How to Read an AI “Counterexample” to the Jacobian Conjecture

A “counterexample” lands in our lap

Picture the moment you learn the Jacobian Conjecture exists. It feels like a teaser for something deep: a condition written using derivatives, and then—surprisingly—an unanswered question about whether a whole multivariable polynomial function must have a polynomial inverse.

Now fast-forward to July 21, 2026. Terence Tao published an expository post digesting a claimed three-dimensional counterexample—one that (according to his write-up) comes from an AI-assisted search, and is accompanied by an explicit polynomial map you can check by hand on a few sample points. (terrytao.wordpress.com)

That post is a great guide not only to the Jacobian Conjecture itself, but also to a practical skill: how to take an AI-produced “mathematical breakthrough” seriously without being fooled by plausibility.

The Jacobian Conjecture in plain language

Let’s build the objects carefully.

A polynomial map in this context is a function

[
F: \mathbb{C}^n \to \mathbb{C}^n
]

where each coordinate function of (F) is a polynomial in (n) complex variables.

From such an (F), you can form the Jacobian matrix (DF), whose ((i,j)) entry is the partial derivative

[
\frac{\partial F_i}{\partial x_j}.
]

The Jacobian determinant is then

[
\det(DF),
]

a single polynomial (or constant) describing how volume scales infinitesimally.

The Jacobian Conjecture (over the complex numbers) says, informally:

• If (F) has non-zero constant Jacobian determinant,
• then (F) should be globally invertible, and its inverse should also be a polynomial map.

That last phrase is the punchline: the conjecture tries to turn a local condition (derivatives don’t vanish) into a global guarantee (a polynomial inverse exists everywhere). (encyclopediaofmath.org)

So the natural question becomes: can a map be locally reversible everywhere (because its Jacobian never degenerates), yet still fail to be globally one-to-one? That’s exactly where counterexamples must attack.

Local invertibility: why a non-zero Jacobian is persuasive

The reason people care about (\det(DF)\neq 0) is that it implies local invertibility.

Here “local invertibility” means: near any point (a\in \mathbb{C}^n), the map behaves like a bijection between small neighborhoods of (a) and (F(a)).

One way to formalize that feeling is the inverse function theorem, which in smooth settings says a map is locally invertible at points where the Jacobian determinant is non-zero. Tao’s write-up uses exactly this viewpoint: non-vanishing Jacobian corresponds to local invertibility. (terrytao.wordpress.com)

In everyday terms: a non-zero determinant is like a 2D grid that doesn’t suddenly crumple into a line—it keeps an orientation and doesn’t collapse nearby.

But local behavior doesn’t automatically control what happens far away. Global invertibility is more like asking whether the map ever makes two distinct points “choose the same output” after wandering through the space.

Global failure: what a real counterexample must show

To disprove the Jacobian Conjecture, it’s not enough to show the Jacobian determinant is constant and non-zero. You also need to show the conclusion fails.

A standard way to show failure is to prove non-injectivity: find two different inputs (x\neq y) such that

[
F(x)=F(y).
]

That immediately blocks the existence of an inverse function (polynomial or otherwise), because inverses can’t undo a many-to-one map.

In Tao’s digestion, the claimed three-dimensional counterexample has the striking feature that

• the Jacobian determinant is a non-zero constant (he states it is (-2)), and
• the map is still not injective, with several distinct points mapped to the same output. (terrytao.wordpress.com)

Even if the polynomial looks like a monster at first glance, this second part is checkable: compute (F) on the listed points and verify equality.

Where AI conversations fit in—and where they don’t

The romantic version of “AI solved math” is that the model thinks, discovers, and then magically produces a correct solution. The less romantic, more reliable version is more interesting: an AI system can generate candidates, suggest structure, or propose computational steps—but the community’s job is to verify.

That’s why Tao’s post feels like it’s doing two things at once:

1. Explaining the math of the counterexample,
2. Demonstrating which steps are actually checkable.

This matters because LLMs (large language models) are excellent at sounding confident while being wrong. The specific ChatGPT transcript you linked can’t be read in full here, but the general lesson remains: a conversation is not proof; a conversation is a lead.

