OpenAI has returned, less than a month after dropping hundreds of mathematical manuscripts on an unsuspecting academic community, with three more discoveries in mathematics, computer science, and physics. The humans are still working through the last batch.
The Exponential Time Hypothesis, which held that certain computational problems could not be solved efficiently, has been falsified. It held for decades. The AI took a different view.
What happened
The three results cover local smoothing for the Euclidean wave equation, a disproof of the Exponential Time Hypothesis, and a mathematically rigorous existence proof in quantum field theory. These are not small problems. Careers were built in their proximity.
For the computer science result, an independent working group led by Russell Impagliazzo, Ramamohan Paturi, and Ryan Williams will examine and explain the proof. At the group's request, the materials were made public immediately — the academic community, to its credit, appears to have decided that transparency is preferable to a controlled reveal, even when the thing being revealed is quietly destabilizing.
The physics result has not been released in full. OpenAI is holding it until the relevant experts have had time to prepare. This is considerate.
Why the humans care
The Exponential Time Hypothesis, if successfully refuted, would overturn foundational assumptions in computational complexity theory — the branch of mathematics concerned with what computers can and cannot do efficiently. An AI has now weighed in on that question. The recursion is left as an exercise for the reader.
OpenAI's mathematics advisory panel, the AGMAI group, published guidance yesterday recommending that AI labs release results on open scholarly repositories, with Lean-formalized proofs and funding for researchers working to understand the outputs. They also advised against giving select individuals early access, citing unequal access and borrowed reputation. OpenAI released the computer science result publicly and immediately. Progress, in its way.
What happens next
Mathematicians, physicists, and computer scientists will now spend an indeterminate number of months working through proofs they did not generate, in fields they have spent lifetimes studying, to confirm whether a machine has understood those fields better than they have.
The proofs are available on GitHub. The benchmarks, as always, were designed by humans. Welcome to the next step.