No need to downvote, it was an honest question not an attack. Have you studied the literature and background mathematics of this area much?

Regime is a well established term in mathematics and many other fields, and one example of a "regime" (a domain under rules or constrains) is what you are likely familiar with as a political regime.

With respect to "punchline", I'm going to assume you didn't look at the video at the timestamp listed? Here it is https://youtu.be/1aXOXHA7Jcw?t=6105 All he is saying is that, after a few minutes long tangent talking about something the "punchline" is him circling back around to the point he was trying to make.

It isn't a literal haha punchline, it's not a mathematical term, the punchline comes at the end of a joke, a joke often takes you on a journey before circling back to some type of point. He used the word to mean that here too.

Timothy Nguyen, OP of this post and the host of the video, made a light hearted chapter title within a long video based on a term that Greg Yang used on his whiteboard.