James,
I've got it now. I remember that I wanted to explain Scott Aaronson's argument in simpler and more accessible arithmetic, and then realized that the explanation grows ever more sophisticated and that I couldn't make it simpler.
So let me try again.
Remember, I claim that the answer to the first question *has* to be "yes," no matter what the question is. Here expressed in arithmetic, is why:
1 - 1/2 1/4 - 1/2 1/4 - 1/2 = 0
The important thing is that no value in the sum of these terms exceeds unity. Consistent with Scott's upper bound of probability .75 for a 2-player cooperative game, the first iteration (where the zeroth iteration is 1 - 1/2) is 1/2 1/4 = 3/4. When continued through the second and third iterations, we find 3/4 - 1/2 = 1/4 and 1/4 1/4 = 1/2. To make this easier to read:
(eqn 1) 1 - 1/2 1/4 - 1/2 1/4 - 1/2 = 0
= .5, .75, .25, .5
0 1 2 3 4 iterations
If we were to sum and average all the values of the results of these iterations -- which is what quantum mechanics does by averaging values of a run of experimental results -- we would get 1/2. This is the upper bound of a coin toss probability for sufficient length of n independent Bernoulli trials. When we revese signs for the above, we get:
(eqn 2) 1 1/2 - 1/4 1/2 - 1/4 1/2 = 2
Which is the CHSH upper bound for Bell-EPR type experiments.
What's the problem? The former case is algebraically closed; zero indicates that the computation halts. The Bell-CHSH case does not halt; it will increase monotonically to infinity. This is interpreted to mean that every experimental event not measured has a vanishing but non-zero probability of happening. The non-probabilistic result (the algebraic equation 1 summing to zero) tells us that no event is probabilistic; perfectly random tosses of a fair coin are precisely determined.
The way that quantum mechanics reconciles its probabilism (implying quantum entanglement and nonlocality), with actual physical reality, is by quantum unitarity -- the average of the set of iteration results in eqn 2 (1.5, 1.25, 1.75, 1.5) is unity, or probability 1.0 that the upper bound of any quantum pair correlation experiment obeys the upper bound of CHSH.
Now this is the simple arithmetic that Scott Aaronson, Richard Gill and many others believe without question is at the foundation of a probabilistic physical reality. Look, though, when expressed as the answers to yes-no questions (pairs of binary values):
(eqn 1) 1 - 1/2 (Yes) 1/4 (No) - 1/2 (Yes) 1/4 (No) - 1/2 (Yes) = 0
(eqn 2) 1 1/2 (No) - 1/4 (Yes) 1/2 (No) - 1/4 (Yes) 1/2 (No) = 2
If we allow the first iteration to answer "No" such that all subsequent iterative values exceed unity, we have loaded the dice in favor of an infinite dimensionless range of values that we ASSUME constitutes the sum of perfect information that we can treat by probability theory. This is reconciled to actual physical experiments (in which probability plays no actual physical role) by normalizing to quantum unitarity for any experimental result.
Joy Christian's inspired realization was that ORIENTATION of the topological initial condition determines the experimental outcome which must be ALGEBRAICALLY CLOSED as eqn 1 shows, and which thereby obviates quantum entanglement, nonlocality and probabilistic measure. This initial condition -- if one follows Joy's objective mathematical argument without imposing one's personal beliefs on it -- clearly produces E(a,b) = - a.b. Very straightforward.
I've been playing with the consequences of algebraic closure for topology, for quite a while. I decided to attach a draft of one such effort; section 4.25 relates to the present argument.
Tom