{"id":"a018fd3c-0379-4279-8c18-725c4bf25b59","arxiv_id":"2508.07386","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Joint and conditional splitting probabilities are derived for decoupled and unidirectionally coupled drift-diffusive systems, yielding a Bayesian scheme for partially inferring a hidden internal state from first-passage exit events.","lead":"This paper derives formulas for the probability that a moving particle exits a region through a given boundary at a moment when a hidden second variable takes a particular value. It then uses Bayes' rule to turn those joint probabilities into posterior guesses about the hidden variable from observed crossing events.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"The reader's verdict is UNVERDICTED because the full text was unavailable, and I agree that soundness cannot be fully verified from the abstract alone. However, the reader identified Y-independence as the weakest assumption, whereas this is an explicit scope condition of both process classes. Since the paper does not claim to handle feedback from X to Y, no hidden assumption is present. The more relevant concern is the technical execution of the spectral expansion for arbitrary Markov Y and the explicit coupled-case derivations, which require checking. My proposed computational test would settle whether the decoupled-class spectral expression is correct in a simple nontrivial case, and a similar Monte Carlo or PDE test would validate the run-and-tumble or resetting formulas. Because I identify no concrete flaw, the reader's UNVERDICTED status remains unchanged; no adjustment to the verdict is needed.","tokens_in":1068,"tokens_out":7700,"duration_ms":92495,"concrete_test":"Verify the decoupled-class formula against an independent derivation: take Y as a two-state Markov chain with rates k1, k2 and X as Brownian motion on [-L,L]. Compute P(X exits at right and Y=state 1 at the exit time) by solving the coupled backward Kolmogorov equations for the joint exit probabilities with absorbing boundaries, then compare to the spectral expression integrated against the Brownian first-passage time density. Agreement to machine precision would validate the generic eigensystem claim; disagreement would pinpoint a hidden assumption in the spectral expansion or the time-integration step.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract states the central claim for two explicitly scoped classes of processes, both requiring Y(t) to evolve independently of X(t). That independence is a stated condition rather than a hidden assumption, so the reader's weakest_assumption is not a correctness risk for the claimed results. The main risk is unverified technical execution: for 'arbitrary Markov Y,' the spectral expansion and the integration against the Brownian first-passage density must be justified; and for the three coupled cases, the explicit derivations need careful checking of boundary conditions and convergence. However, no concrete mathematical flaw can be identified at the abstract-only level. The Bayes step is logically sound, and the conditional inference scheme is a legitimate interpretation of the joint distribution. The exact-first-passage detection requirement is a practical limitation, not an error in the mathematical derivation.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines a joint splitting probability for a two-dimensional Markov process {X(t), Y(t)}: the probability that X exits an interval L through a specified boundary at its first-passage time while Y takes a given value at that same time. The authors compute this quantity for two process classes: (i) X is Brownian motion and Y is a decoupled Markov internal state, for which a generic expression in terms of the Fokker-Planck eigensystem of Y is claimed; and (ii) unidirectionally coupled drift-diffusive processes where X depends on Y but Y evolves independently, with explicit results for run-and-tumble motion, intermittent piecewise-linear potentials, and stochastic resetting. Using Bayes' theorem, the authors then introduce conditional splitting probabilities—the posterior likelihood of the hidden state Y given a specific exit event of X—and propose an inference scheme for partially recovering Y from point-wise detection events. The abstract advertises closed-form analytic results that would constitute a new tool for hidden-state inference from first-passage observables.","tokens_in":1100,"tokens_out":2238,"duration_ms":25677,"significance":"If the derivations are correct, the paper offers a meaningful extension of classical splitting-probability theory to joint state-exit probabilities and provides a principled Bayesian route to hidden-state inference from first-passage data. The generic eigensystem expression for decoupled Brownian X and arbitrary Markov Y is a potentially powerful result, and the three explicit case studies are canonical models with broad applicability in active matter, stochastic thermodynamics, and search processes. The conceptual step of defining conditional splitting probabilities is logically sound and likely to stimulate further work. However, because only the abstract is available for review, the mathematical validity of the spectral expansions, the handling of boundary conditions, and the convergence of the series cannot be independently confirmed. The significance is therefore conditional on a full-text verification of the technical execution.","major_comments":[{"comment":"The central claim for the first process class—a generic expression for the joint splitting probability in terms of the eigensystem of the Fokker-Planck operator of Y—requires an exchange of an integral over the Brownian first-passage time with a spectral sum over Y's eigenstates. This step is not visible in the abstract and is load-bearing. The referee cannot check whether the spectrum is assumed discrete, whether completeness holds for the relevant function space, or whether the convergence is uniform enough to justify the interchange. The full text must be inspected for these justifications before the result can be accepted.","section":"Abstract (and Sections 2-3, not available)"},{"comment":"For the three coupled examples (run-and-tumble, intermittent piecewise-linear potential, stochastic resetting), the abstract states that