The Jacobian Conjecture lasted 87 years, then one morning with Claude
A number theorist spent a morning querying an AI model, posted twelve lower-case words on X, and a problem that had stood since 1939 was finished. The mathematics can be checked in about a minute. Everything around the mathematics is harder.
On 19 July, the number theorist Levent Alpöge posted this:
hello there the jacobian conjecture is false thanx
No paper. No preprint. No press office. Twelve words in lower case, on a platform, in a post format that expires after seven days. He and a friend had spent the morning querying Claude about a problem that had survived since 1939.
By the following morning strangers on three continents had checked it, and it held.
source video: AI Copium, “Claude Just Did the Impossible”, 7 min 35.
What the conjecture actually says
Ott-Heinrich Keller posed it in 1939, and it is one of those problems that sounds like it should be a warm-up exercise.
Take a polynomial map from complex space to itself. Feed in three numbers, get three numbers back, using nothing but addition and multiplication. Now compute its Jacobian determinant, which measures how much the map stretches or squashes space at each individual point. The conjecture says that if that determinant is a non-zero constant, the map must have a polynomial inverse. You can always get back.
The intuition is nearly physical. A non-zero constant determinant means the map never collapses anything to nothing, anywhere, ever. There is no point where space gets crushed flat. It seemed obvious that something which never folds locally cannot end up overlapping itself globally.
It was not obvious. It was wrong. But it was wrong in a way that took 87 years and a great many careers to establish. Stephen Smale put it on his 1998 list of problems for the twenty-first century, at number sixteen. It acquired a reputation for swallowing proofs whole: dozens of serious attempts published in serious journals, then quietly withdrawn. In 1994 Sergey Pinchuk produced a counterexample to the real version, which sharpened the frustration rather than easing it, because the complex version was always the one that mattered.
Three polynomials and one point
Here is what ended it. Three polynomials in three variables, small enough to fit on a beer mat.
the map, as circulated on the Secret Blogging Seminar, 20 July 2026.
The Jacobian determinant of that map is exactly minus two. Constant, non-zero, Keller condition satisfied, no argument available.
And three genuinely different points of complex three-space all land on the same place, which is (-1/4, 0, 0). The map is generically three-to-one. Something that hits the same point three times has no inverse at all, polynomial or otherwise. Conjecture over.
Both of those facts are checkable by exact arithmetic. Not numerically, where rounding could hide a lie. Exactly, in integers and fractions. Anybody with SymPy installed can confirm the determinant and the collision in about a minute, on a laptop, in a café.
The proof of a counterexample is the counterexample. There is nothing to referee. You either compute the determinant or you do not.
Why this is not just a post on X
Verification arrived faster than the news did. Within a day the polynomials had been run through SymPy by several people independently, Qiaochu Yuan had a Sage script doing the same thing linked from MathOverflow, and Jared Duker Lichtman had walked the three-point collision through by hand. The Secret Blogging Seminar put the map up in full on 20 July so everyone was checking the same object.
On 21 July, Terence Tao published a digestion of the counterexample, offering a geometric account that, in his phrase, minimises the miracles required. He was clear that miracles remain. The explicit map still looks to him like a massive one, because of the scale of the cancellation involved: expressions that ought to run to enormous degree collapse to a constant. In the same post he mentioned using an AI chatbot to discuss the problem and to confirm several of the calculations, which is its own quiet data point.
By 22 July there was a preprint. T. Shaska put a paper on the arXiv that treats the counterexample as settled and starts organising the wreckage: the map turns out to be equivariant for a weighting of its three variables, and the sign pattern of that weighting decides what is possible. Where the weights are all positive, no counterexample can exist at all. In two dimensions, no sign pattern works either.
Which means the two-dimensional case is still open, and the field has already moved from arguing about whether to arguing about why. That is what a real result looks like from the outside.
We pointed the same thing at a cipher and it lost
We have run our own version of this experiment, at a much smaller scale, and it is worth putting next to it.
Last month we spent four articles pointing a frontier model at the Farnese cipher of 1539, roughly 6,300 digits of enciphered Italian, published on MysteryTwister at Level X, seven years online with zero solves. We were very confident in part one. By part four we had stopped trusting and started measuring, and the cipher was still standing.
Lab / four parts
We pointed an AI at the Vatican's last unsolved cipher
Part one, in which we are very confident. It ends with part four, in which the cipher remains politely undefeated. The two failures in between, the book that beat the computer and the idea nobody had published, are the useful ones.
Same class of tool. Comparable effort. Opposite outcome. The difference is not intelligence, and pretending it is intelligence is how people end up with the wrong expectations in both directions.
The difference is the shape of the problem.
A cipher is a search where you cannot tell whether you are close. A partial decryption of 6,300 digits looks exactly like noise until the instant it does not. There is no cheap oracle, so every candidate costs a human judgement, and a machine that generates a thousand candidates an hour is generating a thousand hours of work.
The conjecture is the exact inverse. Producing a candidate is brutally hard, which is why it took 87 years. Checking one is trivial, instant and exact. Compute a determinant. Test a collision. No opinion involved, no waiting, no referee.
An 87-year-old problem fell to a tool that could be told, immediately and without argument, every time it was wrong.
The part that should make you cautious
There is no complete prompt transcript. There is no account of how many candidates were produced and discarded before this one. Anthropic has said nothing. Alpöge credited Fable without a version number, and the Fable 5 in most of the headlines is secondary inference rather than anybody's statement. He worked with a friend, and whether the framework came from the humans and the algebra from the model, or the other way round, is not on record anywhere.
The mathematics is fully auditable. The discovery is not. Those are different claims and they are being reported as one.
There is also a detail nobody has a satisfying answer to. The counterexample is low degree, with small integer coefficients. It is not buried in some vast region that classical search could never have reached. It was, in retrospect, sitting fairly close to the surface. So why did 87 years of extremely capable people miss it? Nobody knows yet. That is either evidence the model did something genuinely creative, or evidence that the space was better organised than anyone believed. It is probably both, and the two are harder to separate than either camp wants.
Nor was it one model. In the comments on the Secret Blogging Seminar thread, Will Sawin reported that ChatGPT, prompted by a different mathematician entirely, had produced most of an elegant geometric construction for the same object, which suggests a more conceptual explanation is out there waiting. Whatever happened here was not one lucky run of one system.
What this changes for everyone else
You are not disproving conjectures. The lesson still travels further than it looks.
Before pointing one of these things at anything that costs money, work out how you would check the answer. Not whether the answer looks plausible, because they are all built to look plausible. How you would check it, cheaply, repeatedly, without booking a meeting. If you can state that check, you can let the machine run wide and be wrong most of the time, because being wrong is nearly free and one correct answer pays for all of it. If you cannot state the check, speed is the wrong thing to be buying.
Every deployment of AI we have seen actually work in production has a cheap verifier hiding somewhere inside it. Code that compiles and passes its tests. Draft copy a human reads in nine seconds. A determinant that either equals minus two or does not. The ones that quietly fail are the ones where nobody can say what correct would look like until the invoice arrives, which is most of the reason we spend so long on scope in our guide to custom AI agent development.
The two-dimensional case is still open. Somebody is asking about it right now, and by the time it gets written up we will all have got used to the idea.