{
  "schema_version": 1,
  "problem_number": "OWR-14740-003",
  "title": "Nonadditivity of the Mean Symmetric-Difference Risk for Kernel Level Sets",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We give a smooth planar counterexample to simultaneous bandwidth-independent variance and bias limits for the mean symmetric-difference risk of kernel density level sets. For a standard bivariate normal density, a compactly supported radial C2 polynomial kernel, and critical bandwidths h_n(t) proportional to n^(-1/6), the normalized risk tends to C E|1+tZ|. Its strict convexity rules out universal constants A and B in sqrt(nh^2) I1 -> A and h^(-2) I2 -> B for any exact decomposition of the risk; three bandwidths suffice. An explicitly defined centered comparison gives unequal residual limits. A direct proof uses a triangular-array central limit theorem, an integrable boundary bound and an integrated-variance tail estimate. The general nonlinear risk law is earlier Doss-Weng/Qiao theory, which is fully credited. The result refutes the joint proposal in Chacon/Polonik's OWR32/2016 discussion while retaining its variance premise. The report does not define its I2 explicitly, so our centered comparison is not identified with that term and no whole UnsolvedMath OWR-14740-003 record closure is claimed. This five-page unrefereed preprint was prepared with AI assistance and self-audited; it has no independent referee or proof-assistant verification. Symbolic regression checks accompany the written proof.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.ST",
    "stat.ME"
  ],
  "keywords": [
    "OWR-14740-003",
    "kernel level sets",
    "symmetric difference risk",
    "bandwidth",
    "folded normal"
  ],
  "manuscript_version_date": "2026-10-27",
  "publication_date": "2026-10-27",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-27",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-14740-003/",
  "pdf_url": "https://eulersolve.org/papers/owr-14740-003/paper.pdf?v=805ad01eddf4",
  "doi": "10.5281/zenodo.23682166",
  "zenodo_record_url": "https://zenodo.org/records/23682166",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete smooth planar counterexample to simultaneous bandwidth-independent limits sqrt(nh^2) I1 -> A and h^-2 I2 -> B for any decomposition of mean symmetric-difference risk. Three monotone critical bandwidths force a strictly convex folded-normal risk rather than an affine function. An exact centered comparison has explicitly unequal bias residual limits. Both Lebesgue and probability loss are covered. The report does not specify its I2 formula. The simultaneous universal two-limit proposal is refuted, but the exact intended I2 remains unidentified and no whole frozen-record closure is claimed. Earlier nonlinear risk laws of Doss-Weng and Qiao are explicitly credited. 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": "805ad01eddf44d6c9ba2e0e595757156b0fbf9b0a1476886e7f7d8af8a31435b"
    },
    "source.zip": {
      "sha256": "249f8e770d39d30295e2987c92421a8c4604c5946c9aa51c919d328c6685c1b0"
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    "verification_report.md": {
      "sha256": "33d40818c9f608253554d9374f4e494509208dfc231a2e8fa85a8e40594d7f63"
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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."
}