Lemma: Localized scalar crossed distribution

On $\mathcal{O}$,

Choose $\zeta \in {C_c}^{\infty }(\mathcal{O})$ equal one on a neighborhood of $\mathfrak{f}(\overline{U})$, and define

For sufficiently small $\delta >0$,

on a neighborhood of $\mathfrak{f}(\overline{U})$, with equality of all relative derivatives.

Proof. On $\mathcal{O}$,

By (eq:scalar-kernel-identities),

This proves (eq:localC) as an identity of distributions in independent relative variables.

The compact relative image has a positive distance from the region where $\zeta \ne 1$ or the time signs change. Let $R_{\rho }$ bound the Euclidean radius of $\mathrm{supp}\rho$. For sufficiently small $\delta$, every translated convolution test centered near this image has support in the region where the distributional identity holds and $\zeta =1$. The difference convolves to zero there. The same support argument applies to every profile derivative, proving the jet equality.