  Local flatness is a property shared by all the spin foam models. It ensures
that the theory's fundamental building blocks are flat by requiring locally
trivial parallel transport. In the context of simplicial Lorentzian spin foam
theory, we show that local flatness is the main responsible for the emergence
of geometry independently of the details of the spin foam model. We discuss the
asymptotic analysis of the EPRL spin foam amplitudes in the large quantum
number regime, highlighting the interplay with local flatness.
