Paper

Utilization in Artificial Ego of Quantum Calculations Basedon Anti-Einstein Field Hypothesis
反アインシュタイン場仮説にもとづく量子計算の人工自我への活用

MITSUYOSHI Shunji

TOMONAGA Kosuke

光吉 俊二

The University of Tokyo, Graduate School of Engineering,
Collaborative Researcher
東京大学大学院工学系研究科 共同研究員

The University of Tokyo, Graduate School of Engineering,
Project Associate Professor
東京大学大学院工学系研究科 特任准教授

HASHIMOTO Masatsugu

朝長 康介

TEI Yuichi

橋本 昌嗣

鄭 雄一

The University of Tokyo, Professor
東京大学大学院工学系研究科・医学系研究科（兼任）教授

Digital Hollywood University, Graduate School,
Visiting Professor
デジタルハリウッド大学大学院 デジタルコンテンツ研究科 客員教授

In this paper, the extended Riemann sphere of the anti-Einstein field hypothesis is used to clarify themechanism by which quantum gates emerge. In addition, this paper shows that artificial nerves exhibiting
emergent phenomena are achievable as an engineering application of said mechanism.
In general, the Bloch spheres and Riemann spheres used for quantum gate spinor representation cannot
handle attribute inversion at poles. However, our proposed extended Riemann sphere is capable of handling
attribute inversion at the poles where Riemann spheres and anti-Riemann spheres are connected. Thus, it isanticipated that emergence will be controllable using quantum gates based on the extended Riemann sphere.
With this in mind, we designed an emergence-imitating artificial nerve model in an extended Riemann sphere,
and implemented and tested the model on a computer. In addition to confirming emergence phenomena in theextended Riemann sphere, we discovered that the behavior of transfer functions, which diverge and become
unmanageable in control engineering, gave rise to various sigmoid functions.
Therefore, we use the extended Riemann sphere as a basis to clarify why quantum gates emerge, and, inconsideration of the results obtained by using the extended Riemann sphere to confirm the actions of theemergence-imitating artificial nerve model, mathematically express the implications for black holes, white holes,
emergence, artificial nerves, and artificial ego.
Keywords: Bloch sphere, Riemann sphere, quantum gate, division by zero, emergence, artificial nerve, artificial
ego, quantum computing

1. Introduction

something will emerge due to its influence”, and published, in

As of June 2021, production of the humanoid communication

DHU JOURNAL in 2019, an extended Riemann sphere thatrobot“Pepper”has been halted for temporary adjustments.

connects at the poles where a Riemann sphere (positive world)First released in 2015, Pepper has remained in production

and an anti-Riemann sphere (anti-world) undergo attributefor a long time for a humanoid robot, yet it is not well known

inversion. [3] In the extended Riemann sphere, for example,

that Pepper now comes equipped with an early-stage artificial

the north pole of a Riemann sphere surface (infinity) defining

which, in addition to recognizing human emotions,stereographic projection on a complex plane to which an idealdraws on other sensor information to generate emotions such

point has been added (expanded complex plane) is connected,

as“like”and“dislike”in the robot itself. This was the world’

as in a Klein bottle or Möbius strip, to the south pole of as first attempt at installing an artificial ego in a commercial

converse Riemann sphere surface (anti-infinity, anti-northdevice.[2]pole) that is related to the former sphere by attribute inversion.

ego

[1]

A major problem in this artificial ego project was whether

Obverse and reverse surfaces related by attribute inversion are

the robot would exhibit “free emergence” like a human.

connected to each other in any Klein bottle or Möbius strip, but

Emergence is one of the most important functions of artificial

in the proposed extended Riemann sphere, the connection isego. However, at the time, the mathematical models for

between an infinity and anti-infinity that are related by attribute

emergence were not clear and there were no mathematical

inversion. In this extended Riemann sphere,“emergence”ismodels of morality to control the selfish behavior resulting fromthe phenomenon whereby an infinity formed in a null set of theemergence. It is for these two reasons that emergence waspositive world, and an anti-infinity formed in an anti-null set ofthe anti-world, overlap and produce 1.[3]not then achievable.
In response to this problem, we postulated, as in Figure

We also illustrated basic designs for the artificial ego, and4, that“there is a world we cannot currently perceive, andpresented various calculations required by emergence. More©︎2021 Digital Hollywood University

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precisely, as basic designs for the artificial ego, we provided(inv ) operations for explicitly differentiating and calculating

examples of how homeostatic resilience in the positive world

converse and inverse with respect to an anti-world produced

and anti-world can be expressed in functions, in association

in a partition of unity. [4] Moreover, we have already applied,with an Emotion Map.

[4]

In addition, as calculations required

through the University of Tokyo, for a patent for a quantumby emergence, we published, in DHU JOURNAL in 2020, agate based on the extended Riemann sphere.

cut operation for explicitly and flexibly performing a partition ofThis paper clarifies the operating principles of emergent

unity, a dynamization operation ( ) for linking mathematicalquantum gates based on the extended Riemann sphere. In

phenomena eventuated by the cut operation, a superposition

addition, it applies these operating principles to implement

operation (f) for expressing dynamization operations inemergence-imitating artificial nerves and show that artificial

which emergence occurs, and converse (rev ) and inverse

ego emergence occurs at the artificial nerve level.

Table 1: Novel Calculation Standards
Novel
Calculation

Operator

Cut operation

cut

Result ofCalculation

Special Notes

shiki
識

Encompasses ÷(partitive)

Dynami-zation
operationtai
態Straight lines and flat
planes are also tai .

Superpo-sition
operation

ketsu
結Encompasses ÷ 0

inv 裏
rev 逆

Trialism also
encompasses not .
Definition of not is thesame as for dualism.

Inversion
operation

Definition of Calculation

Dualism: not .
Trialism: inv/rev .

The novelty of this paper rests on the following two points.

can be handled in a unified manner. Clarifying the definitions

The first point of novelty is our mathematical clarification ofof extended Riemann sphere, inversion operation, andthe fact that emergence is caused by the infinity produced

superposition operation makes it possible to recreate andby division by zero. In information engineering, dividing byeven control the emergence phenomenon on existingzero on a computer produces an error. In mathematics,

computers. That is to say, using the inversion operation todividing by zero in a zero ring returns 0=1 and is trivial.

preserve the essence of a function in infinite dimensional

Alternatively, with division by zero in a Riemann sphere, anvector space makes it possible to use the superposition

ideal point on an expanded complex plane is translated byoperation to distort and condense to a single point the space

stereographic projection to infinity on the Riemann sphere,

over which the ideal point stretches and develop the essence

resulting in a tautology. In physics, the approach is that theof the function using anti-space coordinates. Furthermore,

infinity resulting from division by zero does not exist. In thethis approach shows that even if the function disappears atinverse square of distance law and Schwarzschild solutions,infinity, a function is simultaneously generated from infinity.

despite strong suggestion that division by zero is an entity

In other words, this clearly shows a state in which positivethat cannot be ignored, neither Einstein, Penrose, Hawking,and anti-infinite dimensional vector spaces preserve essence

nor any other eminent physicist has ever ventured a clearof function, and the temporal axes of the functions that areinterpretation of division by zero. This paper shows, throughgenerated and disappear in the two spaces are superposed.

concise mathematics, that division by zero is the trigger of the

To give an analogy, a memory preserved in space would be

emergence phenomenon.

an Akashic record, and the novel calculation would indicate a

The second point of novelty is that these newly defined

five-dimensional existence.

calculations make clear that the emergence phenomenon©︎2021 Digital Hollywood University

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2. Assumptions

to not . The reason “inside” is conceived as reverse is, to

2.1 Updates to Novel Calculation Standards

give an admittedly limited example, that in the sentence“youcan’t reverse the clothes because they’re not reversible”,

Using the extended Riemann sphere, wave and particle“reverse”means turning the clothes inside out. In Japanese,mechanisms can be understood as in Figure 1.

