  Motivated by M-theory, we study rank n K-theoretic Donaldson-Thomas theory on
a toric threefold X. In the presence of compact four-cycles, we discuss how to
include the contribution of D4-branes wrapping them. Combining this with a
simple assumption on the (in)dependence on Coulomb moduli in the 7d theory, we
show that the partition function factorizes and, when X is Calabi-Yau and it
admits an ADE ruling, it reproduces the 5d master formula for the geometrically
engineered theory on A(n-1) ALE space, thus extending the usual geometric
engineering dictionary to n>1. We finally speculate about implications for
instanton counting on Taub-NUT.
