On a Sunday night in July, while most of the internet was watching Spain and Argentina play out a World Cup final, a small post appeared on X from a mathematician named Levent Alpöge. He thanked a friend for asking about an old problem, and thanked another friend, an AI model called Fable, for doing the work during the match. What he had found, he said, was a counterexample to the Jacobian conjecture. It had taken 87 years to get here.
AI Generated Illustration
The Jacobian conjecture has sat untouched since 1939, when German mathematician Ott-Heinrich Keller proposed it. Stephen Smale later put it on his own list of the defining math problems of the 21st century, alongside the Riemann hypothesis and P versus NP. Generations of mathematicians tried to prove it. None succeeded, and none could disprove it either, which is its own kind of stubborn.
What makes this moment strange is how small the answer turned out to be. Alpöge, working with Anthropic's Claude Fable 5, produced a counterexample that fits comfortably inside a single social media post. Three lines of polynomials, 216 characters, and a problem that outlasted entire academic careers was suddenly in question. That gap between the size of the answer and the size of the problem is exactly why mathematicians are arguing about it. Did the AI actually solve something, or did it just get lucky finding a needle that was always sitting in the haystack?
To understand the argument, it helps to see what Fable 5 actually built, and then what happened after Alpöge hit post.
How AI Reached the Three-Line Solution
The Jacobian conjecture makes a claim about a specific kind of mathematical machine: a polynomial map, something that takes a point in space and moves it to a new point using a formula built entirely from adding, multiplying, and raising to powers. The conjecture says that if this machine's local stretching factor, its Jacobian determinant, stays a fixed nonzero number everywhere, then the machine must be reversible. Feed in the output, and you should always be able to recover the exact input you started with.
Think of it like a shredder that always cuts paper into the same pattern of strips no matter what you feed it. The conjecture bet that this consistency would guarantee you could always reassemble the original page. Alpöge's counterexample says otherwise. He built a map from three-dimensional space to itself with a Jacobian determinant fixed at negative two everywhere, then showed that three completely different starting points, (0, 0, negative one quarter), (1, negative three halves, thirteen halves), and (negative one, three halves, thirteen halves), all land on the exact same output. Same shredding pattern, three different original pages producing identical scraps.
Where Fable 5 came in was the search itself. Mathematicians had long suspected such an example might exist, since building either property alone is easy and it is only the combination that resists. What the model did was closer to an exhaustive, guided search through that combination space than a single flash of insight, and Alpöge has been clear it took real back and forth rather than one lucky prompt. What matters for credibility is what happened next. Within hours, other mathematicians ran the polynomials through computer algebra systems and checked the arithmetic by hand. Terence Tao published a detailed walkthrough of the construction on his blog. As of this writing, the result sits at the pre-print stage, tracked on an open registry as pending formal peer review rather than fully closed out. The arithmetic checks out. Whether the wider mathematical community treats it as settled is a separate question, and that gap is where the real disagreement lives.
Why Mathematicians Are Still Divided
A three-line disproof of an 87-year-old conjecture sounds like it should end all debate. Instead it opened one. Part of the friction is that a counterexample is a strange kind of mathematical object. It does not need to explain itself. It only needs to exist and be checkable, which this one clearly is. But existing and being checkable is not the same as being understood.
Mathematicians build proofs to communicate reasoning, not just conclusions. A good proof shows why something is true in a way that teaches the reader a technique or pattern they can reuse elsewhere. Alpöge's counterexample, by contrast, is a set of coordinates that happens to work. Nobody, including Alpöge, has fully explained why this particular combination of terms broke the pattern rather than any of the countless others that do not. Fable 5 found it. It did not, in any complete sense, explain it.
That distinction is the heart of the pushback. A correct answer is not necessarily the same thing as a convincing proof. Columbia mathematician Andrew Blumberg has argued the example is real but thin, offering little insight into the deeper structure of the problem, while Fields medalist Timothy Gowers has acknowledged the achievement while noting it does not carry much theoretical weight on its own. Neither is disputing the math. Both are asking what kind of contribution this actually is, and that question does not have a clean answer yet.
Breakthrough or Mathematical Shortcut?
Set the philosophy aside for a moment and look at what actually happened technically. Fable 5 was not just testing random formulas until one worked. Alpöge and other mathematicians who examined the construction afterward found it connects to older, known structures in algebraic geometry, including a link to the separate and equally famous Dixmier conjecture. A pure brute-force hit would be an isolated fact. A result that ties back into existing theory is closer to a genuine discovery, even if the AI did not narrate the connection itself.
This is not the first time a machine has reshaped how mathematicians work. Computers have been doing heavy lifting in proofs since the 1976 four-color theorem, which relied on a program to check thousands of map configurations no human could verify by hand. What is different now is the nature of the labor. The four-color proof ground through an already-understood, finite set of cases. Fable 5 was not checking a known list. It was navigating an open-ended space of possible formulas, guided by something closer to judgment about which directions were worth pursuing.
What would actually settle the debate is not more opinions but process: a completed peer review, independent reproduction of the construction from scratch by a separate team, and ideally a clean explanation of why this counterexample exists that a human mathematician can defend on its own terms. Until those three things happen, the result occupies a strange middle ground: too solid to dismiss, too unexplained to fully claim.
What This Could Change for Mathematics
Strip away the argument over credit and there is a genuinely useful capability underneath. Many open conjectures fail not because nobody has the right idea, but because the space of possible counterexamples is too vast for any one person to search by hand. A mathematician can spend a career testing plausible constructions and never happen across the one that works. That is precisely the kind of search where a model that can generate and check millions of candidates quickly starts to matter.
The payoff would not stay contained to pure math. Polynomial maps and their invertibility show up in robotics, cryptography, and the nonlinear systems engineers model constantly. A tool that can probe those structures faster changes the pace at which adjacent fields test their own assumptions, not just settle abstract puzzles for their own sake.
The same capability carries a genuine catch. If a model becomes good enough to find results that are correct but not fully explainable, mathematics risks accumulating a pile of true statements nobody fully understands. A field built on shared, checkable reasoning does not obviously benefit from a growing stack of verified but opaque facts, even if each one individually passes inspection.
The Bigger Question Is Still Unanswered
It is worth being precise about what has and has not happened here. The Jacobian conjecture is disproven for three dimensions and, trivially, for every dimension above it. The original two-dimensional version, the one Keller actually proposed in 1939, remains completely open. Formal peer review has not concluded. Calling this a closed case is premature, no matter how fast the arithmetic checked out online.
The more interesting story might not be this one conjecture at all. It is whether a system like Fable 5 can do this again, reliably, on problems where the answer is not as easy to verify as a list of three coordinates. One striking result during a soccer match is a headline. A track record of verifiable contributions across different fields of math is a shift in how the discipline works.
That leaves the real question hanging over all of it: if an AI system can hand mathematicians proofs that check out but that no human fully understands, what exactly counts as discovery? The answer that field settles on will shape not just how this particular conjecture gets remembered, but how the next 87-year-old problem gets solved, and by whom.
