{
  "schema_version": 1,
  "problem_number": "AIM-PROBABILITY-0126",
  "title": "A Slice–Riesz–Bochner Characterization of Joint Brown Determinant Functions",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We characterize the functions arising as Fuglede–Kadison determinant transforms of bounded commuting tuples in a fixed II₁ factor. After recovering the log-modulus formula intended by an AIM range question, we show that such functions are exactly the logarithmic potentials of compact probability measures on ℂⁿ. Intrinsically, each complex line must give a normalized subharmonic logarithmic potential with uniformly bounded Riesz support, and the unit-frequency transforms of the slice measures must form a continuous positive-definite function on ℂⁿ. Complex scaling of all slice measures is forced automatically by the single global function; Bochner's theorem then reconstructs the unique joint measure. Every compact probability measure is realized by a commuting normal tuple in any prescribed II₁ factor. In one variable, the representing measure is simply the Riesz measure of one transformed function.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.OA",
    "math.FA"
  ],
  "keywords": [
    "Fuglede–Kadison determinant",
    "Brown measure",
    "II₁ factors",
    "logarithmic potentials",
    "Bochner theorem",
    "joint Brown measure",
    "Riesz measure",
    "logarithmic potential",
    "AIM-PROBABILITY-0126",
    "open mathematics",
    "mathematical proof",
    "math.OA",
    "math.FA"
  ],
  "manuscript_version_date": "2026-09-01",
  "publication_date": "2026-09-02",
  "publication_date_kind": "first public online release",
  "version": "1.0 (typesetting revision 2026-09-05)",
  "date_modified": "2026-09-05",
  "presentation_revision_only": true,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-probability-0126/",
  "pdf_url": "https://eulersolve.org/papers/aim-probability-0126/paper.pdf?v=731c1851ffbd",
  "doi": "10.5281/zenodo.22245595",
  "zenodo_record_url": "https://zenodo.org/records/22245595",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete for the bounded global log-modulus interpretation fixed by the recovered AIM display.",
  "files": {
    "paper.pdf": {
      "sha256": "731c1851ffbd6e451101c1d4e118a1286155d955ea614a1d62ec045f909d1b7a"
    },
    "source.zip": {
      "sha256": "a863a1550bc3ffe57c2b4c4359c50d72a6a4efd63910a194e2fc1e9e1e905fca"
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    "verification_report.md": {
      "sha256": "e69989a47963f16b42dbf9e3168b8f4a3ad780efdb851c79de752ae2bba06e3e"
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  },
  "ai_use_disclosure": "AI-assisted tools supported literature search, computation, proof auditing, and manuscript preparation. The author remains responsible for all claims and the final text."
}