Individual response and first-component jets

Let

Define insertion by its action:

On the outer cone,

Put $f_N=1+a^0+\beta$, ${\zeta }_N=\zeta {|}_N$. Then the individual response coefficient is

in the second component. The cutoff has ${\partial }_v\zeta =0$ on $N$, so there is no additional cutoff jet. Tests $v\psi (\tau )$ detect this nonzero order-one jet.

If ${\Phi }_2=0$, the complete first component satisfies

with the same bounded scale remainder, where

Its normal-jet representation is

The corresponding first cutoff derivatives and mixed derivative vanish on $N$. Tests $v\psi$ and $r^{1}v\psi$ independently detect the two jets.

These are statements about individual coefficients. They do not override the cross-sector identity (eq:tip-cancellation) for the second component.