Anthropic AI Disproves 87-Year-Old Jacobian Conjecture

An AI model developed by Anthropic has resolved the Jacobian conjecture, a mathematical problem that remained unsolved for 87 years. Levant Alpöge, an Anthropic employee, announced the breakthrough on Sunday, and by Monday morning the result had been verified through automated proof-checking systems.

The announcement drew more than 20 million views on X, reflecting both the significance of the achievement and growing public interest in AI's mathematical capabilities. Kevin Buzzard, a mathematician at Imperial College London, called it "a big day" and "a great time to be alive."

Why it matters

This breakthrough represents more than an isolated mathematical achievement. It signals AI's accelerating ability to tackle problems that have resisted human reasoning for generations, forcing the mathematics profession to confront fundamental questions about the nature of understanding, the value of human insight, and the future structure of mathematical research itself.

The problem and the solution

The Jacobian conjecture originated in 1939 with German mathematician Ott-Heinrich Keller. At its core, the problem concerns mathematical "maps" and whether certain conditions allow you to determine inputs from outputs. The conjecture built on work by Carl Gustav Jacob Jacobi from a century earlier.

Alpöge's result demonstrated that the conjecture was false. The AI found a case where the Jacobian determinant remains constant at −2 everywhere, yet three different starting points map to the same destination—a counterexample that disproves the conjecture.

A pattern of rapid progress

The Jacobian breakthrough follows a series of AI-driven mathematical advances. Language models first solved five of six International Mathematical Olympiad problems in mid-2025. In May, an OpenAI model disproved an 80-year-old conjecture by mathematician Paul Erdős. By June, 16 researchers from 15 universities published the Leiden Declaration on Artificial Intelligence and Mathematics, calling for guardrails around transparency and peer review.

The understanding gap

While the result is verifiable, it leaves mathematicians unsatisfied in a crucial way. "One can check out that it's correct," said Akhil Mathew, a University of Chicago mathematician who suggested the problem to Alpöge, "but it would be nice to be able to tell a story."

Current AI models deliver solutions without the explanatory framework that mathematicians value. Traditional mathematical proofs can run hundreds of pages and require months of expert review. They represent not just correctness but understanding—the ability to regenerate results from foundational ideas.

Buzzard, whose career project Lean enables machine verification of proofs, acknowledged that AI doesn't yet create the delicate, multi-hundred-step proofs that define advanced mathematics. Language models still bridge logical gaps with plausible-sounding but potentially incorrect reasoning.

The question of what remains human

Beyond calculation and even logical reasoning, Buzzard suggested that "taste"—knowing which questions to ask—may be mathematics' distinctly human element. "People have tried to get machines to ask questions, and they're abysmal," he said. "All the questions they ask are either boring or obviously true or obviously false."

The field's most celebrated problems are named for those who posed them, not those who solved them. The Riemann hypothesis, Keller's Jacobian conjecture—these represent the insight to identify which questions matter.

Meanwhile, the profession faces practical pressures. Federal funding for mathematics research has dropped roughly 72% under current administration cuts to the National Science Foundation. PhD admissions at top research universities fell 15% this fall, the second consecutive year of decline.

These details were first reported by Fortune.