Claude Fable 5's Jacobian conjecture counterexample in SymPy: det ≡ −2, 3-to-1

On July 20, 2026, Levent Alpöge, a mathematician at Anthropic, posted on X a polynomial map in three variables that he says Claude Fable 5 generated.
According to the post, it is a counterexample to the Jacobian conjecture, open since 1939 — 87 years.

The map has Jacobian determinant equal to the constant $-2$ at every point, and yet it sends three distinct points to the same single point.
Those are the only two claims to check, the formula is short, and I reproduced both on my machine with SymPy.

This article assumes you have never seen a Jacobian matrix.

The Jacobian matrix is the derivative of a multivariable function

For a one-variable function $f(x)$ , the derivative $f'(a)$ gives the slope of the graph at the point $a$ .
Put differently, it is the coefficient you get when you approximate the function by a linear polynomial near $a$ .

$f(x) \approx f(a) + f'(a)(x - a)$

With several variables, both input and output become tuples.
In three variables, a map looks like $F(x, y, z) = (F_1, F_2, F_3)$ : it takes three numbers and returns three numbers.
The object playing the role of the derivative is the Jacobian matrix, built by taking the partial derivative of each component with respect to each variable and arranging them in a 3×3 grid.

$J_F = \begin{pmatrix} \dfrac{\partial F_1}{\partial x} & \dfrac{\partial F_1}{\partial y} & \dfrac{\partial F_1}{\partial z} \\[6pt] \dfrac{\partial F_2}{\partial x} & \dfrac{\partial F_2}{\partial y} & \dfrac{\partial F_2}{\partial z} \\[6pt] \dfrac{\partial F_3}{\partial x} & \dfrac{\partial F_3}{\partial y} & \dfrac{\partial F_3}{\partial z} \end{pmatrix}$

The partial derivative $\partial F_1 / \partial x$ comes from differentiating $F_1$ with respect to $x$ while treating $y$ and $z$ as constants.
The notation changes, but the computation is the same one-variable differentiation.
The role does not change either: the Jacobian matrix approximates the map near a point by a linear map, i.e. matrix multiplication.

One variable	Several variables
derivative $f'(a)$	Jacobian matrix $J_F(a)$
approximate by the tangent line	approximate by a linear map
$f'(a) \neq 0$ means locally invertible near $a$	$\det J_F(a) \neq 0$ means locally invertible near $a$

If matrices themselves are unfamiliar, Vectors and matrices in AI articles, the minimum you need to read them covers just how to read them.
Jacobian matrices also show up on the AI side: the J in J-lens, the read-out tool Anthropic used to inspect model internals (the shared workspace J-space in Claude’s middle layers), stands for Jacobian.

The Jacobian determinant is a volume scaling factor

The $\det J_F$ in the last row of the table is the Jacobian determinant (the “Jacobian”).
Its absolute value gives the volume scaling factor of the linear map (area in 2D, volume in 3D), and its sign tells you whether orientation is preserved or mirror-flipped.

For example, $F(x, y, z) = (2x, 3y, z)$ has a Jacobian matrix with just $2, 3, 1$ on the diagonal, so the determinant is $6$ .
Near every point, this map scales volume by a factor of 6.

At a point where the determinant is $0$ , the linear approximation at that point collapses at least one direction.
Conversely, if $\det J_F(a) \neq 0$ , the linear approximation collapses no direction, and the inverse function theorem guarantees that, restricted to a neighborhood of $a$ , the map $F$ is one-to-one with a smooth inverse.

The catch is the phrase “restricted to a neighborhood.”
Being uncollapsed everywhere locally does not make the map one-to-one globally.

$F(x, y) = (e^x \cos y,\; e^x \sin y)$

This map has Jacobian determinant $e^{2x}$ , which is never $0$ .
Still, $(x, y)$ and $(x, y + 2\pi)$ land on the same point, so globally the map is infinitely many-to-one.
Not collapsing locally and not overlapping across distant regions are different properties, and the inverse function theorem only guarantees the first.

Where Jacobian matrices get used

The most familiar place is the change of variables in integrals: the $r$ in the polar-coordinate $r\,dr\,d\theta$ is (the absolute value of) a Jacobian determinant.
It multiplies in the area scaling factor so the integral comes out right after the coordinate change.
In optimization and machine learning, computing how the output moves when you nudge the input is exactly a Jacobian matrix, and neural-network gradient computation is a stack of per-layer Jacobians.
In robotics, the Jacobian tells you which way the end effector moves when you nudge a joint, and the same matrix is used to solve backwards from a target position to joint motions.

What the Jacobian conjecture claimed

The problem was posed in 1939 by the German mathematician Ott-Heinrich Keller. In modern language, the claim reads:

For a map $F: \mathbb{C}^n \to \mathbb{C}^n$ whose components are all polynomials, if the Jacobian determinant $\det J_F$ is a nonzero constant, then $F$ is a bijection and its inverse is also given by polynomials.

A bijection means every output corresponds to exactly one input.
The standard statement is over the complex numbers $\mathbb{C}$ , but this counterexample has real coefficients and real collision points, so it is a counterexample over the reals as well.

In one variable, the conjecture is true for an immediate reason: if $f'(x)$ is a constant $c \neq 0$ , then $f(x) = cx + d$ is the only possibility, and the inverse $x = (y - d)/c$ is again degree one.

The earlier $e^x \cos y$ example had a determinant $e^{2x}$ that varied from point to point.
The Jacobian conjecture only deals with maps whose determinant is a nonzero constant — the same scaling factor at every point.
The polynomial condition is also restrictive: no periodicity like $\sin$ or $e^x$ is available.
With both conditions in place, it looked like there was no way left to make the local and the global disagree.

