Global Continuation Across the Kerr Ring: Axiom Sets, Junction Data, and Disk-Crossing Geodesics

Abstract

With the stationary Killing field continued from $r>0$, the fixed-$M$ Boyer--Lindquist continuation of Kerr has a second asymptotically flat end with ADM mass $-M$. We separate this metric statement from signed Komar conventions and organize the data required for other continuations across the regular disk bounded by the ring. For normal-reflection gluing of identical positive-$M$ Kerr blocks, we rederive the classical Darmois--Israel disk data. The timelike disk world tube has nonzero second fundamental form. No real smooth tangential identification preserving the first fundamental form gives a $C^1$ vacuum reflection matching; the identity and simultaneous $(t,\phi)$ reversal give the same factor-two jump. The disk layer is defined only on the ring-excised world tube, and the singular ring requires separate treatment. Independently, the Carter equations give $R(0)=-a^2Q$: transverse disk crossing requires $Q<0$, together with polar admissibility, and occurs in finite affine parameter. Deleting the negative-$r$ branch while leaving the regular disk terminal therefore produces timelike and null geodesic incompleteness. These results distinguish analytic signed-$r$ continuation, selected asymptotic generators, and genuinely positive-$M$ block gluings; they do not select a global completion or determine time orientability.