Lemma: Separation of the action identity from the observable identity

On ${\mathfrak{F}}_W$ the following hold simultaneously and are compatible:

1. $\left(S_{\mathrm{ext}},S_{\mathrm{ext}}\right)=0$ and ${s_{\mathrm{ext}}}^2=0$ on ${\mathfrak{A}}_W$ (Theorem thm:sext);

2. $s^2\mathcal{O}=0$ off shell (Proposition prop:kinematic);

3. for $c=2\vartheta {\xi }_{\ast }$ on the detecting tuple, $s_0\mathcal{O}(u_{\ast })={\mathfrak{B}}_{{\xi }_{\ast }}(u_{\ast })\vartheta =-8I\vartheta \ne 0$, where $s_0\mathcal{O}$ is the ${\kappa }^0$ part of $s_{\mathrm{ext}}\mathcal{O}=s\mathcal{O}$ (eq:relO).

Neither (a) nor (b) implies that $s\mathcal{O}$ vanishes on ${{\mathcal{C}}_{\mathrm{sol}}}^{\mathrm{bos}}$; (c) shows that it does not. The master equation constrains the action; it does not decide the invariance of the given lift.

Proof. (a) and (b) are proved above; (c) is copied from Input statement input:ledger. Compatibility: (a) is a statement about $S_{\mathrm{ext}}$ and about $s_{\mathrm{ext}}$ on ${\mathfrak{A}}_W$; (b) is the statement that $s\mathcal{O}$ is $s$-closed, which is automatic for an image of $s$; (c) is the value of that image at one point of ${{\mathcal{C}}_{\mathrm{sol}}}^{\mathrm{bos}}$, outside the intersection (eq:intersection) because $c\ne 0$ there. No statement among (a)–(c) cancels another.