Proposition: Coherence

If ${\sum }_{\lambda }{r}_{\lambda }{q^{\circ }}_{\lambda }=0$ in ${\mathcal{D}}^{\prime }({\mathbb{R}}^4\setminus \{0\})$ with constants $r_{\lambda }$, then

In particular, edge interchange (${q^{\circ }}_{\alpha \beta }={q^{\circ }}_{\beta \alpha }$) is already quotiented by unordered labels, and the wave-operator relation ${\eta }^{\alpha \beta }{q^{\circ }}_{\{0,e_{\alpha }+e_{\beta }\}}=0$ forces ${\sum }_{\alpha ,\beta }{\eta }^{\alpha \beta }{C}_{\{0,e_{\alpha }+e_{\beta }\},\gamma }=0$ for every $|\gamma |\le 2$.

Proof. $\Delta {\mathfrak{R}}_{\mathrm{L}}$ is linear on ${\mathcal{V}}_{\mathrm{adm}}$, so it annihilates every constant-coefficient relation; linear independence of $\{{\partial }^{\gamma }\delta \}$ yields the coefficient form. The two sample relations are Proposition prop:VLincl(1),(3).