Partial results had accumulated.
The conjecture was proved for polynomial maps of degree 2 (Wang, 1980), and a reduction was known that trades dimension for degree (Bass–Connell–Wright, 1982: proving the degree-3 case in all dimensions settles everything).
In 1994, Pinchuk showed that the real version with the weakened condition “nonvanishing everywhere” instead of “nonzero constant” does have counterexamples.
The constant case remained, and it was number 16 on Smale’s 1998 list of 18 mathematical problems for the next century.

The published counterexample

What Alpöge posted is the following $F: \mathbb{C}^3 \to \mathbb{C}^3$ .
It is not an official Anthropic publication or paper but an individual researcher’s post, and the attribution to Fable 5 currently rests on that post.

$F(x, y, z) = \begin{pmatrix} (1 + xy)^3 z + y^2 (1 + xy)(4 + 3xy) \\ y + 3x(1 + xy)^2 z + 3xy^2 (4 + 3xy) \\ 2x - 3x^2 y - x^3 z \end{pmatrix}$

The component degrees are 7, 6, and 4 from top to bottom.
The first claim is that the Jacobian determinant is identically $-2$ (the hypothesis of the conjecture holds).
The second is that the following three points map to the same point (the conclusion of the conjecture fails).

$F\left(0, 0, -\tfrac{1}{4}\right) = F\left(1, -\tfrac{3}{2}, \tfrac{13}{2}\right) = F\left(-1, \tfrac{3}{2}, \tfrac{13}{2}\right) = \left(-\tfrac{1}{4}, 0, 0\right)$

Three distinct inputs go to one output, so the map is not injective.
Far from having no polynomial inverse, it has no inverse at all.

graph TD
    A[Check all components are polynomials] -->|all 3 are| B[Compute the Jacobian determinant]
    B -->|expands to the constant -2| C[Hypothesis holds]
    C --> D[Plug in the three points]
    D -->|all 3 give the same output| E[Not injective<br/>no inverse exists]
    E --> F[Conclusion fails, counterexample stands]

One of the collisions can be followed in your head.
Setting $x = 0, y = 0$ kills every $xy$ term, so the first component is $1^3 \cdot z = -\frac{1}{4}$ , the second is $y = 0$ , and the third is $2x = 0$ .
The other two points involve fractions and more hand arithmetic, but the computations to check never go beyond basic arithmetic.

Reproducing it in SymPy

The determinant side is nine partial derivatives and a 3×3 determinant expansion, so I wrote a script for it.

import sympy as sp

x, y, z = sp.symbols("x y z")

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

F = sp.Matrix([F1, F2, F3])
J = F.jacobian([x, y, z])
print("det J =", sp.expand(J.det()))

pts = [(0, 0, sp.Rational(-1, 4)),
       (1, sp.Rational(-3, 2), sp.Rational(13, 2)),
       (-1, sp.Rational(3, 2), sp.Rational(13, 2))]
for p in pts:
    print("F", p, "=", [f.subs(dict(zip((x, y, z), p))) for f in (F1, F2, F3)])

Output:

det J = -2
F (0, 0, -1/4) = [-1/4, 0, 0]
F (1, -3/2, 13/2) = [-1/4, 0, 0]
F (-1, 3/2, 13/2) = [-1/4, 0, 0]

The expanded determinant loses every variable and really is just $-2$ .
The three substitutions match as well.
That is the entire verification — there is no proof to read and digest.
Checking Wiles’s proof of Fermat’s Last Theorem took years of refereeing; this has the shape of a counterexample, so its truth comes down to whether you accept this computation.

How far the verification has gone

Checks have been running in parallel since the post went up.
Paul Lezeau of Imperial College London formalized the counterexample in Lean and submitted it as a pull request to Google DeepMind’s Formal Conjectures repository.
It has approvals from several reviewers and, as of this writing (July 22, 2026), is still open awaiting merge.
On the mathematicians’ side, there is a Sage-script verification on MathOverflow, and the Secret Blogging Seminar presents Will Sawin’s geometric interpretation of why a map of this shape can exist.

On the other hand, there is no refereed paper and no preprint yet.
The search process by which Fable 5 arrived at the formula has not been published either.

What changes now that a counterexample exists

Everyday computation and engineering do not change.
Coordinate changes, gradient computation, and robot control all use the local property, and the inverse function theorem that backs them is untouched by this.
What changes is the view of multivariable polynomials that said “constant Jacobian means you can also invert globally.”

If the counterexample holds up, the Jacobian conjecture is refuted for all $n \geq 3$ at once.
Appending identity components $w \mapsto w$ raises the dimension while the Jacobian determinant stays the constant $-2$ .
What remains as research is the $n = 2$ case, and understanding how the overlap in this counterexample behaves at infinity.

The Dixmier conjecture about Weyl algebras moves in lockstep.
It is proven that if the Dixmier conjecture holds in a given dimension, the Jacobian conjecture holds in the same dimension, so by contraposition the Dixmier conjecture is also refuted for $n \geq 3$ .
The relationship is not a plain equivalence: in the other direction, the $2n$ -dimensional Jacobian conjecture implies the $n$ -dimensional Dixmier conjecture (Belov-Kanel–Kontsevich’s paper).

Kevin Buzzard of Imperial College London, a long-time advocate of Lean formalization, counts this on his blog as the third AI-produced counterexample to an open problem in 2026 (Erdős’s unit distance conjecture in May, a Grothendieck-era group scheme problem in early July, and now this).
He adds that even with machine verification done, the stage of humans precisely understanding what is going on in this example is still ahead.