u/NutInBobby

OpenAI posted a math result today claiming that one of its general-purpose reasoning models found a construction disproving the conjectured n^{1+O(1/log log n)} upper bound in Erdős’s planar unit-distance problem.

Announcement:

https://openai.com/index/model-disproves-discrete-geometry-conjecture/

Proof PDF:

https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-proof.pdf

Abridged reasoning writeup:

https://cdn.openai.com/pdf/1625eff6-5ac1-40d8-b1db-5d5cf925de8b/unit-distance-cot.pdf

The mathematical claim, as I understand it, is that there are finite planar point sets with more than n^{1+δ} unit distances for some fixed δ > 0 and infinitely many n. That would rule out the expected near-linear upper bound, though it does not determine the true asymptotic growth rate.

What seems especially relevant for this subreddit is the process claim: OpenAI says the solution was produced by a general-purpose reasoning model, then checked by an AI grading pipeline and reviewed/reworked by mathematicians. The proof PDF also includes the original prompt given to the model, but not the full experimental details: no model name, sampling setup, number of attempts, compute budget, hidden system prompt, or full grading pipeline.

Curious how people here read this as an ML result. Is this best viewed as evidence of frontier models doing genuine autonomous research, or as a cherry-picked but still important sample from a large search process? What kind of disclosure would you want before treating this as a reproducible AI-for-math milestone?

OpenAI says one of its general-purpose reasoning models found a construction disproving the conjectured n^{1+O(1/log log n)} upper bound in Erdős’s planar unit-distance problem.

The linked post includes a proof PDF and an abridged chain-of-thought writeup. The proof statement says the original model output was later checked by an AI grading pipeline and by mathematicians, and that Will Sawin simplified and strengthened the argument.

The mathematical claim, as I understand it, is that there are finite point sets in the plane with more than n^{1+δ} unit distances for some fixed δ > 0 and infinitely many n, so the expected near-linear upper bound cannot hold. The true growth rate is still open, with the classical upper bound around O(n^{4/3}).

Curious what people here think of the construction itself, especially the use of Golod-Shafarevich/class-field-tower ideas in what looks at first like a discrete geometry problem.