{"id":"55f6bb11-2a44-4860-b635-ca46c4b00df7","arxiv_id":"2607.20373","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Every autonomous win-martingale on [0,1] is the exact posterior of an explicit binary diffusion experiment, and current-state posteriors exist exactly under a Riccati condition on the drift gap.","lead":"The paper proves that any smooth, non-degenerate martingale diffusion on [0,1] can be exactly realized as the Bayesian posterior of an explicit binary diffusion experiment. It also characterizes when the posterior is a function of only current time and observation, recovering classical sequential-testing filter as the unique non-harmonic autonomous case.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; strict positivity of σ is explicit and the proof steps check out.","rationale":"The reader's weakest_assumption identifies strict positivity of σ as the load-bearing premise. I agree that this is the point without which the Lamperti transform fails, but it is explicitly part of the definition of autonomous win-martingale, not a hidden assumption. The central claim is exactly about this class, so no counterexample with interior zeros would refute it; it would only narrow the scope, which the paper already declares. I independently checked the main proof threads and found them sound: the Girsanov computations in Section 3 are localized correctly at τ_n; the support lemma gives the full support needed for the pointwise upgrade; the range argument in Theorem 4.7(ii) is valid because the intervals Γ((0,∞),x) share a common endpoint for κ≠0. The algebraic consistency of the embedding with the Riccati characterization provides an additional internal check. Therefore, I have no significant objection and recommend the verdict remain unchanged.","tokens_in":16999,"tokens_out":32257,"duration_ms":235106,"concrete_test":"Verify algebraically that for arbitrary σ∈C^1((0,1);(0,∞)), the drift gap δ(x)=σ(G(x))/(G(x)(1−G(x))) from Theorem 3.2 satisfies the Riccati equation δ' + 2ν0δ + δ^2 = 0, with ν0 given by (3.5). This confirms the embedded experiment is a C-S posterior with κ=0 and that the construction nests consistently with the converse classification.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The only premise without which the construction would break is σ>0 on (0,1), which makes the Lamperti transform F in (3.1) a C^2 diffeomorphism and the drifts (3.5) well-defined continuous functions on the open interval I. This is not an unstated gap: Definition 2.1 and the abstract state it as part of the class. I re-examined the key steps: Lemma 2.1's Engelbert–Schmidt well-posedness argument; the Girsanov calculation leading to (3.5); the pathwise posterior identity in Theorem 3.2; the support lemma 4.2 and the pointwise PDE derivation in Theorem 4.3; and the range/intersection argument in Theorem 4.7(ii). No internal inconsistency or hidden circularity surfaced. The only real limitation is the acknowledged one-dimensional binary setting, which the authors explicitly flag in the conclusion. Thus the reader's ACCEPT is justified.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the relationship between time-homogeneous martingale diffusions on [0,1] and Bayesian posteriors in binary sequential experiments. It calls a process Π an autonomous win-martingale if it satisfies dΠ_t = σ(Π_t)dB_t with σ∈C^1((0,1);(0,∞)) and absorbing endpoints. The main positive result, Theorem 3.2, constructs, for every such Π, a binary diffusion experiment whose posterior is indistinguishable from Π: after the Lamperti transform F, the two state-dependent drifts are (3.5), and the posterior is P=G(X). The paper then characterizes the converse. Theorem 4.3 shows that the posterior depends on (t,X_t) only through a smooth function exactly when the drift gap δ satisfies the Riccati equation (4.1); Corollary 4.4 recasts this as a Doob h-transform of the null dynamics. Theorem 4.7 classifies autonomy: for κ=0 the posterior is a time-independent function of X_t, while for κ≠0 autonomy forces δ to be constant and recovers the classical volatility σ(p)=|Δ|p(1-p). Examples include absorbed Brownian motion, the Wright-Fisher diffusion, and the Aldous martingale, with a table of explicit experiments. A short section gives posterior-threshold error probabilities and expected decision times.","tokens_in":17193,"tokens_out":11146,"duration_ms":84001,"significance":"If correct, the results provide a complete and elegant one-dimensional link between bounded martingale diffusions and Bayesian sequential experiments. The embedding theorem is fully constructive and parameter-free, and the converse gives a precise, checkable condition—Riccati equation (4.1), equivalently a space-time harmonic likelihood ratio—for current-state sufficiency. The classification of autonomy is a genuinely informative dichotomy: the κ=0 branch characterizes exactly the time-homogeneous posterior dynamics from the embedding, while the κ≠0 branch forces the classical Shiryaev–Wonham volatility. The paper is technically careful: proofs use standard tools (Engelbert–Schmidt, Girsanov, filtering theorems, support arguments) with proper localizations, and the examples are nontrivial and correctly computed. The one-dimensional binary setting is a limitation of scope, but it is explicitly acknowledged and does not undermine the paper's claims.","major_comments":[],"minor_comments":[{"comment":"The drifts ν_i have denominators G(y) and 1−G(y), which vanish as y approaches the endpoints of the interval I. It would be helpful to state explicitly that continuity is claimed only on the open interval I, as required by the model in Section 2.1, and not up to the boundary; the current wording 'C(I;R)' is clear