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Four-form flux in F-theory compactifications not only stabilizes moduli, butgives rise to ensembles of string vacua, providing a scientific basis for astringy notion of naturalness. Of particular interest in this context is theability to keep track of algebraic information (such as the gauge group)associated with individual vacua while dealing with statistics. In the presentwork, 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 firstapproach exploits the connection between the stabilization of complex structuremoduli 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 $Y_4 \longrightarrowB_3$, which makes this problem complicated. The heterotic-F-theory dualityindicates that elliptic fibration morphisms should be identified moduloisomorphism. 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 exploitedheavily. With applications to more general four-folds in mind, we also clarifyhow to use Ashok-Denef-Douglas' theory of the distribution of flux vacua inorder to deal with statistics of sub-ensembles tagged by a given set ofalgebraic/topological information. As a side remark, we extend theheterotic/F-theory duality dictionary on flux quanta and elaborate on itsconnection to the semistable degeneration of a K3 surface.

The Noether-Lefschetz Problem and Gauge-Group-Resolved Landscapes: F-Theory on K3 x K3 as a Test Case