Proposition: Global positive-frequency upper bound

For every $0<\alpha <1$ and $0\le \beta <1$,

In particular, there are no one-zero-endpoint covectors and no additional pure angular wavefronts at an image boundary.

Proof. The proof separates the real values of $Q$ in (eq:reducedboundary).

{1. The regions $Q>1$ and $Q<-1$.} For $Q>1$, $\eta =\mathrm{arcosh}Q$ is real and positive, and the summed expression is smooth. For $Q<-1$, its two boundary values have

The geometric denominators in (eq:Sgeom) are nonzero because $e^{-v/\alpha }<1$. The factor $\sinh \eta$ is also nonzero. Thus each boundary value is smooth.

{2. The interval $-1<Q<1$.} Write

The only poles in (eq:Sgeom) occur when

They are precisely $Q_k=0$. Each of the two geometric fractions has a simple pole in a smooth defining function for the corresponding front. Even where two fronts cross in the base, the two fractions provide the separate conormal descriptions. Consequently the wavefront is contained in their conormals.

The half-plane boundary-value orientation in Lemma lem:boundaryvalue gives a nonnegative coefficient of $\mathrm{d}Q$ for $T>0$, and a nonpositive one for $T<0$. Since

the first temporal covector is negative. This is exactly the orientation $\lambda T>0$ in (eq:Gcoordinates).

{3. The branch point $Q=1$.} At an angular value not congruent to zero, the evenness in $\eta$ makes (eq:reducedF) analytic in $Q$. Near $\delta =0(\mathrm{mod}2\pi )$, the factorization in the proof of Lemma lem:local gives the direct geometric conormal. At a diagonal point the complete local wavefront is the usual positive-frequency diagonal null relation by (eq:localHadamard).

{4. The branch point $Q=-1$, including $\delta \in {\mathcal{B}}_{\alpha }$.} Here $T\ne 0$. The map

is a submersion, and (eq:reducedboundary) expresses the kernel as the pullback of a two-variable distribution, multiplied by the smooth factor $1/(8{\pi }^2\alpha r{r}^{\prime })$. The pullback theorem therefore gives

for every possible wavefront covector at this branch point. There is no contribution from passive coordinates.

On $F_{\mathrm{ap}}=0$,

The first endpoint part of $\mathrm{d}Q$ is null and has no angular component. Hence

The bisolution equation (eq:bisolution) and microlocal elliptic regularity force $p(X,\kappa )=0$, so $\nu =0$. The total covector is nonzero, therefore $\lambda \ne 0$; both endpoint covectors are nonzero. Lemma lem:boundaryvalue fixes its positive-frequency orientation. This is precisely ${\mathcal{D}}_{\mathrm{ap}}^{+}$.

This last argument applies to the total summed kernel. It does not assign or differentiate separate values of moving image-sector indicators.

5. Exhaustion. At $T=0$, the only possible singular points are on the diagonal: indeed $Q\ge 1$, with equality only when $r=r^{\prime }$ and $Z=0$, and a singular angular value then requires $\delta =0(\mathrm{mod}2\pi )$. The preceding cases cover all of $M\times M$.