= Solution
Locally write the <null hypersurface> as $F=0$. Its generators are tangent to its raised normal, so on it $U_a=h\nabla_aF$ for a nonzero scalar $h$. The antisymmetric derivative is
$$
\nabla_{[b}U_{a]}=(\nabla_{[b}h)(\nabla_{a]}F),
$$
since the Hessian of $F$ is symmetric for the torsion-free <Levi-Civita connection>. Each term contains $\nabla F$, which is proportional to $U$ and is killed by the <screen-space projector>. Hence
$$
\boxed{\omega_{ab}=P_a{}^cP_b{}^d\nabla_{[d}U_{c]}=0.}
$$
This is the null version of hypersurface orthogonality in the <Frobenius theorem>. A <null hypersurface> has no independent normal direction outside its tangent space: its <null vector> normal also generates it. That is why the same argument applies to the generators' <null twist>.
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