= Solution
Take the screen trace of the <optical matrix of a null congruence> equation. The cross-contractions of symmetric <null shear> with antisymmetric <null twist> vanish, and
$$
\operatorname{tr}(\widehat B^2)=\frac12\theta^2+\widehat\sigma_{ab}\widehat\sigma^{ab}-\widehat\omega_{ab}\widehat\omega^{ab}.
$$
The minus sign in the <null twist> contribution follows from tracing the square of an <antisymmetric matrix>. The tidal trace is $P^{ef}R_{cedf}U^cU^d=R_{cd}U^cU^d$: the additional null terms in $P^{ef}$ vanish by curvature antisymmetry. Thus the <Null Raychaudhuri equation> is
$$
\boxed{U\cdot\nabla\theta=-\frac12\theta^2-\widehat\sigma_{ab}\widehat\sigma^{ab}+\widehat\omega_{ab}\widehat\omega^{ab}-R_{ab}U^aU^b.}
$$
In particular, <null shear> focuses the <null geodesic congruence> while <null twist> enters with the opposite sign. With vanishing <null twist> and the <null energy condition>, the <Einstein field equations> imply $U\cdot\nabla\theta\le-\theta^2/2$. These signs and the factor $1/2$ depend on using the full-trace <null expansion> in a two-dimensional screen.
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