"To be more precise, what we shall show is that the particles’ response∗ to a certain type of experiment is not determined by the entire previous history of that part of the universe accessible to them."
John Conway, Simon Kochen: Free Will Theorem https://arxiv.org/abs/quant-ph/0604079
Always keep in mind that Bell himself was quite a fan of Bohm's theory, which is the canonical hidden variable theory really. The question becomes one of characterising that 'strangeness': is it the violation of locality ('that part of the universe accessible to them'), is violating outcome independence palatable, but not parameter independence (this was Shimony's idea originally)? Lots of curious stuff.
> We consider experimenters A and B performing the pair of experiments described in the TWIN axiom on separated twinned particles a and b, and assert that the responses of a and b cannot be functions of all the information available to them.
I don’t get it — this sounds like they assumed their conclusion.
There’s also a couple places they make strong assumptions about information geometry I’m not sure I agree with — namely, if you want to refute modern Bohm-inspired models, you need to account for radically non-Euclidean spacetime. (Where you have a much harder time with “space like separated”.)
> FIN is not experimentally verifiable directly, even in principle (unlike SPIN and TWIN3).Its real justification is that it follows from relativity and what we call “effective causality,” that effects cannot precede their causes. […]
> Not all information in the universe is accessible to a particle a. In the light of FIN, information that is space–like separated from a is not accessible to a. The information that is accessible to a is the information in the past light cone of a.
That is why there was a lot of interest in doing cosmic Bell-like experiments: it is very hard to define a mechanism whereby the hidden property of the particle that left the quasar many billions of years ago also determined that Alice and Bob would set their apparatuses to measure these specific angles today. Theoretically this doesn't rule out the possibility, but it does mean at least that the theory would have to be significantly non-intuitive in its own right (whereas if Bell-like inequalities only happened for small-scale experiment, the theory that explained them could have been very simple indeed - such as some new wave that affected the measurement apparatus or some aspect of how we choose measurement angles).
The exact alternative to non-determinism is super-macro quasiparticles/correlations. Which this seems to assume away.
To take it to a more human realm - we can all agree that if Caesar weren't killed by Brutus, the world would be so different that it's very unlikely both Biden and Putin would be presidents of their respective countries today - this is an almost trivially true statement I would argue. However, if we observe that at every public appearance Biden and Putin wear costumes of the same color, it's very hard to come up with a theory that explains that their choice of costume is caused by Caesar killing Brutus.
And this is essentially what the super-determinism argument gets at: the same thing that caused this photon emitted by a quark to be polarized up 1 billion years after the Big Bang also caused Alice to measure the polarization of that particle along 30 degrees 13 billion years after the Big Bang.
Eg, that particle being a particular state is correlated to Alice picking a particular measurement because the total system must maintain that quantum number. The particle which emitted the photon and the particles which give way to Alice have carried that information since the inflationary period — and so the photon and Alice share that correlation now.
The conclusion seems to be “the correlation must be at least this old!” — but that’s exactly the claim being made.
So I’m not sure I understand the problem for Bohm-derived models.
Also, not sure what particular quantum number you think would have to be conserved and would influence Alice's decision of which way to configure her measurement apparatus - this would definitely require some new quantum property.