{"id":"26361dec-918a-47b7-a057-388483e03b75","arxiv_id":"2608.12291","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a broad class of win-martingale belief processes, the paper characterizes the optimal stopping rule by a free boundary, proves the boundary is smooth before the horizon, and derives a unique integral equation for it.","lead":"This paper works out when to stop watching a live probability estimate and commit to a binary call, balancing the cost of waiting against the cost of being wrong. It gives a general mathematical recipe, with proofs, for the best stopping rule in a broad class of belief models used in prediction markets and sequential testing.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Core proof appears coherent, but the claimed applicability to the Aldous, Bass, and binary sequential-inference examples rests on unverified checks of Assumption B, especially B.(v) for Bass and B.(ii) for cross-entropy.","rationale":"The reader's verdict is CONDITIONAL, and I agree with that verdict, but I do not think the early-termination limitation flagged as the weakest assumption is the main load-bearing concern: the paper explicitly and transparently restricts attention to processes satisfying the Feller condition, so the exclusion of Wright-Fisher and absorbed Brownian motion is a stated scope choice, not a gap in the argument under Assumption A. The more consequential issue is that the theorems are conditional on Assumption B, while the paper's headline claim that Aldous, Bass, and binary sequential-inference martingales are covered is supported only by assertions such as 'It can easily be verified that Assumption B holds.' The most delicate part of Assumption B is the true-martingale condition on the stochastic flow (B.(v)), which is not verified in detail for the Bass martingale and is not obvious there because σ_B' is unbounded; the authors cite a Lamperti-drift sufficient condition but omit the calculation. Assumption B.(ii) for the cross-entropy terminal cost is also asserted rather than shown. This does not undermine the internal validity of the proofs under the assumptions, but it does mean the scope of the central claim is not fully established. A concrete verification of B.(ii) and B.(v) for the listed examples would settle the matter. My recommended verdict is therefore UNCHANGED: the conditional verdict is appropriate, with the requested verification being the natural condition for acceptance.","tokens_in":48680,"tokens_out":30597,"duration_ms":299340,"concrete_test":"For each canonical pair (σ,g) in Section 8, compute Lg(x)=1/2 σ(x)^2 g''(x) and verify analytically or numerically that ∂_x Lg(x)<0 on (0,1/2), ∂_x Lg(x)>0 on (1/2,1), and that Lg(1/2)+c(t)<0 for the stated running costs. For Assumption B.(v) in the Bass case, write the Lamperti drift μ(l)=l/2 and verify the flow derivative is a true martingale by checking E[∂_x X^x_t]=1 for t>0, e.g. using the identity E[φ(L_t)] = φ(l)e^{-t/2}; repeat the flow-martingale check for σ_A(x)=sin(πx)/π and σ_I(x)=x(1-x). If any check fails, restrict the claimed applicability to the verified subclasses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The main theorems (4.1, 4.2, 4.4, 4.5) are conditional on Assumption B, and the paper's motivating claim is that these theorems cover the Aldous, Bass, and binary sequential-inference martingales. That applicability is asserted, not demonstrated. The most delicate condition is Assumption B.(v), which requires the stochastic flow derivative ∂_x X^x_t to be a true martingale on finite time intervals; it is used for the derivative representations (6.6), (6.21), and for the concavity argument in Lemma 6.2, hence for smooth fit and the C^1 regularity of the boundary. For the Bass martingale σ_B(x)=φ(Φ^{-1}(x)), the function σ_B' is unbounded at the endpoints, so the standard Novikov condition does not apply directly. The paper notes in §3.2 that linear growth of the Lamperti drift μ(l) suffices via a Beneš-type argument, but it does not display μ or the verification that this condition holds. Likewise, §8 states that the cross-entropy examples satisfy Assumption B, but the required monotonicity of ∂_x Lg in Assumption B.(ii) is not checked for g_CE with σ_A, σ_B, or σ_I. Because these verifications are load-bearing for the claimed scope, a reader cannot currently determine whether the canonical examples are inside the theorem's hypotheses. This is a scope gap rather than a detected contradiction in the proofs; the internal argument under Assumptions A and B appears substantially sound.