F4: what the sum is not

Equation (eq:plateauzero) is a classical coefficient identity on off-shell plateau parameters $\xi \in {\mathfrak{g}}_U$, for a free scalar held fixed. It is not nilpotence of an interacting differential, not a quantum master equation, and not a quantum Ward identity. No ${\hat{s}}_{\mathrm{int}}$, antibracket, or antifield term is constructed; XI-B's antifields remain slots.

The co-transformation ${\mathcal{L}}_{\xi }\gamma$, now generated by the response rather than inserted by hand, is still not free: on the witness its spatial wave operator contains $-a^{\prime \prime }{\delta }_{ij}$. It is not an automorphism of ${\mathcal{A}}_0$, and the ghost rule $s_0\gamma =\frac{1}{2}Kc$ is not that shift.

Because ${\mathcal{C}}_3$ is not admitted (Theorem thm:F1), there is no element of ${\mathcal{E}}_H$ whose symbol is ${{\mathcal{A}}_2}^{(3)}$, and no quantum version of (eq:ledger) exists in the frozen algebra. The ledger is classical.

The two affine numbers are untouched: the bare insertion still fails by $S_{\epsilon }(t)$, the complete coordinate word still leaves $\epsilon /2$. Neither enters the cubic ledger; neither is erased by it.

No mixed loop is evaluated; no state is selected; no $\epsilon \to 0$ limit is taken; ${\mathcal{J}}_g({\varphi }_m)=\varnothing$ stands.