The widened class, frozen before extension

This widening is a new definition. It does not retroactively modify XIII-B. Nor does it assert that every formal contact expression defines a distribution.

The spectator coefficient in a partial-diagonal change can remain singular in its tangential variables. For example, a contact at $y=v$ can retain a retarded coefficient linking $v$ to $x$. Calling all such coefficients smooth would exclude the graph that motivated the widening. The smoothness requirement concerns the external weights, not the entire spectator distribution.

Differentiating a diagonal distribution can differentiate its multiplying weights upon pushforward. This is not permission to add arbitrary independent higher-jet coefficient functions.