F4: the interface of a prospective cancelling rule

Theorem thm:obstruction supplies the nonzero premise for this section. It obstructs ${{\Phi }^{(2)}}_{\mathrm{D}}$ alone. It does not obstruct every possible further observable completion.

A proposed rule on the existing multiplet would have to specify

and its dependence on the parameter, fields, coupling order, and Cauchy data. If it is intended to act on the carrier, preservation of the equation ideal and compatibility with the inherited product must also be supplied. Merely naming its auxiliary component does not meet these requirements.

At the level of the named lift, a cancelling interface under a uniform causal plateau hypothesis is

For a finite rule, the corresponding finite coefficient law must replace this infinitesimal display. If the proposed correction acts at a lower coefficient in the full relational expansion, that coefficient and its coupling order must be printed separately. It cannot be added to the already assembled second metric slot a second time.

There is an elementary slot restriction: the off-shell formula ${{\Phi }^{(2)}}_{\mathrm{D}}[\gamma ,\varphi ]$ has no independent $b,c,\bar{c}$ argument. Therefore

Changing only auxiliary or ghost variables cannot cancel (eq:main-value) in this formula. A more general proposal would need an induced action on $\gamma$ or $\varphi$, or a separately defined observable term involving the additional slots.

XII-A's free rule

does not supply (eq:cancelling-interface). It has ghost number one and is not the even finite parameter transformation used in the calculation. Substituting a compact plateau parameter for a free ghost also changes the equation domain; the nonzero compact residual intersection is absent by XI-B.

No cancelling map is selected here. In particular, $b$ is not replaced by an antifield, no contact term is chosen after the witness is known, and no symbol ${\hat{s}}_{\mathrm{int}}$ is introduced. The existence of a suitable additional completion is Unresolved. This separate status does not weaken the proved counterexample to invariance of ${{\Phi }^{(2)}}_{\mathrm{D}}$ alone.