Remark: Sharp-step comparison, not a transfer theorem

The preceding proof uses smooth multiplication by ${\sigma }_{\epsilon }$. A sharp step has $\mathrm{WF}(\theta (t))=\{(t=0;\lambda \mathrm{d}t):\lambda \ne 0\}$; it is not a smooth multiplier in the split calculus.

The two sharp prescriptions must also be distinguished. The sharp action-level adjoint corresponding to the divergence-form coordinate source is schematically $\theta C^{\ast }b$. The multiplied-equation prescription has adjoint $C^{\ast }(\theta b)=\theta C^{\ast }b+\delta (t)\mathsf{B}(b{|}_{t=0})$. A nonzero boundary coefficient can therefore introduce a timelike conormal; the latter prescription explicitly has a delta term. The fixed ramp has neither sharp multiplier.

This explains the removal of the sharp initial-conormal mechanism at fixed $\epsilon$. It does not identify the ramp object with VII-B's insertion, prove a sharp-limit theorem, or construct a retarded product with a quantum metric.