While billions of people were still arguing about penalty kicks from the FIFA World Cup Final, a quiet bombshell landed in a very different arena. Levent Alpöge, a mathematician at the AI company Anthropic, posted a message on X so casual it almost felt like a typo. He had found a counterexample to the Jacobian conjecture — one of the most stubbornly unsolved problems in algebraic geometry — and he had done it with help from Claude Fable 5, Anthropic's latest large language model, released only weeks earlier. So what exactly had been cracked?

The Jacobian conjecture is about polynomial functions — think of them as rules that move points around in mathematical space, like a cosmic reshuffling of coordinates on a map. A special measurement called the Jacobian determinant tells you whether that reshuffling is "nice": no folding, no crushing, no two points smashing into one. The conjecture, first proposed by Czech mathematician Ludwig Kraus in 1884 and later generalised to any number of dimensions by German mathematicians, asked: if your reshuffling is always "nice," can you always reverse it perfectly? For 140 years, nobody could say yes or no with certainty.

Alpöge said no — and showed why. What makes this moment feel different from earlier AI-mathematics breakthroughs is the nature of the result. Previous achievements often involved verifying or extending what humans already suspected. A counterexample, by contrast, overturns an assumption that generations of brilliant mathematicians had quietly believed was true.