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Centrality, Noncentral Selection, and the Scope of Bell–CHSH:

Emmerson, Parker

Published Jan 1, 2025• 0 citations
Publisher

Abstract

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, 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 areweighted 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 selectionrequires $\delta\ge (2\sqrt2-2)/4\approx0.2071$.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 $\delta$ 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.

DOI
10.6084/m9.figshare.30908252

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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 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 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
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 Sobs≤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 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.
For local hidden-variable accounts of the CHSH experiment that are faithful -- reproducing the observed singles and coincidence rates and unbiased marginals of a polarization singlet with symmetric detector efficiency ηeffη_{\rm eff}ηeff​ -- I determine the minimal measurement dependence MMM (Hall's measure) needed to reach a CHSH value SSS. Linear programming over all local strategies yields M(S,ηeff)=max⁡{0,ηeff((S+2)ηeff−4)/6}M(S,η_{\rm eff})=\max\{0,η_{\rm eff}((S+2)η_{\rm eff}-4)/6\}M(S,ηeff​)=max{0,ηeff​((S+2)ηeff​−4)/6}, whose edges reproduce Hall's bound, the Garg-Mermin threshold, and the postselection ceiling. The quantum point is certified exactly: M(22,9/10)=(272−33)/100M(2\sqrt{2},9/10)=(27\sqrt{2}-33)/100M(22​,9/10)=(272​−33)/100, by primal and dual certificates in Q(2)\mathbb{Q}(\sqrt{2})Q(2​). I solve the detection profile under which a deterministic sign model reproduces the singlet exactly, and prove exact quantum correlations and angle-independent coincidence rates jointly impossible in this class. Mixtures of this profile with measurement dependence trade MMM against a cos⁡4(a−b)\cos 4(a-b)cos4(a−b) modulation of the fourfold coincidence sum, up to 12.4% in relative amplitude, identical at the two CHSH angles; existing data exclude its full amplitude and bound its mixtures with flat detection to a detection share of order 15%; its sub-percent content is unbounded. A two-setting cross-ratio and a settings-torus log-Fourier protocol, calibrated against channel-efficiency imbalance, reach 5σ5σ5σ at 0.1% within hours; a flat result excludes the solved family to a detection share of order 1%. A local mechanism, moiré phase locking, with deterministic phase evolution and quenched randomness, attains the certified floor exactly at ηeff=1η_{\rm eff}=1ηeff​=1 and, with the solved profile, reaches M=0M=0M=0 at ηeff≈0.80η_{\rm eff}\approx 0.80ηeff​≈0.80. Its floor-attaining census shares a low-order Fourier skeleton across independent solutions and is not realizable by station-local offsets on the terrains tested.
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.
I demonstrate that Bell's theorem is based on circular reasoning and thus a fundamentally flawed argument. It unjustifiably assumes the additivity of expectation values for dispersion-free states of contextual hidden variable theories for non-commuting observables involved in Bell-test experiments, which is tautologous to assuming the bounds of ±2 on the Bell-CHSH sum of expectation values. Its premises thus assume in a different guise the bounds of ±2 it sets out to prove. Once this oversight is ameliorated from Bell's argument by identifying the impediment that leads to it and local realism is implemented correctly, the bounds on the Bell-CHSH sum of expectation values work out to be ±2√2 instead of ±2, thereby mitigating the conclusion of Bell's theorem. Consequently, what is ruled out by any of the Bell-test experiments is not local realism but the linear additivity of expectation values, which does not hold for non-commuting observables in any hidden variable theories to begin with.
I demonstrate that Bell's theorem is based on circular reasoning and thus a fundamentally flawed argument. It unjustifiably assumes the additivity of expectation values for dispersion-free states of contextual hidden variable theories for non-commuting observables involved in Bell-test experiments, which is tautologous to assuming the bounds of ±2 on the Bell-CHSH sum of expectation values. Its premises thus assume in a different guise the bounds of ±2 it sets out to prove. Once this oversight is ameliorated from Bell's argument by identifying the impediment that leads to it and local realism is implemented correctly, the bounds on the Bell-CHSH sum of expectation values work out to be ±2√2 instead of ±2, thereby mitigating the conclusion of Bell's theorem. Consequently, what is ruled out by any of the Bell-test experiments is not local realism but the linear additivity of expectation values, which does not hold for non-commuting observables in any hidden variable theories to begin with.