Mathematical Koans and Cartan Convexity: $\Gamma$-Convex Hulls and Butterfly Realizations

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 $\tau$ and $\lambda=\max\{\tau,\aleph_0\}$, every bounded real free set has a universal direct sum on a Hilbert space of dimension at most $2^\lambda$; 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 $\Gamma$. A local convex lift through $\Gamma$ extends to a neighborhood of $\Gamma^{-1}(\operatorname{co}_{\mathrm{nc}}\Gamma(K))$ and admits a butterfly realization in the $\Gamma$-coordinates. We distinguish this lift condition from intrinsic $\Gamma$-convexity. For $\Gamma(x,y)=(x,y,y^2)$, the intrinsically $\Gamma$-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.