  Jim Propp recently introduced a variant of chip-firing on a line where the
chips are given distinct integer labels. Hopkins, McConville, and Propp showed
that this process is confluent from some (but not all) initial configurations
of chips. We recast their set-up in terms of root systems: labeled chip-firing
can be seen as a root-firing process which allows the moves $\lambda \to
\lambda + \alpha$ for $\alpha\in \Phi^{+}$ whenever
$\langle\lambda,\alpha^\vee\rangle = 0$, where $\Phi^{+}$ is the set of
positive roots of a root system of Type A and $\lambda$ is a weight of this
root system. We are thus motivated to study the exact same root-firing process
for an arbitrary root system. Actually, this central root-firing process is the
subject of a sequel to this paper. In the present paper, we instead study the
interval root-firing processes determined by $\lambda \to \lambda + \alpha$ for
$\alpha\in \Phi^{+}$ whenever $\langle\lambda,\alpha^\vee\rangle \in
[-k-1,k-1]$ or $\langle\lambda,\alpha^\vee\rangle \in [-k,k-1]$, for any $k
\geq 0$. We prove that these interval-firing processes are always confluent,
from any initial weight. We also show that there is a natural way to
consistently label the stable points of these interval-firing processes across
all values of $k$ so that the number of weights with given stabilization is a
polynomial in $k$. We conjecture that these Ehrhart-like polynomials have
nonnegative integer coefficients.
