1. Introduction

The universe! One of the deepest questions: how the universe was created and how was the system “Space/Time/Matter/Energy/Gravity/Electromagnetism” is homogeneous, connected and unified? There exists a “tiny”, concise equation, as most scientists have predicted, which will connect all physical laws of the universe. This unlocked mathematical numerical puzzle model is the key to understanding the universe and reaches beyond traditional physics when attempting to explain the physical laws of the universe. The equation will answer all unknown physical, biological, philosophical and spiritual questions! This powerful equation will provide the answers to some of the most mysterious questions that have ever been found: What is the nature of “time”? How does it function? What are its properties? What are dark energy and dark matter? How speed of light was defined? Do we live in a simulated universe? How gravity works?

1.1. Equation Generality

To begin, I will be providing a step by step, detailed explanation that will outline the methods that were utilized to construct and prove my conjecture/equation which based on a numerical model puzzle that describes the dynamical of a finite discrete system related to the space, time and the quantum circuit! A Mathematical Model with mixed numerical and theoretical methods that proves the universe has a mathematical structure! In this paper I will be proving with one common equation based on a hidden numerical system related to Pascal’s Triangle the architecture of the universe and its quantum structure including biological systems from cells, chromosomes, genes, DNA a set of connected systems that develop from a simple form into a complex form (Fractal) such as “Combinatory Game Theory”. One of the interesting properties I used in this model is the algorithm of strings based on the trans-palindrome numbers to explain the super-symmetry in the universe and to describe also the theory of entanglement in higher dimension and show that the universe is generated through a super-computer that codes, decodes and corrects error codes, a special automata language program based on the discrete system! The equation will provide us with some numerical proves such as speed of light and how time is defined. Mathematically I will be unveiling the hidden secret of the representation, configuration and the dynamical system of the primes, composites, palindrome and trans-palindrome numbers “Dynamical of a lattice related to a special elliptic curve” a trefoil torus knot. My system is based on two important operations: Through singularity “fixed point properties” and through its periodicity.

1.2. Discrete Mathematics

Discrete systems are characterized by integers, including rational numbers in contrast to continuous systems which require real numbers. Discrete mathematics is the study of mathematical entities with discrete structure, with the property that do not vary smoothly, dealing with integers, graphs, with countable set in the fields of combinatory theory, graph theory, operations research, number theory, theory of computation that includes the study of algorithms and its implementations. Integers are absolute abstract entities independent from space and time that function to define abstract and concrete things. To understand the behavior and dynamics of the discrete system we need to analyze its fields, its flow “ramification”, decompositions “how it split” and its group representations. Each integer is defined with its algebraic, analytic and geometric identity. Integers have a solid fundamental foundation and are considered as the primary mathematical and automata language of the universe and its atomics structure.

2. Salahdin Daouairi’s Equation for the “Theory of Everything” is Defined as

Figure 1. Geometrical interpretation of the equation

Figure 2. Dynamical of the torus embedded in the hyper-sphere

2.1. Important Numerical Equations

Those numerical equations are part of the equation of everything and are very important to describe my system!

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

2.2. Descriptions

In this paper I will be decoding the mystery and the mathematical beauty of the number 666 which is related to the equation of everything! I will simply show that is a hexagonal torus of dimension 5 which is space/time with dim3/dim2 embedded in a hyper-sphere of dimension 6 which is the extra dimensions discovered in String Theory and show how the system is connected, unified and homogenous. The Super-symmetry and the Theory of Entanglement also explained through trans-palindrome numbers and their super-partners, while the quantum circuit described by the flow of the trans-palindrome numbers through field’s ramification. This hidden numerical system is related to a special known Automorphic form! Dynamical system of a particular elliptic curve that is related to the spiral of Fibonacci, “a special case of Langland Program”.

To begin, I will be given some useful elementary definitions of some geometric shapes that I will be based on.

2.3. Definitions of Lattice / Torus / Elliptic Curve [8]

2.4. Interpretation Mathematical of the Equation

The equation will describe the dynamical of a particular elliptic curve a “Trefoil Knot” resulting from the transformation of a hexagonal lattice that describes a “torus” of 5 dimensions embedded in a Hyper-Sphere of 6 dimensions with radius . To show how the torus and the hyper-sphere are connected, we will be studying through an asymmetric transformation (Helix) the distribution (packing Spheres) and dynamical system of the primes, composites, palindromes and trans-palindromes of a finite set of discrete numbers considered as ‘spinors’, mathematical entities to define the quantum space and its circuit. These entities represent the vertices of a lattice which correspond to the set where the edges represent the strings / loops. The dynamical of those vertices will describe circles through an oscillation harmonic and will map a hyper-sphere with a radius through the point while the integers are consecutive prime’s elements of the set . The whole system is related to Pascal’s triangle and will be evaluated upon its singularity and periodicity through modular representations. In this mathematical model I will be proving:

- The configuration, representation and dynamical system of primes, composites, palindrome and trans-palindrome numbers.

- Convergence related to singularity “fixed point” and periodicity of the system.

Physically the interpretation can be seen a priori from a lattice field theory, grid composed with cells and charged (+/-), that interchange information with a phenomena of creation and annihilation (+/_). The dynamical of the particle charged + or - describes a cyclical helical electromagnetic wave a “vortex ring model” that transports matter and energy along a solenoid through an asymmetrical transformation. What we will be showing are: - Existence of multi-verse and particles charged +/- : Interaction through electromagnetism creates the dynamical of the multi-verse “concept of entanglement”, while the dynamical of the multi-verse and particles induces the gravity!

2.5. Representation of Triangular Numbers

Geometrically represents the total number of tours: when the circle turns 1 time the circle turns n times, with an arithmetic progression equals to 1 tour between two successive circles.

2.6. Notion of Cardinality

In the set we have 25 primes, 73 composites and number 1. From the special triangular sequences see “Equations (3) and (4)” and correspond to the orbital of the 25 prime numbers and the 73 composite numbers.

Card P(primes) = 25 and Card C(composites) = 73. If we enumerate the set of primes by . The same for composite numbers . From the property of the cardinality there exists a bijection between the set and respectively between and .

The primes and composites represented as objects or mathematical entities.

