Centrality, Noncentral Selection, and the Scope of Bell–CHSH: A Measure-Theoretic Critique, Sharp Inflation Bounds, Canonical Reweighting Constructions, and an Experimental Protocol

Centrality, Noncentral Selection, and the Scope of Bell–CHSH: A Measure-Theoretic Critique, Sharp Inflation Bounds, Canonical Reweighting Constructions, and an Experimental Protocol Parker Emmerson December 18, 2025 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: selection centrality (fair sampling), i.e. that the detected sample is not a settingand hiddenvariable–dependent reweighting of the hidden prior. We give a precise measure-theoretic separation between (i) the central sector, where Eobs = Efull and CHSH ≤ 2 holds under measurement independence and locality, and (ii) the 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 := sup a,b Z|w(a, b, λ)−1|ρ(dλ), so reproducing Tsirelson’s 2 √2 by strictly local measurement-independent models via selection requires δ≥ (2 √2− 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. Contents 1 Introduction (what is being critiqued) 2 2 Framework 2 2.1 Local outcomes ..................................... 2 2.2 Local detection and selection ............................. 2 2.3 Observed vs. unconditional correlations ....................... 3 2.4 A quantitative noncentrality statistic ......................... 3 3 The central sector: where Bell–CHSH actually lives 3 4 Noncentral selection: sharp inflation bounds 3 5 Constructive noncentral models: what becomes possible once centrality is dropped 4 5.1 Canonical RN-density recipe (measure-theoretic) .................. 4 5.2 Explicit local factorized construction for arbitrary finite correlation tables .... 5 1 6 PV as a physical motivation: fibered noninjectivity (optional) 5 7 Experimental protocol to estimate/bound δ5 8 Conclusion 7 A Appendix: a common mistaken “local model” and the correct integral 7 1 Introduction (what is being critiqued) Bell’s theorem is a mathematically correct statement: under measurement independence (MI), local response functions, and bounded outcomes, the unconditional correlations obey CHSH ≤ 2. Modern experiments violate CHSH and therefore exclude the central local hidden-variable class. The critique addressed here is narrower and more operational: CHSH violation implies nonlocality only after one has controlled/justified that the observed correlation is the same as the unconditional correlation, i.e. that selection is central (fair sampling) or that any deviation from centrality is bounded below the amount needed to inflate CHSH above 2. This is not a semantic quibble: in any real experiment, “the observed sample” is produced by hardware gates, thresholds, timing rules, coincidence identification, and analysis windows. All of these can be expressed as a settingand hidden-variable–dependent selection weight. 2 Framework Let SA, SB be the (finite or continuous) setting sets for Alice and Bob. Let (Λ ,F, ρ ) be a probability space of hidden variables λ .Measurement independence (MI) means ρ does not depend on (a, b). 2.1 Local outcomes Outcomes are bounded and local: A(a, λ)∈[−1,1], B(b, λ)∈[−1,1], with A depending only on ( a, λ ) and B only on ( b, λ ). The deterministic {± 1 } case is included. 2.2 Local detection and selection Let ηA(a, λ), ηB(b, λ)∈[0,1] be detection probabilities (or acceptance gates). Define Z(a, b) := ZηA(a, λ)ηB(b, λ)ρ(dλ)∈(0,1], and the normalized selection weight w(a, b, λ) := ηA(a, λ)ηB(b, λ) Z(a, b),Zw(a, b, λ)ρ(dλ) = 1.