Cairo kept reading and thinking. Eventually, she found a way to construct a strange, complicated function out of waves whose frequencies all lay on a curved surface — the type of surface the conjecture required. Usually, when you add these kinds of waves together, they interfere, canceling each other out in some places and reinforcing each other elsewhere.

But Cairo showed that in her function, they didn’t cancel out as expected. Instead, their interference created uneven patterns, causing the function’s energy to spread out over some areas and concentrate in others in a fractal-like way that the Mizohata-Takeuchi conjecture prohibited. She found herself staring at a mathematical construction that by many accounts shouldn’t exist.

At first it made her wary. “This is something that happens to me often,” she said. “I come to something that looks like a proof, and I think I have a proof, but [then] I’ve actually been wrong.”

Then two things happened. The first was that she realized she could replace her complicated construction with a much simpler one and achieve the same result.

The other was that she convinced herself, and Zhang, that the result was right.

“Cairo’s paper is a great example of how natural and elegant conjectures can fail in ways we didn’t think of,” Oliveira said. “But to see that, we need to look through the right lenses.”