Additivity of derived limits in the Cohen model

We show that, in the model constructed by adding sufficiently many Cohen reals, derived limits are additive on a large class of systems. This generalizes the work of Jeffrey Bergfalk, Michael Hru\v s\'ak, and Chris Lambie-Hanson which focuses on the system $\mathbf{A}$. In the process, we isolate a partition principle responsible for the vanishing of derived limits on collections of Cohen reals and reframe the propagating trivializations results of Bergfalk, Hru\v s\'ak and Lambie-Hanson as a theorem of ZFC. In light of results of the author, Jeffrey Bergfalk, and Justin Moore, the additivity of derived limits also implies additivity results for strong homology.