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a sequential decision problem in which a decision maker observes a binary-outcome win-martingale, modeled as a [0,1]-valued martingale posterior, and chooses a stopping time and terminal declaration under a running cost and a terminal loss. After reducing the Bayes risk to an optimal stopping problem for the posterior, the authors apply a deterministic time change to convert separable volatility into a time-homogeneous diffusion with a time-inhomogeneous running cost. Under structural Assumptions A and B, they characterize the continuation region as a symmetric interval with a free boundary, prove C^1 regularity of the value function and C^1 regularity of the boundary on [0,T), derive a nonlinear integral equation uniquely characterizing the boundary, and treat finite and infinite horizons without imposing monotonicity of the running cost. The claimed applications include the Aldous, Bass, and binary sequential-inference martingales, with an example where the optimal boundary oscillates and has no limit in calendar time.","tokens_in":48930,"tokens_out":7513,"duration_ms":67573,"significance":"If the main theorems hold under their stated hypotheses, the paper would provide a substantial and unified free-boundary theory for a broad class of posterior martingale stopping problems, extending earlier Bayesian sequential testing results and accommodating nonmonotone running costs and nonmonotone boundaries. The proof strategy is generally careful and self-contained: the reduction to a free-boundary problem, the use of De Angelis–Peskir and De Angelis–Lamberton regularity results, and the derivation of the integral equation are coherent. The paper also makes a useful decision-theoretic connection between martingale-distance models of \"exciting games\" and concrete sequential prediction. The central theorems are conditional on Assumption B, however, and the advertised applicability to the canonical examples is asserted rather than demonstrated, which is a significant scope gap. The paper is honest about its main limitation, the exclusion of finite-time absorption at 0 and 1, but the internal verification of the assumptions for the motivating examples is missing.","major_comments":[{"comment":"The paper claims that the Aldous, Bass, and binary sequential-inference martingales satisfy Assumption B, but the verification of Assumption B.(v) for the Bass martingale is not supplied. The text in §3.2 states that linear growth of the Lamperti drift μ(l) suffices via a Beneš-type argument and that this indeed holds for the Bass win-martingale, but μ is never displayed and the growth or Lipschitz check is not carried out. Since Assumption B.(v) is used in Lemma 6.2, in the derivative representations (6.6) and (6.21), and in the smooth-fit arguments of Propositions 6.22 and 6.23, the reader cannot currently verify that the Bass example is inside the hypotheses of Theorems 4.1–4.5.","section":"§8 and §3.2, Assumption B.(v)"},{"comment":"Example 8.5 uses the cross-entropy terminal cost g_CE with the Aldous martingale and states that Assumption B holds, but Assumption B.(ii) is not checked. For g_CE one has Lg = -σ^2/(2x(1-x)), so the required sign change of ∂_x Lg on (0,x0) and (x0,1) is not immediate for σ_A or σ_B and is not demonstrated in the paper. Because B.(ii) is load-bearing for the interval structure of the continuation region in Proposition 6.13 and for the strict positivity used in Lemma 6.15, the oscillatory-boundary example of §8.5 is not rigorously supported as stated.","section":"§8.5, Assumption B.(ii)"},{"comment":"The text says \"It can be verified that all of the examples satisfy Assumption B\" and \"It can easily be verified that Assumption B holds,\" but no verification or reference to supplementary material is provided. This is a load-bearing point because the abstract and introduction claim a common solution theory for the Aldous, Bass, and binary sequential-inference martingales, while the main theorems are conditional on Assumption B. I would ask the authors to either provide explicit checks of B.(ii), B.(iii), and B.(v) for each claimed example, or state clearly which examples are only heuristic.","section":"§8.4, Examples 8.4 and 8.5"}],"minor_comments":[{"comment":"The definition of the stochastic flow derivative ∂_x X^x_t is deferred to equation (3.14), which appears after Assumption B.