Proposition: Conditional class-obstruction transfer

Suppose $u_0\in \mathcal{K}$ satisfies

If a constructed prepared kernel $u_{\mathrm{D}}$ obeys $u_{\mathrm{D}}-u_0\in {\mathcal{I}}_{\mathrm{sol}}$, then

A proof that $u_{\mathrm{D}}$ belongs to a different class removes this transfer argument; it does not establish that the new class intersects ${\mathcal{K}}_{\mathrm{mc}}$. Admission of that class requires a separate proof.

Proof. Since ${\mathcal{I}}_{\mathrm{sol}}$ is a subspace,

The assumed empty intersection therefore transfers.

The calculation uses equality of classes, not a criterion for a different class. Merely distinguishing classes contains no wavefront or representative-existence assertion, so it supplies no admission conclusion.