# Experimental Non-Violation of the Bell Inequality
## The Bell Inequality  
Consider now the relationship between (1) and (2) from the perspective of Invariant Set Theory. As above, let \(X=0,1\), \(Y=0,1\) denote four random points on the sphere, three of which (relevant to the discussion below) are shown in Figure 5a. Let \(\theta_{XY}\) denote the relative orientation between an \(X\) point and a \(Y\) point. Recall that complex Hilbert states can represent uncertain trajectory segments on \(I_{U}\) providing squared amplitudes are rational. Hence, \(\text{Corr}(X,Y)=-\cos\theta_{XY}\) requires \(\cos\theta_{XY}\in\mathbb{Q}\). Suppose [PERSON] freely chooses \(X=0\) and [PERSON] \(Y=0\) when measuring a particular entangled particle pair. Then, it must be the case that \(\cos\theta_{00}\in\mathbb{Q}\). Could [PERSON] and [PERSON] have chosen \(X=1\) and \(Y=0\) when measuring this same particle pair, given that they actually chose \(X=0\), \(Y=0\)? In other words, does the state \(U^{\prime}\) of the universe in which this counterfactual experiment takes place also lie on \(I_{U}\)? To answer the question in the affirmative, we additionally require \(\cos\theta_{10}\in\mathbb{Q}\). However, as with the Stern-Gerlach analysis, applying the cosine rule for spherical triangles, we have  
\[\cos\theta_{10}=\cos\theta_{00}\cos\alpha_{X}+\sin\theta_{00}\sin\alpha_{X}\cos\gamma, \tag{5}\]  
where \(\alpha_{X}\) is the angular distance between \(X=0\) and \(X=1\). Now, it is always possible for [PERSON] to send the particle which she has just measured in the \(X=0\) direction, back into her measuring apparatus to be again measured in the \(X=1\) direction. Hence, \(\cos\alpha_{X}\) must be rational. Now, we also require the angle \(\gamma\) to be a rational multiples of \(2\pi\). This would be so if the three points \(X=0,1\) and \(Y=0\) lay on a great circle _exactly_, so that \(\gamma=180^{\circ}\)_precisely_. However, as before, because of ubiquitous unshieldable gravitational waves, this cannot be the case. Hence, \(\cos\theta_{01}\) is the sum of two terms, the first a rational and the second the product of three independent terms, the last of which is irrational. Being independent, these three terms cannot conspire to make their product rational. Hence, \(\cos\theta_{01}\) is the sum of a rational and an irrational and must therefore be irrational. Hence, the state of the universe \(U^{\prime}\) in which the counterfactual experiment takes place is not realistic and is \(g_{p}\)-distant from worlds on \(I_{U}\). Hence, the counterfactual question cannot be answered in the affirmative: \(\mathrm{Corr}(1,0)\) is undefined. In general, it is never the case that all four correlations in (1) are definable on \(I_{U}\)--the Bell inequality is always undefined in Invariant Set Theory. An experimenter might ask how one could set up an experiment with sufficient care to ensure that the corresponding Hilbert state descriptors were rational rather than irrational. The answer is that the experimenter need take no care: if an experiment is performable, i.e., corresponds to some \(U\in I_{U}\), then by construction the descriptors must be rational. Physical perturbations (e.g., gravitational waves in space-time) only introduce uncertainty in the values of the rational descriptors and not in the fact that they are rational. Conversely, if the descriptor of a counterfactual state is irrational, then no amount of noise that respects the primacy of \(I_{U}\) can change it into an ontic state. This property provides an attractive finitist feature that is missing in conventional physical theories based on \(\mathbb{R}\) or \(\mathbb{C}\) and hence in theories that utilise the Euclidean state-space metric.