= Solution
For the radial null pair above, the <screen-space projector> selects the angular tangent plane, with metric $q_{AB}=r^2\gamma_{AB}$. The ingoing covector is $U_a=-E(dv)_a$, and $\Gamma^v{}_{AB}=-r\gamma_{AB}$. Hence the <optical tensor> is
$$
\widehat B_{AB}=\nabla_BU_A=-Er\gamma_{AB}=-\frac Erq_{AB}.
$$
It is purely symmetric and proportional to the screen metric, giving
$$
\boxed{\theta=-\frac{2E}{r},\qquad\widehat\sigma_{ab}=0,\qquad\widehat\omega_{ab}=0.}
$$
For $E=1$, the <null expansion> is $-2/r$. The area check $A\propto r^2$ and $dr/d\lambda=-E$ gives $A'/A=-2E/r$, in agreement with the <spherical null expansion in ingoing coordinates>. In vacuum, $d\theta/d\lambda=-2E^2/r^2=-\theta^2/2$ also satisfies the <Null Raychaudhuri equation>; the ingoing beam focuses at the singularity rather than remaining future complete.
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