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Fix 12<β<(q,1) and work on the annulus A= 1<|y|<4 with x=Ry . The scaled fields are g_R(y)=g(Ry), k_R(y)=RK(Ry), and likewise for the reference data. Their constraint densities are the original densities multiplied by R^2 . If h_R=g_R-δ , then \|h_R\|_ C^2( A) +\|k_R\|_ C^1( A) +\|h_ *,R \|_ C^2( A) +\|k_ *,R \|_ C^1( A) ≤CR^ -β . Choose χ∈C^∞( A) with 0≤χ≤1 , equal to one near |y|=1 and zero near |y|=4 , with all its derivatives supported in a fixed compact subannulus. Blend both fields by this cutoff. Write the constraints in the unnormalized form C(g,k) =(_g+(_gk)^2-|k|_g^2, _g(k-(_gk)g)). Their Euclidean linear part is ( Lh, Dk) . On fields satisfying Equation, the nonlinear remainder is bounded in supremum norm by CR^ -2β . Thus the blended constraints equal the blended original constraint sources, plus a quadratic error and the compact commutators f_R, = L(χ(h_R-h_ *,R )) -χ L(h_R-h_ *,R ),; V_R, = D(χ(k_R-k_ *,R )) -χ D(k_R-k_ *,R ). They have C^ 0,1 norms at most CR^ -β . In detail the scalar commutator applied to a symmetric tensor h is , 2∂_iχ(∂_jh_ ij -∂_i h) +(∂_i∂_jχ)h_ ij -(Δχ) h; one derivative of this expression uses at most two derivatives of h . The vector commutator is (∂_jχ)(k_ ij -( k)δ_ ij ) ; one derivative uses at most one derivative of k . No derivative of the complete matter sources is needed. Every affine moment of f_R and every Killing-field moment of V_R is o(R^ -1 ) . For the constant moments, integration by parts on A gives R^ -1 times the difference between the physical translation fluxes on its two boundary spheres, less the integrals of the blended linear sources. The flux difference tends to zero because the reference and original ADM vectors agree. Replacing the physical momentum integrand by its Euclidean trace reversal costs O(R^ 1-2q ) , which tends to zero. By Lemma 32 ([Integrable sources give sublinear first moments]), the physical linear-source integrals on this receding annulus tend to zero as well. For a linear scalar test or a rotational vector test the corresponding factor is R^ -2 . Its physical boundary fluxes are o(R) by the same lemma, while the weighted source integrals are o(R) by Equation. This proves the claim. Remove these small moments by fixed smooth bumps. Explicitly, for a basis p_1,…,p_4 of affine functions and a fixed nonnegative bump ω positive on a ball inside the annulus, the Gram matrix G_ ab =∫ωp_ap_b is positive definite. Subtract ω∑_ a,b p_a(G^ -1 )_ ab ∫p_bf_R from f_R . For the vector moments use in exactly the same way the Gram matrix ∫ωY_a·Y_b of a basis of the six Euclidean Killing fields. It too is positive definite, since a nonzero Killing field cannot vanish on an open ball. Both bump corrections have every fixed smooth norm o(R^ -1 ) . Apply Lemma 31 ([Compact scalar and symmetric divergence inverses]) with the negative of the moment-free commutators. Since their W^ 1,p norms are O(R^ -β ) , choosing p>3 gives metric and tensor corrections of size O(R^ -β ) in W^ 3,p and W^ 2,p , respectively. Sobolev embedding gives the needed C^ 2,α and C^ 1,α bounds. Their supports stay in a fixed compact subset of A . Extend them by zero and undo the scaling. The metric remains positive definite for large R . To check DEC, denote the original scaled sources by (A_R,B_R) , so A_R≥2|B_R|_ g_R . The reference sources vanish. The corrected sources therefore have the form (χA_R,χB_R) +o(R^ -1 )+O(R^ -2β ) in supremum norm. The norm of B_R changes by at most CR^ -β |B_R|≤CR^ -2β when the metric is changed, since the pointwise constraint formula gives |B_R|≤CR^ -β . All other nonlinear correction terms have the same quadratic bound. Because 2β>1 , the scaled DEC deficit is o(R^ -1 ) . Multiplication by R^ -2 gives Equation. Outside the correction annulus the data are either the original DEC data or the exact vacuum reference data. Smoothness follows from smoothness of the original fields and of the compact inverse constructions, irrespective of the sizes of their unneeded higher derivatives.
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