{"id":"80574a35-60d5-4a72-bb63-a27a880a4c1f","arxiv_id":"2607.13213","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For N once-reinforced (saturating) random walkers, the k-walker splitting probability is a generalized binomial/Beta formula and simultaneous and sequential launching give identical splitting probabilities.","lead":"This paper derives exact formulas for the odds that k out of N one-dimensional walkers sharing a trail reach one wall before the other, and for how long at least k walkers survive. The striking result is that for a broad 'saturating trail' class, these odds are identical whether the walkers start together or one after another.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The exact claim rests on an unproved sequential/simultaneous equivalence for mixed absorption; SM §2.2.1 concedes the inhomogeneous local-time process is unsolved.","rationale":"The paper's most valuable and least secure claim is protocol invariance for SATW_phi. The all-at-one-wall case has an induction proof using additivity of squared Bessel processes, but the mixed-absorption case does not: Eq. (4) is derived by constructing the sequential-start local-time field and then invoking an unproved equivalence. The reader's conditional verdict identifies exactly this gap, and the manuscript itself contains the relevant admissions: SM §2.1.1 supposes the urn scheme, and SM §2.2.1 says the mixed-absorption process is not solved. The numerical simulations in Fig. 3 are supportive but cannot establish exactness for all phi, N, and x0. Independent support includes correct N=1 and phi=1 limits and internal normalization checks, but those do not cover the N>1 mixed-absorption case. I therefore see no reason to change the reader's conditional verdict: the central claim is plausible and well-supported numerically, but the simultaneous-start formula (4) is not rigorously established. The proposed exact finite-lattice enumeration would settle whether the asserted equivalence is a theorem or a conjecture.","tokens_in":32462,"tokens_out":14730,"duration_ms":144533,"concrete_test":"Perform exact Markov-chain enumeration on a finite lattice (L sites, e.g., L=10,15,20) for N=2 and N=3 SATW_phi with phi in {0.25,0.5,1.5,2} and several x0, using a precisely specified simultaneous-update rule (e.g., random sequential order within each discrete step) and the sequential-start protocol. Compute P(+k|x0) for both protocols and extrapolate to L→∞. Compare with Eqs. (4)–(5). A discrepancy beyond finite-size error would falsify the claimed protocol invariance; agreement would support it but would not supply the missing proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central formula (4) for 0<k<N is derived only for the sequential-start protocol (SM §2.2.2–2.2.3), then asserted to hold for simultaneous starts. SM §2.2.1 explicitly states that the inhomogeneous mixed-absorption local-time process is not solved, and that protocol independence 'should not change the description'; SM §2.1.1 says 'We first suppose that the Pólya urn scheme can still be used' for N>1, with rigorous convergence cited only for N=1. The SM §2.1.2 blindness argument shows SATW_phi transition probabilities are insensitive to which particle draws from an urn, but it does not prove that simultaneous and sequential launches give the same law of the collective local-time field at absorption when both walls are hit. The event (+k) involves exactly such mixed-absorption histories, so the exactness of Eq. (4) for simultaneous starts is unsupported. If this asserted equivalence fails, Eq. (4) and the protocol-invariance headline collapse; the paper would still contain a valid sequential-start result, but not the advertised simultaneous one.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies N one-dimensional self-interacting random walks that share a common trail (local-time) field, focusing on first-passage observables. For the saturating class SATW_phi, it claims exact generalized persistence exponents theta_{k,N}=(k+phi-1)/2 for the k-th survival event and exact splitting probabilities in the interval [0,1], Eqs. (4)-(5), for both simultaneous and sequential launching protocols. For generic nonsaturating self-interacting walks it claims theta_{1,N}=theta_{1,1} for the last survivor. The derivations extend Toth's Ray-Knight/BESQ local-time representation to N walkers and use the sequential-start construction to compute mixed-absorption splitting probabilities, invoking protocol independence for SATW_phi. Numerical simulations are presented for SATW_phi and for TSAW/PSRW/SESRW models.","tokens_in":32790,"tokens_out":3888,"duration_ms":42463,"significance":"If the central claims are correct, the paper provides the first exact collective first-passage statistics for strongly history-dependent interacting walkers, including a striking and nontrivial protocol-invariance property. The explicit formulas are simple and falsifiable, and the paper correctly reproduces the known limits phi=1 (independent Brownian walkers) and N=1 (single SATW_phi). It also carefully uses existing rigorous single-particle results (Toth 1996; Bremont et al. 2025) as inputs, and the numerical support in Figs. 2-3 and the supplementary tables is substantial. The weakness is that the N-particle Ray-Knight extension and the simultaneous/sequential equivalence are not rigorously established; the supplementary text