User blog comment:Hyp cos/SCG(n) and some related/@comment-10429372-20140812170828/@comment-10429372-20140815083336

First, I want to say this is not a formal proof. You are completely right about that.

Second, notice a similiar thing happens at the Goodstein function. G(n) is not even close to f_e_0(n) for relatively small n. But when we look to say, G(2^^^^4) = G(2^^^(2^^^^3)) = G(2^^(2^^^(2^^^^3-1))) ~ f_e_0(2^^^(2^^^^3-1)) ~ f_e_0(2^^^^4), we see that for large n G(n) ~ f_e_0(n) holds. Note that we only looked at G(2^^^^4), so let alone G(G(G(...))). Which would be still much larger than f_(e_0+1)(n-1).