Proof of Work: Mathematicians and AI

What happened. In July 2025, several AI models solved five of six problems at the International Mathematical Olympiad, and Terence Tao, one of the world’s leading mathematicians, described 2025 as the year the tools became broadly useful. In October 2025, Ernest Ryu, then a mathematician at UCLA, spent roughly 12 hours with ChatGPT and solved a problem that had been open for 42 years. Tao, Javier Gómez-Serrano of Brown, and two DeepMind mathematicians ran Google’s AlphaEvolve on 67 open problems, beating the best known answer on 23 and matching it on 36 in a November 2025 paper. In January 2026, Ravi Vakil, president of the American Mathematical Society, and colleagues proved a new result using two Gemini-based systems.

Why it matters. A year of AI development separates high-school competition problems from conjectures that had defeated professional mathematicians for decades. Gómez-Serrano estimates that roughly two-thirds of his working time now involves AI tools, and notes that a specialist might have obtained any one of the AlphaEvolve results in a few months, while his team obtained comparable results across unfamiliar fields in a day or two. Grant cycles, tenure review, and hiring committees are calibrated to a rate of output that is being exceeded.

What happened. On July 20, Levent Alpöge, a number theorist who works at Anthropic, posted a counterexample to the Jacobian conjecture, crediting Claude Fable 5 for “working during the World Cup final.” Ott-Heinrich Keller proposed the conjecture in 1939, and T. T. Moh predicted in 2008 that a human solution might take another century. Other mathematicians checked it by hand and by machine within hours, and Tao published an explanation the following day. The work settles most of the conjecture and has not cleared journal peer review.

Twelve days later, OpenAI released ten results in mathematics and theoretical computer science produced by an internal version of Astra, its next major model, each problem open for at least a decade. OpenAI published a 249-page manuscript and a machine-checkable certificate for every result on GitHub under an open licence, at a compute cost Greg Brockman put at roughly $2,000. Eleven weeks earlier, when a model overturned a long-standing Erdős conjecture, validation required nine external mathematicians signing off.

Why it matters. Both results were accepted on the merits that a third party could check independently. Alpöge’s counterexample is short enough that any competent reader can confirm it in an evening with free software. Astra’s proofs are up to hundreds of pages, so OpenAI supplied certificates that a proof assistant can check line by line. The question for any other institution is what its own version of a machine-checkable certificate would be, and whether one exists. In law, medicine, engineering, and government, the answer is still that a reviewing authority reads the document and forms a judgment.

What happened. On June 2, an international group published the Leiden Declaration on Artificial Intelligence and Mathematics, and it was endorsed by the International Mathematical Union on the same day. It addresses verification, attribution, disclosure, research autonomy, funding, and unequal access. The Society for Industrial and Applied Mathematics’s editorial policy on AI requires authors to be human and able to defend every aspect of a submission directly, assigns responsibility for correctness to the authors, and bars reviewers from entering manuscripts into generative AI tools on confidentiality grounds. Thomas Bloom’s Erdős Problems database publishes categories separating cases where a model did all the work from genuine collaboration and ordinary assistance.

The limits are visible too. Some mathematicians have documented problems reported as solved autonomously that had in fact been solved years earlier, and cautioned that visible successes sit alongside a massive volume of unreported failures. Igor Pak doubts these systems will touch the hardest problems for a very long time, while Joel David Hamkins describes journals being overwhelmed by low-quality AI-generated submissions. No Millennium Prize problem has been solved.

Teaching changed faster than research. Hamkins stopped assigning homework because a substantial share of submissions now arrive AI-written. Robert Talbert of Grand Valley State concluded that he cannot meaningfully grade anything completed outside the classroom and now counts only in-class work. Princeton’s economics department moved to in-person-only final exams after students petitioned for the change. Oral examinations are returning across US campuses. Cornell’s Chris Schaffer, who introduced an oral defence in his biomedical engineering course, puts the rationale simply: “You won’t be able to AI your way through an oral exam.”

Why it matters. None waited for legislation. A professional body set norms, a publisher set submission conditions, and individual departments changed how credentials are earned. Each part is available to copy. Take-home work stopped functioning as evidence of learning for the same reason a plausible proof stopped functioning as evidence of a theorem, and both issues got the same answer: check in a setting where the output cannot be produced elsewhere.