itself labels the key steps as 'supposed' or 'should not change'. Thus the paper's exactness claims go beyond what is currently proven, and the significance of the results is contingent on closing that gap or clearly restating the claims as conjectural.","major_comments":[{"comment":"The central formula (4) for mixed-absorption events with 0<k<N is derived in the supplementary material for the sequential-start protocol only. SM §2.2.1 explicitly states that the inhomogeneous local-time process for mixed absorption is unsolved and that protocol independence 'should not change the description'. The subsequent calculation in §2.2.2–2.2.3 computes P(+k|x0) by ordering the particles sequentially and then combining events (m-+...) and (m+-...). Since the advertised simultaneous-start result is the headline of the paper, and Eq. (4) is asserted to hold for both protocols, the exactness of the simultaneous-start formula is not established. The numerical agreement in Fig. 3(a) is suggestive but is not a proof. Either a proof of the protocol-invariance for mixed-absorption events must be supplied, or the claims must be weakened to state that Eq. (4) is derived for sequential s","section":"SM §2.2.1–2.2.3; Eq. (4)"},{"comment":"Even the all-at-one-wall result Eq. (5) rests on an unproved N-particle extension of Toth's Ray-Knight theorem. SM §2.1.1 begins 'We first suppose that the Pólya urn scheme can still be used' for N>1, and §2.1.3 notes that rigorous convergence in law is given in [1] only for N=1. The recurrence (2.2) and the resulting BESQ representation (2.5)-(2.6) are thus heuristic for N>1. Since Eq. (5) and the derivation of Eq. (8) depend directly on this representation, the exactness of the N-particle splitting probabilities for all-at-one-wall events is not rigorously supported. The paper should either provide a proof or reference a proof of the N-particle local-time limit, or explicitly mark this step as a conjecture.","section":"SM §2.1.1, §2.1.3; Eq. (5)"},{"comment":"The claim that the last-survival exponent for generic SIRWs equals the single-walker exponent, Eq. (3), is argued in the text by asserting that the last walker spends negligible time in the region previously visited by other walkers, because the probability of staying in the finite interval R decays exponentially. This argument is not made rigorous, and it is not obvious that it holds for all SIRWs, in particular for nonsaturating attractive interactions where a walker might be effectively trapped in an already-visited region. The numerical support in Fig. 2(b) covers TSAW, PSRW, and SESRW, which are repulsive or nonattractive models. The claim should either be restricted to the class for which the argument is valid or replaced by a precise statement of the needed assumptions, with a proof or at least a more explicit controlling argument.","section":"Main text: 'Determination of θ(SIRW)1,N'; Eq. (3)"}],"minor_comments":[{"comment":"The reported point estimate 2.220 and the 95% CI [1.188,1.272] are mutually inconsistent, and the CI does not contain the theoretical value 2.25. This appears to be a transcription or data-entry error; the table should be corrected or the text should explain the discrepancy.","section":"SM Table 3(b), k=2 row"},{"comment":"The 'another way' of recovering theta_{k,N} uses phrases such as 'We can reasonably consider' and 'it is very likely that' to argue that the leftmost particle behaves as a single SATW and the others as Brownian walkers in its wake. This is heuristic and should be labeled as such, not presented as an independent derivation.","section":"SM §2.4"},{"comment":"The normalization of Eq. (4) is asserted ('Although not obvious from Eqs. (4)-(5), these probabilities are normalized') but not demonstrated. A short verification or a reference for the generalized binomial identity would be helpful.","section":"Eq. (4) and text after Eq. (5)"},{"comment":"There are several typos: 'persistence expontents' (SM §3.4), 'deacease' (Annex A.5), 'praticaly' (Annex A.5), 'cumpute' (SM §2.2.1). These should be corrected.","section":"SM §3.4 and Annex A"},{"comment":"The caption shows numerical symbols but does not specify the number of lattice sites or the simulation time for the splitting-probability data. For reproducibility, the simulation parameters used in Fig. 3 should be stated in the caption or in the text.","section":"Fig. 3 caption"}],"recommendation":"major_revision","confidential_remarks":"The paper is competently written and the numerical work is careful, but the central exactness claims for N>1 simultaneous-start splitting probabilities and for generic SIRW last-survival exponents are not proven; the supplementary text itself flags the missing steps. This is a load-bearing gap rather than a presentation issue. I would recommend major revision: the authors should either supply rigorous proofs for the N-particle Ray-Knight extension and protocol invariance, or explicitly reframe the paper's claims as conjectures supported by numerics. With such a change, the paper could be publishable; in its current form, the advertised 'exact' simultaneous-start results are not established."