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Bell–CHSH Under Setting-Dependent Selection: Sharp Total-Variation Bounds and an Experimental Audit Protocol

Parker Emmerson

Published Jan 23, 2026• 0 citations
Publisher

Abstract

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 $E_{\mathrm{obs}}=E_{\mathrm{full}}$ and CHSH$\le 2$ holds under measurement independence and locality, and (ii) the \emph{noncentral} sector, where the observed correlations are computed under a reweighted measure $d\nu_{a,b}=w(a,b,\lambda)\,d\rho$. We prove a sharp CHSH inflation bound \[ S_{\mathrm{obs}}\le 2+4\delta, \qquad \delta:=\sup_{a,b}\int |w(a,b,\lambda)-1|\,\rho(d\lambda), \] so reproducing Tsirelson's $2\sqrt2$ by strictly local measurement-independent models via selection requires $\delta\ge (2\sqrt2-2)/4\approx0.2071$. 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 $\delta$ 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.

DOI
10.3390/quantum8010008

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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=EfullE_{\mathrm{obs}}=E_{\mathrm{full}}Eobs​=Efull​ and CHSH≤2\le 2≤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) dρd\nu_{a,b}=w(a,b,\lambda)\,d\rhodνa,b​=w(a,b,λ)dρ. We prove a sharp CHSH inflation bound [ S_{\mathrm{obs}}\le 2+4\delta, \qquad \delta:=\sup_{a,b}\int |w(a,b,\lambda)-1|,\rho(d\lambda), ] so reproducing Tsirelson's 222\sqrt222​ by strictly local measurement-independent models via selection requires δ≥(22−2)/4≈0.2071\delta\ge (2\sqrt2-2)/4\approx0.2071δ≥(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\deltaδ 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--CHSH is frequently paraphrased as ``local realism is impossible'' after experimentalviolations 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 asetting- and hidden-variable--dependent reweighting of the hidden prior.We give a precise measure-theoretic separation between (i) the \emph{central} sector, whereEobs=EfullE_{\mathrm{obs}}=E_{\mathrm{full}}Eobs​=Efull​ and CHSH≤2\le 2≤2 holds under measurement independence and locality,and (ii) the \emph{noncentral} sector, where the observed correlations are computed under areweighted measure dνa,b=w(a,b,λ) dρd\nu_{a,b}=w(a,b,\lambda)\,d\rhodνa,b​=w(a,b,λ)dρ.We prove a sharp CHSH inflation boundSobs≤2+4δ,δ:=sup⁡a,b∫∣w(a,b,λ)−1∣ ρ(dλ),S_{\mathrm{obs}}\le 2+4\delta,\qquad\delta:=\sup_{a,b}\int |w(a,b,\lambda)-1|\,\rho(d\lambda),Sobs​≤2+4δ,δ:=supa,b​∫∣w(a,b,λ)−1∣ρ(dλ),so reproducing Tsirelson's 222\sqrt222​ by strictly local measurement-independent models via selectionrequires δ≥(22−2)/4≈0.2071\delta\ge (2\sqrt2-2)/4\approx0.2071δ≥(22​−2)/4≈0.2071.We provide (A) a canonical Radon--Nikod\'ym construction (piecewise constants on sign regions plusa small positive remainder), (B) an explicit local factorized detection-loophole construction thatreproduces arbitrary finite correlation tables, and (C) an implementable optics protocol to estimateor upper-bound δ\deltaδ from time-tag data via window dithering, threshold and spectral sweeps, andauxiliary-tag binning.This paper critiques not the validity of Bell's theorem, but the common overstatement of what CHSHviolations imply without an explicit, quantitative centrality audit.
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: selection centrality (fair sampling), i.e. that the detected sample is not a setting-and hiddenvariable-dependent reweighting of the hidden prior. We give a precise measure-theoretic separation between (i) the central sector, where E obs = E full and CHSH≤ 2 holds under measurement independence and locality, and (ii) the noncentral sector, where the observed correlations are computed under a reweighted measure dν a,b = w(a, b, λ) dρ. We prove a sharp CHSH inflation bound
