  We construct a category of Breuil-Kisin $G_K$-modules to classify integral
semi-stable Galois representations. Our theory uses Breuil-Kisin modules and
Breuil-Kisin-Fargues modules with Galois actions, and can be regarded as the
algebraic avatar of the integral $p$-adic cohomology theories of
Bhatt-Morrow-Scholze and Bhatt-Scholze. As a key ingredient, we classify Galois
representations that are of finite $E(u)$-height.
