This work derives an exact and operational tradeoff between Bell–CHSH violation and measurement dependence in local hidden-variable models. Measurement independence is relaxed via a single parameter MMM, defined as the maximum total-variation distance between hidden-variable distributions conditioned on different setting pairs. Under locality, we prove the sharp inequality ∣S∣≤min(2+6M,4)|S| \le \min(2 + 6M, 4)∣S∣≤min(2+6M,4), valid for all admissible MMM, and construct an explicit two-state local model saturating the bound for every M∈[0,1/3]M \in [0, 1/3]M∈[0,1/3]. The result is invertible: any observed CHSH value ∣S∣=2+Δ|S| = 2 + \Delta∣S∣=2+Δ certifies a minimum required measurement dependence M≥Δ/6M \ge \Delta/6M≥Δ/6. The parameter MMM has a direct hypothesis-testing interpretation, placing Bell violations and setting predictability in the same operational units used in statistical distinguishability and cryptography. The paper also provides a linear-programming template that reproduces the analytic frontier and can be adapted to other Bell functionals.
This work derives an exact and operational tradeoff between Bell–CHSH violation and measurement dependence in local hidden-variable models. Measurement independence is relaxed via a single parameter MMM, defined as the maximum total-variation distance between hidden-variable distributions conditioned on different setting pairs. Under locality, we prove the sharp inequality ∣S∣≤min(2+6M,4)|S| \le \min(2 + 6M, 4)∣S∣≤min(2+6M,4), valid for all admissible MMM, and construct an explicit two-state local model saturating the bound for every M∈[0,1/3]M \in [0, 1/3]M∈[0,1/3]. The result is invertible: any observed CHSH value ∣S∣=2+Δ|S| = 2 + \Delta∣S∣=2+Δ certifies a minimum required measurement dependence M≥Δ/6M \ge \Delta/6M≥Δ/6. The parameter MMM has a direct hypothesis-testing interpretation, placing Bell violations and setting predictability in the same operational units used in statistical distinguishability and cryptography. The paper also provides a linear-programming template that reproduces the analytic frontier and can be adapted to other Bell functionals.
Bell--CHSH is frequently paraphrased as ``local realism is impossible'' after experimental violations of CHSH. The theorem is correct, but the paraphrase hides a structural assumption: \emph{selection centrality} (fair sampling), i.e.\ that the detected sample is not a setting- and hidden-variable--dependent reweighting of the hidden prior. We give a precise measure-theoretic separation between (i) the \emph{central} sector, where Eobs=Efull and CHSH≤2 holds under measurement independence and locality, and (ii) the \emph{noncentral} sector, where the observed correlations are computed under a reweighted measure dνa,b=w(a,b,λ)dρ. We prove a sharp CHSH inflation bound Sobs≤2+4δ,δ:=supa,b∫∣w(a,b,λ)−1∣ρ(dλ), so reproducing Tsirelson's 22 by strictly local measurement-independent models via selection requires δ≥(22−2)/4≈0.2071. We provide (A) a canonical Radon--Nikod\'ym construction (piecewise constants on sign regions plus a small positive remainder), (B) an explicit local factorized detection-loophole construction that reproduces arbitrary finite correlation tables, and (C) an implementable optics protocol to estimate or upper-bound δ from time-tag data via window dithering, threshold and spectral sweeps, and auxiliary-tag binning. This paper critiques not the validity of Bell's theorem, but the common overstatement of what CHSH violations imply without an explicit, quantitative centrality audit.
