{"id":"81451832-e041-4c53-a604-9d272442c187","arxiv_id":"2412.05019","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For Markov jump processes, current-current susceptibility equals a difference of mean first-passage times, yielding an affine, non-additive relation between multiple steady currents.","lead":"This paper derives a new formula for how one steady current in a random hopping network changes when another current is altered: the response coefficient is a difference of mean first-passage times. The result extends a recently discovered linear relation between currents to multiple simultaneous changes, where the effects do not simply add.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the affine current relation (IV.2) is exact within the stated irreducible-network scope.","rationale":"The reader's conditional verdict is driven by the irreducibility scope and textual inconsistencies. My stress-test confirms the central theorem within its stated assumptions: the derivation of (III.1) and (IV.2) is algebraically exact for finite irreducible Markov jump processes, and the multigraph extension is coherent. The irreducibility assumption is explicit and not a mathematical flaw; it is a scope condition. The only real errors I found are non-central: the rate table in Example V.1 is internally inconsistent and Eq. (V.8) contains a sign typo in an intermediate equality. Because neither affects the proof of Eq. (IV.2) or the nonadditivity conclusion, I do not change the reader's verdict and do not escalate the concern.","tokens_in":8415,"tokens_out":37293,"duration_ms":334705,"concrete_test":"Recompute Eq. (V.8) on a three-state graph with one added edge, using the paper's corrected first-passage times: evaluate V(a)-V(b) from the Poisson equation and compare with -lambda0(ab,(x,y)); if the sign differs from the printed equality, confirm that Eq. (V.5) still holds with V(b)-V(a).","verdict_should_be":"UNCHANGED","load_bearing_attack":"I read the derivation of Eq. (IV.2) as internally sound. The key step (III.1) follows from the adjoint relation and the first-passage Poisson equation, and substituting it into the unperturbed channel current gives (IV.2) without approximation. The only mathematical boundary is the explicit assumption in Section II that irreducibility is preserved: if a perturbation removes a bridge and disconnects the state graph, mean first-passage times diverge and the formula cannot be evaluated. This is stated, not a hidden error. The remaining defects are presentational: Example V.1 contains inconsistent rate assignments (k46 appears with values 1 and 2 in Eq. (V.9)), and the displayed equality in Eq. (V.8) has a sign slip (V(a)-V(b) equals -lambda in a three-state check, while the final relation (V.5) uses V(b)-V(a) and remains correct). Neither defect threatens Eq. (IV.2).","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper derives an exact, nonperturbative affine relation between steady currents in a finite-state continuous-time Markov jump process when transition rates are perturbed. Starting from the Poisson equation for mean first-passage times and an adjoint identity, Eq. (III.1) expresses the change in the stationary distribution in terms of the perturbing currents and mean first-passage time differences in the unperturbed process. Substituting this into the expression for an unperturbed channel current gives the main result (IV.2): the current is affine in the perturbing currents, with coefficients determined solely by the unperturbed process. The authors then specialize to the case of one added edge, give a quasipotential representation (V.5), extend the result to multiple perturbed edges, and illustrate the non-additivity of the multi-edge relation. An appendix supplies a Kirchhoff-tree representation of the susceptibility.","tokens_in":8584,"tokens_out":24794,"duration_ms":233547,"significance":"If correct, the result is a clean and exact generalization of the mutual linearity discovered by Harunari et al., with a shorter derivation and a new first-passage-time interpretation. The derivation is self-contained and requires no fitting, simulation, or back-reference to the target result; the extension to multiple simultaneous perturbations and the explicit demonstration that the affine relation is not additive are genuine additions. The graphical Kirchhoff representation in Appendix A is also useful for identifying which network features control current response. The main claims are mathematically sound within the stated irreducible-network setting, and the paper properly acknowledges the prior results in [1] and [9].","major_comments":[],"minor_comments":[{"comment":"The displayed rate assignments are inconsistent: the edge k46 is assigned the values 2, 1, and 1 in the three lines of Eq. (V.9), and k65 and k54 are each assigned both 2 and 1. As a result, the numerical values in Eqs. (V.11)-(V.13), including j(4,6)=1/9, cannot be reproduced from the stated rates. Please correct the rate list and re-run the example so that the non-additivity illustration is verifiable.","section":"Section V, Eq. (V.9)"},{"comment":"The chain of equalities in Eq. (V.8) proves V0_xy(a)-V0_xy(b) = -lambda0(ab,(x,y)), not +lambda0 as printed; the final relation (V.5), which uses V0_xy(b)-V0_xy(a), is nevertheless correct. Please fix the sign in the displayed equality and in the sentence that follows it.","section":"Section V, Eq. (V.8)"},{"comment":"There is a factor-of-two convention that should be stated explicitly: Eq. (IV.1) defines lambda0_i(zz',(x,y)) for an ordered pair (z,z'), while Eq. (V.3) defines the coefficient for an unordered added edge as twice the ordered-pair value, because both (a,b) and (b,a) contribute to the sum in (IV.2). Without an explicit remark, a reader