Anthropic的数学家Levent Alpöge利用AI模型Claude Fable 5发现了Jacobian猜想的反例，证明该猜想在三维及更高维度上并不成立[1]。这个反例是一个简洁的多项式函数，具有常数Jacobian行列式-2，但能将多个不同的输入点映射到同一个输出点[1]。
Jacobian猜想由捷克数学家Ludwig Kraus于1884年提出，后由德国数学家Ott-Heinrich Keller在1939年推广至任意维度[1]。这一猜想已经困扰数学家超过一个世纪之久，甚至被Fields奖得主Stephen Smale列入1998年的《下个世纪的数学问题》中[1]。Alpöge利用Claude Fable 5——这款AI模型仅在数周前向公众发布——找到的反例简洁到可以装入一条X帖子[1]。该发现证明猜想在所有大于2的维度上都是错误的，尽管二维版本仍未得到解决[1]。
Levent Alpöge, a mathematician at Anthropic, has used the AI model Claude Fable 5 to find a counterexample that disproves the Jacobian conjecture in three dimensions and higher [1]. The counterexample is a remarkably compact polynomial function with a constant Jacobian determinant of -2 that maps multiple input points to a single output point, demonstrating that the conjecture fails in all dimensions greater than two [1].
The Jacobian conjecture, originally proposed by Czech mathematician Ludwig Kraus in 1884 and later generalized to arbitrary dimensions by German mathematician Ott-Heinrich Keller in 1939, has challenged mathematicians for over a century [1]. The problem was significant enough to be included by Fields Medal winner Stephen Smale in his 1998 list of mathematical problems for the next century [1]. The newly discovered counterexample is so concise that it can fit within a single X post [1]. While the discovery resolves the conjecture for all dimensions greater than two, the two-dimensional version remains unsolved [1]. Claude Fable 5 was released to the public only weeks before this discovery, marking another major breakthrough in applying artificial intelligence to mathematical research [1].