User blog comment:Kyodaisuu/Mashimo function/@comment-1605058-20140706065714/@comment-25418284-20140706184324

By the way, it is NOT true that O(3 * 5^11) = w^2. I picked that in a very contrived ordering of partial recursive functions for the sake of the example only. In reality, we have to pick a non-arbitrary ordering of all Turing machines, which makes it extremely difficult to determine exact values of FS's.

A definition for the fundamental sequence of the Church-Kleene ordinal is given on its page, and I've also concocted definitions for all recursive ordinals (at Talk:Church-Kleene ordinal), so yes, f_{w_1^CK}(n) is defined.

As for defining fundamental sequences for w_2^CK, you need to be much more explicit than that.