Corollary: Compatibility with the frozen obstruction

Suppose the matching frozen XI-G kernel $u_0$, for data on which its obstruction theorem applies, satisfies

Then, under Theorem thm:admission,

Thus the admitted prepared class is not that excluded frozen class.

Proof. If the difference were in ${\mathcal{I}}_{\mathrm{sol}}$, the affine classes would coincide. The admitted representative $t_{\mathrm{D}}$ would then contradict the frozen obstruction.