  Recent kinetic simulations sparked a debate regarding the emission mechanism
responsible for pulsed GeV $\gamma$-ray emission from pulsars. Some models
invoke curvature radiation, while other models assume synchrotron radiation in
the current-sheet. We interpret the curved spectrum of the Vela pulsar as seen
by H.E.S.S. II (up to $\sim$100 GeV) and the $Fermi$ Large Area Telescope (LAT)
to be the result of curvature radiation due to primary particles in the pulsar
magnetosphere and current sheet. We present phase-resolved spectra and
energy-dependent light curves using an extended slot gap and current sheet
model, invoking a step function for the accelerating electric field as
motivated by kinetic simulations. We include a refined calculation of the
curvature radius of particle trajectories in the lab frame, impacting the
particle transport, predicted light curves, and spectra. Our model reproduces
the decrease of the flux of the first peak relative to the second one,
evolution of the bridge emission, near-constant phase positions of peaks, and
narrowing of pulses with increasing energy. We can explain the first of these
trends because we find that the curvature radii of the particle trajectories in
regions where the second $\gamma$-ray light curve peak originates are
systematically larger than those associated with the first peak, implying that
the spectral cutoff of the second peak is correspondingly larger. However, an
unknown azimuthal dependence of the $E$-field as well as uncertainty in the
precise spatial origin of the GeV emission, precludes a simplistic
discrimination of emission mechanisms.
