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.
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.
Bell inequalities have long served as the standard criterion to verify local hidden variable theories. Experimental violations of these inequalities are commonly interpreted as fundamental quantum nonlocality at the microscopic scale. This paper proves rigorously that Bell inequalities hold only under three prerequisites: countable discrete variables, countably additive probability measures, and classical absolute simultaneity. The state space of entangled quantum systems forms an uncountable infinite continuum. By the property of Vitali non-measurable sets in measure theory, the integral deduction of the CHSH inequality collapses naturally over continuous state spaces. Furthermore, special relativity excludes universal global time synchronization, so the absolute time embedded in Bell’s theory is an artificial assumption inconsistent with physical reality. Inspired by the framework dependence of probability revealed in Bertrand’s paradox, we demonstrate that the excess correlation observed in experiments is an apparent effect caused by the mismatch between classical measurement frameworks and the intrinsic continuum property of quantum systems, rather than evidence against local hidden variables. Statistical correlation between entangled particles originates from their shared evolutionary sequence defined at pair generation, with no superluminal interaction, fully complying with relativistic causality. This paper does not explore the ultimate origin of entanglement, but treats it as a given primitive physical law. Two falsifiable predictions are proposed to distinguish the continuum model from standard quantum mechanics. The conclusion is that hidden variable theories have never been falsified by experiments; they only need to be extended from discrete forms to continuum models compatible with relativity.
Violations of Bell inequalities are commonly interpreted as evidence for nonlocal influences or as constraints on realist descriptions. We show that the failure of Bell-type factorizability arises naturally when observable outcomes are obtained through a non-injective mapping from an underlying configuration space. In this setting, the standard factorization assumption can be viewed as an implicit requirement that observable variables admit a jointly factorizable completion at the underlying level. We demonstrate that this requirement need not hold when the mapping from underlying configurations to observables is many-to-one. The resulting breakdown of probabilistic factorization does not rely on superluminal dynamics or hidden causal influences, but follows from information loss under projection. Observable outcomes correspond to equivalence classes of underlying configurations, preventing the assignment of independent local variables. We illustrate this mechanism with an explicit toy model producing Bell–CHSH violations while preserving operational no-signalling and statistical independence of measurement settings. The model is not intended to reproduce quantum correlations quantitatively, and may exceed the Tsirelson bound; its role is to isolate the structural origin of the violation. This analysis does not contradict Bell’s theorem, but identifies a class of effective descriptions for which its factorizability assumption does not apply. The framework preserves locality at the underlying level, introduces no additional hidden-variable dynamics, and does not modify quantum mechanics. It clarifies how classical factorization is recovered in regimes where the effective mapping becomes approximately injective. In the operator language of quantum theory, the same mechanism admits a natural reformulation in terms of reduction to an effective observable subalgebra by a noncommutative conditional expectation.
The term Bell's theorem refers to a set of closely related results which imply that quantum mechanics is incompatible with local hidden variable theories. Bell's inequality is the statement that if measurements are performed independently on two space-like separated particles of an entangled pair, the assumption that outcomes depend on hidden variables implies an upper bound on the correlations between the outcomes. Quantum mechanics predicts correlations which violate this upper bound. The CHSH inequality is a specific Bell inequality in which classical correlation (i.e. if local hidden variables exist) can take the maximum value of 2. Quantum mechanics violates this bound, allowing for a higher bound on the correlation, which can take the maximum value 22, known as the Tsirelson bound. Popescu and Rohrlich showed that the assumption of relativistic causality allows for an even higher bound on the CHSH correlation, this value being 4. Why is the bound coming from causality higher than the Tsirelson bound? Are there relativistic causal dynamical theories which violate the Tsirelson bound? In the present paper we answer this question in the affirmative. We show that the pre-quantum theory of trace dynamics, from which quantum theory is emergent as a thermodynamic approximation, permits the CHSH correlation to take values higher than 22. We interpret our findings to suggest that quantum theory is approximate, and emergent from the more general theory of trace dynamics.
The Bell inequality constrains the outcomes of measurements on pairs of distant entangled particles. The Bell contradiction states that the Bell inequality is inconsistent with the calculated outcomes of these quantum experiments. This contradiction led many to question the underlying assumptions, viz. so-called realism and locality. The probability model underlying the Bell inequality is generally left implicit. We propose an explicit probability model for the CHSH version of the Bell experiment. This model has only two simultaneously observable detector settings per measurement, and therefore does not assume realism. The quantum expectation now becomes a conditional expectation, given the two detector settings. This probability model is in full agreement with both quantum mechanics and experiments. As a result, the model satisfies the Bell inequality; there are no so-called violations. We extend this model to include a hidden variable. This extended model is not Bell-separable. This non-separability implies that the model is non-deterministic or non-local (or both).
Since the experimental observation of the violation of the Bell-CHSH inequalities, much has been said about the non-local and contextual character of the underlying system. However, the hypothesis from which Bell's inequalities are derived differ according to the probability space used to write them. The violation of Bell's inequalities can, alternatively, be explained by assuming that the hidden variables do not exist at all, that they exist but their values cannot be simultaneously assigned, that the values can be assigned but joint probabilities cannot be properly defined, or that averages taken in different contexts cannot be combined. All of the above are valid options, selected by different communities to provide support to their particular research program.
Bell nonlocality is the experimentally testable incompatibility between quantum correlations and explanations based on local hidden variables. Its significance now extends beyond the foundations of quantum mechanics: a Bell-inequality violation can certify entanglement, randomness, measurement incompatibility, and cryptographic security without requiring a detailed model of the devices producing the observed data. This article develops the mathematical structure of Bell experiments and explains its role in device-independent quantum information. Conditional probability distributions are classified into local, quantum, and no-signalling sets; the Clauser-Horne-Shimony-Holt inequality is derived, and its classical, quantum, and algebraic limits are distinguished. Tsirelson’s bound is obtained from the CHSH operator, while Popescu-Rohrlich correlations demonstrate that no-signalling alone does not identify the quantum set. The relation between entanglement and nonlocality is examined through local entangled states, hidden nonlocality, and activation. Nonlocal games, semidefinite relaxations, self-testing, certified randomness, and device-independent quantum key distribution are treated as operational consequences of Bell violation. Particular attention is given to entropy accumulation, finite-data security, loophole-free experiments, and the limitations of device independence. Bell nonlocality emerges as both a foundational separation between classical and quantum causal models and a quantitative resource for cryptographic and informational tasks.