-cr}})^{\rm rig}$. We relate ${\mathfrak{X}}_{\overline \rho_{x,v}}^{\square,{\bf k}_v{\rm -cr}}$ to $X_{\rm tri}^\square(\overline \rho_{x,v})$ by introducing a third rigid analytic space $\widetilde {\mathfrak{X}}_{\overline \rho_{x,v}}^{\square,{\bf k}_v{\rm -cr}}$ finite over ${\mathfrak{X}}_{\overline \rho_{x,v}}^{\square,{\bf k}_v{\rm -cr}}$ parametrizing crystalline deformations $\rho_v$ of $\overline \rho_{x,v}$ of Hodge-Tate weights ${\bf k}_v$ [*together with an ordering*]{} of the Frobenius eigenvalues on $D_{\rm cris}(\rho_{v})$, see §\[variant\] for a precise definition. The space $\widetilde {\mathfrak{X}}_{\overline \rho_{x,v}}^{\square,{\bf k}_v{\rm -cr}}$ naturally embeds into $X_{\rm tri}^\square(\overline \rho_{x,v})$ and contains the point $x_v$ (and is smooth at $x_v$). We prove that there is a unique irreducible component $Z_{{\rm tri}}(x_v)$ of $X_{\rm tri}^\square(\overline \rho_{x,v})$ containing the unique irreducible component of $\widetilde {\mathfrak{X}}_{\overline \rho_{x,v}}^{\square,{\bf k}_v{\rm -cr}}$ passing through $x_v$ (Corollary \[defnZtri(x)\]). Let $Z_{\rm tri}(x):=\prod_{v\in S_p}Z_{\rm tri}(x_v)$, which is thus an irreducible component of $X_{\rm tri}^\square(\overline \rho_{x,p})$ containing $x$. Then Theorem \[mainintro2\] easily follows from the following theorem (see (i) of Remark \[uniqueirredcompo\]):

\[mainclasscrit\] Assume that ${\mathfrak{X}}_{{\overline{\rho}}_x^p}\times{\mathbb{U}}^g\times Z_{\rm tri}(x)\subseteq X_p(\overline\rho_x)$ via (\[maininclusion\]). Then the point $x$ is classical.

Let us finally sketch the proof of Theorem \[mainclasscrit\] (in fact, for the same reason as above, we have to sharpen Theorem \[mainclasscrit\], see Theorem \[classicalitycrit\]). Let $R_\infty$ be the usual patched deformation ring of ${\overline{\rho}}_x$, there is a canonical morphism 