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When the Undergrad Beats the Experts
Here's something wild: for decades, mathematicians believed a certain famous conjecture about reversing mathematical functions was probably true. They just couldn't prove it. Then someone used an AI to search for a counterexample — and found one that's literally short enough to fit in a tweet.
I don't know about you, but that story absolutely fascinates me.
Levent Alpöge, working at Anthropic (the company behind the AI Claude), recently announced on X (formerly Twitter) that he'd used Claude Fable 5 to find a counterexample to the Jacobian conjecture. And when I say "found," I mean literally discovered proof that the conjecture is wrong — for certain dimensions, at least.
This isn't just another "AI does something cool" story, though. There's something almost poetic about how this played out.
So What Exactly Is the Jacobian Conjecture?
Let me try to break this down in a way that actually makes sense — because honestly, the Wikipedia explanation makes my head spin.
Imagine you have a function, which is basically a machine where you put in numbers and get out other numbers according to some rule. Now, the Jacobian conjecture is about whether you can always reverse this machine — whether there's another function that undoes what the first one did.
Think of it like this: if your function moves points around in space, the conjecture says that as long as the function doesn't crush or fold space in a certain way (mathematicians measure this using something called the "Jacobian determinant"), you should always be able to find a reverse function that puts everything back where it started.
Sounds reasonable, right?
The problem is, nobody had ever been able to prove this was always true. And now, thanks to some help from AI, we know it's actually false in certain cases.
The Plot Twist: Simpler Than Anyone Expected
Here's what's really got mathematicians buzzing: the counterexample that Alpöge found is tiny. Like, embarrassingly small. It fits in a single social media post.
You see, the tricky part about finding a counterexample to this conjecture is that you need to construct a very specific kind of function — one that both doesn't fold space AND can't be reversed. Those requirements sound contradictory, but somehow they can coexist in certain dimensions.
And here's the kicker: experts had thought such examples might not exist, or would be incredibly complex. Instead, Claude found something so short and elegant that, as one Math Stack Exchange user noted years ago, "some smart undergraduate could simply write a formula" that would break the whole thing.
Well, turns out that user was right. The formula is short enough that other mathematicians verified it almost immediately after Alpöge posted it.
Why Should You Care?
I know what you're thinking: "Great, some math thing that probably doesn't affect my life. Why should I care?"
Fair question. Here's my take: this story isn't really about the math. It's about what it represents.
We're living through a moment where AI is becoming a genuine partner in human creativity and discovery — not just a tool that does narrow tasks, but something that can actually help us think in new ways. The fact that an AI model released just a few weeks ago could help crack a problem that's nearly 90 years old? That's remarkable.
This also follows a pattern we're seeing more and more: AI making genuine contributions to fields that seemed immune to automation. Mathematicians, physicists, programmers — these aren't professions people expected to be "disrupted" by AI. But here we are.
The original two-dimensional version of the conjecture (from 1884!) still hasn't been resolved, by the way. So there's still work to do. But for dimensions three and higher? Case closed.
The Bigger Picture
What I find most interesting is what this tells us about intuition in mathematics. For decades, experts believed the Jacobian conjecture was probably true. They built careers around that belief. And now, a quick search by an AI has shown that intuition was wrong — at least for higher dimensions.
There's a lesson there about how even brilliant people can miss something obvious when they're looking in the wrong direction. Maybe sometimes you just need a fresh pair of eyes — or a fresh set of neural networks — to point out what's right in front of everyone.
Whatever your take on AI, I think we can all agree: these are strange and exciting times to be alive.
Source: ScienceDaily