explicit derivations are carried out, but no details are given. Each of these models has nontrivial boundary conditions at the interval endpoints and at switching/resetting events. Incorrect treatment of boundary terms or of the joint density at the first-passage time would invalidate the resulting formulas. The referee cannot verify these points from the abstract alone; a careful check of the full derivations is required.","section":"Abstract (case-study derivations)"},{"comment":"The Bayes step converting joint splitting probabilities into conditional splitting probabilities is definitionally sound. However, the inference scheme as described assumes that detection events coincide with true first-passage times of X. This is a practical limitation that may severely restrict applicability, since experimental or numerical detection often occurs at sampled times rather than at the exact boundary-crossing instant. The abstract does not discuss how the results degrade under coarse or noisy detection. While this is not a mathematical error in the derivation, it is a load-bearing assumption for the proposed inference scheme and should be clearly stated and, ideally, quantified in the full text.","section":"Abstract (Bayesian inference step)"}],"minor_comments":[{"comment":"The phrase 'arbitrary Markov Y' is ambiguous: it is not specified whether Y is continuous-time, discrete-state, or has a generator with a discrete spectrum. A brief clarification in the abstract would help.","section":"Abstract"},{"comment":"The term 'conditional splitting probabilities' is new; the authors should ensure it does not conflict with existing usage in the first-passage literature, and define it formally in the introduction.","section":"Abstract"},{"comment":"The constraint that Y(t) evolves independently of X(t) is explicitly stated, which is good. The authors might add a note in the abstract indicating that feedback from X to Y is outside the current scope, to preempt overgeneralization by readers.","section":"Abstract"}],"recommendation":"uncertain","confidential_remarks":"This review is based solely on the abstract; the full text was not accessible. The conceptual framework is promising and no obvious logical error is evident, but the mathematical claims cannot be verified without the derivations. I recommend that the editor obtain the full text or an independent verification before a substantive decision. If the full text is available to the reviewer, a major-revision or accept decision may be appropriate depending on the technical execution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: I can't judge the math because only the abstract is in front of us. But the idea is clearly framed and the scope is honestly stated, which is more than most abstracts manage. The new bit is the conditional splitting probability defined via Bayes' theorem and used for hidden-state inference from first-passage events. That framing looks genuinely absent from the literature the abstract cites, and the three worked examples (run-and-tumble, intermittent piecewise-linear potential, stochastic resetting) are canonical enough that a careful referee can check them without building new machinery.\n\nWhat's good: The authors state the main assumptions up front. Both process classes require Y(t) to evolve independently of X(t), and they say so. That weakens the reader's worry about unidirectional coupling being a hidden assumption—it isn't. The Bayes step is logically safe. And the eigensystem expression for the decoupled case is a natural product structure, which is elegant. Credit where due: the paper is not overclaiming.\n\nSoft spots: Everything rests on derivations we can't see. Two steps are where first-passage papers usually accumulate hidden assumptions: conditioning on the exact exit time and justifying the spectral expansion's convergence. The abstract doesn't address either. Also, the inference scheme requires detection events to coincide with true first-passage times, not coarse samples; that's a practical limitation, and the abstract hints at it but doesn't say how severe it is. For the decoupled class, the joint splitting probability may factor into P(Y=y) times a classical exit distribution (Poisson kernel for Brownian intervals), so the novelty there might be packaging rather than a new mathematical object. That's worth the referee probing, but it isn't a flaw.\n\nThe stress-test note is right: the independence condition is a stated scope, not a hidden risk. The main risk is technical execution, which we can't assess.\n\nBottom line: This deserves a serious referee. The derivations are checkable and the application is potentially useful. I'd send it out, asking the referee to verify the spectral expansions and the conditioning carefully, and to compare the decoupled-case results against classical hitting distributions to pin down what's actually new.\n\nFor my own work: I wouldn't cite it until it's refereed and updated, but I'd put it on the reading list if the full text comes out.","headline":"Abstract-only, so unverdictable; but a clean, honestly scoped idea that deserves referee time.","tokens_in":1657,"tokens_out":1489,"would_cite":false,"duration_ms":17739,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60J60","60J65","82C31"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes closed-form joint splitting probabilities for drift-diffusive processes and uses Bayes' theorem to turn them into conditional posterior probabilities for a hidden internal state at first-passage exit.","keywords":["splitting probabilities","first-passage time","hidden state inference","Brownian motion","run-and-tumble","stochastic resetting","Fokker-Planck eigensystem","Bayes theorem"],"falsifier":"Simulate a Brownian particle with an independent two-state Markov internal state, record the internal state exactly at the first passage to either boundary across many realizations, and compare the empirical joint exit-boundary/internal-state distribution with the closed-form expression from the Fokker-Planck eigensystem; any systematic mismatch falsifies the generic formula. A second test would repeat this for a run-and-tumble process with position-dependent tumbling that makes $Y$ depend on $X$, which should fall outside the paper's second class and violate