbitrary dependence must be affine with one globally summable weight budget; renormalizing weights when new hypotheses arrive is impossible within this class.","On a countably infinite universe, no nontrivial symmetric admissible pointwise merger family is coherent under neutral padding; only the trivial merger remains.","The results are not specific to the false discovery proportion: bounded configuration-monotone local losses inherit the same closure, representation, and minimality results."],"supporting_citations":[{"why":"Supplies fixed-time e-closure and the up-to-containment representation used as the template for the dynamic theorems.","marker":"[Xu et al., 2025]"},{"why":"Characterizes admissible arbitrary-dependence e-merging functions, giving the affine form used in Theorems 4.2 and 4.8.","marker":"[Wang, 2025]"},{"why":"Introduces the online closure predictability condition on which future-extension coherence is modelled.","marker":"[Fischer et al., 2024a]"},{"why":"Establishes the stopped e-BH procedure and the global stopping-time validity criterion that this paper generalizes.","marker":"[Wang et al., 2025]"},{"why":"Provides the filtration-lifting construction used to turn local-filtration e-processes into globally valid ones.","marker":"[Choe and Ramdas, 2026]"},{"why":"Gives the adjuster theorem for running maxima used to build time-monotone persistent e-processes.","marker":"[Dawid et al., 2011]"},{"why":"Defines e-merging functions and the neutral value one under padding, the backdrop for neutral-padding projectivity.","marker":"[Vovk and Wang, 2021]"},{"why":"Develops the one-axis online SupFDR e-closure recovered here as a special case and serves as the main comparison point.","marker":"[Xu et al., 2026]"},{"why":"Formulates persistent testing via running maxima and the setwise persistence property used in the paper.","marker":"[Tavyrikov et al., 2026]"}],"fun_headline_variants":["Dynamic e-closure tames online FDR with arbitrary stopping times","One coherence rule gives simultaneous stopped-FDR and SupFDR","Coherent e-mergers must be affine, banning renormalized averages","On infinite hypotheses, no nontrivial symmetric coherent e-mergers","Future-extension coherence locks in FDR control under arrivals"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole theory assumes the active hypothesis set at every time is deterministic and finite, and that every coordinate or intersection e-process is valid in one common global filtration; if hypothesis arrivals are random or processes are valid only in local filtrations, the localization step that converts the random true-null intersection into a fixed terminal intersection fails.","fun_headline_variants_meta":{"raw":{"variants":["Dynamic e-closure tames online FDR with arbitrary stopping times","One coherence rule gives simultaneous stopped-FDR and SupFDR","Coherent e-mergers must be affine, banning renormalized averages","On infinite hypotheses, no nontrivial symmetric coherent e-mergers","Future-extension coherence locks in FDR control under arrivals"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000288,"raw_usage":{"total_tokens":1683,"prompt_tokens":931,"completion_tokens":752,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":547,"completion_tokens_details":{"reasoning_tokens":663}},"tokens_in":547,"tokens_out":752,"duration_ms":7742,"temperature":1.0,"reasoning_tokens":663,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T04:13:21.341711+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete counterexample would be a future-extension coherent dynamic intersection e-process collection, with deterministic finite active sets and all processes valid in one global filtration, whose dynamic closure at an almost surely finite global stopping time has expected FDP greater than alpha; such an example would refute Theorem 3.6, while Proposition 3.2 already shows that without coherence the failure is real.","supporting_citations":[],"review_version":1}