State-Dependent CHSH Inequality: A Reproducible Demonstration of Angular Configuration Requirements for All Four Bell States

This work provides an explicit, reproducible demonstration of how theClauser-Horne-Shimony-Holt (CHSH) inequality depends jointly on thechoice of Bell state AND measurement configuration in the spin-1/2qubit convention used by gate-based quantum processors. Main result. The combination of CHSH Form I, S = | E(a,b) - E(a,b') + E(a',b) + E(a',b') |, with the commonly-cited angular configuration {0, pi/2, pi/4, -pi/4}yields S = 0 for ALL FOUR Bell states in the spin-1/2 Bloch-sphereconvention, not selectively for one. This frequently misleadspractitioners who prepare |Phi+> on gate-based hardware and applytextbook angles, obtaining S = 0 and misdiagnosing the issue as ahardware fault, code bug, or absence of entanglement. Corrections provided. For the two states whose correlation functionin the XZ plane has argument theta_a - theta_b (|Phi+> and |Psi->),two minimal one-parameter fixes recover the Tsirelson bound 2*sqrt(2): 1. Retain the angles and apply CHSH Form II: S = | E(a,b) + E(a,b') + E(a',b) - E(a',b') |. 2. Retain Form I and replace b' = -pi/4 with b' = 3*pi/4. For the remaining two Bell states (|Phi-> and |Psi+>, whose correlationargument is theta_a + theta_b), neither single-parameter fix suffices;the optimal angles b and b' must be exchanged. A complete optimal-angletable for all four states is given in Section 4 of the manuscript. Contents. - paper.pdf / paper.tex: 8-page manuscript with full derivation, tables, code listing and appendix. - chsh_verification.py: self-contained Python script (NumPy only) reproducing all numerical claims of the paper. Its standard output is the verbatim text reproduced in Section 5, and the script additionally cross-checks the closed-form correlation expressions against explicit 4x4 matrix calculations on random angles (azimuthal angles included) to machine precision. - CHANGELOG.md: complete history of corrections relative to earlier versions of this deposit. Reproducibility. Running 'python3 chsh_verification.py' produces theverification tables displayed in Section 5 of the paper, byte for byte.