Thanks Rich and Jos   
      
   Yes, I know that QM explains average outcomes perfectly and, yes, I am   
   trying to think of matters in microscopic detail using hidden variables so   
   as to calculate an event-by-event simulation.   
      
   I follow Rich's formula for QM (and laboratory) giving probability ratio:   
   (1+cos(theta))/2:(1-cos(theta))/2 for +1:-1 probabilities.   
   That is OK.   
      
   An equivalent ratio for classical/ hidden variables is   
   (1+theta/180)/(1-theta/180) {where theta is measured in degrees}   
   which is based on the sawtooth curve of correlational outcomes.   
      
   The results for Rich's example of theta = 85 deg is   
   for QM  54.4/45.6 which agrees with Rich's figures   
   and for Hidden Variables is 52.8/47.2 which represent the Bell  inequality   
   cut-off value.   
      
   So QM at 54.4 exceeds the inequality cut of value of HV.   
   which means that, in order to make a hidden variable simulation more   
   realistic, more +1 readings are needed than is given by HV.  That is more +1   
   readings are needed than is given by sign(e.a)   
      
   Jos's formula (1 + e.a) / 2 is equivalent to Rich's value of   
   (1 + cos(theta))/2.  But when using a single electron with a hidden variable   
   represented by vector e, The usual HV calculation for the outcome   
   measurement is given by the sign of e.a  not by the exact value of e.a.   
   This is the nub of quantisation of the measurement.  It is obviously wrong   
   to use sign(e.a) which is why I am trying alternatives.   
   For an individual electron, (1 + e.a) / 2  is in general neither +1 nor -1.   
   But if (1 + e.a) / 2 can be built into my HV measurement, in the form of   
   adding chance to the outcome, I will try it.  This has given me an idea for   
   another simulation which will take a few days.  Using e.a rather than   
   sign(e.a) will make the electron angle not a single constant value and hence   
   likely not a proper hidden variable.   
      
   I hope I never need to resort to a multiverse to bring chance into the   
   calculations. Once the electron is in the detector magnetic field it will be   
   turned away from vector e to point at vector a.  This turning can bring   
   dynamism into the spin direction and hence add chance.  My original question   
   was worrying that the initial value of e determined the outcome exactly.   
   Period.  But that might imply infinitesimal effects.  At what point in its   
   time of flight after leaving its source is the electron measurement decided?   
   If it is decided initially and infinitesimally (I mean a very weak magnetic   
   field far from the detector) then we have an HV measurement, which is known   
   to be unreal.   
      
   Thanks again for your help.   
      
      
   Austin Fearnley   
      
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