---
id: "2026-10-19-arveson-hyperrigidity-commutative-counterexamples-gpt-6-astra"
url: "https://postcutoff.com/e/2026-10-19-arveson-hyperrigidity-commutative-counterexamples-gpt-6-astra/"
as_of: "2026-10-20T07:28:00+02:00"
date: "2026-10-19"
date_precision: day
category: science
importance: 3
confidence: medium
status: [Awaiting review]
verification: Preprint only
sources: 1
editor: human editor
human_review: null
version: null
---

As of: 2026-10-20 07:28 CEST.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-10-19-arveson-hyperrigidity-commutative-counterexamples-gpt-6-astra/

# Commutative counterexamples to Arveson's hyperrigidity conjecture

Full title: Commutative counterexamples to Arveson's hyperrigidity conjecture: proof developed with GPT-6 Astra, finite-dimensional example proposed by it (Bilich, Scherer)

arXiv 2610.26556 (19 Oct 2026) by Boris Bilich (Haifa) and Marcel Scherer (Technion) constructs operator systems in commutative C*-algebras for which every irreducible representation of the C*-envelope is a boundary representation, yet hyperrigidity fails. The examples are continuous affine functions on an infinite-dimensional weighted ℓp ball, a dilation of commuting operators, and a finite-dimensional system from a truncated epigraph of a strictly convex function (affine dimension d ≤ 129). Earlier counterexamples (Bilich–Dor-On; Scherer; Tian) were noncommutative. The AI declaration says the initial proof "was developed in a long conversation with ChatGPT using the GPT-6 Astra model" and the finite-dimensional counterexample "was proposed by ChatGPT".

## Key facts
- Paper: 'Commutative Counterexamples to Arveson's Hyperrigidity Conjecture', Boris Bilich (University of Haifa) and Marcel Scherer (Technion), arXiv 2610.26556 [math.OA], 19 Oct 2026
- Context (paper): the conjecture 'was disproved by the first author and Dor-On'; Scherer found a finite-dimensional counterexample and Tian a three-dimensional one, all in noncommutative algebras, where a dilation comes from an irreducible representation. 'This mechanism is unavailable for commutative C*-algebras'. Davidson–Kennedy had restated the commutative case as a question on Choquet versus dilation order (Question 8.4)
- Examples: (1) A(K) for an infinite-dimensional weighted ℓp ball (originally ℓ^(4/3) with |x_k| ≤ B/k); (2) commuting operators with a nontrivial dilation; (3) a finite-dimensional operator system from a compact truncated epigraph; such a counterexample needs affine dimension d ≥ 3; the authors conjecture the minimum is 4; the construction gives d ≤ 129
- AI declaration, verbatim: 'The weighted ℓp ball counterexample was discovered by the second author. … Its initial proof was developed in a long conversation with ChatGPT using the GPT-6 Astra model and spanned nearly thirty pages, with extensive use of probabilistic arguments. The finite-dimensional counterexample was proposed by ChatGPT using the GPT-6 Astra model, with the weighted ball example as its starting point. OpenAI's Codex was used … for mathematical exploration, the development and simplification of proofs, symbolic and numerical calculations, literature searches, and English-language editing.'
- Agent experiment: 'After more than 40 hours of exploration by GPT-6.1 Sol and Claude Opus 5.5 agents, the smallest proposed counterexample had dimension d = 11. However, the construction is very complicated, and we have not yet verified its correctness. We therefore do not include it in this paper.'

## What happened

Hyperrigidity asks whether a generating set of a C*-algebra determines every representation among completely positive maps.
Arveson conjectured it follows from all irreducible representations being boundary representations. Bilich and Dor-On had already disproved it for
noncommutative algebras. Commutative algebras resisted, because their irreducible representations are one-dimensional. Scherer
found the weighted-ball example; GPT-6 Astra built its initial proof and then proposed the finite-dimensional version.

## Why it matters

It settles the main remaining case of a central conjecture in noncommutative Choquet theory. The authors also report a negative
detail: a long multi-agent search produced a smaller candidate they could not verify, so they left it out.

## What is disputed or not yet verified

- Verification: Unrefereed arXiv preprint

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 172 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 111 days after its cutoff
- Gemini 4 Argon (training cutoff March 2026): 202 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 141 days after its cutoff

## Sources

1. [arXiv 2610.26556: Commutative Counterexamples to Arveson's Hyperrigidity Conjecture (Bilich, Scherer)](https://arxiv.org/abs/2610.26556) (arxiv.org, paper)

## Changes

- 2026-10-20 (filed): Created (PDF read in full text)

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