## 1. Introduction

## 2. Invariant Set Theory

**Theorem**

**1.**

**Proof.**

- ${g}_{p}(x,y)$ is Euclidean,
- ${g}_{p}(x,y)\le 1$,
- ${g}_{p}(x,z)={g}_{p}(y,z)=p.$

## 3. The Sequential Stern-Gerlach Experiment

## 4. The Bell Inequality

## 5. Conspiracy and Free Will

#### 5.1. Nullifying the Notion of Conspiracy

#### 5.2. Free Will and Inaccessible Determinism

## 6. Relations to Other Approaches

#### 6.1. Bohmian Theory

#### 6.2. The Cellular Automaton Interpretation of Quantum Mechanics

#### 6.3. p-Adic Quantum Theory

## 7. Discussion

## Acknowledgments

## Conflicts of Interest

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**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 $\overline{)a}$ 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 jth-iterate disks, and each of these comprises 17 $(j+1)$th-iterate disks. An element of $\mathcal{C}$ can be represented by a sequence $\{{\varphi}_{1},{\varphi}_{2},{\varphi}_{3},\dots \}$, where ${\varphi}_{i}/2\pi =n/N\in \mathbb{Q}$.

**Figure 3.**Here, $N=16$ classical state-space trajectories diverge into two distinct regimes labelled a and $\overline{)a}$. In this example, seven of the 16 evolve to the $\overline{)a}$ regime and the other nine evolve to the a regime. In terms of the parameter $\theta $ described in the text, here $\mathrm{cos}\theta =1/8\in \mathbb{Q}$.

**Figure 4.**(

**a**) a sequential Stern-Gerlach experiment where a particle is sent through three Stern-Gerlach devices, A, B and C; (

**b**) A, B and C shown as directions on the celestial sphere. Although to experimental accuracy A, B and C may be coplanar, they are not coplanar precisely. In invariant set theory, we demonstrate the non-commutativity of spin observables by number theory.

**Figure 5.**(

**a**) in general, it is impossible for all the cosines of the angular lengths of all three sides of the spherical triangle to be rational, and the internal angles rational multiples of $2\pi $. (

**b**) what actually occurs when (2) is tested experimentally. Here, the cosines of the angular lengths of all sides are rational. In a precise sense, (b) is ${g}_{p}$ distant from (a).

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