{"id":"bc1c42c7-6e8f-4c66-8f96-3669e8562817","arxiv_id":"2607.22528","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For many interacting Brownian searchers, bounded interactions cannot beat the independent 1/ln N fastest-search time, while coherent pushes and random pairwise kicks can reach 1/N and 1/(N ln N), respectively.","lead":"This paper asks how interactions between many Brownian searchers change the time until the first one finds a target. It proves that bounded interactions cannot beat the independent-searcher speed limit, identifies two interaction mechanisms that can, and gives a single bound covering both.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant mathematical objection identified; the reader's concern about Prop. 1(ii) appears to rest on a sign error in the direction of repulsive forces.","rationale":"The reader's weakest assumption is Proposition 1(ii), specifically the claim that repulsive interactions cannot delay the extreme first-passage time relative to independent searchers. The stated reason—that a constant long-range repulsion can push a near searcher away from the target—is physically backwards: repulsion between a particle closer to the target and one farther away pushes the closer one targetward and the farther one away. Hence the proposed counterexample does not threaten Eq. (10). The paper's main theorems are plausible and the scaling claims (logarithmic no-go, 1/N deterministic speedup, and 1/(N lnN) stochastic speedup) are consistent with the sketched mechanisms. Two genuine but non-fatal issues remain: (1) all rigorous proofs are deferred to an unavailable Supplemental Material, so full verification is impossible from the manuscript alone; (2) Eq. (15) is dimensionally inconsistent as printed, although the correct expression follows from Eq. (25) and the surrounding text states the intended asymptotic branches. These do not overturn the central argument, so the reader's CONDITIONAL verdict stands, but not for the reason identified in the weakest-assumption analysis.","tokens_in":9398,"tokens_out":33921,"duration_ms":349298,"concrete_test":"Run a 1D two-particle simulation with strong finite-range repulsion (e.g., truncated Lennard-Jones or hard-core no-passing) and target at z=0, starting at z=(1,2), D=1. Estimate P(T_N > 1) over 10^6 trajectories and compare with erf(0.5)*erf(1) ~ 0.4386. If the simulated probability exceeds the product, Prop. 1(ii) fails; if it remains below (as expected), the reader's counterexample is refuted. Separately, re-derive Eq. (15) from End Matter Eq. (25) to confirm the typo and its corrected asymptotic branches.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claims—Theorem 1 no-go, Proposition 1 logarithmic upper bound, and Theorem 2 acceleration limits—are internally consistent. The specific concern raised about Proposition 1(ii) does not land: for a planar target z=0, a purely repulsive interaction pushes the current minimum-z searcher toward the target, because every other searcher has z_j >= z_i and the z-component of the repulsive force on i is (z_i - z_j)/|r_i-r_j| <= 0. Thus the reader's proposed 'near pushed away, distant pushed toward' scenario reverses the sign of the interaction. The minimum-z process therefore has non-positive extra drift relative to the independent-particle minimum; this is the natural route to Eq. (10), and the stated hard-core/no-passing limit is consistent with an ordered-wedge probability no larger than the product of one-particle survivals. The real limitations are verification: proofs of Theorem 1, Proposition 1, and Theorem 2 are all deferred to SM [41], which is not included, and Eq. (15) contains an obvious dimensional typo (the correct threshold from Eq. (25) is t* = l0^2 / (sqrt(D_eff lnN + B_N l0) + sqrt(D_eff lnN))^2). Neither limitation invalidates the central argument, but both justify keeping the verdict conditional until the SM is available and the typo is fixed.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the extreme first-passage time T_N of N interacting overdamped Brownian searchers toward an absorbing target. It states three main results: (i) a no-go theorem (Theorem 1) showing that bounded force budgets cannot beat the 1/ln N timescale of independent searchers, with a corollary for bounded finite-range repulsion under initial coordination control; (ii) matching logarithmic upper bounds (Proposition 1), including the strong comparison inequality P(T_N>t) ≤ ∏_i Q0_i(t) for reciprocal purely repulsive interactions; and (iii) a unified acceleration limit (Theorem 2) yielding N^{-p} and (N^q ln N)^{-1} speed limits for deterministic (p+1)-body and stochastic (q+1)-body forcing, together with two explicit solvable models attaining the pairwise cases. The paper also discusses mechanisms that escape the logarithmic class and presents the claimed results as establishing an exact logarithmic universality class for generic bounded interactions.","tokens_in":9728,"tokens_out":30669,"duration_ms":297740,"significance":"If the statements are correct, the paper would establish a