User blog comment:Bubby3/Ridiculously strong ordinal hyper-operations/@comment-30754445-20170612070222/@comment-24725252-20170612094359

The reason w^^(w^2+1) is that is because w^^n is equivalent to a stack of 1+n height in Bashicu matrix system in my function, and when we add 1, we add a head with a 0 to the top. So w^^(w^2) = (0,0)(1,1)(2,1), so w^^(w^2+1) should be (0,0)(1,1)(2,1)(3,0), which has level phi(w,0).