Pierce–Birkhoff conjecture disproved by a multi-agent GPT + Claude harness
Counterexamples Lean-verified
Event confirmedAwaiting review
Importance: major (4 of 5)The 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).
Status
- Claim
Event confirmedAwaiting review
- Our reporting
- High confidence
- Verification
- Lean proof, not yet reviewed
- Importance
- Major (4 of 5)
- Last verified
- 19 October 2026
Your AI and this story
- GPT-6 Astra132 days after its cutoff
- Claude Opus 5.571 days after its cutoff
- Gemini 4 Hydrogen162 days after its cutoff
- Grok 4.7101 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 71 days before it.
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.’
What happened
Lek-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.
Why it matters
Pierce–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.
What is disputed or not yet verified
| Verification | 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 |
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Sources
2 sources from 2 sites. Numbers match the chips in the text.
2 sources: 2 primary
Primary
- arXiv 2609.10420: Pierce-Birkhoff conjecture is false (Lai, Lim, Ren)arxiv.org, paper
- GitHub: 7pocheR/Pierce-Birkhoff (Lean formalisation)github.com, code
Changes
- Status confirmed → pending (unrefereed preprint; the Lean statement is the authors’ own and not independently audited); added the abstract’s AI disclosure verbatim
- Filed