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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 ) .

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证明2026-10-21

Lemma 9 ([The decaying Neumann problem and strictification]). Assume g-δ=O_2(r^…Lemma 8 ([Compact inverses with the required moments]). Fix nested compact…观点

A smooth radial majorant on the end can be patched to a sufficiently large smooth function on the core; it need not reproduce higher derivatives of K . Differentiate and . The derivative of the overall factor e^ -2δφ multiplies the active value zero. The drift term is O(r^ -n-q ) , and the electric term divided by ρ_0 tends to zero. The assertions now follow from , , and .

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