  The Fermi Large Area Telescope has revolutionised the $\gamma$-ray pulsar
field, increasing the population to over 250 detected pulsars. The majority
display spectra with exponential cutoffs in a narrow range around a few GeV.
Models predicted cutoffs up to 100 GeV; it was therefore not expected that
pulsars would be visible in the very-high-energy ($>$100 GeV) regime.
Subsequent surprise discoveries by ground-based telescopes of pulsed emission
from four pulsars above tens of GeV have marked the beginning of a new era,
raising important questions about the electrodynamics and local environment of
pulsar magnetospheres. Detection of the Vela pulsar by H.E.S.S. (20-120 GeV)
and Fermi provides evidence for a curved spectrum. We posit this to result from
curvature radiation via primary particles in the pulsar magnetosphere and
current sheet. We present energy-dependent light curves using an extended slot
gap and current sheet model and invoking a two-step accelerating $E$-field as
motivated by kinetic simulations. I include a refined calculation of the
curvature radius of particle trajectories, impacting the particle transport,
predicted light curves, and spectra. The model reproduces the decrease of flux
of the first light-curve 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 fundamentally explain the first of these
trends, since I found 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 a
correspondingly larger cutoff for the second peak. An unknown azimuthal
dependence of the $E$-field as well as uncertainty in the precise emission
locale preclude a simplistic discrimination of emission mechanisms.
