Lemma 10 ([A positive shell in the strong-decay class]). Assume all hypotheses of Theorem 1.1 ([The enclosing-area Penrose inequality]) except possibly E>|P|_δ . For every λ>0 there is a smooth function U_λ≥1 , constant near S , such that …

For U>0 put a=U^ 2/p and (g_U,K_U)=(a^2g,aK) . Substitution of s=2p^ -1 U into Equation gives, as scalar and covector identities, a^2μ_U=μ-(2k)/(pU)Δ_gU, aJ_U=J+(2k)/(pU)K(∇U,·). Indeed, the scalar terms combine as -kΔs-(kp)/(2)|ds|^2=-(2k)/(pU)ΔU . Taking the new-metric covector norm introduces another factor a^ -1 , so a^2(μ_U-|J_U|_ g_U ) ≥μ-|J|_g+(2k)/(pU) (-Δ_gU-|K|_g|dU|_g). The boundary identity is aθ_ +,U =θ_++(2k)/(pU)∂_νU. Thus a factor constant near S preserves its expansion sign, and -Δ_gU≥|K|_g|dU|_g preserves the dominant energy condition. Choose 0<ξ<(q,1) . In a sufficiently distant part of the coordinate end set T(r)=r^ -p -r^ -p-ξ . There T>0 and T'<0 . Since p=n-2 , Δ_δr^ -α =α(α+2-n)r^ -α-2 . The original metric and tensor estimates therefore give -Δ_gT-|K|_g|dT|_g =ξ(p+ξ)r^ -n-ξ +O(r^ -n-q )>0 after increasing the radius. Choose a nondecreasing smooth switch β from zero to one in this region, constant near its endpoints, and put G(r)=-∫_r^∞β(t)T'(t)\,dt. Extend G constantly through the interior. Then G≥0 , it is constant near S , and G=T sufficiently far out. Because dG=β\,dT , -Δ_gG-|K|_g|dG|_g =β(-Δ_gT-|K|_g|dT|_g)-β'T'|dr|_g^2≥0. Use U_λ=1+λG . Equation and the boundary formula prove the required signs for every fixed λ>0 , without a smallness condition. The factor is bounded for each fixed λ and is at least one, so completeness with S included is preserved. The manifold and its unique coordinate end are unchanged. On the tail G=O_j(r^ -p ) at every fixed derivative order. Since q<p , the added metric tail is faster than the original decay, and g_λ-δ=O_6(r^ -q ), K_λ=O_5(r^ -1-q ). Moreover, Δ_gU_λ=O(r^ -n-ξ )+O(r^ -n-q ), |K|_g|dU_λ|_g=O(r^ -n-q ). These terms are integrable against dV_g r^ n-1 dr\,dω . The scalar and covector identities above, and the bounded conformal factors for fixed λ , show that the new constraint densities are integrable against dV_ g_λ . For the charges write f_λ=U_λ^ 4/p . Then f_λ=1+(4λ)/(p)G+O_1(r^ -2p ). In the metric flux of Equation, the leading change is -k∂_i(f_λ-1) . Terms involving g-δ or its first derivatives have integrated flux O(R^ -q ) , and the nonlinear remainder has flux O(R^ -p ) . Since _ R→∞ ∫_ S_R ∂_rG\,dA_δ =-pω_ n-1 , the current normalization gives E_λ-E =-2λ pω_ n-1 _ R→∞ ∫_ S_R ∂_rG\,dA_δ =2λ. If π=K-(_gK)g , its transformed tensor is exactly π_λ=U_λ^ 2/p π . The additional momentum flux in Equation is O(R^ -q ) , because U_λ^ 2/p -1=O(r^ -p ) and π=O(r^ -1-q ) . Thus P_λ=P , and the new ADM limits are finite. Finally, the admissible domains in Definition 2 ([Full enclosing cuts]) are unchanged. For their entire intrinsic boundaries, _ g_λ (∂D) =∫_ ∂D U_λ^ 2k/p \,dA_g ≥_g(∂D). This comparison includes every component and boundary-coincident portion. Taking the infimum proves the claimed full-cut area inequality.