{
  "schema_version": 1,
  "problem_number": "AMR-030-0030",
  "title": "Klein Quartics as Eigenschemes of the Fano Adjacency Tensor",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For the Cooper-Dutle normalized adjacency tensor of the Fano plane, we explicitly identify the entire projective eigenscheme at each root of lambda^2+lambda+2 with the bicanonical Klein quartic. A Fourier transformation turns the seven incidence equations into an adjugate equation and a quadric. A displayed integer identity excludes all off-hyperplane components, while a local adjugate argument removes infinitesimal thickening. The proof gives an explicit isomorphism over Q(zeta_7), including scheme structure, and a portable exact rational checker. The earlier anonymous preprint DOI 10.5281/zenodo.22850084 already reported degree-eight genus-three eigencurves and a full computational spectrum; those assertions are credited. The contribution here is their Klein-quartic identification and direct certificate-based proof. The full characteristic polynomial and resultant multiplicities are not recomputed, and AMR-030-0030 is not counted as a newly closed whole-spectrum question. AI-assisted, self-audited preprint; no independent review, formal verification or absolute-priority certification is claimed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO",
    "math.AG"
  ],
  "keywords": [
    "AMR-030-0030",
    "Fano plane",
    "Klein quartic",
    "tensor eigenvalues",
    "projective eigenscheme",
    "Veronese embedding",
    "exact polynomial certificate"
  ],
  "manuscript_version_date": "2026-10-25",
  "publication_date": "2026-10-25",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-25",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-030-0030/",
  "pdf_url": "https://eulersolve.org/papers/amr-030-0030/paper.pdf?v=dc53c55a1d0c",
  "doi": "10.5281/zenodo.23645516",
  "zenodo_record_url": "https://zenodo.org/records/23645516",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For the Cooper--Dutle normalized adjacency tensor of the cyclic Fano plane, the complete projective eigenscheme at each root of lambda^2+lambda+2 is explicitly isomorphic over Q(zeta_7) to the bicanonical Klein quartic a^3 c+a b^3+b c^3=0. Fourier coordinates, a degree-six polynomial certificate and local adjugate identities prove support and scheme structure. No full spectrum or resultant multiplicity is asserted. This explicitly identifies the two projective eigenschemes at the roots of lambda^2+lambda+2 as bicanonical Klein quartics. Earlier degree-eight genus-three assertions and the reported full spectrum are credited to Anonymous2026, DOI10.5281/zenodo.22850084. The full spectrum and resultant multiplicities are not recomputed; no new whole-source closure is claimed. 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": "dc53c55a1d0c1ccbd63a7bb814fde191ac6d9b8773879fd4727e86dce2223198"
    },
    "source.zip": {
      "sha256": "9f5639db993d119b6fd0e45b9eec633ce6b7206380545d984c80d5fa7ca02aba"
    },
    "verification_report.md": {
      "sha256": "7b53ab9b3ed7de317c334f4eed9f228527c1b3f4528616aad707fa1c146eae88"
    }
  },
  "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."
}