{
  "schema_version": 1,
  "problem_number": "AMR-022-1032",
  "title": "Iterated-Logarithmic Size of Gross Exceptional Sets",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For each q>1, we construct a transcendental entire function and one regular inverse germ whose exceptional directions in Gross's star theorem contain a compact set of positive finite Hausdorff measure for h_q(t)=t[log log(e^e/t)]^q. The same inverse branch escapes at unit radius in every chosen direction. This single example has infinite measure for every gauge t[log(e/t)]^epsilon, epsilon>0, while remaining of length zero. The six-page proof refines Sina Nadi's positive-capacity construction using slowly varying corridors, a nonatomic Bernoulli shell sum and an exponential holomorphic approximation weight. Nadi's capacity result and Eremenko's public dimension-one attribution are credited; neither is claimed as new here. The endpoint q=1, equality with the entire exceptional set and a full exceptional-set classification are excluded. The package includes 16,878 supporting exact/symbolic and high-precision regression checks; these do not replace the written analytic proof. Developed with AI assistance and self-audited, this is an unrefereed preprint without independent review, formal verification or certified absolute priority.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CV",
    "math.CA"
  ],
  "keywords": [
    "AMR-022-1032",
    "Gross star theorem",
    "Hausdorff measure",
    "exceptional directions",
    "entire function",
    "inverse branch"
  ],
  "manuscript_version_date": "2026-10-28",
  "publication_date": "2026-10-28",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-28",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-022-1032/",
  "pdf_url": "https://eulersolve.org/papers/amr-022-1032/paper.pdf?v=ea0adfd94dae",
  "doi": "10.5281/zenodo.23713665",
  "zenodo_record_url": "https://zenodo.org/records/23713665",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For each q>1, one transcendental entire function and one regular inverse germ have a compact exceptional set of positive finite Hausdorff h_q measure, h_q(t)=t[log log(e^e/t)]^q. The same branch escapes at unit radius in every chosen direction. This gives infinite t[log(e/t)]^epsilon measure for every epsilon>0 in one example. Nadi already established the capacity counterexample; Eremenko credits dimension one to Nadi. This manuscript establishes the stated iterated-logarithmic gauge refinement, with zero new original-source closures. The endpoint q=1, equality with the full exceptional set and a classification of all exceptional sets are excluded. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
  "files": {
    "paper.pdf": {
      "sha256": "ea0adfd94daec3fde22d57fa2c6be3876429a5396dba12b6a369c015a4b606d7"
    },
    "source.zip": {
      "sha256": "d59bc8a08d699f6e854603169b6f4e45ce68d0ccc07c484f7844ea38798fdc93"
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    "verification_report.md": {
      "sha256": "8a3eb592954bbad7f8f4c5135b087ab6f3a22ef6c2bcc59e872221bf16b0cbc6"
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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."
}