turning clothes inside out typically means flipping the insideand outside, and does not mean “reverse” in the sense of
forward and backward. On the other hand, the reason inverse
is conceived in terms of“back”is that the not P→not Q concept
of converse of contrapositive (inverse) and the concept ofback (backside) are closely related linguistically in Japanese.
If we think of bits as coins, the 1 on the front/obverse surface(foreside) is flipped over and becomes a 0 on the back/reverse surface (backside). In other words, the concept of
converse of contrapositive (inverse) is connected to the

Figure 1: Relationship of cut operation, superposition
operation, and dynamization operation in wave function

concept of back surface (backside) via this concept of flipping(inversion). Thus, as shown in Figure 2, when we think in

Various kinds of information can be obtained from wave

terms of a planar two-dimensional cut operation, front/obverse

functions depending on the measurement and interpretation,and back/reverse can be obtained. Likewise, when we think inand wave-motion-ness and particle-ness are examples of suchterms of a volumetric three-dimensional cut operation, front/

information. It is the action of measurement or interpretation

obverse, inside, and back/reverse are obtained.

that defines the frame for obtaining outputs that are limited towave-motion-ness and particle-ness from among any numberof nature. The frame is defined by the input to the function
resulting from these actions (action input), and by performing
a cut operation of a wave function in such frame, it is possible
to obtain a superposition of wave-motion-ness and particleness. In mathematics, the cut operation can be expressed
as a partition of unity which is defined in the topologicalspace theory. In physics, because the sum E of physicalquantity corresponding to

Figure 2: Difference between two-dimensional cut operationand three-dimensional cut operationa unity need not be 1, the cut

operation can be expressed as a partition of E by using basevectors or base functions. Consequently, the superposition

Just as“inverse”means different things in logic, algebra,

of wave-motion-ness and particle-ness resulting from the cut

and geometry, caution is needed with the extended Riemann

operation acts complementarily while preserving the sum.

sphere and novel calculation standards to ensure there is no

The polynomial expression for maintaining the sum obtained

confusion regarding inv . In the extended Riemann sphere,

in the cut operation is referred to as“shiki”(識) , and the set

the con (converse) of the Riemann sphere is the converseof solutions satisfying shiki is referred to as“tai”(態) . In thisRiemann sphere, and the inv (inverse) of the Riemann sphere

context, we will define positive and anti in order to describe

is the inverse Riemann sphere. Meanwhile, in the novel

the details of shiki . “Positive” (orthodox) is defined as thecalculation standards, the inv (inverse) of the Riemann sphereproperty of the space spanned by each vector of shiki thatis the converse Riemann sphere, and the rev of the Riemannrelate to observation extend, on the basis of the projection

sphere is the inverse Riemann sphere.

hypothesis. Likewise,“anti”is defined as the property of the

There are three major differences between division andspace spanned by each vector of shiki that do not relate to

cut operations. The first is that division is limited to partitive

observation extend. Furthermore,“superposition operation”isdivision (equal division), whereas cut operations are not so

defined as the calculation for superposing positive and anti inlimited. The second difference is that partitioning of concepts

infinite dimensions. The sum produced by the superposition

cannot be represented in division, but can be represented inoperation is referred to as“ketsu”(結) . Finally, the“inversion

cut operations; when partitioning of concepts is represented inoperation” is defined as the operation for calculating thea cut operation, the terms of the cut operation a0N0,a1N1,...,an-terms of the anti. The cut operation, dynamization operation,1

N n-1,a nN n are distributed representations of words. The thirdsuperposition operation, and inversion operation are detailed

difference is that the functions (roles) exercised by numbers,in Table 1. It should be noted that results of addition,

volumes, concepts, subjects, acts, and results are difficult

subtraction, multiplication, and division operations are

to explain using division, but can be explained according toreferred to as“sum”,“difference”,“product”, and“quotient”

definitions in cut operations. For example, if a cut operation isrespectively; likewise, the results of the cut, dynamization andviewed in terms of such functions, the subject (input) of thesuperposition operations are referred to as“shiki”,“tai” andfunction may be not only a number represented by a discrete“ketsu”respectively.

quantity, a volume represented by a continuous quantity but a

In dualistic logic, only“not”is anti, but in pluralistic logic,

concept. In addition, the function of the cut operation can beinside (rev , reverse) and back (inv , inverse) are anti in addition

explained as the partitioning of subjects; this partitioning function©︎2021 Digital Hollywood University

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can be considered a partitioning action; and the result (output)a machine will choose its own behavior via emergence.

of the function will be an equation (shiki ) representing dynamic

Thus, as the foundation of such a logic, we turned

equilibrium that partially changes while preserving a sum is

our attention propositional logic and considered using

output.

propositional logic to express a code of values for behavior. In
propositional logic, if x is a behavior and P and Q are sets of
behaviors, a code of value for the behavior can be expressed

2.2 Computers and Consciousness
For the extended Riemann sphere to give computersas“(x∈P) →(x∈Q)”. Below, the proposition“(x∈P) →(x∈Q)”

consciousness, the calculation processes of those computers

is represented simply as (P→Q). Likewise,“code of value formust expand from a dualistic logic model to a pluralistic

behavior”is shortened to“behavior code”.

model. This is because, when looking from the perspective

As a concrete example, let us consider the behavior codeof wave or particle, just as nature does not make a dualisticof the samurai. If the set P is“acts of justice”and the set Q

choice between wave and particle, but a pluralistic choice inis“means of living”, then the samurai behavior code can bewhich wave-motion-ness and particle-ness are superposed,

expressed by the thesis (P→Q). According to the samurai

consciousness, which is a natural phenomenon, similarly does

behavior code, any of the acts of justice exemplified by fidelity

not make a dualistic choice. For example, when humans andto one’s lord will be a“means of life”. In other words, justiceother advanced primates believe that a thesis A→B holds,

is the means by which life is fulfilled.
The converse of the samurai behavior code can bethey have a biased tendency to believe that its converse B→A
In other words, it is possible for superposition of

expressed as (Q→P). This converse holds that anything that isthesis and converse to occur in consciousness. However, ina“means of living”is“just”. This converse can be thought oflogic based on dualism, only propositions whose truth values

as expressing an“egoistic set (β)”behavior code of people

match, such as contrapositives, are treated as equivalent. It

who do not choose the means by which they live. In thisis not clear what causes us to conceive of the superposition

context, people who always adopt the converse behavior codealso holds.

[5]of propositions. Thus, because thesis and converse havewould be called demons [oni ]. The converse is a demonic

inconsistent truth values, they are simply treated as different

behavior code. The demons always seem to expand thethings. There are some humans who tend to distinguishconcept of justice to secure their means of survival.
The inverse obtained by inverting the samurai behavior code

thesis and converse in their thinking, but this behavior isandcan be expressed as (not-P→not-Q). In other words, this is a

insects. [7] Therefore, the above-described tendency toward

behavior code where nothing that is not an act of justice canbiased thinking does not exist in insects and computers, andbe a means of living. We can think of the inverse proposition

can be considered a bias that is observed in humans andas expressing a culture in which a samurai that has acted

similar to the behavior model of primitive animals[5][6]

other advanced primates precisely because they have theunjustly commits seppuku out of shame. In this case, thesophisticated function of consciousness. Further, although,shame is an egoistic act.

for example, memory and consciousness reciprocally form

The contrapositive of the samurai behavior code can be

one another, in logic based on dualism, the superposition

expressed as (not-Q→not-P). This is a behavior code inof memory and consciousness cannot be represented.

which nothing that is not a means of living can be a just act.

