Stationarity of the adopted total action is equivalent to the Einstein equation with the framework's coupling

(Einstein-Hilbert part for arbitrary compact support.) Let $g_s$ be an arbitrary family in the Statement, equal to $g$ off a compact set $K$, and put $h={\mathrm{\partial }}_s{g}_s{\mathrm{\mid }}_0$, $q^{ab}=-g^{ac}{g}^{bd}{h}_{cd}$. By [L3] choose finitely many smooth cutoffs ${\chi }_i$ with compact chart supports and ${\sum }_i{\chi }_i=1$ near $K$. Put $h_i={\chi }_{i}h$, so $h={\sum }_i{h}_i$ and $q={\sum }_i{q}_i$ with $q_i={\chi }_{i}q$. The derivative at $g$ of the top form $(S-2\mathrm{\Lambda }){\mathrm{v}\mathrm{o}\mathrm{l}}_g$ is linear in $h$ and its first and second derivatives: this follows directly from the inverse-metric, connection and Ricci variation formulas of First variation of the Einstein-Hilbert density under compactly supported metric variations, and of the timelike length functional clauses (a)-(d), applied locally: near any point one may multiply $g_s-g$ by a compactly chart-supported cutoff equal to $1$ on a neighborhood of that point. For small parameter values the resulting metric remains Lorentzian by compactness and agrees with the original family on that neighborhood, so the same local variation jets and formulas apply. Thus this derivative for $h$ is the sum of its derivatives for the $h_i$, including all derivatives of the cutoffs; none is discarded. Each $h_i$ is realizable by a Lorentzian family after restricting its parameter interval about $0$, by [L1]. For each such chart-supported family [L1] gives the integrated derivative ${\int }_M(G_{ab}+\mathrm{\Lambda }{g}_{ab})q_i^{ab}{\mathrm{v}\mathrm{o}\mathrm{l}}_g$, where chart and global integrals agree by [L3]. Differentiation of the original global action difference is legitimate by the same finite localization: in each compact chart support its smooth coefficient and parameter derivative are uniformly continuous on a compact parameter interval times that support, so difference quotients converge uniformly to the derivative and hence their integrals converge. Finite linearity and ${\sum }_i{q}_i=q$ therefore give $\frac{d}{ds}\mathrm{\Delta }{S}_{\mathrm{E}\mathrm{H}}{\mid }_0=\frac{c^3}{16\pi G_N}{\int }_M(G_{ab}+\mathrm{\Lambda }{g}_{ab})q^{ab}{\mathrm{v}\mathrm{o}\mathrm{l}}_g\mathrm{.}$ No single chart containing $K$ and no partition on the whole noncompact manifold are required.

(Matter part.) By [L2], $\frac{d}{ds}\mathrm{\Delta }{S}_m{\mathrm{\mid }}_{s=0}=-\frac{1}{2c}{\int }_M{T}_{ab}{q}^{ab}{\mathrm{v}\mathrm{o}\mathrm{l}}_g$.

(Total variation.) Adding steps 1.1 and 1.2 and using $c^3\mathrm{/}(16\pi G_N)=(1\mathrm{/}(2c))\cdot c^4\mathrm{/}(8\pi G_N)=(1\mathrm{/}(2c)){\kappa }^{-1}$ by [L5], $\frac{d}{ds}\mathrm{\Delta }{S}_{\mathrm{t}\mathrm{o}\mathrm{t}}{\mid }_0=\frac{c^3}{16\pi G_N}{\int }_M(G_{ab}+\mathrm{\Lambda }{g}_{ab}-\kappa T_{ab})q^{ab}{\mathrm{v}\mathrm{o}\mathrm{l}}_g\mathrm{.}$

(Coefficient vanishes.) Suppose $g$ is stationary in the sense of clause (a)(1): the derivative in step 2.1 vanishes for every smooth family with compactly supported variation. Every smooth symmetric test field $q$ compactly supported in a chosen chart is realizable by [L1], so ${\int }_U{F}_{ab}{q}^{ab}w{d}^{4}x=0$ for the continuous symmetric field $F_{ab}:=G_{ab}+\mathrm{\Lambda }{g}_{ab}-\kappa T_{ab}$ and every smooth symmetric compactly supported $q$; [L4] gives $F=0$ on every chart, that is, $G_{ab}+\mathrm{\Lambda }{g}_{ab}=\kappa T_{ab}$. Conversely, if the equation holds, the integrand of step 2.1 vanishes identically for every family, so the derivative vanishes and $g$ is stationary. This proves the equivalence (a).

(Identification of the presentations.) Clause (b) is now immediate: clause (iv) of Adopted classical gravitational action and variational field law states the variational principle whose equation form is, by step 3.1, exactly $G_{ab}+\mathrm{\Lambda }{g}_{ab}=\kappa T_{ab}$, the equation adopted in Adopted Einstein field equation in the direct-equation presentation clause (i); the prescribed-$T$ sectors adopt that same equation directly by clause (iii) of the action postulate. Since each model adopts exactly one presentation, no model uses both as independent premises.