Lemma 64 ([The original invariant mass is the static ADM energy]). The metric g' is complete up to B₀ , has one strongly asymptotically flat end, and in its stationary rest coordinates g'_ij -δ_ij =O₂(r^-1 ), E'=O₁(r^-2 ), E_ADM …

Smoothness through B_0 , compactness of the core and completeness of the end follow from Proposition 60 and Lemma 62 ([The maximal hypersurface as an orbit section]). The symbol bounds for the stationary end in Proposition 57, together with Equation, give the stated strong asymptotic flatness and field bounds. Fluxes were checked in the proof of Proposition 63 ([Staticity and vanishing of matter]). We spell out why the mass is that of the original data. In the unimproved stationary coordinates the spacetime perturbation of its limiting constant Minkowski metric has differentiated decay O_2(r^ -q_0 ) , q_0>1/2 . It is electrovacuum near infinity. The skew-slot flux in Lemma 14 ([Asymptotic Lorentz flux]) therefore has spacetime divergence O(r^ -2-2q_0 )+O(r^ -4 ). Stokes' Theorem between cuts in the original asymptotic plane and cuts in a plane orthogonal to the limiting Killing direction has error O(R^ 1-2q_0 )+O(R^ -1 )→0 : the connecting chains have volume O(R^3) and remain at radii comparable to R . The same estimate permits replacing comparable large cuts by coordinate spheres. With the positive- K convention, Lemma 14 ([Asymptotic Lorentz flux]) pairs a translation (b^0,b^i) with b^0E+∑_i b^iP_i . Here the limiting unit Killing translation furnished by Proposition 50 is b^0= E m, b^i=-(P_i)/(m). Its flux on the original section is therefore b^0E+∑_i b^iP_i =(E^2-|P|^2)/(m)=m. On the orthogonal rest plane that same flux is its ADM energy. Its spatial metric and h_s differ by the quadratic shift term, which is O_1(r^ -2q_0 ) in this chart and contributes O(R^ 1-2q_0 ) to the ADM flux. Finally the stationary harmonic correction has spatial coordinate displacement O_2(r^ 1-q_1 ) , q_1>1/2 , as in Proposition 57. Its leading metric change is a symmetric flat gradient 2∂_ (i a_ j) . The ADM divergence of this term is Δa_i-∂_idiva , whose divergence vanishes identically. Extend a smoothly across the inside of a coordinate sphere; the flux of this leading term is exactly zero by the Euclidean divergence theorem. Products of coordinate and metric errors have flux O(R^ 1-2q_1 )→0 , with a smaller decay exponent chosen if necessary. The energy is consequently still m in the strong asymptotic coordinates. This argument uses the original invariant ADM flux, not the mass of a deformation.