More specifically, if the proposition variable X is “formation

According to such a behavior code, suicide is not a just actof consciousness state A”and the proposition variable Y is

because it is not a means of living. We might also think of“formation of memory B”, the thesis X→Y can be defined as“ifthis code, which regards suicide as unjust, as expressing

consciousness state A is formed, then memory B is formed”.

the widely known principle of monotheistic religions that

Conversely, the converse Y→X is the proposition“if memory“any human who commits suicide will go to hell”. However,

B is formed, then consciousness state A is formed”. Generallyaccording to traditional Japanese codes of value, the seppuku

speaking, the truth claims of a thesis and its converse do notof the samurai is not mere suicide, and needless to say, ismatch, and dualism thus regards these two propositions as

not evil. In this we find a depth that cannot be expressed

distinct. Such an approach lacks the depth that would allow

dualistically. Even monotheism allows for thinking thatit to use rev to treat propositions as the same while also

distinguishes egoistic suicide from altruistic suicide, so thedifferentiating them.

story is not as simple as is commonly thought.

In response to this problem, we have constructed theThus, an “altruistic set (α)” can be conceived as an

new logical model necessary to give computers deep andantipode of the“egoistic set (β)”of the converse, and we

sophisticated consciousness. The foundation of our new logicalcan consider this α code of value to exist within the samurai.

model is not the dualism rooted in the Old Testament that

When rev is used, α=rev-(P→Q) and β=rev-(Q→P). Whenregards existences as entirely separate, but a pluralism rooted in

rev is used to add depth to the contrapositive, (not-Q→not-P)

Japanese culture that regards existences as superposed.

attains a dual nature comprising α=rev-(not-Q→not-P) and
β=rev-(not-Q→not-P). Thus, the suicide that is regarded
as evil in typical monotheistic religions can be interpreted as

2.3 Construction of a Pluralism-Based Logic

evil in the sense of being the execution of“egoistic set (β)”.

The foundation of von Neumann computers is dualism.
The foundation of current quantum computing is a dualism

Likewise, samurai seppuku can be interpreted as good in thein which an original multiaxial superposition is turned into

sense of being the execution of“altruistic set (α)”. In othera superposition of 0 and 1. This happens because thewords, in a single contrapositive, there are different essences

foundation of logic is dualism.

(rev of the proposition), and we understand this difference

Our goal, however, is the development of an emergentof essence to be determined according to lineage – in otherartificial ego. Codes of value for behavior are necessary whenwords, the state before inverse and converse were calculated.

©︎2021 Digital Hollywood University

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Figure 3: Using propositional logic to explain the difference between dualism and pluralism
We also believe that this relationship between rev and

2.4 Pluralistic Model Necessary to Achieve Morality

not provides a hint to the uniquely human thought principle

and Emotion

which assumes the converse to be true. In dualism, a thesis

In this section, we refer to the pure and fundamental motives

and its converse are different entities, but in a pluralism thatpreceding the reception of external influences as“emotion”,

accounts for rev , the parameters implicit in the propositions

and consider the representation of robot emotion using

can match. Thus, we believe there is a principle of thought

vectors. Vectors representing the fundamental motives ofwhere, for example, when a thesis and the rev of its converse

robots are referred to as“emotion vectors”, which are defined

match, a person will at once mix up and seek to unify these.

as expressing, as one point in an n-dimensional vector space,

The existence of such a principle can also be predicted fromthe direction of the next state to be achieved in a robot’s

the fact that most invention and emergence is born of humanstate space.

preconceptions, and comes into the world by accident.

A robot behaving in accordance only with emotion vectors

In light of the foregoing, the difference between dualism,

will be in danger of inclining towards self-centeredness andwhich represents behavior codes using only not , andegoism. Mechanisms for controlling emotion vectors arepluralism, which represents behavior codes using both nottherefore necessary; for example, mechanisms restricting theand rev , is clear, as shown in Figure 3. The distinction canvery generation of emotion vectors are one such possibility.

be drawn according to the essential difference whereby, forPepper, which was mentioned in the introduction to thisexample, suicide falling under the inverse is always evil underarticle, is equipped with an emotion vector generation function

dualism, whereas under pluralism, egoistic suicide is evil but

and has emotions such as like and dislike for people andaltruistic suicide is good. Similar claims may be made about
the converse of the inverse (contrapositive). Consideredenvironments. However, functions that impulsively exhibit

pluralistically, the behavior code holding that something ishatred for an owner are likely to be deemed industrially uselessnot a just act if it is not a means of living can be interpreted,and commercially dangerous by manufacturers. By“industriallyaccording to the “altruistic set (α)”, as a standard thatuseless”, we mean that a robot is of no use if it does notcounsels mercy for the poor and the weak. Any justice that

follow instructions, and by “commercially dangerous”, wewould deny the weak in their struggle for life can thus bemean that commercial value will be destroyed if, for example,

denied on altruistic grounds. Meanwhile, according to theimpulsive behavior by the artificial ego is spread online through“egoistic set (β)”, this behavior code can be interpreted only

malicious videos. As a result, systems have been adopted inin terms of clinging to life. This set expresses a self-centeredoperational apps in which emotion vectors are deactivatedview according to which no behavior is just which is not aand then activated temporarily in only certain communicationmeans of securing one’s own life. Dualism cannot expressor game apps where impulsive expression is permitted; thethis difference of fundamentally latent parameters. Therefore,reason that it was not well known that the early-stage artificial

unless a pluralistic model rooted in pluralism is constructed inegos had been installed is that operations were restricted.

a behavior code, morality and emotion will be unrepresentable,

Another option worthy of consideration is a mechanism thatand as a result, behavior codes that can empathize only withwould use morality vectors to control emotion vectors – thatsome people will be all that can be implemented in machines.

is to say, a mechanism defined by“emotion vector + morality
vector = desire vector”, where the robot acts on the basis of©︎2021 Digital Hollywood University

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the desire vector. This method would allow the robot to be

However, no interpretation of infinity is ventured in a Riemann

controlled by the morality vector such that the emotion vector

sphere, and because there is nothing after infinity, theis oriented toward sympathizing emotionally with others.

collapsed matter information must, for example, return along

Incidentally, as discussed above, when behavior codeswith the Hawking radiation. [10] In other words, a Riemann

are defined dualistically with consideration only for not , one

sphere can be regarded as treating ∞ as the reciprocal of

encounters a problem in that, for example, altruism and egoism

zero and division by zero as the same as multiplication by

are not separable when those behaviors are superficially theinfinity, and z∈C\0, in which 0 is excluded from the complex

same. Thus, it is necessary to conceive of behavior codes onplane, defines z⁄0=∞ and z・∞=∞. In addition, stereographic

the basis of a pluralism that also considers rev . Research into

projection is defined such that points on a complex plane towhether“only-not ”behavioral codes or behavior codes thatwhich an ideal point has been added correspond one-to-one

also consider rev lead to moral or emotional judgments, andto points on the Riemann sphere; specifically, stereographic

to sympathy with others, is therefore also necessary.

projection is defined such that the ideal point on the complex
plane corresponds to the north pole of the Riemann sphere.

3. Emergent Quantum Gates

However, no interpretation of the north pole – for example,

We postulate, as in Figure 4, that“if there is a world weof the height of the north pole – is ventured, and nothing iscannot currently perceive (anti-world), something will emerge

connected to the north pole. Thus, what comes after infinity

due to its influence”.

must also be considered in Riemann spheres.
At the same time, Riemann spheres also present the problem
of being incapable of conceiving a partition of infinity. In what
follows, we will begin by explaining quotative and partitive
division, and then show that in a Riemann sphere, a partition of
infinity cannot be conceived in the manner of quotative division.
To begin, in arithmetic education, partitioning of a whole
can be conceived in terms of quotative or partitive division.
Quotative division is division to determine how many a whole
becomes when divided by a certain number. For example,
quotative division is used when calculating the answer to,“If 6 total candies are divided into groups of two, with how

Figure 4: Existence of imperceptible anti-world

many people can the candy be shared?” Partitive division
The mathematical model that explains the anti-world is

is used to determine the number per group when a whole

an extended Riemann sphere that connects a Riemannis divided among a given number of groups. For example,

sphere and anti-Riemann sphere at the poles where attributepartitive division is used when calculating the answer to,“If 6

inversion has occurred. [3] In the extended Riemann sphere,

total candies are divided among 3 people, how many candies

the anti-world is represented as the converse Riemann sphere

will each person receive?” In both quotative and partitiveand inverse Riemann sphere. In addition, “emergence” candivision, the whole is a discrete quantity, but for the purposes

be defined as the phenomenon whereby an infinity formedof this paper, the whole may be a continuous quantity as well.

in a null set (KU) of the positive world, and an anti-infinity

However, in quotative division, the quotient must be a discrete

formed in an anti-null set (anti-KU) of the anti-world, overlap

quantity, and the custom is to treat fractions as remainders.

and produce 1.[4] By means of the extended Riemann sphere,

For example, when 5.5 apples are divided into groups of 2.5

black holes and white holes are sketched as entities whoseby quotative division, the apples can be shared with 2 people

obverse and reverse are stuck back-to-back to each other,and there is a remainder of 0.5. The divisor in partitive division

and appear as mechanisms related to emergence.

is generally a discrete quantity.

