External inputs are not matter-defined clocks

For example, let

with $\chi$ smooth and nonconstant. Held fixed, ${\sigma }_{ϵ}$ has zero variation with respect to the metric and matter fields. Its differential vanishes on both plateaux. It therefore does not satisfy Definition def:clock as a dynamical clock on the full slab.

A strictly monotone transition segment may still reparametrize an interval. This is an externally prescribed parametrization, not the construction of a matter-defined clock.

For X-A-conv, the corresponding non-clock entry ${\mathsf{R}}_{\mathrm{conv}}$ means the fixed convolution prescription: kernel, cutoff, width, and placement. It is an operation, not a scalar clock field. The field dependence of an observable formed with that operation is not asserted to vanish.

These distinctions do not prove that no separately constructed smoothed scalar could be a clock. They prevent identification of the external regulator data with that scalar.