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Many thanks, Gordon.
In the RSOS paper I am working with 3-dimensional physical space only, with gravity neglected and time not playing any significant role for the issue of strong correlations at hand.
However, the 3-space I am working with is compactified. Usually the physical space is modeled by R^3. That is no good, because R^3 runs off to infinity along infinity of its radial directions. But the physical space does not run off to infinity in infinity of radial directions. Thus R^3 is not a valid model for the physical space. The problem with it can be fixed by identifying all of these infinitely many infinities with a single additional point. The resulting space is then S^3, which is equal to R^3 U {oo}. Note that S^3 is still a 3-dimensional space, but it is a compact space without infinities. Thus it is a much better model of the physical space. But it is still not a good enough model of the physical 3-space until it is parallelized. I quote from one of my 2011 papers to explain this:
The triviality of the tangent bundle TS^3 means that the 3-sphere is parallelizable. A k-dimensional manifold is said to be parallelizable if it admits k vector fields that are linearly-independent everywhere. Thus on a 3-sphere we can always find three linearly-independent vector fields that are nowhere vanishing. These can then be used to define a basis of a tangent space at each of its points. As a result, a single coordinate chart can be defined on a 3-sphere that fixes each of its points uniquely. Informally, a manifold is said to be parallelizable if it is possible to set all of its points in a smooth flowing motion at the same time, in any direction. Rather astoundingly, this turns out to be possible only for the 0-, 1-, 3-, and 7-spheres. Thus parallelizability of these spheres happens to be an exceptionally special topological property. One way to appreciate it is by considering a manifold that is not parallelizable. For example, it is not possible to set every point of a 2-sphere in a smooth flowing motion, even in one direction. However you may try, there will always remain at least one fixed point --- a pole --- that will refuse to move. This makes it impossible, for example, to cover the Earth with a single coordinate chart. For similar reasons, parallelizability of the 3-sphere, or equivalently the triviality of its tangent bundle, turns out to be indispensable for respecting the completeness criterion of EPR. And since this criterion is the starting point of Bell’s theorem, understanding the parallelizability of 3-sphere turns out to be indispensable for understanding the topological error involved in all Bell-type arguments.
When all this is done, the resulting space is a quaternionic 3-sphere, which is written as S^3 in the mathematical literature. S^3 provides the most adequate model for the physical space.
So, in short, I am working with only 3D physical space (without involving gravity or time), modeled by a 3-sphere that does not harbor any kind of undesirable infinities or singularities.
This still does not answer your question: What is the significance of S^7. Well, S^7 I am working with is the algebraic representation space of the quaternionic 3-sphere described above. It is essential to understand the local origins and strengths of ALL quantum correlations, not just EPR-Bohm type singlet correlations. See theorem 3.1 of my RSOS paper.
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