Commutative counterexamples to Arveson’s hyperrigidity conjecture
Proof developed with GPT-6 Astra, finite-dimensional example proposed by it (Bilich, Scherer)
Awaiting review
The takeaway
Arveson’s hyperrigidity conjecture had already failed for noncommutative C*-algebras. The commutative case, where the known mechanism cannot work, stayed open. Bilich and Scherer now give commutative counterexamples. GPT-6 Astra developed the 30-page initial proof and proposed the finite-dimensional example, and a 40-hour agent search found only an unverified smaller one.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Preprint only
- Importance
- 3 of 5
- Last verified
- 20 October 2026
Your AI and this story
- GPT-6 Astra172 days after its cutoff
- Claude Opus 5.5111 days after its cutoff
- Gemini 4 Argon202 days after its cutoff
- Grok 4.7141 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 111 days before it.
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 |
|---|
Sources
1 source from 1 site. Numbers match the chips in the text.
1 source: 1 primary
Primary
Changes
- Filed (PDF read in full text)