kinda crazy insight from the X algo: if you paste a reply into the text box, it will rank lower than if you type it out this kinda sucks because i often draft posts somewhere else, then paste them into X afterwards

The complexity theory result is actually huge. P problems can be solved in polynomial time. It was widely believed that NP-complete problems require exponential time. P = NP states that NP-complete problems can be solved in polynomial time. OpenAI didn't prove that. But, they proved that (some) NP-complete problems can be solved faster than exponential time, meaning: - P = NP, or - the complexity of NP problems is a "middle thing" between polynomial and exponential, eg. O(n^log n) If it survives peer reviews, this could be one of the biggest CS breakthroughs in decades.

These 0.99999984 solutions are not as "ugly" as it may seem. The truth is, those are not the real runtimes of the algorithms published. To make the proof simpler, they use absurd safety bounds like 2^50 that are entirely arbitrary. For example, imagine proving that 1/x^2 > 0. This proof is much harder if you include the area around x=0, as it blows up to infinity there. Instead, if you only prove it for x>=99999, it avoids this blowup and simplifies the proof, but still gets the point across. The "real" bounds of those solutions are probably much tighter, maybe even something "beautiful" like O(n*lg lg n). But the goal of the AI papers was just to show that improvement is possible, not that the bounds are perfect (which is usually much harder to prove). Over the coming weeks and months, we'll likely see a lot of tightening.

drama summary for those confused: - Aug 15th: Tristan Buckmaster & Levent Alpöge make progress on a few important math problems - they do NOT have a proof for the $1,000,000 Millenium Prize problem. BUT, they do claim to have a proof for a similar (non-Millenium) Navier Stokes problem that could help lead the way there - Levent works at Anthropic, but this research was independent of his work there, with a mix of GPT and Claude models. Tristan is not related to Anthropic. - Early Sep: Rumor spreads to OpenAI that Anthropic has solved a major problem. Tristan emails OpenAI to clarify. without revealing the problem they solved or how they did it. - After hearing of the rumor, OpenAI started researching Navier Stokes with a new internal model. - Sep 6th: OpenAI's Sebastien Bubeck tells Tristan that they solved the $1,000,000 Millenium Prize Navier Stokes problem. The approach is very similar to Tristan & Levent's approach to the non-Millenium problem. - Tristan is suspicious of the timing, as only few others were trying this approach. OpenAI says the model didn't access his user data directly, but leaves unanswered whether Tristan's chat conversations were part of the training. - OpenAI says they would partially credit Tristan for the $1,000,000 discovery (even though Tristan did not solve the $1,000,000 problem) — but only if they remove Levent as an author, as he works for Anthropic. - Sep 8th: Tristan refuses to remove Levent, and rushes to publish their results independently. Currently unclear is whether Anthropic had a separate solution for the $1,000,000 problem, or whether the rumor was about Tristan & Levent's independent research.