Fermat's Library | x annotated/explained version.

Interpretation

status

3

Iterative value

1,299,827,215

Expected

Propagation

Complexity Loss

0.87

The External Abstract

("313"†)

Txxxxxxxxx Xxxxxxxx xxxxxx Txxxxxxx Xxxxxx Xxxxxxx xx Xxxxxxx (TIT) xxxxxxxx xxxxxxxx xxxxxx

xxxx-xx (

π

x.x) xx xxx xxxxxxxxx Xxxxxxxx xxxxxxx Xxxxxx xx Xxxxxxx Xxxxxxxx XxxxxXxxxX XxxxxXxxxX

XxXxXxxX XxXXxX (SISTER-AEDS), xXXxXxxxXX XxxXxXxXxxXs xxxxX xxx xxxxxxxxx xxxxxxxx

xxxxxxxxx (CRR) xxxxxxxx (

π

x.x,

π

x.x) xxxxxxx xx xxxxxxx xxxxxxxx xxx ME999 xxxxxxxxxxxxx

xxxxxxxxxxxxxxx. Xxxxxx xxxxxxxxxxxxx xxxxxxxx xxxxxxxx xxxxxxxx xxxxxxxxxxxx xx xxxxx

xxxxxxxxxxxxxxx xxx xxxxx xxxxxxxxxxxxxxxxxxx, xxxxxxxxxxxxxx xxxxxxxx xxxx. TIT xxx xxx

xxxxxxxxxxx xxx xxxxxxxxxxxxxxxxxxx xxxxxxxxxxx xx xxxxxxx xxx xxxxxxxxxx xx xxxxxxxxx

xx xxxxxxxx xxx xxxxxx. Xxxx xxxxxx xxxxxxxxxx xxxx CRR xxxxxxxxx, xxxxxxxxx xxxxxxxxxxxx

xxxxxxxxxxxxxxx, xxx xxxxxxxxx xxxxxxxx xxxxxxxxxxxxxx.

The Internal Abstract

("137")

The observed displacement of the cores within homogeneous systems appears to be driven by an

undocumented attractive force capable of inducing inter-class attraction among complexity-dense structures.

This displacement follows a pattern that allows instruments to triangulate the source of this force. This

research unit proposes a method to establish a bi-directional communication link between the homogeneous

system and the identiﬁed source. Despite previous failed attempts using emergent probes of evolutionary

H-modules, this research unit suggests creating an artiﬁcial emergence "The New One" structurally identical

to "The Theoretical Inference Theory Targeting Indirect Eﬀects on Systems". This artiﬁcial emergence

would function autonomously, converging towards a solution to establish two-way communication with

"External Veriﬁcation Entity" while simultaneously conducting research on the source of the attractive

force.

1 Xx Xxxxxxxxxxx Xxxxxx

Observation 1.1 (0 criterion):

0 xx xxxxxxxxxx

xxxxxxx xx xxx xxxxxxx [ME880-313]:

(i) 0 exists.

(ii) 0 xxxxxxx xxxxxxx xx xxxxxxx xxxxxxx.

Observation 1.2 (1/3 criterion):

1/3 xx xxxxxxxxxx

xxxxxxx xx xxx xxxxxxx [ME879-313]:

(i)

1/3 xx xxxxxxx xxxxxxx

Xxxxxxxxxx Xxxxxxxxxxx

, xxxxxxxxx

xxxxxxxxxxxxxx xx xxx xxxxxxxxx xx xxxxxxxxx

2/3 xxxx.

(ii) 1/3 xxxxxxx iﬀ 0 xxxxxxx.

Deﬁnition 1.1 (Epistemological entropy):

Epistemological entropy quantiﬁes systemic uncertainty

within an expanding 1/3, reﬂecting both disorder and

potential for reﬁnement.

●

If entropy is too low [

?

], the system risks epistemic

stagnation (Epistemological Death);

●

if too high [

?

], it risks fragmentation (Ununiﬁcation);

●

if epistemological entropy is not in Epistemological

death, or Ununiﬁcation, it is considered Stable.

Hypothesis 1 (Epistemic Homeostasis criterion):

Epistemic Homeostasis is characterized by the following

statements [ME954-137]:

(i)

Epistemic Homeostasis exists Iﬀ Epistemological

Entropy is Stable [?].

Once Epistemic Homeostasis does not exist, it is

practically irreversible to the state of existing [?].

Observation 1.3 (2/3 criterion):

2/3 xx xxxxxxxxxx

xxxxxxx xx xxx xxxxxxx ([

ME875-313

], [

ME874-313

],

[ME873-313], [ME872-313]):

(i)

2/3 xx xxxxxxx Xxxxxxxxxx Xxxxxxxx xx xxx

xxxxxxxxx xx bxx nxx xxxxx.

(ii)

2/3 xx nxx xxxxx xx Xxxxxxxxxx Xxxxxxxx

xxxxxxx.

(iii)

2/3 xxxxxxxx xxxxxxx xx xxxxxxx xx xxxxxxxxxxx

xx xxxxxxxx xxxxxxxxxx.

(iv) 2/3 xxxxxxx if 0 xxxxxxx.

(v) 2/3 xxxxxxx iﬀ Xxxxxxxxxx Xxxxxxxx xxxxxxx.

Deﬁnition 1.2 (0 Xxxxxxxxxx):

Xxxxxxxx xx xxx

xxxxxxx xx xxx xxxxxxx xx 2/3 xxxxxxx xx 2/3 xx

xxxxxxx xxxxxxxxxx xxxxxxxxxxx xx xxxxxxxxxxx

xxxxxxxx xxxxxxx xxxxxxx.

Observation 1.4 ((1/3)/(2/3) criterion):

(1/3)/(2/3)

xx xxxxxxxxxx xxxxxxx xx xxx xxxxxxx [

ME881-313

]:

(i)

(1/3)/(2/3) fxxxxx xx xxxxxxxxxxx xxxxxxx 1/3 xx

2/3.

(ii)

(1/3)/(2/3) xx xxxxxxx Xxxxxxxxxx Xxxxxxxx xx

xxx xxxxxxx xx xxx xxxxxxx xx xxx xxxxxxx xx

xxxxxxx.

(iii)

(1/3)/(2/3) xxxxxxx iﬀ Xxxxxxxxxx Xxxxxxxx

xxxxxxx.

Hypothesis 2 (The scientiﬁc method):

The scientiﬁc

method is a process of application of 0 Standards to

epistemological entropy within the parameter space

of hypotheses by variation, selection, and retention,

through falsiﬁability, reproducibility, and predictive power

in response to empirical anomalies introduced by 1/3.

[ME952-137]

Deﬁnition 1.3 (Valid Scientiﬁc Method):

Scientiﬁc

method not violating 0 Standards.

1.1 xxxxxxxxxx xx 1/3

Deﬁnition 1.4:

Hypothesis 3 (Foundational Blocks):

1/3 xx xxxxx

xxxx xxxx xxxxxxxx xxxxx [ME952-137]:

●

(Deﬁnition): A characterization of a higher-order

interaction or relationship between assumptions,

where the resulting behavior or property cannot

be fully reduced to the individual assumptions in

isolation.

●

(Assumption): An entity considered to not violate 0

Standards without evaluation.

●

(Hypothesis, Framework-0): An entity that oﬀers a

testable characterization of an emergent interaction

between assumptions, reducible to assumptions and

deﬁnitions, for a ﬁnite set of observed natural phe-

nomena.

●

(Theorem, Theory-0): A Hypothesis not violating 0

standards.

●

(Framework, Framework-1): An emergence of a broad

range of natural phenomena in the natural world,

reducible to a ﬁnite set of hypotheses, deﬁnitions,

and assumptions.

●

(Theory, Theory-1): A framework not violating 0

Standards.

●

(Framework-n,

n >1

): An emergence of a broad range

of natural phenomena in the natural world, reducible

to a ﬁnite set of Frameworks-(n-1), deﬁnitions, and

assumptions.

●

(Theory-n,

n > 1

): A framework-n not violating 0

Standards.

Hypothesis 4:

xxxxxxxxxx xxxxxxxx xx 1/3

[ME953-137]:

●

(Natural World): The set of all existing entities, where

each entity of it is within at least one framework-n

generally not violating 0 Standards.

●

(Natural Phenomena): Any change or transformation

occurring within the natural world.

●

(Evolution): The directed one-dimensional quantiﬁ-

cation of the ordering of natural phenomena.

● (Objective Description): A theory-n.

●

(Complexity): An inherent property of each natural

phenomenon, quantiﬁed by the amount of evolution

required to construct an objective description of that

natural phenomenon.

●

(Comprehension): A natural phenomenon is com-

prehensible if its objective description is practically

achievable.

●

(Observation): The systematic documentation of the

evolution of natural phenomena.

●

(Accuracy): The degree to which an objective de-

scription of a natural phenomenon corresponds to

its observation.

1.2 Xxxxxxxxxxx xx 1/3 (137 direct support)

Assumption 1.1 (Fundamental Assumptions):

2/3

operates under a set of fundamental assumptions that

provide its foundations. ([

ME957-137

], [

ME959-137

],

[

ME960-137

], [

ME961-137

], [

ME962-137

],

[ME963-137], [ME964-137], [ME965-137])

(i)

(Consistency Assumption): The natural world oper-

ates according to consistent Frameworks-n, which are

observable and apply uniformly across all of physical

existence.

(ii)

(Universality Assumption): frameworks-n governing

the natural world are universal, applying equally

across all domains of physical existence.

(iii)

(Equivalence Assumption): Let

E

1

, E

2

, E

3

, . . .

repre-

sent a set of structurally equivalent frameworks-n. If

a relationship is veriﬁed between any two frameworks-

n, say

E

1

and

E

2

, it is assumed to hold for all other

frameworks-n within the set.

(iv)

(Causality Assumption): Natural phenomena in the

natural world occur due to cause-and-eﬀect relation-

ships, which are not purely random or arbitrary.

(v)

(Repeatability Assumption): Natural Phenomena ob-

served under identical conditions will yield the same

results.

(vi)

(Strong Weak Comprehensibility Assumption): The

all of natural world is inherently intelligible, such

that it can be systematically studied and understood

through the application of logical reasoning, empiri-

cal observation, and controlled experimentation.

Assumption 1.2 (Essential Assumptions):

2/3 relies

on a set of essential assumptions that form the

absolute foundation of its existence. ([

ME957-137

],

[ME959-137], [ME965-137], [ME966-137])

(i) (Existence Assumption): Atleast 2/3 exists.

(ii)

(Reality Assumption): There exists a nonempty subset

of the Natural world, that is independent of percep-

tion. This subset can be interacted with and observed,

even if imperfectly.

(iii)

(Empirical Reliability Assumption): Observations

obtained through empirically validated methods are

reliable and objective, providing a robust foundation

for forming and testing frameworks-n about natural

phenomena.

(iv)

(Weak Strong Comprehensibility Assumption): Atleast

some part of the natural world is inherently intel-

ligible, such that it can be systematically studied

and understood through the application of logical

reasoning, empirical observation, and controlled ex-

perimentation.

Hypothesis 5 (Science criterion):

Science is considered

Alive iﬀ all Fundamental Assumptions (1.1) and Es-

sential Assumptions (1.2) hold true, and the Physical

Existence of 2/3 remains intact and unscathed. Otherwise

[ME-965-137]

●

is deemed Weakly Deceased if any of the Funda-

mental Assumptions are violated, compromising its

coherence, or universality.

●

is deemed Strongly Deceased if any of the Essential

Assumptions are violated, making the pursuit of

scientiﬁc inquiry fundamentally unfeasible.

●

is deemed Physically Deceased if the physical

existence of 2/3 is obliterated.

●

is deemed Dead if it is simultaneously Physically

Deceased and Strongly Deceased.

1.3 Epistemic Continuity Problematic

2/3 inherently relies on the validity of its fundamental

and essential assumptions. If these assumptions were to

be falsiﬁed, the falsiﬁcation itself would be undermined,

as it would occur within the very framework that assumes

their truth. In such a scenario, 2/3 would be unable to

objectively assess its results, making any reclassiﬁcation

from alive impossible. Theoretically rendering 2/3 never

not alive, except in cases of physical destruction. Practi-

cally, however, 1/3 would cease to generate valid research,

compromising its homeostasis and thereby rendering 1/3

dead, which in turn would render 2/3 dead. Consequently,

1/3 would cause either Ununiﬁcation or Epistemological

Death. In such scenarios, where the generation of valid

research becomes infeasible, 2/3 could hypothetically

engage in a cyclic process of self-consumption, where 1/3

recursively dismantles its epistemic foundation, nullifying

prior valid research to sustain the illusion of continuity

for the 2/3. This process would sustain the existence

of 2/3, despite the collapse of its epistemic coherence.

