Why no additional boundary field is hidden in the proof

The variables

are used only through the identity

Their compatibility is inherited from the original smooth configuration $h$. No independent integration over boundary data or source variables is introduced.

Similarly, $S_{\epsilon }h$ and $S_{\epsilon }\varphi$ in (eq:Beps) are ordinary smoothings of the original fields. The resulting $B_{f,\epsilon }$ are regular elements of the old domain and serve only to test continuity at the exact sharp kernel. The proof does not declare any fixed $\epsilon >0$ observable to be the sharp insertion.