The Global Möbius Manifold: A Complete Topological Blueprint for a Universe Without Beginnings or Singularities

This paper presents the first complete global construction of the Möbius Universe: a smooth, four-dimensional, non-orientable spacetime whose topology removes the need for a cosmological beginning and eliminates curvature singularities. Building on earlier local results (junction behavior, twist-origin dynamics, inter-layer continuity, and finite-curvature black hole interiors), this work establishes the full manifold architecture — including the global atlas, transition maps, layer-index function, and orientation structure.

The geometry is governed by the Wyatt Coupling Tensor WabW_{ab}Wab, the Ashley Junction Tensor AabA_{ab}Aab, and the EJ Twist Constant SEJS_{EJ}SEJ. Together they satisfy the Mize Stability Equation

∥W∥2+α∥A∥2≤SEJ,\|W\|^{2} + \alpha \|A\|^{2} \le S_{EJ},∥W∥2+α∥A∥2≤SEJ,

which enforces finite curvature across all twist-induced transitions.

This global framework unifies the local and dynamical layers of Möbius Cosmology, showing how the manifold supports continuous curvature flow, analytic continuation across orientation reversals, inter-layer memory effects, and large-scale cosmological structure. The result is a complete topological blueprint for a universe without beginnings or singularities.