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the paper in one breath: it reports exact-looking formulas for the k-th arrival and splitting probabilities of N trail-interacting walkers with saturating reinforcement (SATW_phi), plus a protocol-invariance claim — simultaneous vs sequential starts give identical splitting probabilities. If true, that invariance is a genuinely surprising result for a non-Markovian many-body system. The formulas pass obvious sanity checks: phi=1 recovers independent Brownian walkers, N=1 reduces to known single-particle results, the distributions normalize, and the numerics in Figs. 2 and 3 match well across phi and N. The paper also makes a nice empirical extension of the effective-phi mapping to generic saturating walks, and shows protocol invariance breaks down for TSAWs. That is real content, and it's a substantive advance within a relatively small but active subfield.\n\nNow the load-bearing weak spot. The central claim for 0<k<N — mixed absorption events — is proven only for the sequential-start protocol. The simultaneous protocol is then asserted to give the same law. The SM is honest about the gap: §2.1.1 says the Pólya urn scheme is 'supposed' to work for N>1; §2.2.1 says the inhomogeneous mixed-absorption process 'cannot be solved' and protocol independence 'should not change the description'. Rigorous convergence is only cited for N=1. That is not a proof. The paper's title and abstract say 'exact', but for simultaneous starts and mixed absorption the claim is a conjecture supported by numerics. If the equivalence fails, Eq. (4) would remain true for sequential starts but the advertised simultaneous result would collapse. This is not a small technicality; it's the main conceptual headline.\n\nAlso, the discrete-time update rule for simultaneous starts is unspecified, which matters for how first crossings of unvisited edges are defined when two walkers approach the same edge together. Minor relative to the proof gap, but it will need pinning down.\n\nI think the results are most likely correct — the sequential derivation is coherent and the numerics are consistent — but the authors should either prove the N-particle Ray-Knight extension and protocol invariance under a precise update rule, or explicitly restate the simultaneous mixed-absorption formulas as a conjecture. I'd send this to a serious referee who knows Tóth's theory. It deserves the time; as written it should not be desk-rejected, but the 'exact' framing needs revision before publication.","headline":"Clean formulas and a valuable protocol-invariance claim, but the simultaneous-start mixed-absorption results are labeled exact while the SM concedes the key process is unsolved.","tokens_in":33233,"tokens_out":2944,"would_cite":true,"duration_ms":29422,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"For N walkers sharing a saturating trail, the probability that exactly k of them are absorbed at a given wall is an exact closed form, and it is the same whether the walkers start together or one after another.","keywords":["trail-interacting random walks","saturating self-interacting walks","once-reinforced random walk","splitting probabilities","persistence exponents","Ray-Knight theory","squared Bessel processes","first-passage statistics"],"falsifier":"The most direct check is a high-statistics lattice simulation of N=3 or N=4 SATW_phi walkers with phi≠1, comparing simultaneous versus sequential launches for mixed-absorption events (0<k<N): any systematic difference in P(+k|x0) beyond Monte Carlo error would falsify the protocol-invariance claim. A second, sharper test is to verify Eq. (4) numerically near x0=0 or x0=1, where the exponent k+phi-1 controls the behaviour; the exact BESQ integration in Eq. (8) should be validated independently of the urn extension.","tokens_in":32315,"feed_emoji":"🐜","tokens_out":9825,"duration_ms":84713,"temperature":0.7,"pith_summary":"This paper sets out to compute, exactly, the first-passage statistics of N one-dimensional random walkers that interact through a shared, saturating trail field—the once-reinforced SATW_phi family. Its central claims are a closed-form splitting probability (the chance that exactly k of N walkers are absorbed at a chosen boundary is a binomial-like expression in the starting position), explicit all-at-one-wall formulas in terms of incomplete Beta functions, and generalized persistence exponents theta_{k,N} = (k+phi-1)/2. The paper's key claim is that these splitting probabilities are identical whether the walkers explore simultaneously or one after another, despite the strongly history-dependent correlations the shared environment creates. If correct, the results give quantitative, ready-to-use predictions for competitive target search in biological and synthetic active systems with persistent environmental memory, and establish a rare case where collective first-passage statistics with memory are known exactly.","feed_headline":"N trail-memory walkers: exact hit odds, launch order doesn't matter","feed_subtitle":"Exact formulas give the odds that k of N trail-sharing walkers hit each wall; launch protocol makes no difference.","key_machinery":"The load-bearing object is the cumulative local-time field L(x) at the stopping time when all N walkers are absorbed. For SATW_phi, the Ray-Knight/Pólya-urn representation of a single self-interacting walk is extended to N walkers: in the scaling limit, L(x) is a squared Bessel process BESQ^{2(N+phi-1)} on the explored side and an absorbed/reflected BESQ^{2(1-phi)} on the unexplored side. The proof of protocol equivalence uses additivity of squared Bessel processes—the sum of