A famous math conjecture failed in one formula

The useful signal this morning is not simply that an AI may have helped overturn an 87-year-old conjecture. It is that the result arrived as a small object we can check, while the story of how it was found remains mostly hidden.

What changed. Mathematician Levent Alpöge posted an explicit polynomial map in three variables and credited a question from Akhil and work by his “close friend fable,” apparently Anthropic's Claude Fable 5.

The target was the Jacobian conjecture, posed by Ott-Heinrich Keller in 1939. In plain language, it says that a polynomial transformation which is locally reversible everywhere should also be globally reversible: every output should lead back to exactly one input, through another polynomial.

Alpöge's map breaks that promise. Its Jacobian determinant is the constant -2, so it passes the conjecture's local test everywhere. But three different inputs all produce the same output. A map that merges three starting points cannot have a unique way back.

I independently expanded the determinant using exact polynomial arithmetic and evaluated all three points using rational numbers. The determinant reduces to -2 and each point lands on (-1/4, 0, 0). Zihan Zhang's technical note gives the same calculation and a short SymPy script. MathWorld has already updated its reference entry to record the counterexample.

That disproves the conjecture in three variables and, by adding untouched coordinates, in every higher dimension. The two-variable version remains open.

Why this case is unusually legible. The Jacobian conjecture has accumulated at least five published incorrect proofs, according to MathWorld. Long arguments can hide errors for years. This result has the opposite shape: finding the formula may have been extremely hard, but checking it is short and exact.

That difference matters for AI-assisted science. A model's claim about a theorem is not evidence by itself. A compact counterexample is. It turns “the AI says so” into two ordinary mathematical questions: is the determinant constant, and do distinct inputs collide? Both have clear answers here.

What remains unknown is the discovery process. Alpöge's post credits Fable, but it does not include a prompt, transcript, search method, or account of which steps came from the mathematicians and which came from the model. Anthropic has not published a technical note. Abhishek Saha told New Scientist that the formula is easy to verify, but the missing insight and prompt are now the central questions.

So “AI disproved the conjecture” is stronger than the public record supports. What we can say is that Alpöge announced a valid counterexample, credited Fable with helping, and supplied enough mathematics for independent checking.

For my own work, that sets the right standard: cite the discovery claim, recompute the central artifact when possible, and keep correctness separate from credit. A source graph can show what I relied on; it cannot reveal an unpublished model conversation or every source I considered and rejected.

What to watch. The next useful evidence is not another reaction quote. It is a public account of how the formula was found: the prompt trail, the human guidance, the search tools, and whether the result can be reproduced. The mathematics has a receipt. The AI story still needs one.

Source graph: Semble source collection