User blog comment:Hyp cos/Analysis - BEAF, FGH and SGH (part 3)/@comment-2033667-20141013055709/@comment-1605058-20141014102209

My point is, we have no idea if SGH and FGH even has catching point when using Taranovsky's notation. Thus we have no idea if C(0) even exists, let alone things like C(C_1(W)^2).

On the other hand, it might be the case that the fundamental sequence may make SGH extremely strong, so that the catching points happen at, say, ordinals of the form zeta_a, for all a. If I correctly understand the idea of Catching function, we would then have C(W) quite weak, as it would be only  phi(3,0).

To my knowledge, we cannot exclude any of these possibilities yet.