In this section, we will make clear, on the basis of the

Next, we will show that, when a case where distance is zeromathematical model of our proposed extended Riemannin an equation containing terms dividing mass by distancesphere, that black holes and white holes are related to(hereinafter, “zero distance”), as in the law of universal

emergence mechanisms, and will propose a basic design for

gravitation, is considered in terms of quotative division, aan emergent quantum gate.

partition of infinity is inconceivable even using a Riemannsphere. Because a Riemann sphere defines ∞⁄0=∞, conceiving∞⁄ from the perspective of quotative division would mean0

3.1 Relationship between Black Holes, White Holes,
and Emergencethat infinity is divided by the quantity of zero, and thus can be

In Einstein’s general relativity theory, [8] the event horizon

divided into infinity boxes. Conversely, although infinity should

and singularity are known problems in relation to black holes.[9]be the result when zeroes are extracted as quantities fromSpecifically, when a black hole dies due to evaporation fromin a vacuum does not contain matter information, the black

infinity boxes and added together, in a Riemann sphere, 0・∞is undefined, and even if ∞⁄0=∞ is established as in quotative
division, 0・∞=∞ is not established. Alternatively, if we think,hole dies without it being known where the matter informationgeometrically, of a Riemann sphere with a diameter of 1, thenwithin the event horizon went.

even allowing for continuous change such that a continuous

Hawking radiation, because Hawking radiation originating

In a Riemann sphere, the infinity resulting from division

quantity z∈

by zero can be handled as one point on a sphere surface.

©︎2021 Digital Hollywood University

on the complex place is

z →∞, the northpole, acting as a terminus of stereographic projection of the12

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Riemann sphere, corresponds to the discrete quantity 1, andor discrete quantities, the superposition operation on time inany partition thereof is inconceivable. A partition of infinity is

the forward direction and the backward direction will result

thus inconceivable using current Riemann spheres. Moreover,in the disappearance of time and the overlap of the infinite

in physics, division by zero is regarded as an operation withand zero occurs mathematically. The distance of the infinity

no relation to physical phenomena, and zero distance issuperposed in this superposition operation will be the diameter

regarded as not being a physical quantity.

of (length across) the extended Riemann sphere, and the 1

Nonetheless, we believe that the problem of thethat is the discrete quantity arising after emergence due todisappearance of matter information within the event horizon,

the cut operation will correspond to this diameter. In otherand the problem of the partition of infinity, can be explained

words, what becomes 1 after emergence differs depending

using the extended Riemann sphere.

on vantage point. Moreover, the post-emergence 1 is

We postulate that in the extended Riemann sphere, apartitioned by the cut operation in the anti-world and gives

converse Riemann sphere and inverse Riemann sphere arerise to concepts in the anti-world. The concepts thereof are,present as anti-Riemann spheres. The south pole of thefor example, the extended complex plane of the anti-world

converse Riemann sphere is connected to the north pole of(obtained by adding the ideal point to the complex plane) and

the Riemann sphere, and the polarity of the converse Riemannthe function represented thereon.

sphere is opposite that of the Riemann sphere. In other words,

Thus, the problem of the disappearance of matterin the converse Riemann sphere, it is the south pole (anti-northinformation in the event horizon can be solved by interpretingpole) that corresponds to infinity (anti-infinity), and the norththe collapsed matter information not as returning to thepole (anti-south pole) that corresponds to 0 (anti-0). Thus, if

Riemann sphere, but as being translated through emergence

we imagine an axis connecting the north pole of the Riemannto the anti-Riemann sphere. Likewise, the problem of thesphere and the anti-north pole of the converse Riemannpartition of infinity can be solved by interpreting the 1 that issphere on a straight line, that axis will represent infinity at

an emergent discrete quantity as generating concepts throughthe Riemann sphere terminus (north pole), and anti-infinity

the cut operation in the anti-Riemann sphere.

at the converse Riemann sphere terminus (anti-north pole).

In general relativity theory, the distance near the singularity

Meanwhile, the anti-south pole surface of the inverse Riemannresulting from division by zero is interpreted as being infinitely

sphere is connected on its obverse to the north pole of theextended. Thus, if we consider this in terms of falling matter

Riemann sphere, and the polarity of the inverse Riemannin the extended complex plane of the Riemann sphere, matter

sphere is flipped relative to that of the Riemann sphere on

having a center of gravity at the ideal point would fall toward

its obverse. For example, in the inverse Riemann sphere on

a 0 far from infinity. The interpretation that the information

the reverse side of the converse Riemann sphere, it is theof matter that had fallen to 0 would radiate backwards overanti-south pole surface that corresponds to 0 and the anti-

an infinite distance is unnatural. However, if we consider thisnorth pole that corresponds to infinity. In the inverse Riemannin terms of falling matter in the Riemann spheres comprisingsphere, the axis connecting the anti-south pole and anti-

the extended Riemann sphere, the distance to 0 from thenorth pole enjoys a degree of freedom in regard to its rotation

singularity resulting from division by zero can be deemed to be

direction. Thus, on the above-described axis connecting the

1 on the basis of the diameter of the Riemann sphere. Matter

north pole of the Riemann sphere and anti-north pole of thehaving a center of gravity at infinity (north pole) in the Riemannconverse Riemann sphere, the termini of the two spheres

sphere falls to 0, which is separated by a distance of precisely

represent infinity and anti-infinity, and in addition, the interval

1, and falls to anti-infinity (anti-north pole) in the converse

between those termini is a slider along which infinity moves.

Riemann sphere, which is separated from 0 by a distance ofIn the extended Riemann sphere, emergence can be defined

precisely 1. A black hole in the Riemann sphere and whiteas the phenomenon whereby an infinity formed in a null set

hole in the converse Riemann sphere will compete back-to-(KU) of the positive world, and an anti-infinity formed in an

back with the Schwarzschild radius as a boundary between

anti-null set (anti-KU) of the anti-world, overlap and produce

them, and, as in a Möbius strip or Klein bottle, will undergo

1. In other words, if the diameter of the Riemann sphere in theattribute inversion and thereby connect infinity to anti-infinity.

positive world is 1, then this 1 can be defined as a discrete

There will be a gap at the connection resulting from attribute

quantity on the anti-Riemann sphere in the anti-world after

inversion; KU will be the boundary with the connection on theemergence. Alternatively, if, on a larger scale, the length ofinfinity side; anti-KU will be the boundary with the connection

the above-described axis connecting the north pole and anti-on the anti-infinity side; and MU will be the center of the gap.

north pole is 1, then as in the superposition operation of Table

Matter at infinity on the Riemann sphere will pass through the

1, this 1 can be interpreted as a post-emergence discrete

black hole and the white hole and then be reborn on the anti-

quantity. In such case, in the Riemann sphere, time flows

Riemann sphere as the set concept, space-time, and function

in a forward direction and the function undergoes addition

having anti-infinity as the sum. As a result of the foregoing,via Σ towards infinity at the north pole, and in the converse

general relativity and quantum theory can be reconciled, andRiemann sphere, time flows in a backward direction and theas a result of the cut operation, dynamization operation, andfunction undergoes addition via Σ towards the infinitesimal atsuperposition operation, numbers, volumes, and concepts can

the anti-south pole. If the superposition operation is executed

be handled naturally within emergence phenomena. Also, thewith attention to the fact that the sum up to the infinite of aextended Riemann sphere allows not and rev to be handledfunction and the sum up to the infinitesimal of the function

explicitly. Furthermore, as detailed below, the observer

differ depending on whether they are continuous quantities

problem of quantum theory can also be solved.

