  We give a framework to produce C*-algebra inclusions with extreme properties.
This gives the first constructive nuclear minimal ambient C*-algebras. We
further obtain a purely infinite analogue of Dadarlat's modeling theorem on
AF-algebras: Every Kirchberg algebra is rigidly and KK-equivalently sandwiched
by non-nuclear C*-algebras without intermediate C*-algebras. Finally we reveal
a novel property of Kirchberg algebras: They embed into arbitrarily wild
C*-algebras as rigid maximal C*-subalgebras.

Operator Algebras;Dynamical Systems