  Four-form flux in F-theory compactifications not only stabilizes moduli, but
gives rise to ensembles of string vacua, providing a scientific basis for a
stringy notion of naturalness. Of particular interest in this context is the
ability to keep track of algebraic information (such as the gauge group)
associated with individual vacua while dealing with statistics. In the present
work, we aim to clarify conceptual issues and sharpen methods for this purpose,
using compactification on ${\rm K3} \times {\rm K3}$ as a test case. Our first
approach exploits the connection between the stabilization of complex structure
moduli and the Noether-Lefschetz problem. Compactification data for F-theory,
however, involve not only a four-fold (with a given complex structure) $Y_4$
and a flux on it, but also an elliptic fibration morphism $Y_4 \longrightarrow
B_3$, which makes this problem complicated. The heterotic-F-theory duality
indicates that elliptic fibration morphisms should be identified modulo
isomorphism. Based on this principle, we explain how to count F-theory vacua on
${\rm K3} \times {\rm K3}$ while keeping the gauge group information.
Mathematical results reviewed/developed in our companion paper are exploited
heavily. With applications to more general four-folds in mind, we also clarify
how to use Ashok-Denef-Douglas' theory of the distribution of flux vacua in
order to deal with statistics of sub-ensembles tagged by a given set of
algebraic/topological information. As a side remark, we extend the
heterotic/F-theory duality dictionary on flux quanta and elaborate on its
connection to the semistable degeneration of a K3 surface.
