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  "slug": "bounded-product-transposition-cutoff",
  "title": "Cutoff for bounded product-weight random transpositions",
  "shortTitle": "Biased shuffles still cut off",
  "url": "https://evidencepress.org/releases/bounded-product-transposition-cutoff/",
  "oneLine": "Arbitrary bounded product weights admit a cutoff proof with mixing order n log n and window upper bound n log log n.",
  "abstract": "For independent label draws with fixed bounds c/n ≤ p_i ≤ C/n, a conjugacy-invariant reference gradient gives an all-time constant-factor comparison. Entropy estimates yield total-variation cutoff and an O(n log log n) window on the n log n scale. Averaging and binomial smoothing recover the original discrete-time chain. The result is a product-branch proof candidate, not a sharp-location or nonproduct theorem.",
  "datePublished": "2026-09-08",
  "dateModified": "2026-09-08",
  "version": "1.0.0-candidate",
  "doi": "10.5281/zenodo.22662125",
  "doiUrl": "https://doi.org/10.5281/zenodo.22662125",
  "conceptDoi": "10.5281/zenodo.22662123",
  "pdfUrl": "https://github.com/ipitchford/bounded-product-transposition-cutoff/releases/download/v1.0.0-candidate/bounded-product-transposition-cutoff-1.0.0-candidate.pdf",
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  "zenodoUrl": "https://zenodo.org/records/22662125",
  "repoUrl": "https://github.com/ipitchford/bounded-product-transposition-cutoff",
  "releaseUrl": "https://github.com/ipitchford/bounded-product-transposition-cutoff/releases/tag/v1.0.0-candidate",
  "markdownUrl": "https://evidencepress.org/releases/bounded-product-transposition-cutoff/index.md",
  "bibtexUrl": "https://evidencepress.org/releases/bounded-product-transposition-cutoff/cite.bib",
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  "authors": [
    "Anonymous"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-candidate",
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    "detail": "Unrefereed proof candidate for the product-weight branch only. Producer finite checks and internal model-assisted editorial review are not independent reproduction, unaffiliated specialist review or formal verification. No explicit cutoff location, optimal window, limiting profile, nonproduct resolution or historical priority is claimed."
  },
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      "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?",
      "state": "passed",
      "note": "Finite exact checks on S_2 through S_5 and five rejection controls. These do not certify the all-n theorem.",
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      "note": "Producer-coordinated model review and replay do not establish this external dimension."
    },
    {
      "dimension": "independentReimplementation",
      "label": "Independent reimplementation",
      "question": "Has someone else reached the result from an independent implementation?",
      "state": "not-assessed",
      "note": "Producer-coordinated model review and replay do not establish this external dimension."
    },
    {
      "dimension": "formalVerification",
      "label": "Formal verification",
      "question": "Is a formalised statement machine-checked, and over which trusted base?",
      "state": "not-assessed",
      "note": "Producer-coordinated model review and replay do not establish this external dimension."
    },
    {
      "dimension": "specialistReview",
      "label": "Specialist review",
      "question": "Has a domain specialist assessed the argument?",
      "state": "not-assessed",
      "note": "Producer-coordinated model review and replay do not establish this external dimension."
    },
    {
      "dimension": "editorialPeerReview",
      "label": "Editorial peer review",
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      "state": "not-assessed",
      "note": "Producer-coordinated model review and replay do not establish this external dimension."
    },
    {
      "dimension": "dataEnvironmentReproducibility",
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      "state": "partial",
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      "note": "Standard-library diagnostics on local macOS and Linux CI; no independent reproduction inferred."
    }
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  "provenance": {
    "aiGenerated": true,
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      "OpenAI Codex under human direction",
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    "humanRole": "Research selection, direction, supplied review and publication authorization; scholarly creator Anonymous.",
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  "attributionBoundary": null,
  "problem": {
    "name": "AIM-PROBABILITY-0050 — product-weight branch",
    "url": "https://aimath.org/pastworkshops/markovmixing.html"
  },
  "orderedProductAverage": null,
  "corrections": [],
  "publicCorrections": [],
  "publicCorrectionIndexes": null,
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  "metadataProvenance": "Producer-coordinated proof assessment and finite diagnostics are separate from canonical publication; availability does not upgrade external validation.",
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  "keywords": [
    "random transpositions",
    "cutoff",
    "Markov chains",
    "entropy",
    "gradient comparison"
  ],
  "keyResults": [
    "Cutoff for every sequence with fixed bounds c/n ≤ p_i ≤ C/n.",
    "Mixing order n log n, with O(n log log n) window upper bound.",
    "All-time gradient comparison pays the distortion constant once.",
    "Binomial smoothing removes averaging despite holding probability of order 1/n."
  ],
  "reviews": [],
  "evidencePackage": "Written proof, standard-library exact finite diagnostics, five semantic rejection controls, normal/optimized clean replay and five internal role reports. https://github.com/ipitchford/bounded-product-transposition-cutoff/blob/v1.0.0-candidate/EDITORIAL_DECISION.md",
  "openProblems": [
    "Determine an explicit cutoff location for general bounded vectors.",
    "Improve the window bound and investigate limiting profiles.",
    "Fix normalization and periodicity conventions before extending to nonproduct rates.",
    "Independently assess the proof, contribution-specific novelty and priority."
  ],
  "mathObjects": [],
  "relatedWorks": [
    {
      "citation": "AIM Markov chain mixing times workshop (2016), originating problem context.",
      "url": "https://aimath.org/pastworkshops/markovmixing.html",
      "doi": null
    },
    {
      "citation": "Nestoridi–Yan (2024), Cutoff for the biased random transposition shuffle: sharper two-class comparator.",
      "url": "https://arxiv.org/abs/2409.16387v1",
      "doi": "10.48550/arXiv.2409.16387"
    },
    {
      "citation": "Pedrotti–Salez (2025), A new cutoff criterion for non-negatively curved chains: entropy method and normalized inputs.",
      "url": "https://arxiv.org/abs/2501.13079v1",
      "doi": "10.48550/arXiv.2501.13079"
    },
    {
      "citation": "Pedrotti–Salez (2026), The local product condition implies cutoff: general subcommutation context.",
      "url": "https://arxiv.org/abs/2607.05345v2",
      "doi": "10.48550/arXiv.2607.05345"
    },
    {
      "citation": "Hermon–Peres (2017), The power of averaging at two consecutive time steps.",
      "url": "https://doi.org/10.1214/16-AIHP782",
      "doi": "10.1214/16-AIHP782"
    }
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