# Experimental Non-Violation of the Bell Inequality
## 2 Invariant Set Theory  
Results below summarise more detailed analysis given in [6]. As mentioned above, we posit some primal compact fractal-like geometry \(I_{U}\) in cosmological state space, on which the universe \(U\) as a self-contained locally causal deterministic system evolves [4; 5]. Figure 1 illustrates the local fractal structure of \(I_{U}\). On the left is shown, at some \((j-1)\)th fractal iterate of \(I_{U}\), a single state-space trajectory segment ('history') in some three-dimensional subspace of state space. At the \(j\)th iterate, this trajectory segment comprises a helix of \(N\gg 0\) fine-scale trajectories and an additional \(N+1\)th trajectory (not shown) at the centre of the helix. The winding frequency \(\omega\) of a \(j\)th iterate helical segment is assumed proportional to the energy \(E\) associated with the subsystem described by this subspace. In this sense, the deBroglie relationship \(E=\hbar\omega\) reflects a key element of the invariant set postulate: that the laws of physics in space-time are manifestations of the geometry of the more primal \(I_{U}\). At the \((j+1)\)th iterate (not shown), the helical trajectory segments are themselves found to be helical. In general, a cross section through a \((j-1)\)th trajectory segment comprises a Cantor set \(\mathcal{C}\) comprising \(p=N+1\) iterated disks (Figure 2). Figure 1: A state-space trajectory segment, which appears to be a simple line on some coarse scale, is in fact found to be, on magnification, a helix of trajectories. On further magnification, each of these helical trajectory segments is itself a helix of trajectories, and so on. A cross section through the original coarse-scale trajectory segment is a Cantor Set as illustrated below. At any particular level of magnification (i.e., fractal iterate), the trajectory segments can be labelled \(a\) or \(\
ot a\) according to the regime to which they evolve. Figure 2: A Cantor Set \(\mathcal{C}\), comprising \(p=17\) iterated disks: \(N=16\) iterated pieces around the edge of a disk and \(1\) at the centre of a disk. Here, a single disk at the \((j-1)\)th fractal iteration comprises \(17\)\(j\)th-iterate disks, and each of these comprises \(17\)\((j+1)\)th-iterate disks. An element of \(\mathcal{C}\) can be represented by a sequence \(\{\phi_{1},\phi_{2},\phi_{3},\dots\}\), where \(\phi_{i}/2\pi=n/N\in\mathbb{Q}\). We will now suppose that, at any fractal iterate, the \(N\) helical trajectory segments in Figure 1 can be labelled according to a process illustrated in Figure 3, associated with the divergence and nonlinear clustering of trajectories into two distinct state-space regimes or clusters labelled \(a\) and \(\
ot{a}\) (the central \((N+1)\)th trajectory is assumed to lie on the basin boundary and hence to evolve neither to \(a\) nor \(\
ot{a}\)). This divergence reflects the generic phenomenon of decoherence (essentially the butterfly effect) as the sub-system interacts with its environment. These clusters correspond to the measurement eigenstates of quantum theory. Within this geometric framework, complex Hilbert vectors can be used to provide some incomplete probabilistic description of reality. For example, consider the \((j+1)\)th iterate disks inside a \(j\)th iterate disk and labelled \(\phi_{2}\) in Figure 2. Suppose that \(N\cos^{2}(\theta/2)\) of these \((j+1)\)th iterate disks are labelled \(a\), and that _reality_ corresponds to one of the \(N\) disks and is therefore either labelled \(a\) or \(\
ot{a}\). Then, as discussed in [6] in more detail and with \(\phi=\phi_{2}\), an incomplete representation of _reality_ can be given probabilistically by the complex Hilbert vector  
\[\cos\frac{\theta}{2}\ |a\rangle+e^{i\phi}\sin\frac{\theta}{2}\ |\
ot{a}\rangle. \tag{3}\]  
In particular, it is necessary that \(\phi/2\pi\) and \(\cos^{2}(\theta/2)\) (and hence \(\cos\theta\)) are rational numbers.