The Jacobian conjecture has sat unresolved since mathematician Ludwig Kraus first proposed it in 1884, later generalised by Ott-Heinrich Keller in 1939. It asks whether polynomial functions with a constant, non-zero Jacobian determinant are always reversible. For 87 years, nobody could prove it, and nobody could disprove it either. Anthropic's Claude Fable 5 just found a counterexample.
ScienceDaily reported that the AI model navigated an enormous search space of polynomial mappings to land on a specific formula that breaks the conjecture for three dimensions and higher, though the original two-dimensional version of the problem remains unsolved. What made the discovery notable was not just that it happened, but how simple the resulting counterexample turned out to be, short enough to fit into a social media post.
Who Confirmed It
Levent Alpöge, a mathematician at Anthropic, announced the discovery publicly, and Monash University senior lecturer Melissa Lee wrote up an analysis of its significance for The Conversation, later covered by CoinDesk in the context of its unexpected relevance to cryptography, given how some cryptographic assumptions lean on related mathematical structures.
A Prediction That Aged Strangely Well
A 2017 post on Math Stack Exchange, quoted in coverage of the discovery, speculated: "For all what we know, some smart undergraduate can simply write a formula that will be a counter-example." Nine years later, that is essentially what happened, just with an AI model doing the search instead of a person.
Why Mathematicians Are Paying Attention to This Specifically
Most AI contributions to mathematics so far have involved constructing long, complex proofs or verifying existing ones. This is different. It is a case of an AI model finding an unexpected object hiding in a search space too large for a human mathematician to comb through manually, then producing something so compact that any working mathematician could verify it by hand in minutes. That combination, machine scale search producing human scale simplicity, is what has people in the field paying closer attention than they might for a more conventional AI assisted proof.
How Fast It Was Checked
A separate technical writeup pinned the counterexample down to just 216 characters, a polynomial mapping with a constant Jacobian determinant of negative two that is nonetheless not globally invertible, short and clean enough that any mathematician can check it by hand in minutes, even though finding it in the first place required searching a space of polynomial mappings far too large for a person to work through manually.
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