The declared metric-sign transport

Lower the response with ${\eta }_{+}$:

The comparison metric coefficient on VI's background is

This is the transport induced by

For an actual real metric, ${\square }_{-q}=-{\square }_q$. The same identity holds for the formal jets because the determinants agree in four dimensions and the inverses differ by a minus sign. The harmonic equation and ordinary coordinate-time data are therefore transported without changing the coordinate scalar functions.

Complex coefficients are used only through complex linear extension. Real witnesses will be obtained by taking $\mathrm{Re}(\zeta \cdot )$, with $\left|\zeta \right|=1$.