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.

Update, 25 July 2026: Terence Tao has digested it — and states it as a theorem

The caveat this piece was built around has substantially lifted.

On 21 July, Terence Tao published a full mathematical digestion of the counterexample on his blog. He does not frame it as a claim awaiting checking. He states it as a result: "It was recently shown (using the Fable AI) that the conjecture is false in three dimensions (and thus in higher dimensions as well)" — and then sets it out formally as a theorem, that there exists a polynomial map with non-zero constant Jacobian which is not invertible.

Three things in his write-up are worth carrying over.

The verification is trivial; finding it was not. Tao notes that checking the example takes "an extremely quick verification," but that presented bare, "the construction presented in this fashion appears like a massive miracle." His reasoning is the interesting part: the polynomial has degree seven, so the Jacobian ought to be a degree-large polynomial in three variables, and every non-constant coefficient vanishing "looks like a massive cancellation" requiring far more equations than a generic degree-seven map of three variables has degrees of freedom. His conclusion: "finding such a polynomial looks highly unlikely to be located by brute force." That is a direct answer to the obvious sceptical reading — that a machine simply searched hard.

It has since been explained structurally. "The example has since been retroactively explained in more geometric terms," Tao writes, and his own post is an attempt to render that explanation "with relatively little use of algebraic geometry, in a manner that minimizes the amount of 'miracles' required." An object that first appeared as an inexplicable coincidence now has a human account of why it exists.

The scope is narrower than "the Jacobian conjecture is dead." Tao is precise where headlines have not been: "The conjecture remains open in two dimensions, and is easy to establish in one dimension." The counterexample settles three dimensions and above. The two-dimensional case — the one most people picture — is still open.

What we get wrong, and what we got right. This piece was written in a deliberately hedged register, and we said we would follow what the mathematicians did with it. We were four days late doing so: Tao's post went up on 21 July and this update is dated the 25th. The hedging itself we stand by — at publication there was no paper, no formal proof and no independent expert treatment, and the correct posture was caution. But a watch is only worth setting if it is actually checked, and this one was not checked often enough.

For readers tracking the wider question this raises — who verifies mathematics that a machine produced, and what happens to a profession when that becomes routine — Tao personally spending an evening digesting an AI's counterexample is not a footnote. It is the shape of the thing.