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I also wanted to give you some feedback on your paper. Unfortunately, I don’t have that much to add, because I think your discussion is very much on point. 🙂 The Schulmann model is quite interesting (I didn’t know it before) and your arguments concerning symmetry are, of course, correct.
It’s not so much a factual critique, rather a personal feeling, that you’re still giving too much credit to the Wood-Spekkens argument, though. To me, it’s just one of those meta-results that seem deep but are actually quite irrelevant. In a toy-model, where any postulation of probability distributions is ad hoc and where you might have to introduce some artificial variables, the “fine-tuning” objection seems to have some bearing. In any more serious theory, where the probability distribution is either part of – or better – derivable from the fundamental postulates/law of the theory, the Wood-Spekkens argument amounts to the claim: if the theory was different, it’d be wrong. I mean: if the Boltzmann distribution was different, pigs might be able to fly. But who cares?
Still, the Schulmann model is nice as an “intermediate step”, because it demonstrates how the “correct” (i.e. non-signalling) distributions can be justified by deeper principles (e.g. symmetry).