Theorem: Row-by-row substitution

The coefficient kernel of (eq:insertion) is exactly (eq:K47)–(eq:K9m). For compact smooth inputs,

The $y$-variable is inactive internally; adjoining it means tensoring with the constant distribution one.

The substituted source, density, root, and crossed array are respectively (eq:q-extraction), (eq:Didentified), (eq:root), and (eq:actualC).

Proof. Insert the $L$-kernels in the first line of the insertion, then transfer its scalar derivative to $b$. This gives $K_{47}$.

In the next two insertion lines, transferring the source divergence to the incoming $R$-kernel gives a minus sign; transferring the scalar derivative gives a second minus sign. The two tensor placements give $K_{5,+},K_{5,-}$. The local mixed density gives $K_6$. Its $a$-derivative is on the retarded line alone, not on the incidence delta.

For the fourth line,

The insertion sign and scalar transposition therefore give $K_8$. On its supported incidence, the separated future assumptions above justify the derivative reduction for $B$.

The last insertion line gives the derivative and mass root rows, with their displayed signs. Causal properness justifies the intermediate integrations. The explicit distributional kernels and equality on compact product tests determine the equality on joint tests.