Under the hypotheses from 3.6, the model function $f$ is $\theta$-psh if and only if $F$ satisfies for all $x\in\Delta\coloneqq S(\mathscr{X})$

(3.1)

$\displaystyle\sum_{\nu\in T_{x}(\Delta)}w_{x}(\nu)\lambda_{x,\nu}(F)+\deg(\theta|_{\mathcal{C}_{x}})\geq 0,$

where $\nu$ ranges over the set $T_{x}(\Delta)$ of outgoing tangent directions at $x$. Here, $\lambda_{x,\nu}(F)$ denotes the slope of $F$ at $x$ along $\nu$ and we have the weight $w_{x}(\nu)\coloneqq[\tilde{K}(p_{\nu}):\tilde{K}]$ for the singularity $p_{\nu}$ of $\mathscr{X}_{s}$ corresponding to the edge of $S(\mathscr{X})$ at $x$ in the direction of $\nu$. Moreover, if $x$ is a vertex of $S(\mathscr{X})$, then $\mathcal{C}_{x}$ denotes the corresponding irreducible component $\mathcal{C}_{x}$ of $\mathscr{X}_{s}$ and if $x$ is not a vertex, then $\deg(\theta|_{\mathcal{C}_{x}})\coloneqq 0$.