Conditional Spacetime Branching from Non-Unique Black-Hole Interiors

In the standard description of gravitational collapse, the formation of an event horizon is accompanied by the replacement of the regular interior spacetime by a trapped vacuum region terminating at a curvature singularity. This replacement, however, is not dictated by the Einstein equations once the characteristic nature of the horizon and the admissibility of continuous (C^0) extensions are taken into account. Using rigorous results on the non-uniqueness of C^0 extensions of the Einstein equations across null hypersurfaces, it is shown that fixing the exterior spacetime and the intrinsic geometry of a smooth event horizon H^+ does not determine a unique future development. Besides the familiar focusing continuation containing a trapped interior (the Singular Branch), there exist inequivalent non-focusing continuations in which the collapsing interior spacetime remains regular and dynamically evolving (the Regular Branch). It is further shown that, if the interior spacetime manifold is required to persist beyond horizon formation, a non-Hausdorff spacetime branching is a necessary consequence of classical general relativity at C^0 regularity. The resulting branches are causally decoupled after formation, the black-hole mass parameter is fixed by the intrinsic geometry of H^+ at formation and remains invariant under subsequent evolution, and the Regular Branch admits a distinct exterior geometry compatible with smooth vacuum evolution. The branching framework provides a classical resolution of the black-hole information problem by causal sequestration of information and admits a natural cosmological interpretation in which expanding universes arise as Regular Branch interiors of black holes. The analysis applies to both Schwarzschild and Kerr collapse and relies entirely on classical general relativity, without invoking quantum or semiclassical effects.