The Riemann hypothesis, proposed in 1859 by the German mathematician Georg Friedrich Bernhard Riemann, is fascinating. It seems to fit these ideas on various subjects in physics very well. This text refers to imaginary numbers. “Imaginary” does not mean the numbers don’t really exist. Imaginary numbers are the numbers that give a negative number when squared e.g. *i*^2* = -1*.

Most people aren’t accustomed to thinking that the universe is literally composed of mathematics (binary digits, Mobius strips, figure-8 Klein bottles, Wick rotation). I developed these ideas after reading about several professors - John Wheeler, Max Tegmark, Erik Verlinde, Ed Fredkin, and Rafael Sorkin. The Introduction will provide understanding that the Riemann hypothesis doesn’t just apply to the distribution of prime numbers but can also apply to the fundamental structure of the mathematical universe’s space-time. When applied to the universe, it explains - the static universe, dark matter, dark energy, the Higgs boson/field, and aspects of particle physics like the electroweak interaction.

For the purposes of this explanation, “Riemann zeta function” interacts with the term “Hamilton quaternion function” which has already been incorporated into the final paragraph of the following physics hypothesis. Elements of the Riemann hypothesis addressed here include its real and imaginary parts, the upper and lower half-planes, and the knowledge that there are infinitely many zeros on the critical line.

## KEYWORDS:

*Riemann Hypothesis; Dark Matter; Higgs Boson; Electroweak Interaction; Static Universe; Quaternions*

## Introduction

The first part of my article is a proposal that we live in a mathematical universe (though they might or might not agree with the details I propose, this opinion is shared by the scientists Max Tegmark, Erik Verlinde, Ed Fredkin, John Wheeler, and others).

There are 4 scientists I know of that support the idea of the universe being composed of information/mathematics:

a) In 1990, John Wheeler (1911-2008) suggested that information is fundamental to the physics of the universe. According to this "it from bit" doctrine, all things physical are information-theoretic in origin. (1)

b) Erik Verlinde says gravity is not a fundamental force of nature, but an emergent phenomenon. In the same way that temperature arises from the movement of microscopic particles, gravity emerges from the changes of fundamental bits of information, stored in the very structure of spacetime. (2)

c) Cosmologist Max Tegmark hypothesizes that mathematical formulas create reality (3)

d) “Pioneered (in the late 1980's) by Rafael Sorkin, a physicist at the Perimeter Institute in Waterloo, Canada, the theory (causal sets) postulates that the building blocks of space-time are simple mathematical points that are connected by links, with each link pointing from past to future." (4)

Quaternions were first described by Irish mathematician William Rowan Hamilton in 1843. Hamilton defined a quaternion as the quotient of two vectors. In this case (see the section below titled **VECTOR-TENSOR-SCALAR (VTS) GEOMETRY**): the quotient of two vectors is 1/2, the division of the electromagnetic vector (photonic quantum spin of 1) by the gravitational vector (gravitonic quantum spin of 2). Similarly, the real part of every nontrivial zero of the Riemann zeta function is 1/2.

In mapping the distribution of prime numbers, the Riemann hypothesis is concerned with the locations of “nontrivial zeros” on the “critical line”, and says these zeros must lie on the vertical line of the complex number plane i.e. on the y-axis in Fig. 3. Besides having a real part, zeros in the critical line (the y-axis) have an imaginary part. This is reflected in the real +1 and -1 of the x-axis in Fig. 3, as well as by the imaginary *+i *and *-i *of the y-axis. In the upper half-plane of Fig. 3, a quarter rotation plus a quarter rotation equals a half while in the lower half-plane, a quarter rotation minus a quarter rotation equals zero: 1/4 + (-1/4) = 0. The final sentence states, “this concept of dark energy invalidates (dark energy’s) role as the cause of an expanding universe … which could be static.” In the Riemann zeta function, it is known that there are infinitely many zeros on the critical line. This suggests the y-axis is literally infinite. To truly be infinite, it cannot be restricted to the up-down direction but must include all directions. That means it would include the horizontal direction and interact with the x-axis.