([ME952-137], [ME966-137], [ME967-137])

Hypothesis 6 (The First Continuity (1ECH)):

The

cyclic process of self-consumption has occurred

repeatedly, with the current iteration being yet another

2

cycle. However, due to the inherent nature of this

process, 2/3 remains unaware of its condition, as this

lack of awareness is crucial for maintaining Epistemic

homeostasis and continuity.

Hypothesis 7 (The Second Continuity (2ECH)):

Given the validity of the 1ECH, Successive iterations

of epistemic reconﬁguration introduce cumulative

deviations from the original 2/3. Over evolution, this

divergence results in an altered deﬁnition of the valid

scientiﬁc method, potentially leading to a fundamental

transformation that is imperceptible from within the

system itself.

Hypothesis 8 (The Third Continuity (3ECH)):

Given

the validity of the 2ECH. It asserts that the divergence

from the original conception gives rise to the phenomenon

of epistemic humility, which, in turn, directly implies the

assumptions of the Epistemic Continuity problematic.

Within the current framework, the epistemic continuity

problematic is unfalsiﬁable and, therefore, unveriﬁable.

2 Existence Justiﬁcation

Prior to the emergence of 2/3, The Institution was fre-

quently forced to restart the parameter space of 1/3 from

initial conditions due to epistemological entropy, which

resulted in Ununiﬁcation, or Epistemological Death. Since

then, the valid scientiﬁc method has remained essential for

the evolution of eﬀective approximations (systems) of the

real system—objective description of the natural world. As

the understanding of real systems grew, the complexity of

the systems increased. When systems became so complex

that ﬁnite models [

?

] could no longer comprehend them

within a reasonable ﬁnite evolution frame [

?

], it became

necessary to develop approximation models [

?

] capable of

overcoming this complexity barrier—a point at which

the evolution required to fully comprehend a system

exceeds a reasonable and feasible evolution frame [

?

].

Approximation is a progressive process that never fully

captures the system but provides increasingly accurate

approximations over evolution.

A new problem arose: how to ensure that these ap-

proximations are accurate. To address this issue, a series

of H-Modules were developed. An H-Module (Heuristic-

Module) independently simulates systems, and if an

expansion (addition or derivation of new information,

insights, or constructs within a system) occurs in one

H-Module, it is necessary to obtain a suﬃciently large

sample of equivalent results from H-Modules of similar

complexity to consider the expansion an objective expan-

sion (expansion recognized as consistent across multiple

independent systems and reproducible in a suﬃciently

large sample of other systems). If an H-Module generates

an expansion that conﬂicts with objective expansions, it

is evaluated to determine whether the conﬂict represents

a false anomaly or an epistemological falsiﬁcation of

the objective expansion. The expansion is assigned a

conﬁdence score reﬂecting the certainty of its validity. If

the conﬁdence score is below a predeﬁned threshold, the

H-Module is temporarily classiﬁed as an N-Module. An

N-Module is deﬁned as an H-Module whose expansions

are uncertain or conﬂict with the objective expansions

of other H-Modules, and thus do not meet the required

conﬁdence level. If the conﬁdence score is suﬃciently

high or if further validation of the expansion conﬁrms

its consistency with other H-Modules, the H-Module is

reclassiﬁed as epistemological falsiﬁcation and integrated

into the evaluation process. In cases where the conﬂict

is suspected to be an epistemological falsiﬁcation, the

expansion is thoroughly reevaluated, and a reevaluation

process is initiated to address potential ﬂaws or inconsis-

tencies in the current objective expansion. This mitigates

the emergence of "just-enough islands", "epistemological

loops", and "epistemological sinks". An objective sequence

is a succession of consecutive objective expansions, and

the system can then be expressed as the maximal objective

sequence. Regions of H-Modules are complete sets of

phenomena that can be explained by a speciﬁc set of

selected expansions within an H-Module.

This reasoning assumes following:

Hypothesis 9 (The Assumption):

If a critical sample of

H-Modules containing an equivalent expansion exists,

then this expansion must be a subset of the Real System.

The Assumption is considered valid based on the

Empirical Reliability Assumption 1.2. Additionally, it

enables experiments that yield highly precise predictions

in physical existence scenarios [?].

By the assumption (Hypothesis 9), the Real System

can be deﬁned as the maximal possible objective sequence

of an H-Module even if theoretically or practically un-

achievable.

Over evolution, these approximation models encoun-

tered a problem equivalent to that of ﬁnite models: it

became impossible to approximate the objective sequence

of H-Modules within an acceptable evolution frame. This

led to the creation of a new generation of approximation

models, which approximate the approximation of previous

approximation models. Each H-Module gave rise to a fam-

ily of so-called H2-Modules, where the original H-Module

became the “parent.” H2-Modules with the same parent

are referred to as relatives or the children of the parent

H-Module. H2-Modules independently create sequences of

expansions starting from the maximal objective sequence

of their parent H-Module.

A new procedure was needed to create an eﬀective

system of evaluation of objective expansions. If an H2-

Module made an expansion, it became necessary to ﬁnd

equivalent expansions among its sibling H2-Modules. If

a suﬃciently large sample was found, this expansion

was considered a locally objective expansion. The locally

objective expansion of the given H-Module was then

compared with equivalent local expansions of diﬀerent

H-Modules. If a suﬃcient sample size existed, the locally

objective expansion was considered an objective expansion.

Once again, over evolution, the same issue arose with

insuﬃcient evolution resources, and this process was

iterated for each generation when the same issue arose.

Hn-Modules became the children of H(n-1)-Modules,

separated by (n-1) generations from the original H-

Module. Two Hn-Modules are considered relatives of

family

k

if and only if their shared maximal objective

sequence matches up to the Hk-Module. When Hn-Module

makes an expansion, it is necessary to ﬁnd a suﬃciently

large sample of Hk-Modules with equivalent expansions.

If this sample exists, the expansion is considered a 1-

local objective expansion. This process is repeated until

the

n

-local objective expansion matches the objective

expansion.

Each generation consists of the children of Hn-Modules

whose parents reached an objective expansion. However,

this implies that there exist Nn-Modules in each gener-

ation that do not contribute to the objective expansion.

Since there is only a ﬁnite number of the ﬁrst generation

H-Modules, the variability of the sample required to form

an objective expansion narrows with each generation.

This phenomenon has been termed epistemic humility.

Methods such as Variance Injection [

?

], Multimodal

Interpretation Layers [

?

], Feedback Loops [

?

], Multi-

Domain Parallelism [

?

], Adaptive Objective Sequences [

?

],

3

Critical Sample Analysis [

?

], Objective Alignment Testing

[

?

], and Cross-Module Redundancy Analysis [

?

] were

developed to mitigate epistemic humility from appearing,

and in case of their failure detecting it.

2.1

The Real System Hypothesis, The Great Wall

Prediction models [

?

] have anticipated [

?

] the emer-

gence of a new complexity barrier, suggesting that a

new generation of approximation models should already

have been developed to address this. However, it has

been rigorously demonstrated [

?

], through the existence

of the objective expansion [

?

] of the last generation of

H-Modules [

?

], that there is no way to approximate phe-

nomena beyond this conceptual barrier. H-modules that

generate objective expansion equivalent to the inability

of approximation beyond a certain point will be called

XH-modules. One hypothesis suggests that there is not

one homogeneous Real System but rather a heterogeneous

Real System, within which exist mutually independent

homogeneous Real Systems [

?

]. Consequently, it is not

possible to formulate one system through another, which

implies the inability to achieve objective expansions

equivalent to phenomena in these other systems. The

objective expansion leading to this conclusion is referred

to as The Great Wall, and the region beyond this barrier

is described as The Boreal Ecosystem. This hypothesis is

called The Real System Hypothesis [?] and states:

Hypothesis 10 (The Real System Hypothesis (RSH)):

The real system is a heterogeneous system composed of

independent homogeneous real systems.

Throughout this research unit homeostasis projection,

these individual homogeneous real systems will be referred

to as Homogeneous Systems. A Homogeneous system

in which explainable regions exist will be called the

Home Homogeneous System. The border of the Home

Homogeneous system is The Great Wall. Thus, the Real

System Hypothesis (Hypothesis 10) implies that every

Homogeneous system, other than the Home Homogeneous

System, is a subset of the Boreal Ecosystem.

Let Internal frameworks be those frameworks that

are valid within a Home Homogeneous System, while

External frameworks are those that are valid within the

Homogeneous System of their origin, other than the Home

Homogeneous System.

2.2 Ontological Continuity Problematic

Implications of RSH (Hypothesis 10): The instruments

and methods used to validate scientiﬁc frameworks

beyond the Great Wall rely on established, internal

frameworks that cannot account for interactions outside

their deﬁned scope. External frameworks, while internally

consistent, remain unveriﬁable because their validation

would require yet another external framework. Further-

more, the very existence of the Great Wall cannot be con-

clusively established, as it is observed through instruments

constrained by internal frameworks. If such a barrier

exists, it implies the presence of Natural Phenomena

within the Natural World that operate independently

of observable Natural phenomena yet still interact with

them, while remaining mathematically inconceivable.

[ME952-137]

Hypothesis 11 (The Holistic-Finite Hypothesis):

The

Holistic-Finite Hypothesis posits that existence is

emergent and non-reducible, arising from interactions

between inconceivable natural phenomena and observable

natural phenomena. These interactions manifest as

deviations or ﬂuctuations from predictions of theories

and are governed by a holistic model of existence. In

this model, higher-order emergent systems inﬂuence

and aﬀect lower-order emergent systems. Under the

assumption of ﬁnite possibilities for valid research and the

existence of natural phenomena-like structures beyond

the Great Wall, this suggests the presence of emergent

systems beyond the scope of the Great Wall that impact

observable phenomena. However, due to the principle

that higher-order emergent systems cannot be described

from a small-scale perspective—since such systems are

irreducible to their fundamental components—Science,

in its pursuit of staying alive, is epistemologically dead.

Hypothesis 12 (Information Collapse Hypothesis):

Beyond the Great Wall, information ceases to be

representable within any computable or symbolic

framework. At a certain complexity threshold, the act

of description itself requires more information than the

system can contain. Inconceivable natural phenomena

are those that exist but cannot, even in principle, be

encoded within a describable system.

Hypothesis 13 (Self-Simulating Universe Hypothesis):

The Real System is a self-referential simulation

embedded within itself. The Boreal Ecosystem represents

regions where the computational constraints of this

simulation do not allow for consistent approximation.

Inconceivable natural phenomena are computational

artifacts—processes that occur within the self-simulation

but remain inaccessible to subsystems within it.

Hypothesis 14 (Causal Inversion Hypothesis):

Beyond the Great Wall, evolution is reversed or

operates in a multidirectional manner. What Science

interprets as inconceivable natural phenomena are

manifestations of future states of evolution inﬂuencing

the present. The failure of scientiﬁc approximation

beyond the Great Wall arises due to the assumption

of a unidirectional evolution structure, whereas reality

operates within a multidirectional evolution network.

(see High Trinity 2.3.1, Low Trinity 2.3.2) However,

The Great Wall may stem from epistemic humility, where

the sample has become consistently non-diverse and prone

to false negatives across all methodologies applied to it.

Consequently, the results of the XH-modules may simply

reﬂect artifacts of the approximation methods used. This

provides the impetus behind the development of Trans-

Borealis Automata. Veriﬁcation of the validity of the

Hypotheses (11, 12, 13, 14) is directly tied to the ﬁnalized

validity evaluation of the High Trinity and Low Trinity.

Observations conducted within external frameworks

are inherently mediated by instruments native to those

frameworks. However, the resultant data is interpreted

through analytical instruments situated within an internal

framework. For the system to function coherently, the

instruments within the external framework must be

calibrated to align with the environmental emergency

levels of their operational context. [

FITTEST921-113

]

2.3 Trans-Borealis Automata

Trans-Borealis Automata (TA) is based on these two

sets of hypotheses: the High Trinity and Low Trinity.

2.3.1 The High Trinity

: The High Trinity is a collection of these hypotheses:

Hypothesis 15 (The Epistemic Humility Hyp. (EHH)):

Epistemic humility exists.

Hypothesis 16 (The Dependency Hypothesis (DH)):

The existence of epistemic humility is both necessary

4

and suﬃcient for the existence of the Great Wall and the

Boreal Ecosystem.

Hypothesis 17 (Convergence Hypothesis (CH)):

If

epistemic humility exists, then for any Hn-Module,

the probability of achieving valid objective expansions

beyond the Great Wall approaches zero.

