  We continue our study of when topological and measure-theoretic entropy agree
for algebraic action of sofic groups. Specifically, we provide a new abstract
method to prove that an algebraic action is strongly sofic. The method is based
on passing to a "limiting object" for sequences of measures which are
asymptotically supported on topological microstates. This limiting object has a
natural poset structure that we are able to exploit to prove a max-min
principle: if the sofic approximation has ergodic centralizer, then the largest
subgroup on which the action is a local weak$^{*}$-limit of measures supported
on topological microstates is equal to the smallest subgroup which absorbs all
topological microstates. We are able to provide a version for the case when the
centralizer is not ergodic. We give many applications, including show that for
residually finite groups completely positive (lower) topological entropy (in
the presence) is equivalent to completely positive (lower) measure entropy in
the presence.
