An OpenAI model has disproved a central conjecture in discrete geometry

TL;DR

An OpenAI general-purpose reasoning model disproved the Erdős unit distance conjecture (planar unit distance problem), which had remained open for 80 years. This is the first case of AI independently solving an open problem in pure mathematics.

Authors & Org

• OpenAI (internal reasoning model team — specific authors not disclosed)
• External verification: Will Sawin (Princeton University) — refinement of lower bound (δ = 0.014)
• Published: 2026-05-20

Method

1. Chain-of-thought reasoning: The model generated long-running reasoning transcripts by "thinking out loud."
2. Application of algebraic number theory: Used Golod-Shafarevich theory + infinite class field towers to construct an infinite family of point configurations that surpass the square grid.
3. Counterexample discovery: A structure achieving at least n^(1+δ) unit-distance pairs across configurations of n points (with δ > 0 fixed).
4. Human mathematicians extracted the core: Expert mathematicians identified the key ideas in the long reasoning transcripts and rewrote them into a standard proof format.
5. External verification: Princeton mathematician Will Sawin refined the bound to δ = 0.014.

Results

• Disproved the central conjecture of the planar unit distance problem, posed by Erdős in 1946.
• Discovered an infinite family of point configurations that surpass the square grid structure, previously believed optimal for 80 years.
• A general-purpose model (not math-specialized) solved a frontier math problem — achieved through chain-of-thought reasoning alone, without a specialized system.

Significance

• First autonomous AI solution of an open math problem: Until now, AI's mathematical achievements were limited to solving known problems (IMO, AIME) or assisting verification. This is the first case of autonomous solution of a genuinely open problem.
• Scientific contribution of a general reasoning model: A general reasoning model, not a math-specialized model, made a frontier math contribution — suggesting broad scientific applicability of AI reasoning capabilities.
• "Year of Science" milestone: The strongest empirical case in OpenAI's 2026 strategic theme.
• Comparison with DeepMind Gemini Deep Think: DeepMind has led in math AI with an IMO gold medal, solving 18 open problems, and more — OpenAI's first peer-review-grade independent contribution in pure mathematics.

Open Questions

• Is δ = 0.014 a tight bound, or can it be improved further?
• Can this methodology generalize to other combinatorics/number theory problems?
• Will the collaboration model of "AI generates ideas and human mathematicians formalize them" become standardized?

Cite

"An OpenAI model has disproved a central conjecture in discrete geometry." OpenAI, 2026-05-20. https://openai.com/index/model-disproves-discrete-geometry-conjecture/

Related

• OpenAI — "Year of Science" strategy, reasoning model capabilities
• Reasoning Models — solving pure math problems via chain-of-thought reasoning
• Gemini 3.1 Deep Think — DeepMind's autonomous math research agent (Aletheia, IMO gold medal) — comparison target
• Co-Scientist (Google DeepMind) — scientific AI hypothesis generation (different domain)