Corollary: Classical pairing confirmation

The normalization to the inherited classical relational coefficient, together with X-A-ST's on-shell classical identity, independently confirms $\mathbf{\varphi }({e^{\prime }}_2-e_2)=0$ on ${\mathcal{G}}_p$. This is not the proof of record for Theorem thm:ward.

Proof. By Proposition prop:completed, pairing $e_2$ with a smooth free solution gives the complete classical second coefficient. The inherited classical identity therefore makes ${e^{\prime }}_2-e_2$ annihilate every such solution.

To recall the relevant separation argument, let $d\in \mathcal{D}$ annihilate every spatially compact free solution. Since $({\mathsf{E}}_{\mathrm{A}}-{\mathsf{E}}_{\mathrm{R}})a$ is one for every $a\in \mathcal{D}$,

Then ${\mathsf{E}}_{\mathrm{A}}d={\mathsf{E}}_{\mathrm{R}}d=:v$, whose support lies in the compact causal diamond

Thus $v\in \mathcal{D}$ and $P_{m}v=d$. Applying this to $d={e^{\prime }}_2-e_2$ proves the claimed free-algebra equality.