Persistence modules stratify their underlying parameter space, a quality that makes persistence modules amenable to study via invariants of stratified spaces. In this article, we extend a result previously known only for one-parameter persistence modules to grid multiparameter persistence modules. Namely, we show the
K
K
-theory of grid multiparameter persistence modules is additive over strata. This is true for both standard monotone multi-parameter persistence as well as multiparameter notions of zig-zag persistence. We compare our calculations for the specific group
K
0
K_0
with the recent work of Botnan, Oppermann, and Oudot, highlighting and explaining the differences between our results through an explicit projection map between computed groups.