{
  "schema_version": 1,
  "problem_number": "AIM-TOPOLOGY-0276",
  "title": "A Genus-Two Algebraic Family with Vanishing Real Toledo Numbers and Nonzero Generic Witt Class",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We give four explicit matrices defining a genus-two surface-group representation over R[s,t,1/(st(1+t^2))], where R denotes the real numbers. Every real specialization has zero Witt class and zero Toledo number. Over R(s,t), however, the Witt class is the anisotropic Pfister form <1,-(1+t^2)> tensor <1,-s>, of additive order two. A nonzero second residue and a trace pole detect bad reduction at the nonreal divisor 1+t^2=0. The calculation shows concretely that the real Toledo function does not determine the generic Witt-valued invariant. General realization results of Dymara and Januszkiewicz already imply existence of such examples; this note provides a compact matrix presentation and its direct verification. It is not a solution of the full open-ended AIM programme, and no priority claim is made.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.KT",
    "math.GT",
    "math.GR"
  ],
  "keywords": [
    "Witt group",
    "Toledo invariant",
    "surface group representation",
    "Pfister form",
    "real specialization",
    "algebraic family",
    "bad reduction",
    "AIM-TOPOLOGY-0276",
    "math.KT",
    "math.GT",
    "math.GR"
  ],
  "manuscript_version_date": "2026-10-04",
  "publication_date": "2026-10-04",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-04",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-topology-0276/",
  "pdf_url": "https://eulersolve.org/papers/aim-topology-0276/paper.pdf?v=a173076b91b9",
  "doi": "10.5281/zenodo.23129499",
  "zenodo_record_url": "https://zenodo.org/records/23129499",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete proof of the displayed explicit genus-two family, including anisotropy, exact additive order two, vanishing real specializations and a trace obstruction to integral conjugacy. General existence is already implied by Dymara and Januszkiewicz, Theorem 11.6; this is not a new existence or range theorem. The full open-ended AIM-TOPOLOGY-0276 programme remains unresolved and is not counted as a source-problem closure. Novelty of the compact presentation is undetermined; no absolute-priority claim. AI-assisted, self-audited and unrefereed; no independent review or proof-assistant verification is claimed.",
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      "sha256": "a173076b91b9d1752584b6802e787fb7a72bbb0e3c00bd780c1088ac9aecf581"
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    "source.zip": {
      "sha256": "cc913952ccb382283d4e44f4ca779935c0d92f1301e96c496d2b978968fc95b7"
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    "verification_report.md": {
      "sha256": "83156d54bb1137ad7687364897d64057ace5e3ec01a0fdb3dec09a034f5e3b49"
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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."
}