While much of the world had its eyes on Argentina and Spain, a lone line of algebra surfaced online claiming to settle a problem mathematicians had left hanging since 1939.
That’s the compact version. The fuller account is odder — and far more disputed.
In a tone so offhand it nearly read like a gag, mathematician Levent Alpöge said he’d used Anthropic’s Claude Fable 5 to uncover a counterexample to the Jacobian Conjecture. He posted it in the middle of the final. And the announcement pulled off something math news almost never manages: it went viral, racking up over 24 million views on X.
This is precisely what he posted:
“hello there the jacobian conjecture is false thanx to my close friend akhil for asking about it and my other close friend fable for working during the world cup final ((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x – 3 x^2 y – x^3 z): C^3to C^3,…”
What the conjecture actually asks
Rooted in algebra, the Jacobian Conjecture earned a spot on Stephen Smale’s 1998 roster of mathematics’ most important unsolved problems.
Peel back the notation and the idea is intuitive. When is a function reversible? Given an output, can you always work back to one and only one input?
For 87 years, the default answer was yes. Under certain conditions, leading mathematicians repeatedly attempted to prove that it always held.
What Fable 5 turned up
The model surfaced a case that violates the rule. Its function carries exactly the trait the conjecture hinges on — a Jacobian determinant that is constant and nonzero.
Yet it maps three distinct starting points onto a single output.
And that settles it. When three inputs collapse to one output, no single inverse function can exist. In mathematics, a single clean counterexample suffices to topple a general conjecture. Preliminary verification by mathematicians and by symbolic computation software indicates the counterexample stands, and researchers have already begun dissecting how it’s built.
Who Alpöge is
This wasn’t some anonymous user who got lucky with a prompt. Alpöge is a practicing mathematician with a genuine research pedigree and a past appointment to Harvard’s Society of Fellows. He also appears to be employed by Anthropic, the firm behind Fable 5.
Fable 5 stands as Anthropic’s most capable broadly available model, engineered for heavy-duty reasoning, knotty problems and extended tasks. It’s the public, more restrained edition of the company’s Mythos technology, encased in dedicated safety measures.
The skeptic’s case
Not everybody is applauding, and the criticism carries more weight than the excitement.
Andrew Blumberg, a professor of mathematics and computer science at Columbia University, is holding back from the party. By his account, there’s a meaningful difference between a deep proof that generates fresh insight and a counterexample that merely demonstrates a statement is false.
A machine can churn through a staggering volume of polynomial combinations — the sort of grinding search that would exhaust a person. But that isn’t evidence the AI comprehends the underlying meaning of the problem.
It’s a reasonable distinction. Locating the one function that breaks a rule isn’t the same as grasping why the rule seemed credible in the first place.
Why it still matters
Caveat aside, this is one more indication that AI has moved into actual scientific research — not the marketing gloss of it.
Back in May, an internal OpenAI model overturned a notable conjecture by Paul Erdős in discrete geometry. Now this one. Two genuine results, only months apart, out of two separate labs.
The question is no longer whether AI will participate in mathematical discovery. It’s how quickly — and in what shape — researchers figure out how to collaborate with it. For a mathematician, the practical takeaway is to treat something like Fable 5 as a search engine for counterexamples you’d never have the stamina to chase manually, while keeping Blumberg’s caution pinned to your monitor: surfacing the exception is the easy part.