OpenAI model disproves central conjecture in discrete geometry

OpenAI announced that an internal reasoning model has disproved a longstanding conjecture tied to the planar unit distance problem, a well-known question in combinatorial geometry first posed by Paul Erdős in 1946.

The problem asks how many pairs of points can be exactly one unit apart when n points are placed in a plane. For decades, mathematicians broadly believed that square-grid-style constructions were essentially optimal. OpenAI says its model produced an infinite family of examples that gives a polynomial improvement over that construction, disproving the expected n to the 1 plus o(1) upper-bound behavior.

The company says the proof was generated by a general-purpose reasoning model rather than a math-specialized system or a model scaffolded specifically for this problem. OpenAI also says the proof has been checked by external mathematicians, who wrote a companion paper explaining the argument and its mathematical context.

The result includes configurations of n points with at least n to the 1 plus delta unit-distance pairs for infinitely many values of n. OpenAI says the original AI proof did not provide an explicit value for delta, but a forthcoming refinement by Princeton professor Will Sawin shows delta can be taken as 0.014.

OpenAI published the proof, a companion paper, and an abridged version of the model’s chain of thought. The company describes the result as the first time an AI system has autonomously solved a prominent open problem central to a mathematical subfield.

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