Proposition: Known partial assembly

The six-row family with $5z$ removed satisfies

with the same $O_H({\delta }^{-2})$ bound for the scale derivative of the remainder.

Proof. The $47$ adjoint becomes

Only $D^2{\partial }_{r^0}M$ contributes at inverse-cubic order. Multiplication by $-\mathrm{i}$ gives $C_{47}$.

The $5a$ adjoint becomes

Its first term gives $C_{5a}$. No estimate divides by the inner radius: after the adjoint, inner derivatives act on the smooth $r$-weight. The cone measure is finite through its vertex.

In (eq:response-adjoint), only $-f{D}^{2}M$ contributes at inverse-cubic order, giving $C_{\mathrm{resp}}$. Every other response term is $O({\delta }^{-1})$.

The first-component action (eq:first-action) is also $O({\delta }^{-1})$. Uniform master estimates and finite compact seed measures give the stated remainder bounds, including their direct scale derivatives.