AI Disproves 87-Year-Old Jacobian Conjecture in a Single Tweet

An 87-Year Mystery Resolved in One Line

On July 19, 2026, Harvard University mathematician Levent Alpöge posted a short message on X (formerly Twitter) that sent shockwaves through the global mathematics community. With a 216-character counterexample, he demonstrated that the Jacobian conjecture — a problem mathematicians had spent decades trying to prove true — is, in fact, false.

The Jacobian conjecture was first formally posed by German mathematician Ott-Heinrich Keller in 1939. It asks whether a particular type of polynomial mapping, one whose Jacobian determinant is a nonzero constant, must necessarily be bijective (one-to-one and onto). For 87 years, the problem resisted every attempt at proof or disproof, ultimately landing on Stephen Smale's famous 1998 list of 18 unsolved problems for mathematicians to tackle in the 21st century.

Now, thanks to a human-AI collaboration, the answer is confirmed: the conjecture is false.

"My Close Friend Fable"

Alpöge didn't take full credit alone. In his tweet, he attributed part of the breakthrough to his "close friend fable" — widely interpreted as a reference to Anthropic's flagship model, Claude Fable 5. He thanked Fable for working through the night during the FIFA World Cup final, underscoring the increasingly personal and collaborative nature of human-AI partnerships in research.

Anthropic has not publicly commented on the collaboration, and Alpöge has not yet published a full technical report detailing the methodology. The precise mechanics of how the AI contributed remain undisclosed. But experts note that the human-AI pairing appears to have unlocked an approach that neither could have reached independently.

What Experts Are Saying

Abhishek Saha, a mathematician at Queen Mary University of London, called it the most consequential conjecture AI has ever played a role in resolving. "Probably this is the biggest conjecture that AI has played a significant role in proving or disproving so far in mathematics," he said. "This is a pretty big deal. AI has made remarkable progress in the last year."

He noted that the counterexample, while short, was not the kind of result that could be found by brute-force search. "I don't know how he did it, what exactly was the prompt to give Fable, because if one were to search everything, it wouldn't quite work, so obviously there was some insight also which is not currently published." This suggests Alpöge's human mathematical intuition was essential in guiding the AI toward the right approach.

AI vs. the Grand Challenges of Mathematics

This is not the first time AI has cracked a famous open problem. Earlier in 2025-2026, an OpenAI model helped disprove the Erdős unit distance conjecture. But the Jacobian conjecture is regarded as a significantly harder and more historically significant target — and its refutation through human-AI collaboration is raising fundamental questions about the future of mathematical research.

Chris Bowman-Scargill at the University of York draws an important distinction between counterexample-finding and the deeper creative work of mathematics: "If you look at Fermat's last theorem, you had to create a hundred pages of new mathematics — you had to build a whole big theory in order to solve a conjecture. And often the interesting stuff in maths isn't 'oh, we've ticked off this conjecture, yay', it's more the stuff you have to build along the way."

Still, he acknowledges: "I think AI has sort of proven that it can do this, so you're now like, OK, what next?"

Is AI Coming for Mathematics PhDs?

Ivan Fesenko of Westlake University offers a sweeping forecast: "Right now, AI can already produce master's degrees in mathematics. In one year, they will produce PhD degrees in mathematics. And then the question arises, do we really need so many mathematicians around if AI can do such things so nicely? So basically we're talking about fundamental change in mathematics."

The vision Fesenko sketches is not of AI replacing human mathematicians outright, but of a profound restructuring — where human creativity operates at a higher level of abstraction, setting the directions while AI handles the exploration and verification.

Key Facts at a Glance

AI Milestones in Mathematics

The counterexample to the Jacobian conjecture is just 216 characters long — but it carries 87 years of mathematical history. More than a single result, it signals that AI is no longer merely a tool mathematicians use for computation or literature review. It is becoming a genuine intellectual collaborator capable of reshaping which problems are solvable, and how fast. In this new era of human-AI partnership, the boundary between what machines can explore and what humans must invent is shifting faster than most mathematicians expected.