Bell's theorem is a conditional statement about theoretical models specified by a complete physical state
λ. It shows that models satisfying measurement independence and a stochastic Bell-locality (factorizability) condition obey Bell inequalities. Bell experiments, however, do not directly observe
λ or the
λ-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
λ-conditioned factorization identity cannot be directly evaluated at fixed
λ from the operational record alone. Nevertheless, by marginalizing over
λ 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
u0=u(0) and
u1=∂tu(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.
Among the various "no-go theorems" in the foundations of quantum mechanics, Bell inequalities are among the few, which do not implicitly rely on the mathematical structure of quantum mechanics. Instead, Bell inequalities are commonly understood to rest on Kolmogorov's theory of probability and a "locality" assumption. Physical evidence supporting their violation is hence taken as evidence against at least one of these two assumptions.
This work provides a comprehensive analysis of the physical debate on Bell inequalities from the point of view of Kolmogorov's theory of probability, focusing on the Bell-Clauser-Horn-Shimony-Holt inequality (BCHSH inequality). The central contribution is the motivation and mathematical study of the notion of contextuality in Bell experiments: in so-called contextual Bell models the detectors themselves behave randomly and the distribution of the random detector parameters depends on the chosen detector settings. It is shown how even "classical" Bell experiments motivate such a description. The need to account for contextuality was pointed out before by de la Peña, Cetto, Brody, Lochak, Bohm, Hiley and others. A broad class of contextual models is proven to satisfy the BCHSH inequality, thereby generalizing a previous result by Gill and Lambare. Nonetheless, even under strong locality requirements, contextual Bell models may violate the BCHSH inequality: based on work by Popescu and Rohrlich, we consider a "proof of concept" example exhibiting the maximum possible violation.
Contrary to the much debated "loopholes" in empirical tests of Bell inequalities, contextuality thus reveals a conceptual misunderstanding in the physical conclusions commonly drawn from Bell inequality violations.
Bell's inequality is widely regarded as the death knell of local hidden variable theories. However, this paper points out that Bell's argument implicitly relies on an overly narrow assumption about what constitutes a "local hidden variable"—namely, that hidden variables must assign measurement outcomes to each particle independently at the moment of measurement. This paper proposes that the deterministic correspondence established between entangled particles at the moment of their creation, and determined by initial conditions, can itself serve as a local hidden variable. As long as the polarization direction of the photons is not altered, this correspondence persists indefinitely. It is independent of whether a measurement is performed, and requires no superluminal communication to sustain. From this perspective, quantum entanglement experiments do not violate special relativity; the violation of Bell's inequality merely rules out a specific class of hidden variable models, not all forms of local realism.
Bell–CHSH inequalities are commonly interpreted as demonstrating the incompatibility between quantum mechanics and local hidden-variable theories based on pre-existing measurement outcomes. In standard derivations of the CHSH inequality, however, it is implicitly assumed that all possible measurement settings can be assigned simultaneous values within a single global hidden-variable structure. In this paper, we reconsider the Bell–CHSH framework from the viewpoint of the Intrinsic Observability Framework (IOF). We argue that the central issue is not the experimental violation of the inequality itself, but the applicability of the global value assignment assumed in its derivation. In particular, the simultaneous definition of outcomes associated with incompatible measurement settings presupposes a common operational structure shared across all observational configurations. Within IOF, observable outcomes are not treated as globally pre-existing values independent of observational accessibility and decomposition structure. Instead, measurement outcomes are associated with operationally accessible configurations determined through observable–hidden relations. From this perspective, the global assignment required for the CHSH derivation is not operationally justified. We further show that this viewpoint remains fully consistent with the standard quantum-mechanical correlations observed in Bell-type experiments. The present analysis suggests that the Bell–CHSH inequality may be interpreted not as a direct contradiction between quantum mechanics and locality itself, but rather as a limitation of globally defined pre-existing value assignments across incompatible observational configurations.
It is not generally known that the inequality that Bell derived using three random variables must be identically satisfied by any three corresponding data sets of ±1’s that are writable on paper. This surprising fact is not immediately obvious from Bell’s inequality derivation based on causal random variables, but follows immediately if the same mathematical operations are applied to finite data sets. For laboratory data, the inequality is identically satisfied as a fact of pure algebra, and its satisfaction is independent of whether the processes generating the data are local, non-local, deterministic, random, or nonsensical. It follows that if predicted correlations violate the inequality, they represent no three cross-correlated data sets that can exist, or can be generated from valid probability models. Reported data that violate the inequality consist of probabilistically independent data-pairs and are thus inconsistent with inequality derivation. In the case of random variables as Bell assumed, the correlations in the inequality may be expressed in terms of the probabilities that give rise to them. A new inequality is then produced: The Wigner inequality, that must be satisfied by quantum mechanical probabilities in the case of Bell experiments. If that were not the case, predicted quantum probabilities and correlations would be inconsistent with basic algebra.
Abstract. For unit vectors a, b, entanglement requires that spin measurements satisfy those just given for a≠b. Bell’s hidden variable quantum mechanical local particle with a=b cannot provide the values for a≠b and, hence, the values for an entangled particle system. The perception that any hidden variable quantum mechanical description of an entangled system of particles is non-local depends on Bell’s inappropriate definition of locality, which excludes entanglement for a distant particle, and the condition that a=b, which does not give the values for a≠b as required to give the quantum mechanical expectation values, when entanglement occurs for systems that are close, not distant, and ignores that entanglement is a local phenomenon. Expecting the condition a=b and a local particle analysis with no distant entangled particle to give the quantum mechanical result is absurd. Any disentanglement of once entangled particles moving apart sets the spin measurements at those given for a≠b if a≠b and those given for a=b if a=b. Any subsequent measurement of the set values agrees with those for a≠b if a≠b and those given for a=b if a=b just as if the particles are entangled still. In examples where Bell’s inequality is supposedly violated, the conditions for which the inequality is true are not met. There is no contradiction.