D3: the necessary cocycle and its consequence

Write ${\bar{\mathfrak{B}}}_{\xi }$ for the class of the admitted auxiliary row in ${{\mathcal{F}}^{\mathrm{sol}}}_{\mathrm{mc},\mathrm{pol}}$. It is an even linear map from ${\mathfrak{g}}_W$ to that vector space. Parameters and the external functions are not varied.