User blog comment:Googleaarex/Bashicu Matrix System Analysis (Part 1)/@comment-11227630-20170725122119

What do you mean "SAN level 1{1{1,,2}2,,1{1,,1,,2}2}2"?

In SAN, separators have "levels", but the levels are not well-founded. e.g. lv(`) > lv({1`2}) > lv({1{1`2}2}) > lv({1{1{1`2}2}2}) > ... but the growth rates of s(n,n`2), s(n,n{1`2}2), s(n,n{1{1`2}2}2), s(n,n{1{1{1`2}2}2}2), etc. are the same.

If your "SAN level A" means the growth rate of s(n,n, A), then "SAN level 1{1{1,,2}2,,1{1,,1,,2}2}2" means the growth rate of s(n,n,1{1{1,,2}2,,1{1,,1,,2}2}2). However, it's not the limit of s(n,n,1{1,,1{1,,1,,2}2}2), s(n,n,1{1{1{1,,1{1,,1,,2}2}2}2,,1{1,,1,,2}2}2), s(n,n,1{1{1{1{1,,1{1,,1,,2}2}2}2{1,,1{1,,1,,2}2}2}2,,1{1,,1,,2}2}2), etc.

To solve s(n,n,1{1{1,,2}2,,1{1,,1,,2}2}2), when you meet {1,,2} in the process, search out for a separator with lower level than {1,,2}, but there is not such separator, so add a pair of {1 ____ 2} at the base layer, then it becomes s(n,n,1{1{1{1,,2}2,,1{1,,1,,2}2}2}2). Next, it'll reduce to s(n,n,1{1{1 {1{1 ... 2,,1{1,,1,,2}2}2} 2,,1{1,,1,,2}2}2}2). Even a s(n,n,1{1{1,,1{1,,1,,2}2}3}2) still grows faster than "the limit of s(n,n,1{1,,1{1,,1,,2}2}2), s(n,n,1{1{1{1,,1{1,,1,,2}2}2}2,,1{1,,1,,2}2}2), s(n,n,1{1{1{1{1,,1{1,,1,,2}2}2}2{1,,1{1,,1,,2}2}2}2,,1{1,,1,,2}2}2), etc.", which can reduce to s(n,n,1{1{1,,2}2{1,,1{1,,1,,2}2}2}2), which is the limit.

Similar things happen in comparisons from {1,,1,2} to {1{1,,1,2}2,,1,2}. You can see it here.