{
  "schema_version": 1,
  "problem_number": "OWR-12697711-006",
  "title": "Irreducibility of the Exposed-Point Variety of a Discotope",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "A discotope is a finite Minkowski sum of linear images of Euclidean unit balls. We prove that, when every summand has intrinsic dimension at least two, the complex Zariski closure E of its exposed points is irreducible, without a genericity assumption. The proof uses the real-analytic support-point map on the connected complement of finitely many subspaces of codimension at least two. Under an explicit generic maximal-rank condition on all summand spans, every boundary point expressible as a sum of summand-boundary points is a limit of exposed points; hence the purely nonlinear part S equals E and is irreducible. This proves Gesmundo-Meroni Conjecture 8.2 (The Geometry of Discotopes, 2022) and Conjecture 1 in Meroni's contribution to Oberwolfach Report 15/2023, exactly frozen UnsolvedMath record OWR-12697711-006. Segment summands are excluded, and S=E may fail for nongeneric representations. The entire complex critical locus, birationality, degree, zonoid recognition and the adjacent rank-one convexity question are not resolved here. This five-page English preprint credits the existing support-face description and standard analytic inputs. It was developed with AI assistance and self-audited; 255 exact regression checks supplement the written proof. It is unrefereed, without independent review, formal proof-assistant verification or an absolute-priority claim.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.MG",
    "math.AG"
  ],
  "keywords": [
    "OWR-12697711-006",
    "discotopes",
    "irreducibility",
    "support functions",
    "Minkowski sums"
  ],
  "manuscript_version_date": "2026-10-27",
  "publication_date": "2026-10-27",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-27",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-12697711-006/",
  "pdf_url": "https://eulersolve.org/papers/owr-12697711-006/paper.pdf?v=6818f70f9665",
  "doi": "10.5281/zenodo.23689704",
  "zenodo_record_url": "https://zenodo.org/records/23689704",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For every finite Minkowski sum of real ellipsoidal discs of intrinsic dimension at least two, the complex Zariski closure E of exposed points is irreducible without genericity. Under the explicit generic maximal-rank condition on every subset of summand spans, S equals E and is irreducible. This proves OWR15/2023 Conjecture1 and Gesmundo-Meroni Conjecture8.2. All summands have intrinsic dimension at least two. The exposed-point variety E is irreducible without genericity; S=E and irreducibility of the purely nonlinear part require the stated generic maximal-rank condition. Segment summands and nongeneric S=E are excluded. No claim is made about the entire complex critical locus, degree, birationality, zonoid recognition or adjacent rank-one convexity questions. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
  "files": {
    "paper.pdf": {
      "sha256": "6818f70f96656911a70ee126e721279aef46e3aacb40fdb74bfa723a3ec8dab9"
    },
    "source.zip": {
      "sha256": "2576b87dfbbae119fa310fd59b62dda616502196c0ed75751f94ca0ad47d6f41"
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    "verification_report.md": {
      "sha256": "d82545987a2dd5faac2a8ff2c88546ff3b50a0c27439a31aabe6227e9568f778"
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  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}