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Hybrid Time Theory: “Euler’s Formula” and the “Phi-Algorithm”

Stephen H. Jarvis.

http://orcid.org/0000-0003-3869-7694 (ORCiD)

EQUUS AEROSPACE PTY LTD

Web: www.equusspace.com

email: shj@equusspace.com

Abstract: Here a proposed time-equation conforms to a spherical wavefront of propagation in space in deriving the

value of π. This will be achieved by first addressing how a time-equation should be most precisely and efficiently

used to explain the notion of space and time in detailing a just as efficient and precise use of numbers describing the

process of linking a 0 reference in space to an infinite reference. Then, that description will uphold the findings of

papers 1-14 [1-14] in this series of papers, most importantly the final premise reached in paper 14 regarding natural

radioactive decay. There it shall be explained exists an associated equation for time, not explicitly the proposed phi-

algorithm for time, yet an associated algorithm of its own explained in Euler’s formula.

Keywords: time; space; golden ratio; pi; phi; Euler’s formula; phi-algorithm

1. Introduction

In this series of papers on the φ-algorithm for time [1-14], the concept of time as the φ-algorithm

has been the core focus of topic. Through the development of the papers, the idea of the φ-algorithm for

time seeking to define has resulted in the development of a vast field of ideas covering what is perceived

and measured of the natural physical world, containing equations that fit all of the thus-far tackled

phenomenal data regarding the field forces for light, mass, and energy and associated constants thereof.

These results took shape from the initial premise of defining space as a “0” 3-d construct associated to

time, “time” as a concept that is tagged to a basic logical construct of consciousness, namely the features

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of time-before, time-now, and time-after, and how that process of time from time-before to time-after can

fit with the basic feature of 3-d 0-space in time-now.

It was initially proposed that time would propagate from any point in 0-space as a spherical

wavefront at a fixed rate. Now, in this paper, that assumption will be fully addressed, namely the

assumption of time seeking to conform to a spherical wavefront and thus to the notion of the value of .

This will be done by first addressing how words should be most precisely and efficiently be used to explain

the notion of space and time in detailing a just as efficient and precise use of numbers describing the

process of linking a zero (0) reference in space to an infinite (ꝏ) reference. Then that description will

uphold the findings of papers 1-14 [1-14], most importantly the final premise reached in paper 14 [14]

regarding the feature of and associated process of natural radioactive decay as per Euler’s

formula [15]. There, it shall be explained exists an associated equation for time, not explicitly the φ-

algorithm for time, yet an associated algorithm of its own explained in Euler’s formula.

This paper will be structured as follows:

1. Introduction.

2. Explaining space and time with mathematics.

3. How time is currently measured.

4. Hybrid time-theory.

5. Conclusion.

The conclusion reached suggests that Euler’s formula as an equation for time implies a “naught”

future, as a natural process of radioactive decay, whereas the φ-algorithm presents a more worthwhile

steady-state process to time and space, yet a compromise can be reached between those two

determinations for time.

2. Explaining space and time with mathematics

In this chapter the assumption of 0-scalar space and associated time-algorithm seeking “”, the

φ-algorithm, as presented in paper 1 [1] shall be addressed. Here, the basic idea of mathematics shall be

explained, why mathematics represents a type of “logic” for our tendency to “adapt” to reality, and what would

represent a most basic wording for the concepts of space and time. It should be noted that mathematics is a tool that

brings measurement as mathematics to gaze upon physical phenomena.

Science has evolved using mathematics in a fashion that standardises certain features of physical

phenomena, key features, as equations with associated constants. One need not be concerned about the

idea of space and time representing certain measurements as an a-priori. Indeed, space and time are

both unfathomable “until defined” with a mathematical tool of measurement, and thus for the "purpose" of

science, mathematics is employed to discuss their relationship, the relationship between space and time,

ideally as follows.

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2.1 Mathematics

Mathematics represents a broad field of interests, interests such as numerical quantity

as number theory, mathematical structure as algebra, space as geometry, and change as

mathematical analysis, employing with those ingredients certain mathematical models to

formulate conjectures, axioms of choice, through abstraction and logic that are proven true or

false through mathematical proof.

Successful proofs are considered as good models for real phenomena. Through such a

process, the notions of counting, calculation, and measurement, have been applied to the

phenomena of physical objects. Mathematics may nonetheless be purely mathematical without

any design or purpose in mind, yet Mathematics has primarily held its purpose in explaining the

world. In a world that was initially thought of a generally flat, the idea of explaining reality using a

flat surface with drawn lines and angles seemed to be intuitive for the ancients.