2.7. Configurations of the Integers

Properties: The set of positive integers is structured from three important subfamilies:

2.8. Definition of Trans-Palindrome and Palindrome Numbers

- A palindrome number is a 'symmetrical' number like 17271 that remains the same when its digits are reversed, and when the number and its reversed digits are not the same then these two numbers called simply trans-palindrome numbers.

In the set the Set of palindrome numbers defined by:

with 9 elements

And set of trans-palindromes by:

with 90 elements.

3. Pascal’s Triangle / Period of the System

We can resume three important sequences in Pascal’s Triangle system:

Triangular sequence, Fibonacci sequence and power of 2 sequences, thus the period of each of those sequences in base modulo 9 are:

3.1. Simplex Polytope

Geometric Interpretation of the Pascal’s triangle for n=6: In geometry a simplex is a generalization of the notion of triangle and tetrahedron to arbitrary dimension. An N-Simplex is an N dimensional polytope which is the convex hull of its N+ 1 vertex. We can interpret the Pascal’s triangle simply by a succession of N simplex which is the process of constructing a N- Simplex from a (N-1)-Simplex by adding a new vertex to the exterior of the (N-1)-Simplex and joining it to all vertices of the (N-1)-Simplex [9], in five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope, the symmetric group S₆. It has 6 vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 pentatope facets. It is a 5 dimensional polytope which is the dim of space/time that coincide with .

Let’s denote by V: vertices, E: edges, Tr: triangles,

Tet: tetrahedral and by Pen: pentatope

Table 1. Representation of a 5-Simplex polytope

Figure 3. Geometrical shape of a 5-Simplex polytope

3.2. Positioning Pascal’s Triangle to Determine the Symmetry of the System in Base Modulo 9

Figure 4. Pascal’s Triangle in base modulo 9

This method of evens and odds separation is very important in the electrons and protons configuration, since the sequence of odds is connected to the squares by: (Total electron per shell is ) and triangular numbers to maximum number of proton. By separating Pascal’s Triangle with odd numbers one side and even numbers to the other side: we notice that is the axis of the system, where is orbiting around , by joining to , and orbiting around and (Helix) by joining to .

Corresponds to a helicoidally trajectory by joining each point of the axis to its oblique diagonals (e.g. Figure 4).

We notice a finite closet string with repeated algorithm: 2664-4662-6642.

Note: for n=99

3.3. Divine Code 6642/ Key to the Equation

Mathematically this divine code found in the repeated following algorithm of the string in the Pascal’s triangle and (e.g. Figure 4) that has the representation of a harmonic oscillation between the two trans-palindrome numbers 2664 and 4662. Let’s project the numbers in a circle; the code is to rotate the key anticlockwise from 6 to 2 to map the two trans-palindrome numbers 2664 and 4662.

2664 ≤ 4662, the string 2664 rotates in the opposite direction of 4662 (harmonic motion) and as a quantity 2664 is including in 4662 which leads to the following representation (e.g. Figure 5 & 6)

Or:

4662 = 7× 666

While:

By adding the two numbers:

Figure 5. Code & Key

Figure 6. Opposite directions of the 2 trans-palindromes

3.4. Interpretations of the Two Equations

- The equation (10): The mean value A of 4662 and 2664 belongs to the diagonal (A is a multiple of 11)

A 4662 and 2664 have the same axis of orbital.

- The equation (11): 73 and 26 are just the total number respectively of the composite numbers and the prime numbers including number 1 in the set

The greatest common factor of 2664 and 4662 is equal to 666, .

73 composites represent 73 vertices with a total of 72 edges, oscillate the circle (e.g. Equation 8).

For each composite move the circle describes a turn with a period of that correspond to:

Surface of the sphere of radius r denoted by .

4. System’s Modeling

4.1. Interpretation of the Numerical Equation (1)

Equation of a Hyper-Sphere, when with radius , and . The points are seven consecutive primes of the set and are coordinates of a point of the hyper-sphere “Sphere ”.

Though dynamical of the primes and composites is related to the Hyper-Sphere , while the primes and composites are orbiting, they are mapping through the point M.

Let’s denote by:

Then:

(2)

(3)

(4)

4.2. Physical System

Let’s denote by:

4.3. Physical Interpretation of the Sequences in Pascal’s Triangle

Let’s prove the following properties:

4.4. Generators of the System

The system is generated through two important notions of singularity which is based on the fixed point properties and the period.

4.5. Singularity

4.5.1. Configuration Numeric / Root System

Let’s denote by defined by the 3x3 matrix with

Det is singular, not invertible (infinity of solutions), with eigen-value equals 0.

By applying non singular linear transformation the base that generates the lattice is of rank 2.

Let’s denote by d time deviation between two integers or strings and , defined by: since , where 9 is simply a constant related to a quadratic equation, with and where x and y and d are elements of . The space is considered a quantum space (It’s a Hilbert space; the space is defined in a district system over a field where the space is measurable). Each number is considered as an object with a space position and with the coordinates in the space and . Each element of with is connected to per +1 or +4 and each element of is connected to the other element of per .

Example: We get the following representation:

Let’s denote by Giving two integers positives , then their spin value correspond to

Figure 7. Each element or vertex has 6 connections or edges, property which is important in knot theory, graph theory, category theory and simplex theory to study the path or the circuit and determine the geometric form of the map or function

Note: for and then: with we have then: .

The transformation with spin+4 is a combination of spin+1 and spin+3.

The spin’s paths of the elements describe a hexagonal lattice for the group acting which is related to the root system of one of the symmetric groups.

4.5.2. Spinors Definition [12]

Spinors are mathematical entities that can be defined as geometrical objects to expand the notion of the vector space under rotation, the notion of spinors have more advantage in the super-symmetry theory in contrast to tensors which are used in the symmetry theory.

Defined by: , where the operator L is the matrix that transforms the angular momentum under the rotation:

Let’s denote by . The orbital of X and Y relatively to their radius and mean value is defined by:

Then since

When X moves toward the fixed position Y, the point M moves toward –R, Eventually the period is reached when X describes 2 full circles or 720 and M describes the 4 small circles. The transformation consists of computing where OMY and OXM (isosceles triangles). M and R are integers when: [2] that when X and Y have the same parity, which will lead us in the future to introduce the bosonic and firmionic fields with the notion of commutation in the Algebra. We have then the 4 following transformations: For reason of symmetry, let’s denote then by: the 2 matrices of transformation of . (e.g. Figure 8).

