Exact minimum measurement dependence for faithful local deterministic models of multipartite GHZ-Mermin correlations

Abstract

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&#39;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&#39;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.