Input statement: XII-F's remainder release

With the inherited roots

put

For compact plateau parameters and a free scalar,

where

The last tensor is supported in $\mathrm{supp}\xi$. The companion has no ramp-layer contribution and vanishes if

There are admissible data satisfying (CP) for which ${\mathfrak{B}}_{\xi }$ is strictly nonzero. The corresponding admitted lift has nonzero class under $Q_H$. An additional cancelling completion was not constructed XIIF,XIID.