Proposition: Primitive parallel-line degrees

Let

Then

Equality holds when the leading differentiated homogeneous tensor product is nonzero. In particular,

Proof. The leading mass-independent term is homogeneous of degree $-2$. Each nonvanishing derivative raises its short-distance degree by its derivative order. The lower-order massive terms do not increase that bound.

For the first example the leading product is nonzero. For the second, evaluate its homogeneous leading coefficient at spacelike $r$ with $r^0=0$. Each temporal second derivative is a nonzero constant multiple of $|\mathbf{r}{|}^{-4}$, and their product is a nonzero multiple of $|\mathbf{r}{|}^{-8}$. This proves the lower bounds as well as the stated upper bounds. An identically vanishing tensor contraction, such as a wave-operator combination away from its source, is excluded from the equality assertion.