Black Hole–Big Bang Conjugate Duality, Apparent Topological Symmetry Breaking, and the Topology–Geometry Hierarchical Observation Principle

Within the ℝ₂ double-sheeted topological architecture of GDUM, the global underlying structure satisfies at all times the orientation-conjugation symmetry induced by the Deck transformation. For an oriented double cover π: M̃ → M, the Deck group is naturally taken to be ℝ₂; this covering automorphism constitutes a theoretical-program-level topological primitive constraint, running through the pre-geometric ontology, the covering lift mechanism, and the four-dimensional spacetime representation. At the ontological level, the topological conjugation symmetry is strictly conserved and never breaks. The Deck transformation enforces an exact pairing of the positively and negatively oriented manifold branches, constituting a unified global bidirectional conjugate base for the cosmos, and providing the deep geometric origin of the topological equivalence between black hole horizons and the initial state of the universe.