Proposition: Scalar-leg representation

Formula (eq:K9) is the complete untested kernel of the inherited $B_9$ root. It retains the derivative and mass terms.

On the separated chronological patch,

It has a finite conormal amplitude obtained from the source incidence and the displayed differential operators. The derivative term has normal amplitude degree at most three; the mass term has degree at most two.

Proof. Substitute the untested response kernel into (eq:root). Transferring the derivative of $\psi (b)$ to the kernel gives the first line of (eq:K9); the second line is its mass term.

On the retained patch,

because $b^0>a^0$ and the endpoints are separated. Equation (eq:responseseed) therefore expresses the response as at most two derivatives of smooth coefficients times $\delta (q){\delta }^{(4)}(r)$.

The first line of (eq:K9) adds at most one derivative. The second adds none. The same regular conormal coordinates as in (eq:Sresp) give the stated orders and amplitudes.