(v) is stated; please add a forward cross-reference at the assumption.","section":"§3.2, Assumption B.(v)"},{"comment":"In Theorem 4.5, for T=∞ the term g(X^x_{T-t}) should be explicitly written as g(X^x_∞), since T-t is infinite; the current notation is formally ambiguous even though the subsequent sentence clarifies it.","section":"Eq. (4.2), infinite-horizon case"},{"comment":"In the proof of Proposition 6.23, the interval notation J=[a,a] appears degenerate because the underline and overline on the endpoints are lost; please typeset the endpoints as a_* and a^* or as a and overline{a} to make the interval nondegenerate.","section":"Proposition 6.23"},{"comment":"The captions in the examples refer to dashed curves as γ and 1-γ, but in the printed text the dashed and solid styles are not always clearly distinguished; please ensure all curves in Figures 1–3 are labeled consistently.","section":"§8.5 and Figure 3"}],"recommendation":"major_revision","confidential_remarks":"The paper is substantial and the main conditional theorems appear coherent, but the advertised scope rests on unverified checks of Assumption B for the canonical examples. I would recommend insisting on explicit verification of B.(ii) and B.(v) for the Aldous, Bass, and sequential-inference martingales before publication, and possibly asking an expert in stochastic flows to scrutinize Lemma 6.2, since the true-martingale property of the flow derivative is doing heavy lifting there."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, this is a real contribution: it gives a unified optimal-stopping/free-boundary treatment for posterior martingales with separable volatility, covering finite and infinite horizons, with C^1 boundaries, smooth fit, and a unique integral-equation characterization, and it does not require monotone running costs. Second, the central proofs are conditional on Assumption B, and the paper's claim that the Aldous, Bass, and binary sequential-inference martingales satisfy those assumptions is asserted rather than checked. That is a genuine gap in scope, not a detected contradiction in the argument.\n\nWhat's new: the structural theorem (continuation region is an open interval around 1/2), the C^1 regularity of the boundary built on De Angelis–Peskir–Lamberton technology, and the nonlinear integral equation with uniqueness. The oscillatory example with a boundary that has no limit as the horizon approaches is neat and illustrates the nonmonotone-cost flexibility. The paper is also honest about what it excludes: absorbing endpoints in finite time (Wright–Fisher, absorbed Brownian motion) are explicitly left out, and the conclusion says so.\n\nWhere it's soft: the examples section says \"It can be verified that all of the examples satisfy Assumption B\" but doesn't do the verification. The delicate condition is B.(v) (the stochastic flow derivative being a true martingale). For the Bass martingale, sigma_B' is unbounded at the endpoints, so Novikov is not immediate; the paper mentions a linear-growth condition on the Lamperti drift as a sufficient condition, but it never displays mu(l) or checks the condition. Similarly, for the cross-entropy loss, Assumption B.(ii) requires strict monotonicity of ∂_x Lg, and that's not checked for g_CE with the three volatility coefficients. Since the abstract sells these canonical examples as special cases, the reader can't currently tell whether the theorem actually covers them. This is fixable: a short appendix or a table with explicit checks would close the gap. The infinite-horizon example as presented might also have some hidden requirements, but I didn't find a problem there.\n\nAlso, the dependency on [9] for the homogeneous result and on [10], [11], [27] is standard and transparent.