Proposition: Complete-source power counting

Let $d:=|a|+|b|+|p|+|q|$ for an ordered placement $(A;a,b;p,q)$ contributing to (eq:QL), and set $N_{\lambda }:=d$ (the declared order; not defined through $\mathrm{sd}-4$). Then ${q^{\circ }}_{a+p,b+q}$ is homogeneous of degree $-4-d$ on ${\mathbb{R}}^4\setminus \{0\}$, and

with equality if the product is nonzero. Consequently, as consequences of Theorem thm:budget and the vertex bound $|p|+|q|\le 2$ (XVI-K, Proposition prop:hess):

Proof. Multi-index additivity: $|a+p|+|b+q|=|a|+|b|+|p|+|q|=d$. Homogeneity of ${\partial }^{\alpha }S$ of degree $-2-|\alpha |$ is XVI-K, Lemma lem:homog; the product of two such factors on ${\mathbb{R}}^4\setminus \{0\}$ is homogeneous of degree $-4-d$ by the same pullback argument as XVI-K, Proposition prop:sd (Lemma lem:Lprod supplies existence before scaling). Scaling degree: Brunetti–Fredenhagen, Example 4, gives $\mathrm{sd}\le 4+d$, with equality for a nonzero homogeneous distribution of that degree. Sector bounds: Theorem thm:budget fixes the source slot order; XVI-K, Proposition prop:hess fixes $|p|+|q|\le 2$; add.