Hi Michael,
You have framed your question very clearly. It reminds me of some passionate discussions I had last year on these pages with Ray B. Munroe, who is sadly no longer with us. He was a supporter of my use of 7-sphere, but he also saw things from the particle physics perspective and I had to explain my foundational perspective to him from scratch. Please allow me to do the same here, if not for you, at least for other readers who might to be interested.
The issue at heart is local causality. This concept has been crystallized by various people over the years, starting with Einstein in his special relativity, and culminating in Bell's analysis of the EPR scenario. Bell used some earlier ideas of von Neumann to frame the concept for any realistic theory, and made it independent of any specific theory of physics, including quantum theory or quantum field theory, and independent even of the specifics of special and general relativities. He thus provided a very general, very reasonable classical, local-realistic framework, which does not depend on the specifics of a given set of observables. It depends only on the yes/no questions the experimentalists may ask and answer. Thus, for example, for the classic EPR-Bohm scenario involving a joint observable AB for observing spin up and spin down at two remote ends of the experiment, he formulated local causality in terms of the following factorizability condition:
AB(a, b, L) = A(a, L) x B(b, L),
where A(a, L) is independent of the remote context b as well as the remote result B, and likewise B(b, L) is independent of the remote context a as well as the remote result A. That is it. As you can see, his formulation of local causality only involves the measurement results A = yes/no and B = yes/no, apart from the measurement contexts a and b (such as the directions of the local polarizers), and the common cause L, which is the "hidden" variable or a complete EPR state.
It should now be clear why the kind of details you have spelt out for more general scenarios involving particle productions etc are irrelevant for the central concerns of local causality. All that matters is how the yes/no answers to relevant questions are correlated, because any experiment in physics can always be reduced to a series of questions that can be answered in a "yes" or "no."
Nevertheless, let us look at things from your perspective. Let us consider a scenario where an EPR 2-particle state is not of the form P^|P_ (in a variant of your notation) but of the form P^|Q _, where Q =/= P. For you, then, there are two sets of observables with quantum correlations, {^,_} and {P,Q,...}, where the first set contains eigenvalues of the rotation group S3, and the second set contains eigenvalues of SU(4)/SU(3) ~ S7. The question then is: Is Bell's local-realistic analysis applicable to this situation? Yes, absolutely. Is my topological correction to Bell's analysis applicable to this situation? Again, yes, absolutely.
But here is a difficulty for you: Your set {^, _} is restricted to S3. It is, however, not possible in general to reproduce quantum correlations using my framework within S3 if the corresponding quantum systems have the spectrum of eigenvalues (or measurement results) more general than that of a 2-level system. So, ironically, there is no problem for the exotic set {P,Q,...}, for which the "group space" within your framework is S7, which is the most general available within my framework. It is the set {^, _} that will cause a locality problem for you, because, for a general quantum field, the spectrum of eigenvalues within {^, _} would be highly nontrivial. Within my framework, on the other hand, both {P,Q,...} and {^, _} fall under the same "group space" S7, and so there is no problem.
So my framework does put the following constraint on the observables: If one restricts to the group space S3, then the only quantum systems for which local causality can be maintained are the 2-level systems. For more general systems S7 is inevitable.
Best,
Joy
