Artificial intelligence Claude Fable 5 solves unsolved math problem for 87 years
Researchers have revealed that “Claude Fable 5” artificial intelligence has identified a crucial counterexample that puts an end to the Jacobian conjecture, a mathematical challenge that has intrigued minds for nearly a century. The discovery, reported by Levent Alpoge, a mathematician affiliated with Anthropic, demonstrates the growing potential of AIs in solving complex and long-standing problems in science.
How artificial intelligence solved an 87-year-old problem
Mathematician Levent Alpoge, from Anthropic, was responsible for reporting the discovery of the counterexample to Jacobi’s conjecture. The solution, which remained unanswered for 87 years, was found with the help of the “Claude Fable 5” AI model. Alpoge was encouraged to pursue this solution by mathematician Akil Matthew, and artificial intelligence worked on the problem during the FIFA World Cup final.
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What is the Jacobian conjecture and why is it important
The Jacobian conjecture, formulated in its general form in 1939 by the German mathematician Otto-Heinrich Keller, is a problem that questions a fundamental property of polynomial functions. It suggests that if the Jacobian determinant of a polynomial map that maps n complex variables into the same n variables is a nonzero constant, then there must exist an inverse map that can also be expressed as a polynomial. Essentially, the question is whether a property of local “reversibility” at all points implies a global polynomial reversibility for the entire function.
Understanding the Role of the Jacobian Determinant in Mathematics
In mathematics, when a function returns the same output for different inputs, the original input cannot be unique. The process of returning from the output to the input is called an inverse function. However, a function with multiple inputs to the same output cannot have a unique inverse function throughout its domain. For multivariable functions, the ability to restore the function in a narrow range is examined by the Jacobian determinant. This value is calculated at each entry point and represents the change in local area or volume, using a linear approximation. A zero Jacobian determinant indicates that information has been lost in some direction, making a one-time restoration of the input impossible. If it is non-zero, the information is preserved, ensuring traceability of input and output locally.
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The Crucial Discovery of the Counterexample by Claude Fable 5
Claude Fable 5 artificial intelligence discovered a polynomial mapping where the Jacobian determinant always takes on a constant value of -2. However, three distinct inputs were found that result in the same output, which contradicts the Jacobian conjecture.
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- Different inputs:
- * (0, 0, -1/4)
* (1, -3/2, 13/2)
* (-1, 3/2, 13/2) - Identical output for all inputs:(-1/4, 0, 0)
This demonstration indicates that the Jacobian conjecture is false for three or more variables, proving that the existence of an inverse map in the entire space is not guaranteed only by the property of the constant Jacobian determinant.
The importance of verification and post-discovery human understanding
Despite the impressive discovery of AI, the simple generation of a formula by an artificial system does not automatically ensure its validity as a counterexample. Small calculation or substitution errors could invalidate the entire conclusion. For this reason, independent validation by human experts or verification software becomes essential.
After the revelation, there was an effort to formalize the proof using the “Lean” theorem proving support system. Paul Rouzeau, a PhD candidate in formalized mathematics at Imperial College London, developed a Lean counterexample and uploaded it to “Formal Conjectures”, a repository of mathematical conjectures maintained by Google DeepMind. Currently, the pull request is in the review process, but it has already been possible to verify the consistency of the Jacobian determinant and the non-injectivity of the mapping, confirming the absence of inverse mappings throughout the space.
The next steps in collaboration between humans and artificial intelligence
The Xena Project, focused on formalized mathematics, points out that the next big challenge is not only confirming the correctness of the counterexample, but also enabling humans to deeply understand the reasons why it is valid. This episode underscores the growing interplay between the computational power of artificial intelligence and human analytical acumen, opening new frontiers in mathematical research and solving problems that have long defied understanding. The collaboration promises to accelerate progress in complex fields by combining the speed of AI with the depth of human reasoning.
