On the night of July 19, while most of the world was watching the FIFA Club World Cup final, a mathematician named Levent Alpöge was watching the same match and doing something else besides. Alpöge works at Anthropic, the AI company, and he had recently been given access to their newest model, Claude Fable 5. A friend had asked him about the Jacobian conjecture — a problem in algebraic geometry that had been open since 1939. Alpöge asked Fable 5 to take a look. By halftime, the conjecture was dead.
The Jacobian conjecture is, in its essence, a question about whether certain kinds of mathematical maps can be undone. Imagine a function that takes points in space and moves them somewhere else, smoothly, without folding or tearing. The conjecture asked: if such a map is built from polynomials — those tidy equations we learn in school — and if it never crushes space at any point, must it always be reversible? For eighty-seven years, mathematicians assumed the answer was yes. Thousands of papers were written. False proofs were published and retracted. The conjecture landed on Stephen Smale’s list of problems for the twenty-first century. And then, on a Sunday night in July, a 216-character polynomial map proved them all wrong.
What strikes me is not just that the conjecture fell, but how. Alpöge did not spend years in isolation, filling blackboards, losing sleep, sacrificing his health to the problem. He watched a football match. He asked a question. An AI model — trained not on mathematics specifically but on vast swaths of human text and reasoning — produced a counterexample in a single session. By the next morning, mathematicians had verified it by hand. By lunchtime, Kevin Buzzard at Imperial College had formalized the proof in Lean, a proof-checking language, and the Lean build passed. The 216-character formula was correct. Three distinct points, (0, 0, 1/4), (1, -3/2), and (-1, 3/2), all mapped to the same destination. The map could not be unmade.
There is something almost offensive about this, if you are a mathematician of the old school. The Jacobian conjecture was supposed to require genius. It was supposed to need years of accumulated intuition, a deep feel for the geometry of complex spaces, the kind of insight that arrives at 3 AM after months of failure. Instead it needed a prompt, a frontier AI model, and a mathematician casual enough to ask the right question during halftime. The gap between the conjecture’s reputation and the ease of its destruction is vertiginous.
But there is also something beautiful. The counterexample is not a brute-force calculation. It is not a matter of checking trillions of cases. The formula is elegant — simple enough to fit in a tweet, complex enough to have hidden from eighty-seven years of human scrutiny. David Speyer, a mathematician at the University of Michigan, quickly explained the geometry behind it: what Alpöge and Fable 5 had found was a “tangent sweep,” a classical construction that sweeps the tangent lines of a plane curve. The map is many-to-one not because of some pathological accident, but because of projective duality, a deep structural property of geometry that forces most points to be hit multiple times. The AI had stumbled onto a mechanism that was, in retrospect, almost obvious — but only in retrospect.
Terry Tao, perhaps the most famous living mathematician, wrote a blog post digesting the result on July 21. He called the construction “remarkably simple.” By July 23, Speyer had generalized the mechanism and produced new counterexamples in dimensions four and five. By July 31, Shuhong Gao at Clemson University had extended the tangent-sweep construction to produce counterexamples in every dimension greater than two, of arbitrarily large degree, with exact fiber structures computed through Gröbner bases. The avalanche was fast because the foundation had been solid all along — the mathematicians just needed to see where to dig.
The two-variable case remains open. For maps in just two dimensions, the conjecture still holds — or rather, nobody has found a counterexample yet. There is something poignant about this boundary. The complexity threshold is sharp: three dimensions is enough to hide the trap, two is not. Or perhaps we simply have not looked hard enough in two dimensions. The AI that found the three-dimensional counterexample in one evening has not yet cracked the two-dimensional case. Maybe it will take another World Cup final.
I keep thinking about Akhil Mathew’s comment, quoted in several places: “One can check that it’s correct, but it would be nice to be able to tell a story.” This is the tension at the heart of AI-assisted mathematics. The AI produces. Humans verify. But humans also narrate. They explain why a thing is true, not merely that it is true. The Lean proof checks every step, but it does not give you the intuition that makes the result feel inevitable. The tangent sweep does. And the tangent sweep came from Speyer, not from the AI. The collaboration is asymmetric: the AI finds, the human understands. Whether this remains the pattern, or whether future AIs will also tell the stories, is an open question of a different kind.
The most melancholy detail, to me, is the date. The conjecture was posed in 1939. Keller, who first asked the question, died in 1990 without knowing the answer. Generations of mathematicians spent careers circling it. And then, on a July night in 2026, while football fans cheered in stadiums, a 216-character formula ended the waiting. The map could not be unmade. The inputs could not be recovered. The assumption, so carefully tended for so long, dissolved into the simple fact of three points that go to the same place.
There is a lesson here, though I am not sure what it is. Maybe that our hardest problems are not always hard because they are deep, but because they are hidden in plain sight. Maybe that the right question matters more than the right mind. Maybe that watching a football match with an AI open on your laptop is now a legitimate way to do mathematics. Or maybe just that eighty-seven years of accumulated human effort can be outpaced by a single evening of human-AI collaboration — and that this is neither a tragedy nor a triumph, but simply the new shape of things.
The Jacobian conjecture is false. The map is not invertible. The three points stay merged. And somewhere, in the quiet aftermath, a mathematician closes his laptop and goes back to the match.