  To any projective pair $(X,B)$ equipped with an ample $\mathbb{Q}$-line
bundle $L$ (or even any ample numerical class), we attach a new invariant
$\beta(\mu)\in\mathbb{R}$, defined on convex combinations $\mu$ of divisorial
valuations on $X$, viewed as point masses on the Berkovich analytification of
$X$. The construction is based on non-Archimedean pluripotential theory, and
extends the Dervan-Legendre invariant for a single valuation--itself
specializing to Li and Fujita's valuative invariant in the Fano case, which
detects K-stability. Using our $\beta$-invariant, we define divisorial
(semi)stability, and show that divisorial semistability implies $(X,B)$ is
sublc (i.e. its log discrepancy function is non-negative), and that divisorial
stability is an open condition with respect to the polarization $L$. We also
show that divisorial stability implies uniform K-stability in the usual sense
of (ample) test configurations, and that it is equivalent to uniform
K-stability with respect to all norms/filtrations on the section ring of
$(X,L)$, as considered by Chi Li.
