What Is the Jacobian Conjecture? Fable 5 Counterexample Explained
The Jacobian conjecture explained from derivatives to Fable 5's verified C³ counterexample, why three colliding points settle it, and what remains open.
You may share Yohei Nakajima's reaction to the July 2026 news: excited that AI helped prove the Jacobian conjecture false, while not really knowing what the Jacobian conjecture is.
That is a reasonable place to start. You do not need algebraic geometry to understand why the counterexample works. The essential story is about the gap between local and global information: a map can look perfectly reversible under a microscope at every point, yet send far-apart inputs to the same destination.
On July 20, 2026, mathematician Levent Alpögeposted an explicit map on X, crediting Akhil Mathew for asking the question and Claude Fable 5 for work during the World Cup final. The formula has since been independently checked and analyzed in a verification preprint. This guide complements explainx.ai's first-news report by teaching the mathematics from the ground up.
TL;DR — what are people asking?
Question
Direct answer
What did the conjecture claim?
If a polynomial map F: C^n → C^n has a nonzero constant Jacobian determinant, then it has a polynomial inverse.
What does the Jacobian measure?
How tiny changes in the inputs change all the outputs near one point.
What does a nonzero determinant mean?
The local linear approximation loses no direction, so the map is locally reversible.
What did the counterexample show?
A map in C^3 has det JF = −2 everywhere but sends three distinct points to (−1/4, 0, 0).
Why is that decisive?
An inverse cannot choose which of three inputs produced one output. The map is not one-to-one.
Yes. The condition is any nonzero constant, not only ±1.
What dimensions are settled?
False for n = 3 and, by adding identity coordinates, every n at least 3. The n = 2 case remains open; n = 1 is elementary.
Did AI prove it alone?
No such autonomous-discovery claim is established. Alpöge credited Fable 5's work; humans independently verified the explicit result.
Is it peer-reviewed?
It has an independent verification preprint and broad mathematical checking, but that is not the same as completed journal peer review.
Start one step earlier: what does a derivative do?
For a familiar one-input function such as f(x) = x², the derivative tells you how much the output changes when you nudge the input:
text
small output change ≈ f'(x) × small input change
At x = 3, the derivative is 6. A tiny move of 0.01 in x produces an output change of about 0.06.
Now imagine a map with several inputs and several outputs. Moving x may change P, Q, and R. Moving y may change all three in different ways. One derivative is no longer enough, so we put every first derivative into a table.
What is a Jacobian matrix?
For the three-output map F(x,y,z) = (P,Q,R), its Jacobian matrix is:
Each column asks what happens when one input moves a tiny amount. Each row tracks one output. Near a chosen point, this matrix is the best linear approximation to the full, curved polynomial map.
Here is the useful grid analogy. Draw a tiny square grid on a rubber sheet. A map may stretch, rotate, shear, or reflect each tiny patch. The Jacobian matrix describes that local deformation.
The analogy has limits: the conjecture is over complex coordinates and in arbitrary dimension, not literally a two-dimensional rubber sheet. But the local-versus-global distinction is exactly the one we need.
What does the Jacobian determinant tell us?
A square matrix has a number called its determinant. For a Jacobian matrix, the determinant measures the signed local volume scaling.
Determinant 0: at least one infinitesimal direction gets crushed. A tiny box flattens.
Determinant 2: tiny volumes are locally doubled.
Determinant −2: tiny volumes are doubled and orientation is reversed in the real-grid analogy.
Most importantly, a nonzero determinant means the local linear map is invertible. The inverse function theorem then says the original map is reversible in a sufficiently small neighborhood of that point.
But this is only a neighborhood guarantee. It does not say that two distant neighborhoods cannot land on the same part of the output space.
What did the Jacobian conjecture actually claim?
The classical conjecture concerns polynomial maps:
text
F: C^n → C^n
Every output coordinate of F must be a polynomial in the n inputs. The conjecture said:
If det JF is a nonzero constant, then F has a polynomial inverse.
This formulation agrees with the Wolfram MathWorld reference. The determinant condition is necessary: if polynomial maps F and G undo each other, the chain rule forces their Jacobian determinants to multiply to 1.
The conjecture asked whether that necessary local condition was also sufficient globally.
