Displayed templates for the three remaining roles

The tensor product in (eq:contacttemplate) is meaningful for a singular tangential spectator: it combines independent $s_I$ and $r_I$ variables. Smooth multiplication and the displayed finite differential operator are then meaningful local distributional operations. This supplies a coherent local formula without pretending that a singular spectator is a smooth function.

A spectator whose inherited realization requires further products or pushforwards is not recovered merely by this local notation. Those products remain subject to the source membership condition. Nor is a coordinate expression on one collision chart a proof of compatibility on intersections of several collision charts.