Lemma 45 ([Strictification of a fixed rest end]). Let (M,g,K) be a smooth connected four-dimensional exterior, complete with its nonempty compact smooth boundary included, with one Euclidean coordinate end and compact complement. Suppose …

The generic strictification lemma, Lemma 9 ([Strictification]), applies to this fixed rest exterior. Indeed the exact static tail has g-δ=O_d(r^ -2 ) for every d , and K is compactly supported. Choose any 1<q_*<2 to express its O_6/O_5 bounds, set n=4 , k=3 , and take β=δ_1 . The condition 0<δ_1<(1,q_*) is exactly the one required by that lemma. Its smooth nonnegative majorant can be chosen compactly supported: write it as b_0≥|K|_g . The complete mixed-boundary continuation, uniform core and tail estimates, exhaustion, and differentiated-decay proof of that lemma therefore give a positive smooth solution of -Δ_gφ=b_0√ | dφ|_g^2+r^ -6 +r^ -4-δ_1 , ∂_νφ=-1, φ⟶0. The exponent -6 is -2k . The all-order clause of the same lemma gives |∂^jφ|≤C_jr^ -2-j (j≥0). This application occurs after rest replacement, so it uses no higher derivatives of the original weakly asymptotically flat end. We now verify the geometric conclusions. Put F=b_0√ | dφ|^2+r^ -6 +r^ -4-δ_1 . The inner outward normal is -ν , and the finite-flux conclusion of the lemma is L_φ=_ T→∞ ∫_ S_T ∂_rφ\, dA_δ =-|∂M|_g-∫_MF\, dV_g<0. The geometric and Euclidean fluxes differ by O(T^ -2 ) on the static tail, by. Apply Lemma 40 ([Conformal constraint and boundary formulas]) with u=e^ sφ . It gives e^ 2sφ (μ_s-|J_s|_ g_s ) ≥3s(r^ -4-δ_1 -s| dφ|^2). The number B=_M r^ 4+δ_1 | dφ|^2 is finite: smoothness controls the compact core and the tail is O(r^ -2+δ_1 ) . Choose sB≤1/2 . This proves the global margin with c_s=(3s/2)e^ -2s\|φ\|_∞ >0 . On each boundary component, Θ_ K_s =e^ -sφ (Θ_K+3s∂_νφ) =e^ -sφ (Θ_K-3s)<0. Furthermore g≤g_s≤e^ 2s\|φ\|_∞ g. Thus completeness with the boundary included is preserved, and the tensor remains compactly supported. On the tail, R_g=0 , K_s=0 and R_ g_s =6e^ -2sφ (sr^ -4-δ_1 -s^2| dφ|^2) =O(r^ -4-δ_1 ). This proves the scalar decay and integrability there; the rest is smooth and compact. All differentiated metric bounds follow from , and momentum is zero because the tensor vanishes on a distant tail. The leading metric change is 2sφδ+O_1(r^ -4 ) , whose energy flux numerator is -6s∂_iφ+O(r^ -5 ) . The latter remainder has vanishing sphere flux. Division by 6ω_3 therefore gives E_ g_s -E_g=-(s)/(ω_3)L_φ⟶0. Every full cut has its three-dimensional area element multiplied by e^ 3sφ ; the cut class itself is unchanged. Taking infima yields A_*(g)≤A_*(g_s)≤e^ 3s\|φ\|_∞ A_*(g). The infimum is finite, since a distant coordinate sphere is admissible. Its convergence as s0 follows without an attained minimizer. Together these estimates prove every asserted output at this fixed rest datum. If it is (g_R,K_R) , every constant above may depend on R ; strictification is removed with R fixed.