  We do three things in this paper: (1) study the analog of localization
sequences (in the sense of algebraic $K$-theory of stable $\infty$-categories)
for additive $\infty$-categories, (2) define the notion of nilpotent extensions
for suitable $\infty$-categories and furnish interesting examples such as
categorical square-zero extensions, and (3) use (1) and (2) to extend the
Dundas-Goodwillie-McCarthy theorem for stable $\infty$-categories which are not
monogenically generated (such as the stable $\infty$-category of Voevodsky's
motives or the stable $\infty$-category of perfect complexes on some algebraic
stacks). The key input in our paper is Bondarko's notion of weight structures
which provides a "ring-with-many-objects" analog of a connective
$\mathbb{E}_1$-ring spectrum. As applications, we prove cdh descent results for
truncating invariants of stacks extending the work of Hoyois-Krishna for
homotopy $K$-theory, and establish new cases of Blanc's lattice conjecture.
