Lemma: Nonzero comparable endpoint covectors

Every wavefront element of $K$, $\overline{K}$, or $S_{\mu \nu }$ has nonzero covectors in both endpoint slots. For the Euclidean norm on coordinate covectors, the two endpoint norms are equal.

Proof. Split each Fourier integral into low and high frequencies with a smooth radial cutoff. The low-frequency part of $K$ is smooth: its amplitudes are locally integrable, and every coordinate derivative inserts an integrable bounded-frequency factor. The same observation applies to ${\Delta }_m^{+}$ and its finite derivatives.

The high-frequency phases of $K$ are

Their endpoint covectors are, respectively,

They are nonzero at both endpoints and have equal norms. The usual integration-by-parts estimate for oscillatory integrals excludes covectors outside these phase-generated cones. Smooth amplitudes and differentiation do not enlarge the resulting wavefront bound.

For the massive scalar kernel the high-frequency phase is

Its conic limiting endpoint directions are opposite nonzero null covectors with equal norms. Differentiation to obtain $S_{\mu \nu }$ does not enlarge this bound. Conjugation reverses covectors and preserves the assertion.