Generic Future Inextendibility with Square-Integrable Connection Near a Fixed Kerr Spacetime

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 $C^0\cap W_{\mathrm{loc}}^{1,2}$ 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.