User blog comment:DrCeasium/Hyperfactorial Notation - Take 2/@comment-5150073-20130330145048/@comment-5150073-20130330223726

Now I believe you reached the w^w order type. Your rule 2 looks like rule 4 in array notation (where all default passengers take base value), rule 3 is like rule 5 (in array notation, plugging back one entry n times desceasing next by one), and rule 4 is like 1-chopping. As for the rule 1, it is the "brake" rule, as in Bowers' notation {a,b} = a^b.