{"schema":"postcutoff/event@1","as_of":"2026-10-20T07:28:00+02:00","url":"https://postcutoff.com/e/2026-10-19-arveson-hyperrigidity-commutative-counterexamples-gpt-6-astra/","md":"https://postcutoff.com/e/2026-10-19-arveson-hyperrigidity-commutative-counterexamples-gpt-6-astra/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-19-arveson-hyperrigidity-commutative-counterexamples-gpt-6-astra","date":"2026-10-19","date_precision":"day","short_title":"Commutative counterexamples to Arveson's hyperrigidity conjecture","deck":"Proof developed with GPT-6 Astra, finite-dimensional example proposed by it (Bilich, Scherer)","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.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.26556: Commutative Counterexamples to Arveson's Hyperrigidity Conjecture (Bilich, Scherer)","url":"https://arxiv.org/abs/2610.26556","type":"paper","group":"primary","domain":"arxiv.org"}],"official":1,"filed":"2026-10-20","updated":"2026-10-20","orgs":["University of Haifa","Technion","OpenAI","Anthropic"],"title":"Commutative counterexamples to Arveson's hyperrigidity conjecture: proof developed with GPT-6 Astra, finite-dimensional example proposed by it (Bilich, Scherer)","summary":"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.'"],"key_numbers":[],"tags":["math","operator-algebras","functional-analysis","hyperrigidity","choquet-theory","counterexample","gpt-6-astra","codex","agents"],"science":{"field":"mathematics","subfield":"operator algebras / Choquet theory","problem":"Arveson's hyperrigidity conjecture for operator systems in commutative C*-algebras (Davidson–Kennedy Question 8.4 setting)","result":"Claimed commutative counterexamples, including a finite-dimensional one of affine dimension ≤ 129.","open_since":"","ai_system":["GPT-6 Astra","Codex","GPT-6.1 Sol","Claude Opus 5.5"],"human_role":"AI-assisted: human found the first example; GPT-6 Astra developed its initial proof and proposed the finite-dimensional example","verification":{"key":"preprint","label":"Preprint only","text":"Unrefereed arXiv preprint"},"status":"pending","shock":"","short":"Arveson hyperrigidity, commutative case"},"body_md":"## What happened\n\nHyperrigidity asks whether a generating set of a C*-algebra determines every representation among completely positive maps.\nArveson conjectured it follows from all irreducible representations being boundary representations. Bilich and Dor-On had already disproved it for\nnoncommutative algebras. Commutative algebras resisted, because their irreducible representations are one-dimensional. Scherer\nfound the weighted-ball example; GPT-6 Astra built its initial proof and then proposed the finite-dimensional version.\n\n## Why it matters\n\nIt settles the main remaining case of a central conjecture in noncommutative Choquet theory. The authors also report a negative\ndetail: a long multi-agent search produced a smaller candidate they could not verify, so they left it out.","disputed":[{"label":"Verification","text":"Unrefereed arXiv preprint","sources":[]}],"related":[{"id":"2026-10-20-ai-assisted-results-october-20-listing","url":"https://postcutoff.com/e/2026-10-20-ai-assisted-results-october-20-listing/","date":"2026-10-20","date_precision":"day","short_title":"AI-assisted results in the 20 Oct 2026 arXiv listing","deck":"Poset dimension proofs found by GPT-6 Astra, Hitchin domination from ChatGPT 5.6 Sol's ideas, the section conjecture for generic curves, Yau's geometric conjecture, Erdős–Gyárfás r = 5, a Claude-built Gray-code reduction, and a Lean harness solving 6/6 IMO 2026","takeaway":"Tuesday's arXiv announcement was the strongest maths batch yet. Seven results have their own entries, among them the Erdős–Szemerédi sunflower conjecture, Erdős–Hajnal for all paths and a non-sofic hyperbolic group. This page catalogues the next tier: about forty papers whose AI disclosures run from 'the proofs were found by ChatGPT-6 Astra' to proofreading.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":23,"official":23,"filed":"2026-10-20","updated":"2026-10-20","orgs":["OpenAI","Anthropic","DeepSeek","AWS","various mathematicians"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-20","type":"filed","text":"Created (PDF read in full text)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-20","run":null,"sources_read":"PDF read in full text","updated":"2026-10-20","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":172,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":111,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":202,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":141,"in_training_data":false}],"short_url":null}