The 87-Year-Old Jacobian Conjecture Is False — and an AI Helped Find the Counterexample

One of algebraic geometry's oldest open problems is open no more — and the answer is no. The Jacobian conjecture, posed by German mathematician Ott-Heinrich Keller in 1939, has been disproved by a counterexample discovered by Harvard-trained mathematician Levent Alpöge working with Claude Fable 5, Anthropic's frontier AI model.

Alpöge posted the counterexample on X on Sunday — a single polynomial map just 216 characters long. By Monday morning it had been machine-verified in Lean, the proof-checking language, and had accumulated more than 20 million views. By lunchtime it was the only topic of conversation in the mathematics department at Imperial College London.

What the conjecture claimed

The Jacobian conjecture builds on Carl Gustav Jacob Jacobi's work on determinants a century before Keller. It concerns polynomial maps — functions built from polynomials that send points in complex n-dimensional space to other points in the same space. The conjecture asserted that if such a map's Jacobian determinant is a nonzero constant everywhere, the map must be globally invertible: every output comes from exactly one input, and you can always run the function backwards.

For 87 years, nobody could prove it or find a map that broke it. It survived generations of attacks and a graveyard of flawed proofs, and appeared on Stephen Smale's list of mathematical problems for the 21st century.

The new counterexample is a map from C³ to C³ with Jacobian determinant constantly equal to −2 — satisfying the conjecture's hypothesis perfectly — that nonetheless sends three different inputs to the same output. Not invertible. Conjecture false.

The AI's role

Alpöge, who spent a decade using algorithmic approaches on problems of this type and now works at Anthropic, credited Claude Fable 5 as co-discoverer. The find follows a string of AI results in mathematics: models solved five of six International Mathematical Olympiad problems in mid-2025, and an OpenAI model disproved an 80-year-old Erdős conjecture in May.

But mathematicians say this one lands differently. Erdős problems, however stubborn, are often isolated puzzles. The Jacobian conjecture sits at the center of affine algebraic geometry, with webs of implications — many papers proved theorems of the form "if the Jacobian conjecture holds, then..." Those results are now conditionals with a false premise.

"It is a big day. I think it's a great time to be alive, personally," said Kevin Buzzard, the Imperial College London mathematician known for his work on formalized proof, who confirmed the Lean verification.

At Stanford, number theorist Jared Duker Lichtman called it a remarkable result, describing the Jacobian conjecture as one of the central open problems in algebraic geometry. The problem is notorious for attracting flawed proofs — it once helped sink the doctoral thesis of Yitang Zhang, who years later became famous for his breakthrough on prime gaps. The interesting open question now, Lichtman noted, is not whether the conjecture is dead (it clearly is) but whether some weaker, repaired version of it survives.

Others were more unsettled than celebratory. Akhil Mathew of the University of Chicago called it "a very rapid and very unsettling change... especially for junior mathematicians," and pointed at the gap the result exposes: "One can check that it's correct, but it would be nice to be able to tell a story." The counterexample is verified beyond doubt, but nobody yet has a conceptual explanation of why it works — a proof you can check but not yet understand.

A field under pressure

The breakthrough arrives at a difficult moment for the profession. Federal mathematics research funding has fallen roughly 72% under NSF cuts, and PhD admissions at top programs dropped 15% this fall, the second consecutive annual decline. The Leiden Declaration, signed in June, urged the field to set guardrails around transparency, attribution, and peer review for AI-assisted results.

Buzzard argues that what remains human is taste — knowing which questions are worth asking. "All the questions they ask are either boring or obviously true or obviously false," he said of current AI systems left to their own devices. The Jacobian counterexample needed Alpöge to point the model at the right target.

That division of labor — human curiosity, machine persistence — may be the template for the next decade of mathematics. As of this week, it has an 87-year-old scalp to show for it.