Black Holes as Structural Residues of Completed Realization: A Unified Linear, Informational, and Topological Framework

This work develops a theory-independent structural framework for understanding black holes as residues of completed realization, rather than as objects defined by inaccessible internal dynamics. Realization is modeled as a mapping from a structural space to an observable event space. When this mapping saturates—such that no further observational differentiation is possible—a nontrivial structural remainder necessarily persists. We show that this residual structure can be formulated in three mathematically independent but equivalent ways: as the kernel of a realization operator in a linear framework, as conditional entropy in an information-theoretic framework, and as equivalence classes or fibers in a topological framework. These formulations define a single unified quantity, termed the mirror invariant, which quantifies structural content that cannot be realized as observable events within a given observational framework. The mirror invariant arises as a mathematical necessity whenever realization has finite capacity or effective dimensionality, independent of assumptions about gravity, spacetime microphysics, or quantum dynamics. Within this framework, black holes correspond to regimes in which realization has saturated and terminated: observable events cease, while a stable structural residue remains. This perspective reframes the black hole problem from one of hidden interior processes to one of structural invariance under completed realization. The absence of internal events is interpreted not as an empirical limitation, but as a necessary consequence of the realization map itself. More broadly, the framework provides a unified language for discussing structural persistence beyond observability in black holes and other systems subject to finite realization capacity.