Proposition: One-parent numerical observation compatibility

For every declared smooth multiplier $a$ and smooth-coefficient differential operator $D$ on $X$,

In particular,

The parameter in these identities is the same $c$ used in the partial product.

Proof. These are distributional differentiation and multiplication identities for the fixed parent (eq:parent). The inherited numerical calibration supplies the declared analytical compatibility. No descent or gluing step changes the parent.