{
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  "slug": "efx-nine-chores",
  "title": "Nine chores for three agents: complete EFX existence with an exact computational certificate",
  "shortTitle": "Nine chores for three agents",
  "url": "https://evidencepress.org/releases/efx-nine-chores/",
  "oneLine": "Any nine indivisible chores can be shared among three agents with a strong fairness guarantee—even when their additive costs disagree and some chores cost nothing.",
  "abstract": "A computer-assisted candidate proof establishes complete EFX existence for three agents and nine chores with arbitrary nonnegative additive real costs, including deletion of owned zero-cost chores. A written reduction gives a canonical 21-variable counterexample domain; exact input reconstruction, rational Farkas certificates and reverse unit propagation refute it. The package includes a rational-input allocation finder, a general minimum-lowering lemma and a supplementary finite representative bound. It does not settle arbitrary chore counts or validate real-world cost elicitation.",
  "datePublished": "2026-10-08",
  "dateModified": "2026-10-08",
  "version": "1.1.0-candidate",
  "doi": "10.5281/zenodo.23234194",
  "doiUrl": "https://doi.org/10.5281/zenodo.23234194",
  "conceptDoi": "10.5281/zenodo.23234193",
  "pdfUrl": "https://github.com/ipitchford/efx-nine-chores/releases/download/v1.1.0-candidate/efx-nine-chores-v1.1.0-candidate.pdf",
  "altPdfUrl": "https://zenodo.org/records/23234194/files/efx-nine-chores-v1.1.0-candidate.pdf?download=1",
  "altPdfLabel": null,
  "zenodoUrl": "https://zenodo.org/records/23234194",
  "repoUrl": "https://github.com/ipitchford/efx-nine-chores",
  "releaseUrl": "https://github.com/ipitchford/efx-nine-chores/releases/tag/v1.1.0-candidate",
  "markdownUrl": "https://evidencepress.org/releases/efx-nine-chores/index.md",
  "bibtexUrl": "https://evidencepress.org/releases/efx-nine-chores/cite.bib",
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      "description": "Synthetic speech; communication, not additional evidence.",
      "transcriptUrl": "https://evidencepress.org/assets/audio/efx-nine-chores.txt"
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  "authors": [
    "Anonymous"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-candidate",
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    "peerReviewed": false,
    "independentlyReproduced": false,
    "formallyVerified": false,
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    "detail": "Unrefereed candidate: written reduction and internally replayed exact computational certificate; no end-to-end formal verification, established external reproduction or specialist peer review. The supplied review recommended minor revisions and did not identify a mathematical defect. Reviewer identity and independence are not established; prospective star ratings are not publication assurance."
  },
  "assurance": [
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      "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/23234194",
      "note": "Published GitHub/Zenodo assets verified against the frozen local archive; availability is not proof 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": "Internal exact replay and semantic/adversarial controls; not an unaffiliated reproduction.",
      "evidenceUrl": "https://github.com/ipitchford/efx-nine-chores/blob/v1.1.0-candidate/REPLAY_RECEIPT.md"
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    {
      "dimension": "independentRerun",
      "label": "Independent rerun",
      "question": "Has someone else run the supplied implementation and obtained the stated result?",
      "state": "not-assessed",
      "note": "No qualifying evidence establishing this dimension."
    },
    {
      "dimension": "independentReimplementation",
      "label": "Independent reimplementation",
      "question": "Has someone else reached the result from an independent implementation?",
      "state": "not-assessed",
      "note": "No qualifying evidence establishing this dimension."
    },
    {
      "dimension": "formalVerification",
      "label": "Formal verification",
      "question": "Is a formalised statement machine-checked, and over which trusted base?",
      "state": "not-assessed",
      "note": "No qualifying evidence establishing this dimension."
    },
    {
      "dimension": "specialistReview",
      "label": "Specialist review",
      "question": "Has a domain specialist assessed the argument?",
      "state": "not-assessed",
      "note": "No qualifying evidence establishing this dimension."
    },
    {
      "dimension": "editorialPeerReview",
      "label": "Editorial peer review",
      "question": "Has a journal or venue run peer review to a decision?",
      "state": "not-assessed",
      "note": "No qualifying evidence establishing this dimension."
    },
    {
      "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/efx-nine-chores/blob/v1.1.0-candidate/ENVIRONMENT.txt",
      "note": "Standard-library checking, pinned input and full proof objects; Darwin skips the unsupported address-space cap. No container or unaffiliated environment replay."
