Lemma: The basic physical coefficient relations

The distributions in the local reconstruction obey

More generally,

When the indicated generators occur in ${\mathcal{I}}_m$, these give local coefficient null presentations. For a larger presentation space they give null presentations provided that its reconstruction uses the same rule and coefficientwise smooth action.

Proof. For a test $\psi$,

which equals the pairing with $-j{\delta }^{(j-1)}$. The case $j=0$ gives $q\delta =0$.

For the third identity, insert $r^2\phi$ in (eq:Rpair). On the cone points $(t,t\omega )$, $r^2=t^2-t^2|\omega {|}^2=0$. Thus the integral vanishes.

Tensoring the one-dimensional identities with the inner distribution and using Lemma lem:linearity yields the corresponding coefficient relations. The conditional assertion for a larger space follows from the stated rule and action there.