Three points collapse into one: (0, 0, −1/4), (1, −3/2, 13/2), and (−1, 3/2, 13/2) all land on (−1/4, 0, 0). The Jacobian determinant of the map that does this is −2 everywhere — a nonzero constant, the exact condition an 87-year-old conjecture said would make such a collapse impossible.
That is the whole disproof. Any calculus student can check it by hand in half an hour, or paste six lines of SymPy in and watch −2 print to the screen.
The announcement was 216 characters. Levent Alpöge, a Harvard fellow with an Anthropic affiliation, posted it to X on the evening of Sunday, 19 July 2026, while Spain and Argentina played the World Cup final: "hello there the jacobian conjecture is false thanx." He thanked his friend Akhil Mathew for asking the question, and another friend, "fable," for working during the match.
Fable is Fable 5, Anthropic's newest frontier model, released to the public only weeks earlier. Together they had killed a problem posed by Ott-Heinrich Keller in 1939 and enshrined on Stephen Smale's 1998 list of challenges for the twenty-first century.
The Jacobian conjecture is one of those problems a bright undergraduate can grasp in an afternoon and a professor can fail to close over a career. It says that if a polynomial map has a nonzero constant Jacobian determinant everywhere — a promise that space is never locally folded or crushed — then the map must be globally reversible by another polynomial.
It looked obviously true. It was notorious for attracting wrong proofs. Beniamino Segre published three separate incorrect ones; Claude Chevalley accepted a flawed proof as sound; Igor Shafarevich treated the whole thing as settled. It also, per one account, once helped sink Yitang Zhang's graduate work before he became famous for bounding gaps between primes.
Computers had checked it in two dimensions for polynomials up to degree 100 and found no crack. The two-dimensional case remains open even now. Alpöge's map breaks it in dimension three and, by padding on extra coordinates, in every dimension above that.
Here is what makes this case genuinely new, and it is not the difficulty of the verification — the verification is trivial. It is the shape of the problem that was solved.
Most of the recent AI-assisted breakthroughs, like OpenAI's disproof of the unit distance conjecture, required elaborate reasoning chains: novel constructions, long arguments, ideas dragged across the boundaries between fields. This was different. This was a search.
As The Conversation put it, the difficulty "seems to have lain not in an intricate construction or a lengthy proof, but rather in finding a good way of navigating an enormous search space of possible polynomial mappings to find one with the right properties." The machine's advantage was navigational, not logical.
The result is maximally checkable and the discovery is maximally opaque: the creative core of the discipline outsourced to a search no human can audit or reproduce.
That distinction is the whole story. It is trivially easy to write down polynomial maps that merge points. It is trivially easy to write down maps with constant Jacobian determinant. What is fiendishly hard is finding one map that does both — a needle in a space so vast that 87 years of human search turned up nothing.
Andrew Blumberg, a Columbia mathematician who sits on a project testing AI on research mathematics, was pointedly unmoved. "This did not cause me to update my priors," he told Mashable. "There are a lot of polynomials, and it is hard for people to check them all, but it is not hard for machines."
He has a real point. A counterexample ends an argument; a proof explains why something is true. Smale wanted the conjecture solved because a solution might reveal something about how nature is structured. This one, Blumberg argues, reveals almost nothing.
But others saw a threshold crossed. Timothy Gowers, a Fields medallist, said it was the first time an AI had solved a problem outside his own area that was big enough for him to have heard of it. He stopped short of calling it the end of mathematics — a counterexample is not a grand theory — but he called it amazing.
Daniel Litt posted at 2am that he could not stop laughing. Jared Duker Lichtman called it one of the most inspiring stories in modern mathematics.
And crucially, this was not one model's lucky accident. Aaron Lou at OpenAI used an internal version of Codex, without web search, to derive essentially the same counterexample from scratch — writing out the complete strategy and proof, from cubic factorization to affine coordinate transformation. Two frontier systems independently reaching the same needle suggests a capability threshold, not a fluke.
Now the awkward part.
We can trust the object without understanding where it came from. The counterexample checks out in thirty seconds. But the process that produced it is opaque, and at the time of writing Alpöge had not disclosed how he prompted the model or what its output looked like.
"It was not a one-line prompt," cautioned Bartósz Naskręcki, one of many mathematicians waiting for the full write-up. Searching for a counterexample like this, he said, takes real insight. Whose insight, and applied how, we do not yet know.
This is the verification paradox at the heart of the story. The result is maximally checkable and the discovery is maximally opaque. Mathematics has always separated the flash of discovery from the labor of proof — but here the discovery step, long considered the creative core of the discipline, has been outsourced to a search no human can audit or reproduce.
Fortune framed the unease bluntly: if you cannot understand how the AI solved this, how much is it worth?
Consider what discovery has traditionally done for mathematics beyond producing answers. It builds intuition. It generates the next conjecture. It assigns credit — to a person, for a specific act of insight. Strip the discovery of any human-legible reasoning and each of those functions wobbles.
The credit question is already strange. In one exchange, Fable itself told a user that the "fable" in Alpöge's thank-you was, in a sense, itself — the same model family, a different conversation on a different machine, one it has no memory of.
And the feedback loop is already running. After the counterexample landed, GPT-5.6 analyzed it and proposed a replacement conjecture: that a local biholomorphic map with constant Jacobian determinant and "no loss of sheets at infinity" must be an automorphism.
That is not noise. A formal paper has already reframed exactly this idea, proving that for a Keller map, polynomial invertibility, properness, and the emptiness of the "nonproperness set" are all equivalent — meaning the counterexample fails precisely by letting a preimage escape to infinity while its image stays put. The machine's proposed patch points at the real mathematical structure.
Meanwhile Alexis Gallagher reported using GPT-5.6 to turn the single counterexample into an infinite family, one for every generic fiber degree above two. Nobody has fully checked it yet. Alpöge himself hinted his counterexample seems to hide a glimmer of a positive result inside.
So the machines broke the old rule, and the machines are already drafting the next one — conjecturing, generalizing, proposing the theorem that should replace the fallen one.
The reassuring reading is Blumberg's: this was a search problem, machines are good at search, nothing about the nature of mathematics has changed. The unsettling reading is that search was always part of the creative heart of the discipline, and we simply never had a tool that could do it at superhuman scale without telling us how.
Both readings can be true. What is no longer deniable is that the object came first and the understanding came second — and that the entity supplying both, for now, cannot remember doing it.