  As established in a prior work of the author, the linear simplicity
constraints used in the construction of the so-called `new' spin-foam models
mix three of the five sectors of Plebanski theory as well as two dynamical
orientations, and this is the reason for multiple terms in the asymptotics of
the EPRL vertex amplitude as calculated by Barrett et al. Specifically, the
term equal to the usual exponential of $i$ times the Regge action corresponds
to configurations either in sector (II+) with positive orientation or sector
(II-) with negative orientation. The presence of the other terms beyond this
cause problems in the semiclassical limit of the spin-foam model when
considering multiple 4-simplices due to the fact that the different terms for
different 4-simplices mix in the semi-classical limit, leading in general to a
non-Regge action and hence non-Regge and non-gravitational configurations
persisting in the semiclassical limit.
  To correct this problem, we propose to modify the vertex so its asymptotics
include only the one term of the form $e^{iS_{\Regge}}$. To do this, an
explicit classical discrete condition is derived that isolates the desired
gravitational sector corresponding to this one term. This condition is
quantized and used to modify the vertex amplitude, yielding what we call the
`proper EPRL vertex amplitude.' This vertex still depends only on standard
SU(2) spin-network data on the boundary, is SU(2) gauge-invariant, and is
linear in the boundary state, as required. In addition, the asymptotics now
consist in the single desired term of the form $e^{iS_{\Regge}}$, and all
degenerate configurations are exponentially suppressed. A natural
generalization to the Lorentzian signature is also presented.

General Relativity and Quantum Cosmology