Smooth root tests and their limited asymptotics

For the reported compact root $a=a_{f,g}[\varphi ]$, this reproduces the fixed-data estimate of XVI-E. It does not establish:

• the same estimate for a frequency-dependent $\varphi$;

• uniformity as $\epsilon \downarrow 0$;

• pointwise decay of the response;

• a kernel wavefront bound or quantum admission;

• identification with an independently constructed auxiliary completion.

In particular, rapid decay of a smooth compact test pairing can coexist with a nonzero source or response contraction.