A non-TP2 Tawn member away from the Gumbel subfamily #
Table 5 marks Tawn density TP2 as no* (numerical). At θ=2, α=β=1/2 and
u=t^m, v=t^n the CDF is t^((m+n+s)/2) when m²+n²=s². The Pythagorean
double pairs (91,60,109), (91,312,325), (25,60,65), (25,312,313) give the ordered
rectangles S=(t^91,t^25], T=(t^25,1], U=(0,t^312], V=(t^312,t^60], which at
t=99/100 violate the rectangle inequality forced by an MTP2 density. Together with the
Gumbel (unit-weight) members, which do have TP2 densities, this shows the Tawn cell is
parameter dependent: the exclusion holds for this member.