Compact-core completeness for a Euclidean end ($K_R=X\cup {\Phi }^{-1}\{1<|x|\le R\}\text{ compact}$)

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.