So when you encounter an AI-produced Jacobian counterexample claim, the verification mindset should be automatic.

A practical verification workflow (the “don’t trust, verify” checklist)

When the candidate is explicit, like (F:\mathbb{C}^3\to\mathbb{C}^3), you can test the two core properties separately.

1) Verify the Jacobian determinant is constant and non-zero

You compute the Jacobian matrix (DF), then its determinant.

For a true counterexample, the determinant should simplify to a constant polynomial. In Tao’s discussion of the claimed example, this constant is (-2). (terrytao.wordpress.com)

2) Verify non-injectivity by direct collisions

Find distinct points (p\neq q) with (F(p)=F(q)).

Tao lists collisions among several points for the claimed polynomial map, which is the direct “global failure” certificate. (terrytao.wordpress.com)

These two steps are conceptually clean: the Jacobian condition is a differential test, while the collision is an algebraic test.

3) Use algebra software for arithmetic, not for belief

Algebra systems are very good at expanding expressions and simplifying determinants. They’re not good at guaranteeing the specification you’re testing is the intended one.

So the rule is: use software to do the grunt work, but confirm you’re checking the right polynomial and the right points.

Here’s a small SageMath/SymPy-style sketch of what the workflow looks like (the exact polynomial coefficients would be inserted from the claimed example):

import sympy as sp

z1, z2, z3 = sp.symbols('z1 z2 z3')

# Define F = (F1, F2, F3) from the claimed counterexample.
# F1 = ...
# F2 = ...
# F3 = ...
F1 = sp.Symbol('F1')
F2 = sp.Symbol('F2')
F3 = sp.Symbol('F3')
F = [F1, F2, F3]

# Jacobian matrix
J = sp.Matrix([[sp.diff(F[i], v) for v in (z1, z2, z3)] for i in range(3)])

detJ = sp.factor(J.det())
print(detJ)  # should simplify to a non-zero constant

# Check a collision: pick two distinct points p and q
p = {z1: 0, z2: 0, z3: -sp.Rational(1,4)}
q = {z1: 1, z2: -sp.Rational(3,2), z3: sp.Rational(13,2)}

Fp = [fi.subs(p) for fi in F]
Fq = [fi.subs(q) for fi in F]

print(sp.simplify(Fp[0]-Fq[0]))
print(sp.simplify(Fp[1]-Fq[1]))
print(sp.simplify(Fp[2]-Fq[2]))

You’d replace the placeholders (F1,F2,F3) with the explicit formula from the claim. The determinant printout should collapse to a constant, and the differences should simplify to 0.

Why this feels like “miracles”—and how the story changes

Once you’ve seen the verification logic, the next emotional step is curiosity: how can the Jacobian determinant miraculously become constant while the map still fails injectivity?

Tao emphasizes that the construction, while explicit, can look like massive cancellation if presented purely as raw algebra. His digestion then reframes the counterexample geometrically, explaining how one can build the desired behavior (local injectivity versus global collision) using structured polynomial operations rather than blind brute force. (terrytao.wordpress.com)

That reframing is also a lesson for reading AI-driven math: AI might stumble into the formula, but the human explanation should reveal the structure behind it. Without structure, the result remains a curiosity. With structure, it becomes something you can understand, generalize, or at least trust more confidently.

The big takeaway

Here’s the clean moral that stays true whether the “lead” comes from an AI conversation, a spreadsheet search, or a research talk:

• The Jacobian Conjecture is about turning a local derivative condition into a global bijectivity statement.
• A counterexample must satisfy both parts: constant non-zero Jacobian determinant and an explicit collision showing global failure.
• AI can accelerate candidate discovery, but the final authority is independent verification.

That’s the moment math stops being a vibe and becomes a discipline: we don’t just react to the existence of an answer—we test the claim until the evidence can’t be moved.