independent BESQ processes of dimensions d1 and d2 is a BESQ of dimension d1+d2—so that N sequentially launched walkers produce the same cumulative local-time law as N simultaneous ones. Splitting probabilities then fol","core_discovery":"The paper's central discovery is an exact distribution for which walls absorb which walkers: for N SATW_phi walkers in the interval [0,1] starting at x0, the probability that exactly k are absorbed at the right wall is binom(N+2(phi-1), k+phi-1) x0^{k+phi-1}(1-x0)^{N-k+phi-1}, with the two extremes given by regularized incomplete Beta functions I_{1-x0}(N+phi-1, phi) and I_{x0}(N+phi-1, phi). These probabilities are normalized and are exactly equal for simultaneous and sequential launch protocols. The same construction yields generalized persistence exponents theta_{k,N} = (k+phi-1)/2, recovering the known single-walker exponent phi/2 and the independent-Brownian k/2 at phi=1. The authors de","pith_inferences":["Editorial inference: Eq. (4) is the probability mass function of a beta-binomial distribution with parameters (N + 2(phi-1), k + phi-1), so one could view the absorption counts as arising from independent draws with a beta-distributed propensity; the paper does not make this connection.","Editorial inference: protocol invariance for saturating trails suggests the launching order is irrelevant for all observables built on the cumulative local-time field, such as order statistics of absorption times; this remains to be tested beyond the splitting probabilities computed here.","Editorial inference: the paper's numerics for many simultaneous TSAWs show a sharply peaked Gaussian splitting distribution with standard deviation growing as N^{1/4}; deriving this scaling from a hydrodynamic or field-theoretic description is a natural next step but goes beyond what the paper proves.","Editorial inference: a rigorous N-walker Ray-Knight convergence theorem (the paper cites only the N=1 case) would place the BESQ representation on firmer ground and could extend the approach to multiple targets or heterogeneous boundary conditions."],"forward_implications":["For any N and any saturating trail strength phi, splitting probabilities and persistence exponents are known in closed form; simulations in the paper confirm them for several saturating models beyond the exact SATW_phi case.","Sequential and simultaneous launching protocols produce identical absorption statistics for SATW_phi, while the paper's numerics show the two protocols genuinely differ for nonsaturating trail interactions (e.g., TSAW and PSRW), so protocol equivalence is a fingerprint of saturation.","Repulsive trails (phi<1) lower the generalized persistence exponents, making long-lived non-reactive trajectories more probable; attractive trails (phi>1) suppress them.","The last-survival exponent theta_{1,N} for generic self-interacting random walks equals the single-walker exponent theta_{1,1}, independent of N—a result the paper derives by a direct argument and verifies numerically.","The small-x0 decay of the splitting probabilities, combined with the scaling argument, yields the full spectrum theta_{k,N} without separately computing survival probabilities."],"fun_headline_variants":["Trail-sharing walkers: exact odds, launch order irrelevant","Binomial law for N walkers: split odds independent of order","Memory trail: identical split odds for simultaneous or sequential","Exact k-wall absorption odds for N trail walkers","N trail walkers: same split probabilities regardless of launch"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The N-walker extension of the Ray-Knight/Pólya-urn construction is load-bearing: SM §2.1.1 explicitly 'supposes' the urn scheme still works for several walkers, §2.2.1 states the mixed-absorption (both walls reached) process is not solved and that protocol independence 'should not change the description,' and a rigorous convergence proof is cited only for N=1; if this extension or the protocol-invariance assertion fails, Eqs. (2)-(5) are not exact.","fun_headline_variants_meta":{"raw":{"variants":["Trail-sharing walkers: exact odds, launch order irrelevant","Binomial law for N walkers: split odds independent of order","Memory trail: identical split odds for simultaneous or sequential","Exact k-wall absorption odds for N trail walkers","N trail walkers: same split probabilities regardless of launch"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000179,"raw_usage":{"total_tokens":1150,"prompt_tokens":771,"completion_tokens":379,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":515,"completion_tokens_details":{"reasoning_tokens":311}},"tokens_in":515,"tokens_out":379,"duration_ms":4400,"temperature":1.0,"reasoning_tokens":311,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T05:51:24.597153+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The most direct check is a high-statistics lattice simulation of N=3 or N=4 SATW_phi walkers with phi≠1, comparing simultaneous versus sequential launches for mixed-absorption events (0<k<N): any systematic difference in P(+k|x0) beyond Monte Carlo error would falsify the protocol-invariance claim. A second, sharper test is to verify Eq. (4) numerically near x0=0 or x0=1, where the exponent k+phi-1 controls the behaviour; the exact BESQ integration in Eq. (8) should be validated independently of the urn extension.","supporting_citations":[],"review_version":1}