Mathematical Koans and Cartan Convexity: Γ-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 τ and λ=max{τ,ℵ0}, every bounded real free set has a universal direct sum on a Hilbert space of dimension at most 2λ; 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(concΓ(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,y2), 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.
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BibTeX
@misc{pascoe2026mathematical,
title = {{Mathematical Koans and Cartan Convexity: $Γ$-Convex Hulls and Butterfly Realizations}},
author = {J. E. Pascoe},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02714},
eprint = {2609.02714},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02714},
}
RIS
TY - UNPB
TI - Mathematical Koans and Cartan Convexity: Γ-Convex Hulls and Butterfly Realizations
AU - J. E. Pascoe
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02714
UR - https://arxiv.org/abs/2609.02714
AB - 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,ℵ_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(co_ncΓ(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.
ER -