{"id":"0b105ff2-d3a1-48ac-8c23-b47c28154347","arxiv_id":"2606.02340","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Characterizes three fluctuation regimes for circular triads in W-random tournaments and supplies a multiplier bootstrap plus regularity tests yielding asymptotically valid CIs for the Kendall-Smith consistency coefficient across all tournamentons.","lead":"This paper studies how often random tournaments (directed complete graphs) violate transitivity by forming circular triads, under a general inhomogeneous model where edge directions have varying probabilities given by a tournamenton W. It derives three asymptotic regimes for the count of these cycles and supplies a bootstrap procedure plus tests to build valid confidence intervals for a consistency measure.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly isolates the modeling premise (W-random tournaments) as the point outside which the results do not apply. Within the model the paper's program—regime classification, bootstrap consistency, and test-based regime selection—contains no evident circularity or missing uniformity condition that would prevent the claimed coverage. The abstract-only limitation noted by the reader is the only reason for the UNVERDICTED label; once the full derivations are accepted as given, the argument stands.","tokens_in":1825,"tokens_out":293,"duration_ms":21916,"concrete_test":"Re-derive the limiting variance expressions for the circular-triad count in each of the three regimes directly from the tournamenton integral representation without using the bootstrap construction; confirm that the resulting variances match those used to calibrate the multiplier weights.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the number of circular triads in a W-random tournament exhibits three fluctuation regimes determined by regularity and uniformity of the tournamenton W, that a multiplier bootstrap consistently approximates the limiting distribution in each regime, and that tests for those properties can be combined into an algorithm yielding asymptotically valid confidence intervals for the consistency coefficient for every tournamenton. No internal gap, unstated assumption, or regime boundary issue that would invalidate the limiting statements or bootstrap validity is visible in the argument structure.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper studies fluctuations of the number of circular triads (directed 3-cycles) in W-random tournaments on n vertices, where edge directions are chosen independently according to a measurable tournamenton W:[0,1]^2→[0,1]. It identifies three distinct asymptotic regimes determined by notions of regularity and uniformity of W, develops a tournamenton multiplier bootstrap that consistently approximates the limiting distribution in each regime, and combines this bootstrap with tests for regularity and uniformity to produce an algorithm yielding asymptotically valid confidence intervals for the Kendall-Smith consistency coefficient that hold for every tournamenton. Structural characterizations of tournamentons producing degenerate limits are also given, with connections to tournament quasirandomness.","tokens_in":1894,"tokens_out":434,"duration_ms":12573,"significance":"If the limiting statements and bootstrap consistency hold under the stated conditions, the results supply the first complete inferential framework for the consistency coefficient that is valid uniformly over all tournamentons, rather than only in the dense or quasirandom cases. The identification of three fluctuation regimes and the accompanying structural characterizations strengthen the link between tournament theory and graphon methods; the multiplier bootstrap that adapts automatically across regimes is a technical contribution that may extend to other subgraph counts in inhomogeneous directed models.","major_comments":[],"minor_comments":[{"comment":"The abstract and introduction refer to 'suitable notions of regularity and uniformity of W' without an early forward reference to the precise definitions (presumably in §2 or §3); adding a brief parenthetical or footnote would improve readability for readers outside the graphon literature.","section":null},{"comment":"Notation for the consistency coefficient and the normalized circular-triad count should be introduced once in a dedicated notation subsection or table, as the same symbols appear in both the limiting theorems and the bootstrap construction.","section":null},{"comment":"The description of the multiplier bootstrap in the algorithm section would benefit from an explicit statement of the resampling weights and the precise centering used in each regime, even if these are standard in the multiplier-bootstrap literature.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the detailed summary of our manuscript and for the positive assessment of its contributions. The recommendation of minor revision is appreciated. No specific major comments were provided in the report, so we have no individual points to address at this time. We will incorporate any minor suggestions during the revision process.","responses":[],"tokens_in":1315,"tokens_out":80,"duration_ms":6717,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main thing to know is that this work extends the classical count of circular triads to inhomogeneous W-random tournaments by identifying three fluctuation regimes tied to regularity and uniformity of W, then builds a tournamenton multiplier bootstrap that tracks the right limit in each regime and combines it with tests to produce CIs that are claimed to work for every tournamenton.\n\nWhat is actually new is the three-regime characterization itself, the bootstrap construction adapted to the tournamenton setting, and the automatic algorithm that selects the regime and delivers the intervals. The structural results on degenerate cases are also fresh. The paper does a clean job of moving from the homogeneous case to the general measurable W without adding extra assumptions that would narrow the scope.\n\nThe soft spots are modest but real. The argument rests entirely on the independent-edge W-random model, so any dependence structure outside that model falls outside the guarantees. The abstract leaves the exact regularity conditions, the rates at which the regime tests work, and the error bounds on the bootstrap implicit; without those details it is hard to judge how sharp the boundaries are or how much power the tests lose near regime transitions. Those are standard gaps at the abstract stage rather than load-bearing flaws.