(1) Equivalently, define the detected-sample measure νa,b by dνa,b(λ)=w(a, b, λ)ρ(dλ). 2 2.3 Observed vs. unconditional correlations Define Eobs(a, b) := ZA(a, λ)B(b, λ)dνa,b(λ) = ZA(a, λ)B(b, λ)w(a, b, λ)ρ(dλ), and the unconditional (“full”) correlation Efull(a, b) := ZA(a, λ)B(b, λ)ρ(dλ). For a CHSH quartet (a, a′, b, b′), define Sobs := |Eobs(a, b)−Eobs(a, b′)|+|Eobs(a′, b)+Eobs(a′, b′)|. 2.4 A quantitative noncentrality statistic Define the L1deviation from centrality δ(a, b) := Z|w(a, b, λ)−1|ρ(dλ), δ := sup a,b δ(a, b).(2) Note that TV(νa,b, ρ) = 1 2δ(a, b). 3 The central sector: where Bell–CHSH actually lives Definition 3.1 (Central selection / fair sampling).Selection is central if ηA ( a, λ ) ≡ηA ( a )and ηB(b, λ)≡ηB(b). Equivalently, w(a, b, λ)≡1for all (a, b). Lemma 3.2 (Pointwise CHSH bound).Fix λ. For any a, a′, b, b′and any A, B ∈[−1,1], |A(a, λ)B(b, λ)−A(a, λ)B(b′, λ)|+|A(a′, λ)B(b, λ)+A(a′, λ)B(b′, λ)| ≤ 2. Proof. This is the standard CHSH algebra; boundedness |A|,|B| ≤ 1 is enough. (Determinism is not required.) Theorem 3.3 (Centrality identity and Bell–CHSH).Assume MI, locality, bounded outcomes, and central selection. Then Eobs(a, b) = Efull(a, b)for all (a, b), hence for any quartet Sobs ≤2. Proof. If w≡ 1 then Eobs = Efull by definition. Lemma 3.2 holds pointwise in λ ; integrating w.r.t. ρyields CHSH≤2. Remark 3.4 (Hiddenness is irrelevant to CHSH).Lemma 3.2 is pointwise in λ . Whether λ is observed or not does not affect CHSH; what matters is whether the observed sample is governed by ρ(central) or by a reweighted νa,b (noncentral). 4 Noncentral selection: sharp inflation bounds Theorem 4.1 (Deviation bound).For each (a, b), Eobs(a, b)−Efull(a, b)≤δ(a, b). 3 Proof. Eobs −Efull =ZAB (w−1) dρ, so by |AB|≤1, |Eobs −Efull| ≤ Z|w−1|dρ =δ(a, b). Theorem 4.2 (Sharp CHSH inflation).Under MI and locality with bounded outcomes, Sobs ≤2+4δ. Equivalently, Sobs ≤2 + 8 sup a,b TV(νa,b, ρ). Proof. Apply Theorem 4.1 to the four pairs in CHSH and use Theorem 3.3 for Sfull ≤2: Sobs ≤Sfull + 4 X j=1 δ(θj)≤2+4δ. Corollary 4.3 (Tsirelson noncentrality threshold).If Sobs = 2 √2 is reproduced by a strictly local, MI model via selection, then necessarily δ≥2√2−2 4≈0.2071. 5 Constructive noncentral models: what becomes possible once centrality is dropped 5.1 Canonical RN-density recipe (measure-theoretic) Fix a pair ( a, b ). Suppose there exist measurable disjoint regions U+ ab, U− ab ⊂ Λ with ρ ( U± ab ) > 0 such that A(a, λ)B(b, λ) = +1 on U+ ab, A(a, λ)B(b, λ)=−1onU− ab. Let Etgt(a, b)∈[−1,1] be a target correlation and set αab := 1+Etgt(a, b) 2. Define the unnormalized RN density ˜wab(λ) := αab ρ(U+ ab)1U+ ab (λ) + 1−αab ρ(U− ab)1U− ab (λ)+ε ψab(λ),(3) where ψab ≥0 has Rψab dρ = 1 and ε>0 is small. Normalize: wab(λ) := ˜wab(λ) R˜wab dρ. Then wab ≥ 0 and Rwab dρ = 1, and Eobs ( a, b ) can be made arbitrarily close to Etgt ( a, b ) by choosing εsmall. Remark 5.1 (Physical realizability vs. pure RN construction).Equation (3) constructs the conditional measure νa,b . To realize wab via local detection functions ηA ( a, λ ) ηB ( b, λ ), one must ensure wab is consistent with a factorized form across all a and all b . The next subsection provides an explicit factorized local construction on a finite setting grid. 4 5.2 Explicit