The mathematical world was shaken on this day.
A super mathematical conundrum that had stumped countless top mathematicians and forced the Chinese genius mathematician Yitang Zhang to toil for seven years, even washing dishes at Subway—the Jacobian Conjecture—was disproven by Fable 5!
Yesterday evening, Anthropic researcher Levent Alpoge posted on Twitter:
Hey everyone, the Jacobian Conjecture is false, thanks to my friend akhil for asking this question, and thanks to my other friend Fable for working during the World Cup final.
Below the caption was a concise mathematical formula.
A core mathematical problem with an 87-year history was thus effortlessly resolved by Fable 5 on a Sunday evening.
The Jacobian Conjecture problem is extremely difficult. Perhaps humanity would need another hundred years to solve it. Your attempts are commendable; perhaps the Olympian gods 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 said, I rarely see the science community on X in such a state of frenzied excitement and immense shock as it is now.
And behind this lies the seven years stolen from Yitang Zhang.
What is the Jacobian Conjecture? A "Seemingly Obvious" Trap
Let's rewind to 1939. That year, German mathematician Ott-Heinrich Keller posed a question:
If the Jacobian determinant of a polynomial map is a nonzero constant, must this map necessarily have a polynomial inverse?
In fact, the intuition behind this problem is simple—in calculus, the inverse function theorem tells us that if the Jacobian determinant of a function is nonzero at a point, 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 map, must the inverse map also be a polynomial?
This is a classic "local to global" problem. It seems like it should obviously hold, yet it has baffled the world's brightest minds for a full 87 years.
The two-dimensional version was proposed as early as 1884, accompanied by a proof that was later found to contain flaws.
A Casual Strike, AI Delivers a God-Tier Counterexample
This problem seems simple but is a veritable black hole of mathematics. Countless mathematicians have published numerous papers claiming to prove the conjecture, but without exception, logical flaws have been found in them all.
Until this weekend, when Claude Fable 5 took the stage.
It didn't follow human reasoning to attempt a proof; instead, it directly threw out a counterexample in three-dimensional space
:
This rather complex-looking polynomial map from C3 to C3 has a Jacobian determinant that is always -2—a nonzero constant, satisfying the conjecture's premise.
In short, this counterexample is frighteningly concise and elegant. Any freshman who has taken university calculus can compute the partial derivatives to find that the Jacobian determinant of this trivariate polynomial is always -2.
Having satisfied the premise, is it invertible?
Fable 5 simply pointed out its non-injectivity—this 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 map to the same result, it naturally cannot have an inverse function.
And just like that, the generalized Jacobian Conjecture was completely disproven.
An 87-year-old conjecture was thus "casually" cracked by AI.
On social media, stunned mathematics professors and graduate students rushed to verify this counterexample.
Some used Wolfram Alpha for a quick check and found the result perfectly correct.
But the peculiarity of this counterexample lies precisely in the fact that it is self-evident: it's simple enough to verify by hand, its elegance and precision breathtaking.
UC Berkeley Associate Professor of Computer Science/Statistics and former Google DeepMind research scientist Jason Lee exclaimed: "Math is solved."
Soon after, GPT-5.6 swiftly analyzed this result and proposed a brand new, revised conjecture: "A constant Jacobian polynomial's locally biholomorphic map, if it has no loss of leaves at infinity, then it is an automorphism."
OpenAI's Aaron Lou, using an internal Codex (without web search), derived an essentially identical counterexample from scratch, fully wrote out the strategy and proof, and provided reproducible mathematical derivations (from cubic factorization to affine coordinate transformations).
It not only found the structural mechanism behind the counterexample but also guided humans on how to redefine the problem.
This proves that AI now possesses genuine mathematical creativity, not just superficial imitation.
Mathematician Lichtman exclaimed, hoping to see how top algebraists would react to the new conjecture proposed by GPT.
The complete derivation process is as follows:
https://aaronlou.com/jacobian_counterexample_derivation.pdf
Data scientist Cal Aldred even used Fable + GPT 5.6 Sol to propose a method for generating infinitely many counterexamples!
The Seven Years Stolen from Yitang Zhang
But the most poignant part of this story involves Yitang Zhang.
Everyone remembers the story of this mathematical genius who worked at Subway.
Yitang Zhang became globally renowned for his breakthrough contributions to the Twin Prime Conjecture, but for decades prior, his experience was an academic tragedy.
In the early 1990s, Zhang was pursuing his Ph.D. at Purdue University under the supervision of T.T. Moh. Moh firmly believed in the correctness of the Jacobian Conjecture and assigned it as Zhang's doctoral thesis topic.
Moh gave Zhang a "lemma"—a mathematical statement believed to be correct—as the foundation for his research. Zhang delved deeply along this line, even claiming in his doctoral dissertation to have solved a weak form of the conjecture.
However, fate played a cruel joke.
Upon peer review, the key lemma used in Zhang's proof was shown to be false. And this lemma originated from his supervisor Moh's own previously published academic work.
The theoretical edifice he had spent years building suddenly revealed its foundation to be shoddy. All his efforts, all his derivations, collapsed.
Moh commented that Zhang was "not suitable for algebraic geometry," believing his doctoral years had "wasted seven years of his own life and also my time." More fatally, Moh did not write a letter of recommendation for Zhang.
This left a Ph.D. graduate from Purdue University with nowhere to go in academia.
Zhang had to leave academia and began a long period of drifting. He took various odd jobs, most famously working at a Subway fast-food restaurant for seven years.
It wasn't until 2013, at age 58, that Zhang published his breakthrough paper on the Twin Prime Conjecture, transforming overnight from an obscure university lecturer into a globally acclaimed mathematician.
But those seven wasted years could never be recovered.
Later, he quoted a line from Du Fu's poetry to describe his feelings during that time: "In all his life Yu Hsin knew most loneliness, / In his later years his poetry moved rivers and passes."
More ironically, in 2018, after Zhang became famous for the Twin Prime Conjecture, Moh specifically added a new version of his "memoir," further strengthening his criticism of Zhang while continuing to downplay his own responsibility.
He still believed in the correctness of the Jacobian Conjecture, even thinking that AI would soon help resolve logical issues in the proof process.
Finally, today, AI effortlessly provided a simple counterexample.
So poignant.
Of course, it must be pointed out that this AI disproof concerns the three-dimensional Jacobian Conjecture, negating the generalized version that "holds for all dimensions."
Zhang's research back then focused on the two-dimensional case, and two dimensions are not a simple lower-order version of three dimensions—on the contrary, the two-dimensional case is the more core and, it seems, the harder part to crack in this problem, with greater mathematical significance, and remains unsolved to this day.
But this in no way diminishes the impact of this event.
Fields Medalist Timothy Gowers pessimistically yet profoundly remarked: "The 2030 Fields Medal might be the last one awarded to a human."
References:
https://x.com/search?q=Jared%20Duker%20Lichtman&src=typed_query
This article is from the WeChat public account "New Zhiyuan," author: ASI Apocalypse, editor: David Aeneas
