Pierce–Birkhoff conjecture (1956) disproved by a multi-agent GPT + Claude harness; counterexamples Lean-verified
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
Science result
- Field
- mathematics / real algebraic geometry
- Problem
- Pierce–Birkhoff conjecture (open since 1956)
- 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.
- 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
- Lean formalisation of the counterexamples (n = 5, 6, 7, 30, 72); preprint, not peer-reviewed
- Status
- confirmed
- Why surprising
- A 70-year-old conjecture fell to a mixed OpenAI–Anthropic agent team after single models failed.
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.
Changelog
- 2026-09-30: created
Related events
- OpenAI releases GPT-6 Astra, its first GPT-6 model ★★★★★
- Anthropic releases Claude Opus 5 — near-Fable-5 intelligence at half the price ★★★★
- Anthropic releases Claude Fable 5.1 and Claude Mythos 5.1 ★★★★★
- Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help (July–September catalogue) ★★★★
Sources (2)
- paperarXiv 2609.10420: Pierce-Birkhoff conjecture is false (Lai, Lim, Ren)
- codeGitHub: 7pocheR/Pierce-Birkhoff (Lean formalisation)
id: 2026-09-09-pierce-birkhoff-conjecture-disproved · updated 2026-09-30 · open in the interactive timeline