\n\nBottom line: the core theory under Assumptions A–B is coherent and valuable. The paper deserves a serious referee; the referee should push for the missing verification of Assumption B in the examples before publication. I'd bring it to reading group and would cite the general theorems, but I'd hold off citing the example coverage until the checks are explicit.","headline":"A genuine free-boundary theory for a broad class of win-martingales under explicit assumptions; the advertised examples need verification, but the core proof is coherent.","tokens_in":49508,"tokens_out":2447,"would_cite":true,"duration_ms":21036,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G40","62L15","62L10","60G35","35R35"],"pacs":[],"model":"deepseek-v4-flash","headline":"For a broad class of win-martingales, the optimal decision time is the first exit from a symmetric interval around 1/2.","keywords":["sequential inference","win-martingales","prediction markets","optimal stopping","free-boundary problems","nonlinear integral equations","smooth fit","time change"],"falsifier":"Solve the nonlinear integral equation (4.2) for the paper's sinusoidal-cost Aldous example and compare the first-exit value with a direct PDE or numerical solution of the optimal stopping problem; if the values differ, the characterization is wrong. A sharper test is to take the same costs with a martingale that can be absorbed at 0 or 1 before the horizon: if the boundary is no longer $C^1$ or the integral equation loses uniqueness, the Feller condition (3.4) is doing essential work.","tokens_in":48416,"feed_emoji":"🎯","tokens_out":7865,"duration_ms":69989,"temperature":0.7,"pith_summary":"Prediction markets, sports games, and elections all pose the same practical question: once beliefs evolve according to a win-martingale, when should you stop watching and declare an outcome? This paper claims that, for a large class of such martingales with volatility that splits into a time factor and a state factor, the optimal rule is always to stop when the posterior leaves a symmetric interval around 1/2. The value function is shown to be once continuously differentiable, the stopping boundary is shown to be smooth before the horizon, and the boundary is uniquely characterized by a nonlinear integral equation. If correct, this gives a common solution theory for the Aldous, Bass, and binary sequential-inference martingales, and it works even when the running cost is nonmonotone, producing boundaries that can shrink, expand, or oscillate.","feed_headline":"The optimal stop time for win-martingales is an interval exit","feed_subtitle":"It covers the classic exciting-game martingales and allows wild, nonmonotone boundaries.","key_machinery":"The load-bearing object is the two-sided free boundary $b_T(t)$ that defines the continuation interval. It is produced by combining a deterministic time change that absorbs time-dependent volatility into the running cost, the Lamperti transform (a coordinate change making the diffusion coefficient equal to one), probabilistic derivative representations of the value function, the smooth-fit principle, and a change-of-variable formula with local time on curves, which turns the optimal stopping problem into the integral equation (4.2).","core_discovery":"Under the paper's structural assumptions, the Bayes risk of a stop-and-declare problem for an autonomous win-martingale reduces to an optimal stopping problem whose continuation region at every time is the interval $(1-b_T(t), b_T(t))$. The first exit from this region is an optimal stopping time, the value function satisfies smooth fit and is $C^1$ globally, the boundary $b_T$ is $C^1$ on $[0,T)$, and $b_T$ is the unique solution of the nonlinear integral equation (4.2). The same structure holds for finite and infinite horizons without discounting and requires no monotonicity of the running cost.","pith_inferences":["One could test the same machinery on asymmetric terminal costs by dropping the symmetry assumption; the paper indicates this needs extra bookkeeping, and the boundary would become two independent curves rather than a symmetric pair.","The absence of a boundary limit near the horizon suggests that finite-horizon numerical approximations of infinite-horizon problems may converge non-uniformly in time, so numerical schemes should track oscillatory boundaries carefully.","A natural extension would allow early absorption at 0 or 1 before the deadline; the paper explicitly leaves this open, and one can conjecture the continuation region remains an interval but requires an additional exogenous termination boundary and rederived regularity.","The difference in value functions between two win-martingales under the same cost could serve as a decision-theoretic measure of excitement, complementing purely probabilistic distances between martingale laws."],"forward_implications":["For every model in the class, the optimal decision rule is known once $b_T$ is computed, so no general search over stopping times is needed.","The nonlinear integral equation gives a constructive numerical route to the