A tree-composition lemma

Covectors are compared in the fixed Minkowski trivialization. For $n\ge 2$, $Z_n=\{(x_1,p_1;\dots ;x_n,p_n):\sum p_i=0\}\setminus 0$; put $Z_1=\varnothing$. A distribution $k\in {\mathcal{D}}^{\prime }(M\times M)$ with finitely many fiber components is a balanced line if

Examples: ${\partial }^{\alpha }{E^{\mathrm{R}/A}}_{\mu }(x-x^{\prime })$ and ${\partial }^{\alpha }\delta (x-x^{\prime })$, as pullbacks of distributions in $x-x^{\prime }$ under a submersion Hormander (Theorem 8.2.4), and their products with smooth functions of $x$ and $x^{\prime }$.