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
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 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(2√2,9/10)=(27√2−33)/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 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 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.
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BibTeX
@misc{alai2026exact,
title = {{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}},
author = {Aaron Alai},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.18886},
eprint = {2608.18886},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.18886},
}
RIS
TY - UNPB
TI - 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
AU - Aaron Alai
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2608.18886
UR - https://arxiv.org/abs/2608.18886
AB - 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_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_eff)=max{0,η_eff((S+2)η_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√2,9/10)=(27√2−33)/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 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_eff=1 and, with the solved profile, reaches M=0 at η_eff≈ 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.
ER -