From such beginnings, certain "functions" in mathematical congress to explain numbers

and their association to one another become apparent, basic functions-operators such as

addition and subtraction, and even then the concept of what those numbers represent, such as

lines in space or length in time, had to be to be explained "with words", otherwise mathematics

was nothing but a collection of symbols with no literary meaning other than numbers alone with

expressed functions.

2.2 Explaining the diversity of equations, formulas, and theories.

To explain how such diversity has evolved in mathematical equations, formulas, and

theories, it can be considered that such diversity has depended on three key things:

1. Our conscious ability to achieve such, our conscious pixilation of intelligence and

drive.

2. The process of applying our intelligence and drive to what appears to be a never-

ending vastly and unfathomable reality.

3. Our diversity as beings, having a different point of view to each other, and thus

a vast well of unknowns based on the difficulty in sharing a common concept.

Each of such things are primary, and each of such things do not represent a mathematical

process in their entirety. Subsequently, the application of mathematics to observed phenomena

has focused primarily on physical objects in space and their motions with each other, and how

mathematics as numbers with associated structures can be related to space as a type of

geometry with an associated feature of change some would interpret as a process of time’s flow.

How mathematics has been applied to reality has depended primarily on the wording, the actual

transcript of context, for the use of mathematics central to the nature of space, the nature of

objects in space, and the nature of the motion of those objects in space.

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2.3 Wording a mathematical axiom for space

It could be suggested that the concept of space is a fairly routine concept to measure as

a mathematical construct requiring only a few words, only a few literary axioms, yet the concept

of time not so. Two key mathematical features used for space and time that appear to be fairly

routine through scientific definition is that space can be measured with a straight line using three

axes as 3 dimensional space, and time can be measured as a single dimension in regard to that

3d space. A few of the basic tenets of this process include:

- Straight lines are generally used to measure 1d space.

- A circle is measured in relation to 2d space as the curved line drawn as an arc

equidistant from a nominated central point of any nominated line.

Given a circle is very different to a line, , the value for the circumference of a circle with

diameter “1”, is why we need words to primarily describe the mathematics of a circle in regard to

space, to explain the connection between the arc of a radius drawn as a circle in reference to a

line (such as its diameter), and how that is achieved; simply, any key irrational number such as

“” needs words to explain exactly what is happening there, how that irrational number comes

into being.

To define why “” could be an irrational number, a key one, in words, is a good way to

set a standard of use of words to then describe other features of space that could be related to

for instance the idea of time; in describing with words alone, why "is" an irrational number,

one need only ask oneself how and why "" is related to a "line". For instance, take a straight

line, real and rational, determined, say length of arbitrary unit “1”, and then go to the midpoint of

that line and draw an arc around that straight line from the midpoint of that line. The proposal is

that length of the circle around that line can never be the concept of a complete number as the

distance of that line could be. Why? If one suggested that the concept of the distance of that line

the circle arcs around can never be determined, then how can that circle be drawn on such an

undetermined length of line? Furthermore, to draw that circle is to use a geometry related to that

line (diameter) that has no actual relationship to the exactness of the number assigned to the

distance of that line other than a value that (as a number related to that line) is forever incomplete

as a description of a number value, as it can only be, in trying to link the beginning of that line

with the end of that line without being that line. In other words, that line could be at any angle in

reference to the circle. That’s intuitive; and so, if the line is known as a determined length, its

circle can never be properly defined, and thus must be irrational. Therefore, how indeed can a

straight line be in a perfect ratio with a circle if the angle of that line in space can never be

determined owing to the nature itself of the circle, no beginning, and no end? The question is how

such improbability of exact definition manifests itself in reality.

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ꝏ

ꝏ

- ꝏ

- ꝏ

A

B

How does this look therefore on paper in the form of a mathematical axiom for space,

with the notion of time being the variable seeking to perfect , as though time is a type of endless

algorithm forever trying to reach the perfect value of ? Consider a total “1” length of a line “A”

that could represent “any” part of an infinite region of 2-d space around a central “0” point, as

follows (figure 1):

Here the blue dotted line “B” is the value “”. Yet what indeed is the value “”? Can it be

represented as a whole number, a fraction even? One thing that is certain is that the circle never

gets to “0”, never breaches the “0” point.

Therefore, as an arbitrary condition of definition here, let it be suggested that in regard

to space for the circle,

represents that “non-0” concept for the circle in that the circle as a type

of curved line in not skewing “0” would somehow constantly “approach” zero from a

(radius)

reference, never meeting it though, as per equation 1.

(1.)

