Yuval Avidani
Author
An 87-year-old mathematical conjecture that nobody managed to disprove since 1939 fell over the weekend - because of a short polynomial that a researcher named Levent Alpoge found together with Claude Fable 5, one that anyone can verify on any computer in seconds. And we're not just trusting the headline. We ran the verification ourselves, and it worked. Here's the full story, and the real lesson it teaches about AI in science.
Let's break this down slowly, because underneath "AI solved a math problem" there's a much more precise and interesting story.
What the Jacobian Conjecture Actually Says
Let's start with the problem itself, in plain language. Imagine a function that takes three numbers and returns three new numbers, where each output is a polynomial (addition, subtraction and multiplication of the inputs, no division). The Jacobian Conjecture, formulated by mathematician Ott-Heinrich Keller in 1939, says one thing: if this "map" has a constant Jacobian determinant that's never zero, then the map must be invertible.
What's a "Jacobian determinant"? Think of it as a local stretch-meter for the function: at every point, it tells you how much the function is stretching or squeezing the space around it. If that meter is constant and never hits zero, the intuition says the function "behaves nicely" everywhere - and therefore, maybe, you can always take a step backward (invert it). That's the intuition the conjecture tried to turn into a theorem.
The problem: this intuition was never proven. The conjecture is easy to state, but it turned out to be one of the most stubborn puzzles in mathematics. It's easy in one dimension, still open to this day in two dimensions, and the new counterexample disproves it in three dimensions.
The Counterexample: Where the "Nice Behavior" Breaks
Here's the idea behind disproving something. To knock down a conjecture, you don't need a complete theory. You just need to find one example that breaks it - a "counterexample." In our case: a function that satisfies every condition (constant determinant, never zero) and yet still isn't invertible.
That's exactly what Alpoge found. His polynomial, only about 216 characters long, gives a constant Jacobian determinant of minus 2 - perfectly legal under the conjecture - and yet three different points all map to the exact same point. And if three different points land on the same output, you can't "go backward" (which of the three do you pick?), so the map isn't invertible. The conjecture is false.
Terry Tao, one of the greatest living mathematicians, wrote on his blog that this construction "looks like a huge miracle": a degree-seven polynomial should have produced a complicated determinant, but all the non-constant terms cancel out exactly. In numbers: you need to solve roughly 1,329 coefficient equations with only 360 degrees of freedom. Finding a needle like that in a haystack by hand is nearly impossible - and that's where AI comes in.
We Don't Trust the AI. We Check.
And here's the point that, to me, is the whole story. When someone claims they proved a new theorem, we have to go through the proof line by line and trust there's no hole in it. But a counterexample is a completely different animal: you can just plug in the numbers and check. It doesn't matter if a human found it, an AI, or a monkey with a keyboard - either the determinant comes out to minus 2 and the three points collide, or it doesn't.
So that's exactly what we did. We took the polynomial, ran it through SymPy (a symbolic math library in Python), and asked it to calculate the determinant and the three points. Here's the code, and you're welcome to run it yourself:
The output speaks for itself: the determinant comes out to exactly minus 2, and the three different points - (0, 0, minus a quarter), (1, minus a half, six and a half) and (minus 1, a half, six and a half) - all land on the exact same point. That's what makes this disproof bulletproof: it doesn't require trust, just a calculator.
So What Did the AI Actually Do Here
Let's be precise, because the headline "Claude disproved the Jacobian Conjecture" is both true and misleading. The AI didn't wake up one morning and decide to solve an 87-year-old problem. Alpoge, a skilled mathematician, framed the problem: he knew what kind of counterexample to aim for, and how to translate the search into equations. The AI served as a search engine over an astronomical space of possibilities - exactly the spot where a human brain breaks down and raw computing power wins.
To me, this is the pattern that defines AI-in-science-that-actually-works: the human asks the right question, the AI searches through a haystack no human could ever search alone, and the math - not "trust" in the model - does the final verification. This also explains why it worked specifically for a disproof: the result is instantly checkable. For a full positive proof, where every single line needs to be correct, it's still a lot harder.
An Important Caveat: It's Not Over Yet
It's important not to get ahead of ourselves here, and we're sticking to the facts. As of now the result is marked as "pending peer review," and it hasn't yet gone through full formalization in Lean (a proof-checking language where a computer verifies every logical step). The two central checks - that the determinant is minus 2, and that the three points collide - have been independently confirmed by many researchers using tools like SymPy and Wolfram Alpha, and Terry Tao himself "digested" it on his blog. Alpoge's original post on X racked up over 20 million views. But "computationally confirmed" and "passed full peer review" are two different things, and it's worth waiting for the second one.
Bottom Line
To me, this is one of the moments that defines 2026 - not because "AI solved math," but because it showed exactly where it's a wonderful tool: searching a massive haystack, with the human framing the question and the math verifying the answer. The beautiful part here isn't that we trusted Claude - it's that we didn't have to. The numbers check themselves.
So if disproving an 87-year-old puzzle comes down to a polynomial that anyone can verify on their laptop in seconds - how many other stubborn puzzles are waiting for exactly this combination of human, machine, and verification?