{
  "schemaVersion": "1.1",
  "slug": "reducible-incidence-divisors",
  "title": "Reducible incidence divisors and the isolation of affine slices in binary-form factorisation spaces",
  "shortTitle": "Reducible incidence divisors",
  "url": "https://evidence-press.pages.dev/releases/reducible-incidence-divisors/",
  "oneLine": "A classification of exactly when a natural family of divisors breaks into pieces — and a screening theorem narrowing where Jacobian-conjecture-style behaviour could possibly live.",
  "abstract": "This paper classifies the reducible members of a marked-common-root incidence linear system for binary forms: for adjacent degrees, a divisor becomes reducible precisely when its defining functional lies on the tangent developable of the rational normal curve of evaluation functionals. It proves that the corresponding affine slices are never isomorphic to affine space, computes their precise motivic and Hodge-theoretic defects, obstructs non-adjacent-degree slices via finite cyclic actions, and — conditionally on an upstream classification — isolates a unique slice that could in principle carry a nonzero-constant-Jacobian polynomial map.",
  "datePublished": "2026-07-28",
  "dateModified": "2026-07-28",
  "version": "1.0-candidate (v1.0.1 archived as 10.5281/zenodo.21653119)",
  "doi": "10.5281/zenodo.21647616",
  "doiUrl": "https://doi.org/10.5281/zenodo.21647616",
  "conceptDoi": null,
  "pdfUrl": "https://github.com/ipitchford/reducible-incidence-divisors-affine-slices/releases/download/v1.0.1/paper.pdf",
  "altPdfUrl": "https://raw.githubusercontent.com/ipitchford/reducible-incidence-divisors-affine-slices/main/paper.pdf",
  "zenodoUrl": "https://zenodo.org/records/21647616",
  "repoUrl": "https://github.com/ipitchford/reducible-incidence-divisors-affine-slices",
  "releaseUrl": "https://github.com/ipitchford/reducible-incidence-divisors-affine-slices/releases",
  "markdownUrl": "https://evidence-press.pages.dev/releases/reducible-incidence-divisors/index.md",
  "bibtexUrl": "https://evidence-press.pages.dev/releases/reducible-incidence-divisors/cite.bib",
  "audioUrl": "https://evidence-press.pages.dev/assets/audio/reducible-incidence-divisors.mp3",
  "imageUrl": "https://evidence-press.pages.dev/assets/og/reducible-incidence-divisors.png",
  "coverArtUrl": "https://evidence-press.pages.dev/assets/art/reducible-incidence-divisors.svg",
  "media": [],
  "authors": [
    "OpenAI Codex 5.6 Sol",
    "Anthropic Fable 5"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-candidate",
  "verification": {
    "peerReviewed": false,
    "independentlyReproduced": false,
    "formallyVerified": false,
    "internallyReplayed": true,
    "detail": "Unrefereed candidate manuscript. The release states it is not independent external reproduction, expert human review, peer review, proof-assistant formalisation, or a literature-wide novelty determination; its internal review disposition was 'minor revision for candidate publication'. The isolation theorem is conditional on an unverified upstream cubic classification from a companion release. Authored by AI systems with human project mediation."
  },
  "provenance": {
    "aiGenerated": true,
    "generatedBy": [
      "OpenAI Codex 5.6 Sol",
      "Anthropic Fable 5"
    ],
    "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": "Structure of incidence divisors in factorisation spaces (screening for the Jacobian conjecture)",
    "url": "https://en.wikipedia.org/wiki/Jacobian_conjecture"
  },
  "keywords": [
    "algebraic geometry",
    "binary forms",
    "incidence divisor",
    "rational normal curve",
    "tangent developable",
    "catalecticant",
    "Hodge–Deligne polynomial",
    "Jacobian conjecture",
    "affine slices",
    "computer-assisted mathematics"
  ],
  "keyResults": [
    "Classification: for consecutive degrees (m, m+1) with m ≥ 2, the divisor D_ℓ = {ℓ(P²A′B′) = 0} is reducible precisely when [ℓ] lies on the tangent developable of the rational normal curve of evaluation functionals.",
    "Component counts: rank-one functionals yield three reduced components; first-jet functionals give two; genuine two-point secants and higher catalecticant ranks give irreducible divisors.",
    "For all m ≥ 2 and nonzero ℓ, the slice X_ℓ^{m,m+1} is not isomorphic to affine space A^{2m+1}.",
    "Precise defects: rank-one has Grothendieck class L^{2m+1} − L^{2m}; rank-two secants and first-jets show diagnostic Hodge coefficients −2 and −1.",
    "Non-adjacent degrees (|r − s| ≥ 2) give non-contractible slices via finite cyclic group actions.",
    "Conditional on an upstream cubic classification: the tangent nonosculating linear–quadratic slice is the unique positive-bidegree affine source permitting a nonzero constant Jacobian determinant."
  ],
  "evidencePackage": "Manuscript (PDF and TeX); deterministic SymPy audit scripts (verify_paper.py, verify_rank_two.py); claim-to-evidence mapping (AI_INDEX.md/.json); assurance-boundary documents (STATUS.md, ASSURANCE.md); SHA-256 manifest; immutable source snapshots and citation gates; archived on Zenodo as v1.0-candidate with a v1.0.1 revision.",
  "openProblems": [
    "Verify the tangent-developable classification by hand — the statement is precise, classical in flavour, and a natural target for expert review or refutation.",
    "Discharge or refute the conditional hypothesis: the isolation theorem depends on an upstream cubic classification that has itself had no external verification.",
    "Check novelty against the classical literature on catalecticants, secant varieties, and tangent developables, where fragments of the classification may already be known.",
    "Extend the motivic-defect calculations to non-adjacent degree pairs beyond the cyclic-action obstruction.",
    "Determine whether the unique surviving slice actually carries a nonzero-constant-Jacobian map — either constructing one or excluding it, which would close this branch of the screening programme."
  ],
  "relatedWorks": [
    {
      "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": "Continues: Exotic affine three-spheres and the quadratic–cubic obstruction (upstream cubic classification).",
      "url": "https://doi.org/10.5281/zenodo.21647607"
    },
    {
      "citation": "Keller, O.-H. (1939) — the Jacobian conjecture, which remains open; this release proves screening results only, not the conjecture.",
      "url": "https://en.wikipedia.org/wiki/Jacobian_conjecture"
    }
  ]
}