{"id":"36a26dbc-2aad-4c18-bd9a-e2f6ca054271","arxiv_id":"2608.11202","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Mean first-passage time response to arbitrary single-rate perturbations is expressed exactly through unperturbed MFPTs and steady-state probabilities, via a fast-reset correspondence.","lead":"This paper derives exact formulas for how the mean first-passage time of a Markov process responds when a single transition rate is perturbed, including nonlinear and arbitrarily strong changes. The key idea is a mapping from the transient first-passage problem to the steady state of an auxiliary system with a fast reset edge.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"SM proof of the key correspondence (Eq. 12) contains an algebraic error: the reset flux is computed as (W0+W*)π(1−τWπ)/(1+τWπ), whose limit is −1/τ, not 1/τ; Eq. (2) and all main results inherit this gap.","rationale":"The reader's weakest_assumption identified the reliance on the authors' prior steady-state response identities [36]; that is a valid domain concern, but the more concrete and more load-bearing problem is inside this manuscript: the proof of Eq. (12), the essential correspondence, contains an algebraic error. Equation (2) is the unique bridge converting steady-state response into MFPT response; if its proof is invalid, the main formulas (3a)-(3b) are not rigorously established, regardless of the correctness of [36]. The error is limited to the SM derivation and appears correctable: the same Lemma 1 yields π'_j = π_j/(1+W*π_jτ_ji), which gives the intended limit. Because the central result is nonetheless supported by internal algebra and by numerical verification on 8-state fully connected networks, the appropriate response is to require the authors to fix the proof (and ideally check the correspondence numerically) rather than to reject the paper. The verdict should therefore be CONDITIONAL on a corrected derivation of the correspondence.","tokens_in":18619,"tokens_out":25766,"duration_ms":178806,"concrete_test":"On a small random irreducible network, add an edge from target k to initial l with rate W*, compute the stationary probability π^*_k numerically for a sequence of large W*, and check whether W*π^*_k converges to 1/τ_kl. Separately, evaluate the SM's erroneous expression (W^0+W*)π_k(1-τ_kl W*π_k)/(1+τ_kl W*π_k) at the same W* values; it will deviate sharply and become negative for large W*, confirming the proof gap. If the numerical check supports the corrected formula, the issue is a repairable but nonetheless real error in the manuscript's proof chain.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central bridge of the paper is the correspondence τ_kl/τ'_kl = lim_{W*→∞} π^{*'}_k/π^*_k (Eq. 2), derived from the MFPT expression τ_kl = 1/(lim_{W*→∞} W*_{lk} π^*_k) (Eq. 12). The only algebraic proof of Eq. (12), in SM Prerequisite 1 (Eqs. (8)-(9)), computes the reset flux as φ^*_ij = lim (W^0_ij+W*_ij) π_j (1 - τ_ji W*_ij π_j)/(1 + τ_ji W*_ij π_j). For large W*, the factor (1 - τ_ji W*_ij π_j)/(1 + τ_ji W*_ij π_j) tends to -1, so the claimed limit 1/τ_ji is unjustified; the expression as written would give -1/τ_ji. The correct application of Lemma 1 to the added edge gives π'_j = π_j/(1 + W*_ij π_j τ_ji), whose flux limit is indeed 1/τ_ji. Because the manuscript does not provide a correct alternative proof of Eq. (12), the derivation of the linear and nonlinear response identities (3a)-(3b) and all corollaries relies on an invalid intermediate step as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript develops an exact response theory for mean first-passage times (MFPTs) in finite continuous-time Markov chains. The central construction is an auxiliary system with a fast reset edge from the target back to the initial state; the MFPT is shown to equal the inverse stationary flux through that edge, so that the response of an MFPT to a perturbation of a transition rate can be mapped onto the steady-state response of the auxiliary system. This mapping yields exact linear and nonlinear response identities for single-rate perturbations, expressed solely through unperturbed MFPTs and steady-state probabilities, together with a factorized physical interpretation, response-curve reconstruction rules, fundamental bounds, higher-order response formulas, and recursive multi-rate updates. Two applications are given: a folding network with a chaperone cycle and a clarification of a Braess-type paradox in target-resolved search efficiency.","tokens_in":18900,"tokens_out":17389,"duration_ms":129699,"significance":"If the results hold, the paper provides a closed-form, parameter-free description of the full nonlinear response of first-passage times to kinetic perturbations, a question of broad interest in statistical physics, biophysics, and network science. The main identities are compact and directly testable, and the numerical verification on 10^5 random 8-state networks is a concrete strength. The factorization into upstream accessibility, downstream gain/loss, and nonlinear screening offers both physical insight and a practical inference method from a single nonlinear measurement. The bounds on relative log-response and the computational shortcuts for sparse perturbations are additional useful