
AI “punches” the mathematical century-old conjecture: a mathematician's tool, or the end?
Source: Observer Network Mind Observatory Original title: 1987's mathematical conjecture was easily overturned by AI, meaning what 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 gives an amazing answer: the Jacobian conjecture, which people have been trying to prove for almost a century, is 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 Jacoby'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 very small area can be crushed (that is, reduced dimensional). 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 a polynomial map? 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. The elements of the Jacobian matrix are the biases of the mapping: the Jacobian determinant can be understood as the rate of change in the 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 there must be a polynomial inverse mapping for this 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. Counterexample of the World Cup final night: How AI disrupted 87's beliefs July 19, 2026, coincided with the night of the World Cup final. Alpöge discussed the Jacobian 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 → ³P = (1+xy) ³z + y² (1+xy) (4+xy) (4+3xy) Q = y + 3x (1+xy) ²z +...










