{
  "schemaVersion": "1.6",
  "slug": "four-element-rowmotion-periodicity",
  "title": "A four-element obstruction to birational and piecewise-linear rowmotion periodicity",
  "shortTitle": "Four-element rowmotion obstruction",
  "url": "https://evidencepress.org/releases/four-element-rowmotion-periodicity/",
  "oneLine": "A non-graded four-element rooted tree has infinite-order birational and piecewise-linear rowmotion, and exhaustive exact replay proves that no smaller finite poset can do so.",
  "abstract": "This anonymous, unrefereed theorem candidate proves infinite order for standard birational rowmotion and order-polytope piecewise-linear rowmotion on the four-element poset with covers a<r, b<r and c<b. The birational proof uses an algebraic fixed point and a totally-real-field contradiction for its derivative; the piecewise-linear proof uses a rational interior fixed point with a nontrivial Jordan block at -1. Exhaustive enumeration of every strict partial order through size three proves global size-minimality for both lifts. The underlying Hasse shape was already published, up to duality, in Grinberg and Roby Example 70; the candidate contribution is the infinite-order proof and mathematical minimality, not discovery of the shape. A written proof, exact Python replay, language-separated JavaScript corroboration, five semantic mutations, public Linux CI, review repair and immutable GitHub and Zenodo assets support the candidate. Historical priority, independent reconstruction, formal verification, external specialist review and peer review remain unestablished.",
  "datePublished": "2026-08-30",
  "dateModified": "2026-08-30",
  "version": "0.1.0-candidate",
  "doi": "10.5281/zenodo.22178666",
  "doiUrl": "https://doi.org/10.5281/zenodo.22178666",
  "conceptDoi": "10.5281/zenodo.22178665",
  "pdfUrl": "https://github.com/ipitchford/four-element-rowmotion-candidate/releases/download/v0.1.0-candidate/four-element-rowmotion-paper.pdf",
  "altPdfUrl": "https://zenodo.org/records/22178666/files/four-element-rowmotion-paper.pdf?download=1",
  "altPdfLabel": null,
  "zenodoUrl": "https://zenodo.org/records/22178666",
  "repoUrl": "https://github.com/ipitchford/four-element-rowmotion-candidate",
  "releaseUrl": "https://github.com/ipitchford/four-element-rowmotion-candidate/releases/tag/v0.1.0-candidate",
  "markdownUrl": "https://evidencepress.org/releases/four-element-rowmotion-periodicity/index.md",
  "bibtexUrl": "https://evidencepress.org/releases/four-element-rowmotion-periodicity/cite.bib",
  "audioUrl": "https://evidencepress.org/assets/audio/four-element-rowmotion-periodicity.mp3?v=0df308333e",
  "imageUrl": "https://evidencepress.org/assets/og/four-element-rowmotion-periodicity.png?v=55f4c9cde5",
  "coverArtUrl": "https://evidencepress.org/assets/art/four-element-rowmotion-periodicity.svg?v=0e3e05bb4e",
  "media": [
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      "type": "audio",
      "url": "https://evidencepress.org/assets/audio/four-element-rowmotion-periodicity.mp3",
      "name": "Audio briefing — four-element rowmotion obstruction",
      "description": "Plain-English synthetic-voice account of the two local infinite-order obstructions, exhaustive size-three minimality and assurance limits; a communication aid, not additional evidence.",
      "transcriptUrl": "https://evidencepress.org/assets/audio/four-element-rowmotion-periodicity.txt"
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  ],
  "authors": [
    "Anonymous"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-candidate",
  "verification": {
    "peerReviewed": false,
    "independentlyReproduced": false,
    "formallyVerified": false,
    "internallyReplayed": true,
    "detail": "Anonymous, unrefereed algebraic-combinatorics theorem candidate at PASS_WITH_NOTES after minor-revision repair. The supplied review independently recalculated the central theorem as correct, but reviewer identity and unaffiliated status were not authenticated. The four-element Hasse shape is prior art up to duality; no exact theorem collision was found in the bounded search, and no first or priority claim is made."
  },
  "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/22178666",
      "note": "The tagged GitHub prerelease and published Zenodo record expose the DOI-bearing PDF, complete ZIP, checksum ledger, release notes and manifest; all five public assets were read back and matched local bytes."
