Claude's Fable 5 Just Solved an 87-Year-Old Math Problem, and It Matters for Bitcoin
Levent Alpége and Anthropic’s Fable 5 found a three-dimensional counterexample, and researchers said AI can search huge polynomial spaces humans cannot.
- On Sunday, July 19, 2026, Harvard fellow Levent Alp announced he used Anthropic's Fable 5 model to disprove the Jacobian conjecture, a problem that defied mathematicians for 87 years.
- First proposed by Ott-Heinrich Keller in 1939, the conjecture asks whether polynomial maps with non-zero Jacobian determinants are reversible; Stephen Smale included it on his influential 1998 list of 21st-century challenges.
- Fable 5 generated a map from three-dimensional space to itself that passes the reversibility check but cannot be reversed because three different inputs yield the same output, disproving the Jacobian rule with one counterexample.
- Columbia University mathematician Andrew Blumberg noted the achievement is significant but the counterexample offers limited insight compared to a full disproof, while Fields medalist Timothy Gowers acknowledged the milestone yet cautioned it lacks theoretical depth.
- This capability leap strengthens the case for pouring capital into AI development, posing a conundrum for investors as AI's steep capability curve draws market risk appetite away from speculative assets.
16 Articles
16 Articles
AI just disproved a major math problem. A mathematician explains.
The Jacobian conjecture has bedeviled math experts for nearly 90 years. While the rest of the world was watching the World Cup on Sunday evening, mathematician Levent Alpöge casually announced on X that he had used Anthropic's Fable 5 to disprove the Jacobian conjecture. The Jacobian conjecture is a long-standing open problem in algebraic geometry …
The mathematician Levent Alpoge announced on 20 July 2026, in a simple message published on social network X, that he was holding proof that one of the great open problems of algebraic geometry was false. This result, obtained in one evening with the help of a model of artificial intelligence, took short a community that had been struggling on the issue since 1939. The Jacobian conjecture claimed that a polynomial function respecting a specific …
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