Exact Measurement-Dependence Cost across the Noisy GHZ–Mermin Threshold

Abstract

We determine the exact measurement-dependence cost of faithful local deterministic models reproducing the full outcome table on the even-parity GHZ–Mermin setting set for Pauli X/Y measurements. For every integer n ≥ 3 and isotropic visibility 0 ≤ v ≤ 1, let R = 2^floor((n−1)/2). The minimum worst-pair total-variation distance between setting-conditioned hidden-variable distributions is F_n(v) = max{0, (R²v−R)/[2(R²−1)]}. Measurement-independent models suffice precisely up to v = 1/R; above this threshold, the minimum cost increases linearly, reaching R/[2(R+1)] at perfect visibility. Quadratic Walsh spectra yield explicit optimal ensembles, a faithful lift restores all vanishing proper-subset correlators without changing the cost, and a nondegenerate R²-setting subsystem supplies a matching lower bound. Relative to version 1, this version extends the perfect-visibility result to the complete isotropic visibility curve and expands the derivation details and literature discussion. The result concerns this specified full-table family and does not assert an exact cost for arbitrary noise.