An internal OpenAI reasoning model generated a geometric construction that disproves Paul Erdős's 1946 unit‑distance

An internal reasoning model from OpenAI has produced a construction that disproves Paul Erdős’s unit‑distance conjecture (1946), a longstanding prediction about how many pairs of points at exact unit distance can appear among N points in the plane. OpenAI released the result along with a companion paper prepared with nine external mathematicians who verified, shortened, and annotated the model’s proof; coverage lists the publication date as May 21, 2026. The finding matters because it breaks a widely accepted candidate for an optimal arrangement and illustrates that AI can suggest non‑intuitive approaches in pure mathematics.

The unit‑distance problem asks for the maximum number of unit‑distance pairs among N planar points; Erdős conjectured a near‑optimal arrangement roughly based on a slightly skewed square grid. OpenAI’s construction produces noticeably more unit‑distance pairs than that classical grid: Will Sawin of Princeton estimates the gain at roughly one percent more pairs per doubling of the point count. Despite the improvement, a longstanding theoretical upper bound from 1984 — proved by Spencer, Szemerédi, and Trotter — still lies above the new construction, so the problem is not closed in the sense of matching upper and lower bounds.

What surprised collaborating mathematicians was the model’s choice of methods. Rather than working in classical planar geometry, the model drew on algebraic number theory and specialized number fields — techniques tied to class field theory — to exploit internal symmetries of complex number systems and produce especially dense point patterns. Those tools are not ones most planar geometers would naturally apply to unit‑distance questions.

The construction’s central technical idea also departs from obvious generalizations. Instead of enlarging a single extended number system, the model kept scale fixed inside each chosen number field and switched to progressively richer number systems at successive steps. Sawin says the reverse strategy — enlarging scale within one field — tends to revert to Erdős’s bound; the specific interleaving of scale and field choice was not anticipated by the collaborating mathematicians.

The companion paper assembled nine external mathematicians to verify the model’s output, shorten its argument, and annotate the steps so human readers can follow the logic. Their work both confirms the result and clarifies the unusual algebraic constructions the model employed, making the proof tractable for specialists who might otherwise dismiss machine‑generated arguments as opaque.