AI Model Helps Disprove 87-Year-Old Jacobian Conjecture

The Jacobian conjecture, a problem that resisted mathematicians for 87 years, has been shown false thanks to a collaboration between mathematician Levent Alpöge and Anthropic's AI model Fable 5. The counterexample, a single polynomial map from three-dimensional space to itself, was posted on X and verified by hand Source: The Next WebSource: CoinDesk.

The Counterexample

The Jacobian conjecture, posed by Ott-Heinrich Keller in 1939, asserts that any polynomial map with a non-zero constant Jacobian determinant must have a polynomial inverse. Alpöge and Fable 5 found a specific map that meets the condition but lacks a polynomial inverse, disproving the conjecture Source: The Next Web. The result is a short list of polynomials that any mathematician can check Source: The Next Web.

Reactions

News of the disproof stirred the mathematical community. Fields medalist Timothy Gowers called it the first time an AI had solved a problem he had heard of outside his area Source: The Next Web. Daniel Litt posted that he could not stop laughing. However, some were unimpressed: Andrew Blumberg said the result did not change his expectations, as finding counterexamples is precisely what AI should be good at Source: The Next Web. The OpenAI researcher Aaron Lou claimed that an internal version of its Codex model found essentially the same counterexample independently Source: The Next Web.

Broader Impact

Beyond mathematics, the breakthrough has implications for Bitcoin and AI-linked markets. Bitcoin miners have pivoted to AI data centers, so demand for computing power ties cryptocurrency to AI developments Source: CoinDesk. The news added to the narrative that AI capabilities are leaping forward, affecting investor sentiment Source: CoinDesk.