Corollary: The covector every representative inherits

For F1 data, every $v\in [u{]}_{\mathrm{sol}}$, microcausal or not, has wavefront points

for some $(y_1,y_2,x)$ and every unit $\mathbf{n}$: two co-oriented null covectors, one per metric slot, with opposite spatial momenta and zero in the scalar slot. Moreover $\mathrm{WF}({\mathsf{W}}_u)$ meets ${\tilde{V}}_{+}^3\cup {\tilde{V}}_{-}^3$.

Proof. The pairing of $v_{{\varphi }_m}$ with the plane waves does not decay along the ray $-\lambda (k_{1♭},k_{2♭})$, so by Hormander (Proposition 8.1.3) this direction lies in $\Sigma (v_{{\varphi }_m})$, i.e. some $(y_1,-k_{1♭};y_2,-k_{2♭})\in \mathrm{WF}(v_{{\varphi }_m})$, and by the inclusion in the proof of Lemma lem:decay there is $x$ with $(y_1,-k_{1♭};y_2,-k_{2♭};x,0)\in \mathrm{WF}(v)$. Since $\beta$ and $\mathsf{q}$ are real, the conjugate plane waves give the same growth, and the $+$ sign. The last statement is Theorem thm:criterion(b) in contrapositive form.