{"id":"33bf3c6a-7da7-453b-b87a-f57dbd13a75a","arxiv_id":"2602.18265","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In the large-network limit, first-passage time distributions become deterministic when many generator eigenvalues contribute, and exponential when a single eigenvalue dominates.","lead":"Large random networks with random reaction rates often produce first-passage times that are either sharply peaked or exponentially spread. This paper derives the spectral conditions that decide which of the two limits appears, and shows the two regimes are not simple mirrors of each other.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (31) is derived only for r=0, not for the finite-r macroscopic forest condition; without an r→0 guarantee in the large-N ensembles, the eigenvalue-only delta/exponential classification is not established.","rationale":"The paper's strongest theoretical result is the spectral cumulant relation Eq. (31), and the reader correctly identifies the macroscopic forest condition as load-bearing. However, the stress-test shows the condition actually needed is stronger: r=0, not merely r<∞. Eq. (25) only guarantees ⟨τ_m⟩ ∼ (1/(1+r)) tr(M); the polynomial term in Eq. (27) survives for any nonzero r and contributes to all cumulants. The exponential limit is protected because Sec. III D chooses m maximizing the mean, so λ_1 dominance forces r=0. The deterministic limit has no such protection: the proof and the random-network simulations do not establish r→0. This does not invalidate the theorems as stated, since the text often explicitly says 'r=0', but it does undercut the abstract's claim that the two limits arise 'generically' from the eigenvalue spectrum. The printed bound Eq. (29) is internally inconsistent at r=0, reinforcing that the polynomial term needs more careful treatment. The paper's simulations and graph-theoretic exact expressions are valuable, and the conditional-acceptance verdict remains appropriate pending an explicit analysis of r in the simulated ensembles or a restriction of the genericity claim.","tokens_in":25364,"tokens_out":24314,"duration_ms":209146,"concrete_test":"Compute the forest ratio r_N in Eq. (24) for the forward-bias random-network ensembles of Fig. 7(a,c), using the All-Minors Matrix-Tree counts from the generated Laplacians for N=50, 100, 200 over at least 100 realizations. If r_N does not tend to 0 while the coefficient of variation tends to 0, then the observed delta limit is not governed by Eq. (31) and the universality claim needs an explicit r→0 condition or proof. If r_N→0, verify directly that the contribution of ln(Σ r_n s^n) to the second normalized cumulant is negligible in the same ensembles.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central formula Eq. (31) is derived only after setting r=0 (Eqs. 29–30), but the macroscopic forest condition introduced in Eq. (24) and used as the structural precondition for universality is r<∞. For 0<r<∞, the logarithmic term ln(Σ r_n s^n) in Eq. (27) does not vanish and contributes fixed amounts to every cumulant of t/⟨τ_m⟩. Thus eigenvalue delocalization alone does not force a delta, and eigenvalue concentration alone does not force an exponential. The exponential-limit theorem in Sec. III D sidesteps this by taking m with maximal mean, where λ_1 dominance forces r=0. No analogous argument is supplied for the deterministic limit: Sec. III C assumes r=0 without showing that the large-N random networks of Sec. III F—or 'generic' networks generally—satisfy r→0. The discussion in Sec. IV presents finite r as the relevant structural condition, but finite r is insufficient. The printed bound Eq. (29) is also inconsistent for r=0 (RHS < 1 while LHS ≥ 1), so the treatment of the polynomial term is not reliable as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the large-N limit of first-passage time (FPT) distributions for continuous-time Markovian networks with a single absorbing target. It uses a graph-theoretic representation of the inverse generator to express the cumulants of the FPT normalized by its mean in terms of sums of inverse eigenvalues of the generator's principal minor, under a 'macroscopic forest condition' defined by a finite limit r of a ratio of two-tree spanning-forest weights (Eq. 24). The main claim is that the limiting FPT distribution is exponential when one inverse eigenvalue dominates the spectral sum, and deterministic (delta) when infinitely many eigenvalues contribute comparably. The paper derives a cumulant formula (Eq. 31) for r=0, gives sufficient conditions for the deterministic limit via conductance and mean residual life, proves a reversible-network backward-bias condition for the exponential limit, presents a counterexample to naive bias intuition (Sec. III E), and supports the theory with Gillespie simulations on random networks.","tokens_in":25709,"tokens_out":25305,"duration_ms":218717,"significance":"If the classification were established as stated, it would provide a spectral explanation for the emergence of delta and exponential