{
  "schema_version": 1,
  "problem_number": "AIM-PROBABILITY-0133",
  "title": "Geometric Sampling Does Not Determine Regularized R-Diagonal Laws",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For every s0>0, q>1 and finite integer K>=1, we construct a continuum of distinct affiliated R-diagonal laws with identical joint star-moments and identical first K alternating cumulants of c(1+s c*c)^(-1/2) at every scale s=s0 q^k, k in Z. All absolute moments are finite. A finite combination of classical lognormal Stieltjes perturbations gives an explicit resolvent difference with prescribed finite-order zeros on the grid and an exact higher separating cumulant. A Blaschke-Jensen argument gives a sufficient uniqueness condition for other sample sets. This is a complete scoped reconstruction counterexample, not a new radial analytic R-transform or closure of the entire AIM question. Classical inputs are credited. The preprint is self-audited, unrefereed and AI-assisted; independent review, formal verification and absolute priority are not claimed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.PR",
    "math.OA",
    "math.CV"
  ],
  "keywords": [
    "R-diagonal operators",
    "free cumulants",
    "moment problem",
    "Stieltjes transform",
    "lognormal distribution",
    "geometric sampling",
    "AIM-PROBABILITY-0133"
  ],
  "manuscript_version_date": "2026-10-14",
  "publication_date": "2026-10-14",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-14",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-probability-0133/",
  "pdf_url": "https://eulersolve.org/papers/aim-probability-0133/paper.pdf?v=e88205bf7112",
  "doi": "10.5281/zenodo.23362668",
  "zenodo_record_url": "https://zenodo.org/records/23362668",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete proof of a finite-order geometric-sampling counterexample and a sufficient nongeometric uniqueness criterion. The full AIM question is not newly closed. No necessity theorem for arbitrary sampling sets, new moment-free radial transform, free infinite divisibility, certified novelty, independent review or formal verification is claimed. Classical Stieltjes, R-diagonal and Jensen inputs are credited.",
  "files": {
    "paper.pdf": {
      "sha256": "e88205bf71128d7173df375247a3b70a958d5ffd37857ce052a49c911f8b818d"
    },
    "source.zip": {
      "sha256": "57d2c26d440e587d955290b56926b066cd1de5723bb2a713e6501aa252a2e17c"
    },
    "verification_report.md": {
      "sha256": "1fe1028cc49f6a972b9aa33c892fa20c62db4b77dccc37db09f499febc8981ea"
    }
  },
  "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."
}