Did Claude Fable 5 Help Find a 3D Counterexample to the Jacobian Conjecture? The 2D Case Remains Open

Core facts

The Jacobian conjecture studies polynomial maps

The conjecture says that if its Jacobian determinant is the same nonzero constant everywhere, then the map should have a polynomial inverse. This traces back to a problem posed in 1939 and is a long-standing famous conjecture in algebraic geometry.

According to current reports, Levent Alpöge has published the following 3-variable polynomial map:

Two key computations shown in the related material are as follows:

and

This means the following:

• 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 cannot have an inverse function, let alone a polynomial inverse.

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

Why this is not the same as saying the Jacobian conjecture has been "proved"

A "proof" usually means showing that a mathematical statement is true, but what is being reported here is a counterexample that could disprove the conjecture. Therefore, the precise wording is as follows:

Claude Fable 5 was involved in the discovery or search process for an explicit counterexample to the 3D Jacobian conjecture; this result still awaits formal mathematical verification.

At present, the available materials do not disclose the full prompt, the search log, or the number of model runs, and they do not clearly distinguish the respective contributions of the model, the researcher, and external computational tools. So far, what can be confirmed is only that the researcher published the formula for the counterexample and attributed the discovery to the assistance of Claude Fable 5; it cannot be concluded that the model completed the mathematical discovery on its own.

Scope of impact: dimensions 3 and above; the 2D case remains open

According to the reports, a counterexample in three dimensions can be used to construct counterexamples in even higher dimensions. If that result holds, then what is disproved is the version of the Jacobian conjecture for

However, the two-dimensional case

remains open. In other words, the existence of a 3D counterexample does not mean the Jacobian conjecture as a whole has been completely resolved; the 2D Jacobian conjecture remains an unsolved problem.

Why this result is drawing 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 respects:

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

However, being able to "recompute the formulas" is not the same as the result having already been formally accepted by the mathematical community. The current reports explicitly say that the result has not yet completed formal peer review and that independent verification is still needed regarding completeness, methods of generalization, and structural explanation.

Why this matters for AI-assisted mathematical research

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

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

Sources and points to verify

The following materials should be verified first:

• ForkLog：Anthropic’s Claude Fable 5 finds counterexample to 1939 Jacobian conjecture
• 网易科技：AI参与找出反例，87年的雅可比猜想在三维及以上被推翻
• 36氪相关报道
• Glitchwire相关报道
• 中文维基百科：雅可比猜想
• 网易相关报道

When verifying, the key points are whether the Jacobian determinant is identically $-2$, whether the substitutions at the three points match exactly, what the concrete construction is for extending the 3D 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 with a complete proof and computational details is published;
2. Whether independent mathematicians or computer algebra systems can recheck the Jacobian determinant and point-collision calculations;
3. Whether the rigorous proof of the extension of the 3D counterexample to higher dimensions is complete;
4. Whether the actual contribution of Claude Fable 5 in the search process is disclosed;
5. Whether the 2D Jacobian conjecture gains a new research path as a result.

Conclusion

At present, the most reasonable assessment is that the related materials describe a 3-variable polynomial map that is said to satisfy a constant nonzero Jacobian determinant while not being injective. If the formulas and calculations are ultimately formally confirmed, that would rule out the version of the Jacobian conjecture in three and higher dimensions; however, the two-dimensional case would still remain open.

Therefore, the headline should not read "Fable 5 proved the Jacobian conjecture." More accurately, Claude Fable 5 was reportedly involved in the discovery of a verifiable 3D counterexample to the Jacobian conjecture, but the result still awaits formal confirmation.

Original source

ForkLog