Antipodal identification of a closed Friedmann universe at a vanishing big bang: a single-sheeted geometric analogue of CPT-symmetric cosmology

This note considers the quotient of a closed Friedmann–Lemaître–Robertson–Walker spacetime by the involution that combines time reflection with the antipodal map of the spatial 3-sphere, with the identification surface placed at a big bang where the scale factor vanishes. The quotient is defined as a smooth manifold with a smooth conformal-completion metric and a physical metric that degenerates on the identification hypersurface. This gives a single-sheeted, boundary-free geometric analogue of two-sheeted CPT-symmetric cosmology, in which the bang is a one-sided hypersurface diffeomorphic to RP³. The underlying topology belongs to the known class of non-time-orientable antipodal quotients (the elliptic interpretation); no new topological result is claimed. The note states the mode-selection rule for a conformally coupled scalar field on the quotient and derives the parity selection of linear scalar and tensor metric perturbations in the closed radiation era. A numerical study of recollapse with a thawing scalar field is included; the crunch is found not to be the time reverse of the bang, and no identification is imposed there. Gravity beyond linear order, massive fields, fermions and quantum field theory on the quotient are left open. Version 2.0 is a complete rewrite of version 1.1, with a new title, a narrower statement of what is claimed, new sections on linear perturbations and on recollapse, three appendices and an extended reference list. Files: the full text (12 pages), a two-page summary, and a code package reproducing the numerical results (five Python scripts with their outputs and a README). Not peer-reviewed. AI tools were used to assist with calculations, numerical work, literature searches and drafting; the author takes full responsibility for the content.