On the Non-Realizability of Structural Limits

This paper develops a foundational theoretical framework for identifying physical structures that cannot be approached, resolved, or parameterized through increased observational refinement. It challenges the widespread but implicit assumption that improving realization—through higher resolution, extended observation, or alternative measurement channels—necessarily corresponds to approaching underlying physical structure. We introduce the concept of structural invariance as an empirical and theoretical criterion: when admissible changes in realization produce no systematic variation in structural observables, local approachability is falsified. In such cases, structure is not accessed through convergence, but revealed through the absence of realizational dependence. Within this framework, black holes are analyzed as the paradigmatic example of a structural limit rather than an object embedded in spacetime. The familiar paradox of black-hole approach is shown to arise from treating a realizational boundary as if it were a locally reachable entity. Once the distinction between structure and realization is made explicit, the paradox dissolves: a black hole is not something that is approached, but a boundary at which the concept of approach itself loses operational meaning. The paper formalizes realization as an admissible measurement proxy and demonstrates that, for structural limits, no finite or intensified realization parameterizes proximity to structure. Instead, structural quantities emerge as realization-invariant limits. This principle is illustrated through geometric considerations of the Schwarzschild horizon, empirical absence of long-term drift in black-hole observables, and a four-stage recursive closure mechanism that parallels structural stabilization in quantum measurement. The resulting picture reframes black holes as structural boundaries of realizability, not as dynamical objects with hidden interior events. More generally, the work provides a domain-independent criterion for distinguishing realized physical systems from structural limits across physics, based on invariance under all admissible realizations. This article serves as a foundational theoretical contribution to a broader framework in which structure is identified by what cannot be varied, rather than by what can be increasingly resolved.