A Logical Distinction Between Bell's Theorem and Bell Experiments: What Bell Tests Do and Do Not Establish

Abstract

Bell's theorem is a conditional statement about theoretical models specified by a complete physical state $\lambda$. It shows that models satisfying measurement independence and a stochastic Bell-locality (factorizability) condition obey Bell inequalities. Bell experiments, however, do not directly observe $\lambda$ or the $\lambda$-conditioned probabilities appearing in that condition. They record analyzer settings, detector outcomes, and experimental metadata, from which they estimate operational joint distributions, marginals, correlations, and Bell statistics. We clarify the logical relationship between these two levels. The $\lambda$-conditioned factorization identity cannot be directly evaluated at fixed $\lambda$ from the operational record alone. Nevertheless, by marginalizing over $\lambda$ under measurement independence, the model class implies observable inequalities that can be empirically violated. A Bell-inequality violation therefore excludes the corresponding Bell-local model class under the assumptions required for the inequality, but it does not by itself identify which latent assumption---measurement independence, parameter independence, or outcome independence---failed, nor does it uniquely determine an ontological interpretation. To illustrate this distinction concretely, we employ the Unified Quantum Mechanics formulation, in which the physical state is real, and the complex wavefunction is representational. We show that all four formulations of the dynamics---Schr\"odinger, Hilbert-transform, complex second-order, and real second-order---require the same two real initial functions $u_0=u(0)$ and $u_1=\partial_t u(0)$; the complex initial state is merely a repackaging of them, not additional physical information. We formulate the distinction as a three-level hierarchy: operational data, model-class exclusion, and ontological interpretation. The hierarchy separates experimental loopholes, which concern assumptions linking an actual experiment to an ideal Bell model, from the conceptual underdetermination of latent-variable conditional probabilities, which remains even in a loophole-free experiment. The result clarifies what Bell experiments establish without claiming that Bell-local models are empirically untestable. A Monte Carlo simulation illustrates the distinction between operational Bell-test results and latent-state factorization.