Lemma: Nonzero comparable endpoint covectors

Every wavefront covector of $K,\overline{K}$, or $S_{\mu \nu }$ has nonzero covectors in both endpoint slots. Their coordinate Euclidean norms are equal.

Proof. Split each Fourier integral into bounded and unbounded momenta. The bounded-momentum part is smooth, since every coordinate derivative remains integrable there.

The two high-frequency mixed phases have endpoint covectors

The massive scalar phase has conic limiting directions given by opposite nonzero null covectors. Oscillatory integration by parts excludes directions outside these phase-generated cones. Multiplication by smooth amplitudes, differentiation, and conjugation preserve the asserted endpoint property.