Proposition: Homogeneous contact form of the scale-variation coefficients

If $u\in {\mathcal{H}}_{4+N}$ is homogeneous of degree $-4-N$ on ${\mathbb{R}}^4\setminus \{0\}$, then $A_{\mu }(u)$ has the contact form (eq:L-Amu-exp), and the scale-variation coefficients satisfy the coherence constraints of Lemma lem:Ldiffmu. In the critical case $N=0$ (so $u$ is a multiple of $S^2$ on ${\mathcal{H}}_4$), one has $A_{\mu }(u)=r_{\mu }(u)\delta$ for a scalar $r_{\mu }(u)\in \mathbb{C}$.

Proof. Lemma lem:Ldiffmu. For $N=0$ the only multi-index with $|\gamma |\le 0$ is $\gamma =0$.