{
  "schema_version": 1,
  "problem_number": "AIM-LINEAR_ALGEBRA-0010",
  "title": "Exact Vanishing Orders for Exponentially Deformed Toeplitz Minors",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let H(z) be a finite product of positive linear numerator factors divided by positive geometric denominator factors, and let T_t be the lower triangular Toeplitz matrix generated by exp(tz)H(z). For every structurally admissible minor, we give an explicit index-sum formula for its order of vanishing at t=0. Every coefficient from that order to the total index displacement is strictly positive. A unique maximal intermediate index tuple gives the leading coefficient as a product of two positive determinants. The proof uses elementary factor supports, interlacing and the Cauchy-Binet identity. The classical zero-pattern classification and qualitative exponential strictification are explicitly credited. This is a quantitative refinement, not a claim to newly solve the already classified AIM source question. The note is unrefereed and makes no absolute-priority claim.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO",
    "math.CA"
  ],
  "keywords": [
    "AIM-LINEAR_ALGEBRA-0010",
    "Toeplitz matrix",
    "total nonnegativity",
    "vanishing order",
    "Edrei factorization",
    "Cauchy-Binet"
  ],
  "manuscript_version_date": "2026-10-23",
  "publication_date": "2026-10-23",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-23",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-linear-algebra-0010/",
  "pdf_url": "https://eulersolve.org/papers/aim-linear-algebra-0010/paper.pdf?v=9aca8c0cd84e",
  "doi": "10.5281/zenodo.23601572",
  "zenodo_record_url": "https://zenodo.org/records/23601572",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Exact vanishing order, contiguous positive coefficient support and endpoint coefficients for every admissible minor of an exponential deformation of a finite-atom Edrei Toeplitz matrix; includes initial shifts and positive scaling. The classical source question was already classified in the literature. This manuscript is an adjacent quantitative refinement, not a new source closure. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
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    "paper.pdf": {
      "sha256": "9aca8c0cd84eb6db61d78582d1184bcb7e30dde7009a46535970ed0802bcae2e"
    },
    "source.zip": {
      "sha256": "17a0542f7dd1d94df119b4d0f243641c11387858fd24dae0c0896d3c9f4d6e92"
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    "verification_report.md": {
      "sha256": "b9c277147cc447512e396189e28058b5c0f62c4787ccc53b4ad964d528147eef"
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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."
}