Proposition: The additional global relations are genuine

The covariance is locally of Hadamard form at every regular point. Nevertheless:

1. outside $\mathcal{Q}$, its global wavefront contains all strictly diffractive pairs in ${\mathcal{D}}_{\mathrm{ap}}^{+}\setminus {\mathcal{G}}^{+}$, which are not linked by a regular null geodesic with the displayed endpoint covectors;

2. even in a finite-character quotient sector, the closed geometric relation can contain limiting rays through the removed apex that are not actual regular geodesics.

Proof. Local Hadamard form is Lemma lem:local. For (i), choose $\delta \notin {\mathcal{B}}_{\alpha }$ on the apex front. Proposition prop:broken shows that its radial covector pair is outside ${\mathcal{G}}^{+}$, while Theorem thm:main places it in $\mathrm{WF}(W_{\beta })$. A regular geodesic carrying that radial covector cannot avoid the apex and connect the two endpoints.

For (ii), the finite image kernel includes a flat null segment whose transverse projection crosses the covering origin whenever the corresponding image endpoints are antipodal. It is the limit of regular quotient geodesics and hence belongs to ${\mathcal{G}}^{+}$, but its limiting radial segment is not in $M$.