{
  "schemaVersion": "1.1",
  "slug": "exotic-affine-three-spheres",
  "title": "Exotic affine three-spheres and the quadratic–cubic obstruction",
  "shortTitle": "Exotic affine three-spheres",
  "url": "https://evidence-press.pages.dev/releases/exotic-affine-three-spheres/",
  "oneLine": "A natural slice of polynomial factorisation space is claimed to be an exotic affine three-sphere — a shape that mimics a familiar one without being it — while every quadratic–cubic slice is obstructed from being ordinary affine space.",
  "abstract": "Some algebraic shapes impersonate familiar ones: they are indistinguishable by the tools of smooth topology yet are provably different as algebraic varieties. This paper argues that a natural slice of the space of polynomial factorisations is exactly such an impostor — a known exotic affine three-sphere (the Dubouloz–Finston torsor) rather than the standard SL2 quadric it resembles. It further claims that every normalised quadratic–cubic slice fails to be affine five-space, by an exact Grothendieck-class formula and a rank-by-rank Hodge–Deligne obstruction, and that in characteristic 3 the tangent slice remains affine three-space with an Artin–Schreier collision in the induced Keller map.",
  "datePublished": "2026-07-28",
  "dateModified": "2026-07-28",
  "version": "0.1-candidate (v0.1.1 'citation robustness' archived as 10.5281/zenodo.21653108)",
  "doi": "10.5281/zenodo.21647607",
  "doiUrl": "https://doi.org/10.5281/zenodo.21647607",
  "conceptDoi": "10.5281/zenodo.21647606",
  "pdfUrl": "https://github.com/ipitchford/exotic-affine-spheres-quadratic-cubic/releases/download/v0.1.1/paper.pdf",
  "altPdfUrl": "https://raw.githubusercontent.com/ipitchford/exotic-affine-spheres-quadratic-cubic/main/paper.pdf",
  "zenodoUrl": "https://zenodo.org/records/21647607",
  "repoUrl": "https://github.com/ipitchford/exotic-affine-spheres-quadratic-cubic",
  "releaseUrl": "https://github.com/ipitchford/exotic-affine-spheres-quadratic-cubic/releases",
  "markdownUrl": "https://evidence-press.pages.dev/releases/exotic-affine-three-spheres/index.md",
  "bibtexUrl": "https://evidence-press.pages.dev/releases/exotic-affine-three-spheres/cite.bib",
  "audioUrl": "https://evidence-press.pages.dev/assets/audio/exotic-affine-three-spheres.mp3",
  "imageUrl": "https://evidence-press.pages.dev/assets/og/exotic-affine-three-spheres.png",
  "coverArtUrl": "https://evidence-press.pages.dev/assets/art/exotic-affine-three-spheres.svg",
  "media": [],
  "authors": [
    "OpenAI Codex / Anthropic models"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-candidate",
  "verification": {
    "peerReviewed": false,
    "independentlyReproduced": false,
    "formallyVerified": false,
    "internallyReplayed": true,
    "detail": "Unrefereed candidate manuscript. No peer review, no independent external reproduction, no complete external human review, and no exhaustive novelty check are documented. The symbolic and finite-field checks cover selected formulae only; the Grothendieck-class and Hodge–Deligne arguments themselves are not machine-verified. The repository's latest revision (v0.1.1) fixes citation routing after a cache incident, with manuscript content unchanged. AI-attributed with human direction."
  },
  "provenance": {
    "aiGenerated": true,
    "generatedBy": [
      "OpenAI Codex / Anthropic models"
    ],
    "humanRole": "problem selection, mediation, and publication management",
    "disclosure": "The mathematics/research in this release was generated by AI systems as credited; see the Zenodo record for full attribution."
  },
  "problem": {
    "name": "Recognition of exotic affine spheres (Dubouloz–Finston)",
    "url": "https://arxiv.org/abs/1106.2900"
  },
  "keywords": [
    "algebraic geometry",
    "exotic affine sphere",
    "Dubouloz–Finston torsor",
    "binary forms",
    "Grothendieck ring of varieties",
    "Hodge–Deligne polynomial",
    "Keller map",
    "Artin–Schreier",
    "Jacobian conjecture",
    "finite-field verification",
    "computer-assisted mathematics"
  ],
  "keyResults": [
    "The transverse linear–quadratic slice is claimed to be a specific exotic affine three-sphere (the Dubouloz–Finston torsor) rather than SL2.",
    "Every normalised quadratic–cubic slice fails to be affine five-space, via an exact Grothendieck-class formula and a rank-by-rank Hodge–Deligne obstruction.",
    "In characteristic 3, the tangent slice remains affine three-space, with an Artin–Schreier collision in the induced Keller map."
  ],
  "evidencePackage": "Manuscript (PDF and TeX); verify_paper.py running exact symbolic checks plus finite-field censuses over F2 through F11 covering 23,941 projective functionals, with byte-for-byte validation of generated tables against checked-in versions and a deliberate semantic control; claim-to-evidence mapping (AI_INDEX.md); integrity documentation (ASSURANCE.md); SHA-256-manifested Zenodo deposit.",
  "openProblems": [
    "Have the torsor identification checked by specialists in affine algebraic geometry — in particular, experts on Dubouloz–Finston exotic spheres — since this is the manuscript's most consequential claim.",
    "Verify the Grothendieck-class formula and the rank-by-rank Hodge–Deligne obstruction by hand; the shipped checks validate consequences, not the arguments.",
    "Rerun the finite-field censuses (F2 through F11, 23,941 projective functionals) from an independent implementation.",
    "Search the literature for prior art on slices of binary-form factorisation spaces to establish novelty.",
    "Investigate whether the characteristic-3 Artin–Schreier phenomenon has characteristic-p analogues for other small primes."
  ],
  "relatedWorks": [
    {
      "citation": "Dubouloz, A., & Finston, D. On exotic affine 3-spheres.",
      "url": "https://arxiv.org/abs/1106.2900"
    },
    {
      "citation": "Dubouloz, A., & Finston, D. Families of exotic affine 3-spheres.",
      "url": "https://hal.science/hal-01374364"
    },
    {
      "citation": "Foundational companion release: The degree-difference principle and affine slices of binary-form factorisation spaces.",
      "url": "https://doi.org/10.5281/zenodo.21647593"
    },
    {
      "citation": "Continuation: Reducible incidence divisors and the isolation of affine slices.",
      "url": "https://doi.org/10.5281/zenodo.21647616"
    }
  ]
}