AI “punches” the mathematical century-old conjecture: a mathematician's tool, or the end?
Source: Observer Network Mind Observatory
Original title: What does it mean when 87's mathematical conjectures were easily overturned by AI
The Jacobian Conjecture was proposed by mathematician Ott-Heinrich Keller in 1939 and was once regarded as one of the most important unsolved puzzles in the field of algebraic geometry. Over the past 87 years, the world's top mathematicians have tried to prove this conjecture, but they have never been able to break through. Now Claude Fable has found a simple three-dimensional polynomial mapping and easily gave an amazing answer: the Jacobian conjecture, which people have been trying to prove for nearly a century,It was actually wrong.
A brief announcement that changed the history of mathematics
On July 20, 2026, Harvard mathematician Levent Alpöge, who is also an Anthropic researcher, posted a brief message on social media announcing that Jacobi's conjecture was wrong, and immediately thanked his “good friend who was still working during the World Cup finals”: Claude Fable.
A string of simple polynomials that anyone can verify is attached at the bottom of the text: a counterexample of the Jacobian conjecture.
One was proposed in 1939 and included in the 21st century math problem list. It stuck the conjectures of generations of mathematicians, and was easily overturned by a social media dynamic with just a few short lines. The news spread rapidly, and many people checked it out. AI was able to present groundbreaking mathematical results in such a random form, causing quite a stir. Abhishek Saha, a mathematician at Queen Mary University of London, said that so far, this is probably the mathematical conjecture with the highest level of AI participation. “This is no small matter. The progress of AI over the past year is truly amazing.
An 80 year old math myth
Let's start with an intuitive question. A rubber film is continuously stretched and distorted, but it is required that no extremely small area can be crushed (that is, reduced in dimension). Mathematicians will ask: if the dimensions of every local area are not damaged, then does this kind of folding necessarily not occur on the entire rubber sheet: that is, the two positions are rubbed into the same point?
This is exactly the question the Jacobian conjecture is trying to answer.
Consider a polynomial mapping:
F: →
What is polynomial mapping? Simply put, it is to move a point in an n-dimensional space to another set of coordinates in the same space through several polynomial formulas. The mapping of two-dimensional space and three-dimensional space is F (x, y) = (f (x, y), g (x, y)) F (x, y, z) = (P (x, y, z), Q (x, y, z), R (x, y, z)), respectively. Mapping F can be viewed as a deformation operation. It can be compressed, stretched, distorted, and pan.
The Jacobian determinant is the determinant of the Jacobian matrix, and the elements of the Jacobian matrix are the biases of the mapping:
The Jacobian determinant can be understood as the rate of change in volume of a tiny space after being mapped. When the Jacobian determinant is equal to -2, it means that the volume of this tiny space is doubled, and the direction is reversed at the same time. However, when the Jacobian determinant is not 0, according to the inverse function theorem, this mapping must have an inverse transformation locally; it does not compress a three-dimensional space into a low-dimensional plane.
But the question is: if it is partially reversible, does it mean that the whole is reversible?
According to the Jacobian conjecture, if the Jacobian determinant of a polynomial mapping is constant equal to a non-zero constant throughout the space, then this mapping must have an inverse polynomial map. In other words, as long as every tiny area of the rubber film is not flattened (reduced in dimension), and the local volume scaling ratio of each point is exactly the same, then overall there should be no folding, and there should be no situation where two different positions on the rubber film are rubbed to the same point.
This is a typical “push from one part to the whole” math problem. Since it was proposed in 1939, the Jacobian conjecture has plagued the mathematical community for nearly 87 years and has become one of the most famous problems in modern algebraic geometry.
The counterexample of World Cup final night: How AI disrupted '87 beliefs
On July 19, 2026, coinciding with the night of the World Cup finals, Alpöge discussed Jacobi's conjecture with his friend Akhil Mathew. Mathew asked a key question: Can we find a counterexample to Jacoby's conjecture? Alpöge gave this task to the AI model.
Amidst the hustle and bustle of the World Cup finals, the AI model continues to work. Instead of following human thought to prove conjectures, it directly threw out a simple counterexample in three-dimensional space.
This three-dimensional polynomial mapping is:
F: ³ → ³
P = (1+xy) ³z + y² (1+xy) (4+3xy)
Q = y + 3x (1+xy) ²z + 3xy² (4+3xy)
R = 2x − 3x²y − x³z
This mapping has two shocking properties.
First, it satisfies the requirements of Jacobi's conjecture. The calculation shows that the Jacobian determinant is always equal to −2. In other words, no matter what values x, y, and z take, there will be no partial dimensional reduction. According to Jacobian conjecture, the inverse of F should exist.
Second, it's not one-to-one. The AI has identified three completely different points:
A = (0, 0, −1/4)
B = (1, −3/2, 13/2)
C = (−1, 3/2, 13/2)
They are all mapped to the same point by F: (−1/4, 0, 0).
The three completely different locations are mapped to the same place, which means F has no inverse function: because, if the inverse of F exists, then the inverse function of F (A) must be equal to A, B, and C at the same time, which is obviously impossible.
This counterexample is so simple and elegant that it is outrageously elegant. Any student who has studied calculus can obtain partial derivative verification on their own. The shocked professor and students scrambled to find evidence and found that the results were absolutely correct.
