Claude Fable produced a counterexample to the Jacobian Conjecture
Claude Fable produced a counterexample to the Jacobian Conjecture
Claude Fable 给出了雅可比猜想的一个反例
Levent (@alpoge) announced that the Jacobian Conjecture is false, thanks to his friend Akhil for the inquiry and his friend Fable for working on it during the World Cup final. The polynomial map $((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z): \mathbb{C}^3\to \mathbb{C}^3$ has a Jacobian determinant of -2, and it maps the points (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) all to (-1/4, 0, 0).
Levent (@alpoge) 宣布雅可比猜想（Jacobian Conjecture）已被证伪。他感谢好友 Akhil 提出这个问题，并感谢另一位好友 Fable 在世界杯决赛期间完成了这项工作。该多项式映射 $((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z): \mathbb{C}^3\to \mathbb{C}^3$ 的雅可比行列式为 -2，且将点 (0, 0, -1/4)、(1, -3/2, 13/2) 和 (-1, 3/2, 13/2) 全部映射到了 (-1/4, 0, 0)。
Grok confirmed the finding: “Yes, this is a valid counterexample. The polynomial map $\mathbb{C}^3\to\mathbb{C}^3$ has a constant Jacobian determinant of -2 but sends three distinct points to the same image (-1/4, 0, 0), so it is not injective. This disproves the Jacobian conjecture.”
Grok 证实了这一发现：“是的，这是一个有效的反例。该 $\mathbb{C}^3\to\mathbb{C}^3$ 的多项式映射具有常数雅可比行列式 -2，但将三个不同的点映射到了同一个像 (-1/4, 0, 0)，因此它不是单射。这推翻了雅可比猜想。”
The discussion has sparked broader interest in the implications for AI. User changgyu Boo noted that this counterexample illustrates how local gradient information in LLM training might fail to prevent “global collisions,” where semantically different prompts are mapped to the same output, potentially explaining certain types of LLM hallucinations.
这一讨论引发了人们对 AI 影响的广泛关注。用户 changgyu Boo 指出，这个反例说明了大型语言模型（LLM）训练中的局部梯度信息可能无法防止“全局碰撞”（即语义不同的提示词被映射到相同的输出），这可能解释了某些类型的 LLM 幻觉现象。
As the mathematical community processes this breakthrough, some users have jokingly suggested that Fable is now “the boss,” while others are speculating on which major unsolved mathematical problems might be the next to fall to AI-assisted research.
随着数学界对这一突破的消化，一些用户开玩笑说 Fable 现在是“老板”了，而另一些人则在推测下一个可能被 AI 辅助研究攻克的重大未解数学难题会是什么。