Here we will present some very unusual mathematics that may change some of your ideas about space-time. It stems from the important theorem about the transformation of a scalar into a vector. From the presented material it will follow, for example, that the time coordinate in our real space is not just a straight line, but a certain probabilistic subspace. Also, here we will get an interesting conclusion about mathematical entanglement, from which the principle of quantum entanglement of particles can follow.

The entire construction will be based on the property of the vector length, which remains unchanged during any of its transformations, including the transition from scalar to vector form.

It is also necessary to remind our readers that according to our hypothesis of a single multidimensional space, our real world is merely its reflection. One of the reflections.

We will call the Euler subspace the plane formed by two coordinates that represent the real and imaginary components of the following function: \[ f(\alpha) = \cos(\alpha) + i \sin(\alpha) \tag{1}\] where: \(i\) is the imaginary unit, \(\alpha\) is the angle [1].

Let's transform this scalar function into a vector according to the theorem, but we will do this separately for its cosine and sine parts. For the sake of brevity of the following formulas, we will further denote this transformation as follows: \[ \cos(\alpha) \to \mathcal{COS}(\alpha) \\ \sin(\alpha) \to \mathcal{SIN}(\alpha) \tag{2}\] Now let's make the transformation itself: \[ \mathcal{COS}(\alpha) = \pm\, \mathbf{j_0} \pm\, \mathbf{j_1}\, i \alpha \pm\, \mathbf{j_2} \frac{1}{\sqrt{3}} \alpha^2 + \ldots \pm \mathbf{j_{n}}\, \sqrt{\large (-1)^{n} \normalsize \frac{2^{2n-1}}{(2n)!}}\, \alpha^{n} \tag{3}\] \[ \mathcal{SIN}(\alpha) = \pm \mathbf{j_1}\, \alpha \pm\, \mathbf{j_2} \frac{i}{\sqrt{3}} \alpha^2 + \ldots \pm \mathbf{j_{n}}\, \sqrt{\large (-1)^{n + 1} \normalsize \frac{2^{2n-1}}{(2n)!}}\, \alpha^{n} \tag{4}\] Where: \(\mathbf{j_n}\) - unit vector coordinate \(n\), the direction of the unit vector can have both positive and negative values ​​with equal probability, and \(n \in 1,2,3,4,\ldots\) Thus, we have made a transformation of the Euler subspace into a global multidimensional space (GMS).

Since the transformation of a scalar into a vector is linear, the following condition is satisfied for it: \[ f_1(\alpha) + f_2(\alpha) \to \mathbf{f_1}(\alpha) + \mathbf{f_2}(\alpha) \tag{5}\] where: \(f_1, f_2\) - arbitrary functions.

First, let's check the main property of the global vector, based on this condition: \[ \cos(\alpha)^2 + \sin(\alpha)^2 = \mathcal{COS}(\alpha)^2 + \mathcal{SIN}(\alpha)^2 = 1 \tag{6}\] According to the same condition, we will make a transformation for formula (1): \[ \cos(\alpha) + i \sin(\alpha) \to \mathcal{COS}(\alpha) + i\, \mathcal{SIN}(\alpha) \tag{7}\] Let's check the modules of these functions, showing the length of the vector, which, obviously, should always be equal to one: \[ |\cos(\alpha) + i\, \sin(\alpha)| = 1 \tag{8}\] It is logical to assume that the length of the vector should also be equal to one: \[ |\mathcal{COS}(\alpha) + i\, \mathcal{SIN}(\alpha)| = 1 \tag{9}\] However, there is one important point here that can turn your understanding of mathematics upside down. Let's get down to getting a completely unexpected result!

Let us expand the vector part of expression (7) using formulas (3, 4): \[ \mathcal{COS}(\alpha) + i\, \mathcal{SIN}(\alpha) = \pm\mathbf{j_0} + \left( \pm\mathbf{j_1}\, i \alpha \pm\mathbf{j_1}\, i \alpha \right) + \left( \pm\mathbf{j_2} \frac{1}{\sqrt{3}} \alpha^2 \pm\mathbf{j_2} \frac{1}{\sqrt{3}} \alpha^2 \right) + \ldots \tag{10}\] Let us just recall that all \(\pm\) have been equally probable here so far. From this we can immediately conclude that the only possible option when condition (9) is satisfied will be as follows: \[ \mathcal{COS}(\alpha) + i\, \mathcal{SIN}(\alpha) = \pm\mathbf{j_0} \tag{11}\] We have obtained a completely non-obvious formula that can tell us about several patterns at once.

1. The amazing ability of formula (11) stems from the property of conservation of energy. If the modulus of the initial vector is equal to one, then no matter how we transform itcalled, its length should not change. And if so, then all \(\pm\) in expression (3) and (4) are pairwise related to each other and are linearly dependent quantities. This state of affairs arises only when these formulas begin to describe one common system (common vector). And doesn't the entanglement of quantum particles work the same way [2]?

2. From formula (11) it follows that the Euler subspace from our reality, in the GMS is a zero unit vector. And this can say a lot. For example, if in our real space a point makes a circular rotation, plus - moves in time, then in the GMS its trajectory will be a straight line of time.

Another example. The wave in our reality, described by the Euler formula - \(A \exp(i \omega t)\), in the GMS will be a straight line with amplitude \(A\). But no one prevents us from imagining the opposite situation, when a straight line of time in the GMS gives a projection onto our real world in the form of numerous lines of time, equally probable scattered over the plane of the complex Euler space. In this case, the angle \(\alpha\) will be a random variable in the range \(0 .. 2\pi\).