local factorized construction for arbitrary finite correlation tables Fix finite settings {a1, . . . , am}and {b1, . . . , bn}and targets Etgt ij ∈[−1,1]. Theorem 5.2 (Finite-table simulation by strictly local detection selection).There exists a probability space (Λ , ρ ), local deterministic outcomes A ( ai, λ ) , B ( bj, λ ) ∈ {± 1 } , and local detection indicators DA(ai, λ), DB(bj, λ)∈ {0,1}such that: (a) (MI) ρis independent of (i, j), (b) (locality) Adepends only on (i, λ)and Bonly on (j, λ), (c) the observed correlations conditioned on coincidence satisfy Eobs(ai, bj) = Etgt ij for all i, j. Construction. Let Λ := { 1 , . . . , m}×{ 1 , . . . , n}× [0 , 1] with ρ the product of uniform counting measure on {1, . . . , m}×{1,...,n}and Lebesgue on [0,1]. Write λ= (u, v, t). Define local detection indicators DA(ai, λ) := 1{u=i}, DB(bj, λ) := 1{v=j}. Then for setting pair ( i, j ), coincidences occur iff ( u, v ) = ( i, j ); the coincidence probability is Zij =ρ(u=i, v =j) = 1/(mn) (setting-independent). Define outcomes A(ai, λ) := +1, B(bj, λ) := (+1, t ≤αij, −1, t > αij ,αij := 1+Etgt ij 2. These are local because A depends only on ( i, λ ) and B only on ( j, λ ). Conditioned on coincidence (u, v)=(i, j), the variable tis uniform on [0,1], hence E[B|coincidence at (i, j)] = (+1)αij + (−1)(1 −αij)=2αij −1=Etgt ij . Since A≡+1 on coincidences, Eobs(i, j) = Etgt ij . Remark 5.3 (What this theorem means and what it does not).This is an explicit detectionloophole (noncentral selection) simulation. It does not contradict Bell’s theorem because it violates centrality/fair-sampling. It also has low coincidence rate 1 / ( mn ); more efficient constructions exist for specific targets (e.g. singlet) but still require noncentrality quantified by δ. 6 PV as a physical motivation: fibered noninjectivity (optional) The mathematics above does not require any PV-specific mechanism; any hidden-variable space with selection dependence suffices. A PV narrative can be made precise as follows. Suppose the hidden state has the form Λ ∼ =V× Φ where V is an observed kinematic parameter space and Φ is an auxiliary PV coordinate. Let L : V× Φ → M be an operational kinematic summary (e.g. a Lorentz parameter) that is noninjective on fibers: distinct ( v, ϕ ) share the same observed ℓ = L ( v, ϕ ). Then inside a fixed observed cell ℓ , the auxiliary ϕ can carry different sign structure for A ( a, λ ) B ( b, λ ), enabling reweighting without changing the coarse observed kinematics. 7 Experimental protocol to estimate/bound δ The inflation bound Sobs ≤ 2+4 δ makes the noncentrality parameter δ experimentally meaningful: if an experiment measures Sobs and independently upper-bounds δ below ( Sobs − 2) / 4, then all strictly local MI models of the noncentral-selection type are excluded. 5 M(observed kinematics) ℓ Λℓ⊂V×Φ U+ U− AB ≡+1 AB ≡ −1 Noninjective fiber: same ℓbut different hidden subregions U± Figure 1: PV-style motivation: noninjective hidden fibers allow internal sign structure inside a fixed observed kinematic cell. Protocol overview (optics / time-tag experiments) P1. Trial definition (event-ready if possible). Use heralding to define trials independent of outcomes; otherwise use fixed clock bins. P2. Measure singles efficiencies vs setting. Estimate ˆηA ( a ) and ˆηB ( b ) for each setting; look for setting dependence. P3. Coincidence-window dithering. If