Fable 5 just solved a math problem that was impossible to solve for 87 years.

Fable 5 just solved a math problem that was impossible to solve for 87 years, called the Jacobian conjecture. Imagine an infinite 3D complex space. You can move every point anywhere new, but you can only use addition, subtraction, or multiplication. Simply put, you can use only polynomial equations to build this entire system called a polynomial map. At any single point, you can zoom in and check something called the Jacobian determinant. Think of it as a local stretch factor that tells you whether the space around a point gets stretched, squeezed, or crushed flat so that different nearby points become impossible to distinguish. Now, suppose that number comes out to be some constant non-zero value at every single point in the space. No matter where you zoom in, nothing ever gets flattened. Does that guarantee the whole transformation is globally reversible, that every final point came from exactly one starting point, or could two different starting points still end up in exactly the same place, even though everything looks perfectly fine up close? That's the core of the Jacobian conjecture. For 87 years, mathematicians believed the answer was yes. They tried to prove that no matter what polynomial equations you came up with, if they followed those rules, the entire space would always be reversible, but nobody could prove or disprove it. But then came Fable. When researchers gave it the problem and let it reason from first principles, it found something nobody was expecting. A polynomial map with a Jacobian determinant of minus 2. A single example where, even though the Jacobian determinant was fixed and non-zero everywhere, one point could be traced back to three different starting points instead of just one. And that was enough for Fable to falsify an 87-year-old mathematical belief. First posted by the researchers, the result still needs to go through peer review, but if it holds up, Fable might just have settled one of the biggest problems in mathematics. The funny part is, you can even talk to Claude about it and watch it lose its mind.