In Part I, we introduce the notion of simplex-integers and show how, in contrast to digital numbers, they are the most powerful numerical symbols that implicitly express the information of an integer and its set theoretic substructure. In Part II, we introduce a geometric analogue to the primality test that when p is prime, it divides \binom{p}{k}=(p(p-1)…(p-k+1))/(k(k-1)…1) for all 0<k<p. Our geometric form provokes a novel hypothesis about the distribution of prime-simplexes that, if solved, may lead to a proof of the Riemann hypothesis. Specifically, if a geometric algorithm predicting the number of prime simplexes within any bound n-simplexes or associated An lattices is discovered, a deep understanding of the error factor of the prime number theorem would be realized – the error factor corresponding to the distribution of the non-trivial zeta zeros. In Part III, we discuss the mysterious link between physics and the Riemann hypothesis. We suggest how quantum gravity and particle physicists might benefit from a simplex-integer based quasicrystal code formalism. An argument is put forth that the unifying idea between number theory and physics is code theory, where reality is information theoretic and 3-simplex integers form physically realistic aperiodic dynamic patterns from which space, time and particles emerge from the evolution of the code syntax. Finally, an appendix provides an overview of the conceptual framework of emergence theory, an approach to unification physics based on the quasicrystalline spin network.

### About Klee Irwin

**Klee Irwin is an author, researcher and entrepreneur who now dedicates the majority of his time to Quantum Gravity Research (QGR), a non-profit research institute that he founded in 2009. The mission of the organization is to discover the geometric first-principles unification of space, time, matter, energy, information, and consciousness. **

**As the Director of QGR, Klee manages a dedicated team of mathematicians and physicists in developing emergence theory to replace the current disparate and conflicting physics theories. Since 2009, the team has published numerous papers and journal articles analyzing the fundamentals of physics. **

**Klee is also the founder and owner of Irwin Naturals, an award-winning global natural supplement company providing alternative health and healing products sold in thousands of retailers across the globe including Whole Foods, Vitamin Shoppe, Costco, RiteAid, WalMart, CVS, GNC and many others. Irwin Naturals is a long time supporter of Vitamin Angels, which aims to provide lifesaving vitamins to mothers and children at risk of malnutrition thereby reducing preventable illness, blindness, and death and creating healthier communities.**

**Outside of his work in physics, Klee is active in supporting students, scientists, educators, and founders in their aim toward discovering solutions to activate positive change in the world. He has supported and invested in a wide range of people, causes and companies including Change.org, Upworthy, Donors Choose, Moon Express, Mayasil, the X PRIZE Foundation, and Singularity University where he is an Associate Founder.**

Klee Irwin

Quantum Gravity Research

Los Angeles, California, USA

Received 9 September 2019

Accepted 3 November 2019

Published 28 November 2019

This paper introduces the notion of simplex-integers and shows how, in contrast to digital

numbers, they are the most powerful numerical symbols that implicitly express the information

of an integer and its set theoretic substructure. A geometric analogue to the primality test is

introduced: when p is prime, it divides

p

k

for all 0 < k < p. The geometric form provokes a

novel hypothesis about the distribution of prime-simplexes that, if solved, may lead to a proof of

the Riemann hypothesis. Speci¯cally, if a geometric algorithm predicting the number of prime

simplexes within any bound n-simplex or associated An lattice is discovered, a deep under-

standing of the error factor of the prime number theorem would be realized the error factor

corresponding to the distribution of the non-trivial zeta zeros, which might be the mysterious

link between physics and the Riemann hypothesis [D. Schumayer and D. A. W. Hutchinson,

Colloquium: Physics of the Riemann hypothesis, Rev. Mod. Phys. 83 (2011) 307]. It suggests

how quantum gravity and particle physicists might bene¯t from a simplex-integer-based qua-

sicrystal code formalism. An argument is put forth that the unifying idea between number

theory and physics is code theory, where reality is information theoretic and 3-simplex integers

form physically realistic aperiodic dynamic patterns from which space, time and particles

emerge from the evolution of the code syntax.

Keywords: Physics; number theory; geometry.

1. Introductory Overview

The value of this paper lies in the following questions the following author hopes are

provoked in the mind of the reader:

1. Is there a geometric algorithm that predicts the exact number of prime-simplexes

embedded within any n-simplex?

2. If Max Tegmark is correct and the geometry of nature is made of numbers, would

they be geometric numbers like simplex-integers?

This is an Open Access article published by World Scienti¯c Publishing Company. It is distributed under

the terms of the Creative Commons Attribution 4.0 (CC BY) License which permits use, distribution and

reproduction in any medium, provided the original work is properly cited.

Reports in Advances of Physical Sciences

Vol. 3, No. 1 (2019) 1950003 (79 pages)

#.c The Author(s)

DOI: 10.1142/S2424942419500038

1950003-1

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3. Would this explain the correspondence between number theory and physics and

support the conjectures of Freeman Dyson2 and Michel Lapidus,3 who posit the

existence of a missing link between number theory and fundamental physics?

4. If nature is made of geometric numbers, how would it compute itself into existence

and is the principle of least action an indication it is concerned with e±ciency?

5. Are quasicrystal codes maximally e±cient?

6. If nature is a symbolic language — a code operating at the Planck scale, can it

exist without choosing or \measuring" entities at that scale, as is required to put

in action the syntactically free steps in all codes?

Ontologically, it seems clear that the fundamental elements of reality are made of

information. We de¯ne information here as \meaning in the form of symbolic

language". And we de¯ne language as \a symbolic code consisting of (a) a¯nite set of

symbols, (b) construction rules and (c) syntactical degrees of freedom. This is in

contrast to deterministic algorithms which also use a¯nite set of symbols and rules

but have no degrees of freedom. We de¯ne symbol here as \an object that represents

itself or something else". And,¯nally, we de¯ne an object as \anything which can be

thought of".

Fundamental particles have distinct geometries and are, in some sense, geometric

symbols. For example, at each energy state, an orbiting electron forms a¯nite set of

shape-symbols p-orbital geometries composed of the probability distribution of

the wave-function in 3-space. There are strict rules on how these fundamental

physical geometric objects can relate, but there is also freedom within the rules, such

that various con¯gurations are allowed. If we speculate that particles are patterns in

a Planck scale geometric quantum gravity code in 3D of quantized space and time,

we can wonder what the most e±cient symbols would be.

Simplexes are e±cient symbols for integers and their set theoretic substructure.

The prime simplexes within any bound of simplexes, n-simplex to m-simplex, are

ordered according to purely geometric reasons. That is, the set theoretic and number

theoretic explanation is incidental to the geometric one. Accordingly, speculations by

Freeman Dyson,2 Michel Lapidus3 and others on a hidden connection between

fundamental physics and number theory are less enigmatic when considering shape-

numbers, such as simplex-integers. This view brings fundamental physics and

number theory squarely into the same regime geometry a regime where

physics already resides.

As symbolic information, simplexes are virtually non-subjective and maximally

e±cient when used to express the information of an integer and its set theoretic

substructure. A quasicrystalline symbolic code made of simplexes possesses

construction rules and syntactical freedom de¯ned solely by the¯rst principles of

projective geometry.

All particles and forces, other than gravity, are gauge symmetry uni¯ed according

to the E6 Lie algebra, which corresponds to the E6 lattice and can be constructed

entirely with 3-simplexes. That is, it can be understood as a packing of 6-simplexes,

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Each simplex can each be constructed from 3-simplexes. More speci¯cally, all par-

ticles and forces other than gravity are uni¯ed according to the standard model SU

(3) SU(2) U(1) Lie algebra. The single gauge groups that contain this algebra

include SU(5) in the form of Georgi–Glashow Grand Uni¯ed Theory (GUT), SO(10)4

and E6.

5 All three are related by the complex octonion projective plane (C O)P2

which is E6 divided by SO(10) U(1) and by the 20-dimensional set of complex

structures of 10-dimensional real space R10, which is SO(10) divided by SU(5). These

algebraic objects are isomorphic to their Euclidean geometric analogues, which are

simple higher dimensional lattices constructed as packings of simplexes. So E6

embeds SO(10), which embeds SU(5), which embeds SU(3) SU(2) U(1). E6

embeds in E8. In 1985, David Gross, Je®rey Harvey, Emil Martinec and Ryan Rohm

introduced Hetronic string theory using two copies of E8 to unify gravity with the

standard model in an attempt to create a full uni¯cation theory. E8 can be con-

structed entirely by arranging, in 8D space, any choice of n-simplexes of equal or

lesser dimension than the 8-simplex. Other approaches using E8 for uni¯cation

physics include those of Lisi6 and Smith and Aschheim.7

Cutþ projecting a slice of the E8 lattice to 4D along the irrational hyper vector

prescribed by Elser and Sloan generates a quasicrystal. This quasicrystal can be

understood as a packing of 3-simplexes in 4D forming super-clusters 600-cells, which

intersect in seven ways and kiss in one way to form the overall 4D quasicrystal.

Because this 3-simplex-based object derived from E8 encodes the E6 subspace under

projective transformation, it also encodes the gauge symmetry uni¯cation physics of the

standard model along with the less well accepted uni¯cation of general relativity via E8.

A projection encodes the projection angle and original geometry of a pre-projected

object. Consider a copy of a unit length line [yellow] and its projection rotated by 60

[blue] as shown in Fig. 1.

Of course, this is a line living in 2D and rotated by 60 relative to a 1D projective

space, where it contracts to a length of 12 (as shown in Fig. 1). Similarly, a projection

of a cube (as shown in Fig. 2) rotated by some angle relative to a projective plane is a

Fig. 1. A unit length line [yellow] & its projection rotated by 60 [blue].

Fig. 2. A cube projected to a plane.

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pattern of contractions of the cube's edges, as encoded in the projection, which

join to form angles with one another. The total projection is a map encoding the

higher dimensional shape plus the rotation of the projector relative to the cube

and plane.

One can decode the projection itself to induce the pre-projected cube and the

possible projection angles. To form a quasicrystal, more than one cell is projected. A

slice of the higher dimensional crystal, called a cut window, is projected to the lower

dimensional space. The coordinate of the cut window can change to generate addi-

tional quasicrystalline projections that form animations. These changes can be made

by translating and or rotating the cut window through the lattice. The coordinates of

the various vertex types of the projection will change as the coordinate of the cut

window changes. The ways these changes can occur are called the phason rules and

degrees of freedom. This¯nite set of geometric angles and lengths and the rules and

freedom are collectively called the code or language of the quasicrystal. Refer to`Free

Lunch Principles-Forces' in Sec. 5.15 for more on the Cut-and-Project method.

The aforementioned E8 crystal can be built entirely of regular 3D tetrahedra

3-simplexes. When it is projected to 4D, the tetrahedral edges contract, but do so

equally so that the tetrahedra shrink under projection but remain regular, generating

a quasicrystal made entirely of 3-simplexes. As will be discussed later, we then

generate a representation of this 4D quasicrystal in 3D.

A quasicrystal is an object with an aperiodic pure point spectrum where the

positions of the sharp di®raction peaks are part of a vector module with¯nite rank.

This means the di®raction wave vectors are of the form

k ¼

X

n

i1

hia

i ;

ðinteger hiÞ:

ð1Þ

The basis vectors a i are independent over the rational numbers. In other words,

when a linear combination of them with rational coe±cients is zero, all coe±cients

are zero. The minimum number of basis vectors is the rank of the vector module. If

the rank is larger than the spatial dimension, the structure is a quasicrystal.8 And

every aperiodic pure point spectrum in any dimension correlates to some quantity of

irrational cutþ projections of higher dimensional lattices.

There is an intriguing connection between quasicrystals, prime number theory

and fundamental physics. Both the non-trivial zeros of the Riemann zeta function

and Eugene Wigner's universality signature, found in all complex correlated systems

in nature, are pure point spectrums and therefore quasicrystals.9 However, with 1D

quasicrystals that possess many nearest neighbor point to point distances, as op-

posed to the two lengths in the simple Fibonacci chain quasicrystal, it is very di±cult

to know what \mother" lattices and projection vectors generate them. In other

words, we can know it is a quasicrystal because it is an aperiodic pure point spec-

trum. But we would have no deep understanding of its phason syntax rules or

information about the higher dimensional crystals and angles that generated it.

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Andrew Odlyzko showed that the Fourier transform of the zeta-function zeros has

a sharp discontinuity at every logarithm of a prime or prime-power number and

nowhere else.10 That is, the distribution is an aperiodic pure point spectrum

a quasicrystal.2 Disorderly or chaotic non-periodic ordering will not generate an

aperiodic pure point spectrum.

Similarly, for any span of n-simplexes through m-simplexes, the density distri-

bution of simplexes with a prime number of vertices (prime-simplexes) is aperiodic

and non-random. Its prime density pattern and scaling algorithm exists for purely

geometric reasons. For example, one may consider the 99-simplex. It contains 25

prime-simplexes that have an ordering scheme that drops in density as the series

approaches the bound at the 99-simplex. The distribution of the 25 prime-simplexes

within this hyper-dimensional Platonic solid is based purely on geometric¯rst

principles and is not fundamentally related to probability theory. Of course, it can be

predicted using probability theory. The distribution of prime-simplexes, as shape-

numbers, within any bound is trivially isomorphic to the distribution of digital

integers within the same bound. The distribution pattern of digital primes is fun-

damentally non-probabilistic because the identical geometric distribution pattern of

prime-simplexes is not probabilistic.

Interestingly, Wigner's ubiquitous universality signature describes the quasipe-

riodic pattern of the zeta zero distribution.11 The pattern occurs in all strongly

correlated systems in nature. In fact, it shows up in the energy spectra of single

atoms. Indeed, nearly all systems are strongly quantum correlated. Terrence Tao and

Van Vu demonstrated universality in a broad class of correlated systems.12 Later,

we will discuss more about this pattern, which Van Vu said appears to be a yet

unexplained law of nature (see Sec. 4.9).

What could possibly correlate the distribution of primes in number theory to

something as ubiquitous as the universality signature? As Dyson recognized, the

distribution of prime numbers is a 1D quasicrystal and the universality signature

ubiquitous in physics is too.

Each prime-simplex integer is associated with a crystal lattice as a subspace of the

in¯nite A-lattice. For example, the 2-simplex is associated with the A2 lattice (see

Fig. 3), which is associated with a crystal made of equilateral triangles and is a

subspace of all A2þn lattices.

Fig. 3. A triangular lattice.

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If we cut þ project a slice of this crystal to 1D with the golden ratio-based angle of

about 52.24, we generate a quasicrystal code with three \letters" of the lengths 1, ’

and 1=’. As we go up to higher dimensional A-lattice crystals associated with a given

simplex-integer and project to 1D, the number of lengths or \letters" of the quasi-

crystalline code increases. Within any irrational projection of a prime-An lattice-

based crystal, there exists the projections of all A lattice crystals less than n,

including a distribution of prime-A-lattices. For example, the projection of the

crystal built upon the A99 lattice built of the simplex-integer corresponding to the

number 100, encodes the distribution of 25 prime-An lattice crystals.

This connection can be summarized thus: (a) The distribution of non-trivial zeta

zeros and the distribution of prime numbers is a 1D quasicrystal. (b) All 1D quasi-

crystals can be derived by irrationally projecting hyper lattice slices. A likely can-

didate lattice is the one corresponding to simplex-integers. (c) Nature is deeply

related to mathematics. (d) The foundation of all mathematics, even set theory, is

number theory. The appearance of the universality signature in all complex systems

and the distribution of primes may relate to the in¯nite-simplex. That is, the crystal

associated with the in¯nite A-lattice and its projective representation in the lowest

dimension capable of encoding information, 1D.

We propose that the missing link between fundamental uni¯cation physics

and number theory is the study of simplex-integer-based lattices transformed under

irrational projection; quasicrystalline code theory.

In Sec. 5, we mention that a general feature of non-arbitrarily generated quasi-

crystals is the golden ratio. Speci¯cally, any irrational projection of a lattice slice will

generate a quasicrystal. However, only golden ratio-based angles generate quasi-

crystals with codes possessing the least number of symbols or edge lengths. We will

discuss how black hole theory, solid state materials science as well as quantum

mechanical experiments indicate there may be a golden ratio-based code related to

the sought after quantum gravity and particle uni¯cation theory. As preparation for

that, it is helpful to understand how deeply the golden ratio ties into simplexes.

Of course, the simplest dimension where an angle can exist is 2D. And the simplest

object in 2D is the 2-simplex. Dividing the height of a circumscribed 2-simplex with a

line creates one long and two short line segments (see Fig. 4). The ratio of the long to

the short segments is the golden ratio.

In fact, this is the simplest object in which the golden ratio exists implicitly,

since simply dividing a line by the golden ratio is arbitrary and not implied by

the line itself. So there is a fundamental relationship between and ’ in the

circumscribed 2-simplex.

Simplexes are the equidistant relationships between an integer quantity of points

and, in their lattice form as packings of tetrahedra, they correspond to periodic

maximum sphere packings. For example, the maximum sphere packing in 3D

includes the FCC lattice, which is a packing of 3-simplexes. The points where the

spheres meet generate the lattice associated with the 3-simplex is called the A3

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lattice. Similarly, the E8 lattice is a packing of simplexes, and its points are the

kissing points of the maximum packing of 8-spheres.

2. Proof by Deductive Argument that Simplex-Integers are the Most

E±cient Number Symbols for Integers and their Set Theoretic

Substructure

2.1. Introduction

A number is a symbol used to measure or label. Generally, a symbol is an object that

represents itself or another object. For example, an equilateral triangle (as the delta

symbol) often represents the object called \change" or \di®erence".

However, an equilateral triangle (or any object) can also serve as a symbol to

represent itself, the equilateral triangle. Symbols can be self-referential and partici-

pate in self-referential codes or languages. An example is a quasicrystal, such as the

Penrose tiling (see Fig. 5) derived by projecting a slice of the 5-dimensional cube

Fig. 4. A triangle in a circle showing the golden ratio as the ratio between the red and the blue edges.

Fig. 5. The Penrose tiling quasicrystal, derived via cut þ projection of a slice of the Z5 lattice, is a code

because it contains a¯nite set of geometric symbols (two rhombs), matching rules and degrees of freedom.

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lattice, Z5, to the plane.

13 It is a language14 because it possesses (a) a¯nite set of

symbols, (b) construction rules and (c) degrees of freedom called phason degrees

of freedom.

It has two \letters" which are two rhomboid shapes. The rhombus symbols can

only be arranged according to speci¯c assembly rules to form seven di®erent vertex

geometries that can be thought of as the \words" formed by the two building block

geometric \letters". But within the syntax constraints, there are also degrees of

freedom, called phason degrees of freedom. The quasicrystalline code is a language in

every sense of the word, conveying the meaning of geometric form such as dynamical

quasiparticle waves and positions. Yet, the geometric symbols, the building blocks of

this code, represent themselves, as opposed to ordinary symbols which represent

other objects.

It has been shown how 3-simplexes15 can be the only shape in non-space-¯lling

quasicrystals in 3D or 4D. All quasicrystals are languages, not just the Penrose tiling.

And the 3-simplexes, i.e., simplex-integers, in these special quasicrystalline symbolic

codes represent themselves.

2.2. Ultra-low symbolic subjectivity

Generally, symbols are highly subjective, where its meaning lies at the whim of the

language users. However, simplexes, as geometric numbers and set theoretic symbols

which represent themselves, have virtually no subjectivity. That is, their numeric, set

theoretic and geometric meaning is implied via¯rst principles. If space, time and

particles are pixelated as geometric code, low subjectivity geometric symbols with

code theoretic dynamism such as these could serve as quanta of spacetime.

2.3. Simplexes as integers

The geometric structure of a simplex encodes numerical and set theoretic meaning in

a non-arbitrary and virtually non-subjective manner. For example, the digital

symbol \3" does not intrinsically encode information about the quantity of three

objects. In fact, any object can serve as a symbol for a number. So, we introduce the

notion of simplex-integers as virtually non-subjective symbols for integers.

2.3.1. Symbolic function 1: Counting

The number of 0-simplexes in a given n-simplex indexes to an integer. For example,

the 2-simplex has three points or 0-simplexes corresponding to the digital symbol \3".

The most basic information of an integer is its counting function. The series of simplex-

integers counts by adding 0-simplexes to each previous simplex-integer symbol.

2.3.2. Symbolic function 2: Set theoretic meaning

Inherent to an integer is its set theoretic substructure. A simplex-integer is a number

symbol that encodes both the counting function and set theoretic substructure of an

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integer. For example, the quantity of four objects can be communicated by the

symbols, 4 or IV. However, when we use a 3-simplex to represent the counting

function of the number 4 we also encode its set theoretic substructure:

. Four sets of one

. Six sets of two

. Four sets of three

. One set of four.

This is geometrically encoded in the 3-simplex as four 0-simplexes (points), six

1-simplexes (edges), four 2-simplexes (faces) and one 3-simplex (tetrahedron).

2.3.3. Symbolic function 3: Binomial expansion

An n-simplex encodes the binomial coe±cient corresponding to a row of Pascal's

triangle (see Fig. 6).

The coe±cients are given by the expression

n!

k!ðnkÞ!.

Pascal's triangle is the arrangement into rows of successive values of n. The k

ranges from 0 to n generate the array of numbers. It is a table of all the binomial

expansion coe±cients.

2.3.4. Symbolic function 4: Sierpinski triangle fractal

Because each simplex is a higher dimensional map of the 2D Pascal's triangle table,

it too encodes the same Sierpinski triangle fractal when the positions of the table

are coded in a binary fashion to draw out the odd and even number pattern, with

fractal dimension log(3)/log(2) (as shown in Fig. 7). This same fractal can be a

cellular automaton generated by Rule 90,16 the simplest non-trivial cellular au-

tomaton.17 Speci¯cally, it is generated by random iterations of the time steps of

Rule 90.

2.3.5. Symbolic function 5: Golden ratio in the in¯nite-simplex

Pascal's triangle is analogous to a matrix representation of the sub-simplex sums

within any n-simplex. Diagonal cuts through Pascal's triangle generate sums that are

successive Fibonacci numbers (as shown in Fig. 8). Any two sequential Fibonacci

Fig. 6. The Pascal triangle until the¯fth row.

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numbers are a close approximation of the golden ratio. The series of ratios converges

to the golden ratio.

2.4. Symbolic power of simplex integers

In graph theory, one can use a graph drawing as a numerical symbol to count

quantities of objects and explore their set theoretic relationships (connections). This

makes a graph diagram analogous to the counting function and set theoretic sub-

structure function of a simplex-integer.18 For example, the complete and undirected

graph of three objects expresses the set theoretic substructure implied by the integer

3. The graph drawing symbol is usually the 2-simplex (as shown in Fig. 9).

A key aspect of this symbol is the equidistance between its points its con-

nections. The complete and undirected network (graph) of three objects has no

magnitudes in its connections. To geometrically represent the complete graph of

three objects, equidistance symbolizes the notion of equal magnitude of the graph

Fig. 7. Sierpinski triangle showing Pascal triangle values.

Fig. 8. Fibonacci numbers as diagonal sums of a Pascal triangle.

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connections. So the 2-simplex is an e±cient or waste-free symbol for the complete

and undirected graph of three objects.

The diagrammatic symbol graph theorists typically use for the complete and

undirected graph of four objects is given in Fig. 10. It encodes the full set theoretic

substructure of the integer 4.

