Abstract
This article reconceptualizes reliability as a theorem derived from the projection geometry of Hilbert space rather than an assumption of classical test theory. Within this framework, the true score is defined as the conditional expectation E(X∣G) , representing the orthogonal projection of the observed score onto the σ-algebra of the latent variable. Reliability, expressed as Rel(X) = Var[E(X∣G)]/Var(X) , quantifies the efficiency of this projection-the squared cosine between X and its true-score projection. This formulation unifies reliability with regression R2 , factor-analytic communality, and predictive accuracy in stochastic models. The operator-theoretic perspective clarifies that measurement error corresponds to the orthogonal complement of the projection, and reliability reflects the alignment between observed and latent scores. Numerical examples and measure-theoretic proofs illustrate the framework's generality. The approach provides a rigorous mathematical foundation for reliability, connecting psychometric theory with modern statistical and geometric analysis.