The Persistence Criterion:Distinguishing Epistemic Obscuration from OntologicalElimination in Physical Theories

A recurring interpretive move in quantum physics treats the loss of observational distinguishability as equivalent to the loss of ontological structure. In this paper, I show that the formal structure of the theory does not support that move. I prove the Structural Persistence Theorem, which identifies precisely which algebraic invariants survive arbitrary completely positive trace-preserving (CPTP) maps. The invariants form a natural and complete set: the C*-algebra of observables A = B(H), its norm, commutation relations, and positivity structure. No CPTP process removes any observable from the algebra. State-space distinguishability contracts; algebraic structure does not. This partition — preserved structure versus contracted epistemic access — is the core formal result of the paper. I identify and formally diagnose a specific class of ontological overclaiming in the decoherence literature: instances where pointer-basis selection and suppression of off-diagonal density matrix elements are treated as eliminative of superposition. I show exactly where these inferences overreach the formalism and provide corrected formulations. I then demonstrate that this criterion places formal constraints on how the major quantum interpretations can consistently attribute ontological change. Many-Worlds and pilot-wave theory are fully consistent with the criterion. Copenhagen's physical-collapse reading carries an unspecified mechanism burden. GRW and CSL objective collapse models are PC-violating by deliberate design — which the framework clarifies rather than criticizes. Any empirical confirmation of GRW/CSL predictions would constitute direct evidence for PC-violation in nature. Two practical applications are discussed. In quantum error correction, the success of the fault-tolerance threshold theorem is indirect empirical evidence for the Persistence Criterion: decoherence degrades distinguishability but leaves the algebraic structure intact, which is precisely what makes error recovery possible. In effective field theory, the persistence of UV degrees of freedom under coarse-graining is consistent with the same structural partition. This work proposes no new experimental predictions. It clarifies what the existing formalism entails and does not entail — drawing a sharp, formal boundary between what the mathematics licenses and what must be supplied by interpretation.