boundary, for example by Picard iteration, with uniqueness guaranteeing that the computed boundary is the right one.","Monotone running costs in calendar time need not imply monotone boundaries; the paper's sinusoidal-cost example has a boundary with no limit as the horizon is approached.","The value functions provide a decision-theoretic ordering of games: under equal costs, the illustrated Aldous martingale is more costly to call optimally than the Bass martingale."],"supporting_citations":[{"why":"Supplies the theorem that upgrades the Lipschitz boundary to $C^1$ after the paper verifies its assumptions.","marker":"[10]"},{"why":"Provides the Green-regularity and smooth-fit results that convert boundary regularity into value-function regularity.","marker":"[11]"},{"why":"Gives the method for proving that the optimal stopping boundary is locally Lipschitz.","marker":"[12]"},{"why":"The change-of-variable formula with local time on curves is used to derive the free-boundary integral equation.","marker":"[27]"},{"why":"Supplies the standard optimal-stopping verification framework used for hitting-time optimality.","marker":"[28]"},{"why":"Earlier Bayesian sequential-testing theory with monotone costs that the present framework generalizes to nonmonotone costs and broader posterior dynamics.","marker":"[14]"},{"why":"The homogeneous free-boundary solution for a Brownian-drift posterior that the paper extends to general win-martingales.","marker":"[9]"}],"fun_headline_variants":["Optimal stop for win-martingales: exit an interval","Win-martingale stop rule: interval exit is optimal","Predicting with win-martingales: stop at the boundary","When to stop: interval exit solves win-martingale game","Stopping win-martingales: smooth boundary, no monotonicity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that the win probability never reaches 0 or 1 before the horizon, so absorption happens only in the limit; if early absorption is allowed, the theory's regularity results and integral equation would need to be rebuilt.","fun_headline_variants_meta":{"raw":{"variants":["Optimal stop for win-martingales: exit an interval","Win-martingale stop rule: interval exit is optimal","Predicting with win-martingales: stop at the boundary","When to stop: interval exit solves win-martingale game","Stopping win-martingales: smooth boundary, no monotonicity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000681,"raw_usage":{"total_tokens":3116,"prompt_tokens":991,"completion_tokens":2125,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":607,"completion_tokens_details":{"reasoning_tokens":2035}},"tokens_in":607,"tokens_out":2125,"duration_ms":12675,"temperature":1.0,"reasoning_tokens":2035,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:10:06.793443+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Solve the nonlinear integral equation (4.2) for the paper's sinusoidal-cost Aldous example and compare the first-exit value with a direct PDE or numerical solution of the optimal stopping problem; if the values differ, the characterization is wrong. A sharper test is to take the same costs with a martingale that can be absorbed at 0 or 1 before the horizon: if the boundary is no longer $C^1$ or the integral equation loses uniqueness, the Feller condition (3.4) is doing essential work.","supporting_citations":[{"cited_title":"De Angelis and D","cited_arxiv_id":null,"evidence_quote":"Supplies the theorem that upgrades the Lipschitz boundary to $C^1$ after the paper verifies its assumptions."},{"cited_title":"De Angelis and G","cited_arxiv_id":null,"evidence_quote":"Provides the Green-regularity and smooth-fit results that convert boundary regularity into value-function regularity."},{"cited_title":"De Angelis and G","cited_arxiv_id":null,"evidence_quote":"Gives the method for proving that the optimal stopping boundary is locally Lipschitz."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The change-of-variable formula with local time on curves is used to derive the free-boundary integral equation."},{"cited_title":"Peskir and A","cited_arxiv_id":null,"evidence_quote":"Supplies the standard optimal-stopping verification framework used for hitting-time optimality."},{"cited_title":"Ekstr¨ om and J","cited_arxiv_id":null,"evidence_quote":"Earlier Bayesian sequential-testing theory with monotone costs that the present framework generalizes to nonmonotone costs and broader posterior dynamics."}],"review_version":1}