Lemma: Coordinate defect

Coefficientwise, $\mathrm{supp}{\Theta }^{(j)}\subset K$, and

For compact generators supported in $\{\epsilon <t<T\}$, ${\Theta }_F=0$.

Proof. Differential operators in a formal pullback do not enlarge the support of each test coefficient; hence the support statement. Expansion of the definition gives (eq:theta).

On the coordinate plateau, the copied off-shell coordinate identity gives $⟨c[q^{\prime }],F^{\ast }\psi ⟩=⟨c[q],\psi ⟩$ for every scalar jet. Formal duality makes the right side $⟨F^{♯}c[q],F^{\ast }\psi ⟩$. Since $F^{\ast }$ is formally invertible, the tests agree. This uses the copied intertwiner, not a new proof of ramp covariance.