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/@comment-35870936-20180813221855

You seem to be operating under the assumptions that:

(a) There's only 1 version of "pair-sequence system". '''But there IS only one version. Out of all the versions of BMS like BM2, BM2.3, BM3, BM3.1, and BM3.2, they all act the exact same with pair sequence. They only start to act differently in trio sequence and beyond. The only BMS versions we have that have different pair sequences are BM1 and BM2.2. Nobody cares about BM1 since it's broken, and nobody really cares about BM2.2 either, since it's behaviour is so weird.'''

(b) This system is well-defined. '''But it IS well-defined. Why exactly is that not a valid assumption? We literally have a calculator that calculates BMS expressions (http://gyafun.jp/ln/basmat.cgi), therefore it MUST be well-defined.'''

(c) This system has been proven to terminate. '''Yes, pair sequence hasn't been proven to terminate. We've been through this before, many times. You don't have to prove every single claim you make. SAN hasn't been proven to terminate, but people still accept SAN as being a crazy powerful notation.'''

(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) '''Yes, they actually are justified by something more than wishful thinking. They have been justified by detailed analyses that people have done. People have analysed pair sequence up to its limit, and all of them look pretty much the exact same.'''

Unfortunately, all four assumptions are false. '''Nope. (a), (b) and (d) are very much true. (c) is false, I'll give you that.'''

There's sometjhing like 5 or 6 versions of BMS right now. '''Very true. '''For most of them, we can only guess the rules (BM1 being the sole exception). '''BM2.3, BM3.2 and (I think) BM2.2, are actually defined with a ruleset, instead of a program. However, you are right about BM2, BM3, and BM3.1. '''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). '''Actually, we still have not found a loop in BM2, BM2.3, BM3.1, and BM3.2. However, BM2 and BM3.1 are believed to have a loop somewhere, but BM2.3 an BM3.2 are likely to terminate.'''

And I haven't seen any reason to believe the "analyses" of BMS that people throw here on a regular basis. '''Why not? See my other comment detailing a connection between pair sequence BMS and UNOCF. '''None of them have proven their assertions or even justified them in any way. '''Ditto my previous message. 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). When people make "claims" like that, what they are actually doing is speculating.  Nobody randomly makes a claim that notation XYZ reaches PTO(Z2). There's absolutely nothing wrong with speculating. In fact, Nish once literally called me the "speculations guy" in discord xD.'''

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. '''It hasn't just worked a couple of times. It's worked EVERY SINGLE TIME we have analyzed it.'''

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). '''Who even cares about BM1 anymore lol. It's ruleset is broken, and BM2 fixed it.'''

I'm really sorry, but there's no nice way to sum up the situation. The whole thing is farce. Seriously? 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? '''Idk if you are right about that, ask Pi Bot himself. I think he's somewhere in between your side and my side in this debate.'''

'''Why do you post comments like this on literally every BMS blog post? All your comments have the same general idea around them. This is the bottom line: We get that you don't like our BMS analyses because they aren't formal enough. But that's not a reason to just write it off as invalid, just because it hasn't been proven.'''