Carrier Classification for Pure-State Ontology in Quantum Mechanics: Projective Descent, Factorization, and Admissible Ontology Classes

This paper establishes a carrier-classification theorem for finite-dimensional pure-state ontology in quantum mechanics. The starting point is carrier identification. Physical pure-state predictions are invariant under nonzero complex rescaling of Hilbert-space representatives. Consequently, the empirical pure-state carrier is projective Hilbert space (equivalently, the rank-one density operators) rather than the unquotiented Hilbert-space vectors themselves. The central question is therefore not which ontology should be preferred, but which ontological assignments are defined on the empirical carrier. The main result is an exact projective descent and factorization theorem. A representative-level ontology assignment is well-defined on the empirical pure-state carrier if and only if it is invariant under all nonzero complex rescalings, equivalently if and only if it factors uniquely through the projective quotient. Admissible pure-state ontology is therefore characterized by descent and unique factorization through the projective carrier. Assignments that distinguish vector representatives lying over the same ray are not invariants of the standard pure-state datum and require an explicitly extended carrier. The theorem yields a positive carrier classification. Rays, rank-one density operators, projective subspaces, histories, records, and explicitly extended structures carrying gauge, branch, measure, pointing, or selection data are all admissible carrier classes when their defining data are supplied. A nontrivial branch decomposition is not selected by the bare Hilbert-space datum alone. Once a branch or measurement context is fixed, the associated probabilities and branch weights are projective quantities: they depend on the ray rather than on a distinguished vector representative. As an application, representative-level universal-state formulations that employ branch weights, typicality measures, observer-location claims, or “our branch” claims cannot remain simultaneously projectively well-defined and bare-Hilbert minimal. Such formulations must either factor through projective or density carriers, supply branch-selection structure (for example through factorization, decoherence, histories, records, or measurement context), or supply additional pointing, conditioning, indexical, gauge, or selection data. More broadly, the paper develops a carrier-first foundation for pure-state ontology. Rather than beginning with an ontological proposal and asking whether it reproduces quantum predictions, the theorem begins with the empirical carrier and classifies the ontology classes naturally associated with it through descent, factorization, and explicit carrier extension. License note: Distributed under CC BY-NC-ND 4.0.