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It bears noting that the symbolin the dynamization

have named this analog-and-digital simultaneous calculation

operation and superposition operation was first introduced by“quantal calculation”.

the lead author in his 2006 doctoral dissertation.[13] Further,
the definitions of the dynamization operation and superposition
operation that use

were first published by the lead author

et al. in DHU JOURNAL in 2020.[4] The same symbol was also
used in the anti-Einstein field hypothesis (Some hypothesis toderive an anti-Einstein field).[3] The“anti”of the anti-Einsteinfield hypothesis is anti in the sense explained in Section 1.1,
and means the anti of attribute inversion. In other words, it
is synonymous with the anti of anti-Riemann sphere (anti-

Figure 6: Explanation of quantal calculation

Riemann field). Thus, it is not an anti that negates something,in the sense of “anti-establishment” or “anti-nuclear”, andIn quantal calculation, quantum bits behave like oracles

so is not a negation of relativity theory. Rather, in relativity

in computability theory, and output time-dependent wave

theory, the anti-field plays the role of reinforcing the positive

functions or the like. In other words, emergence in thefield with respect to the singularity.

anti-world gives rise to an “oracle” in the positive world.

Current quantum bits are represented using a Bloch sphere,

Conventional quantum gates, which handle only relative phasebut the quantum bits of the proposed method are representedof spin in Bloch spheres, cannot become oracles, but quantum

using the extended Riemann sphere. This is illustrated ingates that handle relative amplitude in the extended Riemann

Figure 5.

sphere can become oracles through the connection of infinity
and anti-infinity. Thus, we have undertaken experiments inwhich quantum superposition states are controlled using
lasers, to confirm emergence phenomena in the anti-world.
3.2.2 Nature of Emergence
The essence of the concept output by quantum gate
emergence is made clear by using the inversion operation
to analyze the interval between positive and anti. Thus, thepremise (p’→q’) is divided by cut operation into the thesis(p→q) and its inverse (not・p→not・q), and that which exists
between the thesis and the inverse is analyzed using theinversion operation.

Figure 5: Difference of observation targets in current methods
and proposed method

When the cut operation is performed two-dimensionally,( p ’→ q ’)c u t 2 = (p →q )( n o t・p → n o t・q ) , a n d i f i n v・( p ’→q’)=(not・p→not・q), then rev・(p’→q’)=0. In other words,

3.2 Basic Design of Emergent Quantum Gate

the obverse is (p →q ), inv is (not・p→not・q), and there is no3.2.1 Overview

rev , as indicated in Figure 2.

Simultaneous calculation of continuous quantity and discrete

When the cut operation is performed three-dimensionally,

quantity cannot be envisioned with current quantum bits,(p’→q’)cut 3=(p →q )where the result of simultaneous calculation of a continuousX(not・p→not・q), and if inv・(p’→q’)=(not・p→not・q), then rev・(p’→q’)=X . In other words,

quantity interpreted as a discrete quantity. Therefore, thethe obverse is (p →q ), inv is (not・p→not・q), and rev is rev・(p’emergence of a continuous quantity from a discrete quantity→q’)=X .

cannot be envisioned, nor can the fact that the continuous

When the cut operation is performed in infinite dimensions,quantity becomes a discrete quantity at the limit. Current(p’→q’)cut ∞=(p →q )quantum bits have only limited capability in this regard.

Z(not・p→not・q),Z=Z 1 Z 2 ... Z ∞ ,and rev・(p’→q’)=Z is understood to represent the nature

However, simultaneous calculation of continuous quantities

of emergence in the extended Riemann sphere. Thus, weand discrete quantities can be envisioned with quantum

formulated rev・(p’→q’)=Z and attempted an interpretation.

bits represented using the extended Riemann sphere. The

In a state where the dynamization operation has continuity,

superposition of infinity, which is the limit of the continuous

rev・(p’→q’)=Z was interpreted as follows using a Riemannquantity, and 0 (0≡∞) occurs in the anti-world, and as asphere.

result, the discrete quantity 1 emerges and a continuous
quantity having anti-infinity as a sum is born in the positive
world. In other words, emergence in the anti-world creates the

In the Riemann sphere, Z can be considered to represent

discrete quantity 1, and the 1 that appears in the anti-world

a nature where a concept is attracted to and continuouslyis partitioned in the positive world, giving rise to a continuous

superposed on the north pole (infinity), and it is thus predicted

quantity. This simultaneous calculation of continuous quantitythat rev・(p’→q’)=Z is a nature where gravity is produced afterand discrete quantity is illustrated in Figure 6. Simultaneous

emergence.

calculation of continuous quantity and discrete quantity is
as described in DHU JOURNAL, vol. 06, 2019, p. 10. We©︎2021 Digital Hollywood University

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In addition rev・(p’→q’)=Z was interpreted as follows using
a converse Riemann sphere.

Incidentally, in the inverse Riemann sphere, the inverse
is not conceived in terms of not as in logic, and instead,can be conceived in terms of rev as in the novel calculation
standards. Consequently, the inverse Riemann sphere can be

When interpreted using the converse Riemann sphere,

observed using rev as well. In other words, to use the EarthZ represents a nature where a concept is attracted to andas a metaphor as in Figure 7, the inv -Earth, and not the not -continuously superposed on the anti-south pole (infinitesimal)

Earth, can be observed using rev .

as a continuous quantity, and it is thus predicted that rev・(p’→q’)=Z is a nature where the action of a fundamental force
other than gravity is produced after emergence.

In a state where the dynamization operation lacks continuity
due to discrete quantification, rev・(p’→q’)=Z was interpreted
as follows using a Riemann sphere.
Figure 7: Relationship between rev of novel computation
standards and inverse Riemann sphere
In the Riemann sphere, because a discrete quantity issuperposed at the north pole (infinity), Z can be considered

In the extended Riemann sphere, orientation can be fixed

to represent the nature of a discrete quantity. Thus, if theto the obverse (foreside) or reverse (inside) of the sphere.

nature of a discrete quantity is passed on after emergence,

If obverse and reverse surface orientations are represented

then ∞→a n N n +a n+1 N n+1 +...+a ∞ N ∞ - in other words, Z can

as vectors, the shiki terms represented in base vector

be formulated as a nature where infinity is divided in infinite

decomposition can be divided between obverse (foreside)

dimensions and a discrete quantity is produced afterand reverse (rev ) using the element representing orientation.
Of these, the reverse (rev ) is calculated using an inversion

emergence.

operation.
The nature of an emergence is observed using the inverse
Riemann sphere (inv ). The inverse Riemann sphere is asphere in which energy is present somewhere between thecenter and the sphere surface. The sphere surface is the antiIn addition, rev・(p’→q’)=Z was interpreted as follows using

south pole surface, and the center is the anti-north pole.

a converse Riemann sphere.

Thus, the energy is present between positive and anti. By
analyzing the energy present between positive and anti withan inversion operation, it is possible to observe the nature ofan emergence. Thus, if the obverse Riemann sphere is (p →q ),In the converse Riemann sphere, because a discrete(p →q )cut ∞ = sphere surface

quantity is superposed at the anti-south pole (infinitesimal),Z

center, where Z=Z1

Z2 ... Z∞, rev・(p’→q’)=Z is observed.

here as well Z can be viewed as representing the nature of a

The following formula can be interpreted for the state wherediscrete quantity. Thus, if the nature of a discrete quantity is

the dynamization operation has continuity.

passed on after emergence, then 0→a nN n +an -1N n -1+...+a 0N 0
– in other words, Z can be formulated as the nature where
0 is divided in finite dimensions and a discrete quantity is
produced after emergence.

