This paper presents the formal equation set accompanying Black Holes as Functional Boundaries of SpacetimeDescription. Metric Representability and Regime Structure I 10.5281/zenodo.19116157, establishing spacetime geometry as a conditional representational regime within an admissible relational domain. Building on Penrose’s singularity theorem, geodesic incompleteness is interpreted as a limit of metric representability rather than a physical endpoint, delimiting the domain in which spacetime can be sustained as a causal manifold. The formulation defines an admissible relational domain, a coherence threshold for metric instantiation, and the transition between pre-metric, proto-metric, and metric regimes. Within the metric regime, the Einstein field equations remain valid but are explicitly restricted by prior admissibility conditions. Trapped surfaces and horizon entropy relations are treated as boundary conditions governing the withdrawal of geometric description. This work provides the minimal mathematical structure underlying the regime-based interpretation of spacetime.
William Smith (Sun,) studied this question.