One Tweet, One Counterexample: Did Claude Fable 5 Disprove the 85‑Year‑Old Jacobian Conjecture?

On a morning tweet, Anthropic mathematician Levent Alpoge announced that the Jacobian conjecture is false, crediting a friend Akhil and the Anthropic model Claude Fable 5. The tweet included a short polynomial mapping from \(\mathbb{C}^3\) to \(\mathbb{C}^3\) whose Jacobian determinant is the constant \(-2\), yet it maps the three distinct points \((0,0,-\tfrac14)\), \((1,-\tfrac32,\tfrac{13}{2})\) and \((-1,\tfrac32,\tfrac{13}{2})\) to the same point \((-\tfrac14,0,0)\).

This single collision violates global invertibility, directly contradicting the Jacobian conjecture, which asks whether a polynomial map with everywhere non‑zero constant Jacobian determinant must be globally invertible with a polynomial inverse. The counterexample is elementary enough that anyone can substitute the numbers and verify the failure.

The mathematical community reacted swiftly. Stanford number theorist Jared Duker Lichtman retweeted and dissected the example, noting that the three inputs collapsing to one output proves non‑invertibility. He listed the contributors as “Alpoge, Mathew and Claude Fable 5,” where Mathew refers to algebraic geometer Akhil Mathew of the University of Chicago.

Discussion on Hacker News and X highlighted a mix of awe and absurdity, with comments ranging from jokes about the brevity of the “paper” to speculation that the model’s success stems from having “read” many failed proofs. Some argued that the lack of prior discovery is not due to impossibility but to a historical lack of focused search, as mathematicians traditionally prioritize high‑value theorems over low‑degree counterexamples.

The article also revisits the Jacobian conjecture’s history: posed in 1939 by Ott‑Heinrich Keller, listed as problem 16 in Smale’s 21st‑century problems, and notorious for numerous flawed proofs. Its persistence for 85 years is attributed to the community’s cautious stance—any new proof is initially presumed false.

Beyond the mathematical breakthrough, the episode raises broader questions about AI’s role in research. Some suggest that Claude Fable 5 succeeded because it internalized patterns from past failed attempts; others contend that finding a counterexample is fundamentally different from proving a theorem. The debate touches on whether AI is merely retrieving existing knowledge or genuinely creating new insights.

The narrative also references Zhang Yitang, whose early work on a special case of the Jacobian conjecture collapsed his Ph.D. thesis, leading to a difficult career path before his later breakthrough on bounded prime gaps. The recent disproof echoes his early struggles, illustrating how a conjecture that once hindered his career is now resolved with AI assistance.

Importantly, the result has not yet undergone formal peer review, though the Wikipedia entry on the Jacobian conjecture has been updated to acknowledge the counterexample. Verification is straightforward: the polynomial can be evaluated with any computer algebra system (e.g., Wolfram Alpha) to confirm the Jacobian determinant and the point collisions.

Open questions remain, such as whether a modified version of the conjecture that excludes degenerate behavior at infinity might still hold. The community continues to explore these possibilities while reflecting on the broader implication that advanced language models are beginning to tackle genuine open problems rather than only benchmark tasks.