The correct XI-G transfer statement

Distinct distributions can satisfy (eq:same-class). Conversely, proving $w_D-w_0\notin {\mathcal{I}}_{\mathrm{sol}}$ would remove this particular same-class transfer, but would not prove that the new class meets ${\mathcal{K}}_{\mathrm{mc}}$.

One differing pairing on a triple of free solutions would prove different classes, once both kernels and the compared evaluations exist. Agreement on only one witness family would not prove (eq:same-class), whose quantifier is all free triples.

Neither the source identity (eq:actualsource) nor the calculation of its value on (eq:witness) constructs $w_D$. XI-G applicability is therefore left unresolved, not categorically asserted or categorically excluded.

X-G's source-preserving variational obstruction is a separate statement. Changing the action or preparation is not, without a comparison theorem, an upgrade of the original model XG,XVIA.