{"id":"1d151473-3c4d-48f4-86ea-5facc4a48557","arxiv_id":"2606.30998","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Extreme value limit laws for the shortest distance between orbits of two distinct strongly mixing maps, depending on lengths, codimensions, and an extremal index for dynamical compatibility.","lead":"The paper derives extreme value distributions for the shortest distance between trajectories from two different dynamical systems on the same phase space. A generalist might read it to see how statistical extremes can quantify compatibility between chaotic evolutions.","discovery_kind":"new_application","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption directly matches the conditions the abstract itself flags as necessary. Because the paper does not claim generality beyond those conditions and supplies concrete formulas plus examples inside them, the identified assumption is already the load-bearing one and no further internal gap is apparent.","tokens_in":1657,"tokens_out":233,"duration_ms":28382,"concrete_test":"Take one of the explicit examples in the paper, recompute the co-dimension from the given density formula, and verify that the numerical value inserted into the limit distribution expression reproduces the claimed extremal law for the shortest-distance statistic.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is explicitly conditional on the maps being strongly mixing and on the invariant measures admitting densities with isolated zeros and singularities (for the co-dimension formula). The abstract states that formulas are derived and examples are computed under precisely these hypotheses, with the extremal index likewise restricted to a class of interval maps. Within this scoped setting the argument is internally consistent; no hidden assumption or circularity is visible from the stated results.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript establishes extreme value distributions for the shortest distance between trajectories of two different maps acting on the same phase space. Under the assumption that the maps are strongly mixing, the limiting distribution is shown to depend on the lengths of the two trajectories, the codimension of the associated invariant measures, and an extremal index that quantifies the tendency of nearby points to diverge under the respective dynamics. Explicit formulas are derived for the extremal index in the case of a class of interval maps and for the codimension when the invariant measures admit densities possessing isolated zeros and singularities. Several concrete examples satisfying the hypotheses are worked out and the modulating parameters are computed explicitly.","tokens_in":1718,"tokens_out":421,"duration_ms":24102,"significance":"If the derivations hold, the results supply a new class of limit theorems that compare orbits generated by distinct dynamical systems rather than within a single system. The explicit dependence on trajectory length, codimension, and the extremal index, together with closed-form expressions under stated regularity conditions on the densities, offers a concrete tool for quantifying dynamical compatibility. The provision of worked examples strengthens the applicability of the theory.","major_comments":[],"minor_comments":[{"comment":"The abstract states that formulas are derived, yet the manuscript would benefit from a brief outline of the main steps in the proof of the extreme-value limit (e.g., the role of the mixing assumption in controlling the dependence between the two orbits) already in the introduction.","section":"Introduction"},{"comment":"Notation for the codimension of the invariant measures should be introduced once and used consistently; the current alternation between “co-dimension” and “codimension” is minor but distracting.","section":null},{"comment":"In the examples section, the numerical verification of the extremal index could be accompanied by a short table comparing the theoretical value with the empirical estimate obtained from finite trajectories.","section":"Examples"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were provided in the report, so we have no points to address individually. We will incorporate any minor editorial or presentational suggestions in the revised manuscript.","responses":[],"tokens_in":1170,"tokens_out":71,"duration_ms":9580,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The central result is a limit theorem for the minimal distance between two trajectories generated by different maps. Under strong mixing the distribution converges to an extreme value law whose parameters depend on the trajectory lengths, the codimensions of the measures, and an extremal index that the authors link to the tendency of nearby points to separate under the two dynamics.\n\nThey provide a formula for that index when both maps are chaotic interval maps and a formula for the codimension when the densities have isolated zeros and singularities. The examples section works out concrete cases that satisfy the hypotheses.\n\nThis is a straightforward extension of existing extreme value results in dynamics to the two-map setting. The explicit formulas and the compatibility interpretation are the parts that look new. The calculations appear to follow directly from the mixing rates and the local behavior of the densities.