{"id":"035002d1-5479-4c24-acec-2e20fab40380","arxiv_id":"2605.30629","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Derives the probability distribution and moments of the kth search time for bounded-speed immigrating searchers from the early-time single-searcher distribution, applied to canonical stochastic search models.","lead":"The paper derives the full probability distribution and moments of the kth search time for searchers immigrating into a domain at bounded speed, in the fast immigration limit, using the early-time single-searcher distribution. A smart generalist might read it to understand how to model realistic biophysical search without unphysical infinite-speed assumptions.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest-assumption statement already isolates the precise point that would need to fail for the claim to collapse. Because the paper asserts a rigorous derivation and supplies both analytic applications and simulation comparisons, and because no contradictory detail appears in the abstract or claim, the load-bearing condition is taken as satisfied on present evidence. No adjustment to the UNVERDICTED verdict is warranted.","tokens_in":1628,"tokens_out":271,"duration_ms":12543,"concrete_test":"For the 1D Brownian-motion case, recompute the first two moments of the k=2 search time from the claimed single-searcher formula and compare against direct Monte-Carlo simulation of 10^5 immigrating trajectories at immigration rate λ\to0; agreement within sampling error confirms the reduction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that in the fast-immigration limit the kth search-time distribution and moments are exactly expressible in terms of the single-searcher early-time distribution under bounded speed. The abstract states that rigorous mathematical results are derived and then applied to canonical models with numerical checks. No internal inconsistency, hidden assumption on domain geometry, or failure of the reduction is visible from the given statement of the result.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript studies search times for searchers that immigrate into a bounded domain over time while moving at finite speed. In the fast-immigration limit the authors derive exact expressions for the full probability distribution and all moments of the kth search time, expressed solely in terms of the early-time hitting probability of a single searcher. These formulas are applied to several canonical search models (including different diffusion variants), and the predictions are compared against numerical simulations.","tokens_in":1684,"tokens_out":376,"duration_ms":13483,"significance":"If the reduction holds, the work supplies a rigorous, parameter-free bridge from single-searcher early-time data to the full multi-searcher search-time statistics under bounded speed. This directly resolves the unphysical infinite-speed artifact that appears in fast-immigration regimes and supplies a practical route to moments and distributions for biophysical search problems. The explicit comparison of distinct diffusion models and the simulation checks are additional strengths.","major_comments":[],"minor_comments":[{"comment":"The abstract states that the results are 'rigorous mathematical results,' yet the manuscript should include an explicit statement of the precise technical conditions (domain regularity, speed bound, immigration rate scaling) under which the reduction to the single-searcher early-time distribution is valid.","section":null},{"comment":"In the applications section, the comparison between different diffusion models would benefit from a short table summarizing which moments or quantiles differ and by how much, to make the claim that 'minutiae of searcher dynamics affect search times' quantitatively visible.","section":null},{"comment":"A brief remark on the numerical method used for the simulations (time-stepping scheme, boundary handling, number of realizations) would strengthen reproducibility of the reported agreement between theory and numerics.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive summary and recommendation of minor revision. No major comments were provided in the report.","responses":[],"tokens_in":1126,"tokens_out":42,"duration_ms":8429,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The central result is that in the fast-immigration limit the full distribution and all moments of the kth search time are expressed directly in terms of the single-searcher early-time hitting probability when speed is bounded. This sidesteps the infinite-speed artifact that appears in earlier fast-immigration estimates.\n\nThe derivations are presented as rigorous and are then applied to several standard models. The authors also compare different diffusion processes to identify when the details of the motion actually change the search times. Numerical simulations are used to check the formulas, which is the right check for this kind of work.\n\nThe reduction itself is the new piece; the rest consists of working out the consequences for canonical cases. That is useful within the subfield because it gives exact expressions rather than approximations once the single-searcher early-time quantity is known.