But here, we see a breakdown in e±ciency because we have wasted information in

the drawing or symbol. It does not inherently represent the notion of equal con-

nection magnitude because four connections have one length and two have a longer

length. This super°uous information in the symbol must be ignored by the graph

theorist. It is wasted.

The only way to have equidistance between four points in a geometric symbol is to

extrude an additional spatial dimension to go to 3D. In this case, the tetrahedron

can symbolize in a non-subjective and waste-free manner, the equidistant relation-

ship of four points and their full set theoretic substructure. This is the case for all

simplexes where each encodes a positive integer and its full set theoretic substructure

in the most e±cient manner possible without wasted symbolism and where all

connections are of the same length or magnitude.

Of course, we cannot make symbols in spatial dimensions greater than three.

However, we can work with higher dimensional simplex symbols in the form of their

associated geometric algebras.a

Next, let us sketch out a proof that simplex-integers are the most powerful

numbers to express counting function and set theoretic substructure.

Fig. 10. The three-simplex, a tetrahedron, a complete graph of four elements projected to a square.

aEach simplex is associated with a Lie group of the series An and its Lie algebra an, which corresponds to

geometry because a Lie group encodes geometric operations (mirror re°ections and rotations). Each

simplex is associated to a Cli®ord algebra (commonly named geometric algebra), while each sub-simplex is

a basis element of the same dimension.

Fig. 9. The two-simplex, a triangle.

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2.5. A method for ranking symbolic power

Here, the term symbolic power shall be synonymous with symbolic e±ciency. Our

discussion is concerned only with the e±ciency ranking of symbols that can represent

the meaning of (a) integers and (b) their set theoretic substructure. The rank of

meaning or information content of an integer and its set theoretic substructure

increases with size.

The challenge is to logically rank the magnitude of inherent information of a given

symbol, then we can consider the set of all symbols which might encode the numeric

and set theoretic meaning of integers to see if there is one type with maximal

e±ciency.

Symbolic e±ciency here is concerned with the ratio of:

1. Irreducible sub-symbols

2. Meaning

Because the numeric and set theoretic meaning is established, we are seeking to

understand what symbols are the most minimalistic or elegant the least

complex for this purpose. Accordingly, let us discuss the magnitude of a symbol's

complexity.

If a symbol can be reduced to irreducible sub-symbols, it is a composition of some

quantity of simpler symbols. We will use that quantity as the magnitude rank of

symbolic complexity. For example, the sentence \The dog ran fast" is itself a symbol.

But it can be decomposed into clauses, words and letters. Indeed, the letters them-

selves can be decomposed into simpler subparts as points and connections or lines.

Again, symbols are objects that represent themselves or another object. In the

universe of all objects, the empty set and 0-simplex are equally and minimally simple.

There can be no simpler object. These are the only two to possess the quality of being

non-decomposable into simpler objects. That is, all other symbols are composites of

other objects/symbols.

It is di±cult to conceive building composite symbols and a symbolic language out

of empty sets. Points (0-simplexes), on the other hand, can be arrayed in spaces to

form familiar symbols or can be connected graph theoretically without spaces to form

non-geometric symbols.

The simplest object in n-dimensions is the n-simplex.19 And every n-simplex is a

composite of irreducible 0-simplexes or vertices fv1,v2; ... ;vng, where every subset

in the structure is a simplex of n-m dimensions. Subsets with one element are points,

subsets with two elements are line segments, subsets with three are triangles, subsets

with four are tetrahedra and so on.

The reason a simplex is the simplest object in any spatial dimension is because it is

the least number of non-decomposable symbols (0-simplexes) needed to form a

convex hull occupying all sub-dimensions of a given spatial dimension. Of course, the

simplex need not be regular to possess this quality. However, regular simplexes have

only one edge length, one edge angle, one dihedral angle, etc. Irregular simplexes

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possess far more information, where members of a set must be distinct from one

another. Accordingly, irregular simplexes possess more information or sub-symbols.

For this reason, a regular n-simplex is the simplest object possible in any spatial

dimension.

It is clear that the irreducible 0-simplex is the simplest object that can form com-

posite symbols. It is clear that compositing a set of them to form a series of symbols

called simplex-integers non-arbitrarily and inherently encoding the numeric and set

theoretic meaning, we are concerned with e±ciency and non-arbitrarily symbolizing.

Can we allow, for example, an irregular equilateral triangle to encode the meaning

we are interested in and still call it equally simple because it uses the same number of

points? It is true that the irregular triangle encodes the numeric and set theoretic

meaning. It is also true that a graph diagram in 2D for, say,¯ve objects encodes the

same information we need with the name number of points (as seen in Fig. 11).

However, as a symbol, it inherently possesses two di®erent connection lengths. It

has additional information not required or needed in our attempt to symbolize the

number¯ve and its set theoretic substructure corresponding to equal magnitude of

connections. One must ignore that additional inherent meaning of two connection

magnitudes is implied by the symbol. We may consider this as an equation, where the

right side is the meaning of the quantity of¯ve objects and their set theoretic

substructure. The left side is a package of inherent geometric information in the

symbol we are considering to equal the right side of the equation. We start by

counting the quantity of irreducible 0-simplexes on left side the symbol side.

Counting them gives the value 5. If we allow any irregular 5-simplex, it leaves us with

an in¯nite number of geometric con¯gurations of¯ve points with more than 1 con-

nection magnitude. We wish to minimize the left side further to¯nd the one con¯g-

uration that is simpler or generates the least amount of unneeded inherent geometric

symbolism/information. Speci¯cally, we need all connections between points to be

equal. Otherwise, additional information accumulates on the left side of the equation

the inherent information of the symbol itself. Only an n-simplex can achieve this

task for any quantity of points. Via this logic, simplex-integers are the most powerful

numbers to encode the numeric and set theoretic meaning of the integers.

Fig. 11. The four-simplex, a complete graph of¯ve elements projected to a pentagram.

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2.6. Section 2. Conclusion

The simplest object in n-dimension is the n-simplex. The simplest object in any

dimension and the only non-decomposable or non-composite symbol is the 0-simplex.

The simplest set of composite symbols is the n-simplex series, which adds one

0-simplex to each successive member of the set. The count of 0-simplexes serves as a

number a geometric symbol representing the counting function of an integer and

its inherent set theoretic substructure.

Adding more points in some lower dimension, such as 2D, can also serve as a

simple counting symbol that also encodes the set theoretic substructure in the form

of the connections on a complete graph drawing. However, this symbol deviates from

the pure implied meaning in the simplex-integer series because, without extruding an

additional spatial dimension for each added 0-simplex, the connections of the graph

drawing take on di®erent length values. The implied information and unnecessary

complexity of the symbol breaks down with this more mathematically complex object,

where the equal set theoretic \connections" are no longer intrinsically implied with

virtual non-subjectivity. One must ignore this extra information and subjectively and

arbitrarily interpret the various connection lengths as having equal magnitude.

We prove through this deductive argument that the n-simplex series is the most

powerful set of symbols to represent the integers and their set theoretic substructure.

A given simplex embeds the full set of theoretic and numeric information of all

simplexes within it, including the distribution of isomorphic prime simplexes to the

distribution of prime numbers on the ordinary number line. Accordingly, the in¯nite

simplex is the most powerful representation of the integers, prime distribution and

the set theoretic substructure of each integer.

3. A Geometric Primality Test and the Prime-Simplex Distribution

Hypothesis

3.1. Introduction

It is generally believed that if an exact expression were found which determines the

number of primes within any bound of numbers, it would lead directly to a proof or

disproof of the Riemann hypothesis. This is because the non-trivial zeros that fall on

the critical line on the complex plane in the Riemann hypothesis correspond to the

error terms created by using inexact expressions to estimate the number of primes

within a span of integers. Inexact expressions are all that have been discovered so far

in prime number theory. An exact expression is still outstanding and would lead

directly to a proof of the Riemann hypothesis if discovered.

There are 25 primes in the bound of numbers 2–100. However, the prime number

theorem expression of Carl Friedrich Gauss of the form ðxÞ x= logx20 incorrectly

predicts 21.7 primes in the same bound. The expression of Peter Gustav Lejeune

Dirichlet of the form Li(x) - (x)21 incorrectly predicts 30.1 primes.

The delta between these inexact approximations and Riemann's exact function

result in error terms that can be written in terms of the non-trivial zeros of the

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Riemann zeta function. Number theorists have not proven the Riemann hypothesis

because they do not deeply understand these non-trivial zeros. That is, they do not

understand how to exactly predict the distribution of primes within a bound of

integers.

0ðxÞ ¼ RðxÞ

X

RðxÞ 1

lnx

þ 1

tan1

lnx

:

ð2Þ

Because an analytical expression for the second term in Equation (1) does not exist,

this term quanti¯es the error in prime counting functions. It is a sum over the non-

trivial zeros of the Riemann zeta function () on the critical line. The Riemann

hypothesis states that all non-trivial zeros lie on the critical line. Riemann's formula

is exact if and only if the Riemann hypothesis is true. Again, mathematicians have

not proven the Riemann hypothesis because they do not deeply understand the non-

trivial zeros. If an exact form for 0ðxÞ is found that does not depend on the zeros, it

could be used to shed light on their nature and should lead to a solution of the

Riemann hypothesis.

Mathematics is like an upside down pyramid, where sophisticated math is built

upon a foundation of simple math, and base math is the counting numbers. For

example, before one can think about set theory, one must possess a notion of

counting numbers. A huge collection of proofs of potential new theorem exist in the

literature. Essentially, they state: \We prove that if the Riemann hypothesis is true,

then this theorem is true."22

With the current state of prime number theory, we have somehow missed

something deep. It is trivially true that within the geometric structure of the in¯nite-

simplex and its irrational projection to 1D, the distribution of prime-simplex and

prime digital numbers is encoded. Accordingly, this \deep" aspect of prime number

theory, and therefore all mathematics and mathematical physics which we have

missed is geometric number theory.

3.2. Geometric primality test

We introduce a geometric analogue to the primality test that when p is prime, it

divides,

p

k

¼ p ðp 1Þ; ... ; ðp kþ 1Þ

k ðk 1Þ; ... ; 1

for all 0 < k < p:

ð3Þ

Our geometric form provokes the prime-simplex distribution hypothesis that, if

solved, leads to a proof of the Riemann hypothesis.

Claim. If and only if the quantity of vertices of an n-simplex is evenly divisible into

each quantity of its sub-simplexes is that simplex a prime-simplex and associated with

a prime A-lattice.

p is prime () ð8k 2 N; k < p ) pj#fSk Sp1gÞ;

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p is not prime () ð9k 2 N; k < p; ð#fSk Sp1gÞ½p 6¼ 0Þ:

Note : #fSk Sp1g is the binomial coefficient

p

k

¼

p!

k!ðp kÞ! :

We want to prove that p is prime if and only if p divides into C, where C is given by

Eq. (X). C ¼

p!

k!ðpkÞ! for any k between 2 and p 1. C satis¯es this equation: p! ¼

C k! ðp kÞ!

First, we demonstrate that, if p is prime, p divides C.

Conjecture 1. If p is prime, p divides C.

Proof. Because p divides p!, p also divides one of the three factors on the right side:

C or k! or ðp kÞ!

k < p and k! is a product of numbers smaller than p:Therefore, p does not divide

k!. If k is greater than 1, ðp kÞ! is a product of numbers smaller than p. Therefore, p

does not divide ðp kÞ!. So, necessarily, p divides C.

Conjecture 2. If p is composite, let p ¼ a b, where a

6¼ 1, b

6¼ 1 then at least one of

the coe±cients is not divisible by p.

Next, we demonstrate that, if p is composite, let p=ab, where a

6¼1, b

6¼1 then at

least one of the coe±cients is not divisible by p:

Take

k ¼ a : C ¼

ða bÞ!

a!ðaðb 1ÞÞ! :

ð4Þ

We can rewrite as

C ¼ bða b 1Þða b 2Þ; ... ; ða b aþ 1Þ=ða 1Þ!

ð5Þ

C is not divisible by a b, because none of the factors (a b 1), (a b – 2); ... ;

(a b aþ 1) is divisible by a, and b is not divisible by a b.

[Credit goes to Raymond Aschheim for assistance with the above equations.]

3.3. Prime-simplex distribution hypothesis

When studied as simplex-integers instead of digital integers, there is a

simple formula that separates prime numbers from composite numbers.

That is, there is a non-constant polynomial that takes in only prime values.

There is no known formula that separates primes from composite numbers. However,

there exists a purely geometric reason why a given simplex is prime or why there are,

for example, 25 prime simplexes embedded in the 99-simplex. The reason is not

directly number or set theoretic. Number and set theoretic aspects are merely inci-

dental or secondary to the geometric structure. The reason is based solely on the¯rst

principles of Euclidean geometry.

An extension can be made to A-lattices, which correspond to simplexes. For

example, the 4-simplex corresponds to the A4 lattice, which embeds the A2 and A3

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lattices. Using the 99-simplex example again, the A99 lattice, built as a packing of 99-

simplexes, embeds 25 prime-An lattices and describes their distribution exactly

without probability theory based approximations.

The unknown formula expressing the drop in prime-simplex and prime-A lattice

density within some bound is also purely geometric. In our future work, we intend to

focus on this problem. However, we can state with certainty that the algorithm can

be expressed with quasicrystalline formalisms when studied via the irrational pro-

jective transformation of a slice of the in¯nite-A-lattice.

The vertices of a prime-simplex are evenly divisible (without a remainder) into

each sum of its sub-simplexes. When one considers what this means in terms of shape

analyses, such as symmetry or topology, it becomes clear that there must be special

shape qualities present in prime-simplexes that are not evident in non-prime-

simplexes. For example, the 3-simplex is the¯rst to fail this geometric primality test.

Its sub-simplex quantities are:

4

0-simplexes,

6

1-simplexes,

4

2-simplexes.

Its four vertices does not evenly divide into its six edges. By contrast, when we

look at the simplex-integer associated with the prime number 5, we see sub-simplex

sets of

5

0-simplexes,

10

1-simplexes,

10

2-simplexes,

5

3-simplexes.

For lack of a better term, there is a division symmetry in this simplex with respect to

its geometric parts. The \beauty" of¯ve vertices evenly dividing into the sums of

each sub-simplex inspires the curiosity about what special volumetric, topological or

symmetry qualities this shape possesses.

3.4. Prime number distribution and fundamental physics

As far as impacting science is concerned, the discovery of the actual algorithm pre-

dicting the distribution of prime simplexes within an n-simplex may have important

implications for fundamental physics, shedding light on an equally monumental

outstanding problem: the theory of everything that uni¯es the theory of space and

time (general relativity) with the theory of the quantum world (quantum mechanics).

It is certainly true that nature is deeply mathematical, which means its founda-

tion is built upon counting numbers. But nature is deeply geometric as well. So

geometric counting numbers, like simplex-integers, are interesting there is a

mysterious connection between physics and the distribution of prime numbers.

Inspired by the Hilbert–Polya proposal to prove the Riemann Hypothesis,

we have studied23 the Schroedinger QM equation involving a highly non-trivial

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potential whose self-adjoint Hamiltonian operator energy spectrum approaches the

imaginary parts of the zeta zeros only in the critical line.

Sn ¼

1

2

þ in:

ð6Þ

This is consistent with the validity of the Bohr–Sommerfeld semi-classical quanti-

zation condition. We showed how one may modify the parameters which de¯ne the

potential,¯ne tuning its values, such that the energy spectrum of the (modi¯ed)

Hamiltonian matches all zeros. This highly non-trivial functional form of the

potential is found via the Bohr–Sommerfeld quantization formula using the full-

°edged Riemann–von Mangoldt counting formula for the number N(E) of zeroes in

the critical strip with imaginary part greater than 0 and less than or equal to E.

Our result shows a deep connection between the most foundational model we have

for reality, quantum mechanics, and prime number theory.

Patterns in nature over time or space can only be of three fundamental species:

1. Periodically ordered

2. Aperiodically ordered

3. Random.

There is no solid evidence for randomness in nature. In fact, demonstrating it is

impossible because one cannot write it down, as can be done with periodic and

aperiodic patterns. An experimentalist can only concede she has not been able to¯nd

periodic or aperiodic order. The lack of¯nding order is not good experimental evi-

dence for the theory of randomness. What does have good supporting experimental

evidence is the theory of non-determinism, which¯ts our code theoretic axiom.24 For

example, in 1984, Dan Shechtman reported his observation of code-theoretic aperi-

odic order in a material known by the scienti¯c community to be disorderly-randomly

structured (amorphous).25 The consensus belief was built upon a bedrock of crys-

tallographic mathematical axioms and decades of failure to observe order in this type

of material. And yet, Shechtman observed the signature of aperiodic order in the ma-

terial. A good portion of the scienti¯c community, led by Nobel laureate Linus Pauling,

rejected his¯ndings due in part to the popular theory that randomness is real.26

Similarly, number theorists have no idea why or how the quasiperiodic spectrum

of the zeta zeros possess the universality spectral pattern. Some mathematicians

think there may be an unknown matrix underlying the Riemann zeta function that

generates the universal pattern. Paul Bourgade, a mathematician at Harvard, said,

\Discovering such a matrix would have big implications for¯nally understanding the

distribution of the primes".27

So why would proving the Riemann hypothesis help in the search for a theory of

everything? Because there is a unifying principle in the form of (1) simplex-integers

and (2) quasicrystal codes based on simplexes. The notion of randomness in physics

would become an old paradigm giving way to the new ideas of aperiodic geometric-

language based physics and the principle of e±cient language.24 Non-deterministic

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syntactical choice would replace randomness as the ontological explanation for non-

determinism. Discreteness would replace the older notion of smooth space and time.

Number and geometry would be uni¯ed via the mathematical philosophy of shape-

numbers, where nature is numerical simplex-integers forming the substance of

reality geometry, all within a logically consistent self-actualized code-theoretic

universe.

3.5. Are digital numbers a dead-end approach to prime number theory?

Clearly, the universality aspect of complex physical systems is deeply rooted in the

geometry of particles and forces acting in 3-space. Prime and zeta zero distribution

display the same quasicrystalline pattern.

Non-geometric methods, such as probability theory and brute force computa-

tional methods, are typical tools for modern number theorists working on prime

number problems. If the 2300 years of stubbornness of this prime distribution

problem is a deep geometric challenge, then we have been using the wrong tools for a

long time.

As mentioned, within, for example, the 99-simplex, there are 25 prime-simplexes,

which are simplexes with a prime quantity of vertices. The reasons for why this

bound of simplexes 2–99 has a density of prime simplexes of 25 is a purely geometric

problem, even though the solution is exactly the same as the unknown algorithm

determining the exact quantity of prime digital numbers in the same bound. In other

words, the algorithm determining the distribution of prime-simplexes in some bound

is the missing and correct algorithm that encodes the error term, i.e., it equals the

error term plus the incorrect result of solutions using the prime number theory

algorithm or others.

A fresh and little-focused-on approach is to move prime number theory problems

from digital number theory into the domain of simplex-integer number theory

into the realm of pure hyper-dimensional Euclidean geometry and its associated

geometric algebras and moduli spaces.

The purely geometric algorithm that determines the number of prime-simplexes

within an n-simplex is knowable. It would give the exact number of prime numbers

within any span of integers. But what is so deeply di®erent about the digital versus

simplex-integer approaches? Two core things: (1) non-transcendental convergence

and (2) non-homogeneity of sequential number deltas.

3.6. Non-transcendental convergence

Digital integers do not converge when you add them as a series. And divergences are

unhelpful because they tell you little i.e., they don't give you a number because

they explode toward in¯nity. In order to convert digital integers into a convergent

series, one applies the zeta function. For example, we put a power on each integer and

then invert it. We do this with the next integer and add to the previous solution. We

repeat this with all integers to transform the integers into a convergent additive

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series that tells us something deep about the integers and their fundamental skeleton,

the primes:

1=ð12Þ þ 1=ð22Þ þ 1=ð32Þ; ... ;¼ 2=6:

ð7Þ

What is remarkable is that is a deeply geometric number even though integers do

not appear to be associated with geometry. It is generally believed that is tran-

scendental, although this is debated.28 The two most famous transcendental num-

bers are and the basis of the natural logarithm, e.

Both e and and are deeply geometric. The exponentiation is the fundamental

operation to transform an angle into a complex number Exp(i ) which, multiplied

by a vector, also expressed as a complex number, operates the rotation of this vector

from this angle. The constant e is de¯ned by the choice of radian as a unit for the

angle, which sets to measure a half circle rotation by Expði Þ ¼ ei, or e ¼ Expð1Þ.

This also involves I ¼ p 1.

Both e and fundamentally relate to the Riemann hypothesis but only when

explored via digital numbers.

Speci¯cally, e is a part of the false error generating algorithms that imperfectly

predict the number of primes in a bound thereby generating the error term that

maps to the zeta zeros and perhaps preventing a proof of the Riemann hypothesis.

And, is related by the convergence of the zeta function itself to 2=6. The zeta

function process is how we plot the zero solutions related to the errors onto the

complex number plane.

It may be helpful to inquire, \If the error term generating method using digital

numbers relates to the transcendental numbers and e, is the inverse true, where in

some sense we can say the use of and e generate the error term?" Although this

question is confusing, it cuts deep. In other words, there is little choice when using

digital integers the zeta function using digital integers convergesb to . And e is

deeply related to by similar reasoning associated with the choice to use digital

integers as opposed to simplex-integers.

It is reasonable to realize, though, that and e are deep aspects of the error term.

And the error term is the problem. It is simply the delta between the imprecise

temporary \placeholder" prime density prediction algorithm and the currently

unknown imprecise ones.

So is it as simple as that? Can we avoid the error term by avoiding digital numbers

and, by extension, and e?

We will not see and e when we attack the problem via simplex-integers. The true

algorithm determining the number of primes within a bound is geometric and related

to an algebraic number. Speci¯cally,

p

2. Interestingly, we do see a relationship to 2 in

current prime number theory based on digital integers. The non-trivial zeros are all on

a coordinate at 12 the length of the strip bounded by 12 on the left and 12 on the right.

bMore technically, zeta(x) converges to 0 when x goes to in¯nite, but zeta(2n) for any positive even integer

is expressed as a rational fraction multiplied by at the power 2n.

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As mentioned, the zeta function is a method to uncover something deep about

the primes by expressing them as a convergent additive series. The additive series

of the simplex-integers is naturally convergent without need to invert values. Con-

sider the circumradius of each n-simplex and index each successive one to a cir-

cumradius generated by an n-simplex with a number of vertices equal to that prime

number. The circumradius of the 1-simplex is 12. And the circumradius of the in¯nite-

simplex is 1=

p

2. The circumradii of all simplexes can be related as a series of con-

centric circles, each two with a di®erent distance between them than the distance

between any other two. The distance between each two corresponds uniquely to a

certain prime or non-prime integer such that we may call each delta between se-

quential circumradii a unique integer. And the sum of all deltas is 1=

p

2. Within any

span of such rings, there is a subset that are prime based, wherein the pattern of radii

are neither periodically nor randomly arrayed. They are arrayed as a quasicrystal.

Here, we see the¯rst example of a radically di®erent form of convergence in

simplex-integers, where the convergence value is an algebraic number instead of a

transcendental number like 2=6.

3.7. Non-homogeneity of sequential number deltas

A key di®erence between digital and simplex-integers is the information encoded in

the delta between successive numbers. For example, a few of the geometric deltas

between two successive simplexes are:

. Dihedral angles (series ranges from ArcCos½ to ArcCos(0).