When this Z is interpreted using the inverse Riemannsphere, a first possibility is conceivable in which, when thethesis (p’→q’) is subjected to a cut operation using inv ,attracted to the anti-north pole (infinitesimal), and superposed
thereon while shrinking, the mass superposed as a result ofsuch shrinking is rev・(p’→q’)=Z . This mass shrunk to theThus, the laser experiment described above may be able toinfinitesimal is referred to as“shrink mass”. Meanwhile, there

discern emergent phenomena in even smaller space-time thanis also a second possibility in which a continuous quantity is Z.

the following equation we proposed.

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In light of these two possibilities, it is feasible to think that,when the infinitesimal emerges ubiquitously as a concept in
the extended complex plane of the anti-world, a shrink mass
having a continuous quantity appears as the nature of such

When this Z is interpreted using the inverse of converse

infinitesimal, but is oriented toward growth in the anti-world

Riemann sphere, a state occurs in which, when all concepts

because of attribute inversion.

are attracted to the anti-north pole (infinity) and superposed
thereon while being concentrated, the smallest possible
discrete number is pushed to infinity as a result of suchconcentration. This state in which the smallest possible
discrete number is pushed to infinity is referred to as“shrink

Alternatively, the following formula can also be interpreted

density”. This shrink density can be considered to represent afor the state where the dynamization operation has continuity.

nature where energy is not inverted at the time of emergencebecause it is not attribute-dependent, and infinity begins toshrink after emerging as a concept in the extended complex
plane of the anti-world.

When this Z is interpreted using the inverse of converse
Riemann sphere, a first possibility is conceivable in which,when the thesis (p’→q’) is subjected to a cut operation using

inv , attracted to the anti-north pole (infinity), and superposed
thereon while being concentrated, the mass superposed as

Moreover, based on the observations in this section, it isa result of such concentration is rev・(p’→q’)=Z . This mass

theoretically possible to conceive of antigravity and distantconcentrated at infinity is referred to as “growth mass”.

gravity. To begin, antigravity can be represented by the

Meanwhile, there is also a second possibility in which afollowing formula by defining the nature of gravity as having

continuous quantity is Z. In light of these two possibilities, it isbeen inverted.

feasible to think that, when the infinite emerges as a concept
on the extended complex plane of the positive world, a growthmass having a continuous quantity appears as the nature of
The inversion (anti) of nature occurs in the converse, assuch infinite, but is oriented toward shrinking in the positivewas exemplified using α and β in section 2.3. Thus, whenworld because of attribute inversion.

the operation (con) for calculating a converse Riemann spherecorresponding to the converse is used to reverse the natureof gravity and assign a pole at the point where antigravity iscontrolled in each of the proposition variables p and q, theThe following formula can be interpreted for the state where

result is as follows.

the dynamization operation lacks continuity due to discrete
quantification.

When this Z is interpreted using the inverse Riemann sphere,a state occurs in which, when the thesis (p’→q’) is subjected
to a cut operation using inv , attracted to the anti-north pole

Next, the distant gravity for controlling antigravity can be(0), and superposed thereon while shrinking, this shrinking

represented by the densities of the following formulae.

pushes infinity to the smallest possible discrete number.
This state in which infinity is pushed to the smallest possible
discrete number is referred to as“growth density”. This growthdensity can be considered to represent a nature where energy
3.2.3 Nature of Quanta

is not inverted at the time of emergence because it is not
attribute-dependent, and the infinitesimal begins to grow after

It can be considered that the quantum bits in current

emerging ubiquitously as a concept on the extended complex

methods handle the states of quantum gates using cut

plane of the positive world.

operations. Thus, because of observation, the final state is
fixed at 0 or 1 and an inversion operation is inconceivable,
as shown in Figure 8. The quantum state is therefore (S →N )cut2=a0|0>+a1|1>, which contains no rev information.

Alternatively, the following formula can also be interpreted
for the state where the dynamization operation lacks continuity
due to discrete quantification.

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Figure 8: Current quantum bits

Figure 9: Representation of attribute inversion using extended
Riemann sphereWith the quantum bits of the proposed method, the stateof a quantum gate is handled by cut operation, dynamization

As shown in Figure 9, the extended Riemann sphere isoperation, superposition operation, and inversion operation.

connected at the pole at which a Riemann sphere that has

Thus, the rev of quanta is handled as in the following formula.

undergone attribute inversion. For example, the south pole(S) of a Riemann sphere is connected to the anti-south pole(anti-S) of a converse Riemann sphere that has been rotated
180 degrees. The extended Riemann sphere has been definedso as to correspond to a logical proposition. For example, theRiemann sphere corresponds to the thesis (P →Q ), and theconverse Riemann sphere corresponds to its converse (Q →P ).

In quantal calculation, a quantum state containing rev
information is named a“quantal state”.
Current methods are limited in that, because the state canonly be interpreted digitally, the analysis must begin and end
with the discrete numbers 0 and 1. However, in the proposed
method, in addition to handling quanta as waves as in a|0>b|1 >, the quality of such quanta can be handled using rev .
Thus, in the proposed method, spin can be conceived not
only as |0> and |1>, but as a |↑> oriented in any direction andits opposed |↓>. In such case, the rev of quanta is expressedby the following formula.

Figure 10: Connection of poles where attribute inversion occurs
In such case, when rev is considered with respect to

As illustrated in Figure 10, on the converse Riemann sphere(S →N) cut 3, a “ ” connected to the same infinity as theintermediate term Z of (S →N) cut ∞ appears. The

surface, stereographic projection is defined from the anti-

at thissouth pole with respect to the extended complex plane where

infinity can be interpreted as a term related to emergence insuch anti-north pole is grounded. If the Riemann spherethe extended Riemann sphere as shown in 3.2.2 “Naturesurface is taken as a reference, then attribute inversion resultsof Emergence”. Therefore, an extended Riemann spherein the anti-north pole of the converse Riemann sphere surface

constructed by geometric expansion in logic could be thebeing anti-infinity, and the anti-south pole being the anti-origin

foundation of a novel quantum logic calculation system.

point. Conversely, if the converse Riemann sphere is taken asa reference, then the anti-north pole can be regarded as the

3.2.4 Mathematical Structure of Quantasouth pole and the anti-south pole can be regarded as the

In current quantum computing, quantum states are generally
represented using Bloch spheres,
Riemann spheres

[12]

[11]

north pole. In other words, in general, a Möbius transformation

but representation using

defined on a Riemann sphere surface can be used as-is in theis also a possibility. The proposed method

converse Riemann sphere by inverting the reference. That isuses the extended Riemann sphere.

to say, in the extended Riemann sphere, quantum gate control
can be described with a Möbius transformation, as in the caseof the Riemann sphere surface.

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3.2.5 Mathematical Structure of Emergence
In a Möbius strip, an obverse surface and reverse surface
that are connected to each other by attribute inversion areconnected, but in the proposed extended Riemann sphere,
an infinity and anti-infinity that are related to each other by
attribute inversion are connected. This section will clarify themathematical structure of Möbius strips and extended Riemannspheres.
Figure 13: Infinitesimal gap
As shown in Figure 13, a limit space is present as a gap
at the center of the sea urchin. This gap would be, in outerspace, an entity the size of the Schwarzschild radius in a
black hole, and, in infinitesimal space, an entity no larger than
the smallest unit (infinitesimal) that can be attained by a spaceof roughly the Planck length.
Thus, as an entity determining the inside of the infinitesimal,a string such as a rubber band surrounded by the gap isexpressed as emptiness (KU). Furthermore, as in Figure 14,
the inside (rev ) of KU is expressed as nothingness (MU). As
a result of the directionality of convergence, the orientation of

Figure 11: State of infinite dimensional convergenceon origin point 0

MU is defined by a vector oriented to the center of KU.

