us from directly applying it to $X=X_{\rm tri}^\square(\overline \rho_{x,v})$ and thus directly deducing Theorem \[mainintro\] from Theorem \[mainintro2\]. Hence we have to sharpen Theorem \[mainintro2\], see Theorem \[classicalitycrit\].

Assuming the classical modularity lifting conjectures for ${\overline{\rho}}_x$ (in all weights with trivial inertial type), there is a certain union $\widetilde X_{\rm tri}^\square({\overline{\rho}}_{x,v})$ of irreducible components of $X_{\rm tri}^\square({\overline{\rho}}_{x,v})$ such that $\prod_{v\in S_p}\widetilde X_{\rm tri}^\square({\overline{\rho}}_{x,v})$ is (essentially) described by the patched eigenvariety $X_p({\overline{\rho}}_x)$ defined in [@BHS] (see Remark \[conjvariant\]). In the last section of the paper (§\[modularity\]), we prove (assuming modularity lifting conjectures) that the inequality in Theorem \[mainlocal\] for $X=\widetilde X_{\rm tri}^\square({\overline{\rho}}_{x,v})$ is an [*equality*]{} for all $v\in S_p$, $$\label{mainequality}
\dim T_{\widetilde X_{\rm tri}^\square({\overline{\rho}}_{x,v}),x_v}=\dim X_{\rm tri}^\square(\overline \rho_{x,v}) + \lg(w_{x,v})-d_{x,v}\ \ \ \ \ \ {\rm (assuming\ modularity),}$$ see Corollary \[bonnedim\]. The precise computation (\[mainequality\]) of the dimension of the tangent space is intimately related to (and uses in its proof) the existence of many *companion points* on the patched eigenvariety $X_p({\overline{\rho}}_x)$. These companion points are provided by the following unconditional theorem, which is of independent interest.

\[companionptsintro\] Let $y=((\rho_v)_{v\in S_p},\epsilon)$ be a point on $X_p({\overline{\rho}}_x)$. Let $T$ be the diagonal torus in ${\mathrm{GL}}_n$ and let $\delta$ be a locally ${\mathbb{Q}}_p$-analytic character 