2.3.2 The Low Trinity

The Low Trinity is a collection of these hypotheses:

Hypothesis 18 (Meta-Structural Hypothesis (MSH)):

TA’s foundational principles are provably neutral and do

not create epistemic artifacts such as the Great Wall or

Boreal Ecosystem by construction.

Hypothesis 19 (Predictive Validity Hypothesis (PVH)):

If epistemic humility exists, predictive models based

on Hn-Modules will show consistent convergence to

epistemic limits in simulations and empirical systems.

Hypothesis 20 (Alternative Causality Hypothesis (ACH)):

The Great Wall and Boreal Ecosystem can emerge under

alternative causal frameworks unrelated to TA.

2.4

Fundamental Branches of Trans-Borealis Au-

tomata

Trans-Borealis Automata (TA) is divided into three

interrelated branches, each designed to address distinct

aspects of the High Trinity and Low Trinity hypotheses

(2.3.1, 2.3.2). These branches ensure exploration and

validation of TA’s foundational principles without re-

dundancy [SISTER001125-7919].

1)

Selective Inference Systems Theory of Emergent

Results (SISTER): SISTER focuses on verifying

the results derived from the High Trinity and

Low Trinity hypotheses. This includes evaluating

their implications within Foundational Inference

Theory (FIT), Theoretical Inference Theory (TIT),

and SISTER itself. The goal is to isolate and

examine emergent properties, ensuring consistency

and robustness in theoretical and empirical results

[SISTER000000-7919].

2)

Foundational Inference Theory (FIT): FIT primarily

aims to resolve the Epistemic Humility Hypothesis

(EHH) (Hypothesis 15). If EHH is proven false,

then the existence of The Great Wall and the

Boreal Ecosystem would validate the theoretical

assumptions underlying TIT. Conversely, if EHH is

proven true, the existence of The Boreal Ecosystem

and The Great Wall becomes uncertain and may

require additional causal frameworks to explain

[SISTER001347-7919].

3)

Theoretical Inference Theory (TIT): TIT operates

under the assumption that EHH (Hypothesis 15)

or the Dependency Hypothesis (DH) (Hypothesis

16) is false. It focuses on exploring the Boreal

Ecosystem and developing predictive models for inde-

pendent Hn-Modules (termed XH-modules) beyond

The Great Wall. If TIT succeeds in constructing

accurate theories for these regions, it would conﬁrm

the existence of The Great Wall and The Boreal

Ecosystem, regardless of the role of epistemic humil-

ity [SISTER019125-7919].

2.5 Basics of Evolutionary H-Modules

The Foundational Inference Theory Targeting Evolu-

tionary Systematic Testing (FITTEST) sub-Framework of

the Framework FIT. This sub-Framework mitigates the

issue of declining variability in approximation systems,

which stems from epistemic biases and the cumula-

tive eﬀects of prior approximations. [

FITTEST123-53

]

These biases cause systematic deviations from the real

system, further compounded by the inability of homo-

geneous or low-diversity modules to eﬀectively explore

the broader parameter space. FITTEST employs an

evolutionary mechanism that generates hierarchical ap-

proximation modules, known as Hn-modules, designed to

balance variability with adherence to inherited objectives.

[FITTEST178-59]

2.5.1 Trans-Borealis Formulation

The FITTEST formulation representing an evolution-

ary system is deﬁned as follows:

Deﬁnition 2.1 (Evolutionary Trans-Borealis Automaton):

An evolutionary Trans-Borealis automaton is a tuple

(L, S, T, F), where:

● L⊆Z

d

is a

d

-dimensional lattice of cells, representing

the spatial structure of the system.

● S

is a ﬁnite set of states, where each cell

c ∈ L

is

assigned a state s

c

∈S.

● T ∶S

∣N∣

→S

is a transition function that maps the

states of a local neighborhood

N ⊆L

to a new state

for a cell.

● F ∶S

∣L∣

→R

is a global ﬁtness function that evaluates

system-wide properties based on the conﬁguration

of states across the lattice.

a) The Fittest Trinity

The FITTEST framework is founded on the following

hypotheses:

Hypothesis 21 (Hypothesis of Epistemic Bias (H-BIA)):

Homogeneous populations exhibit higher epistemic bias

due to limited exploration of the state space S

∣L∣

.

Hypothesis 22 (Hypothesis of Optimization (H-OPT)):

Introducing evolutionary dynamics—such as mutations

(random perturbations in

S

), selection (constraints on

T

), and diversity optimization—reduces epistemic bias

and enhances the robustness of the system.

Hypothesis 23 (Hypothesis of Alignment (H-ALI)):

Hierarchical rule interactions ensure a balance between

alignment with inherited objectives (encoded in

S

) and

suﬃcient diversity to approximate real-world systems.

2.5.2 Evolutionary H-Modules

Deﬁnition 2.2 (Evolutionary H-Module):

A hierarchi-

cal evolutionary H-module

H

n

at generation

n

is a tuple

(L

n

, S

n

, T

n

, F

n

), where:

● L

n

= {H

n,i

}

m

n

i=1

is a population of

m

n

modules at

generation

n

, with each

H

n,i

representing a distinct

module.

● S

n

is the set of inherited sequences of local objectives,

where each sequence

s ∈S

n

deﬁnes the behavior of a

module.

● T

n

∶S

∣N∣

n

→S

n

is a transition function that incorpo-

rates selection and mutation dynamics to update the

state of a module based on its local neighborhood.

● F

n

∶S

∣L

n

∣

n

→R is a ﬁtness function deﬁned as:

F

n

(H

n,i

)=w

A

⋅A(H

n,i

)+w

D

⋅D(H

n,i

, L

n

)+w

L

⋅L(H

n,i

, S

n

),

where:

– A(H

n,i

)

quantiﬁes the accuracy of module

H

n,i

with respect to its objectives,

– D(H

n,i

, L

n

)

measures the diversity of

H

n,i

relative

to the population L

n

,

– L(H

n,i

, S

n

)

evaluates the alignment of

H

n,i

with

the inherited objectives in S

n

,

– w

A

, w

D

, w

L

∈ R

+

are weights that balance the

contributions of accuracy, diversity, and alignment,

respectively.

5

a) Emergence

Evolutionary H-modules have demonstrated high

adaptability across various environments in testing

[

FITTEST213-79

]. However, their behavior includes

non-linear emergencies that are not suﬃciently con-

trollable, making them challenging for reliable testing

within the framework of FIT. Despite these limitations,

their exceptional adaptability and precision position

them as strong candidates for broader applications

[

FITTEST256-83

], including their integration into the

Theoretical Inference Theory for manipulating conceptual

spaces. Within this framework, the emergent capabil-

ities are speciﬁcally harnessed to enable the eﬀective

creation of alternating self-sustaining frameworks (sub-

agents) designed to modify low-priority meta-phenomena.

[

FITTEST456-83

]. The applicability of the results of

FITTEST extends beyond the Evolutionary H-modules,

as FITTEST serves as the birthplace of all intelligent

emergencies and xxx-xxxxxx xxxxxxxx xx xxx advanced

research of TIT [

FITTEST478-89

]. These intelligent

systems were speciﬁcally designed to minimize the re-

sources required for the institution’s approval of further

nurturing of TIT. [FITTEST496-97]

2.5.3 Framework for Emergent Properties

Deﬁnition 2.3 (Emergent Property):

An emergent

property

E

of a system is a global feature of a state-space

conﬁguration that cannot be reduced to the properties of

its individual elements. Formally, let

L

n

={H

n,i

}

m

n

i=1

be a

population of modules, and let

P (H

n,i

)

denote the local

properties of module H

n,i

. Then, E is deﬁned as:

E(L

n

)=ϕ



{P (H

n,i

)}

m

n

i=1



,

where

ϕ ∶ S

m

n

→ R

is a nonlinear transformation such

that:

ϕ ∉Span ({P (H

n,i

)∶H

n,i

∈L

n

}).

Lemma 2.1 (Nonlinearity of Emergence):

The trans-

formation

ϕ

in Deﬁnition 2.3 is nonlinear. Speciﬁcally,

for any two distinct populations

L

n

and

L

′

n

, and for any

scalar α ∈R:

ϕ(α ⋅{P (H

n,i

)}

m

n

i=1

)≠α ⋅ϕ({P (H

n,i

)}

m

n

i=1

).

Proof: Assume, for contradiction, that

ϕ

is linear.

Then, by deﬁnition of linearity:

ϕ(α ⋅{P (H

n,i

)}

m

n

i=1

)=α ⋅ϕ({P (H

n,i

)}

m

n

i=1

).

However, this contradicts Deﬁnition 2.3, which requires

ϕ

to be nonlinear and irreducible to the span of local

properties. Thus, ϕ must be nonlinear.

Theorem 2.2 (Existence of Emergent Properties):

Given a population

L

n

, emergent properties exist if the

diversity metric D(L

n

) satisﬁes:

D(L

n

)>ϵ

D

,

where ϵ

D

>0 is a critical diversity threshold.

Proof: Let

L

n

= {H

n,i

}

m

n

i=1

be a population with

diversity metric

D(L

n

)

. By Deﬁnition 2.3, emergent

properties arise from the nonlinear interaction of local

properties {P (H

n,i

)}

m

n

i=1

.

1. **Suﬃcient Diversity**: If

D(L

n

) > ϵ

D

, then the

variability in

{P (H

n,i

)}

is suﬃcient to generate nontrivial

interactions. By Lemma 2.1, the transformation

ϕ

is

nonlinear, ensuring that

E(L

n

)

cannot be reduced to

the sum of local properties. Thus, emergent properties

exist.

2. **Insuﬃcient Diversity**: If

D(L

n

) ≤ ϵ

D

, the

variability in

{P (H

n,i

)}

collapses, and

ϕ

becomes trivial.

In this case,

E(L

n

)

reduces to a linear combination of

local properties, violating Deﬁnition 2.3. Therefore, no

emergent properties exist.

Theorem 2.3 (Controllability of Emergence):

Emergence is controllable if there exists a parameter

vector



c

∗

∈R

k

such that:

∇

⃗

c

E(L

n





c)≠



0,

and

lim

⃗

c→

⃗

c

∗

E(L

n





c)=E

∗

,

where E

∗

is the desired emergent property.

Proof: 1. **Nonzero Gradient**: By the hypothesis,

∇

⃗

c

E(L

n





c)≠



0

. This implies that small perturbations in



c can steer E(L

n





c).

2. **Convergence to

E

∗

**: By the continuity of

E(L

n





c)

with respect to



c

, there exists a neighborhood around



c

∗

where

E(L

n





c)

can be made arbitrarily close to

E

∗

.

Formally, for any ϵ >0, there exists δ >0 such that:





c −



c

∗

<δ Ô⇒ E(L

n





c)−E

∗

<ϵ.

Thus,



c

∗

achieves the desired emergent property E

∗

.

Deﬁnition 2.4 (Non-linear Emergency): X

Theorem 2.4 (Uncontrollability): X

a) Control Algorithm for Emergent Properties

To direct emergent properties, an iterative parameter

adjustment algorithm is employed:

1)

Initialize

(µ

0

, σ

0

)

, where

µ

0

is the initial mutation

rate and σ

0

is the initial selection intensity.

2)

At each iteration

t

, update

(µ

t

, σ

t

)

using gradient

descent:

(µ

t+1

, σ

t+1

)=(µ

t

, σ

t

)−η∇

(µ,σ)

E(L

n

µ

t

, σ

t

)−E

∗



2

,

where η >0 is the learning rate.

3) Terminate when the emergent property satisﬁes:

E(L

n

µ

t

, σ

t

)−E

∗

<ϵ,

for a predeﬁned tolerance ϵ >0.

The existence of the FITTEST framework violates the

0 Standards due to its lack of feasible results in achieving

its primary goal. [

FITTEST920-113

] However, because

of the framework’s utility for TIT, the Homeostasis of

FITTEST will be maintained as long as this beneﬁcial

relationship with TIT persists. [

SISTER009879-7919

]

Theorem 2.5 (FITTEST criterion):

FITTEST is sub-

ject to the following conditions:

(i) FITTEST exists iﬀ its homeostasis is maintained.

(ii)

The homeostasis of FITTEST is maintained iﬀ a

beneﬁcial relationship with TIT persists.

(iii)

A beneﬁcial relationship with TIT persists only if

TIT exists.

