OpenAI's reasoning model disproves Erdős unit-distance conjecture

OpenAI announced that one of its general-purpose reasoning models discovered a counterexample to a long-standing conjecture in discrete geometry: Erdős's unit-distance problem. The conjecture posited that the maximum number of unit distances among n points in the plane is near-linear, roughly n^{1+O(1/log log n)}. The model found a construction achieving more than n^{1+δ} unit distances for some fixed δ>0 and infinitely many n, thereby disproving the conjectured bound. The proof was validated through an AI grading pipeline and further refined by human mathematicians. OpenAI released the original prompt, the abridged chain-of-thought, and a formal proof PDF.

However, the announcement lacks critical experimental details: the model name, sampling setup, number of attempts, compute budget, and hidden system prompt were not disclosed. This opacity makes it difficult to assess how much of the discovery was autonomous reasoning versus guided search. The mathematical result itself is significant if verified, but the ML community views it as an intriguing yet incompletely documented milestone. Full reproducibility would require open benchmarks, sampling parameters, and ablation studies to confirm the model's generalization ability beyond this cherry-picked example.