The advanced root and observation domain

The advanced observation root and the retarded internal propagations below have independently specified source and support domains. Their coexistence is not a single interchangeable prescription. Lemma lem:root is a theorem for this application, not an identification of a predecessor root.

Choose a compact set $K$ containing neighborhoods of $b_{\ast },y_{\ast }$. Choose a real compact smooth $g$ equal to one near $K$. The admissible observations are real $f$, compactly supported in the interior of $K$, with

This class is nonempty. Indeed, choose $f_0=1$ near $b_{\ast }$ and a compact smooth $\theta$ equal to $t-b_{\ast }^0$ there. Then

For $f=f_0+c{P}_m\theta$, $f(b_{\ast })=1$, and

Avoid the one cancelling value of $c$. After this choice, $F$ is still determined solely by (eq:root), not independently prescribed.

Shrinking $U$ preserves these nonvanishing conditions near $b_{\ast }$, and $g=1$ near $b_{\ast },y_{\ast }$. This proves support activity, not survival of any complete asymptotic coefficient.