{
  "schema_version": 1,
  "problem_number": "AIM-PROBABILITY-0113",
  "title": "Sharp Norms of Rectangular Majorana Matrices and a Two-Matrix Obstruction to Strong Convergence",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let C_(m,n) be an m by n matrix whose entries are distinct self-adjoint Clifford generators, each squaring to one. We prove that its squared operator norm is 2(m+n-2) for m,n at least 2; the one-row and one-column cases have squared norm max(m,n). The argument uses a vector-spin Casimir identity and an attaining highest-weight summand. The square case gives the optimal uniform bound 2 for the normalized arrays in the explicit CAR subquestion of the AIM Free Analysis problem list. Several arrays of distinct generators converge jointly in star moments to a standard free circular family. For N at least 2, the norm of the sum of d normalized array products X_r X_r^* is exactly 2(d+1)-4/N, while the free-limit norm is at most (1+sqrt(d))^2. Strong convergence therefore fails already for two arrays. The broader q-Gaussian approximation question is not resolved. The imported square calculation and Shlyakhtenko's prior work on a different Hermitian fermionic ensemble are credited. This is a self-audited, AI-assisted, unrefereed preprint; no independent review, formal verification or absolute priority is claimed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.OA",
    "math.RT",
    "math.PR"
  ],
  "keywords": [
    "Clifford algebra",
    "CAR algebra",
    "Majorana matrix",
    "operator norm",
    "spin representation",
    "Casimir operator",
    "free circular family",
    "strong convergence",
    "AIM-PROBABILITY-0113"
  ],
  "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-0113/",
  "pdf_url": "https://eulersolve.org/papers/aim-probability-0113/paper.pdf?v=8e7620e7b6fe",
  "doi": "10.5281/zenodo.23374808",
  "zenodo_record_url": "https://zenodo.org/records/23374808",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The explicit CAR-array boundedness subquestion is answered, with a rectangular extension and a two-array strong-convergence counterexample for the specified ensemble. The broader q-Gaussian approximation agenda and the whole source record are not claimed solved. The imported square calculation and prior work on a different Hermitian ensemble are credited. Self-audited, unrefereed and AI-assisted; no independent review, formal verification or absolute priority is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "8e7620e7b6febd42efb18b2da15096348b2aa316fe052502a969da986893a9f5"
    },
    "source.zip": {
      "sha256": "17415acd13ebd400747d8dd618ede281fa2eb49c8c213c939e29529c7e7f08c5"
    },
    "verification_report.md": {
      "sha256": "b6f3c20916cb559b09605834eeed76adf407dff688af1ae05cc43861e162a397"
    }
  },
  "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."
}