The Einstein-shell term, named

On the gauge-fixed free shell $-\square \mathcal{P}\gamma =0$ one has $A\gamma =-2D^{\ast }D\gamma =-2D^{\ast }b$, which vanishes exactly when $D^{\ast }b=0$; constant $b$ (the witness) and $b=0$ (the de Donder Einstein shell) are examples, and every linearized Einstein solution in any gauge satisfies $A\gamma =0$. For gauge-fixed-shell configurations with $D^{\ast }b\ne 0$ on $U$, Theorem thm:F3 retains the term ${\mathcal{A}}_1[\mathcal{P}{{\mathsf{E}}_0}^{\mathrm{R}}{{\mathcal{L}}^{\mathrm{dens}}}_{\xi }(-2D^{\ast }b);{\varphi }_0]$. It is an explicit classical functional; its value on the plateau is not computed here and is not asserted to vanish. This is XI-A's distinction between the gauge-fixed wave equation and the Einstein equation, met again at second order: the cubic source transforms as a density only modulo $A\gamma$.