The Two-Sheeted Topology of Extended Kerr-Type Spacetimes and a Parity-of-Crossings Property for Ring-Traversing Geodesics

We revisit the two-sheeted structure of extended Kerr-type spacetimes. After excising the singular ring, the signed-$r$ oblate point space realises the unique connected meridional double cover of a separate one-sheet ring complement. The cover is unbranched on the excised space; describing the ring as a branch locus refers only to a topological completion and does not regularise the Kerr metric there. Admissible causal geodesics can cross the regular part of the $r=0$ disk without meeting the ring. The Kerr equations give $R(0)=-a^{2}Q$; a transverse crossing requires $Q