Lemma: An asymptotic coefficient needs a connecting map

Let $t_{\delta }={\delta }^{-r}C+R_{\delta }$ in a topological vector space, where ${\delta }^r{R}_{\delta }\to 0$. If $B$ is a fixed continuous linear comparison map, then

Nonvanishing of $C$ does not imply nonvanishing of $BC$. Without a specified comparison map and matching regulated families, this expansion determines no leading coefficient for a different observable.

Proof. Apply continuity of $B$ to ${\delta }^r{t}_{\delta }=C+{\delta }^r{R}_{\delta }$. A nonzero $C$ may belong to $\ker B$, including when $B$ is a nonzero projection. Thus the source coefficient need not survive the target operation. An expansion in the source space, by itself, contains no equality defining a distinct target family.