OpenAI has announced three results produced by its internal models, in harmonic analysis, complexity theory, and quantum field theory. Each one is being handled differently, with outside researchers acting as shepherds to examine and explain the output. OpenAI says it is not seeking academic credit for the discoveries.

The math result is a proof of Sogge’s 1991 local smoothing conjecture in every dimension, which also settles the Kakeya conjecture and Stein’s restriction conjecture in all dimensions. The natural language writeup and a Lean formalization are on GitHub.

The computer science result disproves the Exponential Time Hypothesis, a conjecture about Boolean formula satisfiability that hundreds of papers have used to derive hardness results. A group led by Russell Impagliazzo, Ramamohan Paturi, and Ryan Williams will verify it. Impagliazzo and Paturi proposed the ETH in the first place.…