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The regular pentagonal system and the Smith--Ward problem

Primarily AI-generated textHuman understanding: all partsmath.OA — Operator Algebrasmath.FA — Functional Analysis

Contributed by Shengwei Qiu, Thomas Sinclair ↗

Version 1 / Oct 26, 2026 / CC BY 4.0

Abstract

The three-dimensional function system of the regular pentagon fails the lifting property, giving a counterexample to the operator-system formulation of the Smith--Ward problem with C∗^*-envelope C5\mathbb C^5. We construct unital kk-positive maps from this fixed system into matrix algebras where distance from every ucp map is bounded below independently of kk. We also prove that every regular polygon system with at least five vertices detects nuclearity, while its dual characterizes the weak expectation property. These tensor-product characterizations follow from a graph dichotomy: a graph operator system and its dual are C∗^*-nuclear when the graph is 22-chordal; otherwise they detect nuclearity and WEP, respectively.

Provenance statement

The authors selected the problems, provided the relevant literature, and directed an extensive exploration of the Smith--Ward problem with Claude Opus 5.5. Within that exploration, the proofs of Theorems A and C, the Cayley transference theorem, and the lifting and Calkin-completion results were generated by Claude Opus 5.5. The argument was reorganized and redrafted with the assistance of ChatGPT-6 Astra. Additional prompting of ChatGPT-6 Astra produced the construction in Theorem B. ChatGPT-6 Astra was then prompted to formalize the construction detailed in Theorem B in Lean, assuming only the existence of a sequence of malnormal matrices with rank tending to infinity. During the formalization process, several simplifications of the original construction were found, which are incorporated into this manuscript. Sustained dialog with ChatGPT-6 Astra was used to improve the exposition and distill general principles.

Tools used

OpenAI
ChatGPTVersion GPT-6 Astra
Anthropic
Claude OpusVersion Opus 5.5
Lean
Lean

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Version history

  1. v1Submitted by Thomas SinclairInitial depositCurrentOct 26, 2026