contributions. The main weakness is that the proof of a key auxiliary identity (SM Prerequisite 1) contains an algebraic error, and the derivation imports central steady-state response identities from the authors' previous work without re-proof.","major_comments":[{"comment":"The flux calculation in Prerequisite 1 is algebraically incorrect. Equation (8) contains the factor (1 - τ_ji W*_ij π_j)/(1 + τ_ji W*_ij π_j); as W*_ij → ∞ this factor tends to -1, so the product with (W^0_ij + W*_ij)π_j diverges rather than approaching 1/τ_ji. The correct application of Lemma 1 gives π*_j = π_j/(1 + W*_ij π_j τ_ji), so the flux is (W^0_ij + W*_ij)π_j/(1 + W*_ij π_j τ_ji), whose limit is indeed 1/τ_ji. Since Eq. (7) is the stated proof of the fundamental correspondence (2), this error must be corrected; the alternative derivation in SM Section II, which applies Lemma 1 twice and is valid, should be promoted to the primary proof.","section":"SM Prerequisite 1, Eqs. (8)-(9)"},{"comment":"The central response formulas (3a)-(3b) and Theorem 1 inherit their validity from Lemma 1 and the multi-rate identity (End Matter Eq. (16); SM Prerequisite 2), which are quoted from Ref. [36] without proof. Because these identities are load-bearing, the manuscript should either reproduce their proofs in the SM or explicitly cast them as assumptions with their precise validity conditions. The numerical test in SM Fig. 1 samples only fully connected 8-state networks with rates in [0.01,10]; the claim of universality for arbitrary sparse, directed, or nearly reducible topologies would be better aligned with the evidence if this limitation were stated.","section":"SM II, Lemma 1 and Eq. (16); main text Eqs. (1) and (3)"}],"minor_comments":[{"comment":"The parenthetical statement about positivity cites the triangle inequality τ_km + τ_mn ≥ τ_kn, which applies to Σ_{k|mn}; the positivity of U_{k←l|n} follows from the triangle inequality τ_nk + τ_kl ≥ τ_nl, so the sentence should be amended.","section":"Main text, after Eq. (3b)"},{"comment":"The definition of α_X is terse; a sentence explaining the three cases (X = A_mn, B_mn, E_m) and how α_X enters the response curve would improve readability.","section":"Main text, Eqs. (10a)-(10b)"},{"comment":"The notation ⟨t_kl⟩ is used in the SM alongside τ_kl in the main text; please unify the notation throughout the paper and the supplementary material.","section":"SM, Theorem 1"},{"comment":"The characterization of the discrete-time work in Ref. [44] as 'either not closed or involve quantities without clear physical meaning' is strong; a specific example or a softer wording would be more constructive.","section":"Introduction"},{"comment":"The complexity estimates in Table II would benefit from a brief explanation of the parameters (K, N) and the assumed data structure (e.g., which matrices are precomputed).","section":"End Matter, Table II"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern about the SM flux calculation is valid: the algebraic error in Prerequisite 1 must be fixed, and the self-containedness of the imported steady-state identities should be addressed. The paper's core results are numerically verified and the error is local and repairable, so I would not recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here’s my read. The paper has a real result: an exact finite-amplitude response formula for any MFPT under a single-rate perturbation, expressed entirely through unperturbed MFPTs and steady-state probabilities. The factorization into upstream accessibility, downstream gain/loss, and nonlinear screening is physically clear and genuinely useful. What’s new is the fast-reset correspondence and the closed-form nonlinear identity (3b); the linear-response part overlaps with known Markov-chain sensitivity results (Refs. [17,44]), so the linear piece is more a unification than a novelty. The numerical verification on 10^5 random 8-state networks is solid evidence that the finite update formula is correct. The two applications (folding network and the Braess-paradox clarification) are appropriate and modest.\n\nSoft spots, in order of severity. First, there is an algebraic error in the SM proof of the central correspondence. In Prerequisite 1, Eq. (8), the reset flux is written as (W0 + W*) π (1 − τ W* π)/(1 + τ W* π). For large W*, that ratio tends to −1, not 1/(1 + τ W* π), so the displayed limit would give −1/τ, not 1/τ. The correct application of Lemma 1 gives π′_j = π_j/(1 + W* π_j τ_ji), from which the flux limit is indeed 1/τ. So the result is repairable, but the proof as written is invalid at that step. This is not a fatal flaw, but it has to be fixed. Second, the paper imports the steady-state response identities from the authors’ own Ref. [36] rather than re-proving them; the main formulas inherit whatever validity conditions those have. Third, the numerical tests cover fully connected random networks with rates in [0.01,10], so sparse, directed, or nearly reducible topologies are not directly exercised. There is no obvious reason the formulas would fail there, but the universality claim is not numerically demonstrated for those cases.