{"id":"560ac91a-8546-4f9e-a3d2-d2e1c45dc651","arxiv_id":"2601.11347","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For bounded-mean testing without absolute continuity, GROW and REGROW optimal e-variables exist and are exactly coin-betting e-values Eα(x)=1+α(x−µ0), with α given by explicit formulas.","lead":"The paper derives explicit formulas for optimal e-values — a modern alternative to p-values — when testing the mean of a bounded random variable against composite alternatives. It also identifies which alternative distributions are hardest to distinguish from the null under the GROW and REGROW criteria.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified (only a minor uniqueness overclaim in the abstract)","rationale":"The reader's weakest-assumption analysis correctly identifies Proposition 1 as the pivotal external result on which the paper's reduction depends. However, a close examination shows the proposition is true and even admits an elementary proof from the linear constraints defining mean-µ0 e-variables. The two-line mixture argument above suffices to establish the existence of a dominating coin-betting e-variable, so the central theorems' optimality proofs are not built on shaky ground. I therefore do not find a load-bearing correctness concern. The one substantive issue I noticed is the abstract's 'unique e-values': the paper proves uniqueness of the optimal coin-betting parameter α, but not uniqueness of the e-variable itself. Because the worst-case distributions in Theorem 1 have finite support ({a,b} and {µ1}), pointwise lowering E_α* away from these points yields another e-variable with identical worst-case growth, so uniqueness in the literal sense fails. This does not affect the validity of the explicit e-variables or their optimality; it is an overstatement that can be corrected by referring to the admissible/maximal e-variable. The recommendation to accept remains unchanged, with a request for the authors to soften the uniqueness claim in the abstract and possibly add a remark on non-uniqueness in the non-AC setting.","tokens_in":17419,"tokens_out":35979,"duration_ms":338150,"concrete_test":"Verify Proposition 1 independently: take any e-variable E for P={mean=µ0} and derive α from sup_{y>µ0} (E(y)−1)/(y−µ0) ≤ inf_{x<µ0} (E(x)−1)/(x−µ0), then confirm E(x)≤1+α(x−µ0) for all x and α∈[αmin,αmax]. Also construct a counterexample to uniqueness by reducing E_α* on a point outside {a,b,µ1} (for Theorem 1) and checking it remains an e-variable with the same worst-case growth.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No load-bearing correctness gap identified. The decisive step is Proposition 1, which reduces all e-variables to the coin-betting class; it is external and self-cited, but it is nevertheless true and can be verified directly from the e-variable constraints. For any x<µ0<y, the mixture pδ_x+(1−p)δ_y with mean µ0 gives (y−µ0)E(x)+(µ0−x)E(y) ≤ y−x. Rearranging yields R(x)+S(y)≤0, where R(x)=(E(x)−1)/(µ0−x) and S(y)=(E(y)−1)/(y−µ0), so sup S ≤ −sup R. Choosing α with sup S ≤ α ≤ −sup R gives E(x)≤1+α(x−µ0) for all x, and α automatically lies in [αmin,αmax]. Thus the reliance on Proposition 1 is sound, and the global optimality arguments in Theorems 1–3 are supported. The only imprecision found is the abstract's phrase 'unique e-values': in this non-AC setting, pointwise-dominated versions of E_α* (e.g. lowering E on points outside the finite worst-case supports) also achieve the same worst-case growth. The presented e-variables are still optimal; the claim should say 'admissible' or 'maximal' rather than 'unique'.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper derives explicit GROW and REGROW e-variables for testing the mean of a bounded random variable against composite alternatives, in settings where the hypotheses are not mutually absolutely continuous. The null is the set of distributions with mean equal to (or bounded by) a fixed value, and the alternative is either a point alternative at a different mean, a one-sided interval of means, or the agnostic set of all other means. Using a reduction to the one-parameter coin-betting class (Proposition 1, borrowed from Clerico 2025a) and a c-envelope argument, the authors obtain: Theorem 1, the GROW parameter alpha*_GW = (mu1-mu0)/((mu0-a)(b-mu0)) and a REGROW parameter defined by the fixed-point equation (13); Theorem 2, the analogous one-sided result with closed form (18); and Theorem 3, the agnostic case where GROW is trivial and REGROW has parameter (19). The paper also identifies