Bell and CHSH

In plain terms

Here is the problem. Prepare two particles together, separate them, and measure each with a choice of two settings. Common sense says every measurement you might have made already has an answer waiting -- fixed when the pair was created, merely revealed by the apparatus. Bell's insight was that this innocent-sounding assumption has teeth: it forces a numerical limit on how correlated the two sides can be.

Quantum mechanics predicts correlations over that limit. So common sense and quantum mechanics disagree about an experiment, and you can run the experiment. It has been run for five decades now, with every loophole closed in turn. The correlations are there. The answer sheet is not.

What is proved in this corpus is the sharp version: on one shared space of hidden states, no assignment of answers to all four settings at once -- none, however contrived -- reproduces the singlet's statistics.

In CSD

CSD is contextual and non-local at the substrate level, on purpose. So it owes a demonstration that its own model cannot be secretly rewritten as the common-sense picture -- otherwise it would be claiming a virtue it had not earned. This theorem is that demonstration.

And it comes paired. The corpus also exhibits an explicit contextual model that does reproduce the singlet, so the claim is two-sided: here is the impossibility, and here is the thing that evades it. Obstruction and existence, cited together as one unit.

There is a lesson in how the theorem got stated, too. For a long time each measurement setting had its own state space in the formalisation, and the Bell question was literally unaskable -- there was nothing to compare a global assignment against. A type distinction was standing where a theorem should have been. Rebuilding on one shared space is what made the question expressible, and the answer turned out to be the theorem above.

Mathematically

Over a single shared measurable state space, no jointly-defined assignment of outcomes to the four CHSH settings reproduces the singlet correlations. Companion results rule out a DIFFERENT class rather than restating this one: no setting-local (product) pair of response functions reproduces the singlet at every setting pair, over any shared probability space. Neither result subsumes the other, and the corpus says so. The GHZ analogue for three parties is proved separately.

The shared-space form is the load-bearing one -- it is the shape the contextuality claim actually posits. The corpus pairs the obstruction with an explicit contextual model reproducing the same correlations, so the negative and positive halves are proved together rather than the no-go standing alone.

The name

John Stewart Bell (1928-1990) was born in Belfast to a working-class family and left school at sixteen. He got into physics by working as a lab technician at Queen's University, where the staff noticed and pushed him into a degree. Doctorate at Birmingham; career at CERN, designing accelerators. Foundations, he insisted, was a hobby for sabbaticals.

Some hobby. The 1964 paper was written while visiting Stanford and appeared in Physics 1, a journal so obscure it paid authors instead of charging them -- and died after a few issues. It showed that the hidden-variables question, which physicists had filed under philosophy for thirty years, was experimentally decidable. Clauser, Horne, Shimony and Holt made it testable in 1969.

Bell died of a stroke in 1990, the year he was nominated for the Nobel Prize. The 2022 prize went to the experimenters who confirmed him.