No, the determinant does not have to be ±1
The hypothesis is nonzero constant. The value −2 satisfies it over C.
Why do readers sometimes see determinant 1? Mathematicians can normalize a nonzero constant by scaling an output coordinate, and integer-coefficient versions introduce additional conventions. Neither changes the classical complex statement. Rejecting this example because −2 is not ±1 would reject the actual premise.
Why n = 1 is easy
With one variable, a polynomial whose derivative is a nonzero constant must be linear:
text
f(x) = ax + b, where a is not 0
Its inverse is (y − b)/a, also a polynomial. Nothing mysterious happens until multiple variables can interact.
Local invertibility is not global invertibility
This is the conceptual heart of the story.
Imagine walking with a flawless local street map. Every intersection near you is unambiguous. That does not prove the entire city has no two streets with the same name or no distant routes that merge at one destination.
As another analogy, the exponential function on complex numbers has a nonzero derivative everywhere but repeats values globally. That illustrates the local/global gap, not a polynomial counterexample to the Jacobian conjecture. In one polynomial variable, no counterexample exists.
The conjecture proposed that polynomial structure was rigid enough to prevent this global collision. The July 2026 map shows that in three variables, it is not.
What is the Fable 5-assisted counterexample?
Define F(x,y,z) = (P,Q,R), where:
text
P = (1 + xy)^3 z + y^2(1 + xy)(4 + 3xy)
Q = y + 3x(1 + xy)^2 z + 3xy^2(4 + 3xy)
R = 2x − 3x^2y − x^3z
There are two claims to check:
The determinant of the Jacobian matrix is the constant −2.
The map is not one-to-one.
Symbolic differentiation and determinant expansion establish:
text
det JF(x,y,z) = −2
for every (x,y,z). The determinant never vanishes, so around every point the map is locally reversible. It never crushes an infinitesimal direction.
Yet these three distinct inputs:
text
A = ( 0, 0, −1/4)
B = ( 1, −3/2, 13/2)
C = (−1, 3/2, 13/2)
all have the same output:
text
F(A) = F(B) = F(C) = (−1/4, 0, 0)
That is the counterexample.
Why does one three-point collision settle the question?
Suppose an inverse G existed. Applying it to the shared output would require:
text
G(−1/4, 0, 0) = A
G(−1/4, 0, 0) = B
G(−1/4, 0, 0) = C
A function can return only one value for one input. Because A, B, and C are different, no inverse function exists—polynomial or otherwise.
The Jacobian conjecture made a universal claim: every polynomial map satisfying the determinant condition is invertible. A single valid map satisfying the condition and failing invertibility is enough to make that claim false.
The collision points even have rational real coordinates. Since real numbers sit inside the complex numbers, they are valid points of C^3; no advanced complex-number calculation is needed to witness the failure.
Can an ordinary reader verify the collision?
Yes. The determinant identity is best checked with symbolic algebra or careful differentiation, but the collision needs only substitution and arithmetic.
For A = (0,0,−1/4), we have xy = 0 and 1 + xy = 1:
The remaining signs cancel differently in Q and R, but direct substitution gives:
Input
P
Q
R
(0, 0, −1/4)
−1/4
0
0
(1, −3/2, 13/2)
−1/4
0
0
(−1, 3/2, 13/2)
−1/4
0
0
This split is why the result traveled so quickly. Finding the formula was hard; checking its decisive properties is finite. In New Scientist's report, Queen Mary University mathematician Abhishek Saha said the one-line mathematics was simple to verify and many mathematicians had already done so.
The evidence became stronger and more organized without becoming journal peer review overnight.
Date
What changed
What it establishes
July 20, 2026
Alpöge published the explicit C^3 map, determinant claim, and collision points on X.
A concrete, falsifiable candidate—not merely “AI solved it.”
July 20–21
Mathematicians independently substituted the points and checked the determinant; New Scientist quoted Saha on broad verification. Timothy Gowers and others reacted publicly once the finite checks held.
The displayed finite algebra withstands independent checking; expert reaction shifted from format skepticism to significance.
By July 21
MathWorld recorded the result and the Ulam verification preprint became available. A separate viral loop: pasting the map into other chatbots to watch training-cutoff confusion.
A stable mathematical write-up verifies the counterexample. "Model freak-outs" demonstrate verifiability, not rediscovery.