Since .

We recognize here the Pauli matrices: And the relation: Pauli / Hadamard:

u nitair with row vectors orthogonal.

The Hadamard’s matrix is a well known transformation used in wide applications such as quantum circuits, transmission, signal processing systems and error correcting codes. With polynomial characteristic:

with values with module relative to a square lattice

Figure 8. Spin representation

For the opposite direction: +6, +8, and +5

Example: We get the inverse following representation

Then: [9] where is related to a circle unity.

Or the mean value of the trans-palindromes 37 and 73 is: . which is also the mean value of . The transformation describes the gravity and is related to the phase transformation of U(1) group unity. The last group of spins remaining related to transformations with d equals: +2, +7, +9. We can resume those spins in the following diagram (e.g. Figure 11). Giving a number N and S=N+1 of the matrix M then the difference of the distances of the strings: where and t=1.

For the set with 99 vertices and each vertex has 2 roots and one resultant, except the last vertex has 2 roots, then the total roots mapped: related to time.

Or simply where 4 is the resultant of 1 and 3 for the first transformation!

This combination of spins results from a simple (helix) transformation that transforms a lattice into a cylinder (curved space of dim2). Since the lattice is periodic (modulo 9), then by joining its extremity, the cylinder is then transformed into a torus. The configuration numeric for the elements of is related to Cartan Algebra for the group acting.

4.5.3. Mathematical Notion of Event Horizon, and Singularity Interpreted from Strings

The numbers YX=37 and XY=73 are among important numbers in the system, I will be then showing the numbers 37 and 73 are the event horizon for black holes, simply represented by the letter G: Gravity, while the number 55 there mean value is the singularity; the question is how the gravity and electromagnetism function in the universe? First I will be describing the properties of those numbers:

- 37 is a Cuban prime, a centered hexagonal number in the form of: with

For then

- 37 and 73 result from a rotation determined by d=+6 and d=+1.

- 37 and 73 are asymmetric with opposite directions (oscillate with harmonic motion) and their mean value equals to 55.

Figure 9. Singularity

- As a mass/quantity, . As a result, the mechanism of attraction from law of gravity is induced.

- As a charge 37 and 73 have opposite charges . And as a result, the mechanism of attraction from law of electromagnetism is induced for a magnetic dipole.

- Mathematically 37 and 73 are primes, two closed strings indecomposable, that split in with 0 knot, invariant under rotation and are of short range that spin continuously.

Figure 10. Representation of the strings 37 and 73

Figure 11. Spin related to singularity

Are there more Spins for Since the singularity corresponds to +9, then the operations are in base modulo 9, all roots are , though we can proceed with the following representation:

(commutative/reversible)

Conclusion: Each number has 5 roots that form a base of rank 5. This lattice has then: roots or edges .

To determine the reduced total number of roots for the system, we need to find the smallest period of its sequences related to the system, and that when . This gives us the total of roots equals to roots, or we now the total roots for the group symmetric then 30 roots in our system correspond to the group symmetric , or dim . Each number has 5 roots that form a base of rank 5.

Conclusion: The rank for the basis of the set is equal to 5.

Let’s denote by of rank 5 a set of vectors that span , then

Dimension of Space/Time the 5 simplex polytope. Or then dim(Space)= 3.

4.6. Gravity and Flow 5

Trans-palindrome primes with mirror image prime of the set have opposite orbital with a total of: 8 elements

Figure 12. Trans-palindrome prime with mirror image a prime

Thus only 31, 71 and 79 are Gaussian prime remain inert in .

Flow 5: Let’s denote the flow relatively to n by , from the diagram (e.g. Figure 13)

We have: And mean value of

.

Now let’s find the pair of elements and of which verify:

oscillate circle of radius r denoted by

As a result, a trans-palindrome prime and its prime partner originate from the flow 5.

Flow corresponds to the generator that generates gravity.

4.7. Numerical Flow of the System

Giving the Matrix . The matrix M is singular since det M=0 (has infinity of solutions) with eigen-value equals zero. I will be doing a simple transformation of M to study .

Let’s denote transpose of . Then If I align and combine the elements of nd and since and are connected by the equation: since

Figure 13. Numerical flow

The new matrix leads to the following characteristics: 55 is a symmetry center and 37 located in the third column and third row! If we multiply the two trans-palindrome numbers 24 and 42 by 3x37=111.

while are related to the code the equation!

4.8. Transformation of M to M’

Through the diagonals of M and M’, let’s denote by and two closet sets of integers, we notice that the number 10 is missing to complete . Or 10 is the mirror image of the number 1. In this transformation the diagonal of and the diagonal of are multiple of 5, 9 and 11.

4.9. Origin of the Gravity / Flow of 5, 9 and 11 in

55 is the mean value or the symmetric center of . The mean value of excluding the diagonal is To connect the matrix M to M’, we need to complete the matrix M by adding 10 which is the super-partner of 1 to the diagonal through 11. To cover the new elements we need a square 4x4 matrix with number 11 on the diagonal: Since

There are five empty positions in this matrix , to have a continuity of the elements of , the numbers must oscillate back and forth automatically in harmonic motion.

Two major flows of 11 and 9 fill those gaps automatically under its connection with the flow 5 related to the flows 7 and 2. Flow 5, 7 and 2 are kind of transformers or generators .

4.10. Dynamics of the Flow

This combination of backward and forward flows result in the oscillations of the following pairs and which oscillate respectively the circles: a consequence of the dynamical of the system through the field see circuit quantum path of the system, the system is reversible.

Figure 14. Harmonic Motion of the Flow 5

5. Structure of the Trans-Palindrome Numbers and Their Flow

Consider now the bracket functions defined by: and , where R and M is respectively radius and mean value of and also the spinor’s components.

With and

Note for where H is Hadamard’s Matrix.

Introducing the concept of modular representation over ring and field, integrated with the Lie Bracket embedded with Clifford Algebra giving by in the following ring and fields: and .- In the field and ring :

Since and commute in ring

And and anti-commute in the Field - In the field :

Then the notion of the -graded Algebra for the commutator implies:

Commute when x and y same parity and anti-commute when they are of different parity.

and Are orthogonal and obey Hadamard’s transformation.

Commute if , which means the super-commutator obeys the super- Jacobi identity. - In the field :

commute in those fields if and anti-commute if - In the field :

and are orthogonal and obey Hadamard’s transformation.

