43. Posterior certificates and hyperedge compression

Let the fine recursion be \[\begin{equation} \tag{PRIQ.9.1} \widehat\mu_k^{\mathrm f} =\Phi_k^{\mathrm f}(\mu_{k-1}^{\mathrm f}), \qquad \mu_k^{\mathrm f} =\mathsf B_k^{\mathrm f}(\widehat\mu_k^{\mathrm f}), \end{equation}\] and the coarse recursion be \[\begin{equation} \tag{PRIQ.9.2} \widehat\mu_k^{\mathrm c} =\Phi_k^{\mathrm c}(\mu_{k-1}^{\mathrm c}), \qquad \mu_k^{\mathrm c} =\mathsf B_k^{\mathrm c}(\widehat\mu_k^{\mathrm c}). \end{equation}\] Define the discrepancies before and after each update by \[\begin{equation} \tag{PRIQ.9.3} e_k^- =\left\lVert C\widehat\mu_k^{\mathrm f}-\widehat\mu_k^{\mathrm c}\right\rVert_1, \qquad e_k^+ =\left\lVert C\mu_k^{\mathrm f}-\mu_k^{\mathrm c}\right\rVert_1. \end{equation}\] At the initial time, set \[e_0^+=\left\lVert C\mu_0^{\mathrm f}-\mu_0^{\mathrm c}\right\rVert_1.\]