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.
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.
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.
A local hidden-variable model for tripartite GHZ states is constructed, in which a single threshold on the local signal magnitude governs detection and violates the fair-sampling assumption. The detection probability depends on the hidden variables, so the detected sample is unrepresentative, while the single-detector efficiency is independent of the polarizer settings. On the postselected sample the model violates the Mermin and Svetlichny inequalities, reaching M3=3.85 and S3=5.66, at a single-detector efficiency of 72.49%, below the threshold eta_crit=3/4 of Cabello et al.; the full ensemble satisfies both inequalities. A necessary upper bound eta <= 74.64% for the exact reproduction of the GHZ correlations by any model in this class is derived.
We investigate the violation factor of the original Bell-Mermin inequality. Until now, we have used an assumption that the results of measurement are . In this case, the maximum violation factor is as follows: and . The quantum predictions by n-partite Greenberger-Horne-Zeilinger state violate the Bell-Mermin inequality by an amount that grows exponentially with n. Recently, a new measurement theory is proposed [K. Nagata and T. Nakamura, International Journal of Theoretical Physics, 49, 162 (2010)]. The values of measurement outcome are . Here we use the new measurement theory. We consider a multipartite GHZ state. We use the original Bell-Mermin inequality. It turns out that the original Bell-Mermin inequality is satisfied irrespective of the number of particles. In this case, the maximum violation factor is as follows: and . Thus the original Bell-Mermin inequality is satisfied by the new measurement theory. We propose the following conjecture: All the two-orthogonal-settings experimental correlation functions admit local realistic theories irrespective of a state if we use the new measurement theory.
We determine the least Greenberger-Horne-Zeilinger (GHZ) extractability compatible with a Mermin-Ardehali-Belinskii-Klyshko (MABK) Bell score for every
n≥3. Extractability is the largest squared overlap with a GHZ state obtainable by separate local quantum channels. Between the biseparable bound and the quantum maximum, its exact minimum is the affine interpolation from one half to one. This proves the bound conjectured for six or more parties. The result holds for arbitrary binary measurements and density operators on tensor products of finite- or infinite-dimensional local spaces. We also determine the exact minimum for qubits, which is strictly higher at every interior score. One qutrit and qubits elsewhere attain the unrestricted minimum with measurements fixed as the score varies and, at interior scores, realize the smallest product of ranks of the local reduced states among states attaining that minimum. For
n≥4, the attaining state with this rank product is unique at each interior score, up to local isometries and party permutations. On one qutrit and qubits elsewhere, for
n≥4 and scores a fixed fraction away from both endpoints, a sufficiently small excess of extractability above the minimum controls trace-norm distance to the attaining state of the same score, up to local unitaries. The dependence is of square-root order, with optimal exponent and explicit constants depending only on that fraction, uniformly in
n. Equality and stability concern actual extractability; identical Bell statistics can accompany different extractabilities.