When is it plausible that a result is optimal?

(Lance and Bill did a vidcast about the OpenAI Document Drop. See here. My spellcheck thinks that vidcast and OpenAI are not words.) The Metric TSP problem is TSP where, for all vertices \(x,y,z\), \(w(x,z) \le w(x,y) + w(y,z) \).In 1976 Christofides and Serdyukov (independently of each other) showed that there is a poly time algorithm for a 1.5-approx to Metric TSP.Some people thought the 1.5 could be improved.Some people thought that 1.5 is optimal. 1.5 is a natural number. Hmm, that's not quite right. 1.5 is a pretty number. It would not be a surprise if someone had proven If there is a poly time (\(1.5-\epsilon\)) approx then P=NP.Indeed, people tried to prove that. In 2020 Karlin, Klein, Gharan, and Oveis proved there is a poly time (\(1.5-\frac{1}{10^{36}}\)) approximation algorithm. Darling thinks this result is hilarious. I tried to tell her that the result is important since a) We now know that 1.5 is NOT optimal (assuming \(P\ne NP\)).b) One of the authors, Nathan Klein, had…