User blog comment:Bubby3/Matrix system analysis (new blog post)/@comment-30754445-20181119125930/@comment-24725252-20181119151540

For the answer to your first question, BM2.3 and BM4 alawys evaulate the same, so it doesn't matter which one I'm using, and it says that here. The blog post I linked to also provides the expansion rules for them.

For your fourth question, it has to do with the process of "upgrading" the level of (n,x,0) terms with (n,x,1), such as in (0,0,0)(1,1,1)(2,1,0)(1,1,1) having level psi((W_w)^2) instead  psi((W_w)*W+W_w), because in (0,0,0)(1,1,1)(2,1,0)(1,1,1), the 1 in the (2,1,0) increases when copying the bad part, which means it has a higher level. However, in (0,0,0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,1,0)(2,2,1), the 1 in the (3,1,0) doesn't increase when copying the bad part, which makes there not be an infinite loop, and it makes  (0,0,0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,1,0)(2,2,1) have level psi((W_w)*W+W_w), and (0,0,0)(1,1,1)(2,1,0)(1,1,1) has level psi((W_w)^2). This pattern continues with all expressions ending in an (n,1,0) term, replacing the W in the ordinal with a W_w. Similar things happen with (1,1,1)(2,1,1), (1,1,1)(2,1,1)(2,1,1), etc.

You can fill in the gaps with inuition if the analysis is detailed enough and provides all possible scenerios for what could happen.

Here's a question for you. What part of the analysis don't you understand? I am going to fill in the gaps and make it more detailed or explain my results so you can understand it.