The Hyperextreme 2-Soliton Vortex: Compact Toroidal Closed Timelike Curves, a Naked Ring Singularity, and Thin-Shell Excision

We give a self-contained classical analysis of the stationary, axisymmetric vacuum vortex ob- tained from the two-soliton Belinski–Zakharov (BZ) inverse-scattering transform on a flat seed—equivalently, the Kerr metric—in its hyperextreme branch a > M . In this branch the spacetime has no horizons, is asymptotically flat with ADM mass M and angular momentum J = Ma, is free of conical struts on the symmetry axis, and carries a single-valued frame-dragging potential. −3 Its only singularity is a naked curvature ring, K ∼ Σ , encircled by a compact toroidal region of closed timelike curves (CTCs), gϕϕ < 0, which we map exactly and which is causally acces- sible from infinity. We then excise a toroidal neighbourhood of the ring by an Israel thin shell, compute the surface stress-energy of the junction in ring-adapted (Kerr–Schild) coordinates, and prove numerically that the surface energy density is negative for every tube radius: the shell that simultaneously regularises the ring and censors the CTC region cannot be built from matter satisfying the weak energy condition. This is a sharp classical statement of chronology protection for the localized toroidal time machine. All curvature, causal, and junction claims are verified numerically to the finite-difference floor ( Rµν /scale 10−⁶), and we include an appendix on the normalisation that any raw BZ reconstruction must satisfy (det g₍₂₎ = ρ²) in order to be a genuine solution.