Fable 5 Disproves Jacobian Conjecture, Researcher Says

Anthropic researcher Levent Alpöge announced that Claude Fable 5 has disproved the Jacobian Conjecture, an 87-year-old problem in algebraic geometry that has resisted solution since Ott-Heinrich Keller posed it in 1939. The result — verified in the Lean proof assistant within hours, Imperial College mathematician Kevin Buzzard told Fortune — marks the second time in two weeks that frontier AI has produced results on problems that defined open questions for generations, Fortune reports.

What happened

On July 20, Anthropic researcher Levent Alpöge announced on X that he had used Claude Fable 5 to find a counterexample to the Jacobian Conjecture (Alpöge, 2026). The post drew over 30 million views. By the next morning, the proof had been verified in Lean.

The conjecture concerns what mathematicians call polynomial maps — specifically, whether a map whose Jacobian determinant is a nonzero constant must be uniquely invertible. Alpöge's counterexample produced three different inputs mapping to the same output while maintaining a constant Jacobian determinant of −2, disproving the conjecture's central claim.

"It is a big day," Imperial College mathematician Kevin Buzzard told Fortune.

Two weeks earlier, OpenAI attributed a proof of the Cycle Double Cover Conjecture — another long-open problem, this one in graph theory — to GPT-5.6 Sol (AI Weekly, 2026), though the claim has not been independently peer-reviewed. The velocity is the story: frontier AI is now producing results on problems that resisted solution for decades.

Why it matters

This is not brute force. OpenAI reportedly used mass parallelism — 64 subagents. Anthropic used a single researcher in a focused session with Fable 5. The diversity of approaches suggests frontier AI is broadly capable across paradigms.

The gap between verification and understanding is real. University of Chicago mathematician Akhil Mathew captured the tension: "One can check that it's correct, but it would be nice to be able to tell a story." The AI produces the proof but cannot explain how it arrived there.

Neither OpenAI nor Anthropic built these models for math. Fable 5 and GPT-5.6 Sol were trained as general-purpose frontier models. Cracking open conjectures as a side effect is the real signal.

What changes for you

This is a capability signal, not a product change. Fable 5's directory verdict stays Conditional.

If you work in research mathematics:

• Explore frontier models for proof discovery. Alpöge's result demonstrates a single researcher can tackle field-defining problems.
• Lean verification changes the trust model. AI-generated proofs checked by machine eliminate months-long peer review bottlenecks.
• Don't expect AI to replace mathematical intuition. Knowing what to ask remains a human skill.

FAQ

What is the Jacobian Conjecture? A problem in algebraic geometry posed by Ott-Heinrich Keller in 1939. Alpöge found a counterexample disproving the conjecture.

Was this solved entirely by AI? No. Alpöge guided Fable 5 through the problem. The AI was a tool, not an autonomous mathematician.

Is this the first time AI solved a major math problem? No. Frontier AI has been producing publishable mathematics for months. The Jacobian Conjecture is the most prominent result yet.