Have as solution in the field the pairs: (1,4), (1,6), (2,7), (3,8),(9,4) and (2,3) in which oscillate respectively the circles: by the relation with a prime that split relatively to . Those circles have important property since the primes 13, 17, 37 and 79 are the only primes in the set with super-partner “ inverse image” a prime.

Interpretation physic: (Prime / Prime mirror image) generates the gravity G, while when the magnetic dipole is neutral (absence of charges) the gravity is very important. The universe is closed and interchanges matter through the axis through the black-holes induced by gravity. Where the axis represents the backbone chain that bonds and holds the multi-verse generated by dark matter (e.g. Paragraph 13). Also “Prime / mirror image composite” represents the electromagnetic while “composite/image mirror composite” represent strong force and (1,6) the weak interaction! (e.g. Paragraph 14.3).

Figure 15. Physical representation of trans-palindromes

5.1. Quadratic Equation

Let’s denote such equation by

Since our elements are integers, then corresponds to either 1, a composite or a prime!

In case of a prime:

known by Gaussian integers, elements of , that describes the splitting of primes in Galois extension.

When then it splits into two different factors and when it remains inert (Gaussian prime number). In case equals to 1 then we have the circle unity.

Resolution:

has as solution

- Case: . With

With period equals to 4. Known by U(1), the circle group unity, the multiplicative group of all complex numbers with absolute value 1, used to represent bosonic symmetries.

In the complex set / are the 4 points that intercepts the lines and the circle unity. From recognize here the inverse square law of physics related to intensity, force, quantity and potential which is proportional to the inverse square of the distance in such phenomenal physics from sound, radiation, magnetism, electric and also in the Newton’s force of gravity.

Figure 16. Timer representation

The spin of a discrete number has the same properties as the spin of a particle, related to the quadratic equation.

: the expected value in probability theory coincides with the notion of the mean value for discrete numbers. These formulations are considered to be useful to determine Time’s properties, gravity’s phenomena and quantum circuit.

5.2. Timer or Counter

This property of circle unity shows the vector unit, spans through a square lattice with a total span equals to and each span is equal to then which characterizes the span of the graviton with a total of . The continuity of its span, results in a state without equilibrium which proves the continuity of the particle’s vibration, due to the orbital periodic of the graviton, by Bertrand's theorem, the force, is the only possible central force field with stable closed orbits. The graviton is the counter for the atomic clock: When the circle 73 moves from one of its point to another, the circle 37 moves with one turn, when the circle 73 maps all the 73 points (there is 72 equidistant paths), then the circle 37 made . While the circle unity spins (that when the circle 37 maps the 37 points, with 36 equidistant paths), i describes 18 circles which represent the 18 primes and for each prime it describes 4 turns relatively to the square lattice then its period orbital total equals to 72.

Note: total number of primes excluding is equal to 18 primes left in .

While related to time and energy level “See distribution of primes, code”.

Conclusion: the graviton is the counter of Time. The graviton is related to the circle unity U(1) phase of transformation which is the counter that describes the orbital period of the multi-verse.

The gravity results from the space/time curvature, while (1,6) are elements that create the gravity, we notice 1 is related to time and 6 is related to space.

Case: while 1, 6 are radius respectively mean value of .

Indeed with Hadamard Matrix.

Then , in and in

- Case: its solution is related to since .

6. Quantum Circuit

Let’s denote the flows defined by the ring and fields by , which are kind of transformers or generators or just logical gates.

It follows that the resulting block gates for input and output are equal

In quantum circuits [10] Hadamard gates are represented by:

The transformation is reversible with harmonic motion, due to the orbital periodic of the particles. Since it is a flow of particles we can then introduce the notion of quantum circuit in which a computation is a sequence of quantum gates with a reversible transformation, that imply the inverse quantum Fourier transform. If we consider the qubits of the input equal to and for the output the qubits equal to and the qubits for the logical gates K = 2 in the middle, then the resulting circuit operates with: qubits with block length , and a message length equals to: , with a minimum distance that correspond to . This linear code over a binary alphabet is a subspace of dim 6 of length 64 generated through the reversible transformation of fields.

. For a 6 qubits reversible gates data in the space which consists of strings of 0 and 1, the input and output each consists of . This transformation results from a transmission of 16 strings into 64 strings. The architecture of the universe is based on a quantum circuit path reversible consisting of a transmission of 16 strings into 64 strings for the automata language, those strings are represented by vertices and edges in graph theory (e.g. Figure 17).

This transformation is a consequence of the Hadamard’s matrix order since:

with 16 elements and with 64 elements.

Numerically this transformation describes the path of the composite and the prime numbers relatively to their super-partners “mirror image” since in the set the total number of:

- (primes/composites) ; - (composites /composites) = 24 /24.

Then for each reversible path we have: in total. While inside the system the:

- (primes/primes) generates the gravity, the particles will commute or anti-commute to form the axis:

(Kernel) related to the field .

6.1. Gates / Wormholes

This map is connected to Pascal’s triangle by:

number of gates (Hadamard and controlled phase gates), though for it corresponds to:

gates (total of wormholes in the multi-verse) This property coincide exactly with the total of 15 composites orbiting around the 7 primes coordinates of the point of the sphere (e.g. Paragraph 7.3).

Conclusion: The architecture of the universe is based from a quantum data information circuit with the resulting path of 6 qubits for the automate language. The universe is generated through a super-computer that codes, decodes and corrects code errors based on the Hadamard operator. Do we live in a real simulated life?

Figure 17. line in Pascal’s triangle

Figure 18. Gates configuration

6.2. Quantum Harmonic Oscillations

The Schrödinger equation for quantum harmonic oscillations is a wave function related to Pascal’s triangle: since is connected to Hermite polynomials [11] by the relation:

Table 2. Hermite polynomial

We notice that the coefficients of the Hermite polynomial are related to and (e.g. Table 2)

Although we know from Pascal’s triangle the power is giving from the binomial theorem:

, and by giving a discrete number p, we have:

with ,

then:

Related to a discrete probability distribution of a random variable X, characterized by a probability mass function, also known by the normalization condition for a wave function .

Where

7. Orbital of Primes, Composites and Palindromes

To prove the orbital of the composites around the primes and palindromes, we need first to locate the primes of , for this reason we need to find a sequence or a function that maps all the 25 primes!

