Topological Regularization of the Inverse Cosmological Problem: Perelman's Theorem in Black Hole Bounce Scenarios

Reconstructing global properties of the Universe from cosmological observations is generically anill-posed inverse problem, primarily due to unresolved degeneracies in spatial topology. Distincttopological classes may produce observationally indistinguishable signatures within a finite observablevolume. We demonstrate that this degeneracy can be eliminated under a physically motivatedscenario in which the Universe originates from a nonsingular bounce inside a parent black hole describedby Einstein–Cartan gravity. Horizon formation and causal disconnection imply that spatialhypersurfaces are compact and simply connected. Under these assumptions, Perelman’s proof of the Geometrization Conjecture enforces a uniquetopological class, allowing the admissible spatial topology to be restricted to the three-sphere. Weformulate this result as a delta-function Bayesian prior over topology, acting as an exact topologicalregularizer with vanishing entropy. This construction restores well-posedness of the cosmological inverse problem in thetopological sector, under explicit physical assumptions. It allows observational data to constrainonly physical parameters, such as curvature radius and anisotropy, without contaminationfrom spurious topological degrees of freedom. We discuss how this framework enables consistentBayesian inference from cosmic microwave background measurements and stochastic gravitationalwaveobservations. The role of geometrization is thus clarified: Perelman’s theorem uniquelydetermines the admissible diffeomorphism class of spatial hypersurfaces once compactnessand simple connectedness are assumed, guaranteeing the internal consistency ofcosmological inference.