  We prove that a four-dimensional Lorentzian manifold that is curvature
homogeneous of order 3, or CH_3 for short, is necessarily locally homogeneous.
We also exhibit and classify four-dimensional Lorentzian, CH_2 manifolds that
are not homogeneous. The resulting metrics belong to the class of null
electromagnetic radiation, type N solutions on an anti-de Sitter background.
These findings prove that the four-dimensional Lorentzian Singer number
$k_{1,3}=3$, falsifying some recent conjectures by Gilkey. We also prove that
invariant classification for these proper CH_2 solutions requires
$\nabla^{(7)}R$, and that these are the unique metrics requiring the seventh
order.
