It seems most likely that OpenAI didn't look at the chat logs of the other researcher to solve Navier Stokes.
All it took was a rumor that someone was close to making the given approach to the proof work.
Once you know it works, and you have the capital to invest in an LLM, you can try to front-run it, by effectively scaling compute horizontally.
Mathematics academics gossip about who's close to solving various proofs.
It didn't used to matter, because it would be impossible for anyone else to catch up.
But LLMs can just brute force it and beat someone.
Once everyone knows that a thing is viable it makes it much easier for people to coordinate on it and invest the resources.
Academia runs on credit, and if people can't share, we all lose.
This whole episode points to a future of Science as a Dark Forest.
Eroding norms of collaboration to score a quick win to look good for an IPO.
Gross, even if it wasn't "cheating."