  Using log canonical thresholds and basis divisors Fujita--Odaka introduced
purely algebro-geometric invariants $\delta_m$ whose limit in $m$ is now known
to characterize uniform K-stability on a Fano variety. As shown by Blum-Jonsson
this carries over to a general polarization, and together with work of Berman,
Boucksom, and Jonsson, it is now known that the limit of these
$\delta_m$-invariants characterizes uniform Ding stability. A basic question
since Fujita-Odaka's work has been to find an analytic interpretation of these
invariants. We show that each $\delta_m$ is the coercivity threshold of a
quantized Ding functional on the $m$-th Bergman space and thus characterizes
the existence of balanced metrics. This approach has a number of applications.
The most basic one is that it provides an alternative way to compute these
invariants, which is new even for $\mathbb{P}^n$. Second, it allows us to
introduce algebraically defined invariants that characterize the existence of
K\"ahler-Ricci solitons (and the more general $g$-solitons of Berman-Witt
Nystr\"om), as well as coupled versions thereof. Third, it leads to
approximation results involving balanced metrics in the presence of
automorphisms that extend some results of Donaldson.