Since each element of for is connected to per +1 or +4 and each element of is connected to the other element of per +3.

We have 4 = 1+3 and with respect to the orientation we would follow this path: 4 + (-1) = 3.

Anti-clockwise: based on the circles and (e.g. Figure 20). Since the pairs (1, 6) and (2, 3) generates those circles. The opposite modules verify well: and related to the flow .

Let’s then define the following sequences defined by: composites of the perfect number 6.

;

;

7.1. The Salahdin Daouairi’s Conjecture

Notice: 1, 13, 37 and73 are the only star numbers of with the form: .

We have or . The diagonal of the sequences is in the form of 6k, 6k+1, 6k+2 and 6k+3. With the value , the sequence or the function maps all the primes of . See Fig.19

Figure 19. Primes Configuration

Figure 20. Primes orbital

Connection of the Sequences:

It shows that when we connect the 4 sequences, the trajectory of the composites spins around the primes and the trajectory of the primes orbits around the trajectory of .

Figure 21. wisted torus orbital

And when we project our sequences to the infinite and by using modulo 99 we get a transformations (e.g. Figure 22) which yield to a twisted torus orbital where each number reconnect with its mirror image.

reconnect with reconnect with reconnect with

Or for represent the vertices of an hexagram, with period equals to 3 relatively to the polynomial Since: in base modulo 9 we have (e.g. Figure 21).

Figure 22. Orbital of integers

7.2. Composites Configuration: Shell / Electrons → (Prime / Composites)

The Set corresponds to 25 primes, 73 composites and number 1.

7 consecutive prime’s coordinates of the point , then the 18 remaining primes left are orbiting inside . By decomposing the remaining number of the primes into three

The 73 composites = 72 composites + Number 6

= 24x3 composites + Number 6

Now each group of 6 primes corresponds to 24 composites, and the remaining number 6 which it will combine with the number 1, therefore the pair (1, 6) oscillates to form the circle . If you draw a Tetrahedral and place in its base for each of its three vertices the correspondent pair of 6/24 which correspond to the number of primes respectively number of composites, then place in the middle of the tetrahedral the number 6, and connect it to the upper vertex (number1). Number1 and number 6 are connected.

7.3. Simplification (e.g. Figure 24)

Since in the middle we have operated through a circle with radius r with including inside its super-partner the circle with radius R, with (2 primes in the middle). Then one of the vertices of the previous triangle must have only 4 primes in stay of 6 primes.

So now our prime distributions are: 2-6-6-4 follow a spiral of Fibonacci (e.g. Figure 23)

From the equation: . We have:

With total: . As a result! The elements are on the circle with the direction opposite to the circle . The point of the sphere rotates with the same direction of the sphere . And by using the notion of packing spheres, those 7 primes related to the point M are connected to 15 composites, while the rest of 58 composites are connected to the 18 primes.Interpretation:

From those equations we deduct that the number of composite numbers at the bounded area of are 15 composites and the number of composite numbers along the torus and the circle are 58 composites, which means 15 composite numbers orbiting around 7 prime numbers: 7/15 and 58 composites orbiting around the 18 remaining primes inside the torus 18/58.

7.4. Distribution of the 58 Composite Numbers in the Torus Embedded in the Sphere

Let’s denote by (primes= shells) and

(composites=electrons) orbiting around shells.

From the property 4: Each element of for is connected to per d=+1, d=+4 and each element of is connected to the other element of per d=+3: in the order (1, 4, and 3) → (1 composite, 4 composites, 3 composites).

- The two shells 37 and 73 correspond to the pair (1, 6)

- The first 6 primes/shells, each shell corresponds to 4 composites, while the last shell corresponds to 3 composites. With total composites composites

Figure 23. Distribution of integers / Graph of time

Note 2664 imply also direction of shell groups 2 shell toward 6 shells then next 6 shells then 4 shells. “spiral trajectory”

- The second 6 primes with similar distribution, for a total also of 23 composites.

- The last 4 primes remaining will correspond to . 1^st shell corresponds to 0 composites, 2^nd shell corresponds to 4 composites and 3rd shell corresponds to 4 composites, while the 4^th Shell corresponds to 3 composites.

Figure 24. Classification of the integers

8. Dynamical System of Time

Let’s introduce the operator S for the discrete system:

are discrete numbers, with parameters (Primes, Composites, Palindromes)

Let’s denote by and consecutive elements of

We have in the form of equation that represents respectively: Distance, speed and time. Let’ define by: and by so Triangular number , where related to the square lattice while the vector field related to U(1).

Time is related to the square lattice. When the pair (1, 6) oscillates 72 times, which corresponds to 72 composites that orbit around the primes, we have then:

The two spheres have the same angular momentum, since 666=18x37.

Then: that proves is related to time and 72 is the period.

8.1. Configuration of Prime Shell (Energy Level) and Composite Electron (Particles)

Notice: the case of prime/prime: when a prime contains composites then its super-partner contains 0 composites. The last shell of each group orbits in the opposite direction to the other shells.

8.2. Arithmetic Progression / Triangular Numbers

- For the first group: Let’s denote by: the composites, for the first 5 shells with palindromes with where Located on the shell has opposite direction to the 5 shells then when the counter 37 maps one tour, the first element moves toward and again with another turn of the shell 37, moves to , this operation continue till . Each time moves to the composite moves to three times (e.g. Figure 23). We have then the arithmetic progression: with the condition .

- Apply the same method for the second group of 6 shells:

- Let’s apply the same method also for the third Group of 4 shells:

8.3. Time

8.4. Origin of Time

The generator of time which is the circle corresponds for each tour to a move of a composite number, which maps a unity of time. corresponds to the oscillation of are component of G (Gravity), time is connected to gravity. Time depends on the gravity and gravity governs the time. While the circle unity spins (that when the circle 37 maps the 37 points, with 36 equidistant paths). Since 72 represents the number of composites in the set, which is equals to the period orbital. The graviton is the counter of Time, thus Time is generated by the gravity.

Chemical interpretation: (1, 6) corresponds to (Hydrogen, Carbone) that generates time.

9. Speed of Light

* 92 vertices correspond to 91 edges

* The remaining 7 primes coordinate of the point M of

orbit with opposite direction to and

Let’s denote by the string of elements of where .

