{
  "schema_version": 1,
  "problem_number": "AIM-PROBABILITY-0108",
  "title": "Polynomial Rigidity and a Sharp Degree Bound for Free Quadratic Variation",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We isolate an elementary highest-degree obstruction to polynomial changes of variables preserving unit free-Brownian diffusion. For every unital moment functional and every nonconstant noncommutative polynomial of degree d, its free quadratic-variation polynomial has degree exactly 2d-2. The leading coefficients form a positive Gram matrix of last-letter derivatives. At any algebraically free self-adjoint tuple, this classifies self-adjoint polynomial unit-covariance maps as affine coisometries. For a standard semicircular m-tuple, the distance of the covariance from the scalars is at least 1/sqrt(m) times the squared norm of the polynomial's highest Wick component. The constant is sharp, and for quadratic polynomials this is a sharp distance-to-affine estimate. The stochastic application concerns a prescribed pathwise transformation in the same filtration; it does not resolve the more general terminal-law steering question in the AIM Free Analysis problem list.",
  "result_type": "COMPLETE_SCOPED_PROOF",
  "categories": [
    "math.PR",
    "math.OA"
  ],
  "keywords": [
    "free probability",
    "free stochastic calculus",
    "quadratic variation",
    "noncommutative polynomials",
    "semicircular variables",
    "polynomial rigidity",
    "Wick chaos",
    "AIM-PROBABILITY-0108",
    "math.PR",
    "math.OA"
  ],
  "manuscript_version_date": "2026-09-28",
  "publication_date": "2026-09-28",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-28",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-probability-0108/",
  "pdf_url": "https://eulersolve.org/papers/aim-probability-0108/paper.pdf?v=3e3557656071",
  "doi": "10.5281/zenodo.23021763",
  "zenodo_record_url": "https://zenodo.org/records/23021763",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The completed theorems concern prescribed polynomial pathwise changes and operator-valued covariance under injective polynomial evaluation. They do not solve general terminal-law steering, nor the entropy and pressure questions fused into AIM-PROBABILITY-0108. The general-degree bound controls only the highest Wick component. Standard tools and the inherited one-variable observation are credited. Self-audited and unrefereed; no independent certification, formal verification or absolute-priority claim. Elementary rigidity may be folklore.",
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    "source.zip": {
      "sha256": "e4172de9bc6e082a6f3df3c696992905a56e40b1eb21f7bdd605713e4d3e618c"
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    "verification_report.md": {
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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."
}