AI for mathematics · A checkable counterexample

A Local Truth, Globally False: Fable 5 and the Jacobian Conjecture

25 July 2026 · ~13 min read

For 87 years, the Jacobian conjecture said that a polynomial map which is invertible in every tiny neighborhood must also be invertible everywhere. In July 2026, a three-variable map credited to Claude Fable 5 broke that bridge between local and global behavior. The verification fits on a page. Finding the map was the hard part.

1. What did the Jacobian conjecture say?

Start with a polynomial map \(F:\mathbb{C}^{n}\rightarrow\mathbb{C}^{n}\). It takes \(n\) complex inputs and returns \(n\) complex outputs:

Its Jacobian matrix records every first-order response of every output to every input:

The determinant \(\det J_F(x)\) measures whether the map crushes a tiny volume to zero near \(x\). If the determinant is nonzero, the inverse function theorem gives a local inverse: close enough to \(x\), no two points are folded together.

The hypothesis is local; the conclusion is global. That leap is the whole conjecture. A determinant can tell us that each small patch is mapped cleanly. It does not obviously tell us whether two far-apart patches can land on the same place.

A friendly example where the conjecture works

Consider \(F(x,y)=(x+y^2,y)\). Its Jacobian is triangular:

And it really does have the polynomial inverse \(F^{-1}(a,b)=(a-b^2,b)\). Examples like this support the conjecture's intuition: never crush local volume, and perhaps the map can always be undone.

2. What did Fable introduce?

On 19–20 July 2026, mathematician Levent Alpöge posted an explicit map ↗, thanking Akhil Mathew for raising the question and crediting Claude Fable 5 ↗, Anthropic's model for long-running knowledge work, for the work that produced it. Write \(u=1+xy\), and define \(F=(F_1,F_2,F_3):\mathbb{C}^{3}\to\mathbb{C}^{3}\) by

These are ordinary polynomials. The highest total degree is \(7\). Two exact checks do all the logical work.

Why this immediately disproves the conjecture

Suppose an inverse \(G\) existed. Applying it to the shared output would give

but the same reasoning would force that one value of \(G\) to equal \(p_+\) and \(p_-\) as well. A function cannot return three different answers for one input. Therefore \(F\) is not invertible. Since its Jacobian determinant is nevertheless a nonzero constant, it is a counterexample.

The geometry hiding behind the formula

The coefficients look conjured from nowhere, but the map has a clean retrospective explanation. Begin with a cubic polynomial having three distinct linear factors, \(C=L_1L_2L_3\). Mark one factor \(L_i\), place the other two in a quadratic \(Q_i=C/L_i\), and then forget which factor was marked:

The raw multiplication map \((L,Q)\mapsto LQ\) has unwanted scaling and root-collision directions, so it is not yet locally invertible. The construction normalizes their resultant to remove those directions, then chooses an affine slice that happens to be polynomially equivalent to \(\mathbb{C}^3\). In the resulting coordinates, “forget the marked root” becomes the degree-\(7\) formula above.

This viewpoint, developed in Terence Tao's digestion ↗, explains the otherwise mysterious three-point fiber. The map is not accidentally three-to-one; three choices were built into its geometry from the beginning.

3. Why did this take 87 years?

The year count is \(2026-1939=87\), not 89. The short answer is that checking a proposed witness and discovering one are radically different tasks. Four difficulties reinforced one another.

3.1 Brute force sees an overdetermined desert

A general degree-\(7\) map \(\mathbb{C}^3\to\mathbb{C}^3\) has \(360\) coefficient slots. Its Jacobian determinant can have degree as high as \(18\), with \(1{,}329\) nonconstant coefficients that must all cancel. Those equations are structured rather than independent, but a blind coefficient search does not know the structure. It mostly sees an enormous nonlinear system with no reason to expect a solution.

3.2 The useful idea looks worse after changing coordinates

“Mark one root, then forget the mark” is simple. Converting the spaces of marked and unmarked cubics into ordinary affine coordinates is not. By the time that geometric operation is written as three polynomials in \(x,y,z\), the idea is buried under engineered cancellations. The final formula is easy to test and hard to reverse-engineer.

3.3 Local information does not control infinity

Many local-to-global theorems become true after adding a condition such as properness: points escaping to infinity cannot have bounded images. This counterexample is not proper. Its different sheets can disappear toward infinity rather than collide at a critical point. That is how the map stays locally invertible while being globally three-to-one. A recent structural analysis describes it as an étale, non-proper map with mixed-sign grading [Shaska, 2026] ↗.

3.4 Earlier reductions changed the shape of the search

Classical work showed that the all-dimensional conjecture could be reduced to cubic maps. That is powerful for proving a universal theorem, but the reduction may increase the number of variables substantially. The compact counterexample instead lives in only three variables at degree \(7\). A search insisting on low degree and low dimension at the same time could therefore miss precisely this kind of construction [Bass–Connell–Wright, 1982] ↗.

There was also a sociological asymmetry: a conjecture usually organizes work around proving it, understanding positive cases, and simplifying it. The Jacobian conjecture accumulated many claimed proofs and at least five published incorrect ones. Once the right counterexample is visible, its brevity can make the previous 87 years look irrational. Before the right coordinates were visible, it was anything but obvious.

4. My comments on the work

4.1 Call it a human–AI result, with incomplete provenance

The defensible public account is precise but limited: Akhil Mathew prompted the question, Levent Alpöge announced and checked the result, and the announcement credits Fable with producing the counterexample. The full model conversation and search trace have not been made public. So “Fable disproved the conjecture” is a useful headline, but not yet a complete account of how the mathematical labor was divided.

4.2 This is an unusually good shape for AI-assisted mathematics

A counterexample is a finite witness. Once a model proposes one, independent systems can differentiate it, substitute exact rational points, and formalize the logic. There is no long proof whose hidden gap depends on taste or interpretation. The Archive of Formal Proofs entry ↗ checks the determinant, the collision, the normalization to determinant \(1\), and the extension to every \(n\geq3\).

This does not make verification optional. It makes the collaboration pattern unusually clean: use a generative system to search for a witness, a computer algebra system to reject mistakes quickly, a proof assistant to certify the finite claim, and human mathematics to explain why the witness exists.

4.3 The explanation matters almost as much as the witness

The raw formula answers “is the conjecture false?” The marked-root construction answers “how could such a map exist?” Those are different achievements. Without the second, the counterexample is a miraculous string of coefficients. With it, the result becomes a new reusable idea about non-proper polynomial maps and finite covers.

4.4 The story is not over

The general conjecture is dead, but the plane case is not. The next questions are sharper: what additional global condition restores invertibility, how small can counterexamples be, how should they be classified, and which conjectures known to be equivalent in all dimensions must now be reformulated? The discovery closes one problem and opens a more structural program.

A nonzero Jacobian determinant guarantees a local way back. The counterexample shows that it does not guarantee that all those local ways back belong to one global road.

My main takeaway is not that an AI performed a long proof faster. It is that Fable appears to have found the right object to test—a compact, exact witness hiding behind an unexpected change of coordinates. That is a different and, for mathematical discovery, potentially more consequential capability.