"There are two ways in which I interpret my S^2 version of the algorithm which Minkwe described and implemented:
(1) The hidden state is (e,t) where e is drawn uniformly at random from S^2 and independently of that, t is drawn from the distribution of 1/2 sin^2 theta, where theta in its turn is drawn uniformly at random from [0, pi/2]. The measurement functions A and B take values in {-1, 0, 1}.
According to common understanding and terminology, model (1) is a model where particles sometimes fail to be detected (coded by "outcome" 0)."
This is very misleading, if not plain wrong. For the state (e, t) we can certainly say mathematically that the measurement functions A and B take values in {-1, 0, 1}.
But physically 0 does not correspond to failure of detecting a particle. We cannot detect what is not there in the first place. If there is no particle to begin with, then there is nothing to detect in the first palce.
"(2) The hidden state is (e,t) where (e,t) are drawn from the joint conditional distribution described above but conditional on A(a; e,t) != 0 and B(b; e,t) != 0.
Model (2) is a conspiracy loophole model: the probability distribution of the hidden variables depends on the measurement settings."
In my model it does not. In my model the probability distribution of the hidden variable does not depend on the measurement settings.
"The difference between (1) and (2) is whether we keep or we discard the occasions when A or B takes the value 0."
For the state (e, t) there are no "occasions" when A or B takes the "value" 0, because physically no such value could "occasion" to begin with.
"Anyone is welcome to come up with new interpretations, for instance
(3) The simulation is an imperfect implementation of Joy Christian's local hidden variable model based on S^3."
Sure, but you have used my papers and my formulae to construct the simulation.
"A possible source of "imperfection" is through the way the condition "for all x" has been implemented. I only check x=a and x=b. So I check two particular values of x but not all values of x. From the geometric picture of what is going on it is clear that if we truly had checked all possible values of x, no state would have survived at all."
True in R^3, but not true in S^3. The set of complete states Lambda is not a null set in S^3, as I have proven in this one-page document. Your confusion arises from not being able to escape the R^3 mentality. Recall that S^3 is a surface in R^4, not in R^3, and that there are infinitely many S^2 within S^3 (see also the comments above by Jonathan).
