AI Finds Simple Counterexample to 87-Year-Old Jacobian Conjecture, Toppling Higher-Dimensional Versions

For 87 years, mathematicians have been locked in a stalemate over a seemingly straightforward question: if a mathematical transformation never squishes space flat, can you always perfectly reverse it? The Jacobian conjecture suggested the answer was yes, provided the transformation was made of polynomials.

But the sheer infinite variety of polynomials made proving it impossible, while finding a single counterexample seemed like searching for a needle in a universe-sized haystack. Mathematicians were stuck in a loop of proposed proofs that inevitably contained subtle errors, leaving the problem in a state of suspended animation.

That stalemate broke on July 19, 2026. While millions watched the World Cup final, an artificial intelligence model named Claude Fable 5 navigated that infinite haystack and pulled out the needle. Anthropic and Harvard mathematician Levent Alpöge posted the result on social media: a short, degree-seven polynomial equation that completely shatters the conjecture for three dimensions and higher.[1][4]

The Jacobian conjecture, formalized by Ott-Heinrich Keller in 1939, deals with vector-valued polynomial maps—functions that take multiple numbers in and spit multiple numbers out. To understand the claim, one must look at the "Jacobian determinant," a mathematical tool that measures how a function stretches or rotates space locally.[3][5]

If the determinant is a non-zero constant, the function is locally reversible everywhere. This means that if you zoom in close enough to any point, the transformation behaves like a simple, reversible grid. The conjecture claimed this local reversibility guaranteed a global, polynomial inverse—meaning you could always trace your steps back to the exact starting coordinates from anywhere in the space.[3][5]

For decades, mathematicians believed this to be true. The problem was considered so foundational that Fields Medalist Stephen Smale included it in his 1998 list of the 18 most important mathematical problems for the 21st century.[1][2]

The AI-generated counterexample dismantles this assumption using a specific function mapping three-dimensional complex space back to itself. The data is unambiguous: its Jacobian determinant is exactly -2 at every single point, fulfilling Keller's requirement perfectly.[3][5][6]

Yet, the function contains a fatal flaw for the conjecture. When you plug in three specific, distinct starting coordinates, the function spits out the exact same destination. Because three different paths lead to the same destination, the function cannot be reversed.[2][3][4]

This does not violate the inverse function theorem, which guarantees local reversibility. The local inverses exist around each of the three input points, but they do not fit together into a single global inverse. Near the shared output, there are several local inverse branches leading in different directions, making a global polynomial inverse impossible.[3]

While the math is verified and undeniable, the evidence has strict boundaries. Because the counterexample operates in three dimensions, it disproves the conjecture for 3D and any higher dimension. However, it does not touch the original two-dimensional version of the problem, first proposed by Ludwig Kraus in 1884, which remains entirely unsolved.[1][2][4]

The mathematical structure that allows the 3D collision relies on a scaling freedom that disappears in 2D. The factorization picture gives a limited but revealing explanation of why the new mechanism does not immediately descend to the plane, leaving the most famous version of the problem intact.[3]

The breakthrough highlights a shift in how AI is used in mathematics. The AI didn't write a 100-page logical proof; instead, it navigated an enormous search space of polynomial mappings to find a single, fatal counterexample.[2][4]

The resulting equation is so brief it fits in a single social media post, allowing human mathematicians to verify the arithmetic by hand within hours. This brevity is a stark contrast to the intricate constructions usually required to break long-standing conjectures.[4][6]

What remains opaque is the exact prompting methodology used to coax the model toward this specific polynomial structure. The difficulty lay not in the complexity of the answer, but in finding a way to search the space effectively.[2]

Furthermore, while AI excels at finding these hidden objects, mathematicians caution that models still struggle to generate reliable, step-by-step logical proofs for problems that require them. The Jacobian conjecture is broken in higher dimensions, but the search for a true mathematical reasoning engine continues.[1]