Requirement: Copied IX-C interfaces

A completed construction must satisfy:

1. Every output belongs to ${\mathcal{P}}_{\Sigma }$, with all required metric derivatives and directional descendants.

2. Order zero is classical pointwise multiplication.

3. Every specified old-bulk overlap agrees with ${\star }_H$.

4. The first stages are locked to (eq:Q) and (eq:QR).

5. The constant kernel is $U_c$, with one real parent parameter transported by declared observation compatibility.

6. No subtraction changes an already fixed old bulk product.

7. Unchanged approximants need not converge sharply in ${\tau }_{\Sigma }$.

8. Every finite relation between actual presentations is respected.

9. Every continuity claim has a named domain and proof. Test-family continuity is not carrier-stage hypocontinuity.