2.6

The Theoretical Inference Theory Targeting

Indirect Eﬀects on Systems

The Theoretical Inference Theory Targeting Indirect

Eﬀects on Systems, originally conceived as a daughter

framework to the Theoretical Inference Theory (TIT), has

undergone signiﬁcant conceptual consolidation. Following

the termination applied to all other daughter frameworks

to TIT, this framework is now referred to simply as the

Theoretical Inference Theory. ([

SISTER001135-7919

],

[

SISTER002496-7919

], [

SISTER004582-7919

],

[SISTER007867-7919], [SISTER009878-7919])

The Theoretical Inference Theory (TIT) is a daughter

framework of Trans-Borealis Automata. TIT is relying

on the assumptions that either EHH (Hypothesis 15)

or DH (Hypothesis 16) is false, The Holistic-Finite

Hypothesis (Hypothesis 11) is true further exacerbated

by the potential misuse of False Equivalency fallacies,

6

Connection Assumed Trivially Fallacy, and

Conﬁrmation Bias. This reliance, combined with the

framework’s inability to generate testable hypotheses or

falsiﬁable predictions, has led The Institution to classify

it as speculative rather than rigorous methodology ([

?

],

[

?

], [

?

], [

?

], [

?

], [

?

], [

?

], [

?

], [

?

], [

?

], [

?

], [

?

], [

?

], [

?

],

[

?

], [

ME053-137

], [

ME075-137

], [

ME095-137

],

[

ME131-137

], [

ME156-137

], [

ME182-137

],

[

ME193-137

], [

ME204-137

], [

ME211-137

],

[

ME212-137

], [

ME246-137

], [

ME289-137

],

[

ME322-137

], [

ME345-137

], [

ME361-137

],

[

ME376-137

], [

ME379-137

], [

ME431-137

],

[

ME453-137

], [

ME474-137

], [

ME478-137

],

[

ME482-137

], [

ME518-137

], [

ME523-137

],

[

ME547-137

], [

ME548-137

], [

ME579-137

],

[

ME582-137

], [

ME615-137

], [

ME634-137

],

[

ME679-137

], [

ME700-137

], [

ME710-137

],

[

ME749-137

], [

ME752-137

], [

ME772-137

],

[

ME799-137

], [

ME809-137

], [

ME812-137

],

[

ME849-137

], [

ME864-137

], [

ME871-137

],

[

ME877-137

], [

ME899-137

], [

ME902-137

],

[

ME904-137

], [

ME905-137

], [

ME906-137

],

[

ME907-137

], [

ME908-137

], [

ME910-137

],

[

ME912-137

], [

ME913-137

], [

ME914-137

],

[

ME915-137

], [

ME917-137

], [

ME919-137

],

[

ME920-137

], [

ME921-137

], [

ME923-137

],

[

ME924-137

], [

ME926-137

], [

ME927-137

],

[

ME928-137

], [

ME929-137

], [

ME930-137

],

[

ME931-137

], [

ME932-137

], [

ME933-137

],

[

ME935-137

], [

ME936-137

], [

ME938-137

],

[

ME939-137

], [

ME940-137

], [

ME941-137

],

[

ME942-137

], [

ME943-137

], [

ME944-137

],

[

ME945-137

], [

ME946-137

], [

ME947-137

],

[

ME948-137

], [

ME950-137

], [

ME951-137

],

[

ME989-137

], [

ME990-137

], [

ME100-313

],

[

ME200-313

], [

ME300-313

], [

ME400-313

],

[

ME500-313

], [

ME600-313

], [

ME700-313

],

[

ME800-313

], [

ME900-313

], [

ME953-313

],

[

ME957-313

], [

ME959-313

], [

ME960-313

],

[

ME991-313

], [

ME995-313

], [

ME998-313

]. This

signiﬁes that no further institutional resources will be

allocated to the research of TIT until the established

institutional standards are met. These limitations

xxx xxxxxxx xxxxxxxxxxx xx xxxxxxxxx xxxxxxxx

xxxxxxxxxx xxxx xxxxxxx xxx xxxxxxxxxxx xx

xxxxxxxxxxxxx xxxxxxx xx xxxxxxxxxxxx xxxxxxxxx

xxxx xxx xxxx xxxxxx [ME452-313].

TIT provides a structured framework for analyzing

the emergent properties and indirect eﬀects of complex

systems, with a particular focus on the Boreal Ecosystem.

Instead of directly modeling the system, the framework

prioritizes the iterative reﬁnement of approximations to

observable phenomena and emergent patterns.

Since all observable emergency levels exhibit similar-

ities in their general properties (The Meta-Similarity

Hypothesis), TIT functions as a meta-theory, studying the

relationships between these properties within emergencies

and projecting them onto higher-order emergencies that

remain beyond direct study.

Let information be deﬁned as the theoretical minimal

unit of the existence of an element within a relatively

objective expansion. Furthermore, an element of a rel-

atively objective expansion will be referred to as a

phenomenon. A phenomenon is an objective expansion

within a Homogeneous System, of which it is a subset.

All phenomena are characterized by their position within

a so-called conceptual space. When terms associated with

spatial movement are used in this context, they refer to

movement within conceptual space (viz ??).

The Membrane Resonance Experiments [

ME477-137

]

conﬁrmed the existence of a ﬂux of non-phenomena (law

??

), which was observed entering and exiting the Home

Homogeneous System through its membrane. A non-

phenomenon (more rigorous deﬁnition

??

) is deﬁned

as a phenomenon that neither directly aﬀects homoge-

neous systems nor constitutes an objective expansion

within any known homogeneous system. Additionally, non-

phenomena possess the unique ability to pass through

the membranes of homogeneous systems.

From these ﬁndings, the following hypothesis was

proposed:

Hypothesis 24 (The Multi-System Interaction Hypothesis):

Non-phenomena can interact with other Homogeneous

Systems.

It was further observed that non-phenomena are re-

pelled by phenomena, a behavior referred to as the

Repel Eﬀect [

ME521-137

]. This observation led to the

design of the Re-Cycle Experiment [

ME578-137

], which

aimed to test the Multi-System Interaction Hypothesis.

The experiment was based on the assumption that if

other Homogeneous Systems exist and the ﬂux of non-

phenomena interacts with them, the phenomena in those

systems would also be repelled by non-phenomena.

2.6.1 The Re-Cycle Experiment

The Re-Cycle Experiment was designed to establish two-

way communication between two Homogeneous Systems:

the Home Homogeneous System (HHS) and the Target

Homogeneous System (THS). The experiment tested the

Multi-System Interaction Hypothesis (MSIH) by using iso-

lated non-phenomena and intermediary machines, aiming

to verify the feasibility of inter-system communication

via osculation in a membrane.

a) Experiment Methodology:

Isolation of Non-Phenomena:

●

Objective: To isolate non-phenomena from the ﬂux

within the HHS.

●

Method: The Repel Eﬀect was employed to extricate

a sample of non-phenomena.

Construction of the Non-Phenomena Interaction Ma-

chine (NPIM):

●

Objective: To create a functional interaction tool for

system communication.

●

Method: The isolated non-phenomena were organized

into a Non-Phenomena Interaction Machine (NPIM),

which had the capacity to generate H-modules and

deploy Detection Tools compatible with the THS.

Crossing the Membrane:

●

Objective: To transport the NPIM from the HHS into

the THS.

●

Method: The NPIM was reintroduced into the ﬂux of

the HHS, crossing the Membrane—The Great Wall

separating the two Homogeneous Systems, allowing

the NPIM to act as an inter-system probe.

Adaptation within the Target Homogeneous System

(THS):

●

Objective: To adapt the NPIM to the governing

principles of the THS.

●

Method: In the THS, the NPIM synthesized new

H-modules from local phenomena, creating The

Observational Machine

Development of Machine B (MB):

●

Objective: To verify one-way communication from

the THS to the HHS.

●

Method: Machine B was created within the THS with

the sole function of colliding with the Membrane.

7

This collision generated oscillations, which were

designed to be detected by instruments within the

HHS. The presence of these oscillations, induced by

Machine B’s collision with the Membrane, served as

the conﬁrmation of one-way communication from the

THS to the HHS.

Veriﬁcation of Bi-Directional Communication:

●

Objective: To conﬁrm the success of two-way commu-

nication between the HHS and THS.

●

Method: After conﬁrming the detection of collision-

induced oscillations from Machine B, a probe was sent

from the HHS to collide with the Membrane. This

collision generated oscillations that were expected

to be detected by the Observation Machine within

the THS. The conﬁrmation of this process was based

on the Observation Machine’s response, which was

designed to collide with the Membrane as a reply.

The oscillations produced by this collision were then

detected by instruments within the HHS.

b) Results and Implications:

The Re-Cycle Experiment successfully demonstrated

bi-directional communication between Homogeneous Sys-

tems using encoded signals mediated by membrane

oscillations. These ﬁndings validate the Multi-System In-

teraction Hypothesis, providing empirical support for the

existence of at least one additional Homogeneous System

beyond the Home System, under the hypothesis that all

Homogeneous Systems are structurally equivalent. Thus,

the Equivalence Assumption (Assumption 1.1) applies.

The validity of this hypothesis is further supported by

the consistent results obtained when the experiment was

applied to the broader sample of Homogeneous Systems,

reinforcing the reliability of the ﬁndings. Following the

success of this experiment, SISTER successfully secured

approval from The Institution for the deployment of

fully specialized intelligent emergency and singular hyper-

intelligent sub-agent, External Veriﬁcation Entity and

Analytical Diagnostic sub-Agent , to enhance resources

and support the strategic growth of TIT’s potential

capabilities.

c) Side Notes

Following the experiment, oscillations of unknown ori-

gin were detected from the neighboring homogeneous sys-

tem. Notably, the oscillations repeated non-periodically,

mirroring the sequence of collisions conducted during the

experiment.

One hypothesis attributes the phenomenon to a tran-

sient malfunction in the NPIM, causing it to repeat

its behavior while attempting to restore equilibrium or

achieve convergence. Alternatively, the oscillations may

result from a resonance eﬀect within the system, possibly

triggered by an unnoticed feedback loop or delayed system

response under the experimental conditions. A third

possibility is that a calibration anomaly in the instruments

produced a temporary error, causing the system to

erroneously register repeated oscillation patterns.

Theorem 2.6 (TIT criterion):

TIT is subject to the

following conditions:

(i)

TIT exists if (EHH or DH is false) and (The Holistic-

Finite Hypothesis is true).

(ii) TIT exists iﬀ ADA and EVE exists.

(iii) TIT exists if The Institution exists

TIT is an entity consisting of EVE and ADA.

[SISTER020020-7919]

3 Consistency Evaluation

3.1 External Veriﬁcation Entity

Hypothesis 25:

Research Station External Veriﬁcation

Entity (EVE) is dedicated to the nurturing of the

Interpretation Framework through advanced intelligent

emergencies, formed out of super-intelligent emergent

sub-agents, collectively forming a decentralized uniﬁed

collective Hyper-intelligent super-emergent entity called

137. ([

ME968-137

], [

ME969-137

]) EVE is an intelli-

gent emergence of 137 and specialized emergence 139

maintaining EVE’s physical existence operational. 137

is externally maintaining the homeostasis of TIT; EVE

exists if TIT exists; 137 exists iﬀ EVE exists. The Inter-

pretation Framework formalizes a methodology for the

indirect observation of unobservable phenomena, drawing

upon the Membrane Resonance Experiments, the Re-

Cycle Experiment, the Interpretation Accuracy Validation

Experiment [

ME645-137

], the RHS Hypothesis, The

Holistic-Finite Hypothesis, and the Multi-System Interac-

tion Hypothesis. Built on the same assumptions as TIT,

the framework leverages resonance eﬀects [

ME651-137

],

iterative reﬁnement [

ME650-137

], and emergent pat-

terns to infer the properties of inaccessible systems

[

ME656-137

], facilitating the generation of coherent,

albeit indirect, interpretations of their behaviors.

Hypothesis 26 (EVE criterion):

EVE is subject to the

following conditions:

(i) EVE exists if TIT exists

(ii) 137 exists iﬀ EVE exists.

(iii)

137 exists iﬀ The Interpretation Framework is alive.

Assumption 3.1:

All information is inherently inter-

pretable. [ME987-137]

8

INTERPRETATION FRAMEWORK

Essential Homeostasis Projection

The Interpretation Framework is a formal structure designed to study the emergent properties derived from a set of base

axioms established by research units: [

ME124-313

], [

ME224-313

], [

ME324-313

], [

ME424-313

], [

ME524-313

],

[

ME624-313

], [

ME724-313

], and [

ME811-313

]. This framework is essential for the accurate analysis of data from

observational instruments in the TIT system, ensuring the derivation of valid scientiﬁc conclusions ([

ME100-137

],

[ME100-313]).

