Einstein hoped the universe was secretly ordinary — that entangled particles carry hidden instructions, decided in advance. Bell found a number that would prove him right or wrong, and CHSH made it testable: any ‘hidden instruction’ world is capped at 2. Quantum mechanics reaches 2.828 — and reality agrees. Slide the angle and break the wall.
For an entangled (singlet) pair the correlation between measurements at angles a and b is E(a,b) = −cos(a−b). The CHSH quantity S = |E(a,b) − E(a,b′) + E(a′,b) + E(a′,b′)| is bounded by 2 for ANY local-hidden-variable theory, but quantum mechanics at the optimal angles (0, 90° and 45°, 135°) reaches 2√2 ≈ 2.828 — Tsirelson’s bound. A fail-loud self-check throws unless the quantum S hits 2√2 at the optimal angles AND simulated local-hidden-variable models never exceed 2 — the gap experiments have closed, ruling out local realism.
The E(a,b)=−cos correlation is the exact quantum prediction for the ideal singlet; real Bell tests fight detector loopholes and noise (landmark loophole-free tests: 2015). The 2 vs 2√2 bounds are exact.