The de Donder variation from the fermion

Under the single-field law ${\gamma }^{\prime }=\gamma +K\xi +\kappa s_{\xi }+O({\kappa }^2)$, $s_{\xi }:={\mathcal{L}}_{\xi }\gamma +\frac{1}{2}{\mathcal{L}}_{\xi }K\xi$ (with $\zeta =0$), Lemma lem:gf and $DK\xi =\square \xi$ give

The order-${\kappa }^0$ term is XI-A's and XI-B's linear breaking; the order-$\kappa$ term is the companion the sheet asked for.