{
  "abstract": "In homage to Hofstadter, we call a compact, citable specification of definitions, theorem, proof mechanism, and examples a mathematical koan when a full paper can be reconstructed and verified from it, including in reader-specific forms generated by AI. Every AI-solvable problem is itself a koan, although reconstruction need not be cheap; the present article is an expansion of one. We introduce Cartan convexity for self-adjoint free functions: locally, such a function agrees with a locally bounded matrix-convex free function. If the variable set has cardinality $τ$ and $λ=\\max\\{τ,\\aleph_0\\}$, every bounded real free set has a universal direct sum on a Hilbert space of dimension at most $2^λ$; every point of the set is a reducing summand of an amplification of this sum. Applying the noncommutative Kraus--butterfly theorem at that universal point yields an extension to an open matrix-convex neighborhood of the noncommutative convex hull. For normal affine pencils, the extension domain also contains the bounded strong closure of the hull. We prove an analogous theorem for a graph embedding $Γ$. A local convex lift through $Γ$ extends to a neighborhood of $Γ^{-1}(\\operatorname{co}_{\\mathrm{nc}}Γ(K))$ and admits a butterfly realization in the $Γ$-coordinates. We distinguish this lift condition from intrinsic $Γ$-convexity. For $Γ(x,y)=(x,y,y^2)$, the intrinsically $Γ$-affine polynomial $xy+yx$ has no convex lift germ at the origin, whereas a single quadratic coordinate suffices to lift every uniformly real analytic germ.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02714",
  "author_total": 1,
  "authors": [
    {
      "name": "J. E. Pascoe"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/drx1q5n06sw8a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02714",
  "fulltext_url": null,
  "id": "drx1q5n06sw8a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02714",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CV"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "openalex",
      "2604|Holomorphic and Operator Theory"
    ]
  ],
  "openalex": "W7207711011",
  "pdf_url": "https://arxiv.org/pdf/2609.02714",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/drx1q5n06sw8a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.02714",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02714"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Mathematical Koans and Cartan Convexity: $Γ$-Convex Hulls and Butterfly Realizations",
  "title_text": "Mathematical Koans and Cartan Convexity: Γ-Convex Hulls and Butterfly Realizations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/drx1q5n06sw8a",
  "venue": "arXiv",
  "versions": []
}