User blog comment:B1mb0w/Growth Rate of the S Function/@comment-25601061-20160708203814/@comment-5529393-20160709121356

You're making the same mistake again... $$f_{n^{n\uparrow\uparrow(m-1)}}(n)$$ is not equal to $$f_{\omega \uparrow\uparrow m}(n)$$. In fact, $$ f_{n^{n\uparrow\uparrow(m-1)}}(n) < f_{\omega}( n^{n\uparrow\uparrow(m-1)}) < f_{\omega} (f_3(n)) < f_{\omega+1}(n)$$, so it's not even in the same galaxy as $$f_{\omega \uparrow\uparrow m}(n)$$.