Proposition: The source incidence is regular

The product of deltas in (eq:Qincidence) is defined. Its support is $v=a=z$, and its underlying conormal relation is

The source kernel has conormal order at most three in the twelve-dimensional $(v,a,z)$ space. Its amplitude is the finite polynomial obtained by applying the displayed derivatives to the two independent diagonal deltas.

Pairing its $z$ slot against a smooth $G$ recovers ${\mathsf{Q}}_G$, but this testing is not used to define the untested relation.

Proof. The map

is a submersion onto ${\mathbb{R}}^8$: variations of $z$ and $a$ can prescribe its two outputs independently. The two-delta product is its pullback of the delta at the origin in ${\mathbb{R}}^8$.

Its conormal covectors have the stated zero-sum form. The defining submanifold has codimension eight in dimension twelve. With normal amplitude order at most two, the conormal order is

Lower derivative monomials belong to the corresponding lower filtered orders.

The pairing formula follows directly from the distributional evaluation of the derivatives of the deltas. It reproduces (eq:sourcepolynomial).