Claude Fable 5 AI Finds a Tiny Formula That Topples an 87-Year-Old Math Conjecture

As millions of people were coming down from the excitement of the FIFA World Cup Final recently, a different kind of excitement was building within the mathematical community. Levent Alpöge, a mathematician working at the artificial intelligence (AI) company Anthropic, made a casual announcement on X that he had found a counterexample to the Jacobian conjecture, a very old and well-known problem in a field of mathematics called algebraic geometry. He had done this using Anthropic’s large language model Claude Fable 5, released to the general public only a few weeks ago. This is just the latest of many striking mathematical breakthroughs made by mathematicians working with large language models. But this one feels a little different to those that have come before. ## What is the Jacobian conjecture? First, what is a conjecture? It's an idea that some mathematicians believe is true but nobody has been able to prove or disprove. Now to the Jacobian conjecture. It's fairly abstract but not too difficult to describe. The conjecture involves functions, which are like little machines which you put one or more numbers into and out pop other numbers according to some rule or equation. In this case, the functions use what are called polynomials. Specifically, it's about situations where the numbers represent points in a space, like coordinates on a map. So we can imagine when the function takes a point in space and moves it to another point, kind of like a transformation. The Jacobian conjecture deals with a special property of these transformations: if the transformation is “locally invertible” (meaning its derivative, or Jacobian, is never zero), then it should be globally invertible (meaning you can reverse the transformation for all points in space). The conjecture, first proposed in 1939 by German mathematician Otto Jacobi, has resisted all attempts at proof for over 87 years. It's considered one of the most important open problems in algebraic geometry, a branch of mathematics that studies geometric properties of solutions to polynomial equations. ## The AI-assisted discovery Alpöge, who has a background in number theory and algebraic geometry, used Claude Fable 5 to explore variations of the conjecture. The AI model, trained on a vast corpus of mathematical literature, helped him identify a surprisingly simple counterexample. In three dimensions, the counterexample is a set of polynomial equations that satisfy the local invertibility condition but fail to be globally invertible. The key insight was a small, unexpected formula that had eluded human mathematicians for decades. “The formula was so simple it could fit in a social media post,” Alpöge said in an interview. “But it completely topples the conjecture in higher dimensions.” This discovery highlights a growing trend in mathematics: AI models are not just assisting with complex proofs but also helping to find elusive patterns and mathematical objects. The approach combines the intuition of human mathematicians with the computational power and pattern recognition of large language models. ## The significance of the result The counterexample shows that the Jacobian conjecture is false in three dimensions and above. However, the original two-dimensional version remains unsolved. This is not uncommon in mathematics, where problems often behave differently in lower dimensions. The result has already generated significant discussion in the mathematical community. Some researchers have expressed cautious optimism, noting that the counterexample will need to be verified by independent experts. Others have pointed out that the discovery demonstrates the potential of AI in mathematical research, particularly in generating new ideas and connections. ## Broader implications for AI in mathematics This is not the first time AI has been used to make mathematical breakthroughs. In recent years, researchers have used machine learning to discover new algorithms, prove existing theorems, and even find new links between seemingly unrelated areas of mathematics. However, the Jacobian conjecture counterexample is notable because it was discovered not by a specialized AI system, but by a general-purpose large language model that was not specifically designed for mathematics. This suggests that such models can serve as powerful tools for creative problem-solving in fields beyond language processing. For mathematicians, the implications are profound. AI models can rapidly search through large spaces of possibilities, identify patterns, and suggest conjectures or counterexamples that might take human researchers much longer to find. This could accelerate the pace of mathematical discovery in the coming years. ## What's next? The mathematical community is eagerly awaiting verification of Alpöge's result. If confirmed, it will be a landmark achievement in both mathematics and AI research. In the meantime, the discovery serves as a reminder of the power of collaboration between human intelligence and artificial intelligence. As AI models continue to improve, we can expect more breakthroughs like this one, pushing the boundaries of what we thought was possible. The Jacobian conjecture in two dimensions remains open, and it's possible that AI will play a role in solving that version as well. For now, the mathematical community is buzzing with excitement about what other hidden gems might be waiting to be uncovered with the help of AI.

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