Requirement: C-I1–C-I9

The construction must satisfy:

1. Every output is admitted, including the required metric derivatives and directional descendants.

2. Order zero is classical multiplication.

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

4. The first stages are locked to (eq:firstL)–(eq:firstR).

5. The constant kernel is $U_c$, using one real common parent parameter and its declared observation compatibility.

6. No subtraction changes a fixed old product.

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

8. All finite relations between actual presentations are respected.

9. Every continuity claim has a named domain and proof; test-family continuity is not actual-stage hypocontinuity.