Figure 11 concisely illustrates a state where, with respect
to a shiki represented by an infinite-dimensional orthogonalbasis, all terms are oriented to an origin point. In this state,
the trajectory traced by each term of the shiki can be sketchedlike a“sea urchin”centered on the origin point 0.

Figure 14: Relationship between KU and MU

If, in order to flesh out the picture of shiki converging on
origin point 0, we were to imagine a spatial coordinate (shiki )Based on this definition, MU is also shiki and can be

represented by a three-dimensional orthogonal basis, thesketched using a dynamization operation as shown inconvergence of shiki in the fourth and subsequent dimensions

Figure 15.

would appear impossible. This would be because, as shown inconnecting the Riemann spheres is a control point and KU is

Figure 12, there would be no room left to add the fourth andpresent at this control point.

is drawn like a control lever because the polesubsequent orthogonal bases.

Figure 15: Relationship between MU andAt the pole, the infinity of the obverse is connected to theanti-infinity of the anti, and it thus appears that there is abreakpoint (gap) in the KU string. Therefore, it is clear, as

Figure 12: Room for orthogonal basis in three-dimensional
orthogonal spacein Figure 16, that where the KU string is broken at a certain
point, infinity is connected to infinity, forming a single loop,
and after the break, aThus, for black holes, a theory in which the fourth andof which both ends approach infinity

can be sketched.

subsequent dimensions converge while crossing orthogonallyat origin point 0 is needed. In other words, a theory thatrealizes a condition such as the space-time-swallowing black
holes described by Prof. Hawking and others is necessary.

Figure 16: Existence of break point on KU
Thus, in order to connect the infinity of the obverse and theanti-infinity of the anti, as shown in Figure 17, one side of©︎2021 Digital Hollywood University

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is twisted by 180 degrees and connected to MU. In otheris the obverse, its inside (rev ) is MU. In

words, if infinity at

addition, the 180-degree twisting deformation is represented
by the function, in which therepresented byfwith added deformation is.

Figure 19: Möbius strip f, which realizes orthogonalcrossing at the infinitesimalCurrent quantum gates based on Bloch spheres andRiemann spheres are limited in that they do not realize
attribute inversion at poles, and thus, to be candid, areincapable of using anything other than |0> and |1> to control

Figure 17: Connection between infinity and MU produced by

the entanglement and superposition of quanta. Quantum gatesbased on the extended Riemann sphere remove this limitation.

Therefore, when a 180-degree rotation is imparted to one
using the function, KU is as depicted in Figure 18. Morespecifically, through the function

4. Comparison with Current Methods, the ∞ of the obverse

We will now discuss the differences between the Blochand the MU of the rev -obverse (reverse) are connected (0≡∞), forming a Möbius strip. In DHU JOURNAL in 2019,spheres [11] and Riemann spheres [12] which are the basis of[3]

current methods and the extended Riemann sphere which is

this Möbius strip was explained using a Klein bottle andthe basis of our proposed method.

represented as a matrix.

4.1 Unnaturalness of Spinor Representation Using
Bloch Spheres
Spinor representation in Bloch spheres has an unnatural
aspect – namely, the relative phase restrictions set to ensure
there are no contradictions. For example, Bloch sphere spinor
representation can be expressed using the following formula.

Figure 18: Formation of Möbius strip
The Möbius stripff

by function

formed by the function

can be used

to construct a theory in which four or more dimensions
converge while crossing orthogonally at origin point 0. ThisWith respect to the relative phase, as a result of theis because: first of all, the smooth twist added to the strip

following restriction, z ↑> is z ↓> when θ=π.

makes it possible to have a vector at infinity which crosses
the strip orthogonally, as shown in Figure 19; and secondof all, inverting and connecting zero and infinity just below
allows a vector that orthogonally crosses KU, i.e ., Möbius
strip, to be present at infinity. In other words, it is possible
for all vectors to cross orthogonally in an infinitesimally small
state, both at micro and at macro levels. The rev of KU is MU
in the form of absolute nothingness. As a result of emergence,
the infinity on the obverse changes into the anti-infinity on thereverse (converse Riemann sphere) after passing through thegap between infinity and MU ofand anti- . On the anti,

it appears as though nothingness is translated to anti-infinity
Figure 20: Spinor representation using Bloch spheres

and being has emerged from nothingness. The foregoing is
the mathematical structure of emergence at the connection of

As shown in Figure 20, when θ =π, z ↑> becomes z ↓>,poles where attribute inversion has occurred.

matching the figure, but when θ =2π, the symbols reverseand do not match the figure. Meanwhile, in a Möbius strip
where the symbols reverse after one trip around the strip, thisreversal of symbols can be expressed naturally, as shown in
Figure 21.

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Figure 24: When z =0 in M g (z )=Figure 21: Spinor representation using Möbius strip

4.3 Resolving Unnaturalness with Extended Riemann sphere
Therefore, the extended Riemann sphere operates according

4.2 Unnaturalness of Spinor Representation Usingto “obverse

Riemann Sphere Surfaces

converse

inversecontrapositive ” on thebasis of the proposed method, and when connected as in

Spinor representation using Riemann sphere surfaces also has

Figure 25, resolves the unnaturalness of spinor representation

an unnatural aspect. To confirm this unnaturalness, let us firstin Bloch spheres and Riemann spheres.

define the Möbius transformation M g using the following formula.

When transformation in a spinor representation is definedusing this Möbius transformation, M g , M g can be treated as arotation operation on the Riemann sphere, as in Figure 22.

Figure 25: Connection of poles in extended Riemann sphere
For reference, the quantum gate mathematical structure
we have applied to patent through the University of Tokyo isshown in Figure 26.

Figure 22: Rotation operation on Riemann sphere surface by
Möbius transformation
For relative amplitude = 1/ 2 as well, the rotation operationon the Riemann sphere surface is represented by the Möbius

Figure 26: Quantum gate mathematical structuretransformation M g . As shown in Figure 23, we can see that
relative phase(θ 1 -θ 2 ) matches the Möbius strip, but for

5. Experiment

relative amplitude, there is nothing beyond relative amplitude

It is necessary to execute division by zero and coordinate, and thus, unnaturally, closure does not occur.

expression of infinity using

calculations on current

computers, using the above-described 2019 physical model[3]and 2020 mathematical model.[4]As a prototype, in 1993, we completed a 3D model toclarify our overall image of quantum gates. Then, in 2010, we
produced a working model having a

dynamization operation

as an operating principle. Further, in 2018, we did a trial runof aFigure 23: Difference between relative phase (left) andrelative amplitude (right) in Riemann spheresfsuperposition operation calculation principle in a gyromodel, and in 2019, we did a trial run of afresonance

principle in a twin gyro model. These models are shown in
Figure 27.

For example, in cases like that shown in Figure 24, there isnothing beyond where relative amplitude undergoes attribute
inversion and is connected at infinity.

Figure 27: Experimental models having dynamization
operation and superposition operation as operating principles©2021 Digital Hollywood University

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We now need to derive basic principles of the abovedetailed quantum gates etc. from the actions of these models,verify the actions on computers, and develop new artificial
nerve connection systems based on the artificial ego.

5.1 Implementation of Artificial Nerves
In the artificial nerves we have produced thus far, sigmoid
functions have not been used for activation functions. Figure
28 summarizes the relationship between quantum bits,
dynamization operations, superposition operations, and the

Figure 30: Connection between positive world andanti-world in quantum gates

extended Riemann sphere.

5.2 Results of Implementation of Artificial Nerves
In this section, we confirm that, as a result of applying theoperating principle of quantum gates based on the extended
Riemann sphere to construct an emergence-imitating artificial
nerve model on current computers, the strip that was not
closed in the relative amplitude direction in Figure 23 wasclosed, like a Möbius strip, in the relative amplitude direction
as in Figure 31.