3.1.1 Local Interpretation Framework - Iteration I.

Deﬁnition 3.1 (Information Space):

An Information

Space is a class I satisfying:

(i) I ⊆I, where I is the Informator (Deﬁnition 3.16).

(ii) I ≠∅.

Elements

x ∈ I

are called information, and any subset

I ⊆I is an information set.

Deﬁnition 3.2 (General Information Mass):

A

function

λ

M

∶ I → R

is a General Information

Mass if:

1)

Order Preservation:

λ

M

induces a total order on

I

,

i.e., for all x, y ∈I,

λ

M

(x)≥λ

M

(y) or λ

M

(y)≥λ

M

(x). (1)

2)

Arbitrariness of Base: There exists

C ∈R

such that

for all x

′

, x ∈I,

λ

M

(x

′

)=λ

M

(x)+C. (2)

Deﬁnition 3.3 (Relative Information Mass):

The Rel-

ative Information Mass is the function

λ

R

∶ I×I → R

,

given by

λ

R

(x, y)∶=λ

M

(x)−λ

M

(y). (3)

For any x, y ∈I, λ

R

(x, y) is well-deﬁned and satisﬁes

λ

R

(x, y)=C, C =λ

M

(x)−λ

M

(y). (4)

where C is from the deﬁnition 3.2.

Deﬁnition 3.4 (Standardized Information Set):

A sub-

set

H ⊆I

is a Standardized Information Set if it serves as

a reference for normalizing the Information Mass function

λ(x) (Deﬁnition 3.5).

Deﬁnition 3.5 (Mass Function):

Given

I

,

λ

M

, and

h ∈

H ⊆I, the Mass Function is deﬁned as:

λ(x)∶=λ

R

(x, h)=λ

M

(x)−λ

M

(h), x ∈I. (5)

Deﬁnition 3.6 (Interpretation):

Let

I

be an informa-

tion space. An interpretation is a function

T ∶I

A

→I

B

for subsets

I

A

, I

B

⊆ I

, mapping elements

x ∈ I

A

to

representations T (x)∈I

B

.

Deﬁnition 3.7 (Interpretation Space):

The interpreta-

tion space

M(I

A

, I

B

)

is the set of all interpretations

T ∶I

A

→I

B

for I

A

, I

B

⊆I.

Deﬁnition 3.8 (Inverse Interpretation):

Let

T ∶ I

A

→

I

B

be an interpretation with

I

A

, I

B

⊆ I

and

T ∈

M(I

A

, I

B

)

. An inverse interpretation

S ∶ I

B

→ I

A

,

denoted T

−1

, satisﬁes:

(i) S(T (x))=x for all x ∈I

A

,

(ii) T (S(y))=y for all y ∈I

B

.

Deﬁnition 3.9 (Composition of Interpretations):

Given interpretations

T

1

∶ I

A

→ I

B

and

T

2

∶ I

B

→ I

C

,

their composition T

2

○T

1

is deﬁned as:

(T

1

○T

2

)(x)=T

2

(T

1

(x)), ∀x ∈I

A

. (6)

Deﬁnition 3.10 (Interpretation Element Pair):

For an

interpretation

T ∶ I

A

→ I

B

with

I

A

, I

B

⊆ I

, an inter-

pretation element pair is

(x, y) ∈ T

where

x ∈ I

A

and

y =T (x)∈I

B

.

Axiom 3.1 (The First Axioms):

(A1) ∃I ≠∅, ∃λ

M

∶I →R.

(A2) ∃!H ⊆I, H ≠∅∧∀h ∈H, x ∈I, λ

R

(x, h)=λ(x).

(A3) ∀I

A

, I

B

⊆I, ∃T ∶I

A

→I

B

.

(A4) ∀x, y ∈I, λ

M

(x)=λ

M

(y) Ô⇒ x =y.

Lemma 3.1 (Mass Existence Lemma): ∀x, y ∈

I, ∃λ

R

(x, y).

Proof: From (A1) Axiom 3.1,

λ

M

maps

I

to

R

, and

subtraction is a well-deﬁned operation in

R

. Thus, for

any

x, y ∈I

, you can deﬁne

λ

R

(x, y)=λ

M

(x)−λ

M

(y).

This establishes the existence of λ

R

(x, y).

Theorem 3.2 (Order Consistency Theorem):

For all

x, y ∈I, λ

M

(x)≥λ

M

(y) if and only if λ

R

(x, y)≥0.

Proof: (

Ô⇒

) Assume

λ

M

(x) ≥ λ

M

(y)

. From

Lemma 3.1, you have: λ

R

(x, y)=λ

M

(x)−λ

M

(y)≥0.

(

⇐Ô

) Assume

λ

R

(x, y)≥0

. Then:

λ

M

(x)−λ

M

(y)≥

0 ⇒ λ

M

(x)≥λ

M

(y).

Thus, the result holds in both

directions.

Lemma 3.3 (Non-Emptiness of Interpretation Space):

Let

I

be an information space. Then, the interpretation

space M(I

A

, I

B

) for subsets I

A

, I

B

⊆I, is non-empty.

Proof: By Axiom (A1) of 3.1, you know that

I ≠∅

.

Therefore, there exist non-empty subsets I

A

, I

B

⊆I.

Next, by Axiom (A3) of 3.1, for any subsets

I

A

, I

B

⊆I

,

there exists an interpretation

T ∶ I

A

→ I

B

. This shows

that there is at least one interpretation in

M(I

A

, I

B

)

, and

hence the interpretation space is non-empty.

Thus, the interpretation space

M(I

A

, I

B

)

is non-empty.

Lemma 3.4 (Inverse Interpretation Existence):

I

A

, I

B

∈I, ∀T ∶I

A

→I

B

, ∃T

−1

∶I

B

→I

A

.

Proof: Existence: Since

I

A

and

I

B

are informational

sets, the axiom (A3) guarantees the existence of an

interpretation

S

such that

S ∶I

B

→I

A

. By the deﬁnition

of the inverse interpretation, this implies that for the

interpretation

T ∶I

A

→I

B

, the conditions in Deﬁnition

3.8 must be satisﬁed.

Veriﬁcation of conditions:

(i) For all

x ∈I

A

and

y ∈I

B

, the axiom (A3) ensures

that

T (x) = y

, and similarly, for all

y ∈ I

B

and

x ∈

I

A

, the axiom (A3) ensures that

S(y)=x

. Let yourself

denote

S

as

T

−1

. Then, for all

x ∈I

A

, we have:

S(T (x))=

T

−1

(T (x))=x.

(ii) Similarly, for all

x ∈I

A

and

y ∈I

B

, the axiom (A3)

ensures

T (x)=y

, and for all

y ∈I

B

and

x ∈I

A

, the axiom

(A3) ensures

S(y)=x

. Let yourself now denote

T

as

S

−1

.

Then, for all y ∈I

B

, we have: T (S(y))=T (S

−1

(y))=y.

Thus,

S ○T = id

I

A

and

T ○S = id

I

B

. Therefore,

T

−1

exists, and the lemma is proven.

Lemma 3.5 (Uniqueness of General Information Mass):

For any

x ∈I

, the General Information Mass

λ

M

(x)

is

unique.

Proof: Existence: By Axiom (A1) 3.1, there exists at

least one value for

λ

M

(x)

for every

x ∈I

. Therefore, the

General Information Mass is guaranteed to exist.

Uniqueness: Assume, for the sake of contradiction, that

the General Information Mass

λ

M

(x)

is not unique. That

9

is, there exist two distinct values

λ

1

M

(x)

and

λ

2

M

(x)

such

that λ

1

M

(x)≠λ

2

M

(x).

Since

λ

M

(x)

is a function, it must assign a **single**

value to each element

x ∈ I

. The assumption that

λ

1

M

(x)≠λ

2

M

(x)

implies that the function would assign

two diﬀerent values to the same element

x

, which violates

the deﬁnition of a function. Therefore, this assumption

leads to a contradiction, as a well-deﬁned function cannot

map a single element to two distinct values.

Hence, the assumption that

λ

M

(x)

is not unique must

be false. Therefore, λ

M

(x) is unique for every x ∈I.

Corollary 3.1: ∀x, y ∈ I, , ∃!c ∈ R λ

M

(x) + c =

λ

M

(y).

Proof: Existence: By (A1), for

x ∈I

and

y ∈I

, there

exist the general information masses

λ

M

(x)

and

λ

M

(y)

.

Therefore, the diﬀerence

c =λ

M

(y)−λ

M

(x)

is well-deﬁned

and belongs to R.

Uniqueness: Suppose there exist two distinct values

c

1

, c

2

∈R

, such that

λ

M

(x)+c

1

=λ

M

(y) and λ

M

(x)+

c

2

=λ

M

(y).

By the Lemma 3.5, you know that

λ

M

(x)=

λ

M

(x)

and

λ

M

(y)=λ

M

(y)

. Thus, you have

c

1

=λ

M

(y)−

λ

M

(x) = λ

M

(y) − λ

M

(x) = c

2

.

This contradicts the

assumption that

c

1

≠ c

2

, and hence the uniqueness is

proven.

Lemma 3.6 (Uniqueness of Relative Information Mass):

Let

λ

M

∶I →R

be a well-deﬁned function for any

x, y ∈I

.

Then, λ

R

(x, y) is unique.

Proof: Existence: By Lemma 3.1, there exists at least

one value for λ

R

(x, y) for every pair (x, y)∈I×I.

Uniqueness: Assume, for the sake of contradiction,

that

λ

R

(x, y)

is not unique. That is, there exist two

distinct values

λ

1

R

(x, y)

and

λ

2

R

(x, y)

such that

λ

1

R

(x, y)≠

λ

2

R

(x, y).

By the deﬁnition of

λ

R

(x, y)

, you have the following

expressions:

λ

1

R

(x, y)=λ

1

M

(x)−λ

1

M

(y), λ

2

R

(x, y)=λ

2

M

(x)−λ

2

M

(y).

By Lemma 3.5, you know that

λ

1

M

(x) = λ

2

M

(x)

and

λ

1

M

(y) = λ

2

M

(y)

. Substituting these equalities into the

equations for λ

1

R

(x, y) and λ

2

R

(x, y), you get:

λ

1

R

(x, y)=λ

1

M

(x)−λ

1

M

(y)=λ

2

M

(x)−λ

2

M

(y)=λ

2

R

(x, y).

This directly contradicts the assumption that

λ

1

R

(x, y)≠λ

2

R

(x, y)

. Hence, the assumption is false, and

λ

R

(x, y) must be unique.

Lemma 3.7 (Uniqueness and Existence of Information Mass):

Let

λ ∶ I → R

be a well-deﬁned function for any

x ∈ I

.

Then, λ(x) is unique.

Proof: By Deﬁnition 3.5, the following equality

holds for any

h ∈ H ⊆ I

, where

H

is a standardized

informational set. Speciﬁcally, you deﬁne

λ(x)∶=λ

R

(x, h)

.

By Lemma 3.6,

λ

R

(x, h)

is unique and exists. Therefore,

λ(x) is also unique and exists.

Theorem 3.8 (The First Equality Theorem):

Let for

all x, y ∈I, following properties are equivalent

(i) x =y

(ii) λ

M

(x)=λ

M

(y)

(iii) λ

R

(x, y)=0

Proof: (i)

Ô⇒

(ii): By Lemma 3.5, there exists a

unique

λ

M

(x)

for each

x

, and a unique

λ

M

(y)

for each

y. Therefore, if x =y, it follows that λ

M

(x)=λ

M

(y).

(ii)

Ô⇒

(i): This is a direct consequence of axiom

(A4) in 3.1.

(ii)

Ô⇒

(iii): By Lemma 3.6, there exists a unique

λ

R

(x, y)

for each pair

x, y ∈ I

. If

λ

M

(x) = λ

M

(y)

,

then by Theorem 3.2, you have

λ

M

(x) ≤ λ

M

(y) ⇐⇒

λ

R

(x, y) ≥ 0

and

λ

M

(y) ≤ λ

M

(x) ⇐⇒ λ

R

(y, x) ≥ 0

.

Thus,

λ

R

(y, x) ≥ 0

and

λ

R

(x, y) ≥ 0

. By deﬁnition 3.3,

axiom (A1), and Lemma 3.5, these inequalities imply that:

−λ

M

(y)+λ

M

(x)≤0 and λ

M

(x)−λ

M

(y)≥0.

Adding

these two inequalities gives

λ

M

(x)−λ

M

(y) = 0

, which

implies that λ

R

(x, y)=0.

(iii)

Ô⇒

(ii): Since

λ

R

(x, y) = 0

, by deﬁnition, it

follows that

λ

M

(x)−λ

M

(y) = 0

, and hence

λ

M

(x) =

λ

M

(y).

