# Experimental Non-Violation of the Bell Inequality
## 5 Conspiracy and Free Will
### Nullifying the Notion of Conspiracy  
Consider a specific and pertinent example. Suppose, in a Bell experiment, the measuring apparatuses are set according to the frequency \(\
u\) of photons emitted by distant stars. If \(\
u<\
u_{0}\), a reference frequency, suppose the \(X=0\), \(Y=0\) directions are chosen; if \(\
u>\
u_{0}\), then the \(X=1\), \(Y=0\) directions are chosen. Let \(\Lambda_{00}\) denote some sample space of hidden variables associated with the choice \(X=0\), \(Y=0\) and so on. In Invariant Set Theory, if \(\lambda\in\Lambda_{00}\), then the outcome of measurements in the \(X=0\), \(Y=0\) directions is well defined, but, by the discussion above, the outcome of measurements in the \(X=1\), \(Y=0\) directions is undefined. Conversely, if \(\lambda\in\Lambda_{10}\), then the outcome of measurements in the \(X=1\), \(Y=0\) directions is well defined, but the outcome of measurements in the \(X=0\), \(Y=0\) directions is undefined. The notion of conspiracy arises because we have two seemingly independent pieces of information that determine the type of measurement made: the frequency \(\
u\) and the hidden variables \(\lambda\). Since the photon is emitted long before the entangled particles, and in a different part of the universe, one could imagine that these quantities can be varied independently of one another. If this is so, then the possibility that \(\
u<\
u_{0}\) but \(\lambda\in\Lambda_{10}\) leads to inconsistency since the latter combination is associated with a state of the universe not lying on \(I_{U}\). Hence, there must be some unpalatable 'conspiracy' between \(\
u\) and \(\lambda\), so the argument goes, to prevent such inconsistency. However, this conclusion is incorrect. Firstly, a thousand years (say) before the experiment is performed, the photon does not and cannot 'know' that its frequency (and not, say, bits from a to-be-made sci-fi movie) will be the determinant of some future measurement settings. If the experimenters decide what whimsical process (photons or movie bits) they will use to set the measurement orientations at space-time event \(D\), then the information needed to determine the nature of this decision will be completely delocalised on spacelike hypersurfaces in the causal past of \(D\). Perturbing any one bit anywhere in the causal past of \(D\) could change the nature of this decision. This is the butterfly effect, generic to nonlinear systems evolving on fractal attractors, and the stuff of numerous sci-fi movies. Similarly, it is incorrect to assume that the entangled particles 'know' whether their hidden variables belong to \(\Lambda_{00}\) or \(\Lambda_{10}\). Consistent with both the quantum field-theoretic notion of a particle as a field excitation, and the invariant set premise that the laws of physics in space-time derive from a holistic geometry in state space, it is incorrect to think of \(\lambda\) as somehow internal and localised to the particles being measured. Consider the following analogy. Babies (almost always) either belong to the set \(\Lambda_{\text{male}}\) or \(\Lambda_{\text{female}}\) of babies who are male or female at birth. Information that determines which of the two sets a particular baby belongs is internal to the baby. This type of analogy describes classical hidden-variable theory (probe the particle to find its hidden variables and determine what spin it will have when measured in a particular direction), but does not describe the situation here. A more accurate analogy is this: at birth, all babies either belong to the set \(\Lambda_{L}\) or \(\Lambda_{L}\) of humans that, as adults, either fall in love or don't. Information that determines to which of \(\Lambda_{L}\) or \(\Lambda_{L}\) a particular baby belongs is clearly not internal to the baby.