. Circumradii

. Hyper-volumes

Again, the deltas index to integers. The delta between the circumradius of the

1-simplex and 2-simplex would index to the integer 3 because the 2-simplex corre-

sponds to 3 vertices or 0-simplexes.

The salient point is that the delta between two successive simplex-integer geo-

metric indexes is unique and di®erent than the delta between any other successive

pair of simplex-integers.

By contrast, the di®erence between any two successive digital integers is always 1.

Accordingly, it is homogeneous and therefore gives absolutely no number theoretic

information. This rich extra information of simplex-integers provides a wealth of

geometric clues to fuel a new approach to search for the exact scaling algorithm for

density distribution of geometric primes as prime-simplexes the deep reason for

why a prime-simplex shows up every so often in a given series of simplexes, and this

answer is trivially isomorphic to the distribution of prime digital integers.

3.8. Extensions to lattices and geometric algebras

Consider a 2-simplex, the equilateral triangle. Around it, there can be an in¯nite

2-simplex lattice, called the A2 lattice.

29 This is a tiling of the plane with equilateral

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triangles. The lattice associated with the simplex-integer 4 is the lattice described by

a maximum density packing of spheres in 3D the way oranges are stacked in the

supermarket. This is called the A3 lattice and is composed by rotating A2 lattices

from one another by ArcCos(1=n), where n is the integer corresponding to the A2

lattice (in this case, 3). This continues ad in¯nitum, where, for example, the lattice

associated with the integer 100, the 99-simplex lattice called A99, is a stack of irra-

tionally rotated parallel lattices A2 through A98.

We can extend the idea of prime-simplex distribution to prime-simplex-A-lattice

distribution. Each n-simplex can pack to form a crystal of n-simplexes

compositing to the An-lattice for a given dimension. We can then algebraically

explore the reasons for why prime A-lattices appear where they do in a given span

or stack.

A given A-lattice is associated with various geometric algebras, such as Lie and

Cli®ord algebras. The geometric algebra of a given A-lattice contains an algebraic

stack of sub-algebras associated with each sub-A-lattice. This algebraic space cor-

responds to a point array and is called a moduli space.30 These geometric algebra

tools can be used to work on the geometric problem of¯nding the actual and precise

scaling algorithm for the density distribution of prime-simplex associated A-lattice

geometric algebras within a larger stack of algebras again, an algorithm identical

to the unknown precise density algorithm for the distribution of prime digital

numbers, which would immediately lead to the proof (or disproof) of the Riemann

hypothesis.

The \writing on the wall" seems clear. The 2300 years of searching for the correct

prime distribution algorithm via digital numbers and the 157 years of mathemati-

cians trying to prove the Riemann hypothesis via digital numbers are impressive.

This apparent roadblock supports the argument that the solution can only be found

within the realm of geometry. In addition, the geometric physical connections and

the geometry of and emake it even more reasonable to surmise that digital number

theory and stochastic approaches will not lead to an answer.

Because only prime simplex vertices divide evenly into all quantities of sub-simplex,

there is an aspect of these special prime shapes that is di®erent than non-prime sim-

plexes. This geometric di®erence has additional aspects other than the pure set the-

oretic qualities of the divisibility of vertices. It is analogous to the fact that an

equilateral triangle contains all the set theoretic information of the complete and

undirected graph of three objects. However, it also contains additional shap-related

geometric information that goes beyond the set theoretic information of the graph.

Prime-simplex-integers possess special group theoretic qualities, topological qualities

and various geometric qualities that set them apart from non-prime-simplexes.

3.9. Graph-drawings

As discussed, one may gain intuition by understanding simplexes and A-lattices via a

growth algorithm that generates a graph drawing, where the quality of the connection

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magnitudes are lengths, making the object a graph drawing because the connection

types are geometric line segments. Abstractly, this can be true without admitting a

smooth spatial substrate, such as R3. In other words, the space is discretized such

that there exists no information or space between the line segments of the network.

This minimalistic graph-drawing space is in some sense a quantized subspace of a

continuous space.

Picture a line segment and rotate a copy on one end by 60 into another¯nite

1D spatial dimension to generate the three points of a 2-simplex and its associated

A-lattice. We can play with semantics by saying we have rotated a copy of the¯nite

length 1D universe into another¯nite 1D universe. The objective of the visualization

is to disassociate ourselves from the idea of a smooth Euclidean space in a network

of¯nite 1-simplexes, where length is real but restricted to the connections of the

network a graph-drawing made of 1-simplexes.

Or we can index the length value to an abstract graph theoretic magnitude.

In any case, the connection relationship between the three points (from the two lines

sharing a point) is isomorphic to a 2-simplex (as shown in Fig. 12).

Next, we rotate a copy of the second line (note that we do not need the green line

in the diagram in order to generate the 3 points of the 3-simplex) into the third

spatial dimension by 60 from the previous line to generate the 4 points of the

3-simplex and so on. Again, we reject the assumption of a smooth 3-space in favor of

an approach that is graph-theoretic.

Each of these 60 rotations from the previous edge is equal to rotating the edge by

ArcCos½ðn 1Þ=ð2nÞ from the total simplex construct below it, where n is the

simplex number.

As mentioned, this iterative process results in the stack of simplexes converging to

a circumsphere with a diameter of

p

2 at the in¯nite-simplex. When the construction

of any simplex, such as the 99-simplex, is visualized with 60 rotations extruding

successive spatial dimensions, one realizes that the lines form a hyper-dimensional

discretized non-Archimedean spiral.

In other words, the circumradii of the simplex series starts at 12 for the 1-simplex

and converges at 1=

p

2 at the in¯nite-simplex. This is in stark contrast to the cubic

series of circumradii, which is divergent, converging at an in¯nite radius for the

in¯nite-cube. So from the simplex-integer 2 (1-simplex) to the in¯nite-simplex, the

increasing radii values must be distributed over a distance of 1=

p

2 12 0:207.

Fig. 12. A blue and a red one-simplex rotated by 60 degrees make a two-simplex.

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But in just the distance from the circumradius of the 1-simplex to the 2-simplex, we

cover about 0.077 or more than 37% of the total 0.207 distance to convergence

a distance that must be distributed over an in¯nite number of simplexes.

Of course, this rapid convergence is massively exponential.

The distribution of prime numbers on the digital number line drops with distance.

For years, number theorists have used the prime number theory algorithm (and

improved versions) related to the natural logarithm number e to predict the number

of primes within a given bound. This correlates to a non-Archimedean spiral called

the logarithmic spiral because the distances between turnings increase in geometric

progression as opposed to an Archimedean spiral.

Is it the natural logarithm number e 2:718 that corresponds to the actual

algorithm for prime distribution the one that does not generate the error term?

As mentioned, e is an artifact of the exploration of the problem via digital numbers

and is fundamentally part of the error term. That is, the algorithm for predicting

prime density that is related to e simply does not work. It is the chosen formalism of

the approximation itself that generates the error term corresponding to the non-

trivial zeros. The logarithmic spiral correlated to the distribution of prime-simplexes

should logically relate to a hyper-spiral corresponding algebraic irrational

p

2. So,

just as the prime number theorem algorithm corresponds to the non-algebraic

transcendental number e, the simplex-integer prime number algorithm for prime

distribution would correspond to the algebraic number

p

2 (and to the golden ratio

by arguments beyond the scope of this paper).

3.10. Creation of a prime number quasicrystal

There is an interesting approach we will explore in a subsequent publication. We will

cut + project various A-lattices to lower dimensional quasicrystals and do spectral

analyses on the Fourier transform of each. We predict prime-A-lattice associated

quasicrystals will possess distinct spectral signatures. If so, a spectral analysis of the

superposition of a span of simplex-integer associated A-lattices projected to lower

dimensional quasicrystals is expected to reveal the signature of a prime A-lattice

distribution scaling algorithm. Such an algorithm would predict the exact prime-

simplex density distribution for any bound of projected A-lattices.

3.11. Section 3. Conclusion

We have established a hypothesis, which should be true by trivial deduction;

an exact algorithm for the distribution of primes exists in the realm of pure

geometry.

4. Simplex-Integer Uni¯cation Physics

If nature were a self-organized simulation, it would be a simplex-integer based

quasicrystalline code derived from E8.

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4.1. Introduction

The digital physics31 view is the idea that reality is numerical at its core [See work by

Ed Fredkin,32 To®oli,33 Wolfram,34 and Wheeler35]. But the numbers need not be

digital. They can be shape-numbers, such as simplex-integers. Because reality is

geometric and has three spatial dimensions, one could surmise the following: If na-

ture were built of 3D bits of information that are also numbers, the most powerful

candidate for a 3D geometric number is the 3-simplex. Power in this context is

synonymous with e±ciency in the manner explained in Part 2.

Higher-dimensional lattices, such as E6 and E8 that are associated with uni¯ca-

tion physics, can be constructed entirely from 3-simplexes. Certain projective

transformations result in the 3-simplexes remaining regular but being ordered into a

quasicrystalline code that encodes the higher-dimensional lattices and associated

gauge symmetry physics.

A hallmark and general characteristic of quasicrystals is the golden ratio.

For example, the simplest quasicrystal possible is the two length Fibonacci chain, as

1 and 1/. Virtually all 3D quasicrystals found in nature are golden ratio based on

icosahedral symmetry.

A self-organizing code on an abstract quasicrystalline substrate is in some sense

like a computer but better described as a neural network. Computer theory is con-

cerned with the e±ciency of creating information in the form of solutions to pro-

blems. Information theory is concerned with the e±ciency of information transfer.

Neural network theory 36 is concerned with both the e±cient creation and transfer of

information in a network. Neural networks operate via codes, i.e., non-deterministic

algorithmic processes languages. Neural networks in nature are spatial (geometric)

arrays of nodes with connections, such as particles connected nonlocally by quantum

entanglement or forces. Clearly nature, like a neural network, accomplishes the dual

task of (a) creating new information (computation) and (b) transferring information.

So the universe as a whole is a neural network in the strictest sense of the term.

A special quality of neural networks that sets them apart from computers is that

they are non-deterministic. If one subscribes to the theory of randomness and does

not require a theory to explain the generator of randomness, one can decide that the

free choices in a neural network code are random. On the other hand, there is a

special cases in Physics where human free will emerges in a biological neural network,

which itself emerges from fundamental particle physics and presumably some un-

known quantum gravity theory. In this case, the free will can act on the syntactical

choices in the code-theoretic neural network, providing an explanation for the syn-

tactical choices that might be more explanatory than stopping at randomness as the

unprovable axiom. It is generally known that physical reality is: (a) non-determin-

istic and (b) that it creates the emergence of non-random free will, at least in the case

of humans. As this is not a philosophical paper, we will simply say that whatever free

will is, it is non-deterministic. It behaves similar to the concept of randomness insofar

as being non-deterministic. The di®erence is that choices, as the actions of free will,

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are made with a blend of subjective meaning, perceptions or opinions combined with

logic and choices of strategy. So symbolic language and meaning are deep principles

embedded in the theory of free will. Conway and Kochen proved that if free will is

real, fundamental particles have some form of non-random free will.37

Another fundamental feature of nature is that, as a network, it is concerned with

e±ciency in the form of the principle of least action and similar laws.38 In fact,

e±ciency may be the most fundamental behavior of reality leading directly to

Noether's second theorem about conservation and symmetries in nature,39 conser-

vation laws and from there to the modern gauge symmetries uni¯cation physics, such

as seen in the standard model of particle physics.

4.2. E8 in nature

The most foundational symmetry of nature uni¯es all fundamental particles and

forces. It can be described as:

All fundamental particles and forces, including gravity, are uniquely

uni¯ed according to the gauge symmetry transformations encoded by

the relationships between vertices of the root vector polytope of the E8

lattice – the Gosset polytope.6

We have also shown cosmological correlations to E8 in Heterotic Supergravity with

Internal Almost-Kahler Con¯gurations and Gauge SO,32 or E8 x E8, Instantons.

40

However, our general approach is to exploit projective geometry as the symmetry

breaking mechanism in a quantum gravity plus particle physics approach, which

recovers particle gauge symmetry uni¯cation.

The simplest polytope in eight dimensions is the 8-simplex. The E8 lattice is the

union of three 8-simplex based lattices called A8. This lattice corresponds to

the largest exceptional Lie algebra, E8. That is, the simplest 8D building block of the

Gosset polytope and E8 lattice is the 8-simplex the most powerful number

that inherently encodes the counting function of 32 = 9 and its full set theoretic

substructure. Nine, incidentally, is the¯rst odd number that is not prime.

4.3. E8 derived quasicrystal code

In Starobinsky In°ation and Dark Energy and Dark Matter E®ects from Quasicrystal

Like Spacetime Structures41 and Anamorphic Quasiperiodic Universes in Modi¯ed

and Einstein Gravity with Loop Quantum Gravity Corrections,42 we show how

quasicrystalline codes can relate to quantum gravity frameworks.

A 4D quasicrystal can be created by projecting a slice of the E8 crystal.

43 This 4D

quasicrystal is made entirely of 600-cells, which are each made of 600 regular

tetrahedra. So the fundamental 3D letters or symbols of this quasicrystal are

3-simplexes. The allowable ways these geometric number symbols can spatially relate

to one another is governed by cut þ project-based geometry,44 which generates the

syntax of this non-arbitrary quasicrystalline code. The term code or language applies

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because the syntax allows various legal con¯gurations that are determined by the

size, shape and position of the cut-window in the hyper-lattice from which the

Quasicrystal is cut and projected from. A language or code must have degrees of

freedom within the rules and a¯nite set of symbols that must be arranged by a code

user in order to create meaning. Geometric codes, such as quasicrystals, generate

geometric meaning, such as waveform and quasiparticle position. The code user may

emerge from the evolutionary complexity of the system itself and can be as sophis-

ticated as a human consciousness and beyond. Or it can be simple, like the guiding

tendency of a tornado to preserve and grow its dynamical pattern for as long as

possible in a new physical ontology based on code theory instead of randomness.

Furthermore, how two or more syntactically legal quasicrystals can be ordered

in a dynamic pattern or animation has a separate syntax scheme based on how the

cut-window moves through the hyper-lattice. That is, all behaviors and rules are

part of a code based solely on geometric¯rst principles with no arbitrary ad hoc

mathematical contrivances.

The caveat is that a free will chooser of some form must execute the free choices in

the code. This is the case with all codes, whether that be a computer language or

song-language of birds. A general quality of codes is that meaning is not maximized

and breaks down when strategic choices in the syntax are replaced by, for example, a

pseudo random number generator.

Discovering a fundamental uni¯cation model of all particles and forces based on

such a geometric¯rst principles code is a worthy but formidable challenge. It would

be a microscopic¯rst principles theory of everything. Currently, there exists no¯rst

principles explanation for the¯ne structure constant, the speed of light, Planck's

constant or the gravitational constant. In other words, there is no known¯rst

principles uni¯cation theory. In fact, the fundamental constants h and G and c are

only known to about four places after the decimal. The CODATA values, which go

out to a few more places after 4, are an averaging of six established experimental

methods that all disagree at the 5th place after the decimal.

Syntactically legal con¯gurations of these quasicrystal-based simplex-letters form

simplex-based words and sentences. In other words, groups of simplexes have

emergent geometric meaning shape and dynamism. Sets of these inherently

nonlocal quasicrystalline simplex sentence frames can be ordered into animated

sequences and interpreted as the physical dynamic geometric patterns of space that

have both wave and particle like qualities, a well understood dualistic quality of

phason quasiparticles in quasicrystal codes.45

While this part of the discussion is a mix of fact and conjecture, the reader may

agree that, if nature is a code or simulation based on maximally e±cient symbolism,

the following three ideas may be at play:

(1) Nature must be an e±cient symbolic code capable of simulating a 3D reality.

Accordingly, the most powerful symbol in 3D— the 3-simplex—may be exploited

because it is the simplest and most e±cient 3D quantum of information.

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(2) Spacetime would be discrete. While in quantum mechanics the spin is quantized

(the quantum of action is the Planck's constant), energy is quantized (the photon

is the quantum of energy), and here we also propose spacetime which itself is

quantized. Nature must have an e±cient geometric \pixel" or foundational

symbol, just as a dynamic image on a high-resolution video monitor is composed

of invisible microscopic building-block pixels or just as a binary computer code is

made of irreducible elements symbols, where bytes can be further decomposed

into bits but no further. In our proposed framework, the simplex-integer is the

irreducible and non-transformable \pixel" in the simulation that composes our

3D reality. It is the fundamental shape-symbol in a geometric code/language. In

short, simplex-integers \switch-hit" as both numbers and spatial building blocks.

(3) It must be a symbolic code derived from E8, which encodes the gauge symmetry

uni¯cation of all fundamental particles and forces. We have generated a 3D

quasicrystal language of 3-simplexes derived from E8ð15Þ.

The causal dynamical triangulation program of Amjorn and Loll46 is encouraging

evidence that fundamental physics can be modeled with aperiodic con¯gurations of

3-simplexes as the only building-block element (see Fig. 13). They have generated

very close approximations of Einstein's¯eld equations.47

4.4. Quasicrystals as maximally e±cient codes

Just as simplex-integers are the most powerful numbers to express counting function

and set theoretic information, quasicrystals are the most e±cient codes possible in

the universe of all codes.

This is a major claim. To understand it, we should¯rst establish the fact that an

n-dimensional quasicrystal is a network of quasicrystals in all dimensions lower than

it. For example, the Penrose tiling, a 2D quasicrystal, is a network 1D quasicrystals.

A 3D Penrose tiling, called Ammann tiling48 is a network of 2D quasicrystals, which

are each networks of 1D quasicrystals.

So the building block of all quasicrystals are 1D quasicrystals. Reducing further,

we should understand that there are an in¯nite set of 1D quasicrystals. The \letters"

of a 1D quasicrystal are lengths. A 1D quasicrystal can have any¯nite number

of letters. However, the minimum is two. The Fibonacci chain is the quintessential

1D quasicrystal. It possesses two lengths related as the golden ratio. In order for a

Fig. 13. Causal triangulation of a surface. The vertical dotted lines are the timelike edges.

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quasicrystal greater than 1D to have only two letters, the letters must be 1 and the

inverse of the golden ratio. Interestingly, this simple object has a deep relation to E8.

When a slice of E8 is projected to 4D according to a non-arbitrary golden ratio-based

irrational angle,43 the resulting quasicrystal is made entirely of 3-simplexes and is the

only way to project that lattice to 4D and retain H4 symmetry. The angle between

adjacent 3-simplexes is 60 þ ArcCosð½3’ 1=4Þ ¼ ArcCosð1=4Þ, where ’ is the

golden ratio. This quasicrystal, fully encoding gauge symmetry uni¯cation physics,

can be described as a network of Fibonacci chains. These are the most powerful 1D

quasicrystals for two reasons. As mentioned, the power of a code relates to how many

building block symbols it has. This de¯nition of power relates to the discussion

earlier, where we spoke of the left side of the equation as being the magnitude of

simplicity of the symbolic system. But a code cannot have fewer than two funda-

mental symbols for obvious reasons. This is what makes binary codes so powerful.

Secondly, Fibonacci chain quasicrystal codes are based on the Dirichlet integers 1

and 1=’, which possess remarkable e±ciency characteristics, such as error detection

and correction abilities and multiplicative and additive e±ciencies. For example,

they are closed under multiplication and division. As with all quasicrystals,

Fibonacci chains are fractal.49

The function of division stands out in the arsenal of powers that the golden ratio

possesses because it relates to measurement, which is the deepest and most enigmatic

aspect of quantum mechanics that is not yet fully understood.

Physics Nobel laureate, Frank Wilczek of MIT said,50 \The relevant literature [on

the meaning of quantum theory] is famously contentious and obscure. I believe it will

remain so until someone constructs, within the formalism of quantum mechanics, an

observer, that is, a model entity whose states correspond to a recognizable caricature

of conscious awareness."

Wilczek is speaking of the need to build measurement into a new quantum me-

chanics mathematical formalism that incorporates an operator capable of measuring

at the Planck scale. His motivation is on solid ground. Quantum mechanics indicates

that the ontology of physical reality must be based on measurement, where all that

exists is that which is measured. Certainly, it is extreme to postulate that physical

reality needs humans to measure it in order to be real. A physical formalism based on

the premise that reality is made of information in the form of a code would require

a quantum-scale mathematical operator capable of actualizing information via

measurement.

A measurement of any form is ultimately a spatial relationship between the

measurer and two additional points in space. This is the case with any detector,

such as a human eye or a Geiger counter. Waveforms are reducible to quantum

particles. All detectors are reducible to component particles that interact with signal

particles, such as photons, that are emitted from another particle being measured

at a distance. For example, a camera takes a photo of a tree by receiving rays of

photons that trace to the camera lens. The irreducible measurement, however, is the

relationship between a detecting particle and two other particles at two other

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coordinates. This forms a triangle, where the detector is one vertex and the two

measured coordinates are the other vertices. The fundamental information being

registered by the detector is a transformation via contraction of the edge length of

the triangle that is not connected to the detection particle. For example, you observe

two friends in the distance. We can conceptualize you and the two friends as three

points in space. There is an actual distance length between your two friends, which

we will call L. Because it is impossible for you to have a perfectly equal distance

between you and each of your friends, you observe or measure the information of L-l,

where l is some contraction value on L.

Your transformation of L gives you information about the relationship of your

two friends and their relationships to you. If they are standing one car length apart

but your angle relative to them is such that you perceive it as ½ a car length, then

you intuitively know how to decode that information to tell you their actual distance

as one car length plus your position relative to them. Similarly, when you look at the

complexity of, say, a tree, the massive package of information from that measure-

ment/observation is merely a composite of these individual length transformations

between pairs of points and the measuring detector, forming a transformation of the

pre-transformed triangle.

So L and l form a relationship in your mind as a ratio. The meaningful information

of your measurement is not l it is the ratio of L to l, which tells you information about

the relationship of the two measured points to one another (their actual length

relationship) and their length relationship to you from your vantage point.

Consider this set of three points that are equally spaced in a line in 3-space. If you

measure them with your eye from a golden ratio-based vantage point equal to a

rotation of the line of three points by a golden ratio angle, then you can divide the

total length into two parts as 1 and 1=’ (as shown in Fig. 14). This is true only in

perspective projection, never in orthogonal projection where the segment sizes would

stay equal.

To review, all measurements are divisions or ratios. And a choice of measurement

(observation) is necessary to actualize or make-real any information. If we consider

that reality is information theoretic or code-based, we must model a mathematical

measuring operator, as discussed. So why would the golden ratio obey the principle of

e±cient language better than any other ratio? Why would it be more powerful in

terms of the ratio of symbolism to meaning?

Fig. 14. Two segments of equal lengths rotated by an angle and seen through perspective projection. Blue

is nearest to the observer and looks bigger than the red segment. The ratio is the golden ratio (beware–that

will not be the case with the orthogonal projection).

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For golden ratio divisions, going down from long to short, the ratios between

successive pairs is the golden ratio. Going up, it is the inverse of the golden ratio. This

quality is known as in°ation and de°ation and the golden ratio achieves it with only

two symbols or numbers 1 and 1=’.

