A1: the classical evaluation identity

No carrier quotient is needed for this argument. Its uniform quantifiers concern evaluations on the stated class; they do not turn evaluation into a homomorphism for ${\star }_H$.

The finite shift does not preserve the shell. If one tries to maintain the auxiliary relation after $\gamma \mapsto \gamma +K\xi$, the required auxiliary change is

This does not generally preserve the homogeneous free equations. The compact residual intersection is zero XIB. Therefore the theorem compares a prescribed shifted off-shell evaluation with its value at an initial shell configuration. It is not a nontrivial residual symmetry of ${\mathcal{C}}_W$.