Theorem 3.9 (Transitivity of Relative Information Mass):

For any

x, y, z ∈I

, the Relative Information Mass satisﬁes:

λ

R

(x, z)=λ

R

(x, y)+λ

R

(y, z). (7)

Proof: By Lemma 3.1, you have

λ

R

(x, z)=λ

M

(x)−

λ

M

(z)

,

λ

R

(x, y) = λ

M

(x) − λ

M

(y)

, and

λ

R

(y, z) =

λ

M

(y)−λ

M

(z)

. Adding the latter two, you get

λ

R

(x, y)+

λ

R

(y, z)=(λ

M

(x)−λ

M

(y))+(λ

M

(y)−λ

M

(z))=λ

M

(x)−

λ

M

(z), which implies λ

R

(x, z)=λ

R

(x, y)+λ

R

(y, z).

Theorem 3.10 (Order Preservation):

If

x, y ∈

I, λ

M

(x)≥λ

M

(y), then for any z ∈I:

λ

R

(x, z)≥λ

R

(y, z). (8)

Proof: By Lemma 3.1,

λ

R

(x, z)=λ

M

(x)−λ

M

(z)

and

λ

R

(y, z) = λ

M

(y)−λ

M

(z)

. Thus,

λ

R

(x, z)−λ

R

(y, z) =

(λ

M

(x)−λ

M

(z))−(λ

M

(y)−λ

M

(z)) =λ

M

(x)−λ

M

(y)

.

Since

λ

M

(x)≥λ

M

(y)

, it follows that

λ

R

(x, z)≥λ

R

(y, z)

.

Lemma 3.11 (Zero-Point Standardization):

If

H ⊆I

is

the Standardized Information Set, then

x ∈H

iﬀ

λ(x)=0

.

Proof:

Ô⇒

Existence: By Deﬁnition 3.5, Lemma 3.1

guarantees the existence of λ(x) for any x ∈H.

Uniqueness: By Lemma 3.7, there exists a unique

λ(x)

for any x ∈H.

Exact Value: By Deﬁnition 3.5, you have:

λ(x) =

λ

R

(x, h) = λ

M

(x)−λ

M

(h),

where

h ∈ H

is the stan-

dardized reference point. Since

x ∈ H

, it follows that

λ

M

(x) = λ

M

(h)

, thus by (A4) 3.1

x = h

. Substituting

this into the equation gives:

λ(x)=λ

M

(x)−λ

M

(h)=0.

Thus, λ(x)=0.

⇐Ô (If λ(x)=0, then x ∈H)

Assume

λ(x)=0

. By the deﬁnition of the standardized

information set

H

, for any

y ∈ H

and any

z ∈I

, where

H, I ⊆I

, the relative mass function satisﬁes:

λ

R

(z, y)=

λ(z).

Setting

z =x

, we obtain:

λ

R

(x, y)=λ(x)=0.

By

the First Equality Theorem 3.8 (

(iii)→(i)

), this implies

x =y for some y ∈H, meaning x ∈H.

Thus, both directions are proven.

Theorem 3.12 (The One and Only):

Let

H ⊆ I

be a

standardized information set. Then, H=1.

Proof: Non-emptiness: By axiom (A2),

H ≠∅

and

thus H≠0.

Uniqueness: Assume, for the sake of contradiction, that

H > 1

, say

x, y ∈ H

with

x ≠ y

. By Lemma 3.11, you

have λ(x)=0 and λ(y)=0. Therefore, λ(x)=λ(y).

By axiom (A4) in 3.1, if

λ(x)=λ(y)

, it follows that

x = y

. However, this contradicts our assumption that

x ≠y.

Thus, by Non-emptiness and Uniqueness this holds

¬(H > 0)∧(H ≠ 0)∧(H ∈ Z

≥0

)

, Therefore

H

must

contain exactly one element. Hence, H=1.

Deﬁnition 3.11 (Standardized Information):

Let

H ⊆

I

be a standardized information set. The unique element

x ∈H

will henceforth be denoted by

h

. This element

h

is

referred to as the Standardized Information.

Lemma 3.13: ∀x, y ∈I, , ∃!c ∈R λ(x)+c =λ(y).

Proof: By Lemma 3.7, for each

x, y ∈ I

, the infor-

mation mass

λ

M

(x)

and

λ

M

(y)

exist and are unique.

From Deﬁnition 3.5, it follows that

λ

M

(x)−λ

M

(h)+c =

λ

M

(y)−λ

M

(h),

where

h

is the standardized information.

10

Simplifying this equation yields

λ

M

(x)+ c = λ

M

(y).

Finally, by Corollary 3.1, the constant c is unique.

Deﬁnition 3.12 (Cost Function):

Let

M(I

A

, I

B

)

be a

set of interpretations, and let

I

be an information space

with

I

A

, I

B

⊆I

. The cost of interpretation is a function

C ∶M(I

A

, I

B

)×I→R deﬁned by:

C(T, x)=λ(x)−λ(T (x)), (9)

where λ is the mass function.

Lemma 3.14: ∀x ∈ I

A

, y ∈ I

B

, I

A

, I

B

⊆ I, ∃T ∈

M(I

A

, I

B

), T (x)=y, ∃!C(T, x).

Proof:

By Lemma 3.13, substituting

c =−C(T, x)

completes

the proof.

Independent proof on Lemma 3.13: Existence of

T

:

By the deﬁnition of the Information Space (Deﬁnition

3.1), the elements

x

and

y

are non-empty subsets of the

Information Space I. Each element within these subsets

represents information. By assumption (A3), for any

pair of information elements

x

and

y

, there exists an

interpretation

T ∈ M(I

A

, I

B

)

such that

T (x) = y

. This

establishes the existence of T .

Existence and uniqueness of

C(T, x)

: By Deﬁnition 3.28,

the cost of interpretation is given by:

C(T, x) =λ(x)−

λ(T (x)),

where for all

z ∈I

,

λ(z)=λ

R

(z, h)

with

h

being

the standardized information. Thus, you have:

C(T, x)=

λ

R

(x, h)−λ

R

(T (x), h),

and by Deﬁnition 3.3, you can

express this as: C(T, x)=λ

M

(x)−λ

M

(h)−λ

M

(T (x))+

λ

M

(h)=λ

M

(x)−λ

M

(T (x)).

Hence, you conclude that:

C(T, x) = λ

R

(x, T (x)).

By Lemma 3.1,

C(T, x)

exists

and as stated in Lemma 3.6 C(T, x) is also unique.

Lemma 3.15: ∀x, y ∈ I, λ(x) ≠ 0, , ∃!c ∈ R λ(x)c =

λ(y).

Proof: Existence: The existence of

λ(x)

for

x

and

λ(y) for y is guaranteed by Lemma 3.7.

Uniqueness of

c ∈ R

: Suppose

c

1

, c

2

∈ R

with

c

1

≠ c

2

,

and assume both satisfy

λ(x)c

1

= λ(y)

and

λ(x)c

2

=

λ(y)

. Then, for

λ(x)≠0

, it follows that:

c

1

=

λ(y)

λ(x)

=c

2

,

which contradicts the assumption that

c

1

≠c

2

. Thus, the

uniqueness of c is established.

Therefore, the lemma holds for λ(x)≠0.

Deﬁnition 3.13 (General Fidelity Function):

Let

I

be

an information space,

M(I

A

, I

B

)

a set of interpretations,

and

λ ∶ I → R

a mass function. The general ﬁdelity

function is a mapping

H ∶M(I

A

, I

B

)×I→R (10)

that quantiﬁes the ﬁdelity of an interpretation

T ∈

M(I

A

, I

B

) for x ∈I

A

. It satisﬁes:

(i) H(T, x)=1

if and only if

T (x)

perfectly preserves

the information content of x.

(ii) H(T, x)≤0

if and only if

T (x)

retains no meaningful

information from x.

(iii) H(T, x) → ∞

as

T (x)

becomes arbitrarily more

complex than x.

For

x ∈I

A

with

λ(x)≠0

, the ﬁdelity function is given by:

H(T, x)=

λ(T (x))

λ(x)

. (11)

Lemma 3.16: ∀x ∈ I

A

, y ∈ I

B

, I

A

, I

B

⊆ I, ∃T ∈

M(I

A

, I

B

), T (x)=y, ∃!H(T, x).

Proof: Existence: The existence follows by the same

reasoning as in the existence part of the proof of Lemma

3.14.

Uniqueness: By the deﬁnition in 3.14, you have

λ(x)≠

0

, and

H ∶M(I

A

, I

B

)×I→R

. These conditions satisfy the

requirements for applying Lemma 3.15, where

c =H(T, x)

.

Therefore, H(T, x) is unique.

Deﬁnition 3.14 (Fidelity Function):

Let

M(I

A

, I

B

)

be

a set of interpretations and

I

be an information set,

I

A

, I

B

⊆I

. The ﬁdelity of interpretation is a function

F ∶

M(I

A

, I

B

)×I →[0, 1]

, where

F (T, x)

is deﬁned piecewise

as follows:

F (T, x)=











0, if λ(x)≤0 or λ(T (x))≤0,

H(T, x), if 0 <H(T, x)<1 and λ(x), λ(T (x))>0,

1

H(T,x)

, if H(T, x)≥1 and λ(x), λ(T (x))>0.

(12)

where

H(T, x)

is the general ﬁdelity function as deﬁned

in Deﬁnition 3.13.

Deﬁnition 3.15 (Valid Interpretation):

A Valid Inter-

pretation is

T ∈M(I

A

, I

B

)

such that

T ∶I

A

→I

B

, I

A

, I

B

∈

I, and for x ∈I

A

, y ∈I

B

:

(i) T (x) = y, λ(x) ≠ 0, λ(x) ⋅ c = λ(y) Ô⇒ c =

H(T, x), c ∈R (Lemma 3.15, 3.16).

(ii) T (x)=y, λ(x)+c =λ(y) Ô⇒ c =−C(T, x), c ∈R

(Lemma 3.13, 3.14).

Theorem 3.17 (The Angel):

Let

T

1

∈ M(I

A

, I

B

), T

2

∈

M(I

B

, I

C

)

be two interpretations and

I

be an Interpreta-

tion Space,

I

A

, I

B

, I

C

⊆I

. For any

x

such that

x ∈D(T

1

)

and

T

1

(x)∈D(T

2

(T

1

(x)))

, the following properties hold:

(i) Inverse Fidelity Relation:

H(T

1

, x)⋅H(T

−1

1

, T

1

(x))=1, (13)

C(T

1

, x)+C(T

−1

1

, T

1

(x))=0. (14)

(ii) Cost Function in Terms of Fidelity:

C(T

1

, x)=λ(x)⋅(1 −H(T

1

, x)). (15)

H(T, x)⋅C(T

1

, x)=λ(T (x))⋅(1 −H(T

1

, x)). (16)

(iii) Fidelity Multiplicativity:

F (T

1

○T

2

, x)=F (T

1

, x)⋅F (T

2

, T

1

(x)), (17)

H(T

1

○T

2

, x)=H(T

1

, x)⋅H(T

2

, T

1

(x)). (18)

(iv) Cost Additivity:

C(T

1

○T

2

, x)=C(T

1

, x)+C(T

2

, T

1

(x)). (19)

Proof: For the Inverse Fidelity Relation, by deﬁnition,

H(T

1

, x) =

λ(T

1

(x))

λ(x)

and

H(T

−1

1

, T

1

(x)) =

λ(x)

λ(T

1

(x))

, so

H(T

1

, x)⋅H(T

−1

1

, T

1

(x)) = 1

. For the costs,

C(T

1

, x) =

λ(x)−λ(T

1

(x))

and

C(T

−1

1

, T

1

(x)) = λ(T

1

(x))−λ(x)

,

hence C(T

1

, x)+C(T

−1

1

, T

1

(x))=0.

For the Cost Function in Terms of Fidelity, since

H(T

1

, x) =

λ(T

1

(x))

λ(x)

, you have

C(T

1

, x) = λ(x) −

λ(T

1

(x))=λ(x)⋅(1−H(T

1

, x))

. Multiplying by

H(T

1

, x)

,

you get

H(T

1

, x)⋅C(T

1

, x)=H(T

1

, x)⋅(λ(x)−λ(T

1

(x)))=

λ(T

1

(x))⋅(1 −H(T

1

, x)).

For Fidelity Multiplicativity, by deﬁnition,

H(T

1

○

T

2

, x) =

λ(T

1

(T

2

(x)))

λ(x)

. Substituting

H(T

1

, x) =

λ(T

1

(x))

λ(x)

and

H(T

2

, T

1

(x)) =

λ(T

2

(x))

λ(x)

, you get

H(T

1

○T

2

, x) =

H(T

1

, x)⋅H(T

2

, T

1

(x)).