But for divisions other than the golden ratio, going down needs two ratios (3/2,

2/1 for example); going up also requires two ratios. Consider the idea that Planck

scale measurement operators in the quantum gravity code use abstract observation

actions in the E8 derived quasicrystalline point space to actualize compact symbolic

objects that are themselves ratios simply ordered arrays of the two Dirichlet

integer values 1 and 1=’. This binary pair of values is maximally e±cient in terms of

the symbolism to meaning ratio.

For the ’ (golden ratio) division, there is only one ratio needed for encoding the

relationships of the consecutive segments going down in length (as shown in Fig. 15).

a

b

¼ b

c

¼ ’;

ð8Þ

Or going up in length

c

b

¼ b

a

¼ 1

’

:

ð9Þ

For example, any other division, such as dividing into thirds, requires two ratios and

therefore more symbolic information to express

a

b

¼ 3

2

;

b

c

¼ 2

1

or

c

b

¼ 1

2

;

b

a

¼ 2

3

:

ð10Þ

The other aspect of the golden ratio that is powerful and may be important for a

simulation code of reality is its fractal nature. In the last 37 years, fractal mathe-

matics has been found to be at play at all scales of the universe from cosmic to the

sub-atomic scales.49 Dividing a line by the golden ratio, if we take the short length

and place it on top of the long length, we are left with a section of the long length that

is left over. That length is even shorter than the short length of the¯rst division and

the ratio of this new short length to the original short length is the golden ratio. This

process can continue to in¯nity in the smaller direction with the ratio of the re-

mainder to the previous length always being the golden ratio. Furthermore, this

process can be applied in the other direction, where we add the long piece from the

original division to the undivided length. The ratio of the new combined length to the

long length from the¯rst division is the golden ratio. This also continues to in¯nity.

The golden ratio is the ultimate recursive fractal, generating the most information

for the least amount of symbolic symbolism and measurement action.

Fig. 15. The golden ratio: B is to A what C is to B.

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A phason °ip in a quasicrystal is a binary state change of a point, where it is

registered as being on or o®. If it is on, it is an active node with a connection to other

points in the quasicrystal. The syntax rules allow legal choices of whether a point can

be on or o®. One can call the total set of points the possibility space. The points that

are chosen to be on by the code user, are active in that frame of the dynamic

quasicrystal. Active or on points have connections and are syntactically legal se-

lection con¯gurations of the possibility space. For example, this is a projection of the

32 vertices of the 5-cube to the plane, where we see 31 total points with an overlapped

32nd point hidden in the middle. Note that the Penrose tiling is made by projecting a

slice of the 5-cube lattice to the plane. These 31 points are a small section of the

possibility space that the dynamical phason code of the Penrose tiling operates on

(see Fig. 16).

The Penrose tiling is a tiling of two types of selection patterns of 16 of the 31 point

decagonal possibility space. The two decagons can overlap other decagons in two

ways or kiss without overlapping. In Fig. 17, we highlight those two selection

Fig. 16. The projection to 2D of a¯ve dimensional cube with 32 vertices projected to 31 giving possibility

space of Penrose Tiles.

Fig. 17. Two types of selection pattern of 16 of the 31 point decagonal possibility space, as they exist in a

larger Penrose tiling Quasicrystal.

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patterns, as they exist in a larger Penrose tiling quasicrystal. You can easily visualize

how to select one of the two 16 point combinations by looking at the projection of the

full 31 point possibility space (shown in Fig. 16).

The empire of a given point consists of points that are forced to be on by the given

point. For example, look again at the 31 points in Fig. 16. Note that if you select the

center point to be on, you are forced to have a certain 15 additional points also be on

and another 15 to be o®.

Those 15 points that were forced to be on are the empire of the one point which

you, the language user, chose to be on in a phason °ip binary action.

Why is this interesting in terms of e±ciency? In computer theory, we try to

conserve binary actions. It costs electricity and time to open or close logic gates in a

computer. So e±ciency is important. We want codes that achieve maximal infor-

mation with as few actions as possible.

Changing a single point to be on or o® in a Fibonacci chain 1D quasicrystal forces

an in¯nite number of additional points throughout the possibility space of the 1D

chain to also change state. This global or nonlocal \spooky action at a distance" is

very powerful in terms of the ratio of action to meaning.

If we live in an information theoretic universe, then, abstractly, the action we are

trying to conserve is binary choice. Speculation of what the substance or entity or

action is that makes the choices in the code is °exible. Our frame work deals with the

math and behavior of the code, not so much who or what the operator of the code

must be. The principle of e±cient language requires the operation of the code to tend

toward maximal meaning for the least number of on–o® choices.

Resources are always used to make a choice in any physical model. For example,

in the neural network of a human brain, choices cost calories and time. In an arti¯cial

neural network, choices require time and electricity. So e±cient neural networks

generate as much meaning as possible with a given number of connection actions

they generate maximal information for as few binary choices as possible.

Each Fibonacci chain is isomorphic to a Fibonacci word, which is a string of 0s

and 1s that encodes a unique integer.51 Of course, the larger the integer, the greater

the magnitude of the information. For example, a one millimeter Fibonacci chain

with Planck length tiles is isomorphic to a Fibonacci word with 1031 0s and 1s and

corresponds to an equally enormous integer. Changing one point on the Fibonacci

chain possibility space from o® to on, changes the state of points along the entire

chain, thereby changing the Fibonacci word to a di®erent integer. The principle of

empires in arti¯cial neural networks consisting of networks of Fibonacci chains

involves enormous e±ciency when one is interested in conserving binary actions or

choices. If nature is a computational language based on neural network theory and

globally distributed computation and connectivity, quasicrystal codes are the most

e±cient possible.

When a network of Fibonacci chains is formed in 2D, 3D or 4D, a single binary

state change at one node in the possibility space changes the Fibonacci chains

throughout the entire 1þ n dimensional network of chains.

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4.5. DNA and quantum computers as examples of 3D

neural networks

Manmade computer code symbols are, in some sense, minimally e±cient. That is, the

binary symbols encode an instruction to do one binary state action in a logic gate to

express only one bit of information. DNA is not quite a computer. It is a neural

network in the sense that it both computes information using its code rules and it

transfers information within its structure. Like a neural network, it achieves its com-

putations and information storage in a distributed manner within 3-space. A single

position in the DNA possibility space of coordinates where one of the four molecules in

the code can exist serves as information in more than one 1D string of code. For

example, an adenine molecule can exist at some location in the DNA coordinate space.

This then forces certain states of the 4-letter code for other positions in the string,

according to syntactical rules of the code. That string of code is wrapped around the

double helix and has an empire of forced coordinate identities from the four letter code.

But the empire is not just in 1D along that single string. Information relative to that

one molecule selection of adenine is also encoded into strings that run in-line with the

axis of the double helix and also diagonal to the axis. It is similar to the analogy of

the game Scrabble, where a choice of a single letter on the grid of the possibility space of

the game can encode information in more than one word. So the choice of the adenine

molecule at that coordinate achieves a great deal of e±ciency by (a) playing a role in

multiple 1D strings and (b) by forcing other syntactically controlled actions of coor-

dinates in the empire of that single registration of adenine in the DNA possibility space.

DNA has quasicrystalline structure. In fact, Erwin Schrodinger¯rst deduced that

DNA has a quasiperiodic structure in his book What is Life, published nine years

prior to Watson and Crick's discovery of DNA in 1953.52 His deduction is in-line with

the theme of this paper. Speci¯cally, crystalline structures are deterministic and have

no degrees of freedom in terms of their abstract construction. They are not inherently

languages because they are too rigid in their construction rules.

On the other hand, amorphous or disorderly materials do not have structural

rules and can have a virtually in¯nite number of microscopic geometric relationships

– geometric symbols. The lack of rules and lack of a¯nite set of geometric symbols

prevent a dynamic code from evolving within amorphous materials. The \sweet

spot" between order and disorder, where a language or code can emerge, is in qua-

sicrystalline order. Only within aperiodically-ordered structure is there a true code

with a¯nite set of geometric symbols, rules and syntactical freedom.

Themost powerful codes are based on the golden ratio because the ratio of symbolism

to geometric meaning output is maximal. For example, DNA ismade of two helices that

have pentagonal rotational symmetry, which is based on the golden ratio. The two

helices themselves are then o®set from one another by a golden ratio related value called

a Fibonacci ratio, which is a rational approximation of the irrational golden ratio.53

Quantum computers are another example of systems where one node serves

multiple roles in various relationships. 3D clusters of atoms, often with golden

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ratio-based icosahedral symmetry54 in a quantum correlated state interact with one

another in various combinations to process and create information as a group

a spatial network of nodes very di®erent than the ordinary notion of a 1D

computational system.

4.6. Symbolic power of Fibonacci chain networks

It is well known that Fibonacci codes have unique and powerful properties in terms of

error correction and detection.51

For example, all sequences in a Fibonacci word end with \11". And that sequence

appears nowhere else in the data stream of that symbolic group object. Changing a

bit corrupts the sequence (the symbolic group object). However, within a few more

symbols, the pattern \11" will appear again, which indicates the end of the string or

group symbol.

The system or user can then simply resume coding with only those few symbols

felt to be incorrect. The power in this is that one bit can only corrupt up to three

symbols. No other code shares this property. Error detection is fast, and errors are

limited in how much damage they can do. Error correction is similarly powerful and

unique. Let us say that a 0 is erroneously changed to a 1 that is adjacent to a correct

1. A 1 that is part of the data stream gets changed to a 0. A 1 that is part of the

ending 11 gets changed to a 0 and so on.

When an error occurs in ordinary codes, it will exist uncorrected in the string

forever.

The power of 3D networks of Fibonacci chains relates to the spatial dimension of

the quasicrystal being able to host objects with icosahedral symmetry. For example,

the 4D analogue of the icosahedron is the 600-cell.55 The icosahedron is one of the

¯ve regular polytopes in 3D the Platonic solids. Three of the solids correspond to

crystal symmetries because their combinations can tile space. These are the square,

octahedron and tetrahedron. The other two are correlated with quasicrystal sym-

metry, the 600-cell and the 120-cell. These correspond to the quasicrystal-based

Platonic solids called the icosahedron and dodecahedron, each possessing icosahedral

symmetry. Again, in 3D there are¯ve regular polytopes. In 4D, there are six. And in

all dimensions higher than 4D, there are only three the analogues of the tetra-

hedron, octahedron and cube the crystal-related polytopes. The quasicrystal-

related regular polytopes are exclusive to dimensions less than 5D. So the special

dimensions for Fibonacci chain-related quasicrystals are 1D, 2D, 3D and 4D. Of these

dimensions, 4D can host the quasicrystal with the densest network of Fibonacci

chains, where 60 Fibonacci chains share a single point at the center of the 600-cells in

the E8 to 4D quasicrystal discovered by Elser and Sloane.

43 In other words, a binary

state change in the possibility space for this quasicrystal changes the state of

many other Fibonacci chains associated with that point. And numerous other

points in the possibility space also change state, not just the ones in the Fibonacci

chains connected to the aforementioned point. All this binary state change the

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empire occurs due to the geometric¯rst principles via the state change of a single

node in the possibility space.

If the universe is a neural network interested in maximal e±ciency, this would use

a substrate like this. The fact that this quasicrystal and its 3D analogue discovered

by our group called the quasicrystalline spin network (QSN) encodes gauge sym-

metry uni¯cation physics may be evidence for the trueness of the conjecture. And

this would be more likely if the universe is a neural network code concerned with

expression of maximal meaning for the minimum number of binary state choices/

actions.

The principle of e±cient language guides the behavior of the code choices in this

framework, where binary actions in the code are chosen such that maximum infor-

mation or meaning is generated for the least number of binary choices.

Meaning comes in two categories:

(1) Physical or ultra-low subjectivity geometric information — the prototiles of the

quasicrystalline code, wherein all particles and forces can be simulated such that

the simulation are one and the same and are themselves in physical reality.

(2) Emergent or virtually transcendent and highly subjective information, such as

Mathematics and Humor. This form of information can never be separated from

the geometric physical information and quasicrystalline code. For example, the

abstract thought of \love" comes with a package of memories and associations

that trigger countless changes in the nonlocal waveform domain of quantum

mechanics, gravity and electromagnetism.

At a physical level, evidence for this tendency toward e±cient code use would exist in

the form of the principle of least action and similar principles and conservation laws.

At a non-physical level, evidence for this would exist in the form of the delayed choice

quantum eraser experiment56 and Bem's retro-causality experiments57 in addition to

well-known experiments of quantum entanglement over space and time. As engines

of abstract meaning generation and perception, humans would be a special case in a

universe obeying the principle of e±cient language, where our perceptions of

meaning and information far exceed the brute simple geometric meaning expressing

physical phenomena in the quasicrystalline code.

The degree 120 vertices of the E8 to 4D quasicrystal appear to be the maximum

possible density of Fibonacci chains in a network of any dimension and therefore the

most powerful possible possibility space for a neural network. 3D quasicrystals or-

dinarily have a maximum of degree 12 vertices with six shared Fibonacci chains.

Fang Fang of Quantum Gravity Research discovered how to create a 3D network of

Fibonacci chains with degree 60 vertices.15

This quasicrystal is made entirely of 3-simplexes, the simplest possible \pixel" of

information in 3D (see Fig. 18). It encodes E8 uni¯cation physics and is derived from

the aforementioned E8 to 4D quasicrystal. Refer to Fang Fang's et al. paper An

Icosahedral Quasicrystal as a Packing of Regular Tetrahedra103 regarding the

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construction of a dense, quasicrystalline packing of regular tetrahedra with icosa-

hedral symmetry.

4.7. Is the error correction code found in gauge symmetry physics a

clue that nature computes itself into existence?

James Gates, the John S. Toll Professor of Physics at the University of Maryland and

the Director of The Center for String and Particle Theory found the widely used

doubly-even self-dual linear binary error-correcting block code embedded in the

network of relationships of the gauge symmetry uni¯cation equations of fundamental

particles.58 These are the exact same codes used in web-browsers and peer-to-peer

network simulations to ensure the consistency of information transfer from client to

client. Furthermore, he found that the error correction codes relate speci¯cally to

geometric symbols, called adinkas,59 which encode the relationship of particle gauge

symmetry equations. This astounding¯nding is one of the most powerful pieces of

evidence in support of the digital physics view that is growing in popularity in aca-

demic circles the view that reality itself is a computation, essentially a simulation.60

Gates himself commented, \We have no idea what these things are doing there".61

4.8. Is there evidence for a golden ratio code in black hole equations,

quantum experiments and solid state matter?

Other compelling evidence used to support the digital physics view includes black

hole quantum gravity theory and an idea known as the holographic principle, which

is derived from the mathematical proof called the Maldacena conjecture.62 It states

that the total amount of binary information from all the mass and energy pulled into

a black hole is proportional to its surface area, where every four Planck areas of its

surface encodes the state of a fundamental particle that fell into it.

It is a distinctly binary code-based framework that comes directly from the¯rst

principles application of general relativity and quantum mechanics at the limit of

Fig. 18. A chiral quasicrystal derived from E8 – a tetrahedral decorated E8 to 4D then to 3D quasicrystal.

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both theories the environment of a black hole. Black hole quantum gravity

equations are one of the best clues we have about what a theory of everything might

look like.

As stated, quasicrystals generally relate to¯ve-fold symmetry and the golden

ratio. The pentagon is the 2D analogue of the icosahedron and the quintessential 2D

quasicrystal is the Penrose tiling with its 5-fold symmetry related golden ratio

structure. Virtually all of the 3D quasicrystals discovered in nature have icosahedral

symmetry. That symmetry is possessed by any object having the combination of 2-

fold, 3-fold and 5-fold rotational symmetry. The E8 to 4D quasicrystal has these

symmetries and is fundamentally based on the golden ratio.

Black hole physics relates deeply to the golden ratio. It is the precise point where a

black hole's modi¯ed speci¯c heat changes from positive to negative.63

M4

J2

¼ :

ð11Þ

It is a part of the equation for the lower bound on black hole entropy.

e

8Sl 2

P

kA

:

ð12Þ

The golden ratio even relates the loop quantum gravity parameter to black hole

entropy.64

2 ¼ :

ð13Þ

In 1993, Lucien Hardy, of the Perimeter Institute for Theoretical Physics, discovered

that the probability of entanglement for two particles projected in tandem is65:

5:

ð14Þ

In 2010, a multinational team of scientists found an E8-based golden ratio signature

in solid state matter. Cobalt niobate was put into a quantum-critical state and tuned

to an optimal level by adjusting the magnetic¯elds around it. In describing the

process, the researchers used the analogy of tuning a guitar string. They found

the perfect tuning when the resonance to pitch is in a golden ratio-based value

speci¯cally related to the geometry of E8. The authors speculated that the result is

evidence in support of an E8-based theory of everything.66

Xu and Zhong's short paper,67 Golden Ratio in Quantum Mechanics, points out

the connections to the golden ratio in various works linking it to particle physics

and quantum gravity (quantized spacetime). The short piece is worth reprinting

here, and we have included their citations in our bibliography.

The experimental discovery of the golden ratio in quantum magnetism68 is an

extremely important milestone in the quest for the understanding of quantum me-

chanics and E-in¯nity theory. We full-heartedly agree with the explanation and dis-

cussion given by Prof. A®leck69 ... ...For this reason, we would like to draw

attention to a general theory dealing with the noncommutativity and the¯ne structure

of spacetime which comes to similar conclusions and sweeping generalizations about

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the important role which the golden ratio must play in quantum and high energy

physics. Maybe the most elementary way to explain this point of view is as follows:

Magnetism is just one aspect of the¯ve fundamental forces of nature. In a uni¯ed

picture where all the¯ve forces melt into one, it is reasonable to suspect that the

golden ratio will play a fundamental role. This fact immediately follows from the work

of the French mathematician Alain Connes and the Egyptian engineering scientist

and theoretical physicist M.S. El Naschie. In Connes' noncommutative geometry, his

dimensional function is explicitly dependent on the golden mean. Similarly, the bi-

jection formula in the work of El Naschie is identical with this dimensional function

and implies the existence of random Cantor sets with golden mean Hausdor® di-

mension as the building blocks of a spacetime which is a Cantor set-like fractal in

in¯nite dimensional but hierarchal space. Invoking Albert Einstein's ideas connecting

spacetime to geometry with energy and matter, it is clear that these golden mean

ratios must appear again in the mass spectrum of elementary particles and other

constants of nature. There are several places where this work can be found.70–72

4.9. Wigner's universality

The universality pattern is another fundamental clue about what a theory of

everything should look like. It is aperiodic but ordered liberally de¯ned as a

quasicrystal. It was¯rst discovered by Eugene Wigner in the 1950s in the energy

spectrum of the uranium nucleus.73

In 1972, number theorist Hugh Montgomery found it in the zeros of the Riemann

zeta function, so it deeply ties into the distribution of prime numbers.11 In 2000,

Krbalek and Šeba reported it in the complex data patterns of the Cuernavaca bus

system.74 It appears in the spectral measurements of materials such as sea ice75 and

human bones. In fact, it appears in all complex correlated system virtually every

physical system. Wigner's hypothesis states that the universality signature exists in

all complex correlated systems.9 Van Vu of Yale University, who has proven with

coauthor Terence Tao that universality exists in a broad class of random matrices,

said, \It seems to be a law of nature".12

Why something as fundamental as the universality signature would relate to both

the distribution of primes and complex physical systems is a mystery unless

somehow number theory and an unknown theory of everything are deeply related. Of

course, that is trivially true since the entire edi¯ce of mathematics is built upon the

counting numbers. And the foundational \skeleton" of the counting numbers are the

primes. Eugene Wigner famously said that nature is unreasonably mathematical.76

So the ultimate foundation of both complex mathematics and nature herself reside in

number theory.

Freeman Dyson de¯nes a quasicrystal as \a [aperiodic] pure point distribution

that has a pure point spectrum". He said, \If the Riemann hypothesis is true, then

the zeros of the zeta-function form a one-dimensional quasicrystal...".2 Andrew

Odlyzko published the Fourier transform of the zeta-function zeros. It showed sharp

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peaks at the logarithm of the primes and prime.10 This demonstrated that the dis-

tribution is not random but is aperiodically ordered. By the same de¯nition, the

universality signature is a quasicrystal. Quasicrystals in nature generally correspond

to the golden ratio. So how might the universality signature correspond to it?

Universality relates fundamentally to matrix math. It de¯nes the spacing between

the eigenvalues of large matrices¯lled with random numbers. This is interesting be-

cause the four-term two-by-two binary matrix is the most fundamental of all matrices.

14 of its 16 possible combinations of 1 and 0 have either trivial or simple eigenvalues as

0, 1 or 2. However, the remaining two eigenvalues are golden ratio based as

þ ¼ and ¼

1

:

ð15Þ

Quantum systems, such as the hydrogen atom, are governed by matrix mathematics.

Freeman Dyson said, \Every quantum system is governed by a matrix representing

the total energy of the system, and the eigenvalues of the matrix are the energy levels

of the quantum system."80

Based on work done by Suresh and Koga in 2001,77 Heyrovska78 showed the

atomic radius of hydrogen in methane to be the Bohr radius over the golden ratio.

rH ¼

a0

:

ð16Þ

The random matrix correspondence to physics is not an indication that actual ran-

domness occurs. The matrices of some correlated systems, like a hydrogen atom, can

be worked out precisely. However, more complicated systems, such as a uranium atom,

are non-computable by current methods. The values of its unknown matrix become

super-imposed like the blur of voices in a crowded conference hall. Although, there is

no randomness in the conversations of the people in the crowd, the super-position of

soundwaves behaves exactly like the solutions to a matrix with random numbers.

Scientists are still trying to¯gure out why universality has the exact pattern that

it does. Vu said, \We only know it from calculations". Because this pattern also

matches perfectly to the distribution of the non-trivial zeros in the Riemann zeta

function, the distribution of primes must relate to a strongly-correlated matrix.

Dynamically, quasicrystals obey random matrix statistics.79 And they are strongly

correlated and nonlocal, due to the empire concept discussed above.

The distribution of prime numbers is encoded in the spectral pattern derived by

an irrational projection of a slice of an An lattice to 1D. The cell types of An lattices

are simplex-integers, n-simplexes, where each An lattice and its n-simplex cell type

embeds the stack of allAn lattices with dimensions lower than it. That is, the series of

simplex-integers, including prime-simplex-integers, are encoded in the projection of a

slice of an A-lattice to 1D.

The salient point for now is that the distribution of primes and, accordingly, the

zeta zeros corresponds to geometric-number theory simplex-integers and their

associated An lattices. We conjecture that our quantum gravity framework based on

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a quasicrystal projected from the 8-simplex based E8 lattice will explain why the

quasiperiodic universality pattern appears both in nature and prime number theory.

That is, the matrix analogue of our quasicrystal may be the missing matrix corre-

lated to the universality signature.

Like all quasicrystals, the dynamical behavior of our E8 derived quasicrystal is

described by a complex matrix.79 Because its complex phason code is strongly and

nonlocally correlated, it will obey random matrix statistics and map to the univer-

sality signature. But the random matrix and universality pattern would be secondary.

We agree with Laszlo Erd€os of the University of Munich, who said \It may happen

that it is not a matrix that lies at the core of both Wigner's universality and the zeta

function, but some other, yet undiscovered, mathematical structure. Wigner matrices

and zeta functions may then just be di®erent representations of this structure".80

4.10. Section 4. Conclusion

This section began with the conjecture:

If nature was a self-organized simulation, it would be a simplex-integer based

quasicrystalline code derived from E8.

