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  "slug": "hilbert-eight-points-reducedness",
  "title": "Reducedness of the Hilbert scheme of eight points in affine four-space",
  "shortTitle": "Eight points without hidden nilpotents",
  "url": "https://evidencepress.org/releases/hilbert-eight-points-reducedness/",
  "oneLine": "A computer-assisted proof candidate claims reducedness of the Hilbert scheme of eight points in affine four-space over algebraically closed characteristic-zero fields.",
  "abstract": "This unrefereed candidate argues that Hilb^8(A^4_k) is reduced for every algebraically closed characteristic-zero field k. A global Borel reduction reaches two monomial local rings. Ambient Pfaffian transport separates components, while integral open charts, mixed-prime finite-module comparisons and a regular parameter identify their full associated graded algebras. The complete archive preserves proof modules, four finite bases, native counterparts and direct comparison records covering 55,557,191 terms. Aggregate replay checks saved receipts and dependencies rather than recomputing every large basis. Reducedness does not imply smoothness or irreducibility. Positive characteristic and general length or dimension are excluded.",
  "datePublished": "2026-09-07",
  "dateModified": "2026-09-07",
  "version": "0.1.0-candidate",
  "doi": "10.5281/zenodo.22643443",
  "doiUrl": "https://doi.org/10.5281/zenodo.22643443",
  "conceptDoi": "10.5281/zenodo.22643442",
  "pdfUrl": "https://github.com/ipitchford/hilbert-eight-points-reducedness/releases/download/v0.1.0-candidate/paper.pdf",
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  "repoUrl": "https://github.com/ipitchford/hilbert-eight-points-reducedness",
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  "markdownUrl": "https://evidencepress.org/releases/hilbert-eight-points-reducedness/index.md",
  "bibtexUrl": "https://evidencepress.org/releases/hilbert-eight-points-reducedness/cite.bib",
  "audioUrl": "https://evidencepress.org/assets/audio/hilbert-eight-points-reducedness.mp3?v=2a65a4b3ec",
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      "url": "https://evidencepress.org/assets/audio/hilbert-eight-points-reducedness.mp3",
      "name": "Audio briefing — eight points without hidden nilpotents",
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  "authors": [
    "Anonymous"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-candidate",
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      "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?",
      "state": "passed",
      "note": "Fresh archive aggregate checks and 18 corruption/mode controls; two full planar Pfaffians and a small fresh native fixture. Large native bases are saved hash-bound evidence, not newly recomputed by the aggregate.",
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      "question": "Is a formalised statement machine-checked, and over which trusted base?",
      "state": "not-assessed",
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    },
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      "question": "Has a domain specialist assessed the argument?",
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      "note": "Producer-coordinated checks and internal AI review do not establish this dimension."
    },
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    },
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      "dimension": "dataEnvironmentReproducibility",
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      "state": "partial",
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      "note": "Tool versions, original inputs and fresh native recipe supplied; no hermetic or unaffiliated full-environment reproduction."
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  "provenance": {
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      "Python, SymPy and FLINT",
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    "humanRole": "Problem selection, research direction, supplied review and publication authority; scholarly creator remains Anonymous.",
    "disclosure": "AI-assisted mathematics, computation, exposition, internal review and publishing. Coordinated AI review is not independent external assurance."
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  "problem": {
    "name": "AIM-ARITHMETIC_GEOMETRY-0071; AIM Hilbert schemes Problem 24",
    "url": "https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf"
  },
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  "metadataProvenance": "Research assurance assessed 2026-09-07T12:10:22.675Z separately from subsequent site publication. Internal AI editorial review is not external assurance.",
  "grandfatheredAtSchemaVersion": null,
  "keywords": [
    "Hilbert schemes",
    "reducedness",
    "commutative algebra",
    "Groebner bases",
    "computer-assisted proof",
    "unrefereed candidate"
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  "keyResults": [
    "Claimed reducedness of Hilb^8(A^4) over algebraically closed characteristic-zero fields.",
    "Global reduction to B15 and B16 with source-specific component separation.",
    "Rank-1907 and rank-9923 finite-module bridges to all-degree local arguments.",
    "Four native finite-basis comparisons and explicit replay levels; finite computations do not replace the universal written argument."
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  "evidencePackage": "Complete archive with exact ring presentations, proof modules, finite bases, native counterparts, direct comparison receipts, 18 wrapper controls, two symbolic planar-arc checks and five producer-coordinated internal editorial reports. Small Git checkout excludes large evidence; use the full ZIP for aggregate replay. Internal disposition: https://github.com/ipitchford/hilbert-eight-points-reducedness/blob/v0.1.0-candidate/EDITORIAL_DECISION.md",
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    "Unaffiliated scrutiny of ambient Pfaffian transport and B15 purity/open coverage.",
    "Independent reconstruction of source encodings and full native computations.",
    "Investigate positive characteristic or other lengths only as new problems with their hypotheses checked anew."
  ],
  "mathObjects": [],
  "relatedWorks": [
    {
      "citation": "Cartwright, Erman, Velasco and Viray (2009). Hilbert schemes of 8 points. Component and Pfaffian antecedents.",
      "url": "https://doi.org/10.2140/ant.2009.3.763",
      "doi": "10.2140/ant.2009.3.763"
    },
    {
      "citation": "Hu (2025). On singular Hilbert schemes of points: Local structures and tautological sheaves. Corrected reduction and local-structure methods.",
      "url": "https://doi.org/10.46298/epiga.2025.12827",
      "doi": "10.46298/epiga.2025.12827"
    },
    {
      "citation": "Kustin. Generic matrix-vector Northcott perfectness input, introduction (0.1)–(0.2); see the citation audit for the precise application boundary.",
      "url": "https://people.math.sc.edu/kustin/papers/compl.pdf",
      "doi": null
    },
    {
      "citation": "The Stacks Project, Tag 00R4. Miracle flatness.",
      "url": "https://stacks.math.columbia.edu/tag/00R4",
      "doi": null
    }
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      "remainingRisks": [
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