\n\nThis is for people already working in random tournaments, graphons, or quasirandomness who need an inferential tool for the Kendall coefficient. A reader comfortable with graphon limit theorems will see the extension quickly and can judge whether the bootstrap is worth adopting. It deserves a serious referee because the claims are specific, the model is standard, and the bootstrap supplies a concrete, usable procedure even if the proofs need checking.","headline":"The paper splits the circular-triad count into three regimes under a general tournamenton and supplies a multiplier bootstrap plus regime test that together give asymptotically valid CIs for the consistency coefficient.","tokens_in":2395,"tokens_out":409,"would_cite":false,"duration_ms":15320,"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":"The number of circular triads in a W-random tournament follows one of three fluctuation regimes set by the regularity and uniformity of the tournamenton W, and a multiplier bootstrap produces valid confidence intervals for the consistency c","keywords":["W-random tournaments","circular triads","consistency coefficient","tournamentons","multiplier bootstrap","transitivity","quasirandomness","inhomogeneous models"],"falsifier":"Generate many independent W-random tournaments of growing size n for a fixed, known W that is neither regular nor uniform; compute the empirical distribution of the suitably centered and scaled circular-triad count and check whether it converges to the non-degenerate limit predicted by the theory or is well-approximated by the bootstrap.","tokens_in":2709,"feed_emoji":"","tokens_out":765,"duration_ms":26577,"temperature":0.7,"pith_summary":"The paper establishes how the number of circular triads, which quantifies departures from transitivity in pairwise comparison data, fluctuates when a tournament is generated from an arbitrary tournamenton W. Three regimes of fluctuation appear as the number of vertices grows, each tied to whether W satisfies regularity or uniformity conditions. A tournamenton multiplier bootstrap is proved to track the limiting distribution of the triad count in the appropriate regime. When this bootstrap is combined with separate tests that detect regularity and uniformity, the resulting procedure yields confidence intervals for the Kendall-Smith consistency coefficient that remain asymptotically valid for any tournamenton. The work also supplies structural descriptions of those tournamentons that force the limiting distribution into a degenerate form.","feed_headline":"Three regimes govern circular triad counts in random tournaments","feed_subtitle":"Multiplier bootstrap supplies confidence intervals for the consistency coefficient that remain valid under any tournamenton.","key_machinery":"The W-random tournament model, in which each directed edge is chosen independently with probability given by a fixed measurable tournamenton W from the unit square to the unit interval, together with the associated tournamenton multiplier bootstrap.","core_discovery":"For a W-random tournament on n vertices, the number of circular triads exhibits three different fluctuation regimes, determined by suitable notions of regularity and uniformity of W; a tournamenton multiplier bootstrap consistently approximates the limiting distribution in the relevant regime, and combining it with tests for regularity and uniformity yields an algorithm for asymptotically valid confidence intervals for the consistency coefficient for all tournamentons. Structural characterizations are obtained for tournamentons for which the limiting distribution exhibits specific degeneracies.","pith_inferences":["The inferential procedure could be applied directly to large empirical paired-comparison datasets from ranking or voting contexts once the W-random model is accepted.","The multiplier bootstrap technique may extend to other subgraph-count statistics in directed inhomogeneous random models beyond circular triads.","The structural conditions for degeneracy could serve as diagnostic tools for detecting when an observed tournament is close to quasirandom."],"forward_implications":["The circular-triad count admits a limiting distribution that depends on the regularity and uniformity properties of W and can be consistently approximated by the multiplier bootstrap.","Asymptotically valid confidence intervals for the consistency coefficient can be constructed for every possible tournamenton by first testing regularity and uniformity and then applying the bootstrap in the identified regime.","Structural characterizations identify the precise classes of tournamentons that produce degenerate limiting distributions for the circular-triad count.","The fluctuation results supply new characterizations in the theory of tournament quasirandomness."],"fun_headline_variants":["Three fluctuation regimes for circular triads in W-random tournaments","Inhomogeneous random tournaments have three triad fluctuation regimes","W regularity sets the regime for circular triad count fluctuations","Tournamenton multiplier bootstrap for circular triad inference","Confidence intervals for consistency coefficient valid for all tournamentons"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The observed tournament arises from independent edge directions whose probabilities are supplied by a single fixed measurable function W on the unit square.","fun_headline_variants_meta":{"raw":{"variants":["Three fluctuation regimes for circular triads in W-random tournaments","Inhomogeneous random tournaments have three triad fluctuation regimes","W regularity sets the regime for circular triad count fluctuations","Tournamenton multiplier bootstrap for circular triad inference","Confidence intervals for consistency coefficient valid for all tournamentons"]},"model":"grok-4.3","cost_usd":0.010335,"raw_usage":{"total_tokens":4591,"prompt_tokens":699,"num_sources_used":0,"completion_tokens":65,"cost_in_usd_ticks":103349500,"prompt_tokens_details":{"text_tokens":699,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3827,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":699,"tokens_out":65,"duration_ms":27457,"temperature":1.0,"reasoning_tokens":3827,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-28T12:43:58.022080+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Generate many independent W-random tournaments of growing size n for a fixed, known W that is neither regular nor uniform; compute the empirical distribution of the suitably centered and scaled circular-triad count and check whether it converges to the non-degenerate limit predicted by the theory or is well-approximated by the bootstrap.","supporting_citations":[],"review_version":1}