Proposition: Numerical observation compatibility

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

In particular,

The inherited numerical finite-part estimate has test-order loss $64\to 64+\mathrm{ord}D$.

Proof. The distributional identities follow from differentiation and linearity. The quantitative descendant estimate is the explicit IX-B transfer premise reported in IX-C, not a new ultraviolet estimate.