{
  "schema_version": 1,
  "problem_number": "OWR-4136-005",
  "title": "Rademacher's Half-Ball and Cone Counterexample in Three Dimensions",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Rademacher proposed a half-ball joined to a cone, with a small tip truncation, as a counterexample to monotonicity of the expected volume of a random tetrahedron under inclusion. We prove that this specific construction works in dimension three, with cone height 10^-5 and truncation height 10^-10. An integer-only outward interval certificate proves that the half-ball expectation exceeds its origin-anchored counterpart 9*pi/1024 by more than 1/6000. A quantitative mixture argument transfers this gap to explicit nested convex bodies. General three-dimensional nonmonotonicity was already proved by Kunis, Reichenwallner and Reitzner. This AI-assisted, computer-assisted, self-audited preprint verifies the particular half-ball construction and makes no absolute priority, independent review or formal-verification claim.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.PR",
    "math.MG"
  ],
  "keywords": [
    "random tetrahedra",
    "stochastic geometry",
    "half-ball",
    "monotonicity under inclusion",
    "computer-assisted proof",
    "interval arithmetic"
  ],
  "manuscript_version_date": "2026-10-13",
  "publication_date": "2026-10-13",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-13",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-4136-005/",
  "pdf_url": "https://eulersolve.org/papers/owr-4136-005/paper.pdf?v=57c59d04da37",
  "doi": "10.5281/zenodo.23350280",
  "zenodo_record_url": "https://zenodo.org/records/23350280",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete answer to the specific half-ball/cone construction question. General dimension-three nonmonotonicity was already known and is credited; an exact expectation formula is not claimed. The proof uses a reproducible integer interval certificate. Novelty, independent review and formal verification are not certified.",
  "files": {
    "paper.pdf": {
      "sha256": "57c59d04da37baa425ace1bde2faae8e0b984e476f0611777bc75d4273f9d584"
    },
    "source.zip": {
      "sha256": "592e07f30478477864860b68312c15b47ced5532e0d2b05a0a3129a92a0883c0"
    },
    "verification_report.md": {
      "sha256": "07a8f1f2d05ad3747b22a3782d343da728c909fc9765bc53c04cb4f663bb1838"
    }
  },
  "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."
}