Bell derivations rest on locality, determinism, and measurement independence. Hall [Phys. Rev. Lett. 105, 250404 (2010)] priced the third assumption exactly for the singlet state, and in the Kochen-Specker analysis of Phys. Rev. A 84, 022102 (2011) priced the four tripartite Mermin perfect correlators at a surrendered fraction of 1/3, leaving open the problem of an optimal model for the Mermin state itself. This paper solves the faithful version of that problem -- every full correlator reproduced and every proper-subset marginal vanishing -- and extends it to thirteen parties. A reduction theorem shows the faithfulness constraints are free, so Hall's correlator-only threshold is promoted to the faithful value, F(3) = 1/3; linear-programming optima, certified exactly by an integer-arithmetic squeeze between a proven lower bound and an explicit construction, then give F(5) = 2/5, F(7) = 4/9, F(9) = 8/17, F(11) = 16/33, and F(13) = 32/65, each value through n = 11 repeated at the following even size. All computed points obey the closed law F = R/[2(R+1)] with R = 2^floor((n-1)/2) the Mermin violation ratio, a proven combinatorial lower bound is tight on every computed core, and a universal ceiling F <= 1/2 shows the statistics never require total abandonment of measurement independence at any size. The optimal hidden-variable densities have a closed physical form: uniform measures on the contextual ground states of the prepared state's frustrated stabilizer Hamiltonian, a structure confirmed out of sample on cluster states in three entanglement classes. The floors constitute counterfeiting thresholds for multipartite device-independent certificates and an exact demand curve that any measurement-dependent account of quantum correlations must fund. Complete proofs of all theorems are given in the main text and appendices.
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9 pages, 1 table; all exact values machine-verifiable from archived code (DOI in paper)
Bell derivations rest on locality, determinism, and measurement independence. Hall [Phys. Rev. Lett. 105, 250404 (2010)] priced the third assumption exactly for the singlet state, and in the Kochen-Specker analysis of Phys. Rev. A 84, 022102 (2011) priced the four tripartite Mermin perfect correlators at a surrendered fraction of 1/3, leaving open the problem of an optimal model for the Mermin state itself. This paper solves the faithful version of that problem -- every full correlator reproduced and every proper-subset marginal vanishing -- and extends it to thirteen parties. A reduction theorem shows the faithfulness constraints are free, so Hall's correlator-only threshold is promoted to the faithful value, F(3) = 1/3; linear-programming optima, certified exactly by an integer-arithmetic squeeze between a proven lower bound and an explicit construction, then give F(5) = 2/5, F(7) = 4/9, F(9) = 8/17, F(11) = 16/33, and F(13) = 32/65, each value through n = 11 repeated at the following even size. All computed points obey the closed law F = R/[2(R+1)] with R = 2^floor((n-1)/2) the Mermin violation ratio, a proven combinatorial lower bound is tight on every computed core, and a universal ceiling F <= 1/2 shows the statistics never require total abandonment of measurement independence at any size. The optimal hidden-variable densities have a closed physical form: uniform measures on the contextual ground states of the prepared state's frustrated stabilizer Hamiltonian, a structure confirmed out of sample on cluster states in three entanglement classes. The floors constitute counterfeiting thresholds for multipartite device-independent certificates and an exact demand curve that any measurement-dependent account of quantum correlations must fund. Complete proofs of all theorems are given in the main text and appendices.
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.
Multipartite Bell tests provide a correlation-only route to benchmarking quantum processors, but their application at large scales is hindered by the rapid decay of many-body correlators under noise and exponentially many terms in conventional Bell expressions. Here we address these scalability obstacles by introducing a finite-setting generalized Mermin family of state-tailored Bell inequalities with analytic certification bounds, in which the measurement-setting number m provides an additional certification dimension complementary to the system size n. We show that, for the powers-of-two setting choices considered here, increasing m leaves the ideal normalized multipartite quantum value unchanged while lowering the relevant classical bounds, thereby strengthening the Bell-violation ratios and yielding an improved noise-robustness scaling compared to the standard Mermin inequality. We test this construction experimentally on a programmable superconducting processor by preparing Greenberger-Horne-Zeilinger (GHZ) states of up to 80 qubits. Using randomized sampling for direct Bell-operator estimation, we observe Bell ratios that grow exponentially with system size, certify a nonlocality depth of 14, and show that increasing m strengthens both the Bell ratio and depth certification. All results are obtained solely from measured correlators and analytical bounds, without readout correction, tomography, or model-based mitigation. Generalized Mermin inequalities therefore provide a sharper Bell benchmark for noisy large-scale GHZ states.
Local hidden-variable models can reproduce Bell violations if settings depend on hidden variables. Using total-variation distance, we determine the minimum dependence needed to reproduce all outcome probabilities for Pauli X and Y measurements on n-qubit Greenberger-Horne-Zeilinger states, for every n ≥ 3, visibility and coherence phase. For odd n, the larger magnitude of the real and imaginary coherence components sets the cost; for even n, the smaller can raise it. States with identical, Bell-violating Mermin data can thus have different costs.
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 ηeff -- 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,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)/100, by primal and dual certificates in 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 M against a 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 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 ηeff=1 and, with the solved profile, reaches M=0 at ηeff≈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.
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.
The assumption of measurement independence is required for a local deterministic model to conduct a Bell test. The violation of a Bell inequality by such a model implies that this assumption must be relaxed. The degree to which the assumption needs to be relaxed to achieve violation of some bipartite Bell inequalities, has been investigated recently in [Phys. Rev. Lett. 105, 250404(2010), Phys. Rev. A 99, 012121(2019)]. In this work, we study the minimum degree of relaxation required to simulate violations of various well-known tripartite Bell inequalities and opens the possibility of relaxation in multipartite scenario. Local deterministic models are also provided to achieve the violations of these Bell inequalities.