Input covectors and output frequency

The phase covectors are the lowered vectors

They are co-oriented future null covectors, one in each metric slot.

Their vector sum is

Therefore

The generated phase is noncharacteristic for the massless wave operator. This does not combine the two metric-slot covectors into a replacement for the three-slot XI-G criterion.

The arithmetic multiplier $-1/(4{\lambda }^2)$ sends a coefficient ${\lambda }^{2}a$ to $-a/4$ and a coefficient $\lambda b$ to $-b/(4\lambda )$. This is only frozen-frequency arithmetic. For time-dependent coefficients it omits coefficient derivatives, initial data, and preparation-generated homogeneous terms. It is not an asymptotic theorem for ${E_0}^{\mathrm{R}}$.

For each fixed $\lambda$, $S_{\lambda }$ is smooth. That does not establish smoothness or microcausality of the distributional kernel of the underlying polynomial functional. Wavefront properties of source evaluations and of functional kernels are different questions BDH.