\n\nOverall, the central argument holds up: the final identity (3b) is verified numerically, and the proof gap is local and easily repaired. This is a within-field advance, not a revolution. I would bring it to a reading group and would cite it once the SM is cleaned up. It deserves a serious referee; an editor should send it out rather than desk-reject. The referee should ask the authors to fix Eq. (8) and to state explicitly what is imported from Ref. [36].","headline":"A genuinely useful finite-amplitude MFPT response theory with a repairable proof gap in the SM; worth refereeing but needs a corrected derivation of the key correspondence.","tokens_in":19407,"tokens_out":4381,"would_cite":true,"duration_ms":34834,"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":"This paper establishes exact linear and nonlinear response identities for mean first-passage times in continuous-time Markov processes, expressing the perturbed MFPT under arbitrary single-rate perturbations entirely in terms of…","keywords":["mean first-passage time","Markov jump process","nonlinear response","transition rate perturbation","steady-state probability","fast-reset correspondence","global mean first-passage time","response bounds"],"falsifier":"Choose an irreducible 4-state directed network with one narrow bottleneck edge, compute all unperturbed MFPTs and steady-state probabilities by solving the linear systems exactly, perturb a single rate by a finite $\\Delta W_{mn}$, and compare the exact $\\tau'_{kl}$ from an absorbing-chain solve with the prediction of Eq. (3b). A mismatch for any $\\Delta W_{mn}>0$ in an irreducible network, or for a sparse or directed topology outside the random fully connected networks tested, would falsify the claimed universality.","tokens_in":18401,"feed_emoji":"⏱️","tokens_out":6517,"duration_ms":57730,"temperature":0.7,"pith_summary":"The paper seeks a systematic theory for how the mean first-passage time (MFPT) between two states of a Markov process changes when transition rates are perturbed. It proves that, for a perturbation of any single transition rate, the exact nonlinear response of any MFPT can be written as a one-denominator rational function of the perturbation strength, using only unperturbed MFPTs and steady-state probabilities. The argument rests on a correspondence: adding a very fast reset edge from the target back to the initial state turns repeated first-passage events into stationary cycles, so the MFPT response equals the steady-state response of an auxiliary system. This yields a physical decomposition of the response into upstream accessibility, downstream gain or loss, and nonlinear screening, as well as bounds, higher-order derivatives, multi-rate recursions, and computational shortcuts. If correct, the theory means that global timing observables can be predicted and controlled from a small set of pre-computed quantities.","feed_headline":"Exact formula predicts first-passage time response to any rate change","feed_subtitle":"One measured nonlinear point plus unperturbed MFPTs and probabilities reproduces the whole response curve.","key_machinery":"The central object is the fast-reset correspondence: adding a unidirectional edge from the target $k$ back to the initial state $l$ with rate $W^*_{lk}\\to\\infty$ in the original Markov chain turns each first-passage event into one cycle of a stationary process, so that the MFPT becomes the inverse stationary flux through the reset edge. This equality $\\tau_{kl}=1/\\lim W^*_{lk}\\pi^*_k$ lets the authors import steady-state response identities for $\\pi^*_k$, in particular Lemma 1 for how a single-rate perturbation shifts steady-state probabilities, to derive the exact finite and linear MFPT-response formulas. The mechanism closes because the perturbation and the fast-reset edge commute in the rate matrix, so the ratio of auxiliary steady-state probabilities equals the ratio of MFPTs.","core_discovery":"For a continuous-time irreducible Markov chain with stationary probabilities $\\pi$, the paper claims that perturbing one transition rate $W_{mn}\\to W_{mn}+\\Delta W_{mn}$ changes the MFPT $\\tau_{kl}$ exactly as $$\\frac{\\tau'_{kl}-\\tau_{kl}}{\\$\\Delta$ W_{mn}} = \\frac{-\\pi_n(\\tau_{kn}-\\tau_{km})(\\tau_{nk}+\\tau_{kl}-\\tau_{nl})}{1+\\$\\Delta$ W_{mn}\\pi_n(\\tau_{kn}+\\tau_{nm}-\\tau_{km})},$$ with $\\tau'_{kl}$ the perturbed MFPT. The numerator is the exact linear response and the denominator is the only finite-perturbation correction; every quantity on the right is from the unperturbed system. The proof uses the correspondence $\\tau_{kl}/\\tau'_{kl}=\\lim_{W^*_{lk}\\to\\infty}\\pi^{*\\prime}_k/\\pi^*_k$ for an auxiliary system with a fast reset edge $k\\to l$, which converts transient first-passage statistics into stationary steady-state response.","pith_inferences":["The rational form of Eq. (3b) implies that the response surface is parameterized by a handful of unperturbed numbers per target pair, suggesting an experimental protocol: measure $\\tau_{kl}$ and its linear response, apply one finite perturbation, and then predict all other perturbation strengths without further simulation.","Because the nonlinear screening factor is independent of the initial state $l$, measuring response saturation from two different initial states should yield the same inferred bottleneck size; that cross-check would be a direct test of the framework