Strong cosmic censorship near two-ended Kerr data

For each fixed rotating subextremal Kerr background with mass M > 0 and rotation 0< |\mathfrak a|< M, we prove that the smooth vacuum data near its complete two-ended bridge whose full maximal globally hyperbolic development admits a future C^0\cap W^{1,2}_{loc} extension form a meagre set in the weighted smooth topology. The extension metric is continuous and nondegenerate, and its weak connection is locally square-integrable. The neighborhood requires smallness of only the tenth weighted seminorm, while the topology tests every finite order.

For each fixed Kerr spacetime with mass M > 0 and rotation 0< \mathfrak a< M, we prove that a dense G_(δ) set of nearby smooth, complete two-ended vacuum data has a maximal globally hyperbolic development with no future C^(1) extension. The neighborhood and genericity are defined in a weighted smooth topology, and ambient extensions may be nonvacuum.

For every fixed rotating subextremal Kerr bridge, we prove that smooth vacuum data admitting a C^(2) future extension of their full maximal globally hyperbolic development form a meagre set in a neighborhood defined by one finite-order seminorm. The topology allows arbitrary symbol-bounded asymptotically flat tails and imposes no symmetry.