Topological Phase Transitions and Dark Sector Decoupling in a $\mathbb{T}^3$ Universe: A Mechanical Model for the Big Rip

While the standard CDM model successfully describes an accelerating universe, treating Dark Energy (DE) and Cold Dark Matter (CDM) as non-interacting entities leaves the ultimate geometric and thermodynamic fate of the cosmos unresolved. Building upon interacting dark fluid models (such as the generalized Chaplygin gas) and multiply-connected topologies, this paper proposes a unified mechanical framework for a universe with a toroidal () geometry. I hypothesize that continuous isotropic expansion exerts escalating volumetric strain on the spacetime continuum, forcing the central hyperspace void of the torus to contract. As this geometry approaches a forced topological phase transition from a genus-1 to a genus-0 state, the resulting topological stress reaches a critical threshold. I argue that this threshold disrupts the phase equilibrium between DE and DM, halting their conversion. The inability of the metric tensor to smoothly seal the central void culminates in localized spacetime ruptures—a Topological Big Rip. Consequently, the rapid disintegration of the Dark Matter scaffold leads to the gravitational decoupling of baryonic structures, subjecting visible matter to viscous macroscopic dissolution and localized collapse in a torn continuum.