The Jacobian Conjecture is now false, an 87-year-old algebraic-geometry problem that Ott-Heinrich Keller posed in 1939. Levent Alpöge, a mathematician at Anthropic, on July 22, 2026 used help from Claude Fable 5 to construct a polynomial map that sends three-dimensional space to itself yet has no inverse. First posed in 1939 by Ott-Heinrich Keller, the conjecture now has a counterexample in every dimension above two, leaving only the original two-dimensional case open. Alpöge announced the result on X with a single line: "hello there the jacobian conjecture is false thanx."

Why a proposition held true for 87 years collapsed

The Jacobian Conjecture states that if a function built from polynomials has a Jacobian determinant equal to a nonzero constant, then a polynomial inverse must always exist. Since Keller first formalized it in 1939, it stayed one of the most famous open problems in algebraic geometry, drawing a long series of false proofs that were later withdrawn. The counterexample Alpöge presented is a polynomial map from three-dimensional space to itself whose Jacobian determinant is a constant −2, yet which sends two distinct points to the same point. Once a function loses injectivity this way, no inverse can exist, so the conjecture is immediately false. In mathematics a single counterexample is enough to bring down a conjecture, and this one was short enough to fit in a single X post.

The crucial point is that this result proves falsity, not truth. Alpöge did not try to show the conjecture holds; he constructed one concrete object that satisfies the hypothesis while violating the conclusion. A counterexample is also clean to check. Whether the stated map's Jacobian determinant is really constant, and whether two inputs really map to the same output, can be verified directly by hand or with a computer algebra system. That the verdict rests on a calculation humans can independently reproduce is what anchors the credibility of this event.

Who Alpöge is and what tool he used

Levent Alpöge is a number theorist who earned his PhD at Princeton and served as a member of Harvard's Society of Fellows, and is now at Anthropic. In 2015 he received the Morgan Prize, the top research award given to undergraduate mathematicians. The tool he used is Claude Fable 5, Anthropic's newest frontier model and the public version of the system the company once called Claude Mythos, describing it as too capable to release. In his X post, Alpöge thanked both the friend who encouraged him to take on the problem and the model.

A caveat matters here: exactly how Alpöge prompted the model and what its output looked like has not been made public. So whether the model produced the counterexample whole, or whether a human set the direction while the model helped with candidate calculations, cannot yet be stated with certainty. What is certain is that the final counterexample exists in a form humans can verify; the precise division of credit awaits a detailed disclosure.

How this differs from earlier AI math news

Counterexample hunting, not theorem proving, is what sets this Claude Fable 5 result apart from earlier AI math news. Where earlier cases focused mainly on aiding proofs, constructing or searching for arguments for propositions that look true, this one was a search for a counterexample to disprove. Finding a counterexample means picking out, from a vast space of candidates, a single unusual object that meets the hypothesis but breaks the conclusion, which favors sweeping wide candidate sets that human intuition rarely reaches. If language models can ease this kind of exploratory work for human mathematicians, AI's contribution to mathematics widens beyond theorem proving into a separate axis of "counterexample hunting."

Still, restraint is needed not to overstate the difference. One counterexample does topple a conjecture, but understanding why it exists and from what structure it arises is a separate matter. A fallen conjecture is usually reformulated in a more refined form, and that is where the real mathematics begins. The true value of this result lies less in the verdict that the conjecture is false and more in opening a new question: in what kind of polynomial maps does injectivity break down.

How to read the numbers

Impressive figures and figures to read carefully should be separated here. Impressive are the "87 years" and the "one" counterexample. That a problem top algebraic geometers could not settle for well over eight decades was resolved into a single verifiable object is meaningful in itself, whatever the tool. What deserves caution is the summary that "AI solved it." What ultimately confirmed the counterexample as correct was human calculation, not the model, and it was a trained mathematician who chose the problem, narrowed the conditions, and posed it to the model.

Another easily misread point is dimension. This counterexample broke the conjecture in three and higher dimensions, but the most-discussed two-dimensional case remains open. The accurate sentence is not "the Jacobian Conjecture is fully solved" but "it is shown false in high dimensions while the two-dimensional case stays open." That gap between the headline's blunt "solved" and the actual mathematical state must be held precisely, through the numbers.

What it means for research and practice in Korea

The Alpöge result sends two practical signals to Korean mathematics and AI researchers. First, frontier language models should now be seen not as machines that hand over answers but as collaborators that widen a trained expert's search radius. The human role of formalizing the problem and verifying the model's candidates does not disappear; if anything, that verification capacity governs how trustworthy the result is. Second, it reaffirms that verifiability is itself the safeguard. On problems where humans can independently reproduce the result, as with a counterexample, a model's errors or hallucinations struggle to contaminate the final conclusion.

That principle extends beyond mathematics. In work whose output can be checked mechanically, such as code, formal specifications, and numerical computation, a language model's contribution is relatively safe. Conversely, when a model is used for narrative conclusions with no verification apparatus, the human burden of confirmation remains intact. Half the reason this achievement is attractive is that the problem came in a verifiable form, and how to build that condition into one's own work is the practitioner's task.

Open questions and remaining limits

Several questions about the Alpöge counterexample stay open, from how much Claude Fable 5 contributed to whether the result reproduces under independent review. Exactly how much the model contributed to deriving the counterexample, and whether the same result was reachable without human intervention, cannot be answered before a detailed disclosure. Reproducibility must also pass community scrutiny, and whether the original post's counterexample survives independent confirmation by fellow mathematicians is the next gate. Above all, the two-dimensional Jacobian Conjecture remains open, so this event is closer to the start of a narrower problem than the end of one. The big picture, that AI is reshaping the terrain of mathematics as a tool, is clear, but drawing the exact boundary of that contribution is the task from here on.

Source: The Conversation, "'hello there the jacobian conjecture is false thanx'" (July 22, 2026), with reporting from Fortune, Fast Company, and The Next Web (Levent Alpöge disproved Keller's 1939 Jacobian Conjecture with a three-dimensional counterexample using Claude Fable 5; a non-invertible polynomial map with constant Jacobian determinant −2; the two-dimensional case remains open), summarized by ASAP.