Definition: Presentation interface; XIV-J

XIV-J Jdescent works with a linear reconstruction $\mathsf{Rec}:{\mathcal{P}}_{\mathrm{int}}\to {\mathcal{D}}^{\prime }(W)$ with image ${\mathcal{K}}_{\mathrm{int}}=\mathsf{Rec}({\mathcal{P}}_{\mathrm{int}})$ and proves: a linear candidate $\tilde{\mathsf{M}}:{\mathcal{P}}_{\mathrm{int}}\to {\mathcal{D}}^{\prime }(W)$ defines a map on reconstructed kernels if and only if

in which case the induced map $\mathsf{M}(\mathsf{Rec}A)=\tilde{\mathsf{M}}(A)$ is unique; and agreement of $\tilde{\mathsf{M}}$ with canonical contraction on $U_{\mathrm{can}}$ proves (eq:descent) there only. XIV-J presents ${\mathcal{P}}_{\mathrm{int}}$ as the finite direct sum of the internal coefficient presentation spaces of the six operational entries; the summands are named and not displayed in the retrieved record (status open for the summands, identified for (eq:descent)).