As they related in a preprint on January 3, 2026, AlphaEvolve had found that the Bruhat intervals in these particular permutation groups had a surprisingly special structure. When the researchers studied the intervals, they found that they formed higher-dimensional cubes called hypercubes. “If you look at what AlphaEvolve was thinking, I was super surprised,” Libedinsky said. “If it was a human, it would be an extremely creative human.”
AlphaEvolve had answered a question they didn’t know they had. “We didn’t ask AlphaEvolve to find big hypercubes,” Ellenberg said. “We asked it to find something else, and we thought about it and realized it was a gigantic hypercube which we had not anticipated was there.”
As Williamson put it, “It’s a structure that’s been sitting there for 50 years in front of our nose. We just hadn’t noticed it.”
Older machine learning methods had previously enabled such serendipitous mathematical discoveries, too — uncovering patterns no one had thought to look for. But in the past, Williamson said, it was a “real engineering effort. … You need to know how to code, spend a lot of time looking at details of neural network training. It was basically extremely difficult for a mathematician with no significant machine learning background to do this.”