We have defended the reasonableness of the conjecture. Now it is up to our

institute and the scientists who work here continue to publish a series of theoretical

and experimental papers that transform the toy framework into a rigorous formalism

worthy of attracting a community of collaborators. The approach is certainly outside

the box. However, an outside the box approach may be what is needed. String theory

is now 50 years old and it has not made a successful prediction. We believe that a

fresh but rigorous new approach such as ours is overdue. It is possible there are

bridges to aspects of the string theory approach. In fact, the most foundational string

theory was¯rst introduced by David Gross et al. in 1985, heterotic string theory.

It exploits the power of E8.

105

However, our primary approach achieves symmetry breaking in an intuitive

manner via projective geometry to lower dimensions, where full recovery of hyper-

dimensional uni¯cation physics can be achieved. The resulting spacetime and par-

ticle code is a simulation, much more similar in form to loop quantum gravity, where

the code itself is the structure of dynamical spacetime.

Appendix A. Overview of Emergence Theory

Emergence theory, developed by our institute over the last eight years, exploits the

ideas discussed above. The program is at an intermediary stage of development.

A.1. Foundational papers

Fang Sadler et al. published the foundational tetrahedral golden ratio rotational

relationships and helical behavior in 2013.81 In 2012, Kovacs et al. introduced the

sum of squares law82 and in 2013, Castro-Perelman et al. proved the derivative

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sum of areas and volumes law.83 In 2014, Fang et al. derived the golden ratio rotation

from the¯rst principles approach of the icosagrid method. In 2016, she and coau-

thors published the construction rules of the 3-simplex based quasicrystalline pos-

sibility space and introduced the term golden matrix (\GM") to describe it along

with its E8 derived sub-spaces.

15

A.2. Conceptual overview

Our program is an Occam's razor approach to physics, where we aim to start with

irreducible¯rst principles and relentlessly question status quo assumptions. Because

nature seems to be governed by rules and beautiful math, it is safer to say that there

exists an analytical expression for the¯ne structure constant, the Planck constant,

the magnitude of the speed of light and the gravitational constant than it is to say

there is not. Put di®erently, either there exists a¯rst principle theory of everything

that explains these values or there is not. However, no such theory has been dis-

covered yet. All theories start with those values and then create equations relating

them and their composite objects.

It is helpful to understand the di®erence between a uni¯cation theory and a

simulation theory. A uni¯cation theory is a network of equations that show how

di®erent things transform into one another. A simulation theory uses geometric building

blocks as the mathematical operators that themselves are physical reality the sim-

ulation instead of merely describing it. Such a framework would spit-out the uni¯-

cation equations while also serving as the \pixels" or functional building blocks of reality.

We want to know what reality is, not just the equations that tell us how it behaves or

how it is uni¯ed. Loop quantum gravity is the most popular simulation theory.

Because reality appears to have three spatial dimensions, we start there and

inquire whether or not it is possible to simulate physics using the simplest building

block or pixel of 3D information, the 3-simplex. The idea is known as a background

independent model because it starts with spacetime building blocks and makes

particles the propagating patterns in that system. The second part of our basic idea is

that we use a quantized irreducible unit of measurement at the Planck scale sub-

structure of our model. We call this operator a quantum viewer. The building block

simplex-integers are made of information. But, they are ontologically real because

they are being actualized by quantized units of primitive measurement the

quantum viewers.

We will now highlight a few of the key components of our framework.

A.3. Ontology and symbolic language \All that exists is that

which is measured"

We would agree with Ilija Barukčić's statement:

\Roughly speaking, according to Bell's theorem, there is no reality

separate from its observation".84

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Classical physics indirectly de¯nes energy as information in the form of an abstract

quantity called the \potential for work". Spacetime is permeated with energy, where

di®erent energetic potentials within it, are equal to local densities of curvature.101

Einstein's mass-energy equivalency reduces matter to the notion of bound up energy.85

Quantum mechanics is more clearly information theoretic, dividing reality into the

abstract possibility space of the wave function and the actualized collapse into a

particle coordinate in the form of measurement data.86 J. A. Wheeler was one of the

¯rst to point out that reality is made of information.35 Max Tegmark and many other

modern physicists hold this view today. Information is real, so ontologically, there is

a division between the potential for information, which is not real, and information

as a product of measurement/observation, that is real.87 The measurement problem

associated with quantum mechanics relates in large part to the choice of ontological

interpretations of what the equations and experiments mean. It is a topic of hot

debate with no broad consensus. Einstein and many others have said that there is

something we are missing and that the formalism is incomplete.88 Some have taken

the bold position that humans or entities at our level must measure something to

actualize it into physical existence. Einstein was one of the¯rst to take issue with this

idea, saying, \I like to think that the Moon is still there even when I'm not looking

at it". So we take the more conservative position that there is some self-actualizing

measurement operator at the Planck scale, where the quantized pixels of reality exist.

We call this operator a \quantum viewer". Its function is to generate a trinary state

change in the 3-simplex quanta of space in a possibility space of such objects. The

possibility space is called the QSN.15 It is an E8 derived space of 3-simplexes, wherein

the trinary state selection actions create syntactically legal quasicrystalline sub-

spaces of the QSN that are physically real frames of space with particle patterns

embedded within it. The trinary quality state choices are: (1) on right, (2) on left and

(3) o®. For example, if a 3-simplex is in the \on right" state in one quasicrystalline

frame and is \on left" in the next frame of a dynamical sequence, the formal action is

a Cli®ord rotor or spin operation on the possibility space. However, there is an

ontological requirement to manifest these actions with an irreducible measurement/

observation operator the quantum viewer action. To understand this, visualize

the idea of standing to the left of a friend and taking a photo. Next, walk to her right

and take a second photo. Each photo is a transformation-symbol. The ordered set of

two photos express the physical information of a discretized rotation of your friend

changing orientation relative to your camera if you are stationary and she rotated

between the two orientations. So as each quantum viewer performs its operation, it

captures symbols which are projective transformations that are equal to a state change

of a tetrahedron as either on-right, on-left or o®. The quantum viewers actualize, via

observation/measurement, the action of a Cli®ord rotor or spin operation on the QSN.

As mentioned in Sec. 4.4, the 3-simplex network can be decomposed as a network of

1D Fibonacci chains with line segments in the golden ratio proportion. The quantum

viewers generate either a right or left-handed rotation of a tetrahedron, which divides a

given edge by the golden ratio on one side or the other (as shown in Fig. A.1).

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Mathematically, the coordinates of the quantum viewers the camera positions

are the edge crossing point set of the QSN. A key geometry of the network can be

understood by taking 20 evenly spaced 3-simplexes that share a common vertex at

the center of the cluster. Rotating each either right or left on an axis running from

the outside face center through the shared inner vertex by the golden ratio based

angle 12 cos

1 1

4

120

creates a 20-group that is either twisted to the right or left

(see Fig. A.2). This is an absolute chirality not relative to one's vantage point.

In summary,

(1) The quantum viewers are the observation or measurement operators.

(2) They make projective transformations based on their position, just as a camera

transforms a 3D image to a 2D image, which is really a network of transforma-

tions of 1D actual lengths (lines) between pairs of points to contracted or

transformed lines. So the irreducible measurements are 1D phason °ips that divide

a line into the golden ratio with the long side on one side of the line or the other.

(3) The transformations are information. They are observations that are equal to

symbols. Because those symbols are ontologically real due to actualization via

observation, they compose the next frame or state change in that region of the

QSN — a physically real region of space and time with particle patterns in it.

(4) Formally, the system is a spin network on a discretized moduli space, where the

operators are primitive measuring entities generating physically real information.

Fig. A.1. Golden division of tetrahedral edges with twisting.

Fig. A.2.

(a) An icosahedron divided into regular tetrahedron which are spaced from the inside sym-

metrically so there is space between the faces. (b) The same tetrahedron with a common vertix in the

center rotated with face kissing.

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(5) Its rules and syntactical degrees of freedom are derived by the geometric¯rst

principles of phason cutþ project dynamics related to the movement of a cut

window through the Elser and Sloan E8 to 4D quasicrystal.

A.4. Quantized space and \Time"

As explained above, space is quantized as 3-simplexes. And time is quantized like a

35 mm¯lm, as ordered sets of individual quasicrystalline frames of 3-simplexes

generated by ordered sets of trinary selection choices of the quantum viewers in

the QSN. Of course, this concept of a universal frame rate is anathema to key

assumptions in special relativity the invariance of the speed of light and

the notion of smooth spacetime. The old relativistic notion is that, because

spacetime is smooth and structureless, nothing can have intrinsic time or motion but

only relative time and motion. The relativistic concept is well supported by experi-

ments, which show that, no matter how fast an observer chases a photon, it

always seems to elude him at the speed of light. Our solution to this is the electron

clock model.

A.5. Electron clock intrinsic time

We reject the assumption of structureless space. The Michelson–Morley experiment

of 1887 was not designed to test for a structure as described herein or any of the other

loop quantum gravity type theory, where spacetime has a discrete substructure. Prior

to 1887, the scienti¯c community presumed a speci¯c °uid type material called the

aether¯lled space.89 When experiments did not demonstrate this substance, a new

axiom was established that there is no substructure to space. Of course, without

substructure, there can be no logical motion relative to space. An object would not

have intrinsic motion but only motion relative to another object °oating in the ocean

of the structure less vacuum. This key axiom undergirds relativity theory. The

second modern assumption is that fundamental particles, like the electron, have no

substructure and are instead dimensionless points. If this were true, such a particle

could not have an internal clock or any concept of rotation. All time or change that

would be ontologically real would be changed relative to another object changing

another clock.

Louis de Broglie¯rst conceptualized the notion of the electron possessing an

internal clock.90 Later, David Hestenes made this idea more rigorous.91 In the

emergence theory framework, massive particles, like electrons are composites of

multiple Planck length 3-simplexes chosen as ordered sets in frames of the QSN.

There are two forms of dynamic pattern:

(1) Stepwise toroidal knot — This is a knot pattern much like a 3D trefoil knot that

has an asymmetric region that cycles around the geometry of the knot (as shown

in Fig. A.3). Multiple quasicrystal frames are required in order to complete a full

cycle around the knot — a tick of the internal electron clock.

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(2) Helical propagation: For simplicity, let us imagine it takes 10 frozen quasicrystal

states chosen by the quantum viewers to compose an animation of one knot cycle.

The entire knot can remain at one coordinate in the QSN or it can propagate

helically forward in a certain direction. However, if any of the 10 frames are used to

propagate the pattern forward, there will be fewer frames available to complete

time cycles of the internal toroidal knot-like clock. There must always be a rational

fraction of frames used for propagation and frames used for clock time. The two

patterns of \time" and propagation would be inversely proportional to one an-

other. And there would always be an absolute and intrinsic ratio of internal clock-

time to propagation with respect to the global frame rate of the QSN.

A photon in this model is a pattern of tetrahedra in the QSN that is only helical, not

toroidal. So the ratio of propagation to clock time in a given number of frames will

always be 100:0. That is, any non-massive particle (particles without internal knot

structure) will always propagate in an invariant manner with the same distance

covered over a given quantity of frames.

The traveler in a spacecraft moving at 99% of the speed of light will shift their

intrinsic clock cycles (as a ratio of total frames) to a very slow rate. This will include

all massive particles moving with it, including the measurement apparatus and the

operation of the brains of the scientists onboard the craft. The clock cycles or ex-

perience of change on the craft will be very slow and the photon will move at the

speed of light from the projector on the ship and will go to a mirror at some distance

before re°ecting back to the measurement apparatus to be compared to some

quantity of clock cycles. Very few clock cycles will have elapsed because time for

these travelers and their massive equipment will slow to a near halt. Accordingly, the

comparison of the distance traveled by the photon to the number of clock cycles will

indicate that the photon moved relative to the traveling craft at the same speed it

moved when the experiment was done while the vehicle was moving at 1% of the

photon's rate of propagation. However, the intrinsic or actual di®erence between the

Fig. A.3. A Hamiltonian path cycle through the 57 centers of 57 tetrahedron.

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speed of the vehicle moving at 99% of the speed of light and photon moving at 100%,

would in truth be 1% of the speed of light. Clearly, this viewpoint is far less enigmatic

and geometrically pleasing than the ordinary interpretation of these experiments via

the smooth spacetime ontology of special relativity.

A.6. Chirality

The conjecture that fundamental particles are dimensionless points without struc-

ture causes intuitive geometric confusion with other indications that particles deeply

relate to handedness or chirality. For example, a current of electrons has a well

understood geometric chirality feature. The right-handed rule of how a magnetic

¯eld is wrapped around the current in a chiral fashion tells us something deep about

handedness in nature. However, the notion of a right-handed or left-handed indi-

vidual particle is replaced by an abstract non-geometric sign value that is distinctly

non-geometric due to the conjecture of the dimensionless point particle identity of

the particle. For example, the point particle mathematical abstraction is one where

helicity is the sign of the projection of the spin vector onto the momentum vector,

where left is negative and right is positive. It is an outstanding mystery as to why the

weak interaction acts only on left-handed fermions such as the positron and not

right-handed ones like the electron.92

Quasiparticle patterns in the QSN have a fundamentally di®erent feature that

relates to chirality. In Fig. A.4, a left-handed group of 20 3-simplexes, where the

states of the tetrahedra by the quantum viewers on the simplexes are all \on-left" is

shown.

Ordinarily, a helix made of 3-simplexes, as shown on the left in Fig. A.5, will have

no periodicity because of the irrationality of the dihedral angle. However, in the

QSN, tetrahedra can only be related by the golden ratio-based angle:

1

2

cos1

1

4

120

¼ ArcCos

2

2

ffiffiffi

2

p

:

ðA:1Þ

Fig. A.4. A left-handed group of 20 3-simplexes, where the states of the tetrahedra by the quantum

viewers on the simplexes are all \on-left".

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The helix on the left is right-handed. So when the rotation of the golden ratio

angle is of opposite chirality, in this case, rotated by that value right-handed

the periodicity become 5-fold. And when it is rotated left, it becomes 3-periodic. The

deep reason for these two periodicities corresponds the E8 to 4D Elser–Sloan qua-

sicrystal, wherein the projection of the Gosset polytopes in the E8 crystal generates

600-cell made of 600 3-simplexes. Each simplex is part of a rings of 30 simplexes, as

shown in this diagram.

The periodicities of the tetrahedra ring in Fig. A.6 are a superposition of 3-fold

and 5-fold, where the orientation of 15 tetrahedra repeats 3-periodically and 15

repeat 5-periodically. The dihedral angle between any two adjacent tetrahedra is

1

2 cos

1 1

4

120

þ 60. In 4D, there is vectorial freedom for the 60 component of

the angle. When the relationships of 3-simplexes are represented in the QSN, we cast

out the 60 component because it is the portion related to the construction of a

simplex series, where each 60 of a new edge on an n-simplex to generate an nþ 1

simplex is 60 into an additional spatial dimension.

Realistic physics would not be able to be done if we projected the E8! 4D to 3D

or projected E8 directly to 3D. The key feature of the QSN is that, by making the

tetrahedra regular by taking a 3D slice of the 4D QC with regular tetrahedra and then

rotating copies of that slice by the same angle that relates adjacent tetrahedra in the 4D

QC but minus the 60 component, we introduce three crucial elements into the object:

(1) It generates an additional sign value necessary for Physics.

(2) It signi¯cantly increases the degrees of freedom in the code. In other words,

it transforms the code from a binary on/o® code to a trinary code of \on right",

Fig. A.5.

(a) The Boerdijk–Coxeter helix showing no Periodicity. (b) The Boerdijk–Coxeter helix

showing 5 Periodicity by same-handed golden twisting. (c) The Boerdijk–Coxeter helix showing 5

Periodicity by opposite-handed golden twisting.

Fig. A.6. 3D projection 30 tetrahedral ring from 600-cell.

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\on left" and \o®" in terms of the registration possibilities for a given tetrahe-

dron in a frame of the QSN.

(3) It improve the ratio of symbolism to meaning by reducing all of the tetrahedra to

the simplest possible 3D pixel of information, the 3-simplex. If the 3D QSN were

generated by projecting the E8 lattice or the 4D QC to 3D, it would generate

seven di®erent shapes of distorted 3-simplexes. It would change the ratio of

symbol simplicity rank to meaning in the code (see Part 3).

A.7. Conservation and the sum of squares law

Conservation is an inherent quality of irrational projection-based geometry. For

example, consider a tetrahedron with four lines running from the centroid to each

vertex. Assuming the edge length of the tetrahedron is one, we can project the four

inner lines to the plane with an in¯nite number of projection angles, such as in in

projection in Fig. A.7.

The sum of squares of each contracted length in the projection is always conserved

as 4 or the integer corresponding to the simplex-integer, in this case the 3-simplex

corresponding to the integer 4. The sum correlates in a mysterious way to the spatial

dimension of a projected polytope, as reported in two Quantum Gravity Research

papers, Julio Kovac's The Sum of Squares Law82 and Carlos Castro Perelman's et al.

The sum of the squares of areas, volumes and hypervolumes of regular polytopes from

Cli®ord polyvectors.83

Based on this same conservation principle, the \letters" or geometric symbol

types of a quasicrystal are conserved. For example, there are seven di®erent vertex

geometries in a Penrose tiling, as shown in Fig. A.8.

Each of their frequencies of occurrence are conserved as follows:

A = 1

B = ’

C = ’

D = ’2

E = ’3

F = ’4

G = ’5.

Fig. A.7. Projection of a tetrahedron.

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Similarly, the various legal particle con¯guration patterns made of relationships

between 3-simplexes chosen on the QSN have conserved quantities. We suggest that

the deep¯rst principles-based explanation for Noether's¯rst theorem, gauge sym-

metries and conservation laws in nature is hyperdimensional projective geometry,

where the full encoding and richness of hyperdimensional structure is transformed

into lower dimensional geometric symbolic code quasicrystal language.

Quasicrystals have the fractal quality that any shape, such as the seven vertex

geometries in the Penrose tiling (see Fig. A.8), repeat according to a scaling algo-

rithm, typically the power series of the golden ratio, ’, ’2, ’3 ....

A.8. Alternative expression of geometric frustration

The term geometric frustration can be thought of as \trans-dimensional pressure"

resulting from a projection of an object to a lower dimension. For example, in 3D,

there is vectorial freedom or space for 12 unit length edges to be related by 90 in the

form of a cube. When projected along an irrational angle to 2D, the reduction of

vectorial freedom compresses or transforms the information into a 2D representation

that requires edges to contract and angles to change. The 2D projection or shape-

symbol is a map encoding (1) the information of the pre-projected object and (2) the

angular relationship of the projection space to the pre-projected object. The trans-

dimensional tension or pressure is immediately released or transformed into the

transformed lengths and angles of the projection.

An alternative form of transformation or transdimensional pressure expression is

rotation and translation. For example, consider the transformation of a 20-group of

tetrahedra sharing a common vertex in the 600-cell in 4D space. If we project it to 3D

along a certain angle, we can generate a group of 20 distorted tetrahedra with a

convex hull of a regular icosahedron and 12 inner edge lengths contracted by

ffiffiffiffiffiffiffiffiffi

ffiffiffi

5

p

q

2

¼ cos 18:

ðA:2Þ

Fig. A.8. The seven vertex con¯gurations of the Penrose tiling.

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We can understand the di®erence of dimensions as a curvature of one dimension into a

higher dimension. For example, a °at piece of paper can be curved into the 2nd dimension

such that it is a curved 2D object that requires three spatial dimensions to exist in.

So if we take our 4D 20-group, we can realize that it is bounded by a 3-sphere (4D

sphere), which is a curved 2-sphere (ordinary sphere). And we can slowly de-curve or

°atten the 3-sphere of space containing the 20-group until it is \°at", at which point

it is an ordinary 2-sphere. In this case, the 20 regular tetrahedra living in 4D that

have unit edge lengths would need to distort such that the 12 shared inner edges

contract to cos 18 (see Eq. A.2). This result is identical to the aforementioned

projection of the 4D object to 3D. An alternative method of encoding the projection

or uncurving action is to anchor the 20 tetrahedra around their common shared

vertex and rigidify them, such that they are not allowed to encode the geometric

frustration via edge contraction and angle change. This will force the tetrahedra to

express the information of their hyperdimensional relationships in lower dimensional

space by rotating along each of their 3-fold axes of symmetry that run through face

centers to opposing vertices (the shared center vertex).

Each of the 20 tetrahedra in the 4D space lives in a di®erent 3D space related to

the adjacent 3D space by ArcCosð½3’ 1=4Þ þ 60 ¼ ArcCosð1=4Þ. If we visualize

this as a gradual uncurving of the 4D space toward °at 3D space, we begin with zero

rotation of each tetrahedron.

As we initialize the uncurving, the faces will begin to rotate from one another such

that their 12 shared inner edges \blossom" into 60 unshared inner edges. As we do

this, we are gradually intersecting or converging the twenty separate 3D spaces into a

single 3D space. At the point where the 3-sphere bounding space is completely

°attened to an ordinary 3D sphere, the rotation value between the kissing inner faces

of the 20-group is ArcCosð½3’ 1=4Þ, which is the angular relationship between

kissing 3D spaces containing tetrahedra in the 4D space of the 600-cell, minus the 60

component that there is no room for in 3D (see`60 Construction' in Sec. 5.9).

Now, we have a curvature value of 0 and a rotational value of ArcCosð½3’ 1=4Þ

and have encoded the relationships of the 20 tetrahedra living in 4D into a geometric

symbol in 3D via rotation instead of edge contraction. We have converged 20 tetra-

hedra from 20 individual 3D spaces related to the other by ArcCosð½3’ 1=4Þ þ 60

into a single 3D space where they are related by the same fundamental irrational

Fig. A.9. The 20 group with the axis of rotation through the center of the face.

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component of their former relationships but without the 60 component that was used

to construct them by extruding successive spatial dimensions by a process of rotating

edge copies into the next spatial dimension by 60.

We can now reverse the process and slowly curve the °attened 4D object that is

now a 3D object back into a perfect 4D 20-group. As we do, the rotational value

decreases and the space curvature value increases. At the point in which the 2-sphere

and its rigid tetrahedra are curved into a perfect 3-sphere, the rotational value is 0

and the curvature value is 1=’. So, there is an inverse proportionality between the

curvature limit and the rotational limit as 0 rotation ! 1=’ spatial curvature and

1=’ spatial curvature ! 0 rotation.

In the QSN, every adjacent tetrahedral relationship is þ= ArcCosð½3’ 1=4Þ,

which is the 4D angular relationship between tetrahedra in the E8 to 4D quasicrystal

minus the 60 component not related to 3D.

This special non-arbitrary rotational value is powerful for modeling quantum

gravity and particle patterns for four reasons:

(1) It encodes the relationships of tetrahedra in a 4D space, which can be useful for

modeling 4D spacetime in three spatial dimensions.

(2) It encodes the relationships of tetrahedra in an E8 derived quasicrystal, which

can be useful to model gauge symmetry uni¯cation of gravity and the standard

model particles and forces.

(3) It introduces a binary sign value, chirality. The edge distortion method of

encoding geometric frustration does not generate the chirality value. This may be

useful for fundamental physics which uses three binary sign values (1) polarity,

(2) spin and (3) charge.

