A Minimal Single‑Field Cyclic Cosmology with Klein Bottle Topology and Finite Lifetime

Here I present a minimal cyclic cosmological model driven by a single scalar field in ahigher‑dimensional spacetime with non‑trivial topology. The global spatial structure ismodeled as a Klein‑bottle–type identification implemented through one compact extradimension. The scalar field initially resides in a metastable false vacuum, effectivelybehaving as dark energy (darken era). Due to a small linear tilt in the potential, the fieldslowly evolves and after ~130–150 Gyr rolls into a true vacuum with negative energy,initiating a contraction phase (rebound/whiteic era). An effective stiff component scaling asa⁻⁶ is introduced within the standard Friedmann framework, motivated by the Klein bottlegeometry. This component dominates at small scales and enables a non-singular bounce.The model remains within standard general relativity, without modifying the Einsteinequations, and instead extends the energy content. The non‑orientable topology alsonaturally leads to time reversal at the bounce, quantum branching, a black‑hole–white‑holetransition, and a reset of entropy, giving a finite number of cycles (~10⁵). This proposal isconceptual; full derivations and numerical analysis are reserved for future work.