数学公案与嘉当凸性：$\Gamma$-凸包与蝶形实现
Mathematical Koans and Cartan Convexity: $Γ$-Convex Hulls and Butterfly Realizations
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中文总结 AI 辅助
本文致敬侯世达定义数学公案，针对自伴自由函数引入嘉当凸性，结合非交换Kraus-蝶形定理推导相关凸性与蝶形实现结论，区分提升条件与内在$\Gamma$-凸性。
中文摘要 AI 辅助
为致敬侯世达（Hofstadter），我们将定义、定理、证明机制与示例的紧凑、可引用规范称为“数学公案”，当可从中重构并验证完整论文（包括由AI生成的读者特定形式）时即符合该定义。每个可由AI求解的问题本身就是一个公案，尽管重构未必廉价；本文是其中一个公案的扩展。我们针对自伴自由函数引入嘉当凸性：局部而言，此类函数与局部有界的矩阵凸自由函数一致。若变量集的基数为$\tau$且$\lambda=\max\{\tau,\aleph_0\}$，则每个有界实自由集在维度至多为$2^\lambda$的希尔伯特空间上都有一个通用直和；该集合的每个点都是此直和放大的约化直和项。在该通用点应用非交换Kraus-蝶形定理，可将其扩展到非交换凸包的开矩阵凸邻域。对于正规仿射束，扩展域还包含凸包的有界强闭包。我们证明了关于图嵌入$\Gamma$的类似定理：通过$\Gamma$的局部凸提升可扩展到$\Gamma^{-1}(\operatorname{co}_{\mathrm{nc}}\Gamma(K))$的邻域，并在$\Gamma$坐标中允许蝶形实现。我们将该提升条件与内在$\Gamma$-凸性区分开。对于$\Gamma(x,y)=(x,y,y^2)$，内在$\Gamma$-仿射多项式$xy+yx$在原点处没有凸提升芽，而单个二次坐标足以提升每个均匀实解析芽。
英文摘要
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.