Bell's inequality was originally derived under the assumption that experimenters are free to select detector settings independently of any local ``hidden variables'' that might affect the outcomes of measurements on entangled particles. This assumption has come to be known as ``measurement independence'' (also referred to as ``freedom of choice'' or ``settings independence''). For a two-setting, two-outcome Bell test, we derive modified Bell inequalities that relax measurement independence, for either or both observers, while remaining locally causal. We describe the loss of measurement independence for each observer using the parameters M1 and M2, as defined by Hall in 2010, and also by a more complete description that adds two new parameters, which we call $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{{M}_{1}}$ and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{{M}_{2}}$, deriving a modified Bell inequality for each description. These ``relaxed'' inequalities subsume those considered in previous work as special cases, and quantify how much the assumption of measurement independence needs to be relaxed in order for a locally causal model to produce a given violation of the standard Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality. We show that both relaxed Bell inequalities are tight bounds on the CHSH parameter by constructing locally causal models that saturate them. For any given Bell inequality violation, the new two-parameter and four-parameter models each require significantly less mutual information between the hidden variables and measurement settings than previous models. We conjecture that the new models, with optimal parameters, require the minimum possible mutual information for a given Bell violation. We further argue that, contrary to various claims in the literature, relaxing freedom of choice need not imply superdeterminism.
This paper presents a structural analysis of Bell–CHSH correlations using a hierarchy of geometric hidden-variable models defined on the unit circle. Bell’s theorem constrains hidden-variable models under the joint assumptions of locality, deterministic response functions, and measurement independence. The paper examines how Bell–CHSH behavior changes when these assumptions are modified one at a time. A minimal local model with setting-independent preparation obeys the classical Bell–CHSH bound |S| <= 2. A setting-dependent update can produce Bell violation but introduces operational signalling. Direct imposition of the cosine correlation is used as a structural reference, reproducing Tsirelson’s bound while preserving no-signalling marginals. The paper then shows that Tsirelson-level correlations can also arise from a geometric, context-dependent sampling rule without inserting the cosine function directly into the preparation law. In that construction, local deterministic response functions and operational no-signalling are preserved, while measurement independence is relaxed. The models are presented as structural illustrations rather than physical hidden-variable proposals. Their purpose is to make the assumption-dependence of Bell correlations explicit in a transparent geometric framework. Internal reference: CGI-RSR-000021.
A local deterministic model can reproduce Bell-inequality violations only if the hidden state depends on the measurement settings. Hall found the minimum such dependence for the singlet state [Phys. Rev. Lett. 105, 250404 (2010)] and posed the corresponding problem for the Mermin state [Phys. Rev. A 84, 022102 (2011)]. This paper solves it, for the n-party GHZ-Mermin scenario at every n≥3, with the marginals reproduced as well as the correlators. The minimum measurement-dependence fraction is Fmin(n)=R/[2(R+1)], with R=2⌊(n−1)/2⌋ the Mermin violation ratio: 1/3,2/5,4/9,8/17,…, always below 1/2. The lower bound follows from Mermin's inequality by a general overlap bound; the upper bound is an explicit model, uniform on the hidden-variable classes that violate the fewest constraints, with closed-form counts; a reduction theorem shows that for Mermin scenarios the marginals impose no additional dependence. The Mermin values are also certified by integer-arithmetic enumeration through n=22. The formula was stated as a conjecture in the first version of this paper and was first proved, by a different route, by Wang, Yao and Chen; the proof here was obtained independently and the two agree. Every minimal contradiction of size k among stabilizer constraints has correlator-only minimum 1/(k−1). With the marginals included the minimum is the same on every core of the C5 ring cluster, on the size-4 and size-5 cores of C4 and on every four-core of C6, but larger on the size-6 and size-7 cores of C4 and on 112 of the 1,920 five-cores of C6, with exact certificates on both sides in every case.
Local hidden-variable models can reproduce Bell violations if settings depend on hidden variables. Using total-variation distance, we determine the minimum dependence needed to reproduce all outcome probabilities for Pauli X and Y measurements on n-qubit Greenberger-Horne-Zeilinger states, for every n ≥ 3, visibility and coherence phase. For odd n, the larger magnitude of the real and imaginary coherence components sets the cost; for even n, the smaller can raise it. States with identical, Bell-violating Mermin data can thus have different costs.
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.