On the Status and Derivability of the Born Rule in Quantum Mechanics

Refinement-Consistent Event Binning Forces the Born Rule Author: Radu Necsanu ORCID: https://orcid.org/0009-0009-1281-9180 DOI: https://doi.org/10.5281/zenodo.18408230 Abstract Within standard quantum mechanics, we model sharp measurement outcomes by a finite projective measurement {Pᵢ} and interpret events as finite unions of outcome bins represented by their coarse-grained projector. Assuming only (i) the ordinary probability axioms on finite event sets, (ii) representation invariance under joint unitary changes of basis and phase redefinitions, and (iii) refinement-consistency—namely, that the probability of a coarse event depends only on its projector and is invariant under internal refinements of the event and its complement—we show that the unique compatible probability rule in dim(H) ≥ 3 is the Born rule p(P|ψ)=⟨ψ|P|ψ⟩. With the standard preparation-mixture consistency condition, the rule extends uniquely to mixed states and POVMs via p(E|ρ)=Tr(ρE).