Still pending
Conventional publication and journal refereeing.
Do not describe the result as journal peer-reviewed unless that later occurs.
The careful status phrase is verified counterexample documented in a preprint. “Unverified tweet” is now stale; “peer-reviewed theorem” overstates the publication record.
What did Fable 5 contribute, and what did humans establish?
Alpöge's primary post credits three parts of the event:
Akhil Mathew asked the question.
Claude Fable 5 did work on it during the World Cup final.
Levent Alpöge announced the explicit construction.
That credit matters, but it does not reveal the full discovery trace. The exact prompt, intermediate attempts, human steering, and division of insight have not been published in enough detail to support “Fable autonomously solved it alone.”
Human verification establishes something narrower and stronger: regardless of how the candidate was found, the written polynomial has determinant −2 and is not injective. Mathematical truth does not depend on whether the discovery came from a model, a notebook, or a conversation.
This is also why Google's Chain-of-Evidence approach is relevant. A fluent origin story is not evidence; reproducible claims and auditable checks are. The Jacobian example has unusually short checks, unlike many long AI-generated proofs.
The distinction also guards against cognitive surrender: you should not trust the result because “Fable said so.” You can trust the displayed claims to the degree that independent mathematical verification supports them.
Why is the conjecture false in every dimension n at least 3?
Start with the counterexample F in three coordinates. For four coordinates, define a new map that leaves the extra coordinate alone:
text
F₄(x,y,z,w) = (P(x,y,z), Q(x,y,z), R(x,y,z), w)
Its Jacobian matrix contains the old three-dimensional block plus a 1 on the new diagonal, so its determinant remains −2. The old colliding points still collide whenever they share the same w.
Repeat this construction with as many unchanged coordinates as needed:
text
Fₙ(x,y,z,t₄,...,tₙ) = (P,Q,R,t₄,...,tₙ)
This move is called stabilization or adjoining identity coordinates. It proves the conjecture false for every n at least 3.
Why does the two-dimensional case remain open?
Stabilization only moves upward in dimension. It cannot squeeze a three-variable counterexample into two variables while preserving the needed properties.
So the status is:
n = 1: true for an elementary reason.
n = 2: still open.
n = 3: false by the explicit counterexample.
n greater than 3: false by stabilization.
The plane case is not a minor footnote. It is now the surviving core problem, and the three-dimensional construction may give researchers new clues without automatically resolving it.
Is this related to Anthropic's J-lens or J-space?
Only through the word Jacobian.
Anthropic's J-lens uses Jacobian information to connect internal neural activations to future token tendencies. J-space is the model-internal workspace that research studies with that lens.
The Jacobian conjecture is a pure-mathematics statement about polynomial maps from C^n to C^n. It does not concern Claude's activations, hidden thoughts, or interpretability. Same calculus tool; different objects and questions.
What does this mean for AI-assisted mathematics?
The strongest lesson is not “models replace mathematicians” or “AI output is proof.” It is that discovery and verification can have radically different costs.
The Fable-assisted construction is compact enough for humans and computer algebra systems to check directly. Compare that with explainx.ai's coverage of the OpenAI planar unit-distance result, where a long argument and expert companion work carry much more verification burden.
That asymmetry favors counterexamples. To disprove a universal claim, a model may need to find one exceptional object. Once found, everyone can test it. Producing a general theorem may require a long chain in which every link must survive review.
human question → model-assisted search → explicit candidate
→ independent calculation → stable preprint → deeper human study
The beginner's takeaway
The Jacobian determinant is a local sensor. A nonzero value says a map does not flatten an infinitesimal direction at that point. A nonzero constant says this remains true everywhere.
The conjecture guessed that polynomial maps were rigid enough to turn that everywhere-local guarantee into a global inverse. The C^3 counterexample breaks the bridge: it is locally reversible everywhere, yet three distant points share one output.
That is why a short formula can settle an 87-year-old question—and why the surviving n = 2 problem may still be difficult.
Status note: This article reflects public evidence available through July 21, 2026. The explicit counterexample has been independently verified and documented in a verification preprint; that is not the same as completed journal peer review. Discovery-credit details and the exact Fable 5 workflow may be clarified by later primary reporting.