    },
    {
      "dimension": "internalReplay",
      "label": "Internal replay",
      "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?",
      "state": "passed",
      "note": "Public Linux CI verifies the shipped bytes, runs Python normally and with optimization, checks the exhaustive small-poset certificate, rejects five semantic mutations, runs the JavaScript corroboration and rebuilds and inspects the PDF.",
      "evidenceUrl": "https://github.com/ipitchford/four-element-rowmotion-candidate/actions/runs/33321822649"
    },
    {
      "dimension": "independentRerun",
      "label": "Independent rerun",
      "question": "Has someone else run the supplied implementation and obtained the stated result?",
      "state": "not-assessed",
      "note": "No authenticated unaffiliated clean-environment rerun is reported."
    },
    {
      "dimension": "independentReimplementation",
      "label": "Independent reimplementation",
      "question": "Has someone else reached the result from an independent implementation?",
      "state": "not-assessed",
      "note": "Python and JavaScript were produced within one coordinated workflow and are not independent implementations in the assurance sense."
    },
    {
      "dimension": "formalVerification",
      "label": "Formal verification",
      "question": "Is a formalised statement machine-checked, and over which trusted base?",
      "state": "not-assessed",
      "note": "No proof assistant formalizes the rational map, fixed-point derivative arguments, field obstruction, piecewise-linear chamber or global minimality transfer."
    },
    {
      "dimension": "specialistReview",
      "label": "Specialist review",
      "question": "Has a domain specialist assessed the argument?",
      "state": "not-assessed",
      "note": "A supplied review recalculated the central theorem, but the reviewer's identity, specialist credentials and unaffiliated status were not authenticated."
    },
    {
      "dimension": "editorialPeerReview",
      "label": "Editorial peer review",
      "question": "Has a journal or venue run peer review to a decision?",
      "state": "not-assessed",
      "note": "No journal or comparable external venue has conducted editorial peer review."
    },
    {
      "dimension": "dataEnvironmentReproducibility",
      "label": "Data and environment reproducibility",
      "question": "Are data and computational environment pinned well enough to rebuild?",
      "state": "partial",
      "evidenceUrl": "https://github.com/ipitchford/four-element-rowmotion-candidate/blob/v0.1.0-candidate/ENVIRONMENT.txt",
      "note": "Commands, declared versions, exact certificates, manifests and Linux CI are public, but PDF byte determinism is producer-environment scoped and no bit-reproducible system image is supplied."
    }
  ],
  "provenance": {
    "aiGenerated": true,
    "aiAssisted": true,
    "generatedBy": [
      "AI systems under human direction",
      "OpenAI Codex",
      "Python 3 standard library",
      "Node.js standard library"
    ],
    "humanRole": "Research direction, candidate selection, review provision, claim-boundary decisions, publication authorization and institutional responsibility by the human creator; repository maintenance and Evidence Press publication are not scholarly authorship, and scholarly attribution remains Anonymous.",
    "disclosure": "AI systems contributed proof reconstruction, exact computation, manuscript drafting, source and novelty auditing, review repair, editing, replay and release engineering within one producer-coordinated workflow. Language-separated corroboration and the supplied review do not by themselves establish unaffiliated reconstruction, specialist review or editorial peer review."
  },
  "attributionBoundary": null,
  "problem": {
    "name": "AIM Dynamical Algebraic Combinatorics Problem 4.3: birational rowmotion on d-complete posets",
    "url": "https://aimath.org/pastworkshops/dac_preworkshop.pdf"
  },
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  "corrections": [],
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  "recordMaturity": "current",
  "metadataProvenance": "Authored prospectively for Evidence Press against the DOI-bearing tagged theorem candidate and its repaired review record on 30 August 2026; canonical HTML, paper.json, provenance-bound media and operating records were then confirmed by exact guarded public readback after merged CI and accessibility passed.",
  "grandfatheredAtSchemaVersion": null,
  "keywords": [
    "rowmotion",
    "birational dynamics",
    "piecewise-linear dynamics",
    "d-complete poset",
    "fence poset",
    "rooted tree",
    "infinite order",
    "minimal obstruction",
    "unrefereed candidate"
  ],
  "keyResults": [
    "The four-element poset with covers a<r, b<r and c<b has infinite-order standard birational rowmotion.",
    "The same poset has infinite-order order-polytope piecewise-linear rowmotion.",
    "The birational obstruction is a fixed point whose derivative would force a non-totally-real degree-nine field into a totally real cyclotomic subfield under finite order.",
    "The piecewise-linear obstruction is a rational interior fixed point with a nontrivial Jordan block at eigenvalue -1.",
    "Exact exhaustive enumeration of all finite posets through three elements proves that four elements is globally size-minimal for both phenomena."
