A simple AI agent built on GPT-5.5 Pro disproves the Erdős unit distance conjecture

What is it about?

The unit distance problem was introduced by Paul Erdős in 1946. It asks how to arrange a given number of points in the plane to maximize the number of pairs separated by a fixed distance. Erdős conjectured a limit on how quickly the maximum number of such pairs can grow as the number of points increases. In May 2026, OpenAI announced that an internal AI model autonomously disproved the conjecture. Here we present a simple autonomous agent built on GPT-5.5 Pro, a model publicly released before OpenAI’s announcement. It runs a three-round conversation: brainstorm strategies for disproving the conjecture, execute a strategy, and critically review the resulting argument. We ran eight independent trials with tools and web search disabled. In each trial, the agent produced a correct or essentially correct disproof of the conjecture. All eight disproofs were independently verified by the author. The resulting disproofs are diverse and do not merely reproduce earlier AI-generated ones.

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Photo by Vaaris Chitkara on Unsplash

Photo by Vaaris Chitkara on Unsplash

Why is it important?

Earlier autonomous disproofs relied on models or agents that are not publicly available, and the amount of computation used remains undisclosed. We release our code and all model outputs, providing the first publicly accessible AI agent to resolve this conjecture autonomously. We also report reasoning-token usage. These disclosures support independent reproduction and analysis by the community. GPT-5.5 Pro was released nearly a month before OpenAI published its disproof, so no public proof refuting the conjecture could have been included in its training data.

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