Black Holes as the Degenerate Limit of Realization Geometry

This work investigates the degenerate limit of realization geometry based on null orientation fields and shows that this limit corresponds naturally to the structure identified as a black hole. Within the framework, physical reality arises from the non-collinear closure of null orientations, producing a timelike invariant that defines mass, proper time, and stable structures. In contrast, when this invariant approaches zero, timelike realization degenerates and the local generation of events ceases. The paper introduces the concept of realization depth as a scalar measure of local proper-time structure. Spatial gradients of this quantity reproduce gravitational acceleration and lead, in the spherically symmetric case, to a realization profile consistent with the Schwarzschild temporal factor. The Schwarzschild radius emerges as the point where realization depth vanishes, marking a boundary beyond which timelike realization is no longer possible. The photon sphere is identified as a critical regime of null closure where light trajectories become marginally bound. This produces a degeneracy in the mapping between null trajectories and the observer’s sky, naturally explaining the bright ring observed in horizon-scale images of black holes. At the realization boundary, observable quantities become invariant under transformations of the inaccessible internal structure. This “mirror invariance” reflects the degeneracy of the realization mapping and provides a geometric explanation for the universality of black-hole observables such as shadow size and ringdown modes. In this interpretation, a black hole is not a material object but a geometric boundary of realizability. Matter corresponds to stable timelike realizations, while black holes arise as the opposite limit, where realization collapses toward the null structure. This work completes the geometric framework by identifying both matter and black holes as complementary regimes of the same underlying realization geometry.