Analysis
An Anthropic mathematician, Levent Alpöge, used the company's Claude Fable 5 model to discover a counterexample disproving the Jacobian conjecture, an 87-year-old open problem in mathematics that had stumped experts since German mathematician Ott-Heinrich Keller first proposed it in 1939. The conjecture asked a deceptively simple question: is a locally invertible polynomial map necessarily also globally invertible? Alpöge's AI-assisted counterexample shows the answer is no, at least in three-dimensional space -- the map found passes the local-invertibility check but still can't be reversed globally, because three distinct inputs all map to the exact same output.
The reception from the pure mathematics community has been notably enthusiastic rather than skeptical. Kevin Buzzard, a pure mathematician at Imperial College London, told Fortune, "This is a big day. I personally feel it's a wonderful time to be alive" -- a strikingly positive reaction from a field that has often been cautious about AI's role in genuine mathematical discovery versus computational assistance.
“A verified counterexample to an 87-year-old conjecture is a much harder thing to fake or overstate than a benchmark score.”
The result matters for AI credibility in domains well outside its usual commercial applications: pure mathematics is a field where correctness is rigorously verifiable and where an AI model can't bluff its way past expert scrutiny the way it might in a more subjective evaluation. A verified counterexample to an 87-year-old conjecture is a much harder thing to fake or overstate than a benchmark score.
That said, researchers are careful to flag the current limitation: the counterexample Alpöge found exists in three-dimensional space, meaning the original two-variable version of the Jacobian conjecture -- widely considered the harder and more central case -- still awaits its own resolution. And more broadly, mathematicians note that current AI models tend to deliver the "how" of a solution -- a valid, checkable answer -- without necessarily producing the "why," the deeper conceptual understanding that mathematicians actually value and that tends to generalize to solving related problems.
What to watch: whether the two-variable Jacobian conjecture falls next, and whether this becomes a template for AI-assisted resolution of other long-standing open problems in pure mathematics, an area that's been comparatively insulated from AI's more commercially visible applications so far.