We already know the elements coordinates of the point M of the sphere orbit in the opposite direction of the remaining numbers of (with total 92 numbers).

The maximum speed for the first element of to reach the last element of is then determined by the maximum distance traveled from with a minimum length of time which corresponds to:

(Consider 1second for each of the 5 groups represented in the figure 23)

Figure 25. Geometrical representation of the speed maximum

The maximum distance traveled is reached when all the 18 circles are equals with the highest radius! Or the radius of each of the circles is less than to the one of the Sphere by , where R is the radius of the Sphere . Then the perimeter maximum of each circle equals . There are 92 vertices inside the sphere (with 91 edges or paths) and 7 vertices are coordinates of the point M of the sphere with 7 paths, each path corresponds to 18 circles turns. While the 7 vertices (primes) orbit with the sphere, the 18 circles will describe 18x 7 turns.

Conclusion:

The distance maximum equals to:

The time minimum corresponds to: (5 groups: sec, mn, hr, day, and week)

Then the speed maximum corresponds to that of an electron:

10. Dark Matter/Multi-verse

From the following sequences:

(Fig.6.A) Distribution of 666

Then

And

Or from the following sum:

we have the difference:

Let’s denote by: and the three types of universes, then the equation shows that 5 universes already formed, and a pair of paralleled universe is under construction (inflation), among the pair is our universe which is under expansion. The total of universes then is 7.

If we denote by , it shows the universes are charged +/- , which introduces the phenomena of electromagnetic between universes. The element 6 is the only composite number that reacts with the graviton related to the circle unity, then the element 6 could be the neutrino, the weakly interacting massive particle (WIMP) related to the weak force. Or from the 2 following equations:

And

The number 288 represents dark matter Q, and (1,6) is the components of gravity G=37.

The equation shows that dark matter provides the elements 1 and 6 to hold the gravity, without dark matter, gravity will collapse, although universes are connected through the gravity. The dark matter generates matter through the axis which is the backbone of the multi-verse that controls the space/time and provides also the first elements to create our universe. As a result, those equations describe a deep relation and show the connection between “Time & Gravity”, “Dark matter & Space” and “Dark matter & Gravity”.

Figure 26. Transformation of Dark Matter

Note the number 24 represents the first elements. (e.g. Paragraph 12).

11. Dimension: 6 Universes and Space/Time

It shows that the dimension of the hyper-Sphere 666 equals to: Dim or the orbital of the elements of emmbeded in the hyper-Sphere define an extra of dimensions which is our curved space / time. Our space/time corresponds to a 5-Simplex polytope, then:

The total dimension of the Multi-verse is equal to:

Dim U=5+6=11.

This proves the extra 6 dimensions in the string theory, which add up from the extra universes that govern the dark energy, although , time with its screw dynamical, curls around the space, and travel sideways.

Figure 27. Multi-verse representation

12. First Elements

I have to study the origin of the elements through the reaction:

For each of this sequence the composite decomposition to primes are:

Since 1 and 6 are the predominant numbers and since 6= 2x3 we will be eliminating then

The decomposition of the composites into power primes leads to classify the first primes:

The primes which are in powers are:

In base modulo 9: the powers of 3 and 6 intercept at the point 9:

Conclusion: the first elements are (1, 2, 6, 7 and 8) which correspond respectively to the following particles of the periodic table: Hydrogen, Helium, Carbone, Nitrogen and Oxygen.

Those 5 elements are connected from the equations:

If we consider the reaction between the elements:

Then the difference between the States is:

That defines geometrically the shape of a torus: (space/ time). Space and time are made from matter (first elements).

13. Big Bang before and after / Dark Matter

Figure 28. Big Bang formulation

From the previous formulas of numerical equations, we can give detailed explanations on how the system or the Multi-verse was formed! Well in the beginning it starts with dark matter, since it provides first elements, controls the gravity, which through it, generates time, connects the multi-universe, forms our universe through the first elements and generates matter through the axis which is the backbone of the multi-verse. Thus the singularity 55 of the black hole belongs to the axis

From the equation:

Dark matter = through the components of the gravity.

We have:

dark matter first elements + gravity space / time.

Time won’t exist if there is no gravity, and gravity won’t exist if there is no dark matter.

The first elements responsible for the Big Bang were continuously vibrating, with a harmonic oscillation, if they were stable, it won’t be any reaction, means the time existed with the existence of the particles, among those particles the graviton which is the counter. Since Dark matter dark matter originates from first elements 1, 2 and 6 (Hydrogen, Helium and Carbone).

13.1. Inventory of the Universe

From this string we can deduct that the dark energy represents the 5 universes surrounding a pair of paralleled universe which is under expansion (inflation), among this pair is our universe.

Figure 29. String of Universes

From the string we can view the inventory proportion of the total of the Multi-verse:

Our homogenous system is formed from a total of matter which represents 99%.

The dark energy is represented by the 5 universes with a total matter proportion: . The percentage then is equal to: (Dark energy).

Dark matter exists everywhere in our universe, connects universes through gravity and provides ordinary matter to form our universe, with a total matter proportion:

The percentage then equals to: (Dark Matter).

Then the proportion of dark matter equals to 288, responsible for providing ordinary matter with a percentage equals to: (Ordinary Matter).

Figure 30. Inventory of the universe

14. Dynamical System of the Multi-verse

;;;;;

Equation (1) become:

Then:

If we denote by S the string of the 7 universes: [3].

We recognize this finite sequence as a string concatenation of the form

14.1. Chirality of the Trans-Palindromes

The vertices of the lattice form edges or strings “loops”, those strings have the form of XY or YX with asymmetric property and represented by trans-palindrome numbers where X and Y are elements of .

14.2. Algorithm of Strings with the Concept of Parity or Chiral Symmetry and Spiral of Fibonacci

Giving two elements X and Y orbiting with an oscillation harmonic describing the following algorithm YXXY, XYYX, related to the quadratic equation with the asymmetry property, defined in an oriented space. (Which is seen in nature, example: a pair of human’s feet or hands)

Figure 31. Trefoil Knot

We recognize this finite sequence as a string concatenation of alphabets X, Y with length 4.

For this regular language or expression, let’s define the product by composing letters of the string: Our string then has the form of: The alphabets orbit with an oscillation harmonic following the regular language which is a combination of the two disjoints regular expressions , with a monoid structure, where the union is represented by +, the concatenation by the product and by using the Kleene’s Star closure operation for this algorithm of strings, where defined by over a disc unity [3]. Then we have the equation (I).

