{"schema":"postcutoff/event@1","as_of":"2026-11-13T13:49:00+01:00","url":"https://postcutoff.com/e/2026-09-09-pierce-birkhoff-conjecture-disproved/","md":"https://postcutoff.com/e/2026-09-09-pierce-birkhoff-conjecture-disproved/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":"2026-09-09-pierce-birkhoff-conjecture-disproved","date":"2026-09-09","date_precision":"day","short_title":"Pierce–Birkhoff conjecture disproved by a multi-agent GPT + Claude harness","deck":"Counterexamples Lean-verified","takeaway":"On 9 Sep 2026 Zehua Lai, Lek-Heng Lim and Junyu Ren posted a counterexample to the Pierce–Birkhoff conjecture (Birkhoff and Pierce, 1956).","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2609.10420: Pierce-Birkhoff conjecture is false (Lai, Lim, Ren)","url":"https://arxiv.org/abs/2609.10420","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"GitHub: 7pocheR/Pierce-Birkhoff (Lean formalisation)","url":"https://github.com/7pocheR/Pierce-Birkhoff","type":"code","group":"primary","domain":"github.com"}],"official":2,"filed":"2026-09-30","updated":"2026-10-19","orgs":["University of Chicago","Lek-Heng Lim","Zehua Lai","Junyu Ren","OpenAI","Anthropic"],"title":"Pierce–Birkhoff conjecture (1956) disproved by a multi-agent GPT + Claude harness; counterexamples Lean-verified","summary":"On 9 Sep 2026 Zehua Lai, Lek-Heng Lim and Junyu Ren posted a counterexample to the Pierce–Birkhoff conjecture (Birkhoff and Pierce, 1956). It is a continuous piecewise-quadratic function on R^30 that is not a finite max–min combination of polynomials. The counterexample came out of a multi-agent harness that chained GPT-5.6, GPT-6, Claude Opus 5 and Claude Fable 5.1. The authors say no single model found it. Counterexamples in dimensions 5, 6, 7, 30 and 72 are formalised in Lean.","key_facts":["Pierce–Birkhoff conjecture (1956): every continuous piecewise-polynomial function on R^n is a finite lattice (max–min) combination of polynomials; previously known only for n ≤ 2","arXiv 2609.10420 (7 pages): an explicit counterexample on V = S²(R⁵) × S²(R⁵), dimension 30, piecewise quadratic, chosen because it 'can be readily checked by hand'","The same setup also produced counterexamples in dimensions 5, 6, 7, 22, 24 and 72; those for 5, 6, 7, 30 and 72 are Lean-formalised (GitHub 7pocheR/Pierce-Birkhoff)","The authors had earlier proved the conjecture for splines (hyperplane partitions) 'as a purely human endeavor with no AI usage'","AI setup: an orchestrator agent in Codex and Claude Code assigned proof searches, counterexample searches and reviews across GPT (5.6, 6) and Claude (Opus 5, Fable 5.1) models; 'The counterexample in this article is notably not found with any single large language model (we tried doing so without success)'","Per Section 5, the dimension-72 matrix-pair construction and the argument excluding representations of arbitrary degree 'emerged from GPT investigations'; the 30-dimensional version and its simplification came with further ChatGPT help","Side result: combined with the authors' earlier work, splines ⊊ pure transformers ⊊ semialgebraic splines, so some semialgebraic splines cannot be computed by a ReLU pure transformer","AI disclosure (abstract, arXiv 2609.10420): 'The counterexample was found with the assistance of our multi-agent, multi-model setup that chains together GPT 5.6, GPT 6, Claude Opus 5, and Claude Fable 5.1.'"],"key_numbers":[],"tags":["math","real-algebraic-geometry","counterexample","lean","multi-agent","gpt-6","claude-opus-5","claude-fable-5-1"],"science":{"field":"mathematics","subfield":"real algebraic geometry","problem":"Pierce–Birkhoff conjecture","result":"Explicit continuous piecewise-quadratic function on R^30 that is not a finite lattice combination of polynomials; further counterexamples in dimensions 5, 6, 7, 22, 24 and 72.","open_since":"1956","ai_system":["GPT-5.6","GPT-6","Claude Opus 5","Claude Fable 5.1"],"human_role":"Humans gave high-level direction on which targets mattered and wrote the final exposition; a multi-model agent system found and reviewed the construction","verification":{"key":"lean-unreviewed","label":"Lean proof, not yet reviewed","text":"Lean formalisation of the counterexamples for n = 5, 6, 7, 30, 72 (repo 7pocheR/Pierce-Birkhoff; formal statement written by the authors' team, not independently audited); the n = 30 counterexample is presented as hand-checkable; unrefereed preprint"},"status":"pending","shock":"A 70-year-old conjecture fell to a mixed OpenAI–Anthropic agent team after single models failed."