= Solution
Use an <affine parameter> $\lambda$ along a generator of a <null hypersurface>. Its <null twist> vanishes by <hypersurface orthogonality>. The screen metric is positive definite, so $\widehat\sigma^{ab}\widehat\sigma_{ab}\ge0$. The <Einstein field equations> and <null energy condition> imply $R_{ab}U^aU^b\ge0$, giving from the <Null Raychaudhuri equation>
$$
\frac{d\theta}{d\lambda}\le-\frac12\theta^2.
$$
Starting with $\theta_0<0$, the <null expansion> stays negative as long as the congruence remains regular. Set $z=-1/\theta>0$. Then $z'=\theta'/\theta^2\le-1/2$, so
$$
0<z(\lambda)\le\frac1{|\theta_0|}-\frac{\lambda-\lambda_0}{2}.
$$
A finite negative <null expansion> cannot persist to $\lambda-\lambda_0=2/|\theta_0|$. Thus the <null focusing theorem> gives
$$
\boxed{\theta\longrightarrow-\infty\quad\text{within affine distance at most }\frac2{|\theta_0|}.}
$$
This is a focal breakdown of the transverse congruence, provided the generator and regular geometry extend that far; it is not by itself a <curvature singularity> or a statement that the individual <null geodesic> terminates there.
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