Abhishek Saha of Queen Mary University of London stated, “Some problems are difficult to solve, but once you have a solution, it's quite easy to verify. That's the counterexample. I don't know how he did it, and what are the tips he gave to the AI, because exhausting everything won't work.”
Jacobi's Conjecture and Zhang Yitang
The reason the Jacobian conjecture is important is that it connects many fields of mathematics: algebraic geometry, commutational algebra, singularity theory, and differential topology. American mathematician Steve Smale (Steve Smale) once included the Jacobian conjecture as one of 18 important questions in 21st-century mathematics.
Numerous studies have shown that in many special cases, mapping is indeed reversible as long as Jacobian conditions are met. Mathematicians continue to discover that low-dimensional situations are established, special types are established, and certain high-level situations are established. These results reinforce people's intuition: the Jacobian conjecture should be true, but it's just too difficult to prove. At the same time, human intuition tends to assume that local dimensions are not lost, and that if the scaling ratio is exactly the same at every location, then there will be no overall folding. So everyone continued down the path of proof.
There is also a special history between the Jacobian conjecture and the mathematician Zhang Yitang. In the 80s, young Zhang Yitang was studying for a PhD at Purdue University in the US, and his research direction was Jacoby's conjecture. His doctoral dissertation “The Jacobian Conjecture and the Degree of Field Extension” attempted to cut into the Jacobian conjecture from the perspective of a husband's case. However, his proof relied on an argument previously published by his mentor Mo Zongjian, and this reasoning was later falsified, proving that the chain could not be established. This setback greatly affected his academic path, causing him not to get a mentor recommendation letter after graduating from his doctorate, and was unable to gain a foothold in academia.
Why hasn't anyone discovered it in 87 years?
So, a question naturally arises: this counterexample is so simple, the number of polynomials is so low, and the coefficient is small. In 87, hasn't anyone done an exhaustive search in a small range? Doesn't anyone know about turning a corner?
Perhaps the answer lies in the inertia of mathematical research. Mathematicians are used to proofs rather than looking for counterexamples. However, AI does not have this kind of inertia; it can explore huge spaces that are difficult for humans to search and explore possibilities in blind spots in human thinking.
If confirmed, what does it mean?
If this counterexample were to be rigorously verified by the mathematical community, the impact would be huge.
First, it shows that the Jacobian conjecture is not difficult to prove; it is wrong. The research direction in '87 needs to be reorganized.
Second, the idea of “local control of the whole” in mathematics will be impacted. In the past, most scholars believed that as long as there were no problems around every point, there would be no problems with the overall structure. This counterexample shows that a more complex folding mechanism may be hidden in a high-dimensional space.
Third, AI has entered a new stage of mathematical research. What people used to talk about was can AI prove theorems? But this incident suggests another possibility: AI can help humans discover whether the questions asked by humans are correct themselves.
However, this does not mean that all versions of Jacobi's conjecture will be declared dead. This counterexample falsifies conjecture in three-dimensional space, but the two-dimensional version may still theoretically be true. Some scholars have warned that finding counterexamples is not tantamount to establishing a new branch of mathematics; the latter still requires human creativity.
Will AI become an experimental machine for mathematicians?
Mathematical proofs still require strict logic. After an AI finds a candidate counterexample, a human mathematician must verify the formula, check the calculation, establish strict proof, and judge its theoretical significance, which is not equivalent to being replaced by AI.
But AI changed one thing: in the past, mathematicians' search space was limited; now AI can explore millions or even billions of structures. This is similar to experiments in physics. Before Galileo, humans could only guess intuitively. After the experiment appeared, the laws of nature began to be tested. In the future, AI may become an experimental device for mathematicians. It doesn't replace mathematicians with deep questions, but it can tell humans that it has been going in this direction for almost a century, but it's actually going astray.
The age of mathematical exploration belonging to AI has arrived
From Euclidean geometry to non-Euclidean geometry, from classical physics to quantum mechanics, every major turning point in the history of science began when something “taken for granted” was crushed.
Over the past 87 years, mathematicians have followed a seemingly straight path: put forward conjectures, find proofs, and establish theories. Everyone believed the Jacobian conjecture was true; after all, it was so intuitive. But Claude Fable's answer on the night of the World Cup finals reversed people's perception. What is shocking about this counterexample is not how complicated it is, but precisely because in '87, no human mathematician wanted to take a look at this simple counterexample corner.
This is the transformation brought about by AI. Mathematical research in the past was like sailing in a known ocean. Mathematicians use their intuition, experience, and deep education to choose the most promising direction to explore. This model has created countless splendors, but it also condones the inertia of the human mind. We always tend to prove what “should be right” rather than question those “obviously true” assumptions.
AI doesn't have this inertia. It doesn't worship conjecture, doesn't fear authority, and isn't influenced by obvious things. AI is willing to sail into any unknown sea, even if it seems empty.
In the future, the relationship between mathematicians and AI may become the relationship between explorers and compasses. Humans are responsible for asking deep questions and guiding the way forward; while AI is responsible for exploring hidden corners that cannot be touched by human intuition, and then telling humans: You are at a dead end; what you think doesn't exist is actually there.
This is a story of Jacobian conjecture; it is also a prediction of future mathematics. The age of mathematical exploration, which belongs to AI, has arrived.
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