Proposition: Exact finite normal form of the response sector

In the chart above, the supported response sector has an exact componentwise representation

The coefficient array is unique in the fixed coordinates.

Every term of $K_6,K_8,K_9$ is retained, including derivatives of their smooth coefficients, the actual source coefficients, and the mass term of $K_9$. No equality or vanishing of these coefficients is assumed.

Proof. The copied response formulas express $K_8,K_9$ as finite differential expressions of order at most three in smooth coefficients times $\delta (q_R){\delta }^{(4)}(a-z)$. Formula (eq:K6) has the same seed and differential order at most two.

Change to the regular coordinates $(s,n)$. The chain rule expresses every derivative as a finite sum of smooth-coefficient tangential and normal derivatives. Tangential derivatives act on coefficients. The remaining normal derivatives of the seed have order at most three.

For a smooth coefficient $c(s,n),$ the finite identity

follows by applying the Leibniz rule to its test-function pairing. It brings every coefficient into the stated normal form without dropping any derivative.

Summing the three kernels gives (eq:normalform). Uniqueness is Lemma lem:jetinsert.