The Unified Toroidal Engine

ABSTRACT This paper provides the formal mathematical proof that a non-orientable Möbius-boundary and an internal compressive vortex are inherently compatible and required to form a stable Toroidal body. We demonstrate that the 180° Möbius-twist is the mechanical feedback mechanism that closes the open expansion of a vortex, preventing systemic collapse (Torus-Break). We apply this Unified Toroidal Formula to Earth, the Sun, and Galactic structures, identifying them as nested residues within the universal neutrino flow. This framework eliminates the need for Dark Matter by identifying the structural tension of the Möbius-twist as the source of orbital stability and the "Möbius-Track" of planetary motion.