Definition: The class (f,g,W)

A structural proposal in ${\mathfrak{K}}_{\mathrm{loc},\mathrm{R}}(f,g,W)$ consists of an even, ghost-number-zero, parameter-linear field rule

possibly accompanied by one fixed compensating functional $F$ of the metric and scalar slots, with the following restrictions.

1. The rule has parameter-local parts and, at most, one new retarded output

   $$r_{\xi }[\Phi ]={G_{\mathrm{gr}}}^{\mathrm{R}}{j}_{\xi }[\Phi ],\mathrm{supp}{j}_{\xi }[\Phi ]\subseteq S_{\xi }.$$

   The source is smooth, compact, polynomial in the fields and linear in $\xi$.

2. Every nonlocal component of $Y_{\xi }$ is a finite differential expression in this same $r_{\xi }$, multiplied by smooth polynomial field coefficients. Thus

   $${Y_{\xi }}^A={{\ell }_{\xi }}^A+{\mathsf{M}}^A[\Phi ]r_{\xi },A\in \{\gamma ,\varphi ,b\},$$

   with zero retarded output permitted.

3. The output in (eq:one-retarded) has zero past data and, on the slab, support in $J^{+}(S_{\xi })\cap M_T$. There is no second new retarded application.

4. A proposed cancellation effect must be an even, ghost-number-zero homogeneous quadratic polynomial with a compact polarized kernel bounded by $Z_2$. For a direct slot correction, that effect is

   $${K_{\xi }}^Y=⟨{{\mathcal{O}}^{\prime }}_{\gamma },{Y_{\xi }}^{\gamma }⟩+⟨{{\mathcal{O}}^{\prime }}_{\varphi },{Y_{\xi }}^{\varphi }⟩.$$

   For an integrated compensating-functional proposal, the effect is ${\delta }_{\xi }F$, where $\delta$ is the declared action below. The functional $F$ is fixed independently of $\xi$.

5. No new Einstein–Hilbert vertex, witness-dependent contact, additional field, or modification of $r_{\mathrm{D}}$ is permitted.

For a successful member, the directional rule must define an even derivation of the ordinary polynomial carrier,

preserve ${\mathcal{N}}_{\mathrm{sol}}$, and induce a representation with

Bars denote the inherited classical shell quotient, not a new quotient by ${\mathcal{N}}_W$.