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Everything up to w^^(w^2) is from Sbiis Sabian's examples.

w^^(w^2+1)= phi(w,0) = w^^(w^w) in Saibain's example.

It is too complicated to exmplain. I don't know how it works between w^^(w^2) and w^^(w^2+1), but here is a guess. 

We now invent the >> operator. (#^^##)>>#= #^^### in Saibian's current system. Here are some comparisons,

(#^^##)>>(#+#)= #^^####

(#^^##)>>(##)=#^^#^#

  1. ^^### = #^^^#

(#^^###)>># = #^^^##

This gets really complicated

By w^^(w^w), we are already at BHO. This is strong.

w^^^3, or w^^(w^^w) has level W_w.

w^^^w is as strong as Bashicu matrix system and we are only at pentation. This is very strong.

I don't know how the higher hyper-operators work.

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