Conserved Reality: An Ontological Reinterpretation of Superselection in Finite-Dimensional Quantum Systems

We propose an ontological reinterpretation of superselection rules in finite-dimensional quantum systems. In the standard interpretation, superselection is a constraint on measurement: the algebra of observables is restricted to the commutant of a conserved charge, and cross-sector coherences are deemed unobservable. We propose instead that superselection sectors constitute the structure of physical reality itself. The kinematic Hilbert space is a ``sea'' of all wavefunctions, but only states lying in a definite spectral sector of a self-adjoint reality operator R are physical. Reality is conserved if $[H,R]=0$. We formalize the requirement that nature forces arbitrary states into definite reality sectors via a completely positive, trace-preserving, idempotent channel Φ that is bimodular with respect to the superselection algebra AR and whose fixed-point set is exactly AR. We prove that these axioms uniquely determine Φ to be the conditional expectation Φ(ρ)=∑ᵣ Pᵣρ Pᵣ, where Pᵣ are the spectral projectors of R. We then show that, within the standard probabilistic framework and for the canonical instrument with Kraus operators Pᵣ, the Born rule pᵣ=Tr(Pᵣρ) is the unique consistent probability assignment. We clarify that this is a uniqueness theorem for the canonical instrument, not a derivation of the Born rule from non-probabilistic assumptions. We illustrate the framework with a finite-dimensional toy model carrying a $U(1)$ charge and discuss implications for quantum foundations. This is an interpretation paper: it makes no new physical predictions and does not modify standard quantum mechanics. Extension to quantum field theory, where the relevant von Neumann algebras are of type III and no trace exists, is left to future work.