A single line of algebra, posted while most of the planet was watching Argentina play Spain, claimed to end a question mathematicians had left open since 1939.
That’s the short version. The longer version is stranger, and a lot more contested.
Mathematician Levent Alpöge announced, in a tone so casual it almost read as a joke, that he’d used Anthropic’s Claude Fable 5 to find a counterexample to the Jacobian Conjecture. He dropped it mid-final. And the post did what good math announcements almost never do: it went viral, pulling in more than 24 million views on X.
Here’s what he wrote, exactly as it appeared:
“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
The Jacobian Conjecture lives in algebra, and it made Stephen Smale’s 1998 list of the most important unsolved problems in mathematics.
Strip away the notation and it asks something you can picture. When can a function be reversed? If you know the output, can you always trace back to exactly one input?
For 87 years, the working assumption was yes. Under specific conditions, top mathematicians kept trying to prove the answer was always positive.
What Fable 5 turned up
The model found a case that breaks the rule. The function it produced has the right kind of Jacobian determinant, constant and nonzero, which is exactly the condition the conjecture cares about.
And yet it sends three different starting points to the same output.
That’s the whole ballgame. If three inputs land on one output, there’s no single inverse function. In math, one clean counterexample is enough to knock down a general conjecture. Early checks by mathematicians and symbolic computation tools say the counterexample holds, and people have already started picking apart its structure.
Who Alpöge is
This isn’t a random poster with a lucky prompt. Alpöge is a working mathematician with a real research record, a former member of Harvard’s Society of Fellows. He also appears to work at Anthropic, the company that built Fable 5.
Fable 5 is Anthropic’s most powerful widely available model, built for demanding reasoning, tangled problems and long-running tasks. It’s the public, more guarded version of the company’s Mythos technology, wrapped in dedicated safety mechanisms.
The skeptic’s case
Not everyone’s cheering, and the pushback is worth more than the hype.
Andrew Blumberg, a math and computer science professor at Columbia University, is keeping his distance from the celebration. As he explains it, there’s a real gap between a deep proof that surfaces new ideas and a counterexample that only shows a statement is false.
A machine can grind through an enormous number of polynomial combinations, the kind of tedious search that would wear a human down. That doesn’t mean the AI grasps what the problem means underneath.
It’s a fair line to draw. Finding the one function that breaks a rule is not the same as understanding why the rule was ever plausible.
Why it still matters
Even with that caveat, this is another sign that AI has moved into actual scientific research, not the marketing version of it.
In May, an internal OpenAI model knocked down a significant conjecture of Paul Erdős in discrete geometry. Now this. Two real results, months apart, from two different labs.
The open question isn’t whether AI will take part in mathematical discovery anymore. It’s how fast, and in what form, researchers learn to work alongside it. If you’re a mathematician, the useful move now is to treat a model like Fable 5 as a search engine for counterexamples you’d never have the patience to hunt by hand, and to keep Blumberg’s warning taped to your monitor: turning up the exception is the easy part.
