User blog comment:Bubby3/Matrix system analysis (new blog post)/@comment-30754445-20181119125930/@comment-32697988-20181119183659

"Then why are they given different names?"

It's related to the rule of naming versions of BMS. Versions without decimal points (BM1, BM2, BM3, BM4) are versions defined by Bashicu, and versions with decimal points (BM1.1, BM2.1, BM2.2, BM2.3, BM3.1, etc.) are versions not defined by Bashicu.

Koteitan is a Japanese googologist and he is the creator of BM2.1, BM2.2, and BM2.3. He was researching about different versions of BMS. On June 18th, he made a system that seems to work like BM2. However, he was not sure if they were completely the same so he said, "If this wasn't the same as BM2, I will call it BM2.3." (link to his tweet)

People began researching his system, and called it BM2.3 when mentioning it to distinguish, even though no one knew whether it is the same as BM2. Some people found matrices that expand differently in BM2 and BM2.3, but these matrices were nonstandard. However, on July 16th, koteitan himself found a standard matrix that expands differently in these versions: (0,0,0,0)(1,1,1,1)(2,2,1,1)(3,3,1,1)(4,2,0,0)(5,1,1,1)(6,2,1,1)(7,3,1,1). On the same day he also understood why it expands differently. On July 24th Alemagno made a blog post about this matrix. This matrix actually is a matrix that does not terminate in BM2 but no one knew that at the time.

On August 28th, Bubby3 made a [https://googology.wikia.com/wiki/User_blog:Bubby3/BM2_doesn%27t_terminate. blog post] about a matrix that does not terminate in BM2 and the proof of this. Soon Bashicu knew about it and decided to make a new official version. On September 1st, he published it in his blog post. On the same day koteitan started doing an analysis of the new code, and soon found that actually it was exactly the same as BM2.3 that he made. Note that he did not find it by letting the computer calculate thousands of expressions and compare them. He looked into the code and figured out what exactly each line is meant to do. As evidence, fish used the computer to calculate thousands of expressions in BM2.3 and BM4, and none of them expanded differently. The link to koteitan's analysis of the code is here. (It's in English)

Overall, the reason BM2.3 and BM4 are named differently is because at the time BM4 was published no one knew if it was the same as BM2.3, so people called it BM4 to distinguish them and to follow the naming system (Bashicu's versions does not have decimal points).

"What is needed is a detailed explanation for how that list was derived."

Can you show me an example of this by proving the strength of primitive sequence system is ε0? (you can do another notation if you prefer) I (and probably some other googologists) want to see how you want this analysis to be done.