(4) The chirality sign value servers as an important degree of freedom in the qua-

sicrystalline code, as opposed to a more restrictive ordinary quasicrystalline. This

degree of syntactical freedom makes the geometric language more powerful.

This fundamental rotational value is the basis of action on the QSN. That is, the

Cli®ord rotor spin operations are this rotation, which will serve as the new }, the

reduced Planck constant or Dirac constant, in our emerging geometric¯rst principles

approach to fundamental physics.

There are three ways to visualize operations on the QSN:

(1) Graph theoretically: The QSN is a network of points and connections (edges).

It is simply an extended construction of the 20-twist discussed below. The 180

possible connections on the 60 points derived via any of the construction methods

discussed are part of the possibility space. So when graphed theoretically, we can

picture the 180 connections as a graph diagram in 3-space. And then we can do

graph operations to turn edges \on" or \o®" in order to make patterns.

(2) Trinary code: We can turn entire tetrahedra \on" or \o®" in which case we can

think of a centroid of a tetrahedron as being selected and designated as either the

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right-rotated or left-rotated version or not on at all, for a total of three possible

choices.

(3) Cli®ord rotor/spin network: We can conceptualize the tetrahedra to rotate

smoothly in a classical sense, such that it is rotated from a left to a right position

via the ArcCosð½3’ 1=4Þ rotation value. We can further decompose these

rotations into individual edge rotations.

A.9. Simplex construction by 60

As mentioned in Sec. 3, the simplex series is constructed by starting with an edge, a

1-simplex, and rotating a copy on a vertex by 60 into the next spatial dimension to

form a 2-simplex or three equidistant points on the plane. A copy of one of those

edges is then rotated by 60 into the 3rd spatial dimension to form an equidistant

relationship of four points and a dihedral angle of ArcCos(1/3). The dihedral angle

series ranges from 60 in the 2-simplex to 90 in the in¯nite-simplex, spanning a total

of 30 and where each dihedral angle in the series between 30 and 90 is irrational as

the ArcCos of a successive fraction from the harmonic series 1/2, 1/3, 1=4 ....

We can think of the 60 component of each dihedral angle as being tied to the

action that extruded an additional spatial dimension necessary for the next point to

be added in such a manner that all points are equidistant. The remaining irrational

component of each dihedral angle is the more \meaningful" part, carrying the key

information of the given simplex-integer. For example, in the case of the 4D simplex,

the two parts of its ArcCosð1=4Þ dihedral angle are ArcCosð½3’ 1=4Þ 15:522

and the 60 component correlated to the extra-spatial rotation that extruded out the

next spatial dimension in the buildout process from 3D to 4D. The relationship

between kissing 3-simplexes in a 4D space is 60 þ 15:522. Accordingly, when one

uses the irrational component of this angle in a 3D construction of regular tetrahe-

dra, such as in our approach, it encodes the relational information between tetra-

hedra as they would have existed in, for example, the 4D Elser–Sloan quasicrystal

derived from E8.

And because 15:522 is inversely proportional to the 1=’ curvature value, as

explained above, it is most deeply a transformation of the information of a¯nite 4D

spaces (a 3-sphere of radius 1) into a¯nite 3D space a 2-sphere of radius cos 18

(see Fig. 18).

This same construction approach can also be used to build out the E8 lattice,

which is a packing of 8-simplexes that leaves interstitial gaps in the shape of 8D

orthoplexes.

A.10. Specialness of 3D and 4D

In 2D there are an in¯nite number of regular polytopes, but they all have rational

angles and are trivial in some sense except for the ones based on the angles 60,

72 and 90 as the equilateral triangle, pentagon and square. These are the

polytopes corresponding to the¯ve Platonic solids, the only regular polytopes in 3D.

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For example, the equilateral triangle is the polytope in 2D corresponding to the

tetrahedron. Only the equilateral triangle and square can tile the plane, making them

the \crystal"-based 2D analogues of the platonic solids. The pentagon cannot tile the

plane and corresponds to the icosahedron and dodecahedron. Of the¯ve platonic

solids, three are based on the crystal group, the cube, tetrahedron and octahedron.

The remaining two, the icosahedron and octahedron, are the quasicrystal regular

solids. That is, they cannot tile space alone or in combination with other Platonic

solids. Virtually all quasicrystals discovered physically have the symmetry of the

dodecahedron and icosahedron called icosahedral symmetry. As we go to 4D, we

have four crystal symmetry polytopes, the 4D tetrahedron, 4D cube, 4D octahedron

and a crystal based polytope called the 24-cell. We also have two quasicrystalline

polytopes, the 4D icosahedron called the 600-cell, and the 4D dodecahedron called

the 120-cell.

With this, the quasicrystalline symmetry ends. It never appears again in any

dimension after 4D. In every higher dimension, the only regular polytopes are the

hyper-tetrahedron (n-simplex), hyper-cube and hyper-octahedron.

Some have wondered why 3D and 4D appear especially related to our physical

universe. If reality is based on quasicrystalline code, then this would perhaps be the

reason.

A.11. Principle of e±cient language

The principle of e±cient language is the guiding law or behavior of the universe in

the emergence theory framework. The old ontology of randomness and smooth

spacetime is replaced by a code-based ontology where symbolic information and

meaning become the new¯rst principles basis of our mathematical universe. As

discussed in Symbolic Power of Fibonacci Chain Networks in Sec. 4.6, meaning

comes in two fundamental categories: (1) ultra-low subjectivity physical meaning,

which is purely geometric and (2) ultra-high subjectivity or virtually transcendent

meaning, which includes things such as the meaning of irony and the myriad layers of

meaning imposed by an experimenter about, say, the notion of a particle being

measured as going through one slit or the other in a double-slit experiment. Inter-

estingly, it is impossible to imagine an instance of ultra-high subjective meaning

being disconnected from the underlying geometric code at the Planck scale. For

example, the experience of humor is always associated with countless changes in

particle position and alterations to the quantum, gravitational and electromagnetic

¯elds associated with that event. All forms of meaning are ultimately composed of

actions of the quantum viewing actions that animate the code. The inherent nonlocal

connectivity and distributed decision making actions of this neural-network like

formalism allow various emergent patterns of intelligent choice and actualization of

abstract meaning to be registered and considered within the degrees of freedom of the

code. Choices will be made in such a manner as to create maximal associations and

meanings where, in systems such as human beings, meaning is highly subjective.

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Consider for example, how a joke can be told and one individual will react with

massive levels of neural activity and associated meaning, while another person may

barely comprehend it. The¯rst person generates a much higher degree of correlated

and physically meaningful actions when considered at the Planck scale level of the

code operations. This feedback between the overall system (the universal emergent

neural-network) and the person generating a larger amount of meaning from the joke

plays a role in syntactically free choices of the code. We call these free choices the

hinge variable steps in the code. On average, physical laws and actions are preserved

because the physical meaning of the code (forces and physical laws) are the emergent

and non-¯rst principles manifestations of the underlying waveform language of the

quasicrystalline quasiparticle formulism.

A.12. Phason code

Phason quasiparticles have both a nonlocal wavelike quality and a local particle-like

propagation aspect called a supercell in crystallographic parlance. As mentioned

previously, there are three general ways matter can be organized: (1) Amorphous or

gaseous materials that have massive degrees of freedom and are therefore not nat-

urally codes. Geometric codes require a¯nite set of symbols, strict syntactical rules

and minimal degrees of freedom. (2) Crystalline materials have no degrees of freedom

unless there are local defects or phonon distortions. There are ultra¯ne scale vibra-

tions allowed, but not organized code-based larger scale oscillations. (3) Quasicrys-

tals are maximally restrictive without being ultimately restricted like in the case of a

crystal. For example, unlike a crystal, the assembly rules for a quasicrystal allow

construction choices within the rules that are not forced. A crystal allows only one

possible type of relationship between atoms. For example, all vertex types in a cubic

lattice are identical. In an amorphous or gaseous material, atoms can have a virtually

in¯nite number of relational objects or vertex types. In a quasicrystal, as with any

language, there is a rather small set of allowed combinations. For example, in the

Penrose tiling, which is found in nature, atoms form seven di®erent allowed vertex

geometries and the construction rules allow a very minimal level of freedom within

the construction syntax.

A.13. Empires and phason °ips

Because all quasicrystals are networks of 1D quasicrystals, understanding a phason

°ip and empires should start with how a quasicrystal is made via the cutþ project

method. An irrational projection of a cut or slice of any crystal to a lower dimension

produces a quasicrystal.

For example (see Fig. A.10(b)), one can select a rectangular cut window rotated

with an irrational angle to the 2D crystalline pointset. One projects the points

captured in the cut window to the 1D projection space to generate our 1D quasi-

crystal. In the second image, we translate the cut window, which projects a di®erent

set of points to the 1D space. When the cut windowmoves to a new coordinate, points

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instantly jump in or out of possible positions in the 1D space that we call the

possibility space. This instant change from one coordinate to the other in the pos-

sibility space is called a phason °ip.

When one point is captured in the cut window, there are an in¯nite number of

other points along the length of the cut window (if considering an ideal in¯nite point

space) that are also captured in the cut window at its new coordinate. This creates an

in¯nite number of phason °ips in the 1D possibility space. An arbitrarily large but

non-in¯nite quasicrystal can be built according to assembly rules instead of the cut +

project method. In this case, a user of the assembly language must choose a single

phason °ip, which is simply the designation of a point from the possibility space to be

\on" or \o®".

In Fig. A.10 note that, when the cut window changed location, some points in the

2D space (1) remained in the window, (2) some departed from the window and (3)

some entered the window. When a quasicrystal code user chooses a point to be \on"

(a)

(b)

Fig. A.10. A schematic diagram showing two ways of interpreting the cut-and-project method for gen-

erating a quasicrystal from a higher dimensional lattice. (a) shows that the points are selected for pro-

jection as long as there is non-trivial intersection between their Voronoi cell and the quasicrystal space.

The black points are the lattice points, and the hexagons are their Voronoi cells, E is the quasicrystal

space, E ? is the orthogonal space, and the solid blue and green segments are the projected tiles in the

quasicrystal space; (b) shows that the points are selected for projection as long as their projection on the

orthogonal space falls inside of W .

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from the possibility point space, it causes a certain group of other points in the

possibility space to also be turned on and other points to say on. These two sets of

points are called the empire of the selected \on" point.

A key concept is conservation. The number of points captured in the cut window is

conserved. As points enter the window, an equal number of points exit. A second key

concept is non-locality, the empire of forced points determined to be \on" or \o®" by

a single point selection of a code user is very large. A third key idea is discrete and

instant coordinate change. When the points are a model for particles, an ontology of

instant coordinate change in the \physical" projection space is recognized, much like

the notion of virtual particles in the Dirac sea, where particles are conserved such

that when one is annihilated, another instantly appears.

Quasicrystals in dimensions higher than 1D are more complex because they are

networks of 1D quasicrystals. So a phason °ip and empire of a single 1D quasicrystal

will have a massive empire that in°uences every other 1D quasicrystal in the net-

work. Figure A.11 shows an image from Laura E±nger-Dean's thesis, which shows

the empire of one of the vertex types of the Penrose tiling. We can see that the density

of the empire drops with distance from the vertex being designated as \on" at the

center. One can think of the possibility space as an aperiodic point space where any

point can be selected to be one of the allowed vertex types. In the Penrose tiling, there

are seven di®erent vertex geometries. As mentioned, once one vertex type is selected

for that vertex on the possibility space, it forces other vertices in the space to be

\on" or \o®" the empire. A key point for physics modeling, where forces drop with

Fig. A.11. The empire of a vertex con¯guration in Penrose tiling.

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distance, is that some empires drop in density with distance. We will connect this

with the idea of empire waves and the free lunch principle shortly.

Another key idea for physics using this formalism would be that the minimum

quantum of action notion of quantum mechanics would be replaced by the action of a

point being registered as \on" or \o®".

Phason quasiparticle behavior in any quasicrystal has two distinct sets of

construction rules:

(1) Quasicrystal Assembly Rules: These construction rules govern how a single

frozen state of selections on the possibility space can exist. The rules are de¯ned

by the angle, size and shape of the cut window in the higher-dimensional lattice.

(2) Ordering Rules for Two or More Quasicrystals: These rules govern the creation

of dynamical patterns generated by ordering two or more di®erent selection

states on the possibility space into a stepwise frame-based animation. The rules

are de¯ned by the way that a cut window can translate or rotate through the

higher-dimensional lattice and whether combinations of those actions is discrete

or continuous.

A.14. Empire waves

Just as the 2D Penrose tiling quasicrystal has empires (see Fig. A.11) that are

circular, with radial lines of higher density tiles evenly distributed from the empire

center point, a 3D quasicrystal has empires with radial lines of higher tile density

penetrating evenly distributed points on a sphere. As explained in Electron clock

Intrinsic Time (Sec. 5.5), a massive particle in our framework is composed of a vertex

type (a supercell of 20 3-simplexes) that dynamically animates over many coordinate

changes or frames to form (1) a toroidal knot cycle internally in the QSN and (2) a

propagation pattern through the QSN that changes coordinate along a stepwise

helical path. The interaction of these two forms of stepwise internal toroidal cycling

and forward propagating helical cycling generate a richly complex dynamical pattern

of empire waves waves which extend to the end of the universal space of the QSN

but drop in density with distance. These waves are the geometric¯rst principles key

to modeling forces in this framework. However, it is helpful to explain that quantum

mechanics does not require the assumption of Bohr's conjecture known as comple-

mentarity the core of the Copenhagen interpretation of quantum mechanics. This

is the view that a fundamental particle, such as an electron, is either a wave or

particle but never both. Neither experiment nor the mathematical machinery of

quantum mechanics compel this interpretation. The Broglie-Bohm theory states that

an electron, for example, is always a wave and particle at the same time and that the

wave aspects guides the particle coordinate, like a pilot wave. The cost of this ab-

solutely rigorous but less popular interpretation is the requirement of the assumption

of inherent nonlocality in nature. Empire waves are nonlocal according to the non-

enigmatic geometric¯rst principles of projective geometry.

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A.15. Free lunch principle forces

With this general overview of empire waves established, it is now possible to

understand how forces can be modeled via geometric¯rst principles. Let us begin

with the analogy of the game Scrabble, where you gain points by making multiple

words diagonally, vertically or horizontally using letters from one or more other

words already on the game board. When you do this, you get \free lunch" by earning

points for each word your letter(s) played a position in. The Scrabble board is

analogous to the QSN possibility space. And the 26 Roman letters are like the¯nite

set of geometric relationships or vertex types in a quasicrystal the geometric-

symbols of the language. The rules and freedom of English are like the rules and

freedom of the phason code in a quasicrystal language. In Scrabble, the commodity

that is to be conserved and used e±ciently is the number of turns each player gets.

Each turn needs to generate as much meaning as possible. In emergence theory, the

same principle applies. There are a certain number of quasicrystal frames or \turns"

that a system of, say, two particles can be expressed in over some portion of a

dynamical sequence. Let us consider that it takes 10 frames to model a cycle of

electron internal clock action or some total length of discrete transitions through the

space. The patterns of this object in the QSN always need some integer ratio of the

given number of total frames used for internal clock cycle steps versus helical

propagation steps.

The physical pattern is expressed as the trinary selections of 3-simplexes in the

QSN: on-right, on-left or o®. And just as the words \cat" and \rat" can share an \a"

for greater e±ciency and synergy, the system of two such patterns moving through

the QSN allow us to save steps. We can model free lunch in this geometric code

thanks to the empires. When the¯rst propagating electron is moving near the second

electron, the two begin to bene¯t from one another's empire waves. In the simplest

example, consider that it would ordinarily take two remotely separated electrons 10

frames each to express a certain amount of clock cycling and propagation. However,

the closer they are to one another, the more free lunch they will enjoy. The system

saves frames when a selected tetrahedron from one particle's empire is in the nec-

essary right or left \on" state to that matches the state necessary to¯ll a position in

the geometric pattern of a second electron, thereby saving a frame in the way that we

saved an \a" in our Scrabble game example.

The result is that the particles require fewer phason °ips or frames of trinary

selections on the universal QSN to express their clock cycles and their given number

of propagation steps along some direction. The physical meaning of this is that they

have advanced a further distance than they would have otherwise with 10+10, where

no free lunch is enjoyed. And because the density of free lunch opportunities

increases with approaching distance, the two particles will accelerate toward one

another as their separation decreases.

The empire wave around a massive particle in this framework is distinctly chiral

and behaves according to the right-hand rule, where the direction of propagation

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determines the direction of the chiral free-lunch empire wave system around it. A

\train" of these objects, such as electrons in the QSN, will fall in-line behind one

another and form a current because that positioning ensures the maximum amount

of free lunch. By all moving along the same helical path, a group empire wave system,

in the form of the chiral magnetic¯eld, emerges around them. However, most elec-

tron models are in either groups of free electrons or are in atomic systems that are

arranged with many di®erent orientations, such that the emergence of a chiral

magnetic¯eld does not occur. In other words, picture our model of the electron

approaching Earth. As it accelerates closer, the probability of¯nding free lunch

frame savings increases. Again, the empire wave¯eld of every massive fundamental

particle on Earth has no general similarity in their various orientations or directions

of propagation. And they are not strongly correlated. Accordingly, around Earth,

there is an enormous superposition of empire waves from every massive particle. One

can say that it is a noisy quantum¯eld of empire waves on the dynamical QSN.

There is a high degree of non-coherence, as compared to a current of electrons, where

there are coherent group patterns in the empire waves like combed °owing hair as

opposed to tangled hair. Nonetheless, there will still be some opportunities for free

lunch around the tangled array of empire waves surrounding large groups of massive

particles for any approaching electron from outer space to enjoy as it nears Earth.

But it will be exponentially less than the free lunch around the current of electrons.

Gravity would logically be orders of magnitude weaker than electromagnetic forces.

And it will be distinctly non-chiral, due to the fact that the average chirality is null,

with an approximately equal quantity of right and left-handed empire waves states

on the QSN around Earth (other than the Earth's magnetic¯eld).

A.16. A non-arbitrary length metric

The nearest neighbor lengths between points in the QSN are the Dirichlet integers 1

and 1. So if our framework is generally correct, it would more deeply explain why

black hole physics corresponds to the golden ratio and why quantum mechanics does

in the form of the ’5 entanglement probability discovered by Lucien Hardy.65

Accordingly, a new length system based on golden ratio values would simplify many

equations in physics. For example, the three most fundamental constants are the

speed of light, c, the gravitational constant, G, and Planck's constant, h. The only

number that uni¯es all three is a length called the Planck length, l, which happens to

be about 99.9% of the golden ratio in the metric system.

l ¼

ffiffiffiffiffiffiffi

}G

c3

r

:

ðA:3Þ

If spacetime had substructure built on our Planck length scale QSN, planetary

systems might evolve overtime to energetically favorable cyclical and length ratios

that approximate simple golden ratio fractions. And if we based our measuring

system on a physical valued tied to a planet, it would be less arbitrary than, for

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example, the yard, which was based on the distance from King Henry I of England's

nose to thumb distance.

Indeed, the metric system is less arbitrary because it is based on 14 the circum-

ference of Earth, where the distance from the Equator to the North pole is 10,002

kilometers, making the metric system unit value of 1, a full 99.98% of that distance

(disregarding where the decimal is). When the system was established, they could

not achieve the full accuracy of measuring this distance on Earth. So today, the

metric system unit is almost that distance. It is not well known, but the metric

system deeply relates to approximations of golden ratio values. The Earth and Moon

system is approximately a quarter of the age of the universe. So it has had a long time

to self-organize into optimal ratios that approximate the golden ratio. To an accu-

racy of 99.96%, the dimensionless ratios are

radius of Earth

radius of Earth

2

þ radius of Moonþ radius of Earth

radius of Earth

2

¼ ’2;

ðA:4Þ

or

radius of Moon

radius of Earth

¼

ffiffiffi

’

p 1:

ðA:5Þ

In other words, this is a double coincidence. It is not just that the sum of the Earth

and Moon diameters in the metric system are almost exactly the golden ratio

1.618..., but the breakdown of the two diameters that sum to that value is ’

ffiffiffi

’

p

for the Moon and

ffiffiffi

’

p

for Earth.

The master dimensionless ratio of fundamental physics is the¯ne structure con-

stant, a. Interestingly, it is also closely approximated with golden ratio expressions as

a ¼ ’2=2; ’2=360 ½to an accuracy of about 99:7%:

ðA:6Þ

A.17. A non-arbitrary \Time" metric as ordered quasicrystal frames

Much of the data we present in this paper includes time-based or planet and moon

cycle \coincidences" that seem to match far too closely to the golden ratio to be

explained away by anything other than the presumption of some unknown sub-

structure of spacetime in a new quantum gravity framework.

By combining both time and length-based values, the critical reader can perhaps

be interested in the following impressive number.

The gravitational constant, G, ties time and length based values together as

G ¼ c2=4:

ðA:7Þ

h ¼ 1:0000026 of the golden ratio as 0:6180382ð1059Þ cubic meters per second

[note 1=’ ¼ 0:61803 ... and ’ ¼ 1:61803 ... are the same ratio]. This deviation at the

millionth place after the decimal is remarkable.

Now, having put forth an argument why it is plausible that spacetime can have a

golden ratio-based substructure as a natural result of the projection of E8 to a lower

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dimensional quasiperiodic point space, we can speculate on the idea that the metric

system is deeply related to ’ and consider the idea of a¯rst principles analytical

expression of the constants c, G, h and a. But clearly, there is a problem. The¯rst

three constants are dimensional and tie into the speed of light. And the speed of light

is based on a length metric and a time metric. The length metric is being proposed

as nonarbitrary, according to this speculative argument related to our projective

approach to E8 uni¯cation physics.

However, the speed of light playing into these corresponding equations appears at

¯rst to be based on an arbitrary metric for time, the second. The QSN is based on the

numbers 2, 3 and the golden ratio because the 3-simplex building blocks are regular

or non-distorted. And the golden ratio is deeply related to 5 geometrically in the form

of the pentagon and to 5 algebraically as 12 of

ffiffiffi

5

p þ 1. From the analytical expressions

of the 3-simplex volume to its length values, such as height and centroid to vertex

distances, it is fundamentally built of the numbers 2 and 3 and their square roots. So

the QSN is deeply related to 2, 3 and 5. Incidentally, these are the symmetries that

de¯ne anything with icosahedral symmetry. And nearly every quasicrystal found in

nature (over 300) possesses icosahedral symmetry. It is interesting, then to note,

therefore, that the constant c (in the metric system) is 99.93% of the number 3,

disregarding where the decimal is placed. And the distance of the Earth to the Sun is

99.73% of 3/2.

The number of (presumably) ’-based meters traveled by a photon in vacuum in

one second is a close approximation of 3/2. Assuming hypothetically, that E8 qua-

sicrystalline physics is a good approach, why is this the case if the second is arbitrary?

The second is not arbitrary, of course. It is based on a cycle of the fundamental

Earth clock system, which itself is fundamentally based on ’, as argued above. It is

based on the clock cycles of the Earth rotating once on its axis, which is gravi-

tationally and electromagnetically tied to the Earth, Moon, and Sun system as a

whole. The number, of course, is 86,400 seconds in one of these non-arbitrary

physical cycles of the Earth clock. That is, 60 s 60min 24 h. Remarkably, the

modern precise average Earth day is 86,400.002 s. So the old number is unexpectedly

close to the accurate measurement. Again, we disregard where the decimal place is in

the context of thinking about the fundamental aspect of a number its factori-

zation. Accordingly, 86,400 becomes 864 ¼ 25 33, a number deeply related to 2, 3

and 5.

