Remark: The sign difference can be detected locally

The sign is not merely a choice of names. Choose a compact $\xi \in {\mathfrak{g}}_U$ which near an interior point is

and put $\gamma =0$. In that neighborhood, writing $F=\lambda (t-t_0)(x^2-{x_0}^2)$ gives $s_{\xi }=-\partial F\otimes \partial F$. At the central point, $(A{s}_{\xi }{)}^{11}=(A{s}_{\xi }{)}^{33}=-2{\lambda }^2$. Equation (eq:finite-source) gives $J_3[K\xi ]=-A{s}_{\xi }$. The proposed positive-two-jet response $s_{\xi }-K{E}_0^{\mathrm{R}}D{s}_{\xi }$ has $L$-image $A{s}_{\xi }$, whereas the prescribed response has $L$-image $-A{s}_{\xi }$. They cannot agree. This diagnoses an incompatible response law, not a failure of kernel admission.