The universe (which includes the real, four-dimensional spacetime of the x-axis plus the large-scale, extra-dimensional imaginary spacetime* of the y-axis) would then be already infinite spatially and temporally - not potentially infinite. There could be no expansion or contraction of the universe as a whole. And as the final paragraph states, “the counterclockwise rotation of the x- and y-axes in Wick Rotation - which represents rotation of gravitational and electromagnetic waves - can be viewed as either rotation into diagonal form or as a quaternion function. It can also be responsible for the gravitational/electromagnetic energies forming all mass in space-time or, as dark energy acting via Wick rotation, forming all mass in imaginary time - which, since time and space can never be separated, is linked to an imaginary space and can be illustrated by the imaginary number* i *and its Wick rotation (this imaginary mass is known as dark matter).

* To introduce you to the idea of extra dimensions, consider this – Professor Itzhak Bars of the University of Southern California in Los Angeles says, one whole dimension of time and another of space have until now gone entirely unnoticed by us.

If physicists like Erik Verlinde and Max Tegmark are correct in thinking the universe has a mathematical foundation, that foundation could be the electronic BITS (BInary digiTS) of 1 and 0, which comprise what is known as base-2 mathematics.

## BITS AND TOPOLOGY

Electronics' binary digits can be used to draw a two-dimensional computer image of a Mobius strip. Two united Mobius strips create a three-dimensional figure-8 Klein bottle, (5), that acts as a building block of space, time, forces’ bosons and matter’s fermions. This creates a supersymmetry (linkage) between fermions and bosons. A recent paper (6) says that in a holographic universe, all of the information in the universe is contained in two-dimensional packages trillions of times smaller than an atom. Therefore, trillions of Mobius strips could form a photon and trillions of more complex figure-8 Klein bottles could form a more complex graviton (suggesting union of electromagnetism and gravitation).

**Figure 1 – the Mobius Strip**, which is two-dimensional and only has one surface (source: http://www.clker.com/cliparts/3/7/ a/9/1220546534781713951lummie_Mobius_Strip.svg.hi.png) The Mobius strip is a closed surface with no distinction between inside and outside. Thanks to quantum mechanics’ entanglement applying on macroscopic scales,* this doesn’t refer only to the surface itself. This results in the space-time of our universe existing everywhere and everywhen. The inside and outside of the universe’s energy and mass are continuous when it's composed of Mobius strips and figure-8 Klein bottles acting macroscopically - there cannot be other universes outside our infinite and eternal universe, and there’s no universe with different laws of physics (such a state of supposed multiple universes is called the multiverse). * “Physicists now believe that entanglement between particles exists everywhere, all the time, and have recently found shocking evidence that it affects the wider, ‘macroscopic’ world that we inhabit.” (7) Though the effect is measured for distances in space, the inseparability of space and time means that moments of time can become entangled too. (8)

The physicist and science historian Abraham Pais wrote that “In 1924 the scientist Wolfgang Pauli was the first to propose a doubling of electron states due to a two-valued non-classical "hidden rotation". (9) Extending the ideas of “doubling”, “two-valued” and “hidden rotation” from the quantum spin Pauli had in mind to the Mobius strip being a basic, fundamental unit of reality; it can be seen that Pauli’s proposal has an analogy to this article. The doubled Mobius strips (doubled to form a figure-8 Klein bottle) could be produced by the two-valued binary-digit system used in electronics. The bottles possess a hidden rotation, now identified as adaptive Wick rotation, which gives a fourth dimension to space-time.

**Figure 2 - MOBIUS DOUBLET (FIGURE-8 KLEIN BOTTLE)** (source: https://upload.wikimedia.org/wikipedia/commons/7/73/KleinBottleFigure8-01.png) Note that the positive curvature fits together with the negative curvature to produce the outline of a doughnut which is technically flat (see the reference to Vanessa Janek (10). When many doublets are placed together, binary digits can fill in any gaps or voids in the same way that computers can morph a picture on a screen and extrapolate a small patch of blue sky to make a sky that's blue from horizon to horizon. Morphing by bits can also delete a single doublet's central "hole". But the doublet doesn’t become multiply connected like the doughnut. Merely the doughnut’s outline is adopted – the doublet retains the property of being simply connected, a property necessary for space-time’s infinity. (Informally, if an object in space consists of one piece - the constituent two Mobius strips now have the outline of one doughnut - and no longer has any "holes" that pass all the way through it, it is called simply-connected. A flat universe that is also simply connected implies an infinite universe. (11)

