  Let $\mathcal{T}$ be an $\mathcal{O}_K$-linear idempotent-complete, small
smooth proper stable $\infty$-category, where $K$ is a finite extension of
$\mathbb{Q}_p$. We give a Breuil-Kisin module structure on the topological
negative cyclic homology $\pi_i{\rm
TC}^-(\mathcal{T}/\mathbb{S}[z];\mathbb{Z}_p)$, and prove a $K$-theory version
of Bhatt-Morrow-Scholze's comparison theorems. Moreover, using Gao's
Breuil-Kisin $G_K$-module theory and Du-Liu's $(\varphi,\hat{G})$-module
theory, we prove the $\mathbb{Z}_p[G_K]$-module $T_{A_{\rm inf}}(\pi_i{\rm
TC}^-(\mathcal{T}/\mathbb{S}[z];\mathbb{Z}_p)^{\vee})$ is a
$\mathbb{Z}_p$-lattice of a crystalline representation. As a corollary, if the
generic fibre of $\mathcal{T}$ admits a geometric realization in the sense of
Orlov, we prove a comparison theorem between $K(1)$-local $K$ theory of the
generic fibre and topological cyclic periodic homology theory of the special
fibre with $B_{\rm crys}$-coefficients, in particular, we prove the $p$-adic
representation of the $K(1)$-local $K$-theory of the generic fibre is a
crystalline representation, this can be regarded as a non-commutative analogue
of $p$-adic Hodge theory for smooth proper varieties proved by Tsuji and
Faltings. This is the full version of arXiv:2305.00292, containing additional
details and results.
