  We study the behaviour of lightcone wavefunctions, in coarse transverse
lattice gauge theory, when one or more parton lightcone momenta are small. This
probes the limit of hadron structure at large and small Bjorken x.
Finite-energy boundary conditions on boundstates allow one to derive the
analytic form of wavefunctions in this region. This leads to simple, universal
predictions for the behaviour of quark generalised parton distributions near
their endpoints. For the first few meson wavefunctions at large N_c, we give
the simplest ansatz that incorporates all the boundary conditions.
