Anthropic researcher uses Claude Fable 5 AI to disprove long-standing Jacobian conjecture

Levent Alpöge, a mathematician employed by Anthropic, revealed that the company's Claude Fable 5 large language model produced a concise polynomial function that serves as a counterexample to the Jacobian conjecture in three dimensions and higher. The function maintains a constant Jacobian determinant of -2 yet maps multiple input points to a single output, demonstrating non-invertibility and thereby falsifying the conjecture beyond the planar case.

While the original two-dimensional version, first posed in one outlet century, stays open, Alpöge's finding adds to a growing list of AI-assisted mathematical results, including recent disproofs and proofs by other models. The simplicity of the counterexample allowed rapid verification by peers, highlighting AI's potential for navigating vast search spaces of mathematical objects. Details of the prompting strategy and the model's exact output have not yet been disclosed. The episode underscores ongoing debates about the role of artificial intelligence in advancing theoretical research.