CHSH Inequality and the Tsirelson Bound

$$\text{CHSH} \le 2 \text{ (local hidden-variable theories)}, \quad \text{CHSH} \le 2\sqrt{2}\approx2.828 \text{ (quantum mechanics)}$$

Status: Proven.

The Clauser-Horne-Shimony-Holt (1969) inequality is the practically-testable form of Bell’s 1964 theorem: any theory built on local hidden variables is bounded at CHSH$\le2$, while quantum mechanics allows violations up to the Tsirelson bound, $2\sqrt2$. See Bell’s Theorem & Quantum Entanglement for the fuller status.

Real, independent confirmation: confirmed repeatedly since 1982, with “loophole-free” tests in 2015 closing the last major objections simultaneously — the 2022 Nobel Prize in Physics went to Aspect, Clauser, and Zeilinger specifically for this body of work. One of the most conceptually significant, rigorously confirmed results in modern physics.

Used in: Paper 5 and Paper 12, which claim PBT’s paired-vortex mechanism produces CHSH$\approx2.82$, close to the real quantum bound. The 2026-07-21 audit rebuilt the described mechanism directly and found it actually produces CHSH$=1.999$ — the classical, local bound, not a quantum violation. A cited “Belle II” measurement supporting the original 2.82 figure also could not be verified (Belle II studies B-meson physics, not Bell/CHSH tests). The real bound above is proven; PBT’s specific mechanism has not been shown to reproduce it.