{
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  "slug": "moment-ideals-unbounded-generator-degrees",
  "title": "Unbounded generator degrees in moment ideals via graph faces",
  "shortTitle": "Moment equations have no uniform degree bound",
  "url": "https://evidencepress.org/releases/moment-ideals-unbounded-generator-degrees/",
  "oneLine": "A constructive proof candidate embeds graph faces into full moment models, forcing equations of arbitrarily large necessary degree and producing an intrinsically nonnormal example.",
  "abstract": "For each r at least three, the full moment ideal with n=r squared and d=((r+1)^(r squared−2r+2)−1)/r cannot be generated by polynomials of degree below r. The proof packs binary edge-selection constraints into moment levels and uses an exposed face to exclude shortcuts through other coordinates. It extends to graph toric ideals and an intrinsically nonnormal moment variety. Exact finite checks support the self-contained argument; both model indices vary.",
  "datePublished": "2026-09-09",
  "dateModified": "2026-09-09",
  "version": "0.1.1-candidate",
  "doi": "10.5281/zenodo.22680540",
  "doiUrl": "https://doi.org/10.5281/zenodo.22680540",
  "conceptDoi": "10.5281/zenodo.22680539",
  "pdfUrl": "https://github.com/ipitchford/moment-ideals-unbounded-generator-degrees/releases/download/v0.1.1-candidate/moment-ideals-unbounded-generator-degrees-0.1.1-candidate.pdf",
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  "altPdfLabel": null,
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  "repoUrl": "https://github.com/ipitchford/moment-ideals-unbounded-generator-degrees",
  "releaseUrl": "https://github.com/ipitchford/moment-ideals-unbounded-generator-degrees/releases/tag/v0.1.1-candidate",
  "markdownUrl": "https://evidencepress.org/releases/moment-ideals-unbounded-generator-degrees/index.md",
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  "authors": [
    "Anonymous"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-candidate",
  "verification": {
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    "detail": "Unrefereed candidate with a complete written argument as both n and d vary. Finite replay is supporting evidence, not universal certification. Fixed-n unboundedness, optimal parameters, historical priority, external review, formal verification and impact are not established."
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      "dimension": "availability",
      "label": "Availability and archiving",
      "question": "Is the evidence package publicly retrievable from an archive under a persistent identifier?",
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      "evidenceUrl": "https://zenodo.org/records/22680540",
      "note": "Public archive availability and exact bytes checked before deployment."
    },
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      "dimension": "internalReplay",
      "label": "Internal replay",
      "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?",
      "state": "passed",
      "note": "Finite r=3..8 and separate r=3,4,5 implementation, thirteen controls; not universal proof certification.",
      "evidenceUrl": "https://github.com/ipitchford/moment-ideals-unbounded-generator-degrees/blob/v0.1.1-candidate/REPLAY_RECEIPT.md"
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      "label": "Independent rerun",
      "question": "Has someone else run the supplied implementation and obtained the stated result?",
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      "note": "Internal producer-coordinated roles and supplied reported calculations do not establish this external assurance dimension."
    },
    {
      "dimension": "independentReimplementation",
      "label": "Independent reimplementation",
      "question": "Has someone else reached the result from an independent implementation?",
      "state": "not-assessed",
      "note": "Internal producer-coordinated roles and supplied reported calculations do not establish this external assurance dimension."
    },
    {
      "dimension": "formalVerification",
      "label": "Formal verification",
      "question": "Is a formalised statement machine-checked, and over which trusted base?",
      "state": "not-assessed",
      "note": "Internal producer-coordinated roles and supplied reported calculations do not establish this external assurance dimension."
    },
    {
      "dimension": "specialistReview",
      "label": "Specialist review",
      "question": "Has a domain specialist assessed the argument?",
      "state": "not-assessed",
      "note": "Internal producer-coordinated roles and supplied reported calculations do not establish this external assurance dimension."
