The Conditional Authority of Geometry: Black Holes, Informational Finitude, and the Limits of Geometric Realism

Abstract This paper establishes a necessary condition on the ontological authority of spacetime geometry: a theoretical structure warrants ontological commitment only where it preserves an injective correspondence with physically distinguishable states. We call this the Representational Adequacy Constraint (RAC). When the RAC fails, formal geometric continuation cannot underwrite ontology — geometry persists as mathematics but loses its license to describe what physically exists. We argue that black hole interiors instantiate precisely this failure. The Bekenstein-Hawking entropy bound imposes a finite upper limit on physically realizable degrees of freedom in any bounded region. Classical volumetric geometry, by contrast, demands an indefinitely fine-grained differentiation of interior points that no physical system can sustain under conditions of informational saturation. This is not an epistemic limitation — it is a structural one. No observer, internal or external, could instantiate the discriminations that continuous interior geometry presupposes. The RAC is shown to be a necessary condition under any theory of reference — causal, descriptive, or inferentialist — which means that geometric terms applied to black hole interiors become referentially indeterminate, not merely empirically inaccessible. This result is independently confirmed by the entanglement wedge structure of AdS/CFT: bulk reconstruction fails for interior regions precisely where the RAC predicts representational breakdown. The recent derivation of the Page curve via island formula methods provides a further convergent line of support. The positive consequence is a regime-dependent structural realism: geometric structure carries ontological commitment where and only where it tracks invariant physical organization under physically realizable conditions. Black hole singularities are reinterpreted not as pathologies awaiting resolution by successor theories, but as representational boundaries that any physically adequate theory — including loop quantum gravity and string-theoretic completions — must encounter at some locus. The framework applies beyond black holes to any regime in which representational demands outstrip physically sustainable distinguishability.