Remark: What the trace records

The map $\mathsf{b}$ records the coefficient responsible for the delta contribution in (eq:qa). It is not an independent boundary field. It is computed from the original off-shell functional.

It does not classify every surface singularity. For example, a kernel with vanishing $b{|}_{\Sigma }$ but nonvanishing normal derivative can still have a step singularity. The present topology uses the trace needed for the stated nonconvergence theorem; it is not claimed to be a complete boundary microlocal topology.