  The goal of spin foam models is to provide a viable path integral formulation
of quantum gravity. Because of background independence, their underlying
framework has certain novel features that are not shared by path integral
formulations of familiar field theories in Minkowski space. As a simple
viability test, these features were recently examined through the lens of loop
quantum cosmology (LQC). Results of that analysis, reported in a brief
communication [1], turned out to provide concrete arguments in support of the
spin foam paradigm. We now present detailed proofs of those results. Since the
quantum theory of LQC models is well understood, this analysis also serves to
shed new light on some long standing issues in the spin foam and group field
theory literature. In particular, it suggests an intriguing possibility for
addressing the question of why the cosmological constant is positive and small.