sharp and surprisingly general universality result: many bounded interactions cannot change the logarithmic extreme-search timescale, while coherent many-body forces and enhanced fluctuations provide algebraically faster but tightly bounded acceleration. The results are parameter-free and carry explicit constants (e.g., ℓ0^2/(4D) in Eq. (6), ℓ0^2/(4κ) in Eq. (14)), so they make falsifiable predictions. The two solvable examples and the interpolation theorem provide a clear physical taxonomy of interaction-driven speedup. The main weakness is not internal inconsistency but verifiability: almost all proofs are deferred to a Supplemental Material that is not part of the submitted text, and the paper's central distributional comparison in Eq. (10) is asserted without a proof in the main text.","major_comments":[{"comment":"All load-bearing derivations are deferred to the Supplemental Material [41], which is not included. This includes the proof of Theorem 1 (SM S2 A–B), Proposition 1 (SM S3 A–F), the sharpness constructions for Theorem 2 (SM S5 D–E), and the mean-time estimate for Example A in Eq. (12). Without the SM, I cannot verify the central claims or the claimed optimality of the scaling exponents. The manuscript must either include the SM or present enough of these proofs in the main text for independent checking.","section":"Theorems 1, Proposition 1, Theorem 2, Example A (SM [41])"},{"comment":"Eq. (10) is a very strong statement: for arbitrary-strength reciprocal purely repulsive interactions, the interacting survival probability is bounded above by the product of independent single-particle survivals. This comparison is the load-bearing step for the exact logarithmic upper bound, but the main text gives no proof, only an intuitive remark. The reader's worry that a near particle could be pushed away is a sign error—the minimal-z particle is pushed toward the target by all particles above it—so the physical scenario is plausible. However, a rigorous proof of the label-switching/minimum-process comparison is still required; this is not a minor omission because it is the core of the upper-bound claim.","section":"Proposition 1(ii), Eq. (10)"},{"comment":"The optimality of the two branches in Table I (Example A attaining N^{-1}, Example B attaining (N ln N)^{-1}) is established only by deferred SM constructions. The main text states that a smooth tagged-leader construction 'exactly attains this scale,' but the construction and its verification are in SM Sec. S5 E. Similarly, Eq. (12) for Example A is justified by 'the argument detailed in [41].' These sharpness claims are essential to the paper's message that the bounds in Theorem 2 are optimal, and they cannot be checked from the current text.","section":"Table I and End Matter C"}],"minor_comments":[{"comment":"The formula in Eq. (15) appears dimensionally consistent with the threshold derived from Eq. (25), so I do not see a dimensional error; however, the formula should be checked in the final typeset version because the radical scope in the plain text is ambiguous.","section":"Eq. (15)"},{"comment":"Reference [41] should be supplied with a URL or included as an attachment. Footnote [46] refers to 'the WCA interaction' without defining the acronym; please spell it out. The 'numerical demonstration' in SM Sec. S8 is not shown; if it supports a claim, include the figure or a reproducible description.","section":"References and footnotes"},{"comment":"The definition of ℓ0 as an infimum over N and i is clear, but the subsequent use of ℓ0 in Theorem 1 as a universal lower bound is not explicitly restated; a small reminder near Theorem 1 would improve readability.","section":"Notation, Eq. (3)"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern about Proposition 1(ii) does not survive contact with the sign of the repulsive force on the minimal-z particle; I do not see a circularity or a fitted-parameter issue. The real obstacle is that the manuscript's central results are unverifiable without the omitted Supplemental Material. If the SM supplies the deferred proofs, especially for Eq. (10) and the optimality constructions, I would view the paper as a strong candidate. The recommendation is major_revision primarily to force inclusion of the missing proofs, not because a load-bearing error has been identified."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague —\n\nRead Bao's paper. The reader's verdict is about right: this is a genuinely new framework, and the central claims are plausible, but you can't verify them because every core proof is in SM [41], which isn't included. So conditional is the honest verdict.\n\nWhat's actually new: Theorem 1's no-go bound and Theorem 2's acceleration limit are not in the independent-searcher literature, and the two examples make the mechanisms concrete. The paper also does a good job separating statistical redundancy from coherent transport and amplified fluctuations. The physical discussion (e.g. what an O(ln N) force budget does on the logarithmic window) is clear.