The regulator does not move the initial surface

The data of ${\mathsf{X}}_q$ are prescribed on $\Sigma =\{t=0\}$ for every $\epsilon$. What ${\sigma }_{\epsilon }$ changes is the coefficient of the coordinate operator on $\{0<t<\epsilon \}$. Lemma lem:kernel shows the consequence for the linear kernel: the hypersurface delta of the sharp prescription is replaced by the smooth term ${\sigma }_{\epsilon }{C}^{\ast }b$, and in the multiplied-equation variant by ${\sigma }_{\epsilon }{C}^{\ast }b+{\sigma }_{\epsilon }^{\prime }\mathsf{B}(b)$. No statement about $\epsilon \to 0$ is made.