Proposition

Corollary 10 ([Strict direction for compact variations]). On the original weakly asymptotically flat exterior, choose a smooth majorant B≥|K| equal to Cr^ -1-q far out and apply Lemma 9 ([The decaying Neumann problem and strictification]). …

Corollary 10 ([Strict direction for compact variations]). On the original weakly asymptotically flat exterior, choose a smooth majorant B≥|K| equal to Cr^-1-q far out and apply Lemma 9 ([The decaying Neumann problem and strictification]). Consider the tangent direction (g',K',α',D_m')=(2φg,φK,0,0) at purely electric data, transporting v by the metric square root. On an active ray μ_m+J_m(v)=0 , the derivative C'=2(μ_m+J_m(v))' satisfies C' =2k(-Δφ+K(∇φ,v)) +4(n-2)c_n^2φ|E|² ≥2kρ₀, C'/ρ_0⟶2k. At a marginal boundary θ₊'=-k . The energy derivative is finite, the momentum derivative is zero, and φ=O₂(r²-n ) .