An aggregator account slapped a siren on it and wrote that AI had proven the Jacobian Conjecture false. That framing is built to be shared and half wrong, which is the usual pattern. The odd part this time is that the thing underneath the hype is real, and I can say so because the math is cheap to check and it checks out.
Here is what actually happened. On July 19, 2026, the number theorist Levent Alpoge posted a single casual message saying the Jacobian Conjecture is false, thanking a friend named Akhil for asking about it and thanking Claude Fable 5, an Anthropic model, for doing the work during the World Cup final. He then wrote down an explicit map from three-dimensional complex space to itself and listed three points that all land on the same image point. That is the whole disproof. No hundred-page manuscript, no press embargo, just the object itself.
What the conjecture claimed
The Jacobian Conjecture is one of those problems you can state to a strong calculus student and then watch swallow decades of effort. Take a polynomial map from n-dimensional complex space to itself. Compute its Jacobian determinant. If that determinant is a nonzero constant, the conjecture says the map must have a polynomial inverse. In plain terms, a locally invertible-looking polynomial map should be globally invertible.
It was first written down for two variables by Ludwig Kraus in 1884 and stated in general by Ott-Heinrich Keller in 1939, which is where the 87 years comes from. Shreeram Abhyankar later made it famous as a trap: easy to say, brutal to settle. It sits on Smale's list of problems, and it has a graveyard of published proofs that later cracked under inspection. True for one variable, open for everything above it, over both the real and complex numbers. That was the state of play until last summer.
One deeper fact makes the word disproof precise instead of loose. By the Ax-Grothendieck theorem, an injective polynomial self-map of complex space is automatically bijective. So the real content of the conjecture is injectivity. If you can build a map with constant nonzero Jacobian that sends two different inputs to the same output, you are done. Injectivity is the whole game, and this counterexample plays exactly there.
The map, and why it is decisive
Alpoge's map takes (x, y, z) and returns three polynomials. The first is (1+xy) cubed times z, plus y squared times (1+xy) times (4+3xy). The second is y plus 3x(1+xy) squared times z, plus 3xy squared times (4+3xy). The third is 2x minus 3x squared y minus x cubed z. The Jacobian determinant works out to minus two, a nonzero constant, which is the whole precondition the conjecture cares about.
Then the collision. Three distinct points, (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2), all map to (-1/4, 0, 0). Three inputs, one output. That is not injective, so it cannot be invertible, so the conjecture is false. I ran the algebra through a computer algebra system myself, symbolically rather than numerically, and both facts came out exact: the determinant is genuinely the constant minus two, and all three points genuinely collapse to the same image. There is a small joke buried in it too, since that image point (-1/4, 0, 0) is just the first preimage (0, 0, -1/4) with its coordinates shuffled.
This is why the story is not a repeat of the usual crank claim. A proof is expensive to verify and easy to fudge in a subtle step. A counterexample is the opposite. Anyone with a laptop can confirm or destroy it in seconds. This one survives the check. What remains is the ordinary work of a formal write-up, peer scrutiny, and probably a Lean formalization, but none of that is going to make three points stop landing on one.
The part worth thinking about
Strip away the siren and you are left with something that should interest anyone who cares about how mathematics actually gets made. A real mathematician, not a demo team, pointed a frontier model at a specific named problem, asked it to hunt for a counterexample in low dimension where such a thing was thought unlikely, and it found one. The model did not prove a sweeping theorem or replace the human. It did the part machines are good at, searching a construction space with relentless bookkeeping, while the human supplied the question and the judgment about what would count as an answer.
Kevin Buzzard, who runs the Xena formalization project, wrote about this the next day under the title "Human mathematicians are being outcounterexampled," and his angle is the right one. The interesting frontier is not whether a model can output a claim. It is whether the human community can state conjectures faithfully and verify machine-produced results by machine. A counterexample you can formalize is worth more than a proof you have to take on trust.
Alpoge himself framed it with a joke that lands harder if you are Indian. The image he attached was Srinivasa Ramanujan, with the caption "it was revealed to me in a dream," the word dream struck through and replaced with "chat session."
Ramanujan credited a goddess for results he could not always justify, and the mathematics turned out right anyway. There is a long tradition of insight arriving before the proof that tames it, and the source of the insight has never been the thing that made it true. The verification did. That is still the case here. The model was the source. The symbolic check was the proof.
A few caveats you should keep straight so you are not the one spreading the distorted version. This is the complex conjecture, not the strong real Jacobian conjecture that Sergey Pinchuk already disproved in the plane back in 1994; those are different statements and people confuse them constantly. The formal paper and independent scrutiny were still pending when the tweet went out, so treat it as announced and verifiable rather than fully processed through the machinery. And the "AI did it alone" reading is the one to drop. A mathematician asked a sharp question, a model did a hard search, and the answer was something you can hold in your hand and test. Eighty-seven years, and it ends with three points that refuse to stay apart.