Its fixed points are solutions of the Alexander polynomial that define a Trefoil Knot:

Or corresponds that yield to and by replacing z by Yields to: .

By the method of comparing the coefficients of .

Maclaurin series with undetermined coefficients

The generating function for (Fibonacci sequence):

The convergence radius of this series then is equal to:

Table 3. String Concatenation

Note:

The equation of this dynamical system:

and Fibonacci sequence.

Let’s denote and then

The characteristic Equation:

when:

Note with matrix respectively:

Thus only: inverse of are modular since its determinants equal to 1.

Eigen-values then are: and Eigenvectors

If , with the initial data: Then

Solution of

Let’s denote by and then , we retrieve then the Hadamard Transformation by: with, that generates

Note the equation has solution in with p prime, only if: is a square

Then the smallest prime verify that is

14.3. Equation of Strings in Knot Theory / Conway Polynomial

The two strings and are chiral symmetric and intercept to form a closed string.

Three possibilities of interception: 0 knot, 1 knot, or a link of n knots.

- 0 knot which corresponds to (prime/prime) where and have their greatest common factor equal to 1, and are not decomposable in ℤ. In this case we have two closed strings with opposite directions with shapes of circles. “case of the gravity”

- For a link with 1 knot which corresponds to (Prime/Composite), the Greatest Common Factor

equals to 1, and only is decomposable, this case corresponds to the composites orbiting around the primes with opposite and orthogonal direction to it “case of electromagnetism”

- For the third possibility we will consider a torus link formed from the 2 strings intercepting n times

(n=6 smallest period of the sequences in Pascal’s triangle) which corresponds to (composite/composite), for this reason, let’s introduce the Conway Polynomial and show how it is related also to Pascal’s Triangle, where the 2 strings are twisted 6 times with characteristic 2 (modulo 2 with n odd or even), then: either it’s a link or a knot, by definition the Conway polynomial is giving by the equation: [1]

We recognize the Pascal’s triangle patterns, where Fibonacci numbers represent the sum of the coefficients of the polynomials for z=1. Since for n=6 the polynomial is of deg 5, the Space then is of dimension 5, which coincide also with 5-Simplex dimension.

The composites/composites elements of have the property of longitudinal orbit along the torus that describes a spiral of Fibonacci trajectory.

Since the torus links depends on the parity of n, knot or unknot with:

Introducing the Fibonacci sequence:

Where: and with the formula:

Since (n-1) and (n-2) are consecutive with different parity! Leads to: and

Interpretation: The notion of parity (odd or even) is an indication of the commutation and the anti-commutation of the bosons and fermions over the Super Algebra . While the particles (Long range) including the axis “Kernel” describe the longitudinal orbital, and the particles and (Short range) describe the latitudinal orbital, thus the longitudinal “strong force” and the latitudinal orbits define the geometrical shape of the space/time, which is a twisted hexagonal torus (Solenoid) that describes a spiral of Fibonacci trajectory that converges into the golden ratio .

Table 4. Conway polynomial and Pascal’s triangle patterns

15. Expansion Power in Base Modulo 9

The reason to study the power of the numbers is to determine its periodicity and uniformity to reduce the system. The notion of cardinality is also important to describe the state level of the system and its dimension.

Each composite number c is a product of a finite prime numbers with power

Let’s denote by and The power expansion of the elements in base modulo 9 are:

converges toward 1(fixed point).

and of period

and Converge toward 9.

and Periodic with period

Periodic with period. Then the:

If we project the elements of each set on a circle, we notice that: The powers of the numbers orbit with a harmonic oscillation. The powers of 2 and 5 belong to and are equal but they orbit in opposite direction, while the power of 4 and 7 belong to and are equal also but orbit in opposite direction.

and . The powers of 3 and 6 intercept at the point 9:

while: and

For and we have: and while and And from the equations (8) & (9) we have: and

Note are simply triangular numbers and powers of 2 modulo 9 of the Pascal’s triangle. The powers of numbers 2,4,8,7,5 are harmonic and for those numbers 3,6,9 are non harmonic since it vanish toward the singularity. While 1 is the fixed point. Time degenerates near singularity generated by over the field and . (e.g. Figure 32)

Figure 32. Singularity and periodicity

Figure 33. Geometrical shape of the universe orbital

Let’s denote then by with then with lead to the following values: , , the fixed points seen before of the modular form: are solutions of the equation: . Those points are points of a Torus, also points of a hexagram, with complex representation: .

15.1. Generating Function f over Disc Unity

, for and its conjugate is Mean and Radius of related to By using Hadamard matrix H: . Although the function involves the mobius transformation with the property to generate inverse circles (loops and strings) and preserves angles of the form: With.

Then . Transformation that corresponds to the special linear group SL (2, ℝ), a simple real Lie group defined by: [6].

This transformation also corresponds to the modular group of lineaire transformations of the upper half of the complex plane: used in the hyperbolic geometry, which is known by Poincare half plane model in non Euclidean geometry for the curved metric. The modular group is isomorphic to the projective special linear group , while the modular group and

a holomorphic function, symplectic group.

With modular form over disc unity, we have then: .

Then the system:

Generated from the polynomial: .

Let’s denote by . Then the equation defined by: with and , leads to the cubic form: then

Or simply by: where:

Which yields to the famous elliptic curve: symmetric of the elliptic curve: related to the Eichler’s generating function: where the number of solutions over a finite field is related to the coefficients of

Parametric System: by changing the variable x over the disc unity

Thus

The very known modular form for its fundamental domain and its cusps, the strings and are homotopic, the function is homeomorphic and tends to form a trefoil knot since

f is analytic thus then modular form with weight k that have the property of an automorphic form. This analytical function has the expansion to infinite:

where z is in the upper half plane, indicate the extension of Galois representation to analysis “waveform” for a harmonic motion.

