{ "schemaVersion": "1.6", "slug": "pass-and-swap-bounded-swaps", "title": "Bounded swaps in pass-and-swap queues: counterexamples to a conjecture of Dorsman and Gardner, and a proof for bipartite swapping graphs with odd caps", "shortTitle": "When capped swaps lock in", "url": "https://evidencepress.org/releases/pass-and-swap-bounded-swaps/", "oneLine": "Capping the swaps in pass-and-swap queues can trap the system in arrangements the product-form theory excludes: a 2024 conjecture fails for every cap of two or more, exactly on rings of 2kw+1 jobs and beyond, but holds for bipartite swapping graphs with odd caps.", "abstract": "Pass-and-swap queues extend order-independent queues, whose stationary laws have a product form, by letting a finishing job hand its place along a chain of compatible jobs. Dorsman and Gardner (2024) capped each service at w swaps, proved product form whenever the placement orientation of the swapping graph has no directed path longer than w edges, and conjectured that, if the swapping graph is (w+1)-colourable, every state with a longer path is transient. This candidate shows that the conjecture fails for every w ≥ 2. The balanced states of the odd cycle C_{2w+1} form a closed class of height w+1 for every admissible rate function, recurrent under positive rates, with (2w+2)!/(w!)² states. Under positive rates, the cycle C_n has a recurrent class of excessive height if and only if n ≡ 1 (mod 2w), and it is then unique. Two odd cycles joined by a path give further counterexamples, and an exact census of all connected graphs with at most seven vertices finds 1355 violating (graph, cap) pairs, none bipartite and none at w = 1. Conversely, the conjecture holds for bipartite swapping graphs with odd caps under positive rates; exact computation finds no counterexample for any cap on bipartite graphs with at most nine vertices. On the counterexample class, homogeneous rates give an explicit product-form law, while for additive per-job rates scaling the original product weights by one constant per orientation is stationary only when rates are constant within each queue. Proofs are written; finite claims are certified by exact, replayable verifiers. AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.", "datePublished": "2026-10-20", "dateModified": "2026-10-20", "version": "1.0.0-candidate", "doi": "10.5281/zenodo.23515841", "doiUrl": "https://doi.org/10.5281/zenodo.23515841", "conceptDoi": null, "pdfUrl": "https://github.com/ipitchford/pass-and-swap-bounded-swaps/releases/download/v1.0.0-candidate/pass-and-swap-bounded-swaps-1.0.0-candidate.pdf", "altPdfUrl": null, "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23515841", "repoUrl": "https://github.com/ipitchford/pass-and-swap-bounded-swaps", "releaseUrl": "https://github.com/ipitchford/pass-and-swap-bounded-swaps/releases/tag/v1.0.0-candidate", "markdownUrl": "https://evidencepress.org/releases/pass-and-swap-bounded-swaps/index.md", "bibtexUrl": "https://evidencepress.org/releases/pass-and-swap-bounded-swaps/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/pass-and-swap-bounded-swaps.mp3", "imageUrl": "https://evidencepress.org/assets/og/pass-and-swap-bounded-swaps.png?v=fc930b23dd", "coverArtUrl": "https://evidencepress.org/assets/art/pass-and-swap-bounded-swaps.svg?v=90caf523ee", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/pass-and-swap-bounded-swaps.mp3", "name": "Plain-English audio briefing", "description": "Communication aid; not additional research evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/pass-and-swap-bounded-swaps.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed AI-assisted mathematical candidate. Written proofs and exact certificates have the stated scope; internal AI-assisted and adversarial review only. External specialist review, independent reproduction and formal verification have not been established." }, "assurance": [ { "dimension": "availability", "label": "Availability and archiving", "question": "Is the evidence package publicly retrievable from an archive under a persistent identifier?", "state": "passed", "evidenceUrl": "https://zenodo.org/records/23515841", "note": "The versioned research package and manuscript are publicly archived on GitHub and Zenodo. Provider asset parity was checked against frozen local files; availability does not establish correctness or peer review." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "All fifteen certificate runs, including the nine-vertex bipartite census, regenerate byte for byte under optimised Python in a separate clean replay; the fourteen shorter runs and the negative controls were replayed again on the frozen release archive." }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "No unaffiliated rerun has been established." