{
  "schema_version": 1,
  "problem_number": "AIM-PROBABILITY-0111",
  "title": "Nonuniformity of Free-Gradient Heat Semigroups under Finite Fisher Information",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let X_1,...,X_m be a nonempty finite bounded selfadjoint tuple with finite joint nonmicrostates free Fisher information, and let Delta be the generator of the closed polynomial free-gradient form. We prove that its heat semigroup cannot converge uniformly to the identity in L2 on the operator-norm unit ball of the generated von Neumann algebra. For each tuple we obtain a strictly positive lower bound valid at every positive time, with no restriction excluding one or two variables. The witnesses are coordinate exponentials. A direct proof of the classical marginal L3-density implication, the conjugate-variable adjoint formula, and a one-sided Fejer/Riesz estimate control their generator norm, while their energy grows linearly. Resolvent duality yields the nonuniformity bound. Existing rigidity results then imply non-L2-rigidity and, for at least two variables, primeness. The result addresses the 2006 AIM Free Analysis uniformity question. This preprint is unrefereed and makes no absolute priority claim.",
  "result_type": "COMPLETE_NEGATIVE_ANSWER",
  "categories": [
    "math.OA",
    "math.PR"
  ],
  "keywords": [
    "free probability",
    "free Fisher information",
    "free-gradient heat semigroup",
    "nonuniform convergence",
    "von Neumann algebras",
    "L2 rigidity",
    "AIM-PROBABILITY-0111",
    "open mathematics",
    "mathematical proof",
    "math.OA",
    "math.PR"
  ],
  "manuscript_version_date": "2026-09-05",
  "publication_date": "2026-09-05",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-05",
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  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-probability-0111/",
  "pdf_url": "https://eulersolve.org/papers/aim-probability-0111/paper.pdf?v=e8656a8393a4",
  "doi": "10.5281/zenodo.22327310",
  "zenodo_record_url": "https://zenodo.org/records/22327310",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": null,
  "files": {
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    "source.zip": {
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    "verification_report.md": {
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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."
}