  This paper concludes our efforts in describing SU(3)-Yang-Mills theories at
different couplings/temperatures in terms of effective Polyakov-loop models.
The associated effective couplings are determined through an inverse Monte
Carlo procedure based on novel Schwinger-Dyson equations that employ the
symmetries of the Haar measure. Due to the first-order nature of the phase
transition we encounter a fine-tuning problem in reproducing the correct
behavior of the Polyakov-loop from the effective models. The problem remains
under control as long as the number of effective couplings is sufficiently
small.
