ChatGPT Formally Verifies Erdős Conjecture, Ushering in a New Era of Automated Mathematical Proofs

In May 2026, ChatGPT stunned the mathematics community by refuting the long-standing open problem in discrete geometry, known as Erdős’s unit distance conjecture. This groundbreaking achievement was followed by an even more remarkable development: OpenAI’s new AI model, “Sol,” fully formalized the entire proof using the Lean theorem prover, ensuring its correctness based solely on mathematical axioms.

According to a report on xenaproject.wordpress.com via Lobsters by sanxiyn, formal verification expert Xenaproject (hereafter referred to as “the author”) described the event as an instance of “human mathematicians being outperformed by counterexamples,” signaling a new phase in the automation of mathematical proofs by AI.

Refutation of the Erdős Conjecture

The Erdős unit distance conjecture poses a question about the maximum number of point pairs in a plane that can share the same distance (specifically, a unit distance). This problem had remained unsolved for decades.

The crux of ChatGPT’s refutation lies in its utilization of a profound theorem in number theory proven by Golod and Shafarevich in the 1960s. The proof of this theorem spans over 100 pages and requires extensive knowledge of global class field theory, a complex area of mathematics developed in the early 20th century. To date, no shorter proof has been discovered, and condensing it has proven extremely challenging.

According to the author’s report, several human mathematicians examined and validated the correctness of ChatGPT’s refutation. The author stated, “I know a few of these mathematicians, and some of them are highly trustworthy. They indicated that they understood the argument.”

Efforts in Formal Verification

The author, a long-time proponent of theorem proving using Lean, initially questioned whether the counterexample generated by ChatGPT had been formalized in Lean. The answer at the time was “no.”

However, the situation quickly evolved. On May 26, 2026, Fields Medalist Mike Freedman, who serves as Chief Scientific Officer at Logical Intelligence, contacted the author via email. Freedman reported that their system had fully formalized the entire proof generated by ChatGPT in Lean.

Upon verification by the author and postdoctoral fellow Thomas Browning, it was confirmed that Logical Intelligence had precisely formalized the proposition—that the profound number-theoretical theorem led to the refutation of the Erdős conjecture. This achievement marked an unprecedented milestone, where mathematics generated by a large language model (LLM) was formalized in real time.

Complete Formalization by Sol

A significant hurdle remained: the complete formalization of the Golod-Shafarevich theorem itself. Understanding this theorem requires knowledge of global class field theory, and as of 2025, formalizing global class field theory was considered “almost an illusion,” the author recalled.

On June 26, 2026, the situation transformed dramatically. OpenAI researcher Boris Alexeev announced on Lean’s Zulip chat platform that the Erdős refutation had been entirely formalized using the new Sol model, guided by ChatGPT. A major breakthrough was also achieved by Edison Xie, a postdoctoral researcher, who completed the formalization of local class field theory.

Alexeev publicly released the code, which the author promptly reviewed. The proof, based solely on mathematical axioms, was fully formalized by AI.

Transforming Mathematical Research

This landmark accomplishment heralds a fundamental shift in mathematical research methods. In the traditional framework, human mathematicians verify proofs generated by AI. However, the increasing length and complexity of proofs threaten to surpass human comprehension. Automated verification using systems like Lean offers a practical solution to this dilemma.

The author, who had previously predicted that “Lean would handle significant mathematical proofs better than humans by 2040,” noted that reality is progressing faster than anticipated. The synergy between AI-generated proofs and formal verification has the potential to revolutionize the mathematical research process.

Editorial Opinion

This achievement demonstrates that the automation of mathematical research by AI has transitioned from theoretical possibility to practical application. In the short term, the adoption of formal verification systems like Lean is expected to accelerate in key areas of mathematics, becoming a standardized tool for validating AI-generated proofs. Fields such as number theory and algebraic geometry, where proofs are often extensive and intricate, are likely to benefit the most as human verification becomes increasingly impractical.

In the long term, the seamless integration of AI-driven proof generation and formal verification could redefine the role of mathematicians—from “discoverers of proofs” to “supervisors and interpreters of AI outputs.” This shift may also influence the content of mathematical education and the nature of doctoral research.

From an editorial perspective, a crucial challenge lies in ensuring the “interpretability” of AI-generated proofs. While formal verification guarantees correctness, it remains a separate question whether humans can understand these proofs and derive new insights from them. Mathematics is not merely a collection of proofs; it is a field that advances through human understanding and insight. The impact of AI-human collaboration on the cultural dimensions of mathematics will be an essential topic for future consideration.

References

• “Human mathematicians are being outcounterexampled”, by xenaproject.wordpress.com via sanxiyn — Lobsters, 2026-07-20T22:16:00.000Z (ARR)
• Source URL: https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/