the worst-case alternative distributions and connects the results to the Bernoulli example.","tokens_in":17723,"tokens_out":14726,"duration_ms":154302,"significance":"The results are significant: they appear to provide the first explicit GROW/REGROW characterisations for non-dominated hypotheses in bounded-mean testing, an open direction noted in the e-value literature. The c-envelope lemma is proved self-containedly, and the derived formulas are simple and falsifiable; in the special case a=0, b=1 they reproduce the Bernoulli solutions of Example 1. The main external input, Proposition 1, is quoted from a paper co-authored by one of the present authors, but it is true and can be verified directly from the e-variable constraints, so I do not regard the reliance as circular. The principal imprecision is the abstract's claim of 'unique' e-values, which the manuscript's own statements do not justify in this non-AC setting.","major_comments":[],"minor_comments":[{"comment":"The abstract says 'the unique e-values with optimal growth rate in the worst case'. In the present non-AC setting the maximisers are not unique: because the worst-case objective only depends on the e-variable through its values on the finite worst-case supports (e.g. {a,b} for GROW in (9), and also mu1 for REGROW), sufficiently small pointwise decreases of E_{alpha*} outside those supports preserve the e-variable property and the same worst-case growth. The theorems identify one optimal e-variable, not a unique one. Please replace 'unique' by 'admissible' or 'maximal' and avoid 'the GROW/REGROW e-variable' where uniqueness is not established.","section":"Abstract; Theorems 1-3"},{"comment":"The global optimality claims in Theorems 1-3 rest on Proposition 1, which is quoted from Clerico (2025a) and not proved here. Because the proposition is both load-bearing and self-cited, I recommend adding a short proof in the appendix or at least stating it as a lemma with proof. The argument is only a few lines: for a mixture p*delta_x+(1-p)*delta_y with mean mu0, the e-variable constraint gives R(x)+S(y) <= 0, yielding the required dominating E_alpha.","section":"Section 2, Proposition 1"},{"comment":"In the case beta in (0, alpha*_RGW), the assertion that beta -> G_{alpha*_RGW,beta}(mu1) has a unique minimizer at beta = alpha*_GW is stated without proof. This is a load-bearing step for (15). Please add the one-line justification: G_{alpha,beta}(mu1) equals L(log E_alpha)(mu1) - L(log E_beta)(mu1), and L(log E_beta)(mu1) is exactly the function maximized at alpha*_GW in the GROW computation.","section":"Appendix, proof of Theorem 1 around Eq. (15)"},{"comment":"In the existence argument for tilde-alpha*_RGW, the expressions 'F_{alpha*_GW,alpha_max}(1)' and 'F_{alpha_max,alpha_max}(1)' should use the right endpoint b of the support, not 1; the support is [a,b]. This appears to be a leftover from the Bernoulli normalization a=0, b=1.","section":"Proof of Theorem 2"},{"comment":"In the chain of inequalities for alpha > alpha*_RGW, the text says 'where we used the monotonicity of alpha -> G_{alpha,alpha_max}(mu1) in the second inequality'. The relevant function is alpha -> G_{alpha,alpha*_GW}(mu1), as in the existence argument. Please correct the typo.","section":"Proof of Theorem 1, Section 4.1"}],"recommendation":"minor_revision","confidential_remarks":"The mathematical core is sound and the central formulas are credible. The only substantive issue is the uniqueness overclaim in the abstract; otherwise the requested changes are local. I would ask the authors to include a short proof of Proposition 1 in the revision, both for self-containedness and to remove any appearance of circularity from the self-citation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does what it says: it gives the first explicit GROW and REGROW e-variables for bounded-mean testing against composite alternatives, outside the absolute-continuity umbrella. The formulas are concrete, the worst-case distributions are identified, and the Bernoulli example is a nice sanity check that the general results reduce correctly. The reduction to the one-parameter coin-betting class is the load-bearing step, and it is borrowed from the second author's earlier paper. I know that is a yellow flag on its face, but I checked the reduction directly: for any e-variable for the mean-equal-to-µ0 null, the two-point mixtures with mean µ0 force a pointwise domination inequality that yields the coin-betting envelope. So Proposition 1 is true, and the reliance on it is not a hole. The paper would still benefit from a self-contained proof, since it is the crux of the global optimality claims and the provenance is self-citational.