In other words, the “reference” of the blue line “A” represents a unit vale per anywhere in

ꝏ “as” what would trace a “circle” if “0” is not being used as the reference for that line. It is just a

statement that dispels the notion of “0” and replaces it with the idea of

for the idea of the circle.

Yet how can ꝏ be defined to give substance to this reference for the circle?

Let it now be suggested that to define this circle one must use an increasing denominator

value from the reference of line A as a fraction central to “0”, in approaching “0” from a

and

value in order to define the “0” reference; more correctly, in approaching the “0” reference,

the length of the circle as an exact number would represent a number not expressed by a perfect

single fraction given ꝏ can never be defined, yet a series of fractions that would employ the use

Figure 1: here, A is a straight line of length

“1” in any infinite region of 2d space effected

by the axes x and y around an arbitrarily

reference point “0” such that from that “0”

point the line extends a length of +½ and –

½ from that “0” reference. “B” is the value

of as the arc around the central point “0”

radius ½.

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of a denominator of the fraction extending to ꝏ through a process of subtraction and addition

around a “0” reference (as technically “0” is being approached, yet never reached, as what the

“circle” wold best represent relevant to this line “A”), as per equation 2:

(2.)

The problem there though is that any fraction as a factor of

cannot be used, as

is

integral to the length of each axis for line A (relevant to “0”) “of the line” being used, and

therefore “unique” numbers NOT integral to

thus must be used. Therefore, the process

would become as equation 3:

(3.)

Why is subtraction and addition used in this manner? The point here is being central to

while approaching the “0” reference, and thus what must be a negative and positive scale around

the “0” reference, the first step clearly being a negative step from the basic “1” line reference

(diameter of the circle), the step following that a positive step, the step after that a negative step,

and so on and so forth. In short, the idea of “0” is best explained central to

by this definition for

the circle.

Thus, we start with 1 as the overall length of the line, and then seek to determine how to

define the circle as a concept that would “approach” a “0” reference therewith, to create a process

of balancing subtraction and addition central to this “0” reference of an overall “1” line, one

length to the

length meeting at a “0” point, from

to

. Once again note that any

factor of

from 1 to ꝏ cannot be used in this sequence owing to

already representing the scale

of each axis in use for line A, as a unique scale is needed “from” that

scale all the way to ꝏ

for the circle.

Yet the next question is, what value, what fraction, of is being calculated through

this process? Is “” being calculated whole or a fraction of “? The value of being calculated

can only be a factor of the axes being used, and here this is as a progression based on one

positive axis of length

and one negative axis of length

, and thus a factor of

. Thus, equation

3 must become as equation 4:

(4.)

Note also that ꝏ is a concept that would exist by default as a very large number, and

thus for the most accurate value for

to be reached this series of fractions must extent to include

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a denominator approaching ꝏ, in successfully demonstrating the reference for the circle of

;

simply,

is the value reached that joins the ends of each axis from “0” to a value of

as the radius

around “0”, as it only can be, as this is not a direct calculation of the ends of the axes together,

yet the value held in the context of ꝏ. The next real question is, “what is the implication of this

curve?”. Is it a feature of space or a feature of time? The thinking is that if space has already

been measured as the straight-line axes incorporating line “A”, space as the “0” construct

anywhere and everywhere, then the value of the curve () in the context of ꝏ would be relevant

to that other fundamental feature of reality, “time”, which shall be discussed shortly.

Nonetheless, this equation is an actual confirmed equation for , as reached through

different axioms of definition as per the work of Gottfried Wilhelm Leibniz [16] and Madhava of

Sangamagrama [17]. Here though, has been reached by this equation in the context of defining

the hypothetical concepts of “0” and ꚙ using straight lines and a circle for 2d space, which can

then be applied to a 3d grid of space. Note that this is not a process of rounding something off

using an infinite progression, this is quite the opposite, this is accepting the nature of what is

being defined and why it is being defined in such a manner. The question though is, “how can

this algorithm as space represent a function of time?”. It’s fine to define this concept of space as

, yet what about defining using a function of time? That has been the quest in paper 1-14 [1-

14] using the φ-algorithm for time, and to demonstrate its worth by deriving known feature of time

and space.

In short, if space is associated to”0”, time is associated to the value of a circle, to , the

emphasis of the preceding papers [1-14]. This is why it was considered very intuitive to use this

idea for the wave-function of light in this series of papers [1-14], namely time as the wave-function

seeking to define “”. The implication here is that if 3d-0 space can be determined with lines

exactly, as an ideal scenario, then light cannot and should not be exactly determinable other than

seeking to define .

