Richard,
To follow up:
As Jonathan noted, "It is true that global properties are not localized, but they need not be scalars either. Global properties in the topological domain can locally appear to be scalar quantities, but they are actually pseudo-scalars or tri-vectors wrt R3."
Some objections that I have seen, toward applying topology to locally real measures have been just this -- topology is global; how can it have anything to do with locality?
There is an arithmetic example (due to Karl Popper), that I have often used to show "at least one" correspondence between local and global properties results in correspondence between all local and global properties, given sign reversibility:
Popper reformulated two (still open) arithmetic conjectures, in this way:
Goldbach Conjecture G: For every natural number x > 2, there exists at least one natural number y such that x y and (2 x) - y are both prime.
Twin Primes Conjecture H: For every natural number x > 2, there exists at least one natural number y such that x y and (2 x) y are both prime.
The Goldbach Conjecture is falsifiable in principle; a program to test the conjecture potentially halts if it hits on a counterexample. The Twin Primes Conjecture is not falsifiable in principle.
It's the same with Bell's theorem formulated in R^3. The theorem is true in that domain because it cannot be untrue -- it nonconstructively assumes what it means to prove.
When we move up a dimension to S^3, which is the same as S^2 X S^2, and R^3 is compactified by a single point at infinity, reversibility under cross multiplication depends on a sign change for pairwise correlated points of the Hopf fibration, which implies analytical continuation.
All best,
Tom
