# Experimental Non-Violation of the Bell Inequality
## 1 Introduction  
Recent experiments (e.g., [1]) have seemingly put beyond doubt the conclusion that the CHSH version  
\[|\mathrm{Corr}(0,0)+\mathrm{Corr}(1,0)+\mathrm{Corr}(0,1)-\mathrm{Corr}(1,1)|\leq 2 \tag{1}\]  
of the Bell Inequality is violated robustly for a range of experimental protocols and measurement settings. As a result, it is very widely believed that physical theory cannot be based on Einsteinian notions of realism and local causality ('local realism'). Here, \(\mathrm{Corr}(X,Y)\) denotes the correlation between spin measurements performed by [PERSON] and [PERSON] on entangled particle pairs produced in the singlet quantum state, where \(X=0,1\) and \(Y=0,1\) correspond to pairs of freely-chosen points on [PERSON] and [PERSON]'s celestial spheres, respectively. Of course, in the _precise_ form as written, (1) has not been shown to have been violated experimentally. In practice, the four correlations on the left-hand side of (1) are each estimated from a separate sub-ensemble of particles with measurements performed at different times and/or spatial locations. Hence, for example, the measurement orientation corresponding to \(Y=0\) for the first sub-ensemble cannot correspond to _precisely_ the same measurement orientation \(Y=0\) for the second sub-ensemble; as a matter of principle, [PERSON] cannot shield his apparatus from the effects of ubiquitous gravitational waves associated for example with distant astrophysical events. Hence, as a matter of principle, what is actually violated experimentally is not (1) but\[|\mathrm{Corr}(0,0)+\mathrm{Corr}(1,0^{\prime})+\mathrm{Corr}(0^{\prime},1)- \mathrm{Corr}(1^{\prime},1^{\prime})|\leq 2, \tag{2}\]  
where, relative to the Euclidean metric of space-time, \(0\approx 0^{\prime}\) and \(1\approx 1^{\prime}\) for \(X\) and \(Y\). Could the difference between \(0\approx 0^{\prime}\), \(1\approx 1^{\prime}\) on the one hand, and \(0=0^{\prime}\), \(1=1^{\prime}\) on the other, actually matter? More specifically, is there a plausible framework for physical theory where (1) is the singular [2] rather than the smooth limit of (2) as \(0^{\prime}\to 0\), \(1^{\prime}\to 1\) and, therefore, where (2) is in some sense physically distinct from (1), no matter how accurate are our finite-precision experiments? Intuitively, it would seem not, as [PERSON] [3] himself argued with a form of 'epsilon' analysis. Indeed, common sense might suggest to even contemplate such a possibility would be to entertain a theory that was not only grotesquely fine-tuned, but one that was inconsistent with the fact that the experimental violation of (2) does not require any precision in setting the polariser orientations. The purpose of this paper is to argue that, in this respect, we are being fooled by our intuition. It is worthwhile beginning the discussion with a close analogy: the Penrose Impossible Triangle (sometimes known as the tribar). The triangle seems impossible because we intuitively assume that any two sides of the triangle necessarily become _close_ at a common vertex. Relaxing this metric assumption makes it possible to construct such Penrose Triangles in 3D physical space: it is the projection into 2D of such a 3D structure that provides the illusion (but not the reality) of inconsistency. The relevance of this example is to draw attention to the notion of distance. There is no doubt that space-time has a locally Euclidean metric. However, should we assume such a metric for state space? In conventional quantum theory based on complex Hilbert Space, this assumption is forced on us. However, motivated by both nonlinear dynamical systems theory and \(p\)-adic number theory, we outline in Section 2 a plausible and robust locally causal framework where the metric on state space is explicitly not Euclidean.