Symbolic reports F-S1–F-S4

The geometric mode sum and time-split expansions reportedly matched the displayed Wick-square and stress formulas. The log keys 2_beta0_matches_lit, _beta0_xi16_matches_lit, and _beta0_general_xi_matches_FS were reported true. The corresponding identities have the independent proof in Lemma lem:expansion.

For $N=2,\dots ,6$, $\alpha =1/N$, and $\xi =0,1/6,1/4$, the full Cartesian image tensor reportedly matched exactly in 15 cases: 60 diagonal and 180 off-diagonal entries. All off-diagonal entries and the coincidence on-shell residual were reported zero.

As $1440{\pi }^2{r}^4(T^t{}_t,T^r{}_r,T^{\phi }{}_{\phi },T^z{}_z)$, the retained examples are

They also follow directly from (eq:T-real).

The ${\csc }^2$ and ${\csc }^4$ identities were reportedly checked exactly for $N=2,\dots ,8$. Proposition prop:quotient proves the stronger all-$N$ identities.

Conservation, trace, and $(g\square -\nabla \nabla )r^{-2}$ were reportedly checked symbolically. They are proved analytically in Lemma lem:ward and Theorem thm:friedrichs.