The question and the three endings

The proposed implication is

None of its arrows is supplied merely by the preceding arrow. In particular, ghost number zero and a compact microcausal kernel are necessary bookkeeping facts, not closure theorems.

The production sheet permits three endings.

1. Constructed. A named quantum extension admits the required symbols, a differential is defined and proved nilpotent through the declared order, and the admitted insertion is closed.

2. Proved obstructed. A stated interface is excluded by an explicit incompatibility. Its quantifiers must identify whether admission, nilpotence, or closure is excluded.

3. Unresolved. An indispensable extension, product, antibracket, or identity remains unconstructed.

Here admission remains unresolved, so the overall ending is the third. Nevertheless, one can prove a necessary-condition obstruction without pretending that an interacting differential has been constructed. Such a result excludes a specified closure request if admission respects its required classical limit.

This distinction matters because the production sheet asks for the closure of the bare dressing ${\mathcal{B}}_f$. That is not the same as closure of a complete perturbative observable whose leading term is $\int f\varphi$. The two coefficient problems have different possible cancellations. Section sec:freeclosure makes this difference explicit.

The standard separation between regular polynomial generators, microcausal functionals, and their quantization is also relevant FR. It does not itself supply the extension required here. Likewise, a BV construction requires more than names for antifields Rejzner. We use neither reference as a substitute for a missing construction.