Fable 5 Refutes the 87-Year-Old Jacobian Conjecture, Shaking the Mathematics Community

On this day, the entire mathematics community was shaken.

The Jacobian conjecture, a legendary mathematical problem that has defeated countless top mathematicians and forced Chinese mathematical prodigy Yitang Zhang to spend seven grueling years washing dishes at Subway, has been disproven by Fable 5!

Yesterday evening, Anthropic researcher Levent Alpoge posted a tweet:

Hello everyone, the Jacobian conjecture is false. Thank you to my friend Akhil for asking this question, and to my other friend Fable for working during the World Cup final.

Below the caption is a simple mathematical formula.

A core mathematical problem that had persisted for 87 years was effortlessly solved by Fable 5 on a Sunday evening.

The Jacobian conjecture is extremely difficult. It may take humanity another hundred years to solve it. Your attempt is admirable; perhaps the gods of Olympus will one day favor you.

The disproof of the Jacobian conjecture may be just the first domino to fall, with multiple conjectures potentially being disproven.

As mathematician Jared Duker Lichtman exclaimed: “This is one of the most inspiring stories in modern mathematics.”

Some say I’ve rarely seen X in the scientific community as excited and stunned as it is now.

Behind this lies the seven years that Zhang Yitang had stolen from him.

Let’s rewind to 1939. That year, German mathematician Ott-Heinrich Keller posed a question:

If the Jacobian determinant of a polynomial mapping is a nonzero constant, does the mapping necessarily have a polynomial inverse?

In fact, the intuition behind this question is straightforward—in calculus, the inverse function theorem tells us that if the Jacobian determinant of a function at a point is non-zero, then the function has a local inverse near that point.

The Jacobian conjecture asks: If this condition holds everywhere and the function is a polynomial mapping, must the inverse mapping also be polynomial?

This is a classic "local to global" problem that seems intuitively true but has puzzled the world's brightest minds for 87 years.

The two-dimensional version was proposed as early as 1884, along with a proof that was later found to contain a flaw.

This problem may seem simple, but it is a genuine mathematical black hole. Countless mathematicians have published numerous papers claiming to prove the conjecture, but none have withstood scrutiny—all were found to contain logical flaws.

Until this weekend, Claude Fable 5 has arrived.

It did not follow human reasoning to prove it, but instead directly in three-dimensional space.

Threw out a counterexample:

This seemingly complex polynomial mapping from C³ to C³ has a Jacobian determinant that is constantly equal to -2—a nonzero constant—satisfying the hypothesis of the conjecture.

In short, this counterexample is so concise and elegant that it is almost terrifying: any first-year university student who has studied calculus can compute the Jacobian determinant of this ternary polynomial and find that it is constantly equal to -2 by taking partial derivatives.

If the prerequisites of the conjecture are satisfied, is it reversible?

Fable 5 clearly demonstrates its non-injectivity—the function maps three distinct points in space: (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2)—all to the same image point: (-1/4, 0, 0).

Since three different points lead to the same result, it naturally cannot have an inverse function.

Thus, the generalized Jacobian conjecture has been completely disproven.

An 87-year-old conjecture has been casually solved by AI.

On social media, renowned math professors and graduate students rushed to verify this counterexample.

Someone quickly verified it using Wolfram Alpha and found the result to be completely accurate.

But the special aspect of this counterexample is precisely that it is self-verifying: it is so concise that it can be verified by hand, and its elegance and precision are remarkable.

Jason Lee, Associate Professor of Computer Science and Statistics at UC Berkeley and former research scientist at Google DeepMind, bluntly stated: "Math is solved."

Shortly after, GPT-5.6 quickly analyzed this result and proposed a new, revised conjecture: “A locally biholomorphic mapping of a constant Jacobian polynomial, if it has no leaf loss at infinity, is an automorphism.”

Aaron Lou from OpenAI used the internal Codex (without web search) to derive from scratch an equivalent counterexample, fully writing out the strategy and proof, along with reproducible mathematical derivations (from cubic factorization to affine coordinate transformations).

It not only uncovered the structural mechanisms behind the counterexamples but also guided humans in redefining this problem.

This demonstrates that AI possesses genuine mathematical creativity, not just superficial imitation.

Mathematician Lichtman expressed amazement, hoping to see how top algebraists would respond to the new conjectures posed by GPT.

The complete derivation process is as follows:

https://aaronlou.com/jacobian_counterexample_derivation.pdf

Data scientist Cal Aldred even proposed a method for generating an infinite number of counterexamples using Fable + GPT 5.6 Sol!

But the most poignant part of this story involves Zhang Yitang.

Everyone remembers the story of the math genius who worked at Subway.

Zhang Yitang, renowned worldwide for his groundbreaking contribution to the twin prime conjecture, endured a tragic career in academia for decades prior to that.

In the early 1990s, Zhang Yitang was pursuing his Ph.D. at Purdue University under the supervision of Zhongjian Mo, who firmly believed in the truth of the Jacobian Conjecture and assigned it as Zhang’s doctoral thesis topic.

Mo Zhongjian gave Zhang Yitang a “lemma”—a mathematical proposition believed to be true—as a foundation for his research. Zhang pursued this line of inquiry deeply, even claiming in his doctoral thesis to have solved the weak form of the conjecture.

However, fate played a cruel joke.

After review by peers, the key lemma used in Zhang Yitang's proof has been proven incorrect—and this lemma was derived from earlier academic work published by his advisor, Mo Zhongjian.

The theoretical structure he had spent years building suddenly revealed its foundation to be shoddy—all his efforts and deductions collapsed at once.

Mo Zhongjian evaluated Zhang Yitang as “unsuited for algebraic geometry,” claiming that his doctoral studies “wasted seven years of his own time and mine.” Even more devastating, Mo Zhongjian refused to write a letter of recommendation for Zhang Yitang.

This leaves a Purdue University PhD graduate unable to make progress in academia.

Zhang Yitang was forced to leave academia and began a long period of wandering. He took on various odd jobs, the most well-known of which was working at a Subway sandwich shop for seven years.

In 2013, at the age of 58, Zhang Yitang published a groundbreaking paper on the twin prime conjecture, instantly transforming from an obscure university lecturer into a globally renowned mathematician.

But those seven wasted years can never be regained.

Later, he quoted a line from Du Fu to describe how he felt during that time: “Yu Xin’s life was always the most desolate; in his later years, his poems and writings stirred the rivers and passes.”

More ironically, in 2018, after Zhang Yitang gained fame for his work on the twin prime conjecture, Mo Zhongjian published an updated version of his memoir, further intensifying his accusations against Zhang while still downplaying his own responsibility.

He still believes in the correctness of the Jacobian conjecture and even thinks that AI will soon help resolve the logical issues in the proof.

Finally, today, AI easily provided a simple counterexample.

How heartbreaking.

Of course, it must be noted that this AI refuted the three-dimensional Jacobian conjecture, thereby negating the generalized version that claims it holds in all dimensions.

Zhang Yitang originally studied the two-dimensional case, and two dimensions are not simply a lower-dimensional version of three dimensions—on the contrary, the two-dimensional case is more central and, as far as we know, even more difficult to solve, carrying greater mathematical significance and remaining unsolved to this day.

But this in no way diminishes the impact of the event.

Fields Medalist Timothy Gowers remarked with pessimism and shock: “The 2030 Fields Medal in Mathematics may be the last one ever awarded to a human.”

Reference materials:

https://x.com/search?q=Jared%20Duker%20Lichtman&src=typed_query

This article is from the WeChat public account "New Intelligence Yuan," authored by ASI Revelation, edited by David Aeneas.