The nonzero carrier class and its exact scope

This assertion does not require a new admission theorem for the cubic source. The only new kernel observation concerns the already displayed auxiliary remainder.

For a merely configuration-dependent instance of (CP), cancellation holds on that configuration. It must not automatically be asserted as an identity on the entire shell quotient. The stronger (UCP) used above supplies the uniform hypothesis needed for that quotient identity.

The compact shift $K\xi$ generally does not preserve ${\mathcal{C}}_{\mathrm{sol}}$. Consequently ${\Delta }_{\xi }$ here is the difference of two prescribed off-shell evaluations, tested on an initial shell configuration. We do not claim it is an automorphism or a derivation of ${\mathcal{E}}_H$. The nonzero result excludes invariance of the named classical lift on all such initial free configurations.