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "partial", "note": "Internal AI reviewers recomputed the main counts with their own simulation code inside the producer workflow; this is implementation diversity, not external reproduction." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "No proof-assistant formalisation; the verifiers are Python programs and the interpreter is in the trust boundary." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "AI-assisted internal and adversarial review is not external specialist review." }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed", "note": "No external editorial peer review." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "passed", "note": "Verifiers use only the Python standard library with exact arithmetic and deterministic output; continuous integration replays them on two Python versions." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted mathematical research", "AI-assisted internal review, verification re-review and adversarial review (including OpenAI gpt-5.6-sol)", "Claude Code: publication packaging and release" ], "humanRole": "Publication authority only; no human research contribution inferred.", "reviewStatus": "Internal AI-assisted and adversarial review, not external peer review." }, "attributionBoundary": null, "problem": null, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Prepared from the approved revised manuscript and its exact certificates. The versioned GitHub and Zenodo research archives were checked against the frozen release assets on 20 October 2026.", "grandfatheredAtSchemaVersion": null, "keywords": [ "pass-and-swap queues", "bounded swaps", "product-form stationary distribution", "order-independent queues", "redundancy scheduling", "acyclic orientations", "promotion on linear extensions", "counterexample" ], "keyResults": [ "Conjecture 1 of Dorsman and Gardner fails for every cap w ≥ 2: the balanced states of the odd cycle C_{2w+1} form a closed class of height w+1, recurrent under positive rates, with (2w+2)!/(w!)² states (proof).", "Under positive rates the cycle C_n has a recurrent class of excessive height if and only if n ≡ 1 (mod 2w), and it is unique (proof).", "Two odd cycles joined by a path fail at an explicit cap (proof); all connected graphs with at most seven vertices give 1355 violating (graph, cap) pairs, none bipartite and none at w = 1 (exact census).", "For bipartite swapping graphs with odd caps and positive rates the conjecture holds (proof); no bipartite counterexample exists for any cap up to nine vertices (exact census), and the even-cap case is conjectured.", "On the counterexample class, homogeneous rates give an explicit product-form law; for additive rates that vary within a queue, orientation-scaled product weights are not stationary (proof)." ], "reviews": [], "evidencePackage": "Written proofs; fifteen exact standard-library Python verifiers whose JSON certificates regenerate byte for byte under python -O, including a census of all connected graphs with at most seven vertices and of bipartite graphs with at most nine; cross-checks of three independent descriptions on all graphs with at most six vertices; negative controls rejecting a wrong cap and a wrong cycle rule and detecting tampered files.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Dorsman and Gardner (2024), New directions in pass-and-swap queues. Queueing Systems 107, 205–256.", "url": "https://doi.org/10.1007/s11134-024-09914-1", "doi": "10.1007/s11134-024-09914-1" }, { "citation": "Comte and Dorsman (2021), Pass-and-swap queues. Queueing Systems 98, 275–331.", "url": "https://doi.org/10.1007/s11134-021-09700-3", "doi": "10.1007/s11134-021-09700-3" }, { "citation": "Gardner and Righter (2020), Product forms for FCFS queueing models with arbitrary server-job compatibilities: an overview. Queueing Systems 96, 3–51.", "url": "https://doi.org/10.1007/s11134-020-09668-6", "doi": "10.1007/s11134-020-09668-6" }, { "citation": "Ayyer, Klee and Schilling (2014), Combinatorial Markov chains on linear extensions. Journal of Algebraic Combinatorics 39, 853–881.", "url": "https://doi.org/10.1007/s10801-013-0470-9", "doi": "10.1007/s10801-013-0470-9" }, { "citation": "Anton and Gardner (2023), The stationary distribution of the redundancy-d model with random order of service. ACM SIGMETRICS Performance Evaluation Review 51(2), 