Figure 28: Relationship between quantum bits,
dynamization operations, superposition operations,and extended Riemann sphereWe believe a neural connective structure can be represented
using the extended Riemann sphere. Therefore, as was
explained in “3. Emergent Quantum Gates”, we envision a

Figure 31: Behavior of relative phase andrelative amplitude of constructed artificial nerve

structure like that in Figure 29, in which the positive world andanti-world are connected as in a Möbius strip.

Figure 29: Neural connection model based on extended
Riemann sphere

Figure 32: Specific behavior of constructed artificial
nerve in experimentThus, we have applied the operating principle of quantum

With regard to the specific behavior in the experiment of thegates based on the extended Riemann sphere and attempted

constructed artificial nerve, as shown in Figure 32, relative

to construct an emergence-imitating artificial nerve model on

amplitude behavior was confirmed using the following formula,current computers.

which in control engineering is considered a transfer function.

As discussed above, division by zero and coordinate
expression of infinity are possible using quantum gates basedon the extended Riemann sphere, and connection between
More specifically, we confirmed that, when M g (z ) is used

infinity and nothingness is possible using the shape of theto translate the point z =0 or z =1 on complex plane

Möbius strip shown in Figure 30. Thus, in our program, weiθ

plane), assuming z= e ,have attempted to realize a process in which infinity and anti-

leaps from 0 to ∞.

In current coordinate displays such as logarithmic graphs,

infinity are connected automatically in division by zero, which

whenconventionally has necessitated error processing.

©︎2021 Digital Hollywood University

∞ (z

21

leaps to the ideal point, an error occurs and the

DHU JOURNAL Vol.08 2021

program stops. However, we have confirmed that, in aby zero, which hitherto has been neglected, ignored, andcanavoided in mathematics, physics, and computer science,

coordinate display using the extended Riemann sphere,

calculation signal processing, a signal

be handled as the north pole of the Riemann sphere, thusis handled using

eliminating the need for error processing. The north pole of aappears that maintains continuity after the topological space

Riemann sphere of radius 1 simply means“1”to a computer.

transition, and a sigmoid waveform appears unintentionally, as
shown in Figure 33.

Further, we have confirmed that infinity and anti-infinity
are connected, as in the graph shown on the right side in

While it is difficult to say that we have succeeded in

Figure 32. More specifically, if we assume the point whererecreating human neural connections, if nothing else, we have=∞ is translated to point 0 on another complex plane∞confirmed that analog control related to long-term memory and(converse z plane), by using a generalized Riemann sphereshort-term memory in LTD/LTP connections, and neural ON/

transformation (mapping ∞ to 1), M g (z ) can be handled as a

OFF digital signal processing, can be recreated simultaneously

continuous function even at an infinite point. As a result, thein a quantal connection. In this sense, it is undeniable thatnature of M g (z ) on the z plane can be passed on as M g (z ) to

our experiment has been an important step forward in thethe converse z plane after dispersal. Moreover, effects fromdevelopment of the artificial ego.

the converse z plane will also appear in the z plane. This
reciprocal interference, which can be seen as noise, can now
be conceived as a basis for controlling the converse z planefrom the z plane.
In light of the foregoing, in confirming the actions of theartificial nerves based on our proposed quantum theory modelin which commonly used sigmoid functions are not used
as activation functions, we have confirmed that division by
zero and coordinate expression of infinity using the novel
calculation standards can be reproduced in calculations
by means of topological space transition. That is to say,when division by zero is performed, an infinity coordinate

Figure 33: Discussion of results of experimentis exceeded in a different topological space (unconnected
output side synapse) and a signal appears. At this time,

6. Wrap-Up

if the input coordinate (synapse acting as input phase) is
set to the blue area in Figure 32, a curve such as that inWhile this may not be a quantum mind theory of a Nobel

the graph appears as a continuous waveform. Then, thePrize winner in Physics, the manifestation of a natural

signal disappears when the infinite dimension M g (z )=∞ issigmoid waveform in an emergence-imitating artificial nerve

reached, and in that instant the signal also appears, having

in which extended Riemann sphere-based quantum gates

maintained the same curve characteristic (function), at anwere implemented as operating principles is nonetheless an

output coordinate (synapse acting as output phase) in the red

important discovery, in that it links black holes, white holes,

area on the right side of Figure 32. In other words, we haveemergence, and neural connections. We have shown that theconfirmed that, in an artificial nerve created using quantalengineering applications of the extended Riemann sphere,

principles, when a stimulus reaches the limit of the infinite, awhich reconciles relativity theory and quantum mechanics, dophase transition occurs to 0, at which the stimulus disappears.

not lie in the distant future and will spark immediate evolution

Furthermore, we have confirmed that, in the coordinate

in the artificial nerve and artificial ego engineering fields.

expression of infinity in the artificial nerve, a monotonically

7. Challenges Going Forward

increasing smooth transfer function that satisfies f(0)=0 in aform that diverges to f(x)=∞ at x=∞ and to f(x)=-∞ at x=-∞ will

With respect to applications in engineering fields, togenerate sigmoid functions. In other words, we have seen

achieve an artificial ego that emerges autonomously, it will

that, when the coordinate expression of infinity in the artificial

be necessary to implement networks incorporating artificial

nerve constructed in the experiment is used, diverse sigmoid

nerves that take the extended Riemann sphere and novel

functions can be generated automatically by means of acalculation standards as an operating principle, and confirm

transfer function that diverges at the ideal point.

their behavior.

5.3 Discussion

schools have noted the feasibility of models that recreate

With respect to applications in medical fields, medical
stage gates for illnesses, and have begun research on them.

The confirmation of a similarity between a neural firing rate
and a sigmoid waveform is a widely-known fact in engineeringWith respect to applications in physics fields, because ourfields. As a result, sigmoid functions are now used astheory predicts the existence of anti-waves, we began laser

activation functions in neural networks.

physics experiments in 2020. We can expect that, if theseIn the experiment, we implemented, on a current computer,experiments confirm infinite dimension orthogonality at the

an artificial nerve model based on a quantum theory model

infinitesimal, we will be one step closer to the elucidation ofthat should be referred to as a quantal connection model, andfield energy. This will be due to the fact that, as shown in

did not use sigmoid functions.

Figure 34, black hole-level energy exists in infinitesimal space
as a matrix vector, in the sense articulated by Feynman.[14]The results of the experiment confirmed that when division©︎2021 Digital Hollywood University

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DHU JOURNAL Vol.08 2021

[11] Michael A. Nielsen, Isaac L. Chuang, "Quantum
Computation and Quantum Information", Cambridge University
Press (2010), p.13-16.
[12] JOSEF GRUSKA,“Ryōshi Konpyūta no Kiso [Fundamentalsof Quantum Computing]”, Morikita Publishing Co., Ltd. (2003),
p. 66.
[13] Shunji Mitsuyoshi,“Onsei Kanjō Ninshiki oyobi Jōdō no Nō
Seiri Shingō Bunseki Shisutemu ni kansuru Kenkyū [Research on
Figure 34: Infinite dimension matrix vectors crossing
orthogonally at the infinitesimal

the phonetic recognition of emotion and a system for emotional
physiological brain signal analysis]”, Doctoral Dissertation,
Tokushima University (2006).

Acknowledgments[14] R.P. Feynman, R.B. Leighton, M. Sands, "The Feynman

Our research has benefited tremendously from the astuteLectures on Physics, Vol. III: The New Millennium Edition:

guidance of Dr. Tadao Nakamura. We wish to express ourQuantum Mechanics (English Edition)", Basic Books (2015),

deepest respect and gratitude to Dr. Nakamura for instantly

p.11-7.

perceiving our research’s deep connection to quantum
phenomena that exhibit Klein bottles and Möbius strips withgaps, and for teaching us that quantum gates acting as
oracles can be expressed mathematically.
We are also deeply thankful to Softbank Group Corp.
Representative Director Masayoshi Son, who entrusted us
with the task of giving Pepper emotion and has supported our
research tirelessly.

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Pāsonaru Robotto‘Pepper’wo 6-gatsu hatsuka yori Hanbai
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©︎2021 Digital Hollywood University

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