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An OpenAI Model Has Disproved a Central Conjecture in Discrete Geometry

Source: OpenAI Blog \ Date: 2026-05-20

TL;DR

An internal OpenAI general-purpose reasoning model has disproved the Planar Unit Distance Problem, a central conjecture in discrete geometry first posed by Paul Erdős in 1946. The model constructed an infinite family of configurations achieving $n^{1+\delta}$ unit-distance pairs (where $\delta$ is a fixed constant), a polynomial improvement over the $n^{1+o(1)}$ that was thought optimal. External mathematicians — Noga Alon, Tim Gowers, Arul Shankar, and Jacob Tsimerman — verified the proof. A refinement by Princeton mathematician Will Sawin shows $\delta$ can be taken as 0.014.

Tim Gowers: "If a human had written the paper and submitted it to the Annals of Mathematics and I had been asked for a quick opinion, I would have recommended acceptance without any hesitation."

The Problem

Place $n$ points in the plane. What is the maximum number of pairs that can be exactly distance 1 apart? This is the function $u(n)$, known as the Planar Unit Distance Problem.

  • Previous best construction: A rescaled square grid achieved $n^{1 + C / \log \log n}$ — the exponent trends toward 1. This was thought essentially optimal for decades.
  • Best upper bound: $O(n^{4/3})$ (Spencer, Szemerédi, & Trotter, 1984).
  • Supporting evidence: Work on non-Euclidean distances showed that "most" distances obey the conjecture, lending weight to the hypothesis.

The AI's Approach

  • Model type: A new general-purpose reasoning model — not a system specifically trained for mathematics or scaffolded for proof searching.
  • Discovery context: The model was evaluated on a collection of Erdős problems as part of a broader test of whether advanced AI can contribute to frontier research.
  • Mathematical surprise: The solution invokes sophisticated algebraic number theory (infinite class field towers, Golod–Shafarevich theory) — tools previously unknown to have implications for discrete geometry. The proof replaces Gaussian integers with more complicated generalizations with richer symmetries.
  • Chain of thought: The model showed a strong predisposition to construct a counterexample rather than prove the upper bound.

Arul Shankar: "In my opinion this paper demonstrates that current AI models go beyond just helpers to human mathematicians — they are capable of having original ingenious ideas, and then carrying them out to fruition."

Why This Matters

  1. First time a prominent open problem has been solved autonomously by AI — a milestone for machine reasoning.
  2. Unexpected mathematical depth — the solution required sophisticated algebraic number theory, not brute-force search.
  3. Cross-pollination potential — number theorists are now expected to look at other open problems in discrete geometry.
  4. Verification by top mathematicians — the proof was reviewed by leading experts who wrote a companion paper explaining its significance.

Thomas Bloom: "What other unseen wonders are waiting in the wings?"