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 is the data set for the published, peer reviewed paper: https://ijqf.org/archives/8027 "Phenomenological Velocity and Bell–CHSH: Exceptional-Locus Semantics, Selection Simulations of − cos, and a Microcausal Realization:" https://ijqf.org/wp-content/uploads/2026/02/IJQF2026v12n2p10.pdf Bell--CHSH is often read as a universal prohibition on \emph{local hidden variables}. Mathematically, however, the CHSH bound S≤2S\le 2S≤2 is a theorem about a particular hypothesis class: a \emph{single} Kolmogorov probability space carrying all counterfactual outcomes, together with measurement independence, Bell locality (factorized response functions), and bounded outputs. On that class the proof is pointwise and unconditional, so no algebraic reformulation of a parameter (including a ``phenomenological velocity'' (PV)) can yield Tsirelson-scale singlet correlations E(a,b)=−cos⁡(a−b)E(a,b)=-\cos(a-b)E(a,b)=−cos(a−b) on a Tsirelson quartet. We then make precise two distinct ways in which PV can remain meaningful as a \emph{local} and in that sense ``hidden'' parameter while coexisting with Tsirelson-scale correlations, without contradicting Bell's theorem. First, we formalize ``exceptional-locus/defined-only'' conventions as setting-dependent acceptance (post-selection): reported correlators are conditional expectations under a setting-indexed family of accepted laws {νab}\{\nu_{ab}\}{νab​}. Within this rung-1 semantics we give an explicit Bell-local deterministic base model whose accepted-sample joint law exactly matches the singlet law (unbiased marginals and correlator −cos⁡(a−b)-\cos(a-b)−cos(a−b)) for all a,ba,ba,b, via an explicit acceptance rule γ(a,b,λ)∈[0,1]\gamma(a,b,\lambda)\in[0,1]γ(a,b,λ)∈[0,1]. Second, we give a PV-indexed microcausal (operator-algebraic) realization: PV parameterizes a canonical SU(2)SU(2)SU(2) unitary that conjugates local observables inside commuting Alice/Bob subalgebras, and in the singlet state the unconditional correlator equals −cos⁡(a−b)-\cos(a-b)−cos(a−b) and attains Tsirelson's value 222\sqrt222​. The results clarify a semantic and structural trichotomy: any ``PV-to-Tsirelson'' account must (i) change the effective ensemble by selection/context, or (ii) leave the Kolmogorov (commutative) model class (e.g.\ via noncommutativity or quasi-probability), or (iii) relax a Bell premise such as measurement independence or Bell locality. In this sense, Bell--CHSH constrains a specific formalization of hidden variables rather than ownership of the terms ``local'', ``real'', or ``hidden'' in physics discourse.
Bell-CHSH measurements with polarization-entangled photons are commonly used as nonlocality witnesses. Here we show that the same coincidence data can also provide a quantitative metrological certificate for optical retardance sensing. For a local birefringent phase encoded on one photon, we derive the classical Fisher information directly from experimentally accessible CHSH correlators and obtain a lower bound expressed in terms of the measured correlations and the phase derivative of the CHSH parameter. In the unbiased-marginal two-qubit regime, the canonical CHSH analyzer settings are shown to be simultaneously Bell-optimal and Fisher-optimal, saturating the quantum Fisher information for retardance estimation in the ideal case. We further analyze reduced visibility, phase noise, analyzer misalignment, and finite coincidence counts, and construct conservative lower confidence bounds that can be evaluated without quantum-state tomography. The results also clarify that Bell violation and metrological usefulness are distinct resources: nonzero Fisher information can persist below the CHSH-violation threshold. The protocol therefore enables a standard Bell-test data set to provide both a nonlocality benchmark and a statistically controlled sensitivity certificate for quantum polarimetry.
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.