I swear this isn’t turning into an AI blog. But did you see the one about the Riemann hypothesis?

Someone at Anthropic did something I’m sure they’re all tempted to do, and tried to use an internal version of their Claude AI system to prove the most famous open conjecture in mathematics. It didn’t work, to be clear, and I get the impression they didn’t expect it to. But out of six hundred or so fruitless tries, one attempt did prove a new bound. Previously, mathematicians had been able to prove that at least 41.6% of the zeroes of the Riemann zeta function satisfied the Riemann hypothesis. Now, the new proof shows that at least 67.2% satisfy it.

Anthropic’s press release is impressively careful. As someone who’s had to think about how to write content that both excites the public and doesn’t piss off experts too much, they do an admirable job walking that line. They even say, straight-out, “We don’t expect that the techniques Claude used will lead to proving the Riemann hypothesis.”

Bounds are like that, sometimes.

I should know. Physicists also find bounds.

Physics has its own conjectures with the fame of the Riemann hypothesis. Dark matter might be made of detectable particles. Protons could decay. There might be extra dimensions, or magnetic monopoles, or cosmic strings. General relativity might be subtly wrong.

It would be an amazing achievement to demonstrate any of these things. But most physicists won’t manage that. Instead, they bound them.

Physicists compete to get better bounds, excluding unusual possibilities with greater and greater care. They find evidence that dark matter can’t be of a specific mass with a specific charge, so the next experiment has to look somewhere else, or find evidence that general relativity holds to even greater precision, so any deviation must be even smaller. Some work to improve experiments with better and better bounds. Others analyze data from older experiments, or find under-appreciated consequences of known facts, and can get even better bounds.

Bounds aren’t typically newsworthy (though occasionally they make it through), so most people don’t hear about them. If you read the news, you hear about positive claims much more often than negative ones: evidence for something new, not evidence that our current knowledge holds. But the nature of physics is that most work supports the status quo. Most work improves bounds.

Do bounds lead, with time, to the positive claims? Sometimes, but not always. Often, bounds are just bounds. They’re attempts to use the methods physicists have to learn something new about the world. Even if the new fact is just “don’t look here”.