Requirement: One-word binary interface

A proposed construction must specify which of the following clauses it verifies:

1. Every polynomial output belongs to ${\mathcal{P}}_{\Sigma }$, including the metric derivatives and directional descendants required by IX-A.

2. The order-zero coefficient is classical pointwise multiplication.

3. On every specified overlap with the old bulk domain, the coefficient maps agree with ${\star }_H$.

4. The first-stage left coefficient is the locked ${\mathcal{Q}}_{f,g}$; the right coefficient is its reverse contraction for the same covariance.

5. The fully contracted joint-test kernel is ${u_{\mu }}^{\rho }+cr$, with $c\in \mathbb{R}$. Declared observation compatibility transports that same parameter.

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

7. Unchanged regular approximants are not required to converge to the sharp insertions in ${\tau }_{\Sigma }$.

8. The maps respect every finite linear relation between actual presentations of their arguments.

9. Every continuity assertion has a named domain and a proof. Fixed-support test-family continuity is distinct from continuity or hypocontinuity on arbitrary carrier stages.