Fable “disproves” the Jacobian conjecture, solving a century-old mathematical puzzle with a single formula!

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A classic mathematical problem that has puzzled the mathematics community for 87 years may be approaching a historic turning point.

On July 20, mathematician Levent Alpoge stated on social media that with the assistance of AI model Fable 5, he constructed a counterexample to the Jacobian Conjecture. If this result is ultimately peer-reviewed, it will mean that this longstanding conjecture, unresolved since 1939, will be officially overturned and may impact several related mathematical conjectures.

However, so far, the result has only been made public on social media, has not been posted as an arXiv preprint, and has not undergone academic peer review. Therefore, in the official sense of the mathematics community, the Jacobian Conjecture remains an unsolved problem.

AI provides a counterexample: multiple independent verifications support the core conclusion

Alpoge published a polynomial mapping from complex space C³ to C³ and claimed that it satisfies the Jacobian Conjecture's key condition—that the Jacobian determinant is a nonzero constant—yet it can map three different points to the same point, thus lacking invertibility and forming a counterexample to the Jacobian Conjecture. He said that this construction was completed with the help of AI model Fable 5, which was inspired by a friend's question raised during the World Cup final.

Subsequently, several researchers independently verified the result on social media. Public responses indicate that, through symbolic calculations, numerical calculations, and exact rational arithmetic, the two core conclusions—that the Jacobian determinant is always constant and that the mapping is not injective—have been confirmed, and Wolfram Alpha's computational results also support these conclusions.

However, the relevant result is still only publicly available on social media, has not been posted to arXiv, and has not undergone peer review, so further verification by the mathematical community is still awaited.

A classic problem that has troubled mathematics for more than 80 years

The Jacobian Conjecture was proposed by German mathematician Keller in 1939 and is one of the most famous open problems in algebraic geometry. Its core content can be summarized as: If a multivariable polynomial map has a Jacobian determinant that is a nonzero constant, does it necessarily have a polynomial inverse mapping?

According to the inverse function theorem in multivariable calculus, a nonzero Jacobian determinant ensures local invertibility of the mapping; but whether this local invertibility can be extended to global invertibility has never been proved. Over the past decades, the Jacobian Conjecture has been found to be deeply connected with multiple mathematical fields and is considered one of the foundational problems in modern algebraic geometry.

This conjecture also has a little-known connection to Chinese mathematician Yitang Zhang. According to public information, Zhang worked on the Jacobian Conjecture during his doctoral studies. His advisor required him to develop the proof based on a certain lemma, but the lemma was later found to be incorrect, rendering the related research invalid and becoming one of the reasons for his long-term career difficulties afterward. Years later, Zhang gained fame by proving the existence of bounded gaps between prime numbers.

If the counterexample stands, the impact may go beyond a single conjecture

Alpoge stated that if this counterexample is ultimately confirmed, its impact may not be limited to the Jacobian Conjecture. In a reply, he mentioned, the Dixmier Conjecture and the Poisson Conjecture, which are equivalent to or closely related to it, may also be affected. However, because the Jacobian Conjecture is more widely known, he chose to announce this result first.

Meanwhile, some researchers are trying to consider how to revise the original proposition.

According to public discussions, GPT 5.6 has proposed a possible revised version of the conjecture: If a polynomial mapping with constant Jacobian determinant is locally bianalytic and there is no “fiber loss” at infinity (for example, it is a proper Keller map), then it may still be an automorphism. This direction currently remains theoretical and has not yet resulted in formal research findings.

The real test remains peer review

Although initial verification results have attracted wide attention from the mathematics community, the real fate of this result will be determined by peer review.

Historically, works that were believed to be close to proofs or counterexamples for the Jacobian Conjecture have been found flawed during strict scrutiny, so many researchers jokingly call the problem the “graveyard of amateur mathematicians.” Alpoge himself also joked about this on social media.

At present, there is no complete paper for this result, nor any arXiv preprint. Although Wolfram Alpha and several researchers have completed preliminary computational verification, none of this can replace a rigorous mathematical proof and academic review.

Meanwhile, the capabilities of AI in formal mathematical research still require further validation. The analysis agency explainx.ai pointed out that there is still a lack of systematic evidence to evaluate the reliability of Fable 5 in higher-order mathematical reasoning, and a single case is not enough to deduce its overall ability.

According to mathematical community norms, if this counterexample is ultimately established, the research team is expected to publish a paper containing the complete derivation process, and its academic status will be formally established after peer review. Until then, the Jacobian Conjecture remains a recognized unsolved problem in the mathematics community.

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