{
  "schema_version": 1,
  "problem_number": "AIM-ANALYSIS-0146",
  "title": "A Gaussian Obstruction to Uniform Mesh Bounds for Differential Operators",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We classify the nonzero finite-order differential operators with real polynomial coefficients that preserve hyperbolicity and retain a positive uniform fraction of additive zero mesh on the whole Laguerre-Polya class: they are exactly the nonzero scalar multiples of the identity. For every other such operator and every prescribed positive input mesh, the infimum of the output mesh is zero, even for inputs with exactly three simple zeros. A strengthening Gaussian factor yields a local Hermite cluster; for an arbitrary fixed operator at a nonsingular center we compute its common drift and next common dilation, without assuming global hyperbolicity preservation. The inherited cubic-Gaussian construction and classical antecedents are explicitly credited. This is a complete scoped theorem, not a full resolution of the broad AIM-ANALYSIS-0146 zero-spacing question. The manuscript is AI-assisted, self-audited and unrefereed, with no absolute-priority or independent-review claim. Exact standard-library Python code and identical normal/optimized outputs cover 450 operator/parameter cases and 9,505 checks; the written proofs establish the general statements.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CA",
    "math.CV"
  ],
  "keywords": [
    "Laguerre-Polya class",
    "hyperbolicity preservers",
    "zero spacing",
    "additive mesh",
    "Hermite polynomials",
    "Gaussian scaling",
    "AIM-ANALYSIS-0146"
  ],
  "manuscript_version_date": "2026-10-10",
  "publication_date": "2026-10-10",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-10",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-analysis-0146/",
  "pdf_url": "https://eulersolve.org/papers/aim-analysis-0146/paper.pdf?v=18de16b5ae95",
  "doi": "10.5281/zenodo.23271793",
  "zenodo_record_url": "https://zenodo.org/records/23271793",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete scalar-only classification of uniform additive-mesh retention for nonzero finite polynomial-coefficient hyperbolicity preservers on the whole Laguerre-Polya class, and a fixed-order local Hermite-cluster expansion at nonsingular centers. This does not resolve the broad AIM zero-spacing question. The inherited cubic-Gaussian construction and older first-derivative examples are credited; only a bounded novelty search was completed. AI-assisted, self-audited and unrefereed; no independent review, formal verification or certified priority is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "18de16b5ae9535aca5c61810f6c704500a74806bf1df3d49a5a233c471074a7a"
    },
    "source.zip": {
      "sha256": "036490aaf8a3aba7158a0df5007951090dde8faab23445a23b6a15395dbe5039"
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    "verification_report.md": {
      "sha256": "0a101f86445d66885f48639ab0327ce2b60f9e6483056688fd2583d92ece301b"
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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."
}