Proposition: Every closed geometric covector occurs

For all parameters,

Proof. Lemma lem:local gives every positive diagonal null covector. Fix a regular null geodesic segment starting at $X^{\prime }$. Propagation of singularities for $P_X$, with the primed variables retained as parameters, transports the first covector along that entire segment HormanderIV. The path remains in the smooth regular host; no passage through the apex is made in this argument. Thus every element of ${\mathcal{G}}_{\mathrm{reg}}^{+}$ occurs. The wavefront set is closed in the punctured cotangent bundle, so it contains ${\mathcal{G}}^{+}$.