The takeaway: A credentialed mathematician, crediting the AI model Claude Fable, has posted what he says is a counterexample to the Jacobian Conjecture — a problem open since 1939. It is a short, finite, checkable claim, so we checked it: On The Wire reproduced the arithmetic with exact symbolic computation, and the map does exactly what he says. But there is no paper, no formal proof, and — as of this writing, hours after the post — no independent expert has confirmed it. Treat this as a striking claim under verification, not a settled result.

Update — 20 July, ~06:30 BST: In the hours since we first published, the counterexample has held up. Multiple mathematicians have now independently checked the arithmetic — one, like us, with exact symbolic algebra — and found no error, and a Lean 4 formalization of the counterexample has been posted (deancureton/jacobian), stating the theorem for a general field, for characteristic ≠ 2 with Jacobian determinant −2, and specialised to ℂ. Because this is a finite, checkable object rather than a sprawling proof, that convergence is most of what verification means here. What is still outstanding is a written-up, refereed account and the wider community's formal sign-off — but the core question, "is the map a genuine counterexample," is now answered the same way by many independent checks: yes. The original caveats below record how the claim stood at first publication.

What the conjecture says

The Jacobian Conjecture, posed by Ott-Heinrich Keller in 1939, is one of the more famous open problems in algebra. In plain terms: take a polynomial map F from complex n-dimensional space to itself. Compute its Jacobian determinant — a single polynomial built from the map's partial derivatives. The conjecture says that if that determinant is a nonzero constant, then F must be invertible (a polynomial automorphism), and in particular it must be one-to-one. It has resisted proof for 87 years, and it is notorious for attracting flawed "proofs" and "counterexamples" — a genuine crank graveyard.

A real counterexample, then, is a single polynomial map whose Jacobian determinant is a nonzero constant but which is not one-to-one — two or more different inputs landing on the same output. Produce one explicitly and the conjecture is false.

The claim

On 20 July, Levent Alpöge posted exactly such a map. Alpöge is a serious number theorist — PhD at Princeton under the Fields Medalist Manjul Bhargava, then a Junior Fellow at Harvard's Society of Fellows — who has since joined Anthropic, the company that makes Claude Fable. He credited the counterexample to "fable" — the model itself — "working during the World Cup final," and gave an explicit map F: ℂ³ → ℂ³ with a claimed constant Jacobian determinant of −2, plus three distinct points that all map to the single image (−1/4, 0, 0). He attached Wolfram Alpha computations as a check. He did not post a paper or a formal (machine-verified) proof.

One thing to flag up front: this is an Anthropic researcher using Anthropic's own model to make the claim. That is not a mark against the mathematics — a counterexample is true or false regardless of who posts it, or what tool produced it — but it is a reason to want independent eyes on it rather than to take a vendor-adjacent announcement on trust. It is part of why we checked the arithmetic ourselves rather than simply reporting the claim.

We reproduced the arithmetic

Because this is a finite computation rather than a hundred-page argument, it can be independently verified — so we did, using exact symbolic algebra (no floating-point approximation). The map, as posted:

F1 = (1 + x·y)^3 · z  +  y^2 · (1 + x·y) · (4 + 3·x·y)
F2 = y  +  3·x·(1 + x·y)^2 · z  +  3·x·y^2 · (4 + 3·x·y)
F3 = 2·x  −  3·x^2·y  −  x^3·z

Our results matched his exactly:

• The Jacobian determinant is −2 — a genuine constant, with no variables left in it.
• The three points (0, 0, −1/4), (1, −3/2, 13/2) and (−1, 3/2, 13/2) are distinct, and all three map to (−1/4, 0, 0).

So the map satisfies the conjecture's hypothesis (polynomial, constant nonzero Jacobian) and violates its conclusion (it is not one-to-one). As stated, it is a valid counterexample. This part is not a matter of opinion: the arithmetic either holds or it doesn't, and for the map as posted, it holds.

What this settles — and what it doesn't

What is real is narrow but genuine: the specific map, as written, has the properties claimed, and anyone can verify it in a minute with a computer-algebra system.

What is not settled is nearly everything around it. There is no peer-reviewed paper. There is no formal, machine-checked (e.g. Lean) proof. No independent mathematician has publicly confirmed it in the hours since it appeared, and the discussion so far — a couple of small Hacker News threads, a wave of excited-but-cautious replies — is reaction, not verification. The claim also comes from the model's own maker rather than a disinterested third party. (We did at least remove one small doubt: we matched the map character-for-character against the posted tweet, so what we verified is exactly what Alpöge published.) Given this problem's long history of counterexamples that dissolved under scrutiny, the responsible reading is: a compelling, self-consistent claim from a serious source, awaiting the checks that turn a claim into a result. If a credible mathematician surfaces an error, that changes overnight.

Why we're covering it anyway

The detail worth sitting with is what it would represent if it holds. A capable research mathematician — now working inside the lab that builds the model — credits that model with producing the counterexample itself, not merely tidying up a human's idea. Earlier this month we noted a separate episode of the same model being pointed at a hard open problem. If this one survives formal checking and outside scrutiny, it marks a step past "AI helps a mathematician write a proof" and toward "AI produces the object that settles an 87-year-old conjecture." The maker's-own-researcher framing is a reason for more scrutiny, not less — and the redeeming feature of a counterexample is that, unlike a sprawling proof, it is small enough for anyone to check. We'll follow what the mathematicians do with it — caveats firmly in frame.

Sources: Levent Alpöge on X (mirror), Jacobian conjecture — Wikipedia, Hacker News discussion.