9–11.", "url": "https://doi.org/10.1145/3626570.3626575", "doi": "10.1145/3626570.3626575" }, { "citation": "Gast and Van Houdt (2024), Approximations to study the impact of the service discipline in systems with redundancy. Proceedings of the ACM on Measurement and Analysis of Computing Systems 8(1), 1–33.", "url": "https://doi.org/10.1145/3639040", "doi": "10.1145/3639040" }, { "citation": "Berezner, Kriel and Krzesinski (1995), Quasi-reversible multiclass queues with order independent departure rates. Queueing Systems 19, 345–359.", "url": "https://doi.org/10.1007/BF01151928", "doi": "10.1007/BF01151928" } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:pass-and-swap-bounded-swaps", "attemptIds": [ "ep-attempt:pass-and-swap-bounded-swaps-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "communication" ], "lineageId": null, "accelerationPrimitives": [ "certificate-first", "counterexample-proxy-first", "adversarial-controls", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "counterexample", "description": "The balanced class of the odd cycle C_{2w+1} refutes the Dorsman–Gardner bounded-swap conjecture for every w ≥ 2; the release adds an exact cycle criterion, further counterexample families and a proof for bipartite swapping graphs with odd caps.", "scope": "Closed tandems of two pass-and-swap queues with one job per class (or the symmetric lift), simple swapping graphs and equal caps; recurrence and transience results assume positive service at every position. Census results hold only in their stated finite ranges." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "The conjecture is quoted verbatim with the source's definitions, and the conventions (partite versus colourable, path length in vertices or edges, one job per class, rates, transition rule) are reconciled in Section 2.3 of the paper; the counterexamples violate both path-length readings.", "remainingRisks": [ "Certificates rely on Python execution and on the verifiers implementing Definition 2.2 of the paper.", "The bounded literature check is not exhaustive priority certification." ] }, "humanJudgmentGates": [ "Publication is authorised; scientific endorsement is not inferred.", "External specialist review and independent reproduction remain open." ], "parentLinks": [], "assuranceTarget": { "dimensions": [ "independentRerun", "specialistReview", "formalVerification", "priorityAssessment" ], "nextAction": "Independently check the binding-move lemma and replay the cycle and census certificates with separate code; examine Conjecture 7.2.", "claimCeiling": "Closed tandems of two pass-and-swap queues with one job per class, simple swapping graphs and equal caps; recurrence results under positive rates; census results in their finite ranges. Internal verification is not external assurance." }, "impactClaims": [ { "id": "pass-and-swap-bounded-swaps-no-impact", "aim": "science", "outcome": "An openly inspectable counterexample, classification and continuation target.", "setting": "AI-assisted mathematical research", "status": "NO_IMPACT_EVIDENCE", "designClass": "none", "comparator": "None registered.", "estimand": "No research-speed or scientific-impact effect is estimated.", "evidenceRefs": [], "registeredDesignRef": null, "independentAssessment": null } ] }, "researchMetrics": { "schemaVersion": "1.0", "policyUrl": "https://evidencepress.org/api/research-metrics-policy.json", "outcomePolicyUrl": "https://evidencepress.org/api/v2/research-metrics-policy.json", "workLedgerUrl": "https://evidencepress.org/api/work-ledger.json", "claimCeiling": "These records describe scoped workflow inputs and outcomes. They support calibration and reference-class construction, but do not by themselves establish research acceleration, theorem correctness, novelty, independent assurance or impact.", "attempts": [ { "attemptId": "ep-attempt:pass-and-swap-bounded-swaps-publication", "status": "published", "resultClass": "positive", "metrics": { "schemaVersion": "1.0", "measurementScope": "publication-only", "scopeBoundary": "Only archive packaging, site integration, guarded delivery and canonical readback from registration are prospective. Earlier research, review, revision and extension work are left-censored.", "forecast": { "frozenAt": "2026-10-20T22:58:20Z", "procedureClass": "full-candidate-publication", "targetOutcome": "Accurately scoped GitHub, Zenodo and canonical Evidence Press release with explanatory media and machine readback, if all publication gates pass.", "expectedActiveMinutes": 150, "plausibleLowMinutes": 105, "plausibleHighMinutes": 