= Compact-core completeness for a Euclidean end
{title2=$K_R=X\cup\Phi^{-1}\{1<|x|\leq R\}\text{ compact}$}
Suppose these sets exhaust a manifold, and its <Riemannian metric> is Euclidean beyond some radius in the end coordinates. A finite-time <geodesic> has fixed speed and cannot increase its exterior radius faster than that speed. It remains in one <compact> core enlargement, with velocity in a <compact> disk bundle. The <geodesic> equation therefore continues beyond every finite parameter endpoint. The <compactness> hypothesis is substantive: the open exterior of a closed Euclidean ball, with empty core, is incomplete at its inner boundary despite having a <Euclidean metric> at infinity.
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