\n\nOn the reader's weakest assumption: I disagree. The proposed counterexample to Proposition 1(ii) reverses the sign of the repulsive force. For a planar target z=0, if particle i has the minimum z, every other particle has z_j >= z_i, so the z-component of the repulsive force on i is proportional to (z_i - z_j), which is non-positive — toward the target. So the minimum-z process has non-positive extra drift, and the product bound in Eq. (10) is the natural conclusion. The stress-test note is right on that. That said, the proposition is still stated too loosely: the main text gives no range/decay conditions and simply says 'no constraints on interaction strength,' which makes the proof more delicate than the statement suggests. The reader was right to flag it as unproven.\n\nThe real soft spots are verification. Theorem 1, Proposition 1, Theorem 2 — all proofs deferred. The printed Eq. (15) has a dimensional typo; the correct threshold follows from Eq. (25). None of this makes the central argument look wrong, but it keeps the paper at 'conditional' until the SM is available. Also, Example B is less novel than it appears: the pair kicks just give each particle effective diffusivity D + κ(N-1), so the 1/(N ln N) scaling is the usual independent-searcher result with rescaled D. The mechanism is 'effective noise enhancement,' which is fine, but not a new statistical law.\n\nWho this is for: people working on extreme first-passage in soft matter, active matter, and cellular search. It deserves a serious referee if the SM is actually supplied; without it, it's a preprint to read skeptically. I'd send it to peer review with a requirement that the SM be included. I wouldn't cite it in my own work until the proofs are visible.","headline":"First general framework for extreme first-passage with interactions, but all core proofs sit in an unavailable SM; the reader's central objection doesn't survive contact with the sign of the repulsive force.","tokens_in":10151,"tokens_out":2153,"would_cite":false,"duration_ms":24494,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60J65","60G70","82C22"],"pacs":["05.40.Fb","05.10.Gg"],"model":"deepseek-v4-flash","headline":"A no-go theorem proves that bounded interactions cannot accelerate the earliest hit of N Brownian searchers below the 1/ln N barrier, with two sharp exceptions that break it.","keywords":["extreme first-passage time","interacting Brownian searchers","no-go theorem","logarithmic timescale","repulsive interactions","coherent transport","stochastic forcing","many-body search"],"falsifier":"Take two searchers on the half-line with constant, unbounded-range repulsive force between them, a planar absorbing target at 0, and initial positions ℓ0 and ℓ0+δ. Compute the joint survival probability numerically: if it exceeds the product erf(ℓ0/√(4Dt))·erf((ℓ0+δ)/√(4Dt)) at any time t, then Eq. (10) fails and the exact logarithmic class is not established. Alternatively, in a many-body simulation with bounded finite-range repulsion but a superlogarithmic number of neighbors per particle, observe whether E[T_N] decays faster than 1/ln N; if it does not, the no-go theorem's domain is wider t","tokens_in":9276,"feed_emoji":"⏱️","tokens_out":5966,"duration_ms":63868,"temperature":0.7,"pith_summary":"This paper asks whether interactions among many searchers can make the first successful arrival—the extreme first-passage time—faster than the classic 1/ln N scaling of N independent Brownian searchers. It answers with a no-go theorem: for broad classes of bounded forces, interactions cannot beat the logarithmic barrier, and for repulsive interactions this barrier is exact at leading order. Two mechanisms do break the barrier: deterministic coherent push can reduce the time to order 1/N, and stochastic pairwise kicks that amplify effective diffusivity reach 1/(N ln N). A unified acceleration limit interpolates between these regimes, cleanly separating acceleration due to statistical redundancy from that generated by coherent many-body transport or amplified fluctuations. This framework matters because most real search systems, from cellular signaling to foraging, involve interactions that break the independent-particle assumption.","feed_headline":"No-go: bounded forces can't beat the 1/ln N search limit","feed_subtitle":"Earliest arrival among N Brownian searchers stays logarithmic unless forces push coherently or amplify noise.","key_machinery":"The proof machinery projects each searcher's motion onto a fixed targetward direction and treats it as a one-dimensional Brownian motion with a drift budget B_N; the reflection principle plus a union bound then convert short-time drift control into a lower bound on the earliest-hit probability. The matching upper bound uses a comparison inequality, P(T_N>t) ≤ ∏ erf(z_i(0)/√(4Dt)), valid for reciprocal purely repulsive interactions or confined systems, which proves that repulsion cannot delay the extreme first-passage relative to independent searchers. The two escape mechanisms are explicit many-body constructions: a screened single-file channel where the leader feels an O(N) coherent