Lemma: Relative-coordinate balance

Let $a\in {\mathcal{D}}^{\prime }({\mathbb{R}}^{4(n-1)})$ and define a numerical vertex distribution by submersion pullback,

Then

Multiplication by smooth external weights and finite differential operators preserves this inclusion.

Furthermore, a distributional limit of unweighted amplitudes (eq:A-reg), when such a limit exists, is invariant under simultaneous translation of all vertices and has this relative-coordinate form.

Proof. In coordinates $s=v_1$ and $r_j=v_j-v_1$, the distribution is $1_s\otimes a(r)$. Its wavefront covectors have zero $s$ component. Changing back to vertex coordinates makes that component the sum of all vertex covectors. This gives the asserted bound by the submersion pullback theorem Hormander.

Smooth multiplication and differentiation do not enlarge the wavefront set.

Each unweighted regularized amplitude depends only on differences, so its derivative in every simultaneous-translation direction is zero. Distributional convergence preserves those equations. In the coordinates $(s,r)$, a distribution whose $s$ derivatives all vanish is constant in $s$, hence is $1_s\otimes a(r)$.