Did Claude Fable 5 help uncover a 3D counterexample to the Jacobian conjecture? The 2D case remains unresolved

Core facts

The Jacobian conjecture studies polynomial maps

\[

F:\mathbb{C}^n\to\mathbb{C}^n.

\]

The conjecture says that if its Jacobian determinant is the same nonzero constant everywhere, then the map should have a polynomial inverse. This is usually traced back to a problem posed in 1939, and it is one of the best-known long-standing unsolved conjectures in algebraic geometry.

According to current reports, Levent Alpöge is said to have published the following three-variable polynomial map:

\[

F(x,y,z)=

\Bigl(

(1+xy)^3z+y^2(1+xy)(4+3xy),

\]

\[

y+3x(1+xy)^2z+3xy^2(4+3xy),

\]

\[

2x-3x^2y-x^3z

\Bigr).

\]

Two important computations shown by the related materials are as follows:

\[

\det(JF)=-2,

\]

and

\[

F(0,0,-\tfrac14)=F(1,-\tfrac32,\tfrac{13}{2})=F(-1,\tfrac32,\tfrac{13}{2})=(-\tfrac14,0,0).

\]

This means:

• the Jacobian determinant is a nonzero constant, satisfying the conjecture’s hypothesis;
• three distinct input points have the same image, so the map is not injective;
• a non-injective map has no inverse function, much less a polynomial inverse.

If the above formulas and computations are independently verified as correct, this map would be a counterexample to the Jacobian conjecture in three dimensions.

Why this does not mean the Jacobian conjecture has been “proved”

“Proof” usually means showing that a mathematical statement holds, but what is being reported here is a counterexample that could disprove the conjecture. Therefore, the accurate wording is as follows:

Claude Fable 5 was involved in discovering or searching for an explicit counterexample to the three-dimensional Jacobian conjecture; this result still awaits confirmation through formal mathematical procedures.

From the materials available at present, the full prompt, search log, and number of model runs have not been made public, and the respective contributions of the model, the researcher, and external computational tools are not clearly distinguished. Therefore, what can be confirmed at this point is only that the researcher published the equations for the counterexample and attributed the discovery to Claude Fable 5’s assistance; it cannot be concluded that the model alone completed the mathematical discovery.

Scope of impact: three dimensions and above; the two-dimensional case remains open

According to reports, a three-dimensional counterexample can be used to construct counterexamples in even higher dimensions. If so, the version of the Jacobian conjecture that would be disproved is the one for

\[

n\geq 3

\]

.

However, the two-dimensional case

\[

F:\mathbb{C}^2\to\mathbb{C}^2

\]

is still unresolved. In other words, the existence of a three-dimensional counterexample does not mean that the Jacobian conjecture as a whole has been completely settled; the two-dimensional Jacobian conjecture remains an open problem.

Why this result has drawn attention

The statement of the Jacobian conjecture itself is relatively simple, yet it has remained difficult to resolve for many years. If the reported result is confirmed, it stands out in several ways:

1. the counterexample is given as an explicit polynomial map rather than as an abstract existence proof;
2. the Jacobian determinant can be checked by direct computation;
3. non-injectivity can be shown by the fact that the images of three concrete points coincide;
4. the relevant computations can be rechecked with computer algebra tools.

However, being able to recalculate the formulas is not the same as the result already being formally accepted by the mathematical community. The current reports make it clear that this result has not yet completed formal peer review, and that further verification is needed regarding completeness, generalization methods, and structural explanations from independent mathematicians.

Why this result matters for AI-based mathematical research

The significance of this event lies not only in the conjecture itself but also in how AI may be involved in mathematical discovery. According to the materials available at present, Claude Fable 5 may have been involved in generating candidate structures, algebraic transformations, exploration, or computational verification, but the exact process has not been disclosed.

This also shows that generating a verifiable candidate counterexample with AI is not the same as constructing an independent, complete theory. Once a counterexample is found, it can usually be verified by substitution and symbolic computation. However, explaining the structure, proving that it can be extended to higher dimensions, and determining the long-term impact on related conjectures still require independent analysis by mathematicians.

Sources and points for verification

The following materials are recommended for priority verification:

• ForkLog: Anthropic’s Claude Fable 5 finds counterexample to 1939 Jacobian conjecture
• NetEase Tech: AI helped find the counterexample, and the 87-year-old Jacobian conjecture is overturned in three dimensions and above
• 36Kr related report
• Glitchwire related report
• Chinese Wikipedia: Jacobian conjecture
• NetEase related report

When verifying, priority should be given to confirming whether the Jacobian determinant is identically \(-2\), whether the substitution results for the three points match exactly, what the specific construction is for extending the three-dimensional counterexample to higher dimensions, and whether a formal paper and independent peer review have already appeared.

What to watch next

1. Whether a formal paper containing the full proof and computational details will be published;
2. Whether independent mathematicians or computer algebra systems can reverify the Jacobian determinant and the point-collision computation;
3. Whether the rigorous proof of the extension from the three-dimensional counterexample to higher dimensions is complete;
4. Whether the actual contribution of Claude Fable 5 in the search process will be disclosed;
5. Whether the two-dimensional Jacobian conjecture will gain new research directions as a result of this finding.

Conclusion

At present, the most reasonable assessment is that the related materials describe a three-variable polynomial map that is said to satisfy a constant nonzero Jacobian determinant while not being injective. If the formulas and computations are eventually formally confirmed, that would disprove the three-dimensional and higher versions of the Jacobian conjecture; however, the two-dimensional case remains unresolved.

Therefore, the headline should not read “Fable 5 proved the Jacobian conjecture.” More accurately: According to reports, Claude Fable 5 was involved in discovering a verifiable three-dimensional counterexample to the Jacobian conjecture, but the result is still awaiting formal confirmation.

Original source

ForkLog