  We establish a new family of Carleman inequalities for wave operators on
cylindrical spacetime domains containing a potential that is critically
singular, diverging as an inverse square on all the boundary of the domain.
These estimates are sharp in the sense that they capture both the natural
boundary conditions and the natural $H^1$-energy. The proof is based around
three key ingredients: the choice of a novel Carleman weight with rather
singular derivatives on the boundary, a generalization of the classical
Morawetz inequality that allows for inverse-square singularities, and the
systematic use of derivative operations adapted to the potential. As an
application of these estimates, we prove a boundary observability property for
the associated wave equations.
