Abstract This paper documents and begins the formalization of the Self-Identifying Universe hypothesis, a cosmological proposal developed within the Schoff Research Program. The central claim is: the observable universe is the interior of a black hole, and every black hole in the universe is that same universe viewed from the exterior. The event horizon of a black hole is the cosmological boundary. Entering a black hole from outside is geometrically equivalent to entering the universe — the same topology, approached from the other side of the boundary. The proposal is strictly a single-universe topology: there are no parent universes, no baby universes, no infinite regress of nested cosmologies. The universe is one closed, self-identifying structure whose interior and exterior are the same region approached from opposite directions — the spatial analog of the temporal loop closure cosmology already established in the Schoff Research Program. The paper develops the theoretical motivation, identifies the existing physics literature that partially supports the proposal, states the specific mathematical requirements for full formalization, and specifies the empirical predictions that would distinguish this topology from standard cosmology. The causal asymmetry between black hole interior and exterior — normally treated as a fundamental obstacle — is reframed as a perspectival feature of the topology: which direction is forward depends on which side of the boundary the observer occupies, consistent with the observer-relative account of temporal direction developed in the program's physics formalization series. Keywords: black hole cosmology, single universe topology, self-identifying boundary, loop closure, event horizon, cosmological boundary, causal structure, BCC cosmology, observer-relative causality --- Quotient Manifold - Abstract The Self-Identifying Universe hypothesis proposes that our observable universe is the white hole interior (Region III) of the Kruskal-Szekeres maximal extension of the Schwarzschild metric, with the black hole interior (Region II) identified as the same spacetime approached from the opposite temporal direction. Formally, this requires the Kruskal manifold to admit a consistent quotient under the identification φ: (T, X, θ, φ) → (-T, X, θ, φ). For this quotient to constitute a valid spacetime, three mathematical conditions must be satisfied: (1) φ must be a smooth diffeomorphism; (2) φ must preserve the metric; (3) the resulting quotient must be a Hausdorff manifold without new singularities introduced by the identification. This paper works through all three conditions rigorously. Conditions 1 and 2 are proven: φ is manifestly smooth and its own inverse, and the Kruskal metric is preserved exactly under φ since (-dT)² = dT². Condition 3 requires careful analysis of the fixed point set of φ — the bifurcation surface T = 0 — where the two horizons cross. The analysis demonstrates that the quotient space is Hausdorff away from the bifurcation surface, but the bifurcation surface itself requires separate treatment. Two candidate resolutions are identified: the bifurcation surface as a genuine boundary of the quotient manifold (in which case the identified topology is a manifold with boundary), or the bifurcation surface as a conical singularity of controlled type that can be resolved by the Poplawski torsion mechanism. The relationship to the Schoff Research Program's constraint renegotiation framework is developed: the bifurcation surface corresponds to the unique moment T = 0 where constraint accumulation has no direction — the zero-renegotiation point — which the Loop Closure Cosmology identifies as Ω itself. This connection suggests that the apparent mathematical difficulty at the bifurcation surface is not a pathology to be eliminated but a geometric reflection of the Loop Closure topology's terminal/initial state. Keywords: quotient manifold, Kruskal-Szekeres, time reversal, bifurcation surface, Hausdorff condition, self-identifying topology, white hole, loop closure, Ω-point
Schoff et al. (Sun,) studied this question.