coincidences are defined by |tA−tB|< W , repeat for Won a small grid. P4. Threshold and spectral sweeps. Repeat CHSH runs for detector thresholds ± 5% ,± 10% and filters (e.g. 1,3,10 nm). P5. Auxiliary-tag binning (best practice). Partition trials by an auxiliary tag X correlated with hidden microphysics (arrival-time residual, pulse energy, frequency bin, etc.) and estimate a binned w. P6. Compute a conservative upper confidence bound ˆ δ(+) .Compare Sobs to 2 + 4 ˆ δ(+) (one-sided test). Concrete estimators Window-derived bound. For each pair (a, b) define ∆E(a, b) := max W ˆ E(a, b;W)−min W ˆ E(a, b;W),ˆ δW:= 1 2max a,b ∆E(a, b). This is conservative: changing the window probes hidden-time selection dependence. Auxiliary-tag binned estimator (recommended). Let X∈ { 1 , . . . , K} be a tag. For each (a, b) and bin irecord trials and coincidences and set bw(a, b |i) := c Pr(coinc |a, b, X =i) PK j=1 c Pr(coinc |a, b, X =j) ˆpj ,ˆpi=c Pr(X=i). Then ˆ δ(a, b) := K X i=1 |bw(a, b |i)−1|ˆpi,ˆ δ:= max a,b ˆ δ(a, b). Bootstrap over trials to get an upper CI ˆ δ(+). 6 Decision rule. If ˆ Sobs >2+4ˆ δ(+) +z1−ασS, then local MI noncentral-selection models with δ≤ˆ δ(+) are excluded at one-sided confidence 1−α. 8 Conclusion Bell–CHSH is correct in the central sector: if the detected sample is not a hidden-variable reweighting, then MI+locality imply CHSH ≤ 2. What CHSH violation rules out depends operationally on whether centrality holds. The noncentral sector is cleanly parameterized by the RN weight w and its L1 distance from centrality δ ; the sharp inflation bound Sobs ≤ 2 + 4 δ gives a quantitative bridge between theory and experimental audits. Rather than asserting a “refutation of Bell,” the defensible scientific program is: (i) formalize noncentral selection precisely, (ii) quantify how much noncentrality is required to mimic a given CHSH violation, and (iii) experimentally bound that noncentrality. A Appendix: a common mistaken “local model” and the correct integral A frequently proposed local deterministic model is A(θa, λ) = sgn(cos(θa−λ)), B(θb, λ) = −sgn(cos(θb−λ)), with λ∼Unif [0 , 2 π ). Its correlation is not −cos ( θa−θb ). A direct geometric computation gives a piecewise linear function of ∆ = θa−θb(mod π): E(∆) = −1−2|∆| π, which cannot exceed CHSH= 2. Obtaining −cos locally requires relaxing centrality (selection dependence), or relaxing MI, or relaxing locality. References [1] J. S. Bell, “On the Einstein Podolsky Rosen Paradox,” Physics Physique Fizika 1(1964), 195–200. [2] J. F. Clauser, M. A. Horne, A. Shimony, R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23 (1969), 880–884. [3] P. M. Pearle, “Hidden-variable example based upon data rejection,” Phys. Rev. D 2(1970), 1418–1425. [4] A. Garg and N. D. Mermin, “Detector inefficiencies in the Einstein-Podolsky-Rosen experiment,” Phys. Rev. D 35 (1987), 3831–3835. [5] P. H. Eberhard, “Background level and counter efficiencies required for a loophole-free Einstein-Podolsky-Rosen experiment,” Phys. Rev. A 47 (1993), R747–R750. [6] B. Hensen et al., “Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres,” Nature 526 (2015), 682–686. 7 [7] M. Giustina et al., “Significant-Loophole-Free Test of Bell’s Theorem with Entangled Photons,” Phys. Rev. Lett. 115 (2015), 250401. [8] L. K. Shalm et al., “Strong Loophole-Free Test of Local Realism,” Phys. Rev. Lett. 115 (2015), 250402. 8