CHSH Bell Inequality: |S| ≤ 2 vs |S| ≤ 2√2

Classical LHV bound |S|≤2. Quantum mechanics (entangled state) can reach |S|=2√2≈2.828, violating Bell. Product states cannot.
S = E(a,b) − E(a,b') + E(a',b) + E(a',b')  |  |Ψ⁻⟩ singlet: E(θ_A,θ_B) = −cos(θ_A−θ_B)  |  |Φ+⟩: E = +cos(θ_A−θ_B)  |  Product: E(a,b)=cos(a)·cos(b)
State & Angles
a (Alice 1)
0°
a' (Alice 2)
90°
b (Bob 1)
45°
b' (Bob 2)
135°
CHSH S value
|S|
0.00
Classical region |S| ≤ 2
Individual correlators
Physics checks:
Bell gauge (semicircle) — needle = |S|
Correlation diagram: angles on Bloch circle
Sweep: S vs angle a (0°→360°)
E(a,b) polar view