  We construct a K-theory version of Bhatt-Morrow-Scholze's Breuil-Kisin
cohomology theory for $\sO_K$-linear idempotent-complete, small smooth proper
stable infinity-categories, where $K$ is a discretely valued extension of
$\Q_p$ with perfect residue field. As a corollary, under the assumption that
$K(1)$-local K theory satisfies the K\"unneth formula for $\sO_K$-linear
idempotent-complete, small smooth proper stable $\infty$-categories, we prove a
comparison theorem between $K(1)$-local K theory of the generic fiber and
topological cyclic periodic homology theory of the special fiber with
$\Bcry$-coefficients, and $p$-adic Galois representations of $K(1)$-local K
theory for $\sO_K$-linear idempotent-complete, small smooth proper stable
$\infty$-categories are semi-stable. We also provide an alternative
K-theoretical proof of the semi-stability of p-adic Galois representations of
the p-adic \'etale cohomology group of smooth proper varieties over $K$ with
good reduction. is a short, This is a short preliminary version of the work
that was later expanded in 2309.13654.
