User blog comment:Plain'N'Simple/Letter Notation Up to Z: Outline and mnemonics/@comment-36984051-20191106024302/@comment-35470197-20191106113323

> (0,0,0)(1,1,1)(2,2,2)

It is an important point for me, because it seems the first time when UNOCF users accepted an upperbound. The awful aspect of UNOCF is that UNOCF users regarded any other actual OCFs as weaker functions than UNOCF without any reasons, and tended to say "if there were an OCF known to be stronger than UNOCF, then we would use it". Since there is no actual definition of UNOCF, OCF users were unable to analyse UNOCF, and hence could not persuade UNOCF users.

On the other hand, since many UNOCF users somewhy respect BMS, they accepted that UNOCF is weaker than (0,0,0)(1,1,1)(2,2,2). So if we create an actual notation beyond (0,0,0)(1,1,1)(2,2,2) under the assumptions of their terminations, which UNOCF users rarely care, then we are free from the awful estimation like "it is weaker than UNOCF".

In my opinion, this is the main reason why googologists love to extend or simplify BMS. We do not have to care about any disappointing issue related to UNOCF.

> More precisely: (0,0,0)(1,1,1)(2,1,1)...(n,1,1) is supposed to roughly correspond to Pi-(n-2) reflections.

It is quite interesting. As Duchhardt and Stegert discovered, Rathjen's indecation that a simple iteration arguments for Π_n-Ref was valid only for n≦3. In other words, they actually needed a new method for n≧4. On the other hand, the "guess" suggests that Π_n-Ref can be "linearly" described through BMS. What a miracle!