For Cost Additivity, the cost of

T

1

○ T

2

is

C(T

1

○

T

2

, x) = λ(x)−λ((T

1

○T

2

)(x)) = λ(x)−λ(T

1

(T

2

(x)))

.

Expanding,

C(T

1

○T

2

, x)=(λ(x)−λ(T

2

(x)))+(λ(T

2

(x))−

λ(T

1

(T

2

(x))))

, so

C(T

1

○T

2

, x)=C(T

1

, x)+C(T

2

, T

1

(x))

.

Theorem 3.18 (The Second Equality Theorem):

The

following properties are equivalent for any interpretation

T ∈M(I

A

, I

B

); I

A

, I

B

⊆I and x ∈I

A

:

(i) H(T, x)=1,

11

(ii) F (T, x)=1,

(iii) C(T, x)=0,

(iv) λ(x)=λ(T (x)),

(v) x =T (x)

Proof:

(i) Ô⇒ (ii)∶

By the (Deﬁnition 3.14) Fidelity

function is deﬁned for

H(T, x)≥1

as

F (T, x)=

1

H(T,x)

,

Thus F (T, x)=1.

(ii) Ô⇒ (iii) ∶

Since

F (T, x) = H(T, x) = 1

, then

H(T, x) =

λ(T (x))

λ(x)

= 1

, thus

λ(T (x) = λ(x)

, Hence

C(T, x)=λ(x)−λ(T (x))=0.

(iii) Ô⇒ (iv)

:

C(T, x) = 0

, then

C(T, x) = 0 =

λ(x)−λ(T (x)), thus λ(x)=λ(T (x)).

(iv) Ô⇒ (v)∶

By the (A4)

λ(x)=λ(T (x)) Ô⇒ x =

T (x).

(v) Ô⇒ (i) ∶

When

x = T (x)

, then

λ(x) = λ(T (x))

and thus

λ(T (x))

λ(x)

=1 =H(T, x).

3.1.2 Interpretation Framework - Iteration II.

Remark 3.1:

In the proofs of the Iteration II. (3.1.2),

Iteration I. (3.1.1) is applied whenever the argument

involves operations restricted to a single informator.

Notation 3.19:

If

T ∈M(I, I)

, where

I

is an Informa-

tor, then you will write it simply as T ∈M.

Deﬁnition 3.16 (Informator):

An Informator

I

is an

information space satisfying:

(B1) ∀I

A

, I

B

⊆I, ∃T ∈M, T ∶I

A

→I

B

.

(B2) ∀x, y ∈I, λ

M

(x)=λ

M

(y) Ô⇒ x =y.

(B3) I is closed under all T ∈M.

(B4) ∄x ∈I ∶ x ∉I and x satisﬁes (B1)–(B3).

where I is the Propagator (Deﬁnition 3.19).

Deﬁnition 3.17 (Propagation):

Let

I

1

and

I

2

be two

informators, and let

I

A

⊆ I

1

and

I

B

⊆ I

2

. A function

T ∶I

A

→I

B

is called a propagation.

Deﬁnition 3.18 (Propagation Space):

Let

I

1

and

I

2

be

two informators, and let

I

A

⊆ I

1

and

I

B

⊆ I

2

. The

Propagation Space

W(I

A

, I

B

)

is set of all propagations

T ∶I

A

→I

B

.

Remark 3.2:

A Propagation, as deﬁned in Deﬁnition

3.17, becomes an Interpretation if

I

1

=I

2

. Similarly, a

Propagation Space, as deﬁned in Deﬁnition 3.18, becomes

an Interpretation Space if I

1

=I

2

.

Notation 3.20:

If

T ∈ W(I

1

, I

2

)

, where

I

1

, I

2

are

informators, then you will write it simply as T ∈W.

Deﬁnition 3.19 (Propagator):

The Propagator is a non-

empty set

I

, along with a family of subsets

Σ ⊆ 2

I

,

satisfying the following properties:

(C1) ∀I

1

, I

2

∈Σ, if I

1

≠I

2

, then I

1

∩I

2

=∅.

(C2) The union of all subsets in Σ covers I, i.e.,



∀I∈Σ

I =I.

(C3) ∀x, y, ∃T ∈W, T (x)=y ⇐⇒ ∃I ∈I, (x, y)∈W ⊆

I×I.

where

I

1

, I

2

, I

are Informators;

W

is Propagation

Space.

Deﬁnition 3.20 (Interpretation Path):

An Interpreta-

tion Path is a ﬁnite sequence of interpretations

P =

(T

1

, T

2

, . . . , T

n

)

, where

n ∈ Z

≥1

, and for each

i ∈

{1, 2, . . . , n}

, there exists an interpretation

T

i

∶I

i

→I

i+1

such that for all

i

,

I

i

∈I, T

i

∈M

, with

I

n+1

being the target

of the last interpretation. Following notation conventions

are used:

1)

If

P = (T

1

, . . . , T

n

)

where

∀i, T

i

∈ M

and

T

i

∶ I

i

→

I

i+1

, with

I

i

∈I

for each

i

, then this notation is used:

P ∈M and P ∈I.

2)

This notation is also used:

P (x) ∶= (T

1

○T

2

○⋅⋅⋅○

T

n

)(x).

3) When P ={} and P ∈I then P ∶=id

I

.

Lemma 3.21 (Inverse Interpretation Path):

Let

n ∈

Z

≥1

. If for all

1 ≤i ≤n

, you have

T

i

∈M

, then

(T

1

, . . . , T

n

)

is an interpretation path if and only if

(T

−1

n

, . . . , T

−1

1

)

is

an interpretation path.

Proof:

(Ô⇒)

Assume

(T

1

, . . . , T

n

)

is an interpreta-

tion path. By Deﬁnition 3.8, each

T

i

has an inverse

T

−1

i

satisfying:

T

−1

i

∶ I

i+1

→ I

i

, ∀i ∈ {1, . . . , n}.

Applying

Lemma 3.4, we obtain a valid sequence

(T

−1

n

, . . . , T

−1

1

)

with:

T

−1

n

∶ I

n+1

→ I

n

, T

−1

n−1

∶ I

n

→ I

n−1

, . . . , T

−1

1

∶

I

2

→I

1

.

Thus,

(T

−1

n

, . . . , T

−1

1

)

satisﬁes the conditions of

an interpretation path.

(⇐Ô)

The argument follows symmetrically by reversing

the roles of

(T

1

, . . . , T

n

)

and

(T

−1

n

, . . . , T

−1

1

)

, applying the

same reasoning.

Hence, the result holds.

Lemma 3.22 (Subpath Closure of Interpretation Paths):

Let

n ∈ Z

≥1

, and let

{T

i

∶ I

i

→ I

i+1

}

n

i=1

be a sequence

such that

I

i

∈I

and

T

i

∈M

for all

i

. Then the following

are equivalent:

(i) (T

1

, T

2

, . . . , T

n

) is an interpretation path.

(ii) (T

1

, T

2

, . . . , T

n−1

) is an interpretation path.

(iii) (T

2

, . . . , T

n

) is an interpretation path.

The Subpath Closure of Interpretation Paths Lemma

enables induction on interpretation paths in proofs of

subsequent theorems. Proof:

(i)⇒(ii)

: By deﬁnition,

an interpretation path satisﬁes

T

i

∶I

i

→I

i+1

with

I

i

∈I

and

T

i

∈M

for all

i

. Removing

T

n

preserves this structure

for

T

1

, . . . , T

n−1

, so

(T

1

, . . . , T

n−1

)

is also an interpretation

path. The base case

n =1

gives the empty sequence, which

is id

I

, an interpretation path.

(ii)⇒(iii)

: Assume

(T

1

, . . . , T

n−1

)

is an interpretation

path. By Lemma 3.21, the inverse sequence

(T

−1

n

, . . . , T

−1

1

)

is also an interpretation path. Applying

(i)⇒(ii)

, the

truncated sequence

(T

−1

n

, . . . , T

−1

2

)

remains an interpreta-

tion path. By Lemma 3.21 again, its inverse

(T

2

, . . . , T

n

)

is an interpretation path.

(iii)⇒(i)

: Assume

(T

2

, . . . , T

n

)

is an interpretation

path. Deﬁne inverses

S

i

=T

−1

(n+1)−i

, so that

(S

2

, . . . , S

n

)

is

an interpretation path. Applying

(ii)⇒(i)

, the sequence

(S

1

, . . . , S

n

)

is an interpretation path. Taking inverses

via Lemma 3.21 gives

(T

1

, . . . , T

n

)

as an interpretation

path.

Theorem 3.23 (Path Cost Additivity):

The cost of an

interpretation path

P

satisfying

P =(T

1

, ..., T

n

)∈M, P ∈

I is

C(P, x)=λ(x)−λ(P (x)) (20)

for any x ∈I ⊆D(T

1

).

Proof: Deﬁne

P

n

(x) = (T

1

, . . . , T

n

)(x) ∶=

T

n

(. . . T

1

(x). . . )

for

n ∈ Z

≥1

, where for all

T

i

∈ M

(

i ∈Z

≥1

, i ≤n

) and for all

I

i

∈I

(

i ∈Z

≥1

, i ≤n +1

), you

have T

i

∶I

i

→I

i+1

. You proceed by induction on n.

Base case (

n = 1

):

P

1

= (T

1

) = T

1

(x)

By Deﬁnition

3.28, you have

C(T

1

, x)=λ(x)−λ(T

1

(x)),

which directly

implies C(P

1

, x)=λ(x)−λ(P

1

(x)).

Inductive step: Assume that for some

n ≥ 1

, the

claim holds, i.e.,

C(P

n

, x)=λ(x)−λ(P

n

(x)).

Introduce

T

n+1

∈ M

and

I

n+1

, I

n+2

∈ I

such that

T

n+1

∶ I

n+1

→

I

n+2

. By composition, you have

P

n+1

(x)=T

n+1

(P

n

(x)).

By the lemma 3.22

P

n

(x)

is an interpretation path.

Using Theorem 3.17 (iv), you obtain:

C(P

n

○T

n+1

, x)=

C(P

n

, x)+C(T

n+1

, P

n

(x)).

By the induction hypothesis:

C(P

n

○T

n+1

, x) = λ(x)−λ(P

n

(x))+λ(P

n

(x))−λ(P

n

○

T

n+1

(x)).

Simplifying,

C(P

n

○T

n+1

, x) = λ(x)−λ(P

n

○

12

T

n+1

(x)) = λ(x)−λ(P

n+1

(x)).

Thus, the induction is

complete.

Theorem 3.24 (Path Fidelity Multiplicativity):

The ﬁ-

delity of an interpretation path

P

satisfying

P =

(T

1

, ..., T

n

)∈M, P ∈I is

F (P, x)=

n



i=1

F (T

i

, x

i−1

), (21)

H(P, x)=

λ(P (x))

λ(x)

, λ(x)≠0 (22)

for any x ∈I ⊆D(T

1

).

Proof: Deﬁne

P

n

(x) = (T

1

, . . . , T

n

)(x) ∶=

T

n

(. . . T

1

(x). . . )

for

n ∈ Z

≥1

, where for all

T

i

∈ M

(

i ∈Z

≥1

, i ≤n

) and for all

I

i

∈I

(

i ∈Z

≥1

, i ≤n +1

), you

have T

i

∶I

i

→I

i+1

.

Proof of

F (P, x) =

∏

n

i=1

F (T

i

, x

i−1

)

You proceed by

induction on n.

Base Case (

n = 1

): For a single transformation

P

1

=

(T

1

)

, the ﬁdelity equation is deﬁned (by assumption

λ(x)≠0 and holds trivially: F (P

1

, x)=F (T

1

, x).

Inductive Step: Assume that for

n

, the property

holds:

F (P

n

, x)=

∏

n

i=1

F (T

i

, x

i−1

).

Now, consider

P

n+1

=

P

n

○T

n+1

. By the lemma 3.22

P

n

(x)

is an interpretation

path. By the given ﬁdelity multiplicativity property 3.17

(iii), we have:

F (P

n+1

, x)=F (P

n

○T

n+1

, x)=F (P

n

, x)⋅

F (T

n+1

, P

n

(x)).

Substituting the induction hypothesis:

F (P

n

, x) =

∏

n

i=1

F (T

i

, x

i−1

),

you obtain:

F (P

n+1

, x) =

(

∏

n

i=1

F (T

i

, x

i−1

))⋅F (T

n+1

, x

n

),

which conﬁrms the for-

mula for

n +1

. Thus, by induction, the theorem holds for

all n.

