Proposition: Pure-gauge diagnostic and the order boundary

Let $\xi \in {\mathfrak{g}}_p$, and take the complete two-jet

Then in the degree-one free sector,

Consequently the incomplete second coefficient in (eq:truncation) is

which need not vanish.

Proof. The inherited laws give $w=\xi$ and $v=\frac{1}{2}{X}_{\xi }\xi$. Hence the bare second coefficient is $\frac{1}{2}\int f{X}_{\xi }^2\psi$. Theorem thm:ward, compared with $h=0$, gives the first identity. Applying

to $a=D_{\xi }f$ gives $M=-\mathbf{\varphi }({D_{\xi }}^{2}f)$, which is the stated integral.

For nonvanishing, in the massless case choose $\xi =a(t,\mathbf{x}){\partial }_t$ compactly supported in $U$, with $a{\partial }_{t}a\ne 0$ somewhere. Choose an admissible observation test supported in a small region where it has fixed sign. Against the smooth free solution $u=t$, ${X_{\xi }}^{2}t=a{\partial }_{t}a$, so the pairing is nonzero. This diagnostic applies when those observation choices are made; it is not a nonvanishing assertion for every previously fixed test.