Have we missed anything obvious? Yes, the Earth distorts along the equator. So if

we adjust for the meter to assume a non-rotating Earth with no distortion, we can see

if our number gets closer or further from the golden ratio being the Planck length.

Realizing that the pole-to-pole diameter of Earth is 12,713 km, and simplifying the

value by moving the decimal to 1.2713, we can calculate that a 14 of a circle inter-

secting a non-distorted sphere of this diameter is 0.9984766. This then is normalized

to 1. Note that this approach is not based on a metric. It is based on the ratio of the

Moon to Earth, where we get the dimensionless value. And here, we are not using the

ideal Phi values mentioned. We are using the actual values in their ratio. So this gives

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the dimensionless ratio value of 0.9984766 in the manner just described. We can then

normalize this to a standard unit of 1. Again the justi¯cation is the conjecture that

the substructure of space is based on a dimensionless ratio of one part being 1 and the

other part being 1=’. Now, what this means is that the Planck length now changes

slightly from the current value of 1.616199, which is based on the meter that is

measured from a distorted Equator to the normalized value based on the new di-

mensionless ratio-based length and based on the actual measurement of the Earth's

pole through pole diameter (not plugging the golden ratio approximation of that

diameter). We get a logically adjusted Planck length of 1.6183412... or 1.0002 of the

golden ratio.

A.18. Mass in the quasicrystalline spin network

We will now combine the following ideas in order to understand mass in the emer-

gence theory framework:

(1) Free lunch and empire waves

(2) Massive particle clock time to propagation inverse relationship

(3) Principle of e±cient language

Obviously, our vision of a geometric¯rst principles uni¯ed quantum gravity theory,

as explained thus far, reduces everything to length. Our formalism is Cli®ord rotor

operations on a spin network made of the two Dirichlet integer values 1 and 1=’.

Mass is the degree of resistance to a change in direction or acceleration of a massive

particle. If space and time are discretized, where space is divided into positions like on

a checker board and time is divided into turns of the players, where a piece can only

move to a connected square, an intuitive understanding of mass emerges.

In Fig. A.12, we see that putting a particle in motion along some direction

in spacetime as the square grid achieves an e±cient diagonal progression across

(a)

(b)

Fig. A.12. Two paths in a spacetime grid. (a) Shortest, most e±cient path between position 1 and 8.

(b) Seven e±cient paths of which one is illustrated.

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the board, using the shortest path between position 1 and 8. In the left grid

(Fig. A.12(a)), there is only one shortest path. In the right grid (Fig. A.12(b)), there

are seven paths of which one is illustrated.

As explained in Sec. 5.3, mass is connected to curvature. In our framework of

discretized spacetime, curvature is derived from the angle 15.522 as explained in

Fang's et al. paper`Encoding Geometric Frustration'.104

A.19. Generation of the quasicrystalline spin network

A.19.1. A fang methods

In Method 1, Fang Fang initially constructed the QSN by modifying the icosagrid

with Fibonacci chain spacing to make it a quasicrystal. In doing this, the alternative

Method 2 was discovered. Packings of tetrahedra in the form of the FCC lattice

are Fibonacci chain spaced. Then¯ve copies are rotated from one another by the

15.522 angle.

A.19.1.1. Method 1. Fibonacci icosagrid

This approach is inspired by the pentagridmethod of constructing the Penrose tiling.

A 3D analogue is the icosagrid construction method for icosahedrally symmetric

quasicrystals. 10 sets of equidistant planes parallel to the faces of an icosahedron are

established with periodically repeating parallel planes in each set that, together, form

the icosahedrally symmetric icosagrid. The intersecting planes segment the 3-space

into an in¯nite number of 3D cell sizes. The icosagrid is not a quasicrystal due to the

arbitrary closeness of its edge intersections and the resulting in¯nite number of

prototile sizes. We converted it into an icosahedral quasicrystal by changing the

equal spacing between parallel planes to have a long and short spacing, L and S with

L/S ¼ golden ratio and the order of the spacing follow the Fibonacci sequence.

Therefore, we call this kind of spacing the Fibonacci spacing. This Quasicrystal

turned out to be a 3D network of Fibonacci chains and we would like to name it QSN.

A.19.1.2. Method 2. Golden composition of the¯bonacci tetragrid

Similar to the icosagrid, a tetragrid is made of four sets of equidistant planes that are

parallel to the faces of a tetrahedron. Applying the Fibonacci spacing to this

structure will also give us a quasicrystal with tetrahedral symmetry (Fibonacci tet-

ragrid) again we focus mostly on the regular tetrahedral cells. In order to obtain

icosahedral symmetry, we need to implant the 5-fold symmetry. We applied a Golden

Composition process to this Fibonacci tetragrid and achieved the same QSN structure.

The Golden Composition is described as follows:

(1) Start from a point in a Fibonacci tetragrid and identify the eight tetrahedral

cells sharing this point with 4 in one orientation and the remaining 4 in another

orientation.

(2) Pick the four tetrahedral cells of the same orientation and duplicate another four

copies.

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Put two copies together so that they share their center point, the adjacent tetra-

hedral faces are parallel, touching each other and with a relative rotation angle of

ArcCosð½3’ 1=4Þ, the golden rotation. Repeat the process three more times to add

the other three copies to this structure. A twisted 20-tetrahedra cluster, 20G, is

formed in the end. Now expand the Fibonacci tetragrid associated with each of the

4-tetrahedron sets by turning on the tetrahedra of the same orientation as the four,

an icosagrid of one chirality is achieved. Similarly, if the tetrahedral cell of the other

orientation are turned on, an icosagrid of the opposite chirality will be achieved.

In either case, there is a 20G at the center of the structure.

A.19.2. Cli®ord rotor induction method

As explained in the paper`Emergence of an Aperiodic Dirchlet Space from the

Tetrahedral Units of an Icosahedral Internal Space'102 an inductive framework has

been established to link higher-dimensional geometry from the basic units of an

icosahedron using spinors of geometric algebra and a sequence of transformations

of Cartan sub-algebra. Spinors are linear combination of a scalar and bivector

components de¯ned (see Eq. A.8).:

s ¼ D þ

De12 þ De23 þ De31:

ðA:8Þ

The subscript D denotes that the spinorial coe±cients live in a Dirichlet coordinate

system, i.e.,

D :¼

1 þ

2, where is the golden ratio. This approach presents, for

the¯rst time to our knowledge, a direct inductive and Dirichlet quantized link

between a three-dimensional quasicrystal to higher-dimensional Lie algebras and

lattices that are potential candidates of uni¯cation models in physics. Such an

inductive model bears the imprints of an emergence principle where all complex

higher-dimensional physics can be thought to emerge from a three-dimensional

quasicrystalline base.

A.19.3. Dirichlet integer induction method

The need for quasicrystalline coordinates brought us naturally to consider a class of

number which are more rich than the rational integers (useful for crystals), but more

constrained than the real numbers, the quadratic integers. From this class, the¯ve-

fold symmetry of our quasicrystal guides our choice to the ring living in the quadratic

¯eld associated to 5, which is sometimes noted as Z½’, the quadratic ring of

\Dirichlet integers" referencing to their use in Dirichlet's thesis and following works,

or in short D.

Then we use a digital space D3, to host triplets of Dirichlets integers, and a digital

spacetime D4, to host quadruplets of Dirichlets integers. D4 could have a quaternion

structure, and written H D. Where D3 the space part, is the imaginary part.

Furthermore, the structure can be complexi¯ed to biquaternion, and also put in

bijection with octonion and sedenion.

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Each point can also be seen as a 4 by 4 matrix of integers, a digital tetrad,M4ðZÞ,

or as HH Z.

The 16 numbers are integers.

1

i

j k

aw bw cw dw

ax bx

cx dx

ay

by

cy

dy

az

bz

cz

dz

1

’

I

I’

:

There is one line per dimension, and the¯rst dimension, indexed by w is hidden in the

space construction. a and b are combined to make the real part of the Dirichlet complex.

The generators satis¯es

ijk ¼ i2 ¼ j2 ¼ k2 ¼ I2 ¼ ’ ’2 ¼ 1.

In a¯rst

approach, the imaginary part will be set to 0 (so all c and d are null). A point in the

realized space will just show three coordinates:

ax þ ’bx;

ay þ ’by;

az þ ’bz;

...where ’ ¼ 1þ

ffiffi

5

p

2

, and is equivalent to the non-golden part made of the a, and the

golden part made of the b.

In a Euclidian spacetime, aw and bw can correspond to time, while it is cw and dw

in a Lorentzian spacetime, and all four are used in a Kaluza–Klein model.

A set of eight integers, (the a and b, or the c and dÞ, can encode a position in an E8

lattice (with a doubling convention).

Let us focus on how these numbers emerge. Our model is from the¯rst principle

built from regular tetrahedral in an Euclidian 3D space, because the simplex is the

simplest geometric symbol, and the space is observed as tridimensional.

We ask the question: which sets of vertices in D3 can hold regular tetrahedra of

the same size having one vertex in the center (0,0,0)? The equation is the equation

of the sphere, written in D3, which holds two equations by separating the golden and

the non-golden parts. From this, the result is

. Amaximal possibility space bigger than the QSN but smaller than Dirichlet space.

. Vertex¯gure: New polyhedron with 108 vertices and 86 faces.

. Tetrahedron centers¯gure: New polyhedron with 32 vertices which is not the

icosidodecahedron.

. 72 possible tetrahedra around a vertex (see Fig. A.17).

Having built a¯rst-principle version of Dirichlet space where 20G emerge naturally

(but as 4 copies), I have the intuition that E8 physics can also emerge naturally as

encoded by the possible tetrahedron con¯gurations. We will focus on rule emergence.

Some come from Physics, like the hadronic rule saying that the combined color of

three quarks in a neutron or photon is neutral, also known as the SU(3) symmetry,

quantum chromodynamics; some from information theory and mathematics; some

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from matching with quasicrystal study when importing CQC simulation with dy-

namic window and observing phason occurrence, to deduce phason rules.

(The Dirichlet Integer Induction Method was introduced in an internal commu-

nication from Raymond Aschheim to Klee Irwin via email on May 19, 2016).

A.19.4. Projection and graph diagram method

To illustrate the correspondence of the 20G twist to the E8 lattice, I developed the

following method:

(1) Project E8 to 4D to generate the Elser–Sloan quasicrystal. It is made entirely of

600-cells. Alternatively, we may project one of the 240 vertex root vector poly-

topes of E8 to 4D to generate two 600-cells scaled by the golden ratio.

(2) Select 20 tetrahedra sharing a common vertex in a 600-cell and project the cluster

to 3D such that the outer 12 points form the vertices of a regular icosahedron.

(3) Induce its dual, the dodecahedron, which has 30 points.

(4) Use the 30 points to create a graph diagram by connecting points separated by a

distance of ’ (Sqrt2) times the dodecahedral edge length. This creates a 3D

graph diagram equal to two superimposed tetrahedron 5-compounds, one right

and one left-handed.

In Fig. A.13, the right chirality 5-compound is shown. The Cartesian coordinates

of the 30 vertices are the cyclic permutations of:

ð1;1;1Þ

ð0;1=’;’Þ

ð1=’;’; 0Þ

ð’; 0;1=’Þ:

(5) Select either the right or left-handed tetrahedron 5-compound. And from it, we

select one tetrahedron and translate a copy of it away from the center of the

cluster along one of its 3-fold axes of symmetry by a distance of Sqrt(3/8) times

its edge length — the distance necessary to translate one of its vertices to be

coincident with the center of the tetrahedron 5-compound. We do the same

Fig. A.13. Compound of 5 tetrahedron with right chirality.

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copy-and-translate action along the other three of the selected tetrahedron's 3-

fold axes of symmetry. This generates four new tetrahedra that share a common

vertex with the centroid of the cluster. Their 12 outer vertices form the points of

a cuboctahedron (as shown in Fig. A.14).

We repeat this 4-step process with the remaining four tetrahedra from the

initial tetrahedron 5-compound (see Fig. A.15). We then remove the original¯ve

tetrahedra as well as the dodecahedron.

Thus far, this induction process generated 20 tetrahedra sharing a common

vertex at their group center. It is the 20G twist with 60 outer vertices equal to a

cuboctahedron 5-compound. It has Cartesian coordinates that are the cyclic

permutations of

ð2; 0;2Þ;

ð’;’1;ð2’ 1ÞÞ;

ð1;’2;’2Þ:

Fig. A.14. One tetrahedron is translated along each of three edges sharing a vertex to give four tetrahedras.

Fig. A.15.

(a) A small tetragrid local cluster with eight tetrahedral cells, four "up" and four "down".

(b)–(f) The golden composition process: (b) 1 tetragrid. (c) 2 tetragrids, shown in red and orange.

(d) 3 tetragrids, shown in red, orange and green. (e) 4 tetragrids, shown in red, orange, green and blue.

(f) 4 tetragrids, shown in red, orange, green, blue and purple.

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(6) Finally, in Fig. A.16, we repeat the entire process with the starting tetrahedron

5-compound of the opposite chirality. The right-handed 20G twist [f6a] and left-

handed one [f6c] are superimposed in the QSN to form the basic building block

set of possibilities from 60+1 points and 180 possible 3-simplex edges or

connections [f6b].

A.20. Hyperdimensional information encoded in 3D

One of the principles of the emergence theory approach is simplicity. We question the

physical realism of hyperdimensional spaces implied by models such as general rel-

ativity and string theory.

However, it is clear that the gauge symmetry uni¯cation of all particles and

forces of the standard model, which is everything except gravity, are described by

the six-dimensional root vector polytope of the E6 lattice. Full gauge symmetry

uni¯cation with gravity seems to be possible with the eight-dimensional E8 lattice,

which embeds E6. This uni¯cation can be achieved with or without geometry by

selecting either the pure algebraic Lie algebras or the geometric analogues of

hyperdimensional crystals and geometric algebras. Even without considering the

gauge theory implications of hyper geometry, general relativity alone relies on a 4D

geometric structure.

Fig. A.16.

(f6a) The right twisted 20G. (f6b) The superposition of the left-twisted and right-twisted 20G.

(f6c) The left twisted 20G.

Fig. A.17. Seventy two possible tetrahedra around a vertex. Four combinations of non-intersecting

tetrahedron subsets of the 72. (a) 2 left twisted tetrahedra (see Fig. A.16 f6c). (b) 2 right twisted tetra-

hedra (see Fig. A.16 f6a). (c) 2 right twisted tetrahedra rotated 90 degrees compared to (b) (see Fig. A.16

f6a). (d) 2 left twisted tetrahedra rotated 90 degrees compared to (a) (see Fig. A.16 f6c). From left to right,

the 2 tetrahedron facing left, right, right, left chirality. Here you see we can mix chiralities.

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Clearly, fundamental physics implies a deep tie to hyperdimensional math.

But we have never measured any geometric dimensions beyond 3D. So the

Occam's razor approach is to see if we can derive all the hyperdimensional infor-

mation from a purely 3D framework without having to adopt the ontological

realism of, for example, curled up spatial dimensions or the 4D spacetime of general

relativity.

Because we can measure reality to be 3D and because it is simpler than hyper-

spaces, we seek to model the implied math of hyperdimensional geometry while

restricting ourselves to Euclidean 3-space.

The quintessential example of a lower-dimensional object encoding the informa-

tion of a higher-dimensional object is irrational angle-based projective geometry-

quasicrystallography.

The vertex Cartesian coordinates are the cyclic permutations of

ð1;1;3Þ

ð’1;ð’2Þ;2’Þ

ð’;ð2’1Þ;’2Þ

ð’2;ð’2Þ;2Þ

ðð2’ 1Þ;1;ð2’ 1ÞÞ:

A.21. Emergence of a self-actualized code operator

Frank Wilczek challenged physicists to develop a conscious measurement operator

that comports with the formalism of quantum mechanics.50 This is daunting for

social reasons. Discussions of consciousness in academic circles of physicists is gen-

erally scorned, with few exceptions. And many physics journals reject such notions

under the unwritten premise that philosophy and physics should not be combined.

However, blunt logical deduction, free of social fears, points to the idea that

consciousness is a fundamental element, as though it is the substrate of reality.

J.B.S. Haldane93 said:

We do not¯nd obvious evidence of life or mind in so-called inert

matter...; but if the scienti¯c point of view is correct, we shall ultimately

¯nd them, at least in rudimentary form, all through the universe.

Erwin Schr€odinger94 said:

For consciousness is absolutely fundamental.

Andrei Linde,95 co-pioneer of in°ationary big bang theory, said:

Will it not turn out, with the further development of science, that the study

of the universe and the study of consciousness will be inseparably linked,

and that ultimate progress in the one will be impossible without progress in

the other?

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David Bohm96 said:

The laws of physics leave a place for mind in the description of every

molecule... In other words, mind is already inherent in every electron,

and the processes of human consciousness di®er only in degree and not in

kind.

Freeman Dyson96 said:

That which we experience as mind... will in a natural way ultimately

reach the level of the wave-function and of the 'dance' of the particles.

There is no unbridgeable gap or barrier between any of these levels... It is

implied that, in some sense, a rudimentary consciousness is present even

at the level of particle physics.

Werner Heisenberg97 said:

Was [is] it utterly absurd to seek behind the ordering structures of this

world a consciousness whose \intentions" were these very structures?

The growing credibility of the digital physics argument still leaves one with the sense

of audacious improbability. These scientists claim that the universe is a simulation in

the quantum computer of an advanced being or society. Although they could be

correct, this has a similar level of outlandishness as the idea that a creator God from

outside the universe is the source of everything. Of course, this is a popular religious

view. But the idea that something from outside the universe created the universe

implies a new de¯nition of the term universe. That term is supposed to mean

everything. The idea of a self-actualized universe may be more sensible.

The mounting evidence that the universe is made of information and is being

computed includes the aforementioned mathematical proof of the Maldacena con-

jecture and the discovery of error correction codes. There are many other pieces of

evidence that add to the argument. But for those new to the thought process, here is

a simple way to deduce that something like a computer or mind is needed: Every-

thing we know about physics, including classic physics, indicates that reality or

energy is information. And information cannot exist without something to actualize

it. It is abstract and relates deeply to a mind-like entity, whether that be a biological

neural network or an arti¯cial intelligence.

However, there is a more plausible explanation than the digital physics computer

simulation hypothesis. In his submission to the FQXi Essay contest, mathematical

physicist, Raymond Aschheim,7 a scientist at Quantum Gravity Research, said:

Can reality emerge from abstraction, from only information? Can this

information be self-emergent? Can a structure be both the software and

the hardware? Can it be ultimately simple, just equivalent to a set? Can

symmetry spontaneously appear from pure mathematical consideration,

from the most symmetric concept, a Platonic \sixth element"? Would this

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symmetry be just structuring all the particles we know? Can all this be

represented? Can standard physics be computed from this model? Eight

questions: eight times yes.

The notion of a self-emergent computational but non-deterministic neural-network

universe is more plausible than the idea of a simulation creator from outside the

universe. In fact, the emergence of free will and consciousness need not be a specu-

lation. It is proven to exist, at least in humans. So it is one of the interesting

behaviors of the universe locally in the region of our physical bodies. In my paper, A

New Approach to the Hard Problem of Consciousness: A Quasicrystalline Language

of \Primitive Units of Consciousness" in Quantized Spacetime,98 I discuss in detail

the plausibility of a self-emergent mind-like universe. The¯rst question is to consider

whether or not physics imposes a limit on self-organizing evolution of consciousness.

In other words, are humans the limit or can intelligence tend toward in¯nity? From

what we know about classic and quantum physics, there is no limit. It can tend

toward in¯nite awareness and intelligence. The next question is, \What percentage

of the energy in the universe can self-organize into conscious systems and networks of

conscious systems?" Of course, the answer is the same as the¯rst question. Physics

imposes no upper limit. So the answer is that, in principle, 100% of the energy of the

universe can self-organize into a conscious network of conscious sub-systems. The

¯nal consideration in the deduction relates to the axiom:

Given enough time, whatever can happen will happen.

By this axiom, somewhere ahead of us in spacetime, 100% of the universe has self-

organized into a conscious system. It certainly need not be anthropomorphized. We

can leave the detail of what this entity would be like out of the deduction. For

example, there is no reason to presume that it cannot exist trans-temporally and

have an extremely di®erent quality than what we conceptualize as consciousness.

The next step of deduction is to question whether or not trans-temporal feedback

loops are disallowed by current physics paradigms.

Stephen Hawking of Cambridge and Thomas Hertog of the European Laboratory

for Particle Physics at CERN say that the future loops back to create the past.99 The

delayed choice quantum eraser experiment also indicates that the future loops back

to create the past. And Daryl Bem of Cornel has published several experimental

results demonstrating retro-causality.57 In 2014, Brierley et al.100 demonstrated

quantum entanglement of particles across time. In fact, there is an old wives tale that

general relativity prohibits trans-temporal feedback loops. This is not true. General

relativity simply states that communication between events cannot occur via photon

mediation. In fact, general relativity predicts wormholes through time and space.

The inherent non-locality of quantum reality does not require signals for things to be

connected; any more than rotating a penny while looking at the heads side requires

time to transmit the torque to the other side of the penny. It is a simultaneous or

null-speed correlation. The truth is that until we have a predictive¯rst principles

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theory that uni¯es general relativity and quantum mechanics, one cannot aggres-

sively invoke interpretations of either of these two place-holder theories to say with

con¯dence what can and cannot occur. Both will turn out to be °awed or incomplete

in certain ways when a full theory of everything is discovered.

So for now, let us develop the most conservative argument as follows:

(1) Like the exponential explosion of any doubling algorithm, high-level forms of

consciousness and networked consciousnesses will envelope the universe. There

are no hypotheses that can reasonably challenge this idea. We have hard evi-

dence that consciousness emerges because our minds are sharing the words of this

sentence. The idea of consciousness exponentially spreading throughout the

universe is plausible due to the extraordinary behavior of doubling algorithms.

For example, if we doubled a penny as fast as we can hit the \x2" button on an

iPhone calculator in 30 s, we would have more pennies than all the atoms in the

entire universe. The reason we do not see doubling algorithms in nature go more

than a few iterations because resources halt the doubling algorithm very early.

(2) The question is whether or not a species with high consciousness and evolving

consciousness can leave their biosphere and continue doubling and staying non-

locally networked. Humans made it to another cosmological body in 1969, when

we landed on the Moon. It is only a matter of time before technology and our built-

in compulsion to explore takes us out into the universe, where resource limitation

halting will not occur until all the energy of the universe is exhausted. Again, the

challenge is not to argue why this will occur. That is established by the axiom

\Given enough time, whatever can happen will happen". The onus of logic falls on

those who guess humans will destroy themselves or that society will collapse or and

that all other potential species in the universe will have the same fate.

