A relation calculation at the cone vertex

We now keep both the diagonal and the limiting flow-out of the internal line in the first $K_5$ placement. This is an internal relation calculation. It is not the matching excluded in XIV-F.

Work where the outer cone frequency $\theta$ is nonzero and the internal covector $\kappa$ is nonzero on one fixed time-oriented null sheet. Write

The diagonal Lagrangian piece is

Here $\kappa$ is unrestricted near the chosen nonzero null covector.

The separated internal flow-out piece can be parameterized by

Choose the covector sheet so that the separation $a-z$ is future directed. Its closure has $\tau =0$ as its boundary.

All covectors in these formulas have zero $y$ component. Unlike the balanced testing example, their outer $b$ covector is nonzero when $\theta \ne 0$.