The complete contact space in the declared class

The almost-homogeneity condition excludes new power-law defect scales such as $L^{-2}{\delta }_{\mathcal{W}}$ in the stress. It does not require strict homogeneity, which will generally be impossible. The class is a prescription, not a classification of every possible singular-background renormalization.

Define the conserved, traceless mixed contact tensor

Indeed, $\mathfrak{D}\mathcal{H}=0$ and $\mathfrak{t}\mathcal{H}=2\Delta {\delta }_{\mathcal{W}}-2\Delta {\delta }_{\mathcal{W}}=0$.