User blog comment:B1mb0w/Strong D Function/@comment-5529393-20150805054856/@comment-5529393-20150805081023

Focusing on n=3 is leading you astray though. I notice that you talk about rules like L1, where you say that $$D(l,0,1) >> f_{\psi + 1}(3)$$ based on the mere fact that $$D(l-1,0,1) \approx f_\psi(3)$$. If you think about it this is clearly an impossible claim, since $$f_\psi(3)$$ can be very small in comparison to a very large growth rate of $$f_\psi(n)$$. Also, $$\psi$$ could have many different values that lead to the same value of $$f_\psi(3)$$, depending on fundamental sequences; for example, using the usual fundamental sequences,  $$\varepsilon_0 [2] = \omega^\omega [2] = \omega^2 [2] = \omega \cdot 2 [2] = \omega + 2$$, so $$f_\psi(2)$$ has the same value for all of these ordinals. If we shifted the fundamental sequences by one, we would have $$f_\psi(3)$$ be the same for all of these ordinals. So, by your way of thinking, all one would have to do is to reach $$\omega+2$$ in the fast-growing hierarchy, and that would automatically bootstrap the function up to $$\varepsilon_0$$. This is clearly incorrect.