FPT regimes in complex biochemical networks, unifying earlier kinetic-proofreading observations. The graph-theoretic approach is elegant and yields explicit, parameter-free conditions. The paper contains genuine strengths: the exponential-limit proof under r=0 and reversibility (Supplement IV) is carefully argued, the counterexample in Sec. III E is instructive, and the authors provide reproducible simulation code and data. However, the central cumulant formula is only proven for r=0, whereas the macroscopic forest condition is r<∞, and this gap currently limits the generality of the paper's main claims.","major_comments":[{"comment":"The cumulant formula (31) is derived only after setting r=0 in Eq. (30). For 0<r<∞, the term ln(Σ_{n=0} r_n s^n) in Eq. (27) does not vanish: r_1=r>0, and the recurrence in Supplement II allows non-vanishing higher r_n. This logarithmic term contributes to every cumulant of t/⟨τ_m⟩. Consequently, eigenvalue delocalization (Σ_i λ_i^{-2}/(Σ_i λ_i^{-1})^2→0) does not by itself force κ_2→0, and eigenvalue concentration (λ_1^{-1}/Σ_i λ_i^{-1}→1) does not by itself force κ_n→(n-1)!. Since the macroscopic forest condition is defined as r<∞ (Eq. 24), the main delta/exponential classification is not established under that condition. The authors need either to prove r→0 for the relevant large-N ensembles, or to extend Eq. (31) by controlling the forest-polynomial contributions for finite r.","section":"§III B, Eqs. (24), (27), (30), (31)"},{"comment":"These equations contain a dimensional/typographical error. The correct relation between the mean FPT and the trace of M is ⟨τ_m⟩/(Σ_i λ_i^{-1}) = 1/(1+r), not '1 − ⟨τ_m⟩Σ_i λ_i^{-1}' as printed. As written, Eq. (23) has a dimensionless left-hand side while the right-hand side has units of time squared (unless rates are artificially dimensionless, which the graph-theoretic weights show they are not). Eq. (25) is therefore not a valid equivalence. The same problem appears in Supplement II, Eq. (S2.4). Please correct these identities and re-examine the derivation of Eq. (31).","section":"§III B, Eqs. (23) and (25)"},{"comment":"The conductance argument for the deterministic limit is not rigorously controlled. The Cheeger/Lawler–Sokal bound applies to the generator of an irreducible Markov chain, whereas the relevant object is the principal minor K with an absorbing target. The paper proposes to introduce a small back-rate ε and then let ε→0, but it does not prove that the spectral gap of the ε-perturbed irreducible generator converges to the smallest eigenvalue of K, nor that the ε→0 and N→∞ limits commute. Since this argument is used as the general proof of the deterministic-limit eigenvalue condition, the theorem as stated is incomplete.","section":"§III C, Eqs. (33)–(35)"}],"minor_comments":[{"comment":"The notation p_{m→{j,N}}(N) is confusing: the argument N is overloaded (system size vs. target vertex). Please define it more explicitly.","section":"§III B, Eq. (26)"},{"comment":"The sentence 'this criterion becomes sufficient' refers to the coefficient of variation converging to 1. Please clarify that sufficiency holds only under the r=0 macroscopic forest condition, as the subsequent proof assumes.","section":"§III D"},{"comment":"Typo: 'behavoir' should be 'behavior'. Also in the caption of Fig. 7(c), 'eigenvalue ratio (histogram)' should likely read 'box plot'.","section":"§III F"},{"comment":"Typo: 'expondistributed' should be 'exponentially distributed'.","section":"Supplement V"},{"comment":"Typo: 'approriate' should be 'appropriate'.","section":"Supplement VI"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and addresses a timely problem. The main issue is that the headline claim—that the FPT limit is generically determined by the eigenvalue distribution under the macroscopic forest condition—is only proven for the r=0 subcase. This is a load-bearing gap, but it is potentially fixable either by showing r→0 for the relevant classes of large networks (e.g., the random forward-bias ensembles used in Sec. III F) or by explicitly restricting the main theorems to r=0 and adjusting the abstract and discussion accordingly. I recommend major revision rather than rejection because the r=0 core is sound and the paper contains valuable tools and numerical evidence."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should read this paper if you care about when first-passage time distributions in large Markov networks collapse to a delta or an exponential. The core idea is genuinely useful: cumulants of the FPT rescaled by its mean are controlled by sums of inverse eigenvalues of the generator, provided a certain graph-theoretic condition holds. The derivation of Eq. (31) via spanning forests is clean, and the authors are honest that it requires r=0, not just the weaker r<∞ of their 'macroscopic forest condition.' The exponential-limit theorem for reversible networks under backward bias (Supp. IV) is the most rigorous part: the interlacing argument is sound and gives a sufficient condition that clearly generalizes the old kinetic-proofreading observations. The counterexample in Sec. III E also earns its place, showing that global forward bias is not enough for determinism. Simulations on random networks are reproducible, with code and data provided, and they illustrate the predicted regimes nicely for the cases they cover.