210, "expectedUnattendedWaitMinutes": 30, "referenceClass": { "label": "Full mathematical candidate publication", "sampleSize": 0, "basis": "Route benchmark planning prior, not empirical calibration." }, "fermiComponents": [ { "component": "Research archive packaging, replay and identities", "count": 1, "lowMinutesPerUnit": 45, "centralMinutesPerUnit": 60, "highMinutesPerUnit": 90, "basis": "One frozen package with a six-minute short replay; the 1.5-hour nine-vertex run is not repeated because verifiers are unchanged." }, { "component": "Page, metadata and media integration", "count": 1, "lowMinutesPerUnit": 30, "centralMinutesPerUnit": 45, "highMinutesPerUnit": 60, "basis": "One reader-first release." }, { "component": "Guarded deployment and canonical readback", "count": 1, "lowMinutesPerUnit": 30, "centralMinutesPerUnit": 45, "highMinutesPerUnit": 60, "basis": "Mandatory site tests, two merge cycles, zero-cost deployment and public checks." } ], "probabilityPositiveSignal": 0.9, "probabilityTargetClosure": 0.85, "probabilityHorizonMinutes": 300, "assumptions": [ "The approved scientific package remains valid and configured providers remain available." ], "reforecastTriggers": [ "A publication gate reveals a scientific or integrity defect, or a provider outage blocks delivery." ], "stopRule": "Do not weaken assurance, integrity or cost gates to meet the forecast." }, "outcome": { "status": "unavailable", "policyVersion": "2.0", "recordedAt": "2026-10-20T23:21:19Z", "reason": "Active, waiting, blocked, rework, compute and token telemetry were not instrumented over the registered publication scope. Earlier scientific work remains left-censored; wall-clock timestamps are not substituted for effort measurements.", "missingFields": [ { "field": "calendarElapsedMinutes", "reason": "Opening and delivery timestamps are retained as milestones; their difference is not active effort and no numeric metrics series is emitted." }, { "field": "activeAgentMinutes", "reason": "Not instrumented; unknown is not zero." }, { "field": "unattendedWaitMinutes", "reason": "Not instrumented; unknown is not zero." }, { "field": "blockedMinutes", "reason": "Not instrumented; unknown is not zero." }, { "field": "reworkMinutes", "reason": "Not instrumented; unknown is not zero." }, { "field": "agentRuns", "reason": "Not instrumented; unknown is not zero." }, { "field": "maxParallelAgents", "reason": "Not instrumented; unknown is not zero." }, { "field": "modelTurns", "reason": "Not instrumented; unknown is not zero." }, { "field": "researchCycles", "reason": "Not instrumented; earlier science is left-censored." }, { "field": "falsificationGatesRun", "reason": "Not instrumented; earlier science is left-censored." }, { "field": "candidateArchitecturesTested", "reason": "Not instrumented; this was a publication-only scope." }, { "field": "candidateArchitecturesRejected", "reason": "Not instrumented; this was a publication-only scope." }, { "field": "substantiveReviewRounds", "reason": "Not instrumented within the registered publication-only scope; earlier review is left-censored." }, { "field": "p0Findings", "reason": "Not instrumented within the registered publication-only scope." }, { "field": "p1Findings", "reason": "Not instrumented within the registered publication-only scope." }, { "field": "prepublicationClaimCorrections", "reason": "Not instrumented within the registered publication-only scope." }, { "field": "resultState", "reason": "Published delivery is observed in statusHistory; incomplete telemetry does not negate publication." }, { "field": "positiveSignalObserved", "reason": "Publication evidence is retained separately; no calibrated positive-signal metric is inferred." }, { "field": "targetReached", "reason": "Delivery is recorded in statusHistory; no calibrated target metric is inferred." }, { "field": "forecastErrorMinutes", "reason": "Active effort was not measured; forecast error cannot be computed from wall time." }, { "field": "forecastRatio", "reason": "Active effort was not measured; wall time is not a valid denominator." }, { "field": "withinForecastInterval", "reason": "Active effort was not measured, so interval membership is unknown." }, { "field": "activeHumanMinutes", "reason": "Not instrumented; unknown is not zero." }, { "field": "computeMinutes", "reason": "Not instrumented; unknown is not zero." }, { "field": "deduplicatedModelTokens", "reason": "Not instrumented; unknown is not zero." }, { "field": "uncachedInputTokens", "reason": "Not instrumented; unknown is not zero." } ], "calibration": null } }, "missingnessReason": null } ] } }