Peter,
i know that we talk about spheres. I only scribbled a unit circle first. For convenience, lets assume the north pole being up, the south being down. These are the maximum points for this 'axis'. It follows that the equator is a horizontal line (the x-coordinate-line of the unit cirlce if you will). At the front of the equator (in the middle of it seen from the front of the unit circle) the momentum property is maximum, since at the other side of the equator it is minimum. Therefore at the front side (middle of equator), the curl is zero, since up and down it is maximum (+1, -1). What's left are the both sides of the sphere (if we visualize the unit circle as a sphere what we want to do). So, left and right (in terms of the unit circle at 90 degree and 270 degree) both curl and momentum are zero (zero means half the way between +1 and -1).
If they wouldn't be zero, how can you explain that for two subsequent polarizers (one for unpolarized light, the other for the result having went through polarizer number 1), orientated relative to each other at 90 degrees, no particle will go trough the second polarizer? But since your interaction mechanism is unclear, it could well be that the particle will go through (but this would be in contradiction to experimental findings). Nonetheless, at the other twin-particle with the same polarizer-settings, it then will not go through. So you need two points at the sphere, distinguished by 180 degree, for which no 'intensity' can interact with the field and lead to 're-emission'.
""I visualized the main ingredients on the unit circle. As in computing, one flaw in input and all that follows is nonsense!! There are NO 'zero' points! Nature is 3D so a circle CAN'T model it. Think 'sphere'. Now you missed a key statement; 'UP/DOWN' and 'LEFT/RIGHT' are equivalent arbitrary labels as the poles may ALSO be either. The key word is; 'ORTHOGONAL'(to the poles, i.e. at 90o)."
I conclude from this that two twin-particles, produced by, say a down-conversion process, are 'only' complementary insofar as their complementary parts do always stand orthogonally relative to each other. Say, if the left particle has north left, the right particle has its south pole right. The latter particle then has its maximum momentum at one point of the sphere, the other particle 180 degrees away (but always on the equator, defined by the position of the poles!). If another particle(-pair) has its north pole up (from the same point of view of an 'observer'), the other particle will have its north pole down, and again the latter then has its maxium momentum at some point on the sphere, namely at the equator, its twin 180 degrees away from that point, also on the equator. So the twin-particles can come in arbitrary spinning directions and orientations of their poles and equators from the twin-source, but always must conserve the above described relationships in relation to each other. If the equator plane is spinning up, the equator plane of the twin particle must spin down. Is this the correct understanding?
"If any theory finds something different it would be WRONG as it doesn't correspond to reality!"
I think this would be only true if we had discovered all the consequences of QM yet. How can we know that we already have? I mean, other causal mechanisms could well lead to other experimental findings. That these findings aren't yet made has then its cause in the mathematics of QM, which does not predict or indicate such experimental findings. So nobody does look for them.
Best wishes,
Stefan Weckbach