    },
    {
      "dimension": "editorialPeerReview",
      "label": "Editorial peer review",
      "question": "Has a journal or venue run peer review to a decision?",
      "state": "not-assessed",
      "note": "Internal producer-coordinated roles and supplied reported calculations do not establish this external assurance 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/moment-ideals-unbounded-generator-degrees/blob/v0.1.1-candidate/ENVIRONMENT.txt",
      "note": "Standard-library exact checks and Linux CI; PDF environment recorded but not hermetically pinned."
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  "provenance": {
    "aiGenerated": true,
    "aiAssisted": true,
    "generatedBy": [
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    "humanRole": "Selection, research direction, supplied review and publication authorization; scholarly creator Anonymous.",
    "disclosure": "AI-assisted proof development and internal editorial review within one coordinated workflow; no unaffiliated whole-proof validation inferred."
  },
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  "problem": {
    "name": "Alexandr–Kileel–Sturmfels full-moment generator-degree conjecture",
    "url": "https://arxiv.org/abs/2301.09068"
  },
  "orderedProductAverage": null,
  "corrections": [],
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  "pageStructureVariant": "reader-first-reference",
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  "recordMaturity": "current",
  "metadataProvenance": "Internal assurance assessment is separate from public readback; publication does not upgrade mathematical assurance.",
  "grandfatheredAtSchemaVersion": null,
  "keywords": [
    "moment varieties",
    "toric ideals",
    "generator degrees",
    "graph faces",
    "nonnormality"
  ],
  "keyResults": [
    "An explicit necessary degree-r obstruction for every r at least three, with both n and d varying.",
    "Every finite simple graph with an edge embeds as an exposed moment face preserving its graded toric ideal.",
    "Nonempty binary equality configurations admit the specified complement-variable realization.",
    "An explicit moment variety is intrinsically nonnormal, not merely non-projectively-normal."
  ],
  "reviews": [],
  "evidencePackage": "Nine-page self-contained proof; two exact finite implementations; thirteen semantic controls; full normal/optimized certificate comparison; non-destructive replay regression; fresh-archive checks; internal editorial reports and bounded novelty audit.",
  "openProblems": [
    "Reduce the very large encoded moment order and position count.",
    "Determine whether necessary generator degrees are unbounded for a fixed number of variables.",
    "Identify further graph-ring properties preserved by the moment-face realization.",
    "Formalize the carry-free encoding, face retraction and full-fiber argument."
  ],
  "mathObjects": [],
  "relatedWorks": [
    {
      "citation": "Alexandr, Kileel and Sturmfels (2023), Moment Varieties for Mixtures of Products: originating full-model conjecture.",
      "url": "https://doi.org/10.1145/3597066.3597084",
      "doi": "10.1145/3597066.3597084"
    },
    {
      "citation": "Aoki, Takemura and Yoshida (2008), Indispensable monomials of toric ideals and Markov bases.",
      "url": "https://doi.org/10.1016/j.jsc.2007.07.012",
      "doi": "10.1016/j.jsc.2007.07.012"
    },
    {
      "citation": "Avis and Tiwary (2013), On the extension complexity of combinatorial polytopes: digit-packing precedent.",
      "url": "https://arxiv.org/abs/1302.2340",
      "doi": "10.48550/arXiv.1302.2340"
    },
    {
      "citation": "Petrović, Thoma and Vladoiu (2019), Hypergraph encodings of arbitrary toric ideals: a distinct universal host.",
      "url": "https://doi.org/10.1016/j.jcta.2019.02.017",
      "doi": "10.1016/j.jcta.2019.02.017"
    },
    {
      "citation": "Mori and Ohsugi (2026), Simplex faces and quadratic toric ideals of lattice polytopes, Lemma 2.2(iv).",
      "url": "https://arxiv.org/abs/2606.20430v3",
      "doi": "10.48550/arXiv.2606.20430"
    },
    {
      "citation": "Ohsugi and Hibi (1998), Normal Polytopes Arising from Finite Graphs: classical odd-cycle obstruction.",
      "url": "https://doi.org/10.1006/jabr.1998.7476",
      "doi": "10.1006/jabr.1998.7476"
    }
  ],
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        "Large encoded orders limit direct computational use."
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