Copied ramp premises

The classical source is the supplied manuscript {Landmark X-A: Classical covariance of one fixed-regulator switched relational scalar through order ${\kappa }^2$}, X-A.tex XA. The following are copied premises from that ramp manuscript. Its covariance proof is not repeated.

Let

Indices are moved with $\eta$. Fix $0<\epsilon <T$ and

Thus ${\sigma }_{\epsilon }$ is smooth and flat at the initial surface, and ${\sigma }_{\epsilon }^{\prime }>0$ on $0<t<\epsilon$.

Fix $g\in {C_c}^{\infty }((0,T)\times {\mathbb{R}}^3;[0,1])$, $P_g=\mathrm{int}\{g=1\}$, and the same nonempty relatively compact open $U$ as in X-A:

Fix a real label-space test $f\in {C_c}^{\infty }((0,T)\times {\mathbb{R}}^3)$. Put

The set ${\mathcal{D}}_f$ is compact.

The formal metric is $q=\eta +\kappa h+{\kappa }^{2}k$, with arbitrary smooth real symmetric $h,k$. Write $h={\mathrm{tr}}_{\eta }h$ inside the following coefficient formulas:

Set

The ramp auxiliary density is

It does not contain the interaction switch $g$.

The coordinate coefficients are

The retarded operator is understood on the smooth past-compact coordinate sources, or locally by finite propagation. These formulas reproduce the zero-data Cauchy prescription.

With

the linear coordinate kernel is

Equivalently, the derivative of the argument $x-z$ gives the same formula with a minus sign. The coefficient maps are smooth and continuous in smooth background topologies at fixed $\epsilon$. Smeared first-derivative kernels are smooth, with no initial hypersurface delta.