User blog comment:P進大好きbot/Please Help me on study of Pair Sequence System (2-rowed Bashicu Matrix System)/@comment-30754445-20180813155654

You seem to be operating under the assumptions that:

(a) There's only 1 version of "pair-sequence system".

(b) This system is well-defined.

(c) This system has been proven to terminate.

(d) The relations between this notation and ordinals that the people here are claiming to be true, have been proven (or at least - justified by something more than wishful thinking)

Unfortunately, all four assumptions are false.

There's sometjhing like 5 or 6 versions of BMS right now. For most of them, we can only guess the rules (BM1 being the sole exception). The well-foundness of none of them have been proven, even if we limit ourselves only to pair-sequences (and all these versions, except one, are known to enter infinite loops later on).

And I haven't seen any reason to believe the "analyses" of BMS that people throw here on a regular basis. None of them have proven their assertions or even justified them in any way. Some of them make claims that are so rediculous (like giving a BMS expression that is "conjectured" to reach PTO(Z2)), that you can tell right there that they are  spouting nonsense).

In fact, it looks like all these "analyses" are based on really superficial similarities between the way a BMS expression looks and the OCF expression. Things like replacing (x,0) with (x,1) whenever they want to replace a small-omega with a Big-Omega. Why on earth would someone do that? Because it worked a couple of times, so they just assume it is going to work a third time.

And of that's not enough, there's the issue with BM1 (the very first version of the notation) which was believed to work properly for years, was recently discovered to enter an infinite loop as early as (0,0)(1,1)(2,1)(3,1)(2,1).

I'm really sorry, but there's no nice way to sum up the situation. The whole thing is farce. And while I've accepted that most people here just don't care whether they make sense or not, I assume that you are different in that respect. Am I right?