{
  "schema_version": 1,
  "problem_number": "AIM-GEOMETRY-0388",
  "title": "Paneitz Resonances and Index Growth on a Sphere Times a Hyperbolic Surface",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For a fixed closed connected hyperbolic surface (Sigma,h) of curvature -1, we determine the full Paneitz spectrum for g_t = g_S2 plus t^(-1)h, where the round sphere has curvature 1 and t>0. The exceptional scales with nonconstant kernel form a locally finite set, with explicit resonances of unbounded nullity. An exact finite weighted count gives the negative index. As t tends to infinity, this index equals 2t times the sum of m_nu sqrt(4/9-2nu) over surface eigenvalues nu<2/9, with an O_h(sqrt(t)) remainder. Eigenvalues at the threshold contribute only through explicit narrow scale intervals. We also obtain square-root nullity bounds and infinitely many conformal classes distinguished by their negative indices. This is a complete result for the specified product family, not a classification on arbitrary manifolds. The Singer-Eastwood equal-scale example and established conformal-index theory are credited. The preprint is AI-assisted, self-audited and unrefereed; historical priority is not certified.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.DG",
    "math.SP"
  ],
  "keywords": [
    "AIM-GEOMETRY-0388",
    "Paneitz operator",
    "product manifold",
    "hyperbolic surface",
    "spectral resonance",
    "negative index",
    "conformal geometry"
  ],
  "manuscript_version_date": "2026-10-16",
  "publication_date": "2026-10-16",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-16",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-geometry-0388/",
  "pdf_url": "https://eulersolve.org/papers/aim-geometry-0388/paper.pdf?v=9ef496da15f0",
  "doi": "10.5281/zenodo.23410174",
  "zenodo_record_url": "https://zenodo.org/records/23410174",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete all-scale theorem for a round sphere times a fixed closed connected hyperbolic surface: spectrum, kernel resonances, exact negative index, index asymptotic and square-root nullity bounds. The unrestricted AIM Problem 28 remains unresolved. The equal-scale Singer-Eastwood example and conformal-index theory are credited. This is an AI-assisted, self-audited, unrefereed preprint, not independently or formally verified; historical priority is not certified.",
  "files": {
    "paper.pdf": {
      "sha256": "9ef496da15f0b414d44ca15370d1da3356e188d7f433daf2c46ba1fca85f2cac"
    },
    "source.zip": {
      "sha256": "819d68ecea9dd8a80f01882e7d17c0506a28a8e165df880af5a682d3ee77de3d"
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    "verification_report.md": {
      "sha256": "03bc39e3ebfd7bceb9a000ccea5d437584f2c3095bf12f8babeec30388f0d072"
    }
  },
  "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."
}