Descent, scalar tests, and coherence equations

Part I does not evaluate this test. Part II does.

Let ${\mathsf{L}}_j,{\mathsf{R}}_j$ denote proposed second-stage coefficient maps on the left and right intermediate spaces, with order zero classical. Write $P=B_f{W}_g$, $Q=Q_{f,g}$, $P^{\prime }=W_g{B}_{f^{\prime }}$, $Q^{\prime }={Q^R}_{g,f^{\prime }}$.