to specialists but could be misread.","section":"§3, Eq. (3.5)"},{"comment":"The use of Picard–Lindelöf for the ODE δ' = −2μ_0δ−δ^2 is slightly compressed. Since μ_0 is only assumed continuous, the standard local uniqueness theorem requires the right-hand side to be continuous in x and locally Lipschitz in δ; this holds here, and the argument then extends globally. A one-sentence explanation would make the step more transparent.","section":"§4.1, proof of Theorem 4.7(i)"},{"comment":"For the first row, the sentence about the textbook presentation with drifts (0,1) changing κ from 0 to −1 is terse. Spelling out the shift X↦X+t/2 and how μ_0 changes from −1/2 to 0 would make the remark self-contained and avoid confusion about the sign convention in (4.1).","section":"§5, Table 1"},{"comment":"The remark is correct but could be more reader-friendly: after defining S_t, a short computation showing that the posterior is C-S for S, e.g. logit P_t = logit p + Δ(S_t−S_0)−Δ^2 t/2, would make the point immediately verifiable.","section":"Remark 4.5"}],"recommendation":"accept","confidential_remarks":"The manuscript is in scope for the journal, the central claims check out on close reading, and there are no concerns about circularity or undisclosed reliance on the authors' companion work. The remaining comments are purely presentational. I support acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper deserves a serious referee. It proves a clean, constructive result: every time-homogeneous martingale diffusion on [0,1] with strictly positive C^1 volatility and absorbing endpoints is the exact Bayesian posterior of an explicit binary diffusion experiment. The converse is equally precise: under smoothness, the posterior is a function of current time and state exactly when the drift gap satisfies the Riccati equation (4.1), and then autonomy forces either the spatially harmonic (κ=0) case or constant drift, recovering the Shiryaev–Wonham posterior. The claims are stated carefully and the proofs hold up in the steps I checked: the Lamperti/Doob construction in Theorem 3.2, the support lemma used to pass from stochastic to pointwise identities, and the range-intersection argument in Theorem 4.7(ii) are all valid.\n\nWhat is actually new is filling the gap left by Brigo–Vrins and Jeanblanc–Vrins: those papers represent bounded martingales but do not identify the sequential experiment that generates them. Here the experiment is explicit, with the conditional laws given by Doob h-transforms and the drifts in (3.5). The examples — absorbed Brownian motion, Wright–Fisher, the time-homogeneous Aldous martingale — are coherent and genuinely illustrative. Corollary 4.11 gives a nice inverse-design procedure. The paper does not overclaim: it repeatedly flags the one-dimensional binary setting as the boundary of the method.\n\nSoft spots are minor. The regularity assumptions (σ>0 on the interior, δ∈C^1, Γ∈C^{1,2}) are explicit and standard; they are not hidden gaps. The posterior-threshold formulas in Section 5 are standard material, clearly presented but not new. There are a few typos, including the title's “Mar tingale,” and the self-citation to [11] is benign since it is not used as an input. The only real limitation is the scalar one-dimensional framework, which the authors acknowledge.\n\nThis is not a sweeping revolution, but it is a well-executed, self-contained paper that fills a real gap and gives useful tools. I would cite it and would welcome it at a journal specializing in stochastic filtering or probability. Send it to peer review.","headline":"A genuinely new embedding theorem: every autonomous win-martingale is the exact posterior of an explicit binary diffusion, with a sharp converse classification; the proofs are solid and the acknowledged one-dimensional limitation is the main caveat.","tokens_in":17698,"tokens_out":1661,"would_cite":true,"duration_ms":15850,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G44","60H10","60J60","62L10","93E11"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that every time-homogeneous martingale diffusion on [0,1] with strictly positive smooth volatility and absorbing endpoints is the exact Bayesian posterior of an explicitly constructed binary diffusion experiment.","keywords":["posterior martingales","binary sequential inference","stochastic filtering","Doob h-transform","Lamperti transform","Riccati equation","martingale diffusion","sequential testing"],"falsifier":"Take a volatility σ in the class, compute δ(x)=σ(G(x))/(G(x)(1−G(x))) from the embedding drifts, and substitute it into the Riccati equation (4.1); a failure of the identity for some x would expose an inconsistency between the forward and converse directions. Alternatively, simulate the constructed experiment and compare the empirical law of the posterior path with the prescribed martingale law.","tokens_in":16874,"feed_emoji":"📈","tokens_out":6231,"duration_ms":48158,"temperature":0.7,"pith_summary":"This paper proves that every time-homogeneous martingale diffusion on [0,1] with strictly positive smooth volatility and absorbing endpoints—called an autonomous win-martingale—can be represented exactly as the Bayesian posterior belief of an observer who sees a scalar diffusion whose drift depends on a hidden binary state. The representation is constructive: place the martingale in Lamperti coordinates, condition its law on the two terminal outcomes, and the resulting conditional drifts define the experiment. Going the other way, the paper characterizes which binary diffusion experiments produce posteriors that are functions of the current time and observation alone: the difference between the two signal drifts must solve a Riccati equation, equivalently