Before revision:  We prove that many variants of the \mathsf{PCSA Flajolet and Martin have \mathsf{Fish H_0/I_0 , where H_0,I_0 are two precisely-defined constants, and that all base-q generalizations of (\textsf{Hyper )\textsf{LogLog are strictly worse than H_0/I_0, but  tend to H_0/I_0 in the limit as q  \infty.
After revision:  We prove that many variants of the \mathsf{PCSA Flajolet and Martin have \mathsf{Fish H_0/I_0 , where H_0,I_0 are two precisely-defined constants, and that all base-q generalizations of (\textsf{Hyper \textsf{LogLog are strictly worse than H_0/I_0, but tend to H_0/I_0 in the limit as q  \infty.