the predicted conditional probabil","tokens_in":885,"feed_emoji":"🎯","tokens_out":4583,"duration_ms":41704,"temperature":0.7,"pith_summary":"Hidden-state inference usually needs continuous observations. This paper claims that for a large family of drift-diffusive processes, a single first-passage event — the particle leaving an interval through one of two boundaries — is enough to update beliefs about an unobserved internal state Y. The tool is a joint splitting probability: the probability that X exits through a given boundary at the same time Y takes a given value. For Brownian X with decoupled Markov Y, the paper derives generic expressions in terms of the Fokker-Planck eigensystem of Y; for unidirectionally coupled X–Y dynamics it gives explicit results for run-and-tumble, intermittent piecewise-linear potentials, and stochastic resetting. Bayes' rule then yields the conditional splitting probability, the posterior of Y given the exit event. If correct, this gives an analytic route to infer aspects of hidden dynamics from exit-only records.","feed_headline":"Exit events expose a hidden state's identity","feed_subtitle":"Joint splitting probabilities give the posterior odds of an unobserved internal state at the exact first-passage time.","key_machinery":"The joint splitting probability is the central object: $\\mathcal{S}_{b,y}(x_0,y_0)=P(X_{\\tau}=b,\\,Y_{\\tau}=y)$ with $\\tau$ the first exit time of $X$ from the interval. For decoupled systems the calculation splits into the first-passage-time density of $X$ and the transition density of $Y$ (expanded in the eigenfunctions of $Y$'s Fokker-Planck operator). For unidirectionally coupled systems the same object is computed explicitly for the three paradigmatic dynamics. Bayes' theorem is the second mechanism: it turns the joint probability into the conditional posterior $P(Y_{\\tau}=y \\mid X_{\\tau}=b)$, the quantity relevant to inference.","core_discovery":"The paper's central claim is that the joint splitting probability $P(X_{\\tau}=b, Y_{\\tau}=y \\mid x_0,y_0)$, where $\\tau$ is the first passage time of $X$ out of the interval, can be computed in closed form for two process classes. In the first class, $X$ is Brownian motion and $Y$ is an independent Markov process; the joint probability reduces to an integral over the first-passage-time density of $X$ and the transition density of $Y$, hence the eigensystem of $Y$'s Fokker-Planck operator. In the second class, $X$ is drift-diffusive with drift depending on $Y$, while $Y$ evolves independently; explicit calculations are carried out for run-and-tumble motion, diffusion in an intermittent piecew","pith_inferences":["The eigensystem expansion for class 1 is generic: any hidden Markov state with a known generator (e.g., a multi-state chemical switch) could be substituted, so the method may extend well beyond the three worked examples without new mathematics.","A natural but untested extension is to higher-dimensional domains or to exit through multiple boundaries, where the first-passage-time density of $X$ is the only part that would need to be recomputed.","Because the posterior is conditioned on exact first-passage times, experimental implementations would need event-triggered detection rather than uniform-time sampling; coarse sampling would introduce an approximation the paper's formalism does not cover.","One could turn the conditional splitting probabilities into a likelihood for observed exit events and use maximum likelihood to estimate the hidden state's transition rates, a parameter-inference extension the paper does not explicitly develop."],"forward_implications":["Observing only which boundary a drift-diffusive particle exits through and when can yield quantitative posterior information about its internal state at that moment.","For Brownian particles with independent internal Markov states, the conditional splitting probabilities are available as soon as the eigensystem of the internal state's Fokker-Planck operator is known.","Run-and-tumble, intermittent-potential, and stochastic-resetting models get explicit posterior formulas that depend on their dynamical parameters, making parameter estimation from exit events possible.","The Bayes-based inference scheme works with point-wise detection events, so it does not require continuous tracking of the observable $X$."],"supporting_citations":[],"fun_headline_variants":["Exit events unmask hidden states","First-passage times reveal hidden states","How exit events expose internal states","Joint splitting probabilities decode hidden states","Conditional probabilities infer hidden states"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"The hidden state $Y(t)$ must evolve independently of the observable $X(t)$; if $X$'s history feeds back into $Y$'s dynamics, the joint splitting probability no longer factorizes and the derived closed forms and the Bayes inference scheme built on them break down.","fun_headline_variants_meta":{"raw":{"variants":["Exit events unmask hidden states","First-passage times reveal hidden states","How exit events expose internal states","Joint splitting probabilities decode hidden states","Conditional probabilities infer hidden states"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000265,"raw_usage":{"total_tokens":1516,"prompt_tokens":886,"completion_tokens":630,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":630,"completion_tokens_details":{"reasoning_tokens":587}},"tokens_in":630,"tokens_out":630,"duration_ms":6117,"temperature":1.0,"reasoning_tokens":587,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T22:08:51.501349+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate a Brownian particle with an independent two-state Markov internal state, record the internal state exactly at the first passage to either boundary across many realizations, and compare the empirical joint exit-boundary/internal-state distribution with the closed-form expression from the Fokker-Planck eigensystem; any systematic mismatch falsifies the generic formula. A second test would repeat this for a run-and-tumble process with position-dependent tumbling that makes $Y$ depend on $X$, which should fall outside the paper's second class and violate the predicted conditional probabil","supporting_citations":[],"review_version":1}