On nilpotent extensions of $infty$-categories and the cyclotomic trace

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 Voevodskys motives or the stable $infty$-category of perfect complexes on some algebraic stacks). The key input in our paper is Bondarkos 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 Blancs lattice conjecture.