  We consider the totally asymmetric simple exclusion process with open
boundaries, at the edge of the maximal current phase. Using analytic
continuations from the known stationary eigenvalue, we obtain exact expressions
for the spectral gaps in the limit of large system size. The underlying Riemann
surface,generated by modified Lambert functions, interpolates between the one
for periodic TASEP and the one for open TASEP in the maximal current phase.
