{
  "schema_version": 1,
  "problem_number": "AIM-PROBABILITY-0131",
  "title": "Fourier Shadows and Tracial Free Infinite Divisibility",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We compare tensor and free addition of bounded tracial tuples whose joint laws are freely infinitely divisible, after projecting both laws to scalar Fourier series. The Fourier shadows obtainable from free sums form a proper subset of products of such shadows in every dimension at least two. A witness is the product of two variance-one semicircle Fourier transforms on separate coordinate axes. Every tracial joint lift of this product fails to have a free fifth root: a fixed sum of Hermitian squares has expectation -2/25 at that formal root. At free exponent t its expectation is 2t(4t-1), so positivity requires t >= 1/4; no endpoint sufficiency is asserted. We also prove commuting free-root rank-one rigidity and construct distinct bounded tracial free compound Poisson laws with identical Fourier shadows.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.OA",
    "math.PR"
  ],
  "keywords": [
    "free probability",
    "free infinite divisibility",
    "Fourier transform",
    "noncommutative joint distributions",
    "tensor independence",
    "semicircular law",
    "sum of squares",
    "commutator",
    "free cumulants",
    "compound Poisson law",
    "AIM-PROBABILITY-0131",
    "math.OA",
    "math.PR"
  ],
  "manuscript_version_date": "2026-10-03",
  "publication_date": "2026-10-03",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-03",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-probability-0131/",
  "pdf_url": "https://eulersolve.org/papers/aim-probability-0131/paper.pdf?v=c23a5c5a92d6",
  "doi": "10.5281/zenodo.23114092",
  "zenodo_record_url": "https://zenodo.org/records/23114092",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "A complete proof for the explicit bounded-tracial, common-coordinate formulation of AIM Free Analysis section 0.10, first question (frozen record AIM-PROBABILITY-0131). The lift-independent sum-of-squares certificate excludes all free roots of orders at least five for the witness. The scalar k=1 comparison, arbitrary nontracial lifts and classification of all obtainable series are not settled. The original AIM source record is not counted as fully resolved. Standard cumulant, conditional-positivity and Fock machinery is credited. AI-assisted, originating-researcher self-audited and unrefereed; no independent human review or proof-assistant verification. Novelty remains undetermined after bounded primary-source search; no absolute-priority claim.",
  "files": {
    "paper.pdf": {
      "sha256": "c23a5c5a92d612819c62feb22b7d9c08fa7cadd5186e5f95c4c6ebe545cbcb8b"
    },
    "source.zip": {
      "sha256": "7a1f0fc605b36da77c4d4f5f5fbe4e371e208c02b161d7b155faf32d271a087d"
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    "verification_report.md": {
      "sha256": "769b0ed88eb03c84bebb4541a368f147ac5afd52742c45f8331a1b1e7716f43d"
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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."
}