2.4 WORDING A MATHEMATICAL AXIOM FOR TIME

Here the idea of time is given the quality of not just our conscious ability (time-before,

time-now, and time-after) as described per paper 1 ([1]: p1-6) (the φ-algorithm for time), yet a

property of an axis associated to space, as an analogue of a sinusoidal wave in space along an

axis whose aim is to properly “define” , as per paper 2 ([2]: p4-12). The quest of that paper and

subsequent papers was to determine how that function for time, that wave-function, would

represent what would be “perceived” of reality. And so each paper built upon that basic wave-

function for time in space, driving the basic features of such an association of time with space,

time “as” the consciousness-related algorithm of time in space, as per what would be the most

logical thing to perceive of reality using that concept of time, that golden ratio signature of time,

based on the basic conscious notions we have for time, namely time-before, time-now, and time-

after. That theory then led to where we are now, using those 14 papers [1-14] to then take a

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general and complete look at what has been achieved and perhaps why the scientific community

has yet to cotton-on with such a theory.

The general criticism of contemporary physics in light of the φ-algorithm for time is

contemporary physics’ use of inertia as a concept of cause and effect as opposed to the natural

cause and effect flow of time of time-before to time-after via time-now, and how such a concept

has limited contemporary physics theory, preventing contemporary physics theory from properly

understanding the nature of light in “pure space” and why there is such a thing as the redshift of

light [13], and how a correct understanding of light in space through a correct understanding of

the axiom of time and space can solve the cosmological constant problem [14]. Yet inertia is not

the fundamental problem for contemporary physics. There is something more fundamental that

seems to be the over-arching problem for contemporary physics theory, and that is how

contemporary physics theory measures the concept of time. The only way to highlight this feature

of how time is measured using inertia theory, the basis of Newton and Einstein’s theories of mass

in space with time, and how ineffective it is, is to make the necessary comparison to the φ-

algorithm for time theory, that equation, with a few key highlights,

3. How time is currently measured

As indicated by the successful results of papers 1-14 [1-14], as summarized in paper 14 ([14]: p

29-30) by comparison to contemporary science, in not setting a suitable standard for the idea of “0” and

ꝏ and as a notion of space, another process was used as per contemporary science, that being the

exponential grid equations introduced by Euler. The force behind Euler’s formula is equally interesting,

for we should ask ourselves how the idea of using ꝏ in an algorithm became sought after in physics. For

Euler at the time, it was the mathematics of financial wealth, namely compound interest (superannuation),

for is the number linked to exponential growth, a key determinant in financial analysis. In mathematics,

Euler’s formula, or Euler's identity, named after its founder the Swiss mathematician Leonhard Euler, is

as (equation 5):

(5.)

where e is Euler's number [18], the base of natural logarithms, is the imaginary unit, which by definition

satisfies i2 = −1, and is the ratio of the circumference of a circle to its diameter.

Euler’s formula is considered to be a standard of mathematical beauty in demonstrating the

profound connection between the most fundamental numbers in mathematics in the way it does.

Nonetheless, Euler’s formula uses the idea of and as a limiting function with ꝏ, whereas the concept

presented in this paper regarding time as the φ-algorithm uses the idea of 0-1-ꚙ and as a primary

definition for space which is then annexed with “” as a time-algorithm. Nonetheless, for an equation such

as Euler’s to be presented so simply and fundamentally, it certainly must represent a key concept for

reality, for instance, space or time. It does, as a type of analogue for time. And such is indeed the case,

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as the idea of time is measured objectively using Euler’s formula regarding the concept of the time-

constant [19]. Here, Euler’s formula is used to explain the radioactive decay of particles, and does this

with the radioactive decay of the Caesium atom [20] central to the idea of a “half-life” [21]. Yet, one may

ask if time is in fact completely a process of radioactive decay? What are the general applications

therefore of Euler’s formula?

Half-life (t1⁄2) [21], the key application of Euler’s formula, is the time required for a nominated

quantity to reduce itself to half of its initial nominated quantity value. It has applications in two key fields

of study:

• Physics

▪ Ernest Rutherford applied the principle of Euler’s formula as the radioactive decay

of an element, as a half-life, to study the age of rocks through measuring the decay

period of radium to lead-206.

▪ In nuclear physics half-life describes how quickly unstable atoms undergo

radioactive decay (or conversely how long they survive, depending on the choice of

view, half full or half empty); nuclear chain reactions as per a uranium nucleus

undergoing fission produce multiple neutrons that each can be absorbed by adjacent

uranium atoms, causing them also to undergo fission reactions, and thus a runaway

exponential explosion.

