{
  "schema_version": 1,
  "problem_number": "AIM-ANALYSIS-0164",
  "title": "A Diagonal-Jet Criterion for Normalized Gaussian Analytic Covariances",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let R be a twice continuously differentiable, positive semidefinite complex kernel with unit diagonal on a connected plane domain. We give a necessary and sufficient condition for R to be the normalized covariance of a centered proper Gaussian analytic function with positive variance everywhere. A nonnegative residual formed from diagonal derivatives must vanish, and a real one-form determined by the first diagonal derivative must be exact. On a simply connected domain the second condition is a local closedness test. A sharp Gram inequality propagates the diagonal residual condition without division by any off-diagonal kernel value. The covariance is reconstructed up to a positive constant. A polynomial example has a zero in every base section, and an annular example separates local conditions from the global period obstruction. This elementary note concerns the full complex normalized covariance, not arbitrary zero-process correlations. It uses classical positive-kernel and Gaussian-series methods and makes no absolute priority claim.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.PR",
    "math.CV"
  ],
  "keywords": [
    "Gaussian analytic function",
    "normalized covariance",
    "positive semidefinite kernel",
    "diagonal derivatives",
    "period obstruction",
    "AIM-ANALYSIS-0164",
    "math.PR",
    "math.CV"
  ],
  "manuscript_version_date": "2026-09-29",
  "publication_date": "2026-09-29",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-29",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-analysis-0164/",
  "pdf_url": "https://eulersolve.org/papers/aim-analysis-0164/paper.pdf?v=5dbfd0556a57",
  "doi": "10.5281/zenodo.23031963",
  "zenodo_record_url": "https://zenodo.org/records/23031963",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete theorem for the full complex normalized covariance of a centered proper Gaussian analytic function, with positive variance everywhere and without a zero-free base-section assumption. Includes reconstruction and the global period obstruction. The meaning of correlation is unspecified in AIM-ANALYSIS-0164; neither the entire ambiguous source record nor arbitrary zero-process correlations are claimed resolved. Classical positive-kernel and Gaussian-series methods are credited. Self-audited and unrefereed, with AI assistance; no independent peer review, formal verification or absolute priority is claimed.",
  "files": {
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      "sha256": "5dbfd0556a576e34e05bb7f9c2b3da0fc5cc714aa49add3df3681173bcb9ec1a"
    },
    "source.zip": {
      "sha256": "9b3533200bb9ff3cee757eac658f64bc546e6ae1fc86684120b0e8217ede4467"
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    "verification_report.md": {
      "sha256": "da540aede28a215e61a0d89c67b9c23569687684871d5b4bf5bfb0969c7bfd5f"
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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."
}