Proposition: Positive upgrade incompatibility

The data (eq:recipdata) give

The background can be supported in the retained regular reserve, and the detecting free inputs can be replaced by free inputs with compact Cauchy data.

Proof. On a free input, (eq:typed) gives $Q_{w}u=-{\mathsf{P}}_{\mathrm{gr}}{\mathcal{L}}_{{\xi }_{\mathrm{on}}}^{\mathrm{cov}}u$. The relevant Lie components are

There are also $0i$ components but no spatial–spatial components. For tensors of this form, $({\mathsf{P}}_{\mathrm{gr}}a{)}^{00}=-\square a_{00}/8$. Spatial Laplacians integrate to zero, and

Thus $W=(5M_{\chi }/8)\int {\sigma }_{\epsilon }{\psi }^{\prime \prime }\mathrm{d}t$. Two integrations by parts yield (eq:W).

Choose an observation event $(t_o,{\mathbf{x}}_o)\in \mathcal{O}$. Since $t_o>2\epsilon$, a small spatial ball $B$ satisfies

Choose ${\chi }_0$ supported in $B$. For compact-Cauchy replacements, choose initial data equal to those of $u,v$ on the initial causal shadow of a neighbourhood of $\mathrm{supp}w$, and compactly cut off the data outside a larger ball. Finite propagation preserves $u,v$ near $\mathrm{supp}w$, so the local pairing is unchanged BGP.