\n\nThe main soft spot is the abstract's promise of \"unique\" e-values. In this non-dominated setting, uniqueness only holds up to pointwise domination: you can lower an optimal e-variable on points outside the worst-case support and still achieve the same worst-case growth. The right claim is 'admissible' or 'maximal' rather than 'unique'. That is a wording fix, not a structural flaw. There are also a few endpoint inequalities used to pin down alpha*_RGW that are asserted rather than shown; the case analysis in the appendix is plausible but not fully rigorous as written. It looks fixable. There is also a minor boundary slip in Theorem 2 (F evaluated at 1 during the existence argument), which does not affect the result.\n\nTheorems 1–3 are otherwise coherent, the worst-case interpretation in Section 4.4 is genuinely informative, and the comparison to the Bernoulli case helps the reader see what is going on. The proofs are mostly self-contained and the paper is honest about what is external.\n\nThis will be of real value to people working on safe testing, e-variable design, and optimality criteria for composite alternatives. It deserves a serious referee. My recommendation: send it out, but ask the authors to fix the uniqueness claim, add a proof or a precise citation-to-appendix for Proposition 1, and tighten the endpoint arguments.","headline":"Solid, genuinely new result in e-value theory: the first explicit (RE)GROW characterizations for a non-AC problem, but the abstract's uniqueness claim is stronger than what is proved.","tokens_in":18177,"tokens_out":1073,"would_cite":true,"duration_ms":13619,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62F03","62G10"],"pacs":[],"model":"deepseek-v4-flash","headline":"A single linear bet gives the optimal e-variable for bounded-mean tests.","keywords":["e-values","GROW","REGROW","bounded mean testing","coin-betting e-class","composite alternative","convex envelope","non-dominated hypotheses"],"falsifier":"Run a finite-dimensional optimization over all valid e-variables on a discrete grid of support points (e.g., X∈{−1,0,1} with null mean 0 and alternative mean 1/2), maximizing the worst-case expected log-growth, and compare the optimum to the paper's E_α* formula; any e-variable with a strictly larger worst-case value would falsify the completeness reduction on which the claim rests.","tokens_in":37,"feed_emoji":"📈","tokens_out":9399,"duration_ms":167359,"temperature":0.7,"pith_summary":"The paper derives, for the first time outside the usual mutually-absolutely-continuous setting, the e-variables that are optimal for testing the mean of a bounded random variable under the GROW and REGROW criteria. Its central claim is that for a null hypothesis fixing the mean at µ0 and an alternative fixing it at µ1, the worst-case growth-optimal e-variable is the coin-betting function E(x)=1+α(x−µ0) with α=(µ1−µ0)/((µ0−a)(b−µ0)), and a unique REGROW e-variable of the same linear form exists, determined by an explicit equation. For the agnostic alternative that excludes only the null mean, the paper shows the GROW criterion degenerates to the constant e-variable 1, while REGROW still yields a non-trivial linear e-variable. These closed forms matter because they turn a previously abstract optimality criterion into concrete betting factors, and they show that REGROW can remain informative where the more pessimistic GROW criterion gives up.","feed_headline":"One linear bet is the optimal e-variable for bounded-mean tests","feed_subtitle":"Closed-form betting factors now exist for GROW and REGROW, including an agnostic case where GROW degenerates to a constant.","key_machinery":"The load-bearing object is the coin-betting e-class, the one-parameter family E_α(x)=1+α(x−µ0), α∈[αmin,αmax], which are valid e-variables for the mean-µ0 null. A completeness theorem quoted from the authors' companion work says this family is minimal complete: every valid e-variable is pointwise dominated by some E_α. That reduction turns the infinite-dimensional search for a (RE)GROW e-variable into a one- or two-dimensional optimization over α and β. The second tool is the convex envelope f^c(x)=inf_{Q:E_Q[X]=x}E_Q[f], which by Lemma 1 is either f itself or the affine interpolation between endpoint values; this converts worst-case expectations over mean-constrained distributions into trac","core_discovery":"The paper establishes explicit