\n\nThe soft spots are real but not fatal. The biggest gap is the step from r<∞ to r=0. The paper defines the macroscopic forest condition as r finite, and uses it as the structural precondition for universality, but the spectral-sum classification is only proved for r=0. For 0<r<∞ the logarithmic polynomial term in Eq. (27) contributes to every cumulant, so eigenvalue delocalization alone need not give a delta, and eigenvalue concentration alone need not give an exponential. The authors sometimes write as if the two are equivalent, especially in the abstract and discussion where the word 'generic' appears. They do not show that their random-network ensembles satisfy r→0; they only check the eigenvalue ratio and CV, which are not enough if r is finite. That is an overclaim in the present form. A second, more technical issue is the deterministic-limit section: the Cheeger argument relies on an ε→0 back-rate limit that is not rigorously controlled. The formal proof in Supp. III gives sufficient spectral conditions but does not bridge to the conductance picture. This should be tightened or clearly labeled as heuristic. One minor point: the stress-test claim that Eq. (29) is inconsistent for r=0 appears wrong—the RHS equals 1 there, so the bound is consistent; that particular complaint does not land.\n\nOverall: a worthwhile paper that deserves a serious referee. It establishes a rigorous r=0 theory, a strong reversible-network theorem, and a nice counterexample. The authors should either restrict the 'generic' language to cases where r=0 or prove that the intended large-network ensembles satisfy that condition. As is, I would accept for review, with revisions expected around the finite-r gap and the deterministic-limit rigor.","headline":"A solid spectral-graph analysis of FPT limits for r=0 networks, but the 'generic' claim outruns the proof for finite r; the reversible exponential theorem is the strongest part.","tokens_in":26101,"tokens_out":3466,"would_cite":true,"duration_ms":36556,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["05.40.-a","02.50.Ga"],"model":"deepseek-v4-flash","headline":"For a large Markovian network, the first-passage time distribution is generically either a delta or an exponential, and the eigenvalue spectrum of the generator decides which one.","keywords":["first-passage times","Markovian networks","master equation","kinetic proofreading","eigenvalue spectrum","matrix-tree theorem","spanning forests","universality"],"falsifier":"Take a family of reversible one-step master equations with rates chosen so that the macroscopic forest condition holds but the eigenvalue-dominance ratio R = λ1^(-1)/Σλ_i^(-1) approaches a limit strictly between 0 and 1 as N grows, then numerically compute the normalized first-passage cumulants. Eq. (31) predicts they converge to a non-exponential, non-delta form; if instead the distribution becomes exactly exponential or exactly delta for all such families, the spectral-sum formula is wrong.","tokens_in":25291,"feed_emoji":"⏱️","tokens_out":5987,"duration_ms":63785,"temperature":0.7,"pith_summary":"This paper tries to prove a universality statement: for a Markovian network with many states, the time to first reach a target state, rescaled by its mean, can converge to only one of two shapes—a memoryless exponential or a sharp deterministic peak—provided the walk starts macroscopically far from the target. The deciding quantity is the spectrum of the generator matrix: a single dominant inverse eigenvalue produces the exponential, while infinitely many comparable inverse eigenvalues produce the delta. The proof connects first-passage moments to spanning-tree and spanning-forest weights and reduces the scaled cumulants to ratios of inverse-eigenvalue sums. If true, this explains why complex biochemical decision networks show simple coarse-grained completion-time statistics, and it shows that the deterministic and exponential limits are not symmetric mirror images.","feed_headline":"One spectral ratio decides: delta or exponential first-passage limits","feed_subtitle":"Inverse-eigenvalue sums set whether completion times become deterministic or memoryless, unifying biochemical timing.","key_machinery":"The key object is the matrix M = (-K^T)^(-1), where K is the generator of the process restricted to non-target states; its entries are ratios of weighted two-tree spanning forests, and its trace is the sum of inverse eigenvalues. The macroscopic forest condition—that the ratio r of forest weights with the starting vertex on the target-rooted tree versus the complementary tree stays finite—guarantees that the mean first-passage time scales as the trace of M. The cumulant formula Eq. (31) then transfers the problem from dynamics