Why a complete observable is a different problem

Let $F_f=\int f\varphi$. For a complete candidate beginning $F_f+\kappa {\mathcal{B}}_f$, the first closure coefficient could involve both

Theorem thm:conditional does not exclude such a cancellation. The production sheet's equation for ${\mathcal{B}}_f$ alone has no leading $F_f$ term on which that correction can act.

There is a further Cauchy-data distinction. For a compact plateau parameter $c=2\xi$, X-A's preflight gives $L_{\epsilon }\mathcal{R}c=c/2$. A proposed scalar co-transformation proportional to $c\cdot \partial \varphi /2$ can then participate in a complete observable's coefficient cancellation.

The free witness above is not such a plateau parameter. Its response is $S_{\epsilon }(t)$, not $t$, because the coordinate data are frozen while the free ghost has nonzero initial time derivative. X-A never asserted covariance for those ghost data. Thus neither its classical theorem nor ghost number bookkeeping proves the requested bare closure.