\n\nThe assumptions are stated clearly: strong mixing plus the density conditions. The results hold inside that class and the paper does not claim more. A potential soft spot is that the codimension formula may be sensitive to the precise location of the zeros and singularities; small changes in the map could alter the value. The abstract gives no error bounds or stability statements.\n\nThe work is written for ergodic theorists who already know the standard machinery for extremes in dynamical systems. Someone in that group can use the formulas for their own calculations. It is too specialized for a general dynamics audience.\n\nThe paper engages honestly with the literature on mixing and extremes. The claims are precise and conditional, so a referee can check them against the proofs. I would send it to peer review.","headline":"This paper gives explicit extreme value limits for the shortest distance between orbits of two different maps, with formulas for the extremal index on interval maps.","tokens_in":2213,"tokens_out":395,"would_cite":false,"duration_ms":30333,"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":"Extreme value distributions describe the shortest distance between trajectories of different strongly mixing maps.","keywords":["extreme value distributions","dynamical systems","strongly mixing maps","extremal index","invariant measures","co-dimension","trajectory encounters"],"falsifier":"For two concrete strongly mixing maps on the interval with computed co-dimensions, the histogram of minimal distances over many pairs of long trajectories fails to converge to the predicted extreme value distribution.","tokens_in":2568,"feed_emoji":"","tokens_out":524,"duration_ms":41078,"temperature":0.7,"pith_summary":"The paper shows that the minimal distance between points on trajectories from two different maps on the same space obeys an extreme value law in the limit of long trajectories. This law depends on the lengths of the two trajectories, the co-dimension of their invariant measures, and an extremal index that measures how quickly nearby points separate under the two dynamics. A formula is provided for the extremal index in the case of chaotic interval maps, and for the co-dimension when the measures have densities featuring isolated zeros and singularities. Examples of such systems are given with explicit computations of the modulating parameters. This framework quantifies the compatibility of distinct dynamical systems through their closest encounters.","feed_headline":"Shortest encounters between map trajectories follow extreme value laws","feed_subtitle":"The limit distribution is set by trajectory lengths, measure co-dimensions, and an extremal index for strongly mixing maps.","key_machinery":"The extremal index modulating the extreme value distribution for minimal inter-trajectory distances, reflecting divergence tendency of nearby points under different dynamics.","core_discovery":"We establish Extreme Value Distributions for the closest encounter between trajectories generated by different maps defined in the same reference phase space. For a class of strongly mixing maps, we show that the limit distribution depends on the length of the different trajectories and the co-dimension of the associated invariant measures. It is also modulated by an Extremal Index, that informs on the tendency of nearby points to diverge along with the evolution of their respective dynamics, serving as an indicator of their compatibility. We give a formula for this quantity for a class of chaotic maps of the interval and for the co-dimension in the case when the respective measures admit de","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Extreme value laws for shortest map trajectory encounters","Closest encounters follow extremes in strongly mixing maps","Extremal index modulates shortest distances between dynamics","Measure co-dimensions shape extreme trajectory distance laws"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The maps are strongly mixing and their invariant measures have densities with isolated zeros and singularities for the co-dimension formula.","fun_headline_variants_meta":{"raw":{"variants":["Extreme value laws for shortest map trajectory encounters","Closest encounters follow extremes in strongly mixing maps","Extremal index modulates shortest distances between dynamics","Measure co-dimensions shape extreme trajectory distance laws"]},"model":"grok-4.3","cost_usd":0.004983,"raw_usage":{"total_tokens":2329,"prompt_tokens":618,"num_sources_used":0,"completion_tokens":55,"cost_in_usd_ticks":49828000,"prompt_tokens_details":{"text_tokens":618,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1656,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":618,"tokens_out":55,"duration_ms":18904,"temperature":1.0,"reasoning_tokens":1656,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-01T01:25:27.411997+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"For two concrete strongly mixing maps on the interval with computed co-dimensions, the histogram of minimal distances over many pairs of long trajectories fails to converge to the predicted extreme value distribution.","supporting_citations":[],"review_version":1}