applying (IV.2) to a single added edge may miss the factor of two.","section":"Section IV-V"},{"comment":"There is a minor typo in Eq. (V.5): the argument should read (V0_xy(b)-V0_xy(a)) j(a,b), with a comma rather than a period in 'j(a.b)'.","section":"Section V, Eq. (V.5)"}],"recommendation":"minor_revision","confidential_remarks":"This is a competent paper whose central result is correct. The revision should fix the inconsistent rate list in Eq. (V.9) and the sign in Eq. (V.8); these are local typos, but they must be corrected before publication because the example is meant to illustrate non-additivity. I see no issue with novelty or scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main result is Eq. (IV.2): perturbing rates elsewhere shifts a channel current by a linear combination of the perturbing currents, with coefficients given by differences of mean first-passage times in the unperturbed process. That is exact and nonperturbative, and the derivation is genuinely short: (III.1) follows from the Poisson equation and the adjoint identity. The extension to multiple perturbed edges (Section V) is affine but not additive, which is a real point and worth having on record.\n\nThe paper is honest about prior work. The single-edge linearity reproduces Harunari et al., and Eq. (III.1) is acknowledged as identical to relation (44) of Harvey et al. The new pieces are the first-passage-time expression (IV.1) and the non-additivity example. The Appendix's spanning-tree representation is a useful complement, though not essential.\n\nThe soft spots are presentational, not load-bearing. Example V.1 has inconsistent rate assignments: k46 appears with values 1 and 2 in Eq. (V.9), which makes the numerical claims hard to check. There is also a sign slip in Eq. (V.8) (V(a)-V(b) is -lambda in a three-state check, while the displayed equality says plus); the final relation (V.5) still uses the correct V(b)-V(a). Both are fixable in revision, and neither touches Eq. (IV.2).\n\nThe one real limitation is the standing assumption that irreducibility is preserved. If a perturbation disconnects the graph, mean first-passage times diverge and (IV.2) is not defined. The authors state this in Section II, so it is not hidden, but readers who want bridge-breaking perturbations will need a different tool.\n\nCitation pattern looks fine: the key prior results are credited, self-citations are to the authors' own Poisson-equation paper, which is relevant.\n\nI found the paper worth a careful read. It is not a breakthrough, but it gives a cleaner route to a known result and adds something genuinely new in the multi-edge case. I would send it to a competent referee. If the authors fix the example's rates and the sign typo, the paper is publishable as is.","headline":"A clean, short proof of the affine current relation with a genuinely new first-passage-time interpretation; the defects are textual, not mathematical.","tokens_in":9100,"tokens_out":1676,"would_cite":true,"duration_ms":15991,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Steady currents in finite Markov networks obey an exact affine relation when rates are perturbed, and the response coefficients are differences of mean first-passage times in the unperturbed process.","keywords":["steady nonequilibrium","mean first-passage time","current susceptibility","Markov jump process","affine current relation","nonadditivity","multigraph"],"falsifier":"Take a small finite network, solve the stationary master equation before and after perturbing one edge exactly by matrix inversion, compute the right-hand side of Eq. (IV.2) from mean first-passage times of the unperturbed process, and check equality entry by entry; any mismatch beyond numerical precision would refute the formula. Alternatively, choose a perturbation that removes a bridge: the mean first-passage time between the two resulting components becomes infinite, so the stated formula cannot be evaluated, pinpointing the irreducible assumption.","tokens_in":8250,"feed_emoji":"⚡","tokens_out":6304,"duration_ms":62162,"temperature":0.7,"pith_summary":"This paper establishes an exact, nonperturbative relationship between steady currents in a finite Markov jump process: if the transition rates on one or more edges are changed while the rest of the graph is untouched, the current on any untouched edge is an affine function of the currents on the perturbed edges. The coefficients are differences of mean first-passage times computed in the original, unperturbed process, which gives a physical interpretation of current-current susceptibilities as the excess transport caused by starting from one vertex rather than another. The paper re-derives a known mutual linearity result by a shorter route and extends it to multiple simultaneously perturbed edges, where the relation remains affine but is not additive. A worked nine-state example shows the nonadditivity explicitly.","feed_headline":"Perturbed steady currents follow a first-passage-time law","feed_subtitle":"Exact nonperturbative formula ties current-current response to mean first-passage-time differences, extended to many edges.","key_machinery":"The central object is the mean first-passage time $\\tau_0(z,x)$ of the unperturbed Markov jump process, solved from the Poisson-type equation $L_0 \\tau_0(z,x)=\\delta_{xz}/\\rho_0(x)-1$ for $z\\neq x$, with $\\tau_0(x,x)=0$. Combining that equation with stationarity of the perturbed process yields Eq. (III.1), showing the stationary distribution change $\\rho(x)-\\rho_0(x)$ is $\\rho_0(x)$ times a sum of mean-first-passage-time differences weighted by perturbed currents. Feeding that into the definition of the steady current gives the affine relation and identifies the susceptibility $\\lambda_0$ as mean-first-passage-time