the likelihood ratio must be a Doob h-transform for the null dynamics. Autonomy of the posterior then forces either the spatially harmonic case or a constant drift difference, recovering the classical two-state sequential-testing filter. This gives concrete learning experiments for a broad family of bounded martingales, including absorbed Brownian motion, the Wright–Fisher diffusion, and the most-exciting-game martingale.","feed_headline":"Every win-martingale is an exact Bayesian posterior","feed_subtitle":"A constructive recipe turns any absorbed martingale diffusion into an explicit binary experiment an observer can track.","key_machinery":"The central machinery is the Lamperti transform F(p)=∫_{1/2}^p du/σ(u), which maps an autonomous win-martingale to a unit-volatility diffusion Y=F(Π); conditioning on the terminal outcome then acts as a Doob h-transform with harmonic functions h_1(p)=p and h_0(p)=1−p, producing the two state-dependent signal drifts. In the converse, the Riccati identity δ′+2μ_0δ+δ^2=−κ serves as the compatibility condition, equivalent to the Girsanov density collapsing to an h-transform; it separates autonomy (κ=0) from forced constancy (κ≠0).","core_discovery":"On its own terms, the paper establishes an exact two-way correspondence between autonomous win-martingales (solutions dΠ_t = σ(Π_t) dB_t on [0,1], σ ∈ C^1((0,1);(0,∞)), absorbed at endpoints) and binary diffusion experiments dX_t = μ_θ(X_t) dt + dW_t. Theorem 3.2 gives the forward direction: with F(p)=∫_{1/2}^p du/σ(u) and its inverse G, setting Y_t=F(Π_t), the conditional laws of Y given Π_∞=1 and Π_∞=0 have explicit drifts ν_1 and ν_0, and the experiment with signal X and drifts ν_0,ν_1 has posterior P_t=G(X_t) with exactly the law of Π. The converse, Theorems 4.3 and 4.7, shows that the posterior is a smooth current-state function Γ(t,X_t) iff the drift gap δ=μ_1−μ_0 solves δ′+2μ_0δ+δ^2=−","pith_inferences":["The embedding suggests a converse design template for other scalar martingales: any bounded martingale with a positive volatility can be given a 'story' as a posterior, and one could ask which cost or optimality criteria select among the many possible experiments for the same posterior law.","Because the autonomy classification forces the classical filter for κ≠0, optimality criteria that reward belief movement will pick among the κ=0 family; the paper's examples hint that the most-exciting-game and a normal-mixture martingale are time-changed members of that family.","The threshold formulas imply that, for fixed thresholds, type I/II errors are volatility-independent, but optimizing thresholds reintroduces dependence on σ; a natural extension is to solve the optimal stopping problem for the autonomous posterior family.","The paper notes that higher-dimensional, multi-hypothesis extensions lack the scalar machinery; a concrete next step is to test whether any multidimensional bounded martingale posterior can be embedded in a similarly explicit multi-hypothesis diffusion experiment."],"forward_implications":["Every autonomous win-martingale, including absorbed Brownian motion, the Wright–Fisher diffusion, and the most-exciting-game martingale, comes with an explicit binary signal that a real observer could track.","The posterior-threshold formulas give closed-form error probabilities and expected decision times for any autonomous win-martingale decision rule.","Inverse design: choosing any null drift μ_0 and any constant C yields an alternative drift δ=s_0′/(C+s_0) whose experiment has autonomous posterior dynamics.","Outside the harmonic class κ=0, the only autonomous posterior is the classical two-state filter with volatility |Δ|p(1−p), so that filter is recovered rather than chosen.","Current-time-and-state sufficiency for the hidden state is equivalent to the Riccati condition, so any experiment violating it must have a posterior that depends on the path history."],"fun_headline_variants":["Every win-martingale is exactly a Bayesian posterior","Binary diffusion experiments realize all win-martingales","From martingale to posterior: a constructive recipe","Wright-Fisher, Aldous martingales are binary posteriors","Posterior martingales match explicit binary experiments"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The entire construction requires the volatility σ to be strictly positive on (0,1); if σ had an interior zero, the Lamperti transform would not be invertible and the explicit experiment drifts could not be defined.","fun_headline_variants_meta":{"raw":{"variants":["Every win-martingale is exactly a Bayesian posterior","Binary diffusion experiments realize all win-martingales","From martingale to posterior: a constructive recipe","Wright-Fisher, Aldous martingales are binary posteriors","Posterior martingales match explicit binary experiments"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000202,"raw_usage":{"total_tokens":1268,"prompt_tokens":841,"completion_tokens":427,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":585,"completion_tokens_details":{"reasoning_tokens":349}},"tokens_in":585,"tokens_out":427,"duration_ms":5229,"temperature":1.0,"reasoning_tokens":349,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T10:00:28.676217+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a volatility σ in the class, compute δ(x)=σ(G(x))/(G(x)(1−G(x))) from the embedding drifts, and substitute it into the Riccati equation (4.1); a failure of the identity for some x would expose an inconsistency between the forward and converse directions. Alternatively, simulate the constructed experiment and compare the empirical law of the posterior path with the prescribed martingale law.","supporting_citations":[],"review_version":1}