  This is the final paper in the series \cite{tao:heatwave},
\cite{tao:heatwave2}, \cite{tao:heatwave3}, \cite{tao:heatwave4} that
establishes global regularity for two-dimensional wave maps into hyperbolic
targets. In this paper we establish the remaining claims required for this
statement, namely a divisible perturbation theory, and a means of synthesising
solutions for frequency-delocalised, spatially-dispersed, or
spatially-delocalised data out of solutions of strictly smaller energy.
  As a consequence of the perturbation theory here and the results obtained
earlier in the series, we also establish spacetime bounds and scattering
properties of wave maps into hyperbolic space.