solutions to the (RE)GROW optimality problem for single-round tests of the mean of X∈[a,b]. For the point-null/point-alternative pair {E[X]=µ0} vs {E[X]=µ1}, the GROW e-variable exists and is E_{α*_GW}(x)=1+α*_GW(x−µ0), with α*_GW=(µ1−µ0)/((µ0−a)(b−µ0)). The REGROW e-variable also exists and is the same linear form with a unique parameter α*_RGW, determined for µ1>µ0 by solving F_{α*_RGW,αmax}(µ1)=G_{α*_RGW,α*_GW}(µ1); the case µ1<µ0 is symmetric. For one-sided hypotheses (mean ≤ µ0 vs mean > µ1), the GROW parameter has the same formula, while REGROW has the closed-form parameter given in the paper. For the agnostic alternative {E[X]≠µ0}, the paper shows that e","pith_inferences":["An implication the authors leave implicit is that the explicit linear factors multiply under independent observations, so they could be plugged into a product e-value to form an anytime-valid confidence sequence for the mean; the paper notes the product form but does not prove worst-case optimality for such a sequential strategy.","A testable extension is to run a finite-dimensional grid optimization over all valid e-variables on a discrete support and compare the worst-case expected log-growth with the paper's formula; this would directly check the completeness reduction in a concrete case.","If the quoted completeness theorem were extended to other linear constraints, the same convex-envelope proof scheme would yield explicit (RE)GROW e-variables for moment-constrained hypotheses; the paper does not pursue this, but the structure of the proof suggests it."],"forward_implications":["If correct, the paper gives explicit, closed-form e-variables for bounded-mean testing: a practitioner can set the betting parameter directly from µ0, µ1, and the support endpoints a,b.","For the agnostic alternative {mean ≠ µ0}, GROW-type worst-case optimality is vacuous, while REGROW still produces a non-trivial linear e-variable, sharpening the case for REGROW as the more useful criterion in this problem.","The optimal e-variables are always of the linear coin-betting form even though the null contains discrete, continuous, and singular distributions, providing a concrete minimal complete class for the stated testing problems.","The worst-case alternatives for the composite alternatives are all supported on at most two points—the boundary atoms δ_a and δ_b for the one-sided and agnostic problems—giving a simple picture of which alternatives are hardest to distinguish from the null.","For the Bernoulli special case (a=0, b=1), the paper's REGROW formulas reproduce the explicitly computed Bernoulli solutions, confirming consistency of the general theory with the simplest example."],"fun_headline_variants":["Explicit optimal e-values for bounded-mean tests","First explicit (RE)GROW e-values for bounded means","Closed-form betting factors for testing bounded means","When GROW fails, REGROW finds the optimal e-variable","Optimal e-values solved for bounded random variables"],"cache_read_input_tokens":19584,"weakest_assumption_plain":"The global optimality of every formula in the paper rests on the quoted completeness theorem (Proposition 1)—that every valid e-variable for the mean-µ0 null is pointwise dominated by some linear coin-betting variable 1+α(x−µ0)—which is stated as a corollary of the authors' companion work and not proved here.","fun_headline_variants_meta":{"raw":{"variants":["Explicit optimal e-values for bounded-mean tests","First explicit (RE)GROW e-values for bounded means","Closed-form betting factors for testing bounded means","When GROW fails, REGROW finds the optimal e-variable","Optimal e-values solved for bounded random variables"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000184,"raw_usage":{"total_tokens":1145,"prompt_tokens":726,"completion_tokens":419,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":470,"completion_tokens_details":{"reasoning_tokens":355}},"tokens_in":470,"tokens_out":419,"duration_ms":4403,"temperature":1.0,"reasoning_tokens":355,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T10:01:25.659986+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a finite-dimensional optimization over all valid e-variables on a discrete grid of support points (e.g., X∈{−1,0,1} with null mean 0 and alternative mean 1/2), maximizing the worst-case expected log-growth, and compare the optimum to the paper's E_α* formula; any e-variable with a strictly larger worst-case value would falsify the completeness reduction on which the claim rests.","supporting_citations":[],"review_version":1}