{
  "schema_version": 1,
  "problem_number": "AIM-PROBABILITY-0118",
  "title": "Symmetric Unimodality for Real Free Convolution Powers",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We derive preservation of symmetric weak unimodality under every real free additive convolution power t >= 1, without support, moment or density assumptions. The essential input is the Cauchy-transform inequality in Anonymous's unrefereed preprint Symmetric unimodality under free additive convolution (2026), DOI 10.5281/zenodo.22649756. We prove an exact transport identity for its inequality defect along power subordination. Consequences include preservation under free compression and an interval description of the unimodal times of a symmetric free Levy process. Combined with an example of Hasebe and Sakuma, this gives a symmetric freely infinitely divisible law whose powers are unimodal exactly for t >= 1. This is a dependent continuation of the cited transform inequality, not a new proof of the original pairwise symmetric-unimodality conjecture. The threshold construction is existential; it does not evaluate an explicit density or onset time. The note is self-audited, AI-assisted and unrefereed.",
  "result_type": "COMPLETE_ATTRIBUTED_DEPENDENT_CONTINUATION",
  "categories": [
    "math.PR",
    "math.OA"
  ],
  "keywords": [
    "free probability",
    "free convolution powers",
    "symmetric unimodality",
    "subordination",
    "free compression",
    "free Levy process"
  ],
  "manuscript_version_date": "2026-10-11",
  "publication_date": "2026-10-11",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-11",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-probability-0118/",
  "pdf_url": "https://eulersolve.org/papers/aim-probability-0118/paper.pdf?v=6929c5ec3d9f",
  "doi": "10.5281/zenodo.23296226",
  "zenodo_record_url": "https://zenodo.org/records/23296226",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Dependent continuation of Anonymous's essential transform inequality, DOI 10.5281/zenodo.22649756, an unrefereed preprint. Complete real-power transport and t>=1 preservation, free compression and onset interval results; sharp-range law exists but its onset threshold is not evaluated. No new proof or closure of the original pairwise source question is claimed. Self-audited, AI-assisted and unrefereed; no independent review, formal certification or absolute priority claim.",
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      "sha256": "6929c5ec3d9f12b866191012304a6e56cfef99fb78b8e17ca2c845c1bc7e1902"
    },
    "source.zip": {
      "sha256": "e4d4be3e9a2745f3dad2651f65c5d22e3c44210516a3c8d360e71e08b3c944f9"
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    "verification_report.md": {
      "sha256": "b2153608eef25cfdf14a2336f4c973ce51b6f73d9be3a50a5b559b133afdc353"
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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."
}