#### Wick Rotation, Causality, And Uniting Time

**Figure 3 – WICK ROTATION**: “The complex plane reveals i’s special relationship with cycles via the circle of i, also known as Wick rotation. Whenever a point on the complex plane is multiplied by i, it moves a quarter rotation around the origin or center of the plane.” [Figure and quote from (12)] Wick Rotation’s applications aren’t restricted to soon-to-be-mentioned time dilation, dark matter and dark energy. Electromagnetic waves are described by science as transverse, as are water waves. If a stone is dropped into a pool of calm water, many circular waves soon cover the surface of the water, and the water appears to be moving outwards from where the stone was dropped in. Actually, the particles of water simply rise then fall – it's the wave motion that moves outward. Similarly, there is little movement of photons - they simply rise and fall like corks bobbing up and down in the ocean’s waves. Energy is transferred from photons in one place to photons in the next place by travelling space-time disturbances, and this wave motion is what is referred to as an electromagnetic wave. This process is described one or two pages after dark matter and dark energy. In Figure 3, photons rise from the positive branch of the x-axis (to the right of the origin or centre) to the positive y-axis (extending above the origin). Then the photons fall to the negative x-axis, and continue falling to the negative y-axis. Then they rise to the positive x and resume their rotation. The conclusions are (1) electromagnetic and gravitational waves composing space-time rotate in a circle, giving rise to other large-scale dimensions, and (2) if there is little movement of photons (and gravitons*), the universe could not be expanding (or contracting) but its space and time is static. The Big Bang has impressive points … leading to the idea that it’s a necessary stepping-stone. For example, the Big Bang’s supposed origin from quantum fluctuations is reminiscent of bits switching between 1 and 0. ”

* Einstein's equations say gravitational fields carry enough information about electromagnetism to allow Maxwell's equations to be restated in terms of these gravitational fields. This was discovered by the mathematical physicist George Yuri Rainich. (13) Therefore, gravitational waves also have advanced components going back in time. 1’s and 0’s composing electromagnetic and gravitational waves would compose both “advanced” waves going back in time and “retarded” waves going forward in time. The retarded components with +x motion in time can obviously cancel the advanced components with –x motion in time, producing entanglement. 17th century scientist Isaac Newton's idea of gravity acting instantly across the universe could be explained by gravity's ability to travel back in time, and thereby reach a point billions of light years away not in billions of years, but in negative billions-of-years. That is; the negative/advanced component of a gravitational wave would already be at its destination as soon as it left its source**, and its journey is apparently instant.

** Arriving at its destination billions of years before it left its source is an absurd impossibility if we cling to the traditional view of time flowing in one direction from cause to effect. But it's plausible if we accept the Block Universe theory which developed from Special Relativity's non-simultaneity of events for different observers. In the Block Universe, all time coexists (the entire past, the present, and every point in the future all exist at once) - in the same way that every period of time in a movie exists simultaneously on a store-bought DVD. Time can be visualized as a Cosmic DVD where our brains and consciousnesses take the place of the DVD player's laser. Everything in the Cosmic DVD's time exists at once*** but we're only aware of an extremely limited number of events at any instant (these make up our present). Gravitational waves arriving billions of years prior to emission can be compared to playing part of the Cosmic DVD in reverse. Waves travel from a later frame of the cosmic movie to an earlier frame.

*** All mass is composed of gravitational and electromagnetic waves, according to vector-tensor-scalar geometry. Both types of waves possess retarded and advanced components which entangle all masses. Wick rotation (time) is built into the Mobius strips and figure-8 Klein bottles composing electromagnetism's photons and gravitation's gravitons. Therefore, all time (the entire past and present and future) is united into one thing just as all space and all mass are united into one thing. VTS geometry says space and mass are united and physics already accepts that space and time are united. If space, mass, time, electromagnetism, gravitation, plus the VTS-derived weak and strong nuclear forces are all aspects of the same thing; that suggests the theory of quantum gravity truly exists. Mathematical equations would be just another aspect of the one thing - a tool - which people deem necessary to prove quantum gravity.

# Under Process (Review Completed)