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arXiv:2608.18886

The exact price of local realism in CHSH experiments: a measurement-dependence-detection trade-off surface, a moiré phase-locking mechanism that saturates it, and an unmeasured fringe in the fourfold coincidence sum

Aaron Alai

quant-phSubmitted Aug 19, 2026• Announced Aug 20, 2026• 0 citations• Revised Oct 28, 2026 (v6)
v1 · Aug 19, 2026v2 · Aug 21, 2026v3 · Aug 27, 2026v4 · Oct 18, 2026v5 · Oct 21, 2026v6 · Oct 28, 2026
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Abstract

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 $η_{\rm eff}$ -- I determine the minimal measurement dependence $M$ (Hall's measure) needed to reach a CHSH value $S$. Linear programming over all local strategies yields $M(S,η_{\rm eff})=\max\{0,η_{\rm eff}((S+2)η_{\rm eff}-4)/6\}$, whose edges reproduce Hall's bound, the Garg-Mermin threshold, and the postselection ceiling. The quantum point is certified exactly: $M(2\sqrt{2},9/10)=(27\sqrt{2}-33)/100$, by primal and dual certificates in $\mathbb{Q}(\sqrt{2})$. 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 $M$ against a $\cos 4(a-b)$ 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 $5σ$ 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 $η_{\rm eff}=1$ and, with the solved profile, reaches $M=0$ at $η_{\rm eff}\approx 0.80$. 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.

Comments
16 pages, 5 figures. Code and certificates at https://github.com/RandomInternetPreson/moire-phase-space-sampler. v6: Sec. 7 adds the bandwidth of the floor-attaining census on a lattice of offsets (band-limited censuses certified by LP), its dependence on the flight-time/coupling-scale cell, and the roles-swapped solve; deposit version 4 at doi:10.5281/zenodo.23537610

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Bell's theorem rests on three assumptions: locality, determinism, and measurement independence. Experiments violate Bell inequalities, so at least one assumption must be abandoned. The first two exits are well explored; the third — allowing lawful, partial correlation between measurement settings and hidden variables — is usually dismissed as "superdeterminism" and assumed to require total conspiracy. This paper computes what that third exit actually costs. Hall (2010) priced the bipartite case exactly: 13.81% of measurement independence must be surrendered to reproduce singlet-state correlations. This work extends that program to multipartite GHZ/Mermin scenarios and computes, via exact linear programming, the minimum measurement dependence required to reproduce the full faithful statistics — every correlator and every marginal — for 3 through 13 parties. The floors climb a lawful staircase: 1/3, 2/5, 4/9, 8/17, 16/33, 32/65, all obeying the closed form F = R/(2(R+1)) with R the Mermin violation ratio. A proven universal ceiling shows the required dependence never exceeds 1/2 at any experiment size: quantum statistics never force total superdeterminism. The optimal hidden-variable distributions take a closed physical form — uniform measures on the ground states of the prepared state's frustrated stabilizer Hamiltonian — confirmed out-of-sample on cluster states. The result answers a question Hall posed and left open in 2011, and supplies an answer key against which every published lawful-measurement-dependence program (invariant-set theory, cellular-automaton interpretations, future-input dynamics, all-at-once retrocausality, and the displacement-condensate framework of the present authors) is graded in its own formalism. Operationally, the floors are counterfeiting thresholds for multipartite device-independent certificates. Every exact value is machine-verified on independent hardware and reproducible in under one minute from the linked repository: github.com/RandomInternetPreson/moire-phase-space-sampler (DST_Bell_MI directory). Also available at ai.viXra.org/abs/2607.0049.Version 2 (Oct 2026): closed form proved for all n; cluster-core census with exact certificates; scripts and run logs added.
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 nnn-party GHZ-Mermin scenario at every n≥3n \geq 3n≥3, with the marginals reproduced as well as the correlators. The minimum measurement-dependence fraction is Fmin⁡(n)=R/[2(R+1)]F_{\min}(n) = R/[2(R+1)]Fmin​(n)=R/[2(R+1)], with R=2⌊(n−1)/2⌋R = 2^{\lfloor (n-1)/2 \rfloor}R=2⌊(n−1)/2⌋ the Mermin violation ratio: 1/3,2/5,4/9,8/17,…1/3, 2/5, 4/9, 8/17, \ldots1/3,2/5,4/9,8/17,…, always below 1/21/21/2. 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=22n = 22n=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 kkk among stabilizer constraints has correlator-only minimum 1/(k−1)1/(k-1)1/(k−1). With the marginals included the minimum is the same on every core of the C5C_5C5​ ring cluster, on the size-4 and size-5 cores of C4C_4C4​ and on every four-core of C6C_6C6​, but larger on the size-6 and size-7 cores of C4C_4C4​ and on 112 of the 1,920 five-cores of C6C_6C6​, with exact certificates on both sides in every case.
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.
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.
Bell--CHSH measurements with polarization-entangled photons are usually reported only as nonlocality witnesses. Here we show that, for retardance sensing, the same coincidence data can also provide a quantitative metrological certificate. For a local birefringent phase ϕ\phiϕ encoded on one photon, we derive the classical Fisher information directly from experimentally accessible CHSH correlators Exy(ϕ)E_{xy}(\phi)Exy​(ϕ) and obtain an explicit lower bound in terms of the measured correlators and the phase slope S′(ϕ)S'(\phi)S′(ϕ) of the Bell parameter. In the unbiased-marginal two-qubit regime, this leads to a dual-use result: the canonical CHSH analyzer settings are simultaneously Bell-optimal and Fisher-optimal, saturating the quantum Fisher information for retardance estimation. We further show that the certificate remains informative under visibility loss, finite counts, and moderate departure from the Bell-violation threshold, so useful retardance sensitivity can persist even when nonlocality certification is weak or absent. Relative to tomography-based entangled-photon polarimetry, the method converts a standard Bell-calibration run into a sensitivity statement with rigorous lower confidence bounds using only coincidence correlators. The scheme is implementable with standard SPDC sources, wave plates, and polarizing beam splitters.
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