Theorem: Necessary support for a larger correction rule

If $Z\subset W$ is closed and a correction family supported in $Z$ makes ${t_{\delta }}^{\zeta }+c_{\delta }$ converge weakly, then

Proof. A point of the displayed stratum outside $Z$ has a small neighborhood disjoint from $Z$. The same nonparallel support isolation applies there. Use the preceding normal-polynomial witness with a nonnegative tangential bump.

All three needed normal derivatives must hit the vanishing polynomial factors. The localization contributes its positive value on the stratum; no cutoff derivative survives. Because $f>0$ and $\kappa <0$, the coefficient is nonzero. The correction pairs to zero with that fixed test, a contradiction.