  In this paper, we derive correlations between core-collapse supernova
observables and progenitor core structures that emerge from our suite of twenty
state-of-the-art 3D core-collapse supernova simulations carried to late times.
This is the largest such collection of 3D supernova models ever generated and
allows one to witness and derive testable patterns that might otherwise be
obscured when studying one or a few models in isolation. From this panoramic
perspective, we have discovered correlations between explosion energy, neutron
star gravitational birth masses, $^{56}$Ni and $\alpha$-rich freeze-out yields,
and pulsar kicks and theoretically important correlations with the compactness
parameter of progenitor structure. We find a correlation between explosion
energy and progenitor mantle binding energy, suggesting that such explosions
are self-regulating. We also find a testable correlation between explosion
energy and measures of explosion asymmetry, such as the ejecta energy and mass
dipoles. While the correlations between two observables are roughly independent
of the progenitor ZAMS mass, the many correlations we derive with compactness
can not unambiguously be tied to a particular progenitor ZAMS mass. This
relationship depends upon the compactness/ZAMS mass mapping associated with the
massive star progenitor models employed. Therefore, our derived correlations
between compactness and observables may be more robust than with ZAMS mass, but
can nevertheless be used in the future once massive star modeling has
converged.
