Thomas Ray: "Why non-locality?" NOT! It goes against everything we have learned about Nature.

That's as good an answer as any, but of course it's not decisive. Everywhere else in Nature influence is limited by speed of light, as far as we know. But apparently in QM the "collapse of the wave function" happens faster, for "entangled" particles, as shown in Bell type experiment. It's a new, different phenomenon, although it doesn't "go against" - i.e., doesn't contradict - any other facts. Often, in science and elsewhere, we run into NEW phenomena. This is one of those cases.

Thomas Ray: Let A = nonlocality Let B = local realism

No. That explains why you didn't get my "logic lesson". In fact,

A = local realism

B = Bell's inequality

Thomas Ray: Local realism (Einstein causality) is constructed explicitly in the measure space of special relativity, and implicitly in general relativity, assuming spacetime is real. ("All physics is local".) If we were comparing apples to apples, context would be supplied by measure space. What is the measure space of Bell-Aspect?

There are (at least) two measure spaces involved. First, as you say, we assume Minkowski space (no need for curved GR space), but we're not really treating it as a measure space. Then there's the space of outcomes for Alice and Bob, which is simply tensor product of two copies of the pair of outcomes {-1, 1} with the obvious atomic PDF.

But the key one you're referring to is, no doubt, the space in which the premised "hidden variable", usually called lambda, lives. Its existence is only an hypothesis, which turns out to be wrong: evidently lambda doesn't exist. That means we can't say definitively what space it's an RV in. This point can cause confusion. Bell's theorem is supposed to be valid no matter what (reasonable) space you hypothesize for this RV! The normal choice would be SO(3), the 3-sphere (which, BTW, in mathematical topology we call the 2-sphere). The PDF would be assumed uniform, although other choices are possible. It doesn't matter whether you suppose this is the space of unit quaternions with zero real part (square roots of -1) or the more standard definition, vectors of norm 1. You could also use O(3) if you want. Even SO(2) could be used with appropriate assumptions. Bell's theorem works regardless. It also makes no difference if you justify the geometric algebra approach by invoking FLRW.

I hope that answers your question. Let me emphasize again the key source of confusion. Since it turns out lambda doesn't exist - the point of Bell's proof by contradiction - there isn't one definitive measure space for this RV. We should be able to assume any reasonable space for lambda. I can't imagine any that would invalidate Bell.

Finally note that what really counts here is the QM correlation function. Other aspects of Bell-Aspect experiment can be modelled differently, in various ways, not this one.

That's as good an answer as any, but of course it's not decisive. Everywhere else in Nature influence is limited by speed of light, as far as we know. But apparently in QM the "collapse of the wave function" happens faster, for "entangled" particles, as shown in Bell type experiment. It's a new, different phenomenon, although it doesn't "go against" - i.e., doesn't contradict - any other facts. Often, in science and elsewhere, we run into NEW phenomena. This is one of those cases.

Thomas Ray: Let A = nonlocality Let B = local realism

No. That explains why you didn't get my "logic lesson". In fact,

A = local realism

B = Bell's inequality

Thomas Ray: Local realism (Einstein causality) is constructed explicitly in the measure space of special relativity, and implicitly in general relativity, assuming spacetime is real. ("All physics is local".) If we were comparing apples to apples, context would be supplied by measure space. What is the measure space of Bell-Aspect?

There are (at least) two measure spaces involved. First, as you say, we assume Minkowski space (no need for curved GR space), but we're not really treating it as a measure space. Then there's the space of outcomes for Alice and Bob, which is simply tensor product of two copies of the pair of outcomes {-1, 1} with the obvious atomic PDF.

But the key one you're referring to is, no doubt, the space in which the premised "hidden variable", usually called lambda, lives. Its existence is only an hypothesis, which turns out to be wrong: evidently lambda doesn't exist. That means we can't say definitively what space it's an RV in. This point can cause confusion. Bell's theorem is supposed to be valid no matter what (reasonable) space you hypothesize for this RV! The normal choice would be SO(3), the 3-sphere (which, BTW, in mathematical topology we call the 2-sphere). The PDF would be assumed uniform, although other choices are possible. It doesn't matter whether you suppose this is the space of unit quaternions with zero real part (square roots of -1) or the more standard definition, vectors of norm 1. You could also use O(3) if you want. Even SO(2) could be used with appropriate assumptions. Bell's theorem works regardless. It also makes no difference if you justify the geometric algebra approach by invoking FLRW.

I hope that answers your question. Let me emphasize again the key source of confusion. Since it turns out lambda doesn't exist - the point of Bell's proof by contradiction - there isn't one definitive measure space for this RV. We should be able to assume any reasonable space for lambda. I can't imagine any that would invalidate Bell.

Finally note that what really counts here is the QM correlation function. Other aspects of Bell-Aspect experiment can be modelled differently, in various ways, not this one.