{
  "schema_version": 1,
  "problem_number": "AIM-OTHER-0095",
  "title": "When Binary-Form Norm Balls Have Finite Haar Volume",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For a nonzero real binary form of degree d, we prove that all norm sublevel sets in SL(2,R) have finite Haar volume if and only if every real projective root has multiplicity strictly less than d/2. For the Bombieri norm, failure makes every nonempty sublevel set have infinite volume. The proof uses a positive-weight criterion for properness and a full-dimensional Haar-measure thickening. The cubic x^2 y has trivial stabilizer but infinite-volume balls at every positive radius in every norm. We compute its exact cutoff volume and give a critical quartic with stabilizer {I,-I} and empty small balls. The cubic example is credited to the frozen UnsolvedMath research report; its Haar-volume calculation is completed here. This is a normalization obstruction in the setup preceding AIM Question 19, not a solution of the corrected equidistribution question. AI-assisted, self-audited and unrefereed; no absolute-priority or formal-certification claim.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.RT",
    "math.DS"
  ],
  "keywords": [
    "binary form",
    "Haar measure",
    "proper orbit map",
    "real projective root",
    "norm ball",
    "AIM-OTHER-0095"
  ],
  "manuscript_version_date": "2026-10-18",
  "publication_date": "2026-10-18",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-18",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-other-0095/",
  "pdf_url": "https://eulersolve.org/papers/aim-other-0095/paper.pdf?v=aac79646c708",
  "doi": "10.5281/zenodo.23447556",
  "zenodo_record_url": "https://zenodo.org/records/23447556",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The finite-volume criterion and explicit normalization obstruction are complete. The cubic x^2 y was already identified in the frozen upstream research report and is credited; its Haar-volume argument is completed here. This does not resolve the corrected equidistribution question in AIM Question 19 or claim a new full source-record closure. AI-assisted, self-audited and unrefereed; no independent review, proof-assistant certification or absolute-priority claim.",
  "files": {
    "paper.pdf": {
      "sha256": "aac79646c7081024ae3c4030f644af51d52b58ada01b6247bb9acf3d287f87db"
    },
    "source.zip": {
      "sha256": "8854388d23914241302e0a3a3db82c21da177664410d1c60534ff4f159f420bc"
    },
    "verification_report.md": {
      "sha256": "f4c6aa27051f853234a47c08b7b74c3b4dc758a59e92623dd330fbae50b3a10c"
    }
  },
  "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."
}