The finite action being retained

The following gives a self-contained coefficient specification of the X-I-ST interface.

For a formal metric $p$, let $J_p=\sqrt{-\det p}$. Use the curvature convention

with the metric Christoffel formula, and put

Let

Define the local variational action

The sole negative-order coefficient is a divergence,

It is removed as a variational class before taking nonnegative coefficients.

Adjoin canonical antifields with

and fix

For a left Hamiltonian differential, use the signed lift

Then $({\mathcal{H}}_s,{\Phi }^A)=s{\Phi }^A$. In particular the antighost term is $-\int {\bar{c}}^{\ast \mu }{b}_{\mu }{\mathrm{d}}^{4}x$.

Retain exactly the nonnegative coefficients through order three:

The order-three gravitational coefficient uses the order-five coefficient of $\mathcal{E}[q]$. No inversion of a nilpotent $\kappa$ is performed.

Fix the inherited nonnegative, nonzero profile

and retain

An ordinary polynomial representative of this expression is denoted ${\tilde{\mathcal{S}}}_{\lambda }$.

Local actions are understood modulo horizontal divergences; ordinary products of compact local functionals belong to the classical multilocal algebra ${\mathcal{F}}_{\mathrm{ml}}(U)$, not necessarily to the single-integral local subspace FRBV,XHST,XIST.