  The success of loop quantum cosmology to resolve classical singularities of
homogeneous models has led to its application to the classical Schwarszchild
black hole interior, which takes the form of a homogeneous Kantowski-Sachs
model. The first steps of this were done in pure quantum mechanical terms,
hinting at the traversable character of the would-be classical singularity, and
then others were performed using effective heuristic models capturing quantum
effects that allowed a geometrical description closer to the classical one but
avoided its singularity. However, the problem of establishing the link between
the quantum and effective descriptions was left open. In this work, we propose
to fill in this gap by considering the path-integral approach to the loop
quantization of the Kantowski-Sachs model corresponding to the Schwarzschild
black hole interior. We show that the transition amplitude can be expressed as
a path integration over the imaginary exponential of an effective action which
just coincides, under some simplifying assumptions, with the heuristic one.
Additionally, we further explore the consequences of the effective dynamics. We
prove first that such dynamics imply some rather simple bounds for phase-space
variables, and in turn, remarkably, in an analytical way, they imply that
various phase-space functions that were singular in the classical model are now
well behaved. In particular, the expansion rate, its time derivative, and the
shear become bounded, and hence the Raychaudhuri equation is finite term by
term, thus resolving the singularities of classical geodesic congruences.
Moreover, all effective scalar polynomial invariants turn out to be bounded.