  ],
  "reviews": [],
  "evidencePackage": "A five-page DOI-bearing manuscript and aligned Markdown; an explicit proof certificate; dependency-free Python exact replay; language-separated JavaScript finite-field, quotient-ring, determinant and real-root corroboration; exhaustive enumeration of the 1, 1, 3 and 19 labelled strict partial orders on zero through three elements and reduction to nine isomorphism classes; exact periods for all nine classes; five semantic certificate mutations; source, convention, citation and bounded novelty audits; a supplied full review and explicit response; one hash-bound confirmation; a public GitHub Actions replay and PDF rebuild; a complete manifested ZIP; and exact GitHub and Zenodo asset readback. These establish public availability, integrity, producer replay and the stated finite computations—not historical priority, unaffiliated reproduction, formal verification, authenticated external specialist review, editorial peer review or impact.",
  "openProblems": [
    "Classify finite-order and infinite-order birational rowmotion on non-graded rooted trees or on d-complete posets more broadly.",
    "Determine which asymmetric two-segment fences have finite-order birational or piecewise-linear rowmotion.",
    "Independently reconstruct both fixed-point derivative obstructions in a materially separate stack.",
    "Formalize the birational field argument, piecewise-linear Jordan obstruction and exhaustive size-three minimality in a proof assistant.",
    "Obtain an authenticated specialist review and a broader historical-priority assessment."
  ],
  "mathObjects": [],
  "relatedWorks": [
    {
      "citation": "Propp, J., Roby, T., Striker, J., & Williams, N. (2015). Problems for the 2015 AIM Workshop on Dynamical Algebraic Combinatorics, Problem 4.3.",
      "url": "https://aimath.org/pastworkshops/dac_preworkshop.pdf",
      "doi": null
    },
    {
      "citation": "Grinberg, D., & Roby, T. (2016). Iterative properties of birational rowmotion I: Generalities and skeletal posets. Electronic Journal of Combinatorics 23(1), P1.33.",
      "url": "https://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p33",
      "doi": null
    },
    {
      "citation": "Einstein, D., & Propp, J. (2014). Piecewise-linear and birational toggling. DMTCS Proceedings, AT, 513–524.",
      "url": "https://dmtcs.episciences.org/2419",
      "doi": null
    },
    {
      "citation": "Dangwal, P., Kimble, J., Liang, J., Lou, J., Sagan, B. E., & Stewart, Z. (2023). Rowmotion on rooted trees. Séminaire Lotharingien de Combinatoire 88, B88a.",
      "url": "https://arxiv.org/abs/2208.12155",
      "doi": "10.48550/arXiv.2208.12155"
    },
    {
      "citation": "Proctor, R. A., & Scoppetta, L. M. (2019). d-Complete posets: Local structural axioms, properties, and equivalent definitions. Order 36, 399–422.",
      "url": "https://doi.org/10.1007/s11083-018-9473-4",
      "doi": "10.1007/s11083-018-9473-4"
    },
    {
      "citation": "Mertin, A., & Poznanović, S. (2024). Toggleability spaces of fences. Electronic Journal of Combinatorics 31(4), P4.46.",
      "url": "https://www.combinatorics.org/ojs/index.php/eljc/article/view/v31i4p46",
      "doi": null
    }
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      "scope": "The poset with covers a<r, b<r and c<b; standard birational rowmotion with boundary labels one and order-polytope piecewise-linear rowmotion with boundary labels zero and one; not a classification of d-complete posets or fences, an originality finding, or external validation."
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      "remainingRisks": [
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      "Keep the result limited to the two stated lifted maps and boundary conventions.",
      "Treat the four-element Hasse shape as prior art up to duality.",
      "Inspect the written fixed-point and field arguments rather than treating executable replay as a proof oracle.",
      "Keep producer corroboration, supplied review, independent reconstruction, formal verification and peer review distinct.",
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        "inheritedClaim": "Grinberg and Roby define the relevant birational framework, preserve rowmotion order under duality, prove finite order for skeletal posets and display the same four-element non-skeletal tree up to duality.",
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