Cyclic Transformational Universe Model: A Mathematical Framework for a Closed Cosmological Model with a Transformational Region

We present a mathematical framework for a cyclic transformational cosmological model in which the universe is treated as a single closed system with no external space or external temporal ordering. The model introduces one transformational region, termed the supersingularity, which converts gravitationally structured states into locally young states with reduced macroscopic gravitational structure while preserving fundamental information. An effective non-invertible macroscopic reset is embedded into a unitary microscopic description by transferring structural information into additional microscopic degrees of freedom. A compact spatial topology S ≃ S1 × S2, a global metric ansatz, and a compact phase field with winding number W = 1 are introduced as a minimal mathematical realization of a single transformational front in a closed geometry. Conservation laws, matching conditions, a phenomenological gravitational-structure reset equation, and local linear stability conditions are formulated. The construction is presently a theoretical framework rather than a complete cosmological solution: existence and stability of a globally self-consistent solution of the coupled gravitational and field equations remain open problems.