▪ Avalanche breakdown is the term given for a dielectric material whereby a free

electron frees up additional electrons as it collides with atoms or molecules of the

dielectric media after becoming sufficiently accelerated by an externally applied

electrical field; the subsequent secondary electrons behave the same way as the

initial free electron. The resulting exponential growth of electrons and ions may

rapidly lead to complete dielectric breakdown of the material

• Biology:

▪ Generally, half-life describes any type of exponential or non-exponential decay; the

biological half-life of drugs and other chemicals in the human body is based on

Euler’s formula.

▪ Converse to the idea of half-life is the idea of “doubling time”, used to describe

biological population growth.

• Studies show that the population of microorganisms in a culture

increases exponentially until an essential nutrient is exhausted,

indicating a constant growth rate.

• So too with the Human population; for instance, the population of the

United States of America is exponentially increasing at an average rate

of one and a half percent a year (1.5%), and thus the doubling time of

the American population is approximately 50 years [22].

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The initial utility of Euler’s formula was in compound interest, yet as is apparent it has been

applied to the idea of a “time-constant” [19] for radio-active decay as the value of the time it takes for

decay to reach

, which technically bases the value of “time” as a type of compound percentage value,

when in fact the concept of time can be something else entirely. How did this happen? It was considered

that the idea of half-life is a “constant” over the lifetime of an exponentially decaying quantity, that radio-

active decay is constant as an accumulative measure of any such quantity.

Such is the contemporary understanding for time as an objective measurement of such processes

that can be relied upon as a standard (as per the radioactive decay of the Caesium atom [20]), time that

uses that process of “decay”, decay of a quantity of an atom no longer being a part of the process of time.

Time though when locked in with space according to the φ-algorithm has a different function as highlighted

in papers 1-14 [1-14]; the issue with exponential growth and exponential decay is one of an atom that is

undergoing a process of exponential growth or decay, radiation, that is no longer a part of the stabilized

form of space and time, yet instead a part of the unhinged and unbound process of time and space, as

highlighted in paper 14 ([14]: p26). Such processes would occur as a manner of disintegration, even

exponential growth, as exponential growth incurs a resource-incursion that takes away from an otherwise

time-correct steady state situation; exponential decay or growth are two sides of the one coin, as even in

a situation of the growth of cellular material, there would be an accountable exponential loss of resources.

In short, as per paper 14 [14], it seems that in the process of time’s flow as the φ-algorithm, there

is an allowance for radioactive decay (as radioactive decay), and thus a distinct allowance for the idea of

Euler’s formula for time and the φ-algorithm for time as one ([14]: p26), held in the one general overall

equation for time of the φ-algorithm. How can such be possible? The answer lies in the code itself, the

presented function, of Euler’s formula according to the φ-algorithm and their relationship to the phi-

quantum wave-function spatial template.

Euler’s formula on the surface is analogous to where and and

, as per equation 6:

(6.)

The idea of is essential to understanding this is a time-equation for particle decay, or quite

simply, a 0-future outcome of particles in regard to energy. The “” component of time as is simply

represented to a power of on a complex number plane, as in a complex number plane.

This is the proposed hybrid time feature, namely , and

, and will be

demonstrated to be essential to a basic understanding of the energy processes for time and space as

timespace, to be proposed in a subsequent paper.

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4. Conclusion

The purpose of this paper has been to present a case for why the φ-algorithm for time seeks to

define “”, and how this happens, to address the assumption of 0-scalar 3d space in paper 1. Yet in doing

so, interesting new conclusions become apparent regarding the current contemporary scientific

measurement process of time, namely via an equation, Euler’s formula [15], that primarily represents a

process of “decay” (or growth, depending on the perspective of interest). In short, if space is associated

to”0”, time is associated to the value of a circle, to , and both are proposed to be to the basic human

perception ability which when reduced to three features of time as time-before to time-now to time-after

brings to bear the φ-algorithm for time, which then brings to bear the phi-quantum wave-function for light

seeking , which then as a process of time in space manifests the theory for time and space and all

associated processes such as electromagnetism, gravity, and mass, to make that work, as per papers 1-

14 [1-14], as summarised in paper 14 ([14]: p29-30). The task ahead is to add further resolution of

understanding to the energy components of these equations for time and space and thence a description

for associated phenomena.

Conflicts of Interest

The author declares no conflicts of interest; this has been an entirely self-funded independent project.

References

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https://www.researchgate.net/publication/328738261_Gravity's_Emergence_from_Electrodynamics,

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