differences, or equivalently as a difference of excess-current quasipotentials satisfying a Poisson equation.","core_discovery":"On a finite multigraph with transition rates $k_i(x,y)$, perturb some channels by $\\Delta_i(x,y)$ while requiring irreducibility. For any channel whose own rates are unchanged, the paper proves Eq. (IV.2): $j_i(x,y)=j^0_i(x,y)+\\sum_{z,z'}\\lambda^0_i(zz',(x,y))\\,j_\\Delta(z,z')$, with $\\lambda^0_i(zz',(x,y)) = \\frac{1}{2}[\\rho_0(x) k^0_i(x,y)(\\tau_0(z,x)-\\tau_0(z',x)) - \\rho_0(y) k^0_i(y,x)(\\tau_0(z,y)-\\tau_0(z',y))]$. Here $\\tau_0$ is the mean first-passage time in the unperturbed process, and $j_\\Delta(z,z')$ is the current carried by the rate changes. The formula is exact for finite irreducible networks and does not rely on being close to equilibrium. The same object can be written as a difference of quasipotentials, and the appendix gives an equivalent Kirchhoff spanning-tree expression.","pith_inferences":["Editorial: Because the coefficients live entirely in the unperturbed process, the relation can be turned into an estimation scheme: sample mean first-passage times from a single simulation of the reference dynamics and predict currents for any single-edge perturbation without re-solving the full network.","Editorial: The nonadditivity in Section V implies that tuning several rates simultaneously is a genuinely joint problem; a control strategy built by adding single-edge responses can be off by an amount controlled by the cross terms $\\lambda_0$ computed with both edges removed.","Editorial: The irreducible assumption excludes bridge removal; a natural extension would treat absorbing or split networks, where the diverging mean first-passage times are replaced by escape-time or boundary-condition quantities, potentially covering open chemical reaction networks with input and output reservoirs."],"forward_implications":["The current on any unperturbed edge is an affine function of the perturbed-edge currents, with coefficients fixed by the original process alone.","The coefficient $\\lambda_0$ is a difference of mean first-passage times, so it can be interpreted as the response of the current to moving the initial condition from one vertex to the other.","The relation is nonperturbative: no small-driving or linear-response approximation is needed.","For several simultaneous perturbations the affine relation holds but is not additive; the susceptibility in that case is computed in the process where all perturbed edges are removed."],"supporting_citations":[{"why":"The earlier proof of mutual linearity of steady currents that this paper re-derives and extends.","marker":"[1]"},{"why":"Supplies the definition and properties of mean first-passage times used in the susceptibility.","marker":"[8]"},{"why":"Source of the earlier relation that is the same as Eq. (III.1), acknowledged in a footnote.","marker":"[9]"},{"why":"Provides the Poisson equation for mean first-passage times in nonreversible Markov jump processes, used in the derivation.","marker":"[10]"},{"why":"Provides the spanning-tree graphical representation used in Appendix A for the Kirchhoff form of the susceptibility.","marker":"[11]"}],"fun_headline_variants":["First-passage times link steady currents","Exact affine law via first-passage times","Current shifts from passage-time differences","Affine currents from first-passage times","Steady currents obey passage-time rule"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The result assumes a finite state space and that the perturbed network remains a single connected component, so every state is still reachable from every other; if a perturbation disconnects the graph, mean first-passage times diverge and the formula is not defined.","fun_headline_variants_meta":{"raw":{"variants":["First-passage times link steady currents","Exact affine law via first-passage times","Current shifts from passage-time differences","Affine currents from first-passage times","Steady currents obey passage-time rule"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001164,"raw_usage":{"total_tokens":4800,"prompt_tokens":906,"completion_tokens":3894,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":522,"completion_tokens_details":{"reasoning_tokens":3831}},"tokens_in":522,"tokens_out":3894,"duration_ms":26981,"temperature":1.0,"reasoning_tokens":3831,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T20:59:18.227022+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a small finite network, solve the stationary master equation before and after perturbing one edge exactly by matrix inversion, compute the right-hand side of Eq. (IV.2) from mean first-passage times of the unperturbed process, and check equality entry by entry; any mismatch beyond numerical precision would refute the formula. Alternatively, choose a perturbation that removes a bridge: the mean first-passage time between the two resulting components becomes infinite, so the stated formula cannot be evaluated, pinpointing the irreducible assumption.","supporting_citations":[{"cited_title":"Harunari, S","cited_arxiv_id":null,"evidence_quote":"The earlier proof of mutual linearity of steady currents that this paper re-derives and extends."},{"cited_title":"Harvey, S","cited_arxiv_id":null,"evidence_quote":"Source of the earlier relation that is the same as Eq. (III.1), acknowledged in a footnote."},{"cited_title":"Khodabandehlou, C","cited_arxiv_id":null,"evidence_quote":"Provides the Poisson equation for mean first-passage times in nonreversible Markov jump processes, used in the derivation."},{"cited_title":"Khodabandehlou, C","cited_arxiv_id":null,"evidence_quote":"Provides the spanning-tree graphical representation used in Appendix A for the Kirchhoff form of the susceptibility."}],"review_version":1}