The $C^0$-inextendibility of Kerr spacetimes

We prove the $C^0$-inextendibility of the full Kerr realizations considered in this paper and all their connected covering spacetimes. The result holds for positive mass and nonzero rotation in the subextremal, extremal, and overextremal regimes. The obstruction at the singular ring is expressed by unbounded proper time in a fixed-endpoint causal homotopy class. It persists under every positive continuous conformal rescaling and under continuous widening of the Kerr causal cones, without any global causality assumption. We give sharp bounds in a fixed coordinate norm for stationary and axisymmetric perturbations and describe complete timelike geodesics that encounter arbitrarily large curvature near the ring.