  We study the dynamics of hardcore spin models on the square and triangular
lattice, constructed by analogy to hard spheres, where the translational
degrees of freedom of the spheres are replaced by orientational degrees of
freedom of spins on a lattice and the packing fraction as a control parameter
is replaced by an exclusion angle. In equilibrium, models on both lattices
exhibit a Kosterlitz-Thouless transition at an exclusion angle $\Delta_{\rm
KT}$. We devise compression protocols for hardcore spins and find that {\it
any} protocol that changes the exclusion angle nonadiabatically, if endowed
with only local dynamics, fails to compress random initial states beyond an
angle $\Delta_{\rm J}> \Delta_{\rm KT}$. This coincides with a doubly algebraic
divergence of the relaxation time of compressed states towards equilibrium. We
identify a remarkably simple mechanism underpinning this divergent timescale:
topological defects involved in the phase ordering kinetics of the system
become incompatible with the hardcore spin constraint, leading to a vanishing
defect mobility as $\Delta\rightarrow\Delta_{\rm J}$.
