Theorem: The defect is a finite-jet functional

Let $T$ extend $T^{\circ }$, and let $K$, $\varrho$, $K^{\prime }$, $k$ be as in Lemma lem:flat. For $a\in H$, the restriction of ${d_0}^T(a)$ to $\mathrm{int}(K)\times V_q$ depends only on the $k$-jets along $Z$ of the functions ${\beta }_{\gamma }(a)$, $|\gamma |\le r$. In particular:

1. flat-annihilator relations carry no defect for any extension $T$;

2. a nonzero defect is produced only by jet-detected relations.

Proof. Let $a,a^{\prime }\in H$ have ${\beta }_{\gamma }(a)-{\beta }_{\gamma }(a^{\prime })$ with vanishing $k$-jet on $Z$ for all $\gamma$. Since ${\beta }_{\gamma }(a-a^{\prime })\in \mathrm{Ann}(T^{\circ })$, Lemma lem:flat gives $⟨{\beta }_{\gamma }(a-a^{\prime })T,\psi ⟩=0$ for $\psi \in \mathcal{D}(X)$ with $\mathrm{supp}\psi \subset K$. For $\varphi \in \mathcal{D}(U)$ supported in $\mathrm{int}(K)\times V_q$, each ${{\partial }_q}^{\gamma }\varphi (\cdot ,0)$ is such a $\psi$, so $⟨{d_0}^T(a)-{d_0}^T(a^{\prime }),\varphi ⟩={\sum }_{\gamma }(-1{)}^{|\gamma |}⟨{\beta }_{\gamma }(a-a^{\prime })T,{{\partial }_q}^{\gamma }\varphi (\cdot ,0)⟩=0$. Taking $a^{\prime }=0$ with a flat $a$ gives (i); (ii) is (i) together with XV-D's result on universal relations.