Lemma 30 ([A reference end with prescribed timelike charges]). Let E>|P| and m=(E²-|P|²)¹/2 . There is a spacelike plane near spatial infinity of Schwarzschild spacetime of mass m whose induced data, in asymptotically Euclidean …

Apply Lemma 6 ([Charges of a boosted Schwarzschild--Tangherlini end]) with n=3 , k=2 , p=1 , and M=m . In static isotropic coordinates (T,y) its spacetime metric is -((1-m/(2r))/(1+m/(2r)))^2 T^2 +(1+(m)/(2r))^4δ_ ij y^i y^j, r=|y|. For P≠0 , put v=P/E and rotate the boost axis into its direction; for P=0 , take v=0 . Then γ=(1-|v|^2)^ -1/2 =E/m , and the reference plane has charges (mγ,mγv)=(E,P) . The common lemma uses the same future-normal convention for K , and its ADM prefactors become 1 2kω_2 =1 16π , 1 kω_2 =1 8π . Thus its charges have precisely the present signs and normalization; the use of physical constraint densities here does not rescale them. For the later bend, write the plane as T=v·y and set y=Ax , where A=(I-v⊗v)^ -1/2 . The induced Minkowski metric in the x coordinates is δ . The perturbation of the metric above from Minkowski space has symbol decay of order r^ -1 on this fixed spacelike plane, and Lemma 6 ([Charges of a boosted Schwarzschild--Tangherlini end]) gives the induced metric and tensor orders r^ -1 and r^ -2 through every fixed derivative order. These estimates concern the exact reference end and impose no additional derivative assumption on the original data.