Let’s denote by:

Then and

while

With the following matrices: or are involutive while with solutions located on a hexagonal lattice points of an even polynomial then cyclic if if relatively to the cyclotomic field for the splitting field of the cyclotomic polynomial P, the Galois extension of the field of rational numbers we have then which is a translation. The fixed point: and defines the group cyclic of order 6. Solutions of the polynomial: are points of a hexagram, or each points of this hexagonal lattice have 5 roots, which yields to a total of 6 vertices 5=30 roots, the product corresponds to the icosahydral group. The dynamical of the vertices is related then to and . Since g is even and and for symmetry reason we simply study the function: or , the polynomial is irreducible. The eigen-values of an element M of verify the characteristic polynomial: where is then a rotation, “Elliptic curve” while is the fixed point, and by using the generators S and ST we have The representation of the modular group Г of this transformation is then isomorphic to , product of two cyclic groups . It’s dynamical is represented then by the product: and where are simply centers of .

15.2. Convergence of the Elliptic Curve

Let’s introduce a symmetrical map g with rotation “reflexion” through Euler formulate: defined by. Since are solutions of

The 3 maps are involutives: and that define loops, consider:

Then per iteration

The condition h is involutive with leads to:

Or the iteration converges toward the fixed point:

Converges then toward the Golden Ratio .

Conclusion: The dynamical of the points of the elliptic curve describes a spiral of Fibonacci.

Note: with with we have from Cassini’s identity, modular if n even. Though n=2p, is related to , and since every matrix in the modular group defines a modular knot, then the iteration define a succession of knot and unknot. Example: are modular knots. In his paper Etienne Ghyes did mention those modular knots and Lorentz knots, except he did not realize that are related to Fibonacci sequence where the determinant is related to Cassini’s identity “Link of knots” [7]. The matrix has also the same properties since

15.3. System’s Symmetry

The reflexion group is spherical of finite type since . It describes the rotational triangle group (2, 3, 5) and corresponds to the icosahedral group with order 60, determined by 5 groups of rotation: identity, with 5 conjugacy classes. Or the Galois group of the field extensions: for n=2 ,3 and 5 is isomorphic to the multiplicative group of units of the rings: , with n=2, 3, 5. This transformation defines a tessellation of the hyperbolic plane by hyperbolic triangles. Or is finitely generated Abelian (a ℤ-module), since is an algebraic integer of degree 2 over solution of the polynomial is a sub-ring of the quadratic field , the unique non trivial Galois automorphism of the real quadratic field . The extension is a quadratic field of rational numbers that corresponds to the cyclotomic field. This ring defines the 5 fold rotational symmetry group used in Penrose tiling relatively to the group cyclic with angle, of order 60 that corresponds to an icosahydron isomorphic to alternating group A5.

15.4. The Icosahedron and the Golden Ratio

Let’s denote by the coordinate of the vertices of a cube circumscribed to an icosahedron, then the coordinates of the vertices of the inscribed isocahedron in a cartesian coordinate system are giving by:

, with: (where is the Golden Ratio)

For the coordinates are: [4]

By using the Welsh bound method for an optimal equiangular line packing based on 5-dimensional simplex, “in our case 6 lines” through the inner product, by a 6x6 square input symmetric matrix M called “conference matrix” with trace equals zero [5].

This matrix has an interesting form; all rows are orthogonal with norms equal , a combination of Pauli matrices and Hadamard matrix H. Let’s denote by + number +1 and by – number -1

Since: and. Where with Eigen-values equal , and . All lines intersecting pairwise at a common acute angle , with the coordinates of its vertices related to the golden ratio. The volume of an icosahedron occupies less volume of a circumscribed sphere comparative to a dodecahedron. As a result! Less energy is used for an icosahedron shape than a dodecahedron.

15.5. Biological Interpretation

This dynamical and geometrical shape is seen also in the icosahedron geometrical shape of the viruses and dynamical of the DNA along the torus through the icosahedron.

16. Conclusions

“Summary of the Equation”

With simple tools of ingenuity, pencil and paper, we discovered the equation for the Theory of Everything, and through it we learned that the architecture of the universe is based on the structure of discrete numbers which reveal the perfection, elegance, and beauty of the universe. We disclosed the solid foundation of the universe’s structure and design through its mathematical framework. With this original Theory of Everything, we have created a new paradigm which finally divulges the secret reality behind the physical properties of our universe.

We live in an 11-dimensional universe, inside a mathematical equation. This mathematical equation will guide us and will lead us in the exploration and discovery of wonders that have never before been imagined. The “Equation of Everything” showed us how the entire system: S= {Space/Time/Matter/Energy/Gravity/Electromagnetism} is homogeneous, unified and connected, explained the most important fundamental physical theories, and revealed the hidden connection between quantum theory and the macro-system which relies on space/time/gravity (where the theory of quantum gravity is finally reached).

We finally through this simple equation have at last disclosed the enshrouded secrets of Time! We revealed Time’s origin and its properties, and we demonstrated how a time machine could be generated from gravity and electromagnetism. We also learned about “dark energy” and “dark matter” neither of which could be described experimentally). While dark matter (as the backbone of the multi-verse) attracts, connects the multi-verse, and controls galaxies through gravity, dark energy acts as the opposite “epellant force” that results from the dynamics of the multi-verse…a new breed of dynamics that possess the power to induce inflation and the expansion of our universe.

We learned in the quantum field, about the spin of a particle and its electro-dynamic behavior which determines the path or the “quantum circuits” for the system through an “automata language” program that create woven networks which constitute the “fabric” of space. Signals of radiation and energy are sent through this fabric “lattice” along a cyclic transformation.

The Equation also gave us the means by which to perceive the geometrical shape of the multi-verse which will allow us to accurately determine the rate of universal entropy. We revealed the origin of the “Big- Bang” and how the universe crystallized and coalesced during those first few micro-seconds of existence. The Equation disclosed the enormity of the universe and allowed us to envision and contemplate the universe’s colossal inventory of matter and energy.

The equation that represents The Theory of Everything is an application of quantum gravity and String Theory from the notion of super-symmetry with the representation of spinors through the theory of harmonic motion, which is based on parity or “chiral symmetry” that appears to be reflected and seen in nature.

The Equation will disclose in the field of biology, an important relationship among DNA, quantum entanglement and electromagnetism due to DNA teleportation, structure and shape. A mathematical model based on electromagnetism embedded with the properties of quantum entanglement will provide the cure for many illnesses and will map out the shortest route for the discovery of new methods and techniques with which to finally defeat and eradicate the viruses that have been plaguing humankind for millennia. The equation will lead us also for the classification and the connection of the elements of the table periodic.