to spectral geometry, and, for reversible networks, the interlacing of eigenvalues of principal minors converts a backward-bias condition into single-eigenvalue domina","core_discovery":"On the paper's own terms, the central discovery is Eq. (31): under the macroscopic forest condition, the n-th cumulant of the first-passage time divided by its mean equals (n-1)! times the ratio of the n-th inverse-eigenvalue sum to the first inverse-eigenvalue sum raised to the n-th power. If the dominant inverse eigenvalue absorbs the entire trace of the inverse generator, all normalized cumulants approach (n-1)!, the exponential's cumulants. If instead infinitely many inverse eigenvalues contribute comparably, the variance ratio vanishes and the distribution collapses to a delta. The paper also establishes that, for reversible networks, a vanishing ratio of reverse to forward mean first-p","pith_inferences":["Eq. (31) suggests a cheap pre-simulation diagnostic: compute R = λ1^(-1)/Σλ_i^(-1) from the rate matrix; R approaching 1 predicts exponential statistics, while R approaching 0 predicts deterministic statistics.","The normalized cumulants offer a sharper experimental signature than the coefficient of variation; measuring third and fourth cumulants of single-molecule first-passage times could distinguish a true exponential from other unit-CV distributions.","The asymmetry between the two limits hints at an information-theoretic ordering: the exponential (max-entropy) regime is easily reached by reversible dynamics, while the delta (min-entropy) regime requires stronger dissipative or conductance structure.","The non-generic irreversible case suggests a path to classification via metastable clustering: networks with multiple macroscopic almost-invariant components should produce phase-type limits that are convex combinations or convolutions of exponentials."],"forward_implications":["For kinetic proofreading and other large biochemical networks, observed delta or exponential completion-time distributions follow from the eigenspectrum, not from model-specific rate choices.","Reversible networks with a strong backward bias relative to the target are guaranteed to show exponential first-passage times if the macroscopic forest condition holds; one-parameter coarse-grained Markov models are then statistically sufficient.","Strongly connected networks with locally bounded rates and non-vanishing conductance have deterministic completion times once the mean first-passage time diverges.","A global forward bias is not enough for determinism; the relevant structural quantities are spectral, so bias criteria based on averages of forward-backward ratios can mislead.","Irreversible networks with backward bias can break both universal limits, e.g., through convolutions of exponentials across metastable clusters, so irreversible bottleneck transitions need separate treatment."],"fun_headline_variants":["Eigenvalue sums set first-passage shape: peak or exponential","Network size tilts first-passage times to deterministic or memoryless","Spectral law reveals universal first-passage limits in Markov nets","Inverse eigenvalues decide: sharp timing or exponential timing","Markovian networks show two universal first-passage regimes"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing assumption is that the initial state is macroscopically far from the target, formalized by the macroscopic forest condition: the ratio of two-tree spanning-forest weights with the start on the target tree versus the other tree must stay finite as the network grows; if the process starts too close to the target, local structure dominates and no universal limit is guaranteed.","fun_headline_variants_meta":{"raw":{"variants":["Eigenvalue sums set first-passage shape: peak or exponential","Network size tilts first-passage times to deterministic or memoryless","Spectral law reveals universal first-passage limits in Markov nets","Inverse eigenvalues decide: sharp timing or exponential timing","Markovian networks show two universal first-passage regimes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000228,"raw_usage":{"total_tokens":1326,"prompt_tokens":776,"completion_tokens":550,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":520,"completion_tokens_details":{"reasoning_tokens":475}},"tokens_in":520,"tokens_out":550,"duration_ms":4975,"temperature":1.0,"reasoning_tokens":475,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T21:58:28.648858+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a family of reversible one-step master equations with rates chosen so that the macroscopic forest condition holds but the eigenvalue-dominance ratio R = λ1^(-1)/Σλ_i^(-1) approaches a limit strictly between 0 and 1 as N grows, then numerically compute the normalized first-passage cumulants. Eq. (31) predicts they converge to a non-exponential, non-delta form; if instead the distribution becomes exactly exponential or exactly delta for all such families, the spectral-sum formula is wrong.","supporting_citations":[],"review_version":1}