that the paper does not perform.","The target-resolved monotonicity of the global MFPT suggests that in biological search or folding models the same added edge can be beneficial for one target and harmful for another, so interventions should be designed target-by-target rather than by global averages.","The stated computational advantage indicates that for large networks with few tunable rates, the response recursion could replace full matrix inversions in sensitivity analysis, a use the authors mention but do not benchmark at scale."],"forward_implications":["Every MFPT is monotone in every transition rate, with a concave response for harmful edges and a convex response for helpful edges, so non-monotone regimes cannot occur.","The full response curve for arbitrary perturbation strength is fixed by the unperturbed MFPT, its linear response, and one additional finite measurement; a single nonlinear data point predicts all $\\Delta W_{mn}$.","Relative log-responses are bounded, with the most negative response realized at the perturbed edge itself, and the finite envelope $\\min(c,c^{-1})\\le \\tau_{kl}(cW_{mn})/\\tau_{kl}(W_{mn})\\le \\max(c,c^{-1})$ holds.","Adding a dynamical shortcut, an edge that moves probability to a state closer in MFPT to the target, always reduces the target-resolved global MFPT, so the reported Braess-type search paradox is attributable to target averaging.","The recursive multi-rate update computes perturbed MFPTs and steady-state distributions after any finite set of rate changes in $O(KN^2)$ or better, enabling sparse perturbation analysis of large networks."],"supporting_citations":[{"why":"Supplies Lemma 1 and the multi-rate steady-state response identity that the main proof imports; the validity of the main results inherits from these identities.","marker":"[36]"},{"why":"Provides the original steady-state flux construction with reset-to-initial transitions, which the paper modifies to obtain the MFPT expression used in the correspondence.","marker":"[58]"},{"why":"Gives earlier discrete-time MFPT sensitivity formulas that are not closed; it is the baseline against which the paper's closed-form result is contrasted.","marker":"[44]"},{"why":"Reports the Braess-type search paradox for global MFPT that the paper explains as a target-averaging artifact using its response identities.","marker":"[55]"},{"why":"Establishes unit response bounds that the present paper strengthens; it uses the reset construction in a different way that does not yield the exact response expressions.","marker":"[48]"}],"fun_headline_variants":["Exact MFPT response formula from steady-state correspondence","One rate change: exact first-passage response from unperturbed data","First-passage response theory: linear + nonlinear, all exact","Exact response of first-passage times to any rate perturbation","Steady-state trick yields exact MFPT response relations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The exact steady-state response identities for Markov jump processes imported from the authors' prior work are not re-derived here, and the numerical verification covers fully connected random networks rather than all topologies, so the MFPT formulas stand or fall with those identities and their domain of validity.","fun_headline_variants_meta":{"raw":{"variants":["Exact MFPT response formula from steady-state correspondence","One rate change: exact first-passage response from unperturbed data","First-passage response theory: linear + nonlinear, all exact","Exact response of first-passage times to any rate perturbation","Steady-state trick yields exact MFPT response relations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000213,"raw_usage":{"total_tokens":1437,"prompt_tokens":979,"completion_tokens":458,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":595,"completion_tokens_details":{"reasoning_tokens":374}},"tokens_in":595,"tokens_out":458,"duration_ms":4588,"temperature":1.0,"reasoning_tokens":374,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:14:26.379412+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Choose an irreducible 4-state directed network with one narrow bottleneck edge, compute all unperturbed MFPTs and steady-state probabilities by solving the linear systems exactly, perturb a single rate by a finite $\\Delta W_{mn}$, and compare the exact $\\tau'_{kl}$ from an absorbing-chain solve with the prediction of Eq. (3b). A mismatch for any $\\Delta W_{mn}>0$ in an irreducible network, or for a sparse or directed topology outside the random fully connected networks tested, would falsify the claimed universality.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the original steady-state flux construction with reset-to-initial transitions, which the paper modifies to obtain the MFPT expression used in the correspondence."},{"cited_title":"Tejedor, O","cited_arxiv_id":null,"evidence_quote":"Reports the Braess-type search paradox for global MFPT that the paper explains as a target-averaging artifact using its response identities."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes unit response bounds that the present paper strengthens; it uses the reset construction in a different way that does not yield the exact response expressions."}],"review_version":2}