    }
  ],
  "provenance": {
    "aiGenerated": true,
    "aiAssisted": true,
    "generatedBy": [
      "AI research agents in the supplied package; exact originating model not established",
      "OpenAI Codex assistant: review actions, internal replay and publication"
    ],
    "humanRole": "Publication authority; no human research or verification contribution asserted.",
    "reviewStatus": "Unrefereed candidate: written reduction and internally replayed exact computational certificate; no end-to-end formal verification, established external reproduction or specialist peer review."
  },
  "attributionBoundary": null,
  "problem": {
    "name": "Three-agent EFX existence for nine chores",
    "url": "https://arxiv.org/abs/2609.10585v2"
  },
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  "corrections": [],
  "publicCorrections": [],
  "publicCorrectionIndexes": null,
  "pageStructureVersion": "2.0",
  "pageStructureVariant": "canonical-theorem",
  "pageStructureWaivers": [],
  "recordMaturity": "current",
  "metadataProvenance": "Supplied research and review, revised and internally checked at publication intake on 8 October 2026. Discovery effort and reviewer identity are not reconstructed.",
  "grandfatheredAtSchemaVersion": null,
  "keywords": [
    "fair division",
    "chores",
    "EFX",
    "additive costs",
    "exact certificates",
    "linear real arithmetic"
  ],
  "keyResults": [
    "Theorem 1: complete EFX allocation exists for every nonnegative additive real-cost instance with three agents and nine chores; every owned chore, including zero-cost chores, is covered.",
    "Lemma 3: lowering one global minimum in each row to zero preserves nonexistence for any finite number of agents and chores.",
    "The exact unbounded 21-variable counterexample formula is refuted by 370,780 rational arithmetic axioms and 172,146 RUP deductions.",
    "For rational costs an exhaustive fixed-size finder returns an allocation with all 18 literal deletion comparisons; bit complexity depends on input size."
  ],
  "reviews": [],
  "evidencePackage": "Written reductions and precise source dependencies; exact SMT input; complete selected arithmetic and Boolean proof objects; standard-library checkers and semantic controls; normal and optimized internal replay; exact allocation finder and worked example; manifest and AI index.",
  "openProblems": [
    "An unaffiliated rerun and a separately implemented checker would strengthen assurance.",
    "Formalise the written reduction and input semantics, not only the arithmetic refutation.",
    "Investigate ten chores after a fresh prior-art check; no claim about that case is made here."
  ],
  "mathObjects": [],
  "relatedWorks": [
    {
      "citation": "Zhang (2026), EFX Allocations for Three Agents and Seven or Eight Chores, v2. Theorem 1.2 supplies the eight-chore baseline; Section 6 leaves nine chores open.",
      "url": "https://arxiv.org/abs/2609.10585v2",
      "doi": "10.48550/arXiv.2609.10585"
    },
    {
      "citation": "Kobayashi, Mahara and Sakamoto, EFX Allocations for Indivisible Chores: Matching-Based Approach. Full-preprint Lemma 4.2 supplies shared-cheapest insertion; Theorem 3.1 covers at most twice as many chores as agents.",
      "url": "https://doi.org/10.1016/j.tcs.2024.115010",
      "doi": "10.1016/j.tcs.2024.115010"
    },
    {
      "citation": "He and Tao (2026), EFX for Additive Chores. Nonexistence for every number of agents at least four does not settle the present three-agent case.",
      "url": "https://arxiv.org/abs/2606.08872",
      "doi": "10.48550/arXiv.2606.08872"
    },
    {
      "citation": "Alkassar, Fouz and Mehlhorn (2026), Complete EFX Allocations Exist for Four Additive Agents and Up to Nine Goods. Computational precedent for goods, not this chore theorem.",
      "url": "https://arxiv.org/abs/2608.08590",
      "doi": "10.48550/arXiv.2608.08590"
    },
    {
      "citation": "Akrami et al. (2026), A Counterexample to EFX via SAT-Solving, v3. Section 9 validates an abstract encoding in Lean; it is not end-to-end implementation verification.",
      "url": "https://arxiv.org/abs/2604.18216v3",
      "doi": "10.48550/arXiv.2604.18216"
    }
  ],
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      "state": "explicit",
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      "remainingRisks": [
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        "setting": "Producer-coordinated mathematical publication",
        "status": "NO_IMPACT_EVIDENCE",
        "designClass": "none",
        "comparator": "None.",
        "estimand": "No acceleration or empirical welfare effect estimated.",
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          "scopeBoundary": "Forecast from this valid registration onward, during preparation. Earlier source checks, replay, manuscript and media work are left-censored in this ledger; no discovery-time or total research-effort claim.",
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