An explicit finite table from actual coupling jets

We now make the finite encoding nonvacuous. Specialize to $m=0$, a unit-integral $f\ge 0$ supported in a small cylindrical plateau $U$, and

where $\chi =1$ near the spatial projection of ${\mathcal{D}}_f$ and $q=1$ near the observation times. All cutoffs are compactly supported within the laboratory.

Let ${\varphi }_0=t$, let $k=2\sigma \eta$ be the tensor of (eq:sourced-example), and for real $\lambda$ put

Here

The metric coefficients satisfy (eq:metric-equations) for every $\lambda$: $h_{\lambda }$ is constant, and $G^{(1)}[k]=T^{(0)}[t]$.

Set

For constant conformal $h_{\lambda }$, direct substitution in (eq:AB) gives

On ${\mathcal{D}}_f$, zero-data uniqueness gives

For example, in the temporally homogeneous portion, $V_g[h_{\lambda }]u=-2\lambda (q\dot{u}{)}^{\prime }$; inserting $u_{1,g}$ proves the $4{\lambda }^{2}J$ term. The cutoff outside ${\mathcal{D}}_f$ cannot affect these evaluations.

Define the finite real calibration constants

Thus

These are evaluations of actual members of the named coupling jet, not merely off-shell placeholders.

For a larger-support generator ${\xi }_{\mathrm{out}}={\chi }_{\mathrm{out}}{\partial }_t$, with ${\chi }_{\mathrm{out}}=1$ near $\mathrm{supp}g$,

Choose also any nonzero ${\xi }_{\mathrm{in}}\in {\mathfrak{g}}_U$. The second-order covariance residual of its complete metric pullback is exactly zero.

Use four samples ${\lambda }_a=a$, $a=0,1,2,3$, and four settings $s=0,1,2,3$:

Rows $0,1$ are descended evaluations. Row $2$ records the failure of a different, larger-support identification; it is not relabelled as a descended observable. Row $3$ records an identity already proved for the restricted group.