push fr","core_discovery":"The central claim is a no-go theorem plus an exact logarithmic class. For any interacting overdamped system in which, up to the first hit, the targetward drift accumulated by each searcher is o(ln N) on the timescale t = c/ln N, the mean extreme first-passage time satisfies liminf (ln N) E[T_N] ≥ ℓ0^2/(4D). When a growing number of searchers start within a fixed distance of a flat target and are either confined in a slab or repel each other reciprocally, a matching upper bound gives equality, so 1/ln N is exact for these classes. The two exceptions are explicit: coherent force accumulation (screened single-file repulsion) gives 1/N, and stochastic pair kicks with enhanced diffusivity give 1/","pith_inferences":["A direct corollary the author leaves unexplored: the same no-go logic should apply to bounded-speed, non-Gaussian, or time-correlated noise whenever short-time tails are exponential; testing this would generalize the logarithmic class beyond Brownian motion.","The comparison inequality suggests a practical diagnostic: measure the ratio of interacting to independent many-body survival probability; any excursion above 1 would mark the onset of interaction-driven slowdown, and the paper's O(ln N)-neighbor threshold gives a concrete design rule for experiments.","For biological search, the 1/N coherent-push branch implies that collectively pushing one leader is the most efficient route to speed up first arrival; quorum-sensing or alignment interactions that produce a global drift could approach this limit, whereas purely local repulsion cannot.","The unified bound could be turned into an optimization principle: among all symmetric interactions with a fixed interaction budget, the fastest extreme search is achieved either by concentrating force on one target-facing coordinate or by injecting zero-mean noise that raises effective diffusivity; mixtures of the two mechanisms interpolate continuously between the limits."],"forward_implications":["For soft, finite-range, or other bounded-force interactions with logarithmically bounded initial local density, the classic 1/ln N search time is the universal fastest scale; interactions alone cannot improve it.","Under purely repulsive reciprocal interactions or with confinement, the asymptotic constant ℓ0^2/(4D) is exact: statistical redundancy, not interaction strength, sets the leading speed of the earliest arrival.","To beat the logarithmic barrier at fixed initial gap, a system must either generate coherent targetward transport (deterministic, giving 1/N^p for (p+1)-body forces) or amplify fluctuations (stochastic, giving 1/(N^q ln N)).","The unified lower bound means any faster search must consume a budget of drift or diffusivity on the short-time window; the two explicit models attain the pairwise limits, so those branches are optimal.","The results separate the three sources of speedup—redundancy (logarithmic), coherent many-body transport (algebraic), and amplified fluctuations (algebraic times logarithmic)—providing a classification for interacting extreme-statistics problems."],"fun_headline_variants":["1/ln N stands unless forced","Interacting searchers: log time unless forced","Two ways to beat log search: coherent or noisy","Extreme search: 1/ln N unless forced"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The claim that repulsive interactions cannot delay the extreme first-passage time relative to independent searchers—the inequality P(T_N>t) ≤ ∏ erf(zi(0)/√(4Dt))—is assumed without a range or decay condition on the repulsion, and its proof is deferred to the supplement; a long-range repulsion that pushes a nearby searcher away from the target while pushing a distant one toward it could make the inequality fail.","fun_headline_variants_meta":{"raw":{"variants":["1/ln N stands unless forced","Interacting searchers: log time unless forced","Two ways to beat log search: coherent or noisy","Extreme search: 1/ln N unless forced"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001464,"raw_usage":{"total_tokens":5717,"prompt_tokens":726,"completion_tokens":4991,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":470,"completion_tokens_details":{"reasoning_tokens":4930}},"tokens_in":470,"tokens_out":4991,"duration_ms":40794,"temperature":1.0,"reasoning_tokens":4930,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T04:25:20.163043+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take two searchers on the half-line with constant, unbounded-range repulsive force between them, a planar absorbing target at 0, and initial positions ℓ0 and ℓ0+δ. Compute the joint survival probability numerically: if it exceeds the product erf(ℓ0/√(4Dt))·erf((ℓ0+δ)/√(4Dt)) at any time t, then Eq. (10) fails and the exact logarithmic class is not established. Alternatively, in a many-body simulation with bounded finite-range repulsion but a superlogarithmic number of neighbors per particle, observe whether E[T_N] decays faster than 1/ln N; if it does not, the no-go theorem's domain is wider t","supporting_citations":[],"review_version":1}