AI claims counterexample to 85-year-old Jacobian Conjecture

Key Takeaways

• A mathematician claims Anthropic's Claude AI helped find a counterexample to the Jacobian Conjecture, open since 1939
• The specific polynomial map has Jacobian determinant -2 and allegedly sends three distinct points to the same output
• If verified, this would mark a historic moment for AI-assisted mathematics, but independent peer review is still needed

Mathematician Levent Alpoge posted on X that a counterexample to the Jacobian Conjecture, one of algebraic geometry's most stubborn open problems, has been found with the help of an AI. The assistant in question: Claude Fable, an Anthropic model. If the claim holds up to scrutiny, it would close a question mathematicians have chased since 1939.

Alpoge shared the specific polynomial map on July 20, 2026. He credited his "close friend Fable" for working on the problem during the World Cup final. The map takes three variables over the complex numbers and, according to Alpoge, has a constant Jacobian determinant of -2. He claims it sends three distinct input points to the same output, which would directly contradict the conjecture.

What does the Jacobian Conjecture actually claim?

Ott-Heinrich Keller proposed the conjecture in 1939. It states that any polynomial map from ℂⁿ to ℂⁿ with a non-zero constant Jacobian determinant must be bijective, meaning it has a polynomial inverse. In plain terms: if the "local stretching" of a polynomial transformation is uniform everywhere, the map should be reversible.

The conjecture sounds simple. It has resisted proof for 85 years. Mathematicians haven't even settled the case for two variables. Hundreds of papers have been published on partial results, reductions, and equivalent formulations. The problem sits at the intersection of algebraic geometry, commutative algebra, and complex analysis.

The alleged counterexample

Alpoge's map is explicit. He wrote out three polynomial expressions in variables x, y, and z, using powers up to degree three. The claimed Jacobian determinant is -2, a non-zero constant. According to his post, the points (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) all map to (-1/4, 0, 0). Three inputs, one output. If true, the map is not injective, which means the conjecture fails.

The calculation is verifiable. Anyone with access to Wolfram Alpha, Mathematica, or a symbolic algebra system can check whether the Jacobian determinant is indeed constant and whether the three points map to the same image. Alpoge's post included a Wolfram Alpha link, inviting independent verification.

Why this matters for AI-assisted research

The phrase "my other close friend fable" strongly implies Anthropic's Claude model. If an AI system contributed meaningfully to disproving a conjecture that has defeated human mathematicians for nearly a century, the implications are significant. This would not be the first time AI has assisted in mathematical discovery. DeepMind's AlphaProof and similar systems have found new proofs and optimized existing ones. But disproving a major open conjecture would be a different milestone.

The claim also raises questions about credit and verification. Who gets authorship when an AI proposes a counterexample? How do peer reviewers handle outputs from systems that cannot explain their reasoning in the way human mathematicians do? These are not hypothetical concerns anymore.

What happens next

A social media post is not a peer-reviewed paper. The mathematical community will treat this claim with appropriate skepticism until independent experts verify every step. Errors in symbolic computation are common. Subtle sign mistakes or domain assumptions can invalidate an entire argument.

That said, the claim is falsifiable in a way that many mathematical assertions are not. You can plug the polynomials into a computer algebra system and check. If the Jacobian determinant is not constant, or if the three points do not map to the same image, the claim collapses. If both hold, the burden shifts to finding an error in the conjecture's statement or the verification itself.

Historical context

The Jacobian Conjecture is not a Clay Millennium Prize problem, but it occupies a similar status in algebraic geometry. Fields Medalist Terence Tao has called it one of his favorite open problems. Multiple false proofs and false disproofs have been announced over the decades, only to be retracted after errors emerged. The community is conditioned to wait for rigorous verification.

If Alpoge's counterexample survives review, it will join a short list of famous conjectures disproved by explicit construction rather than abstract argument. And it will be the first such result where an AI system played a named role in the discovery.

Frequently Asked Questions

What is the Jacobian Conjecture?

The Jacobian Conjecture states that any polynomial map from ℂⁿ to ℂⁿ with a non-zero constant Jacobian determinant must be invertible. It was proposed in 1939 and remains unproven for even two variables.

Who is Levent Alpoge?

Levent Alpoge is a mathematician who posted the counterexample claim on X. He credited Claude Fable, an Anthropic AI model, and colleague Akhil for contributing to the discovery.

Has the counterexample been verified?

Not by independent peer review as of July 2026. The calculations are explicit and can be checked with symbolic algebra software, but the mathematical community has not yet confirmed or refuted the claim.

What happens if the conjecture is disproved?

Disproving the Jacobian Conjecture would close an 85-year-old open problem and require revising assumptions in algebraic geometry, commutative algebra, and related fields.

Is Claude Fable the same as Claude?

Fable appears to be a Claude model from Anthropic. The exact version and capabilities used in this work have not been publicly specified.

Source: Hacker News: Best