Proposition: One compatible parent parameter

Fix $\rho ,\mu$ and define

If observation compatibility is imposed by the actual operation $aD$, the compatible descendant family is

with the same $c$ for every declared derivative and multiplier. An infinite-dimensional residue orbit therefore does not imply one independent normalization constant per derivative.

Proof. This is the linear image under $aD$ of the one-parameter affine parent family. Independently choosing a new constant for a derivative would change, rather than follow from, the stipulated compatibility identity.