Why classical smoothness is not a quantum conclusion

The coefficient kernels proved smooth here concern one explicitly modified off-shell classical functional. Their existence does not identify that functional with the sharp insertion studied in VII-B or IX-A. It also does not construct products with the selected interaction, prove a quantum gauge identity, or supply a causal algebra.

In addition, the regulator-thickened support bound is not exact causal dependence in the original metric arguments. A later causal construction would have to confront this fact explicitly.