Lemma: Characteristic triangle cancellation

Consider the numerical triangle

where $F_{\mu }$ is the scalar time-ordered boundary-value line, up to its fixed nonzero action fiber factor. There are configurations with $u,y,x$ pairwise distinct for which the Hörmander sum condition for (eq:triangle) fails. The same cancellation remains available for derivatives whose principal conormal factors are nonzero.

Proof. Choose a future null vector $\ell$ and points

Let $p$ be a nonzero null covector proportional to the metric dual of $\ell$, with sign chosen to match the time-ordered orientation.

At $(y,u)$ the time-ordered line has a wavefront covector $(p,-p)$. At $(y,x)$ it has $(-p,p)$, because the chronological ordering is reversed. The retarded line at $(x,u)$ permits $(-p,p)$: on its nonzero forward null cone the real retarded singularity has both conormal signs.

The sums at the three vertices are

All line covectors are nonzero, and none of the vertices coincide. Differentiation does not remove this configuration when its principal symbol is nonzero at the selected covectors.