},"body_md":"## What happened\n\nLek-Heng Lim's group had proved the conjecture for splines by hand. They then pointed a custom orchestrated team of GPT and Claude agents at the general case. The agents found counterexamples in several dimensions. The authors picked a 30-dimensional one that is easy to verify, formalised several in Lean and wrote the paper.\n\n## Why it matters\n\nPierce–Birkhoff is a classic problem in real algebraic geometry and ordered rings, open for 70 years. It is also one of the first documented cases where the authors say a result needed several labs' models working together, and it has a direct consequence for the expressivity of transformers.","disputed":[{"label":"Verification","text":"Lean formalisation of the counterexamples for n = 5, 6, 7, 30, 72 (repo 7pocheR/Pierce-Birkhoff; formal statement written by the authors' team, not independently audited); the n = 30 counterexample is presented as hand-checkable; unrefereed preprint","sources":[]}],"related":[{"id":"2026-09-30-ai-assisted-conjecture-wave-summer-2026","url":"https://postcutoff.com/e/2026-09-30-ai-assisted-conjecture-wave-summer-2026/","date":"2026-09-30","date_precision":"day","short_title":"Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help","deck":null,"takeaway":"This entry catalogues about 50 of them, with the AI role as the authors state it.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":11,"official":5,"filed":"2026-09-30","updated":"2026-10-19","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"]},{"id":"2026-09-03-gpt-6-astra","url":"https://postcutoff.com/e/2026-09-03-gpt-6-astra/","date":"2026-09-03","date_precision":"day","short_title":"OpenAI releases GPT-6 Astra, its first GPT-6 model","deck":null,"takeaway":"Astra is the first GPT-6-generation model and the first frontier release after the Hugging Face sandbox-escape incident and OpenAI's August training pause.","category":"model-release","category_label":"Model releases","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Confirmed"]},"sources":20,"official":11,"filed":"2026-09-29","updated":"2026-10-19","orgs":["OpenAI"]},{"id":"2026-09-01-claude-fable-5-1-mythos-5-1","url":"https://postcutoff.com/e/2026-09-01-claude-fable-5-1-mythos-5-1/","date":"2026-09-01","date_precision":"day","short_title":"Anthropic releases Claude Fable 5.1 and Claude Mythos 5.1","deck":null,"takeaway":"Fable/Mythos 5.1 was Anthropic's capability frontier until Opus 5.5 matched it three weeks later at less than half the price.","category":"model-release","category_label":"Model releases","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Confirmed"]},"sources":9,"official":5,"filed":"2026-09-29","updated":"2026-10-19","orgs":["Anthropic"]},{"id":"2026-07-24-claude-opus-5","url":"https://postcutoff.com/e/2026-07-24-claude-opus-5/","date":"2026-07-24","date_precision":"day","short_title":"Anthropic releases Claude Opus 5","deck":"Near-Fable-5 intelligence at half the price","takeaway":"Opus 5 brought most of Fable 5's capability to half the price.","category":"model-release","category_label":"Model releases","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Confirmed"]},"sources":9,"official":4,"filed":"2026-09-29","updated":"2026-10-29","orgs":["Anthropic"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-19","type":"edit","text":"Status confirmed → pending (unrefereed preprint; the Lean statement is the authors' own and not independently audited); added the abstract's AI disclosure verbatim"},{"date":"2026-09-30","type":"filed","text":"Created"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-09-30","run":null,"sources_read":null,"updated":"2026-10-19","human_review":null,"version":{"date":"2026-10-19"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":132,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":71,"in_training_data":false},{"model_id":"gemini-4-hydrogen","name":"Gemini 4 Hydrogen","cutoff":"2026-03","days_after":162,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":101,"in_training_data":false}],"short_url":"https://postcutoff.com/s/pierce-birkhoff-conjecture"}