\n\nThe main limitation is that the result still requires the early-time single-searcher distribution to be computable or known. In domains where that quantity is itself hard to obtain, the practical gain is smaller. The paper stays with standard geometries, so this does not appear to be a problem for the cases treated. No inconsistencies in the reduction or in the limit procedure are visible.\n\nThe work is aimed at researchers who model immigrating searchers in biophysics or applied probability and who want closed-form distributions rather than simulation-only answers. It is mathematically grounded enough and the simulations provide an independent check, so it deserves a serious referee.","headline":"The paper reduces the kth search-time distribution exactly to the single-searcher early-time distribution under fast immigration and bounded speed.","tokens_in":2170,"tokens_out":363,"would_cite":false,"duration_ms":19768,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"In the fast immigration limit with bounded speed, the kth search time distribution and moments are given exactly by the single searcher's early-time probability distribution.","keywords":["bounded speed search","immigration","search times","probability distribution","stochastic processes","diffusion models","moments of search time","fast immigration"],"falsifier":"Numerical simulations of the full multi-searcher system with fast immigration and bounded speed that produce a kth search time distribution different from the one calculated from the single searcher's early time probability.","tokens_in":2514,"feed_emoji":"","tokens_out":600,"duration_ms":27741,"temperature":0.7,"pith_summary":"The paper determines exact expressions for the probability distribution and moments of the kth search time when searchers immigrate quickly but move at finite speed. This matters for biophysical processes where searchers enter progressively, avoiding the unphysical infinite speeds of earlier models. The results are derived rigorously and then applied to several standard search models including variants of diffusion. Comparisons with simulations validate the approach. The work clarifies when the details of individual searcher motion matter for overall search times.","feed_headline":"Single searcher early data fixes all kth search times under fast immigration","feed_subtitle":"Exact distributions and moments derived for bounded-speed searchers, fixing unphysical issues in prior estimates for biophysical application","key_machinery":"The exact mapping from the single-searcher early-time probability distribution to the multi-searcher kth search time statistics in the fast immigration limit under bounded speed.","core_discovery":"We show that in the fast immigration limit the full probability distribution and all the moments of the kth search time can be determined in terms of the early time probability distribution of a single searcher. These mathematical results are applied to canonical models of stochastic search, and different models of diffusion are analyzed to investigate when and how the minutiae of searcher dynamics affect search times.","pith_inferences":["This suggests that experimental focus on early single-particle trajectories could predict collective search performance in fast-immigration settings.","The method might generalize to other speed constraints or time-dependent immigration rates.","Optimization of search processes could use these formulas to adjust immigration based on single searcher properties."],"forward_implications":["The kth search time can be analyzed using only single searcher early behavior data.","Different diffusion models can be compared for their impact on search times.","All moments of the search times become explicitly computable.","The theory applies directly to biophysical search with immigration."],"fun_headline_variants":["Early single searcher data sets kth search times under fast immigration","Fast immigration kth times from single searcher early distribution","Bounded-speed immigration search yields kth times via early single data","Single early distribution determines kth search time moments at fast immigration"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"That the early-time probability distribution of a single searcher is known or computable and that bounded speed permits an exact reduction of the multi-searcher problem to this quantity in the fast immigration limit.","fun_headline_variants_meta":{"raw":{"variants":["Early single searcher data sets kth search times under fast immigration","Fast immigration kth times from single searcher early distribution","Bounded-speed immigration search yields kth times via early single data","Single early distribution determines kth search time moments at fast immigration"]},"model":"grok-4.3","cost_usd":0.011523,"raw_usage":{"total_tokens":4918,"prompt_tokens":564,"num_sources_used":0,"completion_tokens":59,"cost_in_usd_ticks":115228000,"prompt_tokens_details":{"text_tokens":564,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":4295,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":564,"tokens_out":59,"duration_ms":30867,"temperature":1.0,"reasoning_tokens":4295,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-29T05:14:37.286586+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Numerical simulations of the full multi-searcher system with fast immigration and bounded speed that produce a kth search time distribution different from the one calculated from the single searcher's early time probability.","supporting_citations":[],"review_version":1}