(3) Now, if the universe is expanding faster than the speed of light, then exponen-

tially expanding consciousness can never sequester all energy into a universal

scale conscious neural network of quantum entangled conscious sub-systems. As

mentioned, general relativity allows wormholes, and quantum mechanics is in-

herently non-local. So until a predictive theory of everything is discovered, it is

not clear whether or not non-local information exchange or teleportation can

occur, where a consciousness can relocate trans-temporally or trans-spatially in

instant-time (perhaps without atomic form) to in°uence matter and energy in

distant regions of the universe. However, it is worthwhile to play the what-if

game to see where the idea leads. If a new¯rst-principles quantum gravity theory

inspired a technology that allowed consciousness to project into spacetime

coordinates non-locally, where would we go¯rst?What if you were given 100 free

airline vouchers to °y anywhere in the world? Would you explore ballistically by

¯rst traveling 100 miles from your home, then 200 miles and so on until you

explored the far reaches of the world? Or would you make a favorites list and

bounce around arbitrarily depending on whether Beijing, Sydney or Rio made it

near the top of your wish list? If humans or any other intelligent life in the

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universe discovers non-local information exchange that a consciousness can ex-

ploit, we will bounce around the universe and plant consciousness in various

parts of the cosmos. At¯rst, the transplanted consciousness outposts will have

the sparse pattern of a sponge — or neural network — throughout the whole

cosmos. So the expansion rate of the universe would not be a problem for the

deduction that, given enough time, high consciousness will eventually envelop all

energy in the universe. The sparse sponge-like pattern of outposts of networked

consciousness will¯ll-in as they approach maximum density at 100% of the

energy of the universe.

(4) What would this high consciousness be like? It is hard to say. But it would not be

very much like us. We are related to snails and horses and dinosaurs, but we are

not very much like them. However, the one thing we would share in common is

that we would understand the¯rst-principles theory of everything that would be

a prerequisite for the exploitation of non-local mental and physical technology.

When a¯rst-principles theory of everything is discovered, it will not be replaced

by something else. To say otherwise indicates a misunderstanding of what the

term¯rst-principles means in this context. The Pythagorean Theorem is based

on¯rst principles. It will not be replaced. We are not talking about a model of

how the universe works. We are speaking of discovering the simulation code of

geometric symbolism itself and interacting with it.

So, we have told an audacious story, even though it may be logically inevitable.

However, it should be noted that the big bang theory is audacious and probably true

at the same time. The emergence of this very conversation, dear reader, and the

human consciousness that it exists within is audacious. And so too is the notion of

the universe being a simulation from a creator outside the universe. So if auda-

ciousness is evil, then we are seeking the lesser of all evils. The deduction herein is in

fact conservative. And yet it is audacious at the same time. It is not just plausible. It

is inevitable.

The punchline of the deduction is this: Because this is an inevitable outcome, the

simplest answer on how an information theoretic universe can exist and what its

substrate would be if it self-actualized is the entire system reality is a mind-

like mathematical (geometric) neural network. Just as our now limited consciousness

can hold within it the notion of a square, we can allow a self-organizing game or

language of squares to emerge in our mind. A far greater neural network could hold

within it the relatively simple geometry of E8 and the 4D and 3D quasicrystals we

have discussed. Primitive quanta or measuring entities (quantum viewers) at the

Planck scale substructure of the imagined possibility space, which are essentially

vantage points of the universal emergent consciousness, would actualize geometric

symbols by observations (projective transformations) within the quasicrystalline

possibility space. Each observation of a Planck scale quantum viewer generates a

projective transformation equal to a rotation of the 3-simplex it is associated with.

These primitive geometric binary choice states on the possibility space are part of a

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code that forms a neural network based on 3D simplex-integers in an E8 derived

quasicrystal. By geometric¯rst principles, the code has a free-variable called the

phason °ip. And the universal consciousness operating the details of the code obeys

the principle of e±cient language, taking instructions from conscious sub-systems

like us, who are engines of emergent meaning.

The universe would not exist if it weren't for intermediary emergent entities like

us. It would also not exist if it weren't for the maximally simple golden ratio-based

quasicrystalline E8 code that self-organizes quarks and electrons into 81 stable atoms

and into countless compounds and planets and people and societies and overly-wordy

sentences and on up through to the collective consciousness of the universe. And that

primitive starting code and the simplex-integers and the quantum viewer operators

needed to animate the whole thing would not exist without the collective emergent

consciousness. Retrocausality allows the whole idea to be logically consistent, where

the future creates the past and the past creates the future. The simple creates the

complex and the complex creates the simple a cosmic scale evolving feedback loop

of co-creation. This framework is both explanatory and conservative. And it requires

no magical moments that are unexplainable, like the moment of the big bang or a

creator-God. It uses¯rst-principles logic where A co-creates B, which co-creates C,

which co-creates A. Non-linear causality is mathematically and logically rigorous.

The entire framework is based on two fundamental and inarguable behaviors of

nature: (1) emergent complexity and (2) feedback loops.

References

1. D. Schumayer and D. A. W. Hutchinson, Colloquium: Physics of the Riemann

hypothesis, Rev. Mod. Phys 83 (2011) 307.

2. F. Dyson, Birds and frogs, Not. AMS 56(2) (2009) 212–223.

3. M. L. Lapidus, In Search of the Riemann Zeros: Strings, Fractal Membranes and

Noncommutative Spacetimes (American Mathematical Society, Providence, R.I., 2008).

4. H. Georgi and S. L. Glashow, Unity of all elementary-particle forces, Phys. Rev. Lett.

32 (1974) 438.

5. F. Gursey, P. Raymond and P. Sikivie, A universal gauge theory model based on E6,

Phys. Lett. B 60(2) (1976) 177–180.

6. A. G. Lisi, An exceptionally simple theory of everything. 2007. Available at https://

arxiv.org/abs/0711.0770 (accessed 21 August, 2017).

7. R. Aschheim, Is reality digital or analog. 2011. Available at http://fqxi.org/community/

forum/topic/929 (accessed 21 August, 2017).

8. Online Dictionary of CRYSTALLOGRAPHY Available at http://reference.iucr.org/

dictionary/Aperiodic crystal (accessed August 21, 2017).

9. E. P. Wigner, On the distribution of the roots of certain symmetric matrices, Ann. Math.

62(2) (1958) 325–327.

10. A. M. Odlyzko, Primes, quantum chaos and computers, Number Theory, in Proc. Symp.

National Research Council, Washington DC, 1990, pp. 35–46.

11. H. L. Montgomery, The pair correlation of zeros of the zeta function, in Proc. Symp.

Pure Math. 24 (1972) 181–193.

12. T. Tao and V. Vu, Random matrices: The universality phenomenon for Wigner ensembles.

Modern aspects of random matrix theory, Proc. Symp. Appl. Math. 72 (2014) 121–172.

Toward the Uni¯cation of Physics and Number Theory

1950003-75

Rep. Adv. Phys. Sci. Downloaded from www.worldscientific.comby 76.168.132.104 on 12/03/19. Re-use and distribution is strictly not permitted, except for Open Access articles.

13. M. Deza, V. Grishukhin and M. Shtogrin, Scale-Isometric Polytopal Graphs In Hypercubes

and Cubic Lattices: Polytopes in Hypercubes and Zn (Imperial College Press, 2004).

14. J. E. Hopcroft, Introduction to Automata Theory, Languages, and Computation

(Addison Wesley Publishing Company, 1979).

15. F. Fang and K. Irwin, An icosahedral quasicrystal E8 derived quasicrystals (2016).

Available at http://arxiv.org/pdf/1511.07786.pdf (accessed August 21, 2017).

16. S. Wolfram, Statistical mechanics of cellular automata, Rev. Mod. Phys. 55 (1983) 601–644.

17. O. Martin, A. M. Odlyzko and S. Wolfram, Algebraic properties of cellular automata,

Commun. Math. Phys. 93 (1984) 219–258.

18. P. Eades, A Heuristics for graph drawing, Congressus Numerantium 42 (1984) 146–160.

19. Wolfram Mathworld, Simplex, Available at: http://mathworld.wolfram.com/search/?

q=Simplex (accessed August 21, 2017).

20. Gauss, C. Friedrich and E. Schering, Carl Friedrich Gauss Werke (Gedruckt in der

Dieterichschen Universitäts-Buchdruckerei, 1874).

21. A. E. Ingham, Note on Riemann's -Function and Dirichlet's L-Functions, J. London

Math. Soc. 5 (1930) 107–112.

22. B. Fine and G. Rosenberger, The density of primes, in Number Theory: An Introduction

via the Distribution of Primes (Birkhäuser Basel, Boston, 2007), pp. 133–196.

23. R. Aschheim, C. C. Perelman and K. Irwin, The search for Hamiltonian whose energy

spectrum coincides with the Riemann zeta zeroes, Int. J. Geom. Meth. Mod. Phys. 14

(2017) 1750109.

24. K. Irwin, The Code Theoretic Axiom, The Third Ontology (2017). Available at http://

www.quantumgravityresearch.org/portfolio/the-code-theoretic-axiom-the-third-ontology

(accessed August 21, 2017).

25. D. Shechtman and I. A. Blech, The microstructure of rapidly solidi¯ed Al6Mn, Metall.

Trans. A. 16(6) (1985) 1005–1012.

26. L. Pauling, Evidence from x-ray and neutron powder di®raction patterns that the so-

called icosahedral and decagonal quasicrystals of MnAl6 and other alloys are twinned

cubic crystals, Proc. Nat. Acad. Sci. 84 (1987) 3951–3953.

27. N. Wolchover, In mysterious pattern, math and nature converge, Quanta Mag. (2013).

Available at https://archive.li/wFNom (accessed August 21, 2017).

28. A. Baker, Transcendental Number Theory (Cambridge University Press, 1990), pp. 102–107.

29. B. Grunbaum and G.-C. Shephard, Tilings and Patterns (WH Freeman and Company,

New York, 1987).

30. D. Mumford and K. Suominen, Introduction to the theory of moduli, Algebraic

Geometry, Oslo 1970 in Proc. Fifth Nordic Summer-School in Math. (WoltersNoordho®,

Groningen, 1972), pp. 171–222.

31. D. Deutsch, The Fabric of Reality (Penguin Books Ltd, UK, 1998).

32. E. Fredkin, Digital mechanics, an informational process based on reversible universal

cellular automata, Phys. D. Nonlinear Phenomena 45 (1990) 254–270.

33. T. To®oli, Cellular automata as an alternative to (rather than an approximation of) dif-

ferential equations in modeling physics, Phys. D. Nonlinear Phenomena 10 (1984) 117–127.

34. S. Wolfram, A New Kind of Science (Wolfram Media Inc, Champaign, IL, 2002).

35. J. A. Wheeler, Information, Physics, Quantum: The Search for Links. Physics Dept.

(University of Texas, 1990).

36. R. Rojas, Neural Networks: A Systematic Introduction (Springer Science & Business

Media, 2013).

37. J. Conway and S. Kochen, The free will theorem, Found. Phys. 36(10) (2006) 1441.

38. P.-L. M. de Maupertuis, Accord de di®erentes loix de la nature qui avaient jusquìci paru

incompatibles (1744).

K. Irwin

1950003-76

Rep. Adv. Phys. Sci. Downloaded from www.worldscientific.comby 76.168.132.104 on 12/03/19. Re-use and distribution is strictly not permitted, except for Open Access articles.

39. E. Noether, Invariante variationsprobleme, Nachrichten von der Gesellschaft der

Wissenschaften zu G€ottingen, Mathematisch-Physikalische Klasse. (1918) 235–257.

40. L. Bubuianu, K. Irwin and S. I. Vacaru, Heterotic supergravity with internal almost

kähler con¯gurations and gauge SO32, or E8 X E8, instantons. Class. Quantum Gravity

34(7) (2017) 075012.

41. R. Aschheim, L. Bubuianu, F. Fang, K. Irwin, V. Ruchin and S. Vacaru, Starobinsky

in°ation and dark energy and dark matter e®ects from quasicrystal like spacetime

structures (2016). Available at https://arxiv.org/abs/1611.04858 (accessed August 21, 2017).

42. M. M. Amaral, R. Aschheim, L. Bubuianu, K. Irwin, S. I. Vacaru and D. Woolridge,

Anamorphic quasiperiodic universes in modi¯ed and einstein gravity with loop quantum

gravity corrections, Class. Quantum Gravity 34 (2017) 18.

43. V. Elser and N. J. A. Sloane, A highly symmetric four-dimensional quasicrystal, J. Phys.

A. Math. General 20 (1987) 6161–6168.

44. H. Bohr, Zur theorie der fastperiodischen funktionen, Acta Math. 46 (1925) 101–214.

45. T. C. Lubensky, S. Ramaswamy and J. Toner, Hydrodynamics of icosahedral quasi-

crystals, Phy. Rev. B. 32 (1985) 7444.

46. J. Ambjørn, J. Jurkiewicz and R. Loll, Reconstructing the universe, Phys. Rev. D. 72

(2005) 064014.

47. D. Oriti, Approaches to Quantum Gravity: Toward a New Understanding of Space, Time

and Matter (Cambridge University Press, 2009).

48. D. Levine and P. J. Steinhardt, I. Quasicrystals, De¯nition and structure, Phys. Rev. B.

34 (1986) 596.

49. R. F. Voss, Fractals in nature: From characterization to simulation, in the Science of

Fractal Images (Springer-Verlag, New York, Inc., 1988), pp. 21–70.

50. B. Rosenblum and F. Kuttner Quantum Enigma: Physics Encounters Consciousness

(Oxford University Press, 2011).

51. A. S. Fraenkel and S. T. Kleinb, Robust universal complete codes for transmission and

compression, Discrete Appl. Math. 64 (1996) 31–55.

52. E. Schrodinger, What is Life? The Physical Aspects of Living Cell with Mind and Matter

(Cambridge University Press, 1967).

53. J.-C. Perez, Codon populations in single-stranded whole human genome DNA are fractal

and¯ne-tuned by the Golden Ratio 1.618, Interdisciplinary Sci. Comput. Life Sci.

2 (2010) 228–240.

54. M. Burrello, H. Xu, G. Mussardo and X. Wan, Topological quantum hashing with the

icosahedral group, Phys. Rev. Lett. 104 (2010) 160502.

55. H. S. M. Coxeter, Regular Polytopes (Courier Dover Publications, 1973).

56. S. P. Walborn, M. O. Terra Cunha, S. Padua and C. H. Monken, Double-slit quantum

eraser, Phys. Rev. A 65 (2002) 033818.

57. D. Bem, P. E. Tressoldi, T. Rabeyron and M. Duggan, Feeling the future: A meta-

analysis of 90 experiments on the anomalous anticipation of random future events

(2015). Available at https://f1000research.com/articles/4-1188/v1 (accessed August

21, 2017).

58. C. F. Doran, M. G. Faux, S. J. Gates Jr., T. Hubsch, K. M. Iga and G. D. Landweber,

Relating doubly-even error-correcting codes, graphs, and irreducible representations of

N-extended supersymmetry (2008). Available at https://arxiv.org/abs/0806.0051

(accessed August 21, 2017).

59. M. Faux and S. J. Gates, Adinkras: A graphical technology for supersymmetric repre-

sentation theory, Phys. Rev. D 71 (2005) 065002.

60. D. Deutsch, The Fabric of Reality (Penguin Random House, UK, 1998).

Toward the Uni¯cation of Physics and Number Theory

1950003-77

Rep. Adv. Phys. Sci. Downloaded from www.worldscientific.comby 76.168.132.104 on 12/03/19. Re-use and distribution is strictly not permitted, except for Open Access articles.

61. Dr. S. J. Gates, Living in the Matrix: Physicist¯nds computer code embedded in string

theory (2010). Available at https://www.sott.net/article/301611-Living-in-the-Matrix-

Physicist-¯nds-computer-code-embedded-in-string-theory (accessed August 21, 2017).

62. J. Maldacena, The large-N limit of superconformal¯eld theories and supergravity, Int.

J. Theor. Phys. 38(4) (1999) 1113–1133.

63. P. C. W. Davies, Thermodynamic phase transitions of kerr-newman black holes in de

Sitter space. Classi. Quantum Gravity 6 (1989) 1909.

64. J. M. Garcia-Islas, Entropic motion in loop quantum gravity, Can. J. Phys. 94 (2016)

569–573.

65. L. Hardy, Nonlocality for two particles without inequalities for almost all entangled

states, Phys. Rev. Lett. 71 (1993) 1665–1668.

66. R. Coldea, D. A. Tennant, E. M. Wheeler, E. Wawrzynska, D. Prabhakaran, M. Telling,

K. Habicht, P. Smeibidl and K. Kiefer, Quantum criticality in an Ising chain: Experi-

mental evidence for emergent E8 symmetry, Science 327(5962) (2010) 177–180.

67. L. Xu and T. Zhong, Golden ratio in quantum mechanics, Nonlinear Sci. Lett. B. 1

(2011) 10–11.

68. R. Coldea, D. A. Tennant, E. M. Wheeler, E. Wawrzynska, D. Prabhakaran, M. Telling,

K. Habicht, P. Smeibidl and K. Kiefer, Golden ratio discovered in a quantum world,

Science 2010(32) (2010) 177–180.

69. I, A®leck, Solid-state physics: Golden ratio seen in a magnet, Nature 464 (2010) 362–363.

70. A. Connes, Noncommutative geometry year 2000, Vis. Math. (2000) 481–559.

71. M. S. El Naschie, Superstrings, knots, and non-commutative geometry in E-in¯nity

space, Int. J. Theor. Phys. 37 (1998) 2935–2951.

72. M. El Naschie, The theory of Cantorian spacetime and high energy particle physics

(an informal review), Chaos Solitons Fractals 41 (2009) 2635–2646.

73. E. P. Wigner, Random matrices in Physics, SIAM Rev. 9 (1967) 1–23.

74. M. Krbalek and P. Seba, The statistical properties of the city transport in Cuernavaca

(Mexico) and random matrix ensembles, J. Phys. A. Math. Gen. 33 (2000) L229.

75. T. Falco, F. Francis, S. Lovejoy, D. Schertzer, B. R. Kerman and M. Drinkwater,

Universal multifractal scaling of synthetic aperture radar images of sea-ice, IEEE Trans.

Geosci. Remote Sens. 34(b4) (1996) 906–914.

76. E. Wigner, The unreasonable e®ectiveness of mathematics in the natural sciences,

Commun. Pure Appl. Math. 3 (1960) 116.

77. C. H. Suresh and N. Koga, A consistent approach toward atomic radii, J. Phys. Chem.

A 105 (2001) 5940–5944.

78. R. Heyrovska, Golden sections of interatomic distances as exact ionic radii and addi-

tivity of atomic and ionic radii in chemical bonds (2009), Available at https://arxiv.org/

abs/0902.1184 (accessed August 21, 2017).

79. H.-R. Trebin, Quasicrystals: Structure and Physical Properties (John Wiley & Sons,

2006), pp. 212–221.

80. N. Wolchover, In mysterious pattern, math and nature converge, Quanta Magazine

(2013), Available at https://archive.li/wFNom (accessed August 21, 2017).

81. G. Sadler, Periodic modi¯cation of the Boerdijk-Coxeter helix (tetrahelix) (2013),

Available at https://arxiv.org/abs/1302.1174 (accessed August 21, 2017).

82. J. Kovacs, The sum of squares law (2012), Available at https://arxiv.org/abs/1210.1446

(accessed August 21, 2017).

83. C. C. Perelman, F. Fang and K. Irwin, Law of sums of the squares of areas, volumes and

hyper-volumes of regular polytopes, Adv. Appl. Cli®ord Algebras 23 (2013) 815–824.

84. I. Baruk, Causality II: A Theory of Energy, Time and Space (Herstellung und Verlag,

2008).

K. Irwin

1950003-78

Rep. Adv. Phys. Sci. Downloaded from www.worldscientific.comby 76.168.132.104 on 12/03/19. Re-use and distribution is strictly not permitted, except for Open Access articles.

85. A. Einstein, Ist die Trägheit eines K€orpers von seinem Energieinhalt abhängig? Annalen

der Physik 323(13) (1905) 639–641.

86. D. J. Gri±ths, Introduction to Quantum Mechanics (Pearson Prentice Hall, 2004).

87. M. Tegmark, Is \the theory of everything" merely the ultimate ensemble theory? Ann.

Phys. 270 (1998) 1–51.

88. A. Einstein, M. Born and H. Born, The Born-Einstein Letters: Correspondence between

Albert Einstein and Max and Hedwig Born from 1916 to 1955 (MacMillan Press,

Basingstoke, 2005).

89. I. S. Newton, Opticks: Or a Treatise of the Re°exions, Refractions, In°exions and

Colours of Light (Sam. Smith, and Benj. Walford, Printers to the Royal Society at the

Prince's Arms in St. Paul's Church-Yard, London, 1704).

90. L. de Broglie, Recherches sur la theorie des quanta, Reedition du texte de 1924

(Masson & Cie, Paris, 1963).

91. D. Hestenes, Electron time, mass and zitter. FQXI. 2008, Available at http://fqxi.org/

community/forum/topic/339 (accessed August 21, 2017).

92. T.-D. Lee and C.-N. Yang, Question of parity conservation in weak interactions, Phys.

Rev. 104 (1956) 254.

93. C. D. Pruett, Reason and Wonder: A Copernican Revolution in Science and Spirit.

(ABC-CLIO, Santa Barbara, CA, 2012).

94. J. W. N. Sullivan, Interview with Erwin Schr€odinger, The Observer, January 11, 1931.

95. A. Linde, Chaos, consciousness, and the cosmos. FQXI. 2011, Available at http://fqxi.

org/community/articles/display/145 (accessed August 21, 2017).

96. D. Skrbina, The Metaphysics of Technology, Vol. 94 (Routledge, 2014).

97. K. Wilber, Quantum Questions (Shambhala Publications, 2001).

98. K. Irwin, A new approach to the hard problem of consciousness: A quasicrystalline

language of \primitive units of consciousness" in quantized spacetime (part I), J. Consc.

Explor. Res. 5 (2014).

99. S. W. Hawking and T. Hertog, Populating the landscape: A top-down approach, Phys.

Rev. D 73 (2006) 123527.

100. S. Brierley, A. Kosowski, M. Markiewicz, T. Paterek and A. Przysiezna, Nonclassicality

of temporal correlations, APS Phys. Rev. Lett. 115 (2015) 120404.

101. A. Einstein, The Formal Foundation of the General Theory of Relativity, 1914, Koni-

glich Preussische, Akademie der Wissenschaften. Sitzungsberichte in The Berlin Years:

Writings, 1914–1917, Vol. 6.

102. A. Sen, R. Aschheim and K. Irwin, Emergence of an aperiodic dirichlet space from the

tetrahedral units of an icosahedral internal space, Mathematics 5(2) (2017).

103. F. Fang, J. Kovacs, G. Sadler and K. Irwin, An icosahedral quasicrystal as a packing of

regular tetrahedra, ACTA Physica Polonica A 126(2).

104. F. Fang, R. Clawson and K. Irwin, Encoding geometric frustration in tetrahedral

packing with gaps, discrete curvature, distortion or twisting (2017). Avialable at

http://www.quantumgravityresearch.org/portfolio/encoding-geometric-frustration-

in-tetrahedral-packing-with-gaps-discrete-curvature-distortion-or-twisting (accessed

October 21, 2017).

105. D. Gross, J. A. Harvey, E. Martinec and R. Rohm, Heterotic string theory (I). The free

heterotic string, Nucl. Phys. B. 256 (1985) 253–284.

Toward the Uni¯cation of Physics and Number Theory

1950003-79

Rep. Adv. Phys. Sci. Downloaded from www.worldscientific.comby 76.168.132.104 on 12/03/19. Re-use and distribution is strictly not permitted, except for Open Access articles.