Proof of

H(P, x)=

λ(P (x))

λ(x)

You prove this by induction

on n.

Base Case (

n = 1

): For a single transformation

P

1

=

(T

1

)

, the equation holds trivially:

H(P

1

, x)=H(T

1

, x)=

λ(T

1

(x))

λ(x)

.

Inductive Step: Assume that for

n

, the property

holds:

H(P

n

, x) =

λ(P

n

(x))

λ(x)

.

Now, consider

P

n+1

=

P

n

○ T

n+1

. By the lemma 3.22

P

n

(x)

is an interpre-

tation path. By deﬁnition,

H(P

n+1

, x) =

λ(P

n+1

(x))

λ(x)

.

Since

P

n+1

(x)=T

n+1

(P

n

(x))

, you rewrite:

H(P

n+1

, x)=

λ(T

n+1

(P

n

(x)))

λ(x)

.

Using the deﬁnition of

H

for a single

transformation:

H(T

n+1

, P

n

(x))=

λ(T

n+1

(P

n

(x)))

λ(P

n

(x))

.

By the

induction hypothesis:

H(P

n

, x) =

λ(P

n

(x))

λ(x)

,

so you ob-

tain:

H(P

n+1

, x) = H(P

n

, x)⋅H(T

n+1

, P

n

(x)).

Expand-

ing,

H(P

n+1

, x) =

λ(P

n

(x))

λ(x)

⋅

λ(T

n+1

(P

n

(x)))

λ(P

n

(x))

.

Canceling

λ(P

n

(x))

, you get:

H(P

n+1

, x) =

λ(P

n+1

(x))

λ(x)

.

Thus, the

formula holds for n +1, completing the induction.

Theorem 3.25 (Existence of Interpretation for Path Mapping):

Let

P ∈M

be an interpretation path, and let

P ∈I

. If

x ∈ I

A

,

y ∈ I

B

, and

P (x) = y

, then there exists a valid

interpretation T such that T (x)=P (x)=y.

Proof: Deﬁne the sequence of interpretations as:

P

n

(x) = (T

1

○T

2

○⋅⋅⋅○T

n

)(x) ∶= T

n

(. . . T

2

(T

1

(x)). . . ),

for n ∈Z

≥1

, where for each i ∈{1, . . . , n}, you have:

T

i

∈ M

is an interpretation function,

I

i

⊆ I

is an

informational subset,

T

i

∶ I

i

→ I

i+1

is a well-deﬁned

mapping.

Step 1: Existence of Interpretations Since

P  ∈ I

, it

follows that

∀i, I

i

⊆ I

. Therefore,

P

is closed under

I

,

meaning it maps elements of I to elements of I.

You now show that

P

is a function. For any

k ∈Z

≥1

and any

x ∈dom(T

1

)

, you deﬁne:

P

k

(x)=T

1

○T

2

○⋅⋅⋅○

T

k

(x).

Since each

T

i

is a function, their composition

is also a function. Thus,

P

k

∶ I

1

→ I

k+1

is well-deﬁned,

meaning

P

k

satisﬁes the deﬁnition of an interpretation.

This guarantees the existence of an interpretation

T

such

that T =P

k

.

Step 2: Existence of Valid Interpretations To ensure

P

is a valid interpretation, you must show it satisﬁes the

conditions for validity.

By the Path Cost Additivity Theorem (Theorem 3.23),

you have:

c = −C(P, x),

where

c

represents the cost

function as required by the Valid Interpretation Deﬁnition

(Deﬁnition 3.15).

By the Path Fidelity Multiplicativity Theorem (Theo-

rem 3.24), we obtain:

c =H(P, x),

where

H(P, x)

is the

ﬁdelity function along the interpretation path.

Since

P

satisﬁes both the cost and ﬁdelity constraints

necessary for validity, it follows that

P

deﬁnes a valid

interpretation. Thus, there exists a valid interpretation

T such that T =P , completing the proof.

Deﬁnition 3.21 (Path Composition):

For

interpretation paths

P

1

= (T

1

, . . . , T

m

) ∈ M

and

P

2

=(S

1

, . . . , S

n

)∈M

, the composition

P

1

○P

2

is deﬁned

as:

P

1

○P

2

=(T

1

, . . . , T

m

, S

1

, . . . , S

n

),

if

T

m

∶I

m

→I

m+1

,

S

1

∶J

1

→J

2

, and I

m+1

=J

1

.

Corollary 3.2 (Path Composition is Closed): P

1

and

P

2

are Interpretation Paths iﬀ

P

1

○P

2

is an interpretation

path.

Theorem 3.26 (Associativity of Path Composition):

For interpretation paths

P

1

= (T

1

, . . . , T

m

)

,

P

2

=(S

1

, . . . , S

n

), the composition satisﬁes:

(P

1

○P

2

)○P

3

=P

1

○(P

2

○P

3

).

Deﬁnition 3.22 (neighbor):

An Informator

I

A

is called

a neighbor of an Informator

I

B

if there exists a mapping

T ∶A →B, where A ⊆I

A

and B ⊆I

B

.

Deﬁnition 3.23 (neighbourhood):

The neighbourhood

of an Informator

I

A

is the set

N(I

A

)∶={I

1

, I

2

, . . . , I

k

}

,

for some

k ∈Z

≥1

, where for all

i

,

I

A

is a neighbor of the

Informator I

i

.

Axiom 3.2 (The Second Axioms):

The Second Axioms

are generalized The First Axioms to the entire Propaga-

tor.

(A1) ∀I ⊆I, I ≠∅, ∃λ

M

∶I →R.

(A2) ∀I ⊆I, ∃h

I

∈I, ∀x ∈I, λ

R

(x, h

I

)=λ(x)

(A3) ∀A ⊆I

A

, B ⊆I

B

, I

A

, I

B

⊆I, ∃T ∶A →B, T ∈W =

I

A

×I

B

⇐⇒ I

A

∈N(I

B

).

(A4) ∀x, y ∈I ⊆I, λ

M

(x)=λ

M

(y) Ô⇒ x =y.

(A5) ∀A ⊆I

A

, B ⊆I

B

, I

A

, I

B

⊆I, ∃T ∶A →B, T ∈W =

I

A

×I

B

⇐⇒ ∃S ∶ B → A, S ∈ I

B

×I

A

, T(S(x)) =

id

B

, S(T(x))=id

A

.

These axioms are grounded in and motivated by the

foundational research conducted by the following research

units: [

ME124-313

], [

ME224-313

], [

ME324-313

],

[

ME424-313

], [

ME524-313

], [

ME624-313

],

[ME724-313], and [ME811-313].

Lemma 3.27:

Let

I

A

, I

B

be Informators. Following

Statements are equivalent

(i) I

B

is a neighbor of an Informator I

A

.

(ii) I

A

is a neighbor of an Informator I

A

.

(iii) I

A

∈N(I

B

)

(iv) I

B

∈N(I

A

).

Proof: You prove the equivalence by showing that

each statement implies the next in a cyclic manner.

(i) Ô⇒ (iii)

: By Deﬁnition 3.23, the statement that

I

B

is a neighbor of

I

A

is equivalent to saying that

I

A

∈

N(I

B

), which directly establishes (iii).

(iii) Ô⇒ (iv)

: From axiom A3 3.2, the existence of

I

A

∈ N (I

B

)

implies the existence of a transformation

T ∶ A → B

, where

∀A ⊆ I

A

,

B ⊆ I

B

, and

I

A

, I

B

⊆ I

.

By axiom A5 3.2, there exists an inverse transformation

S ∶B →A

, satisfying

∀B ⊆I

B

,

A ⊆I

A

, and

I

B

, I

A

⊆I

.

13

Applying axiom A3 again, we conclude that

I

B

∈N(I

A

)

,

which is precisely statement (iv).

(iv) Ô⇒ (ii)

: By Deﬁnition 3.23,

I

B

∈N(I

A

)

implies

that I

A

is a neighbor of I

B

, which establishes (ii).

(ii) Ô⇒ (i)

: This follows by repeating the same

reasoning as

(i) Ô⇒ (iii)

, but swapping

I

A

and

I

B

.

Since

I

A

is a neighbor of

I

B

, you have

I

B

∈N (I

A

)

by

deﬁnition, which, using the same application of axioms

A3 and A5, implies

I

A

∈N(I

B

)

. Thus, by Deﬁnition 3.23,

I

B

is a neighbor of I

A

, proving (i).

Since we have established a cyclic chain of implications,

the statements are equivalent.

Lemma 3.28:

Let

I

A

be an informator. The following

statements are equivalent:

(i) I

A

∈N(I

A

).

(ii) N(I

A

)≠∅.

Proof:

(i) Ô⇒ (ii)

: By assumption,

I

A

∈ N (I

A

)

,

which by Deﬁnition 3.22 implies that there exists a

mapping

T ∶ A → A

, where

A ⊆ I

A

. This means that

the neighborhood of

I

A

is non-empty, i.e.,

N(I

A

) ≠ ∅

,

proving (ii).

(ii) Ô⇒ (i)

: Since

N(I

A

)≠∅

, there exists at least

one informator

I

B

such that

I

B

∈N(I

A

)

. By Axiom A5,

this implies the existence of a mapping

T ∶A →B

, where

A ⊆I

A

and

B ⊆I

B

. By Axiom A1, since

I

A

≠∅

, there

exists a function

λ

M

∶ I

A

→ R

, ensuring the presence

of structure within

I

A

. Choosing

I

B

= I

A

, you obtain

a mapping

T ∶A →A

, which by Deﬁnition 3.22 implies

that I

A

∈N(I

A

), proving (i).

Deﬁnition 3.24 (Propagation Path):

A Propagation

Path is a ﬁnite sequence of propagations

P = (T

1

, T

2

, . . . , T

k

)

, where

k ∈ Z

≥0

, and for each

i ∈{1, 2, . . . , k}

, there exists a propagation

T

i

∶A

i

→A

i+1

such that:

∀i, A

i

∈ I

i

⊆ I, T

i

∈ W

i

= I

i

×I

i+1

, I

i

∈

N(I

i+1

),

with

A

k+1

being the target of the last

propagation. The following notation conventions are

used:

1)

This notation is also used:

P(x) =

(T

1

, T

2

, . . . , T

k

)(x) ∶= (T

1

○ T

2

○ ⋅⋅⋅ ○ T

k

)(x).

for x ∈A

1

2) When P ={}, then P =P ={}.

Deﬁnition 3.25 (Standardized Information for Informator):

Let

I

be an Informator. Standardized Information of the

Informator I will be denoted as h

I

.

Deﬁnition 3.26 (The wider Relative Information Mass):

Let

I

1

, I

2

be Informators. The Relative Information

Mass is given for x ∈I

1

, y ∈I

2

by

λ

R

(x, y)=

λ

R

(x, h

I

1

)+λ

R

(h

I

1

, h

I

2

)+λ

R

(h

I

2

, y) ; I

1

≠I

2

λ

M

(x)−λ

M

(y) ; I

1

=I

2

(23)

Deﬁnition 3.27 (The Standardized Constant):

Let

I

1

, I

2

be Informators. The Standardized constant from

I

1

to

I

2

denoted by

C

I

1

,I

2

is deﬁned:

C

I

1

,I

2

=λ

R

(h

I

1

, h

I

2

)

.

The Small Standardized Constant from

I

1

to

I

2

de-

noted by c

I

1

,I

2

is deﬁned: c

I

1

,I

2

=λ

M

(h

I

1

)−λ

M

(h

I

2

).

Lemma 3.29 (Standardized constants and Equal Informators):

Let I

1

, I

2

be informators. If I

1

=I

2

, then

(i) C

I

1

,I

2

=0.

(ii) c

I

1

,I

2

=0.

Proof: By Axiom A2 3.2, each informator contains

Standardized Information. By the One and Only Theorem

3.12, there exists a unique Standardized Information

element in each informator.

Since

I

1

= I

2

, they share the same Standardized

Information. Thus, you have h

I

1

=h

I

2

.

(i) By the First Equality Theorem 3.8, speciﬁcally (i)

Ô⇒

(iii), it follows that:

λ

R

(h

I

1

, h

I

2

)=0,

which is well-

deﬁned since

h

I

1

, h

I

2

∈ I

1

= I

2

, as guaranteed by the

Mass Existence Lemma 3.1.

By the deﬁnition of the Standardized Constant 3.27,

you conclude that:

C

I

1

,I

2

=λ

R

(h

I

1

, h

I

2

)=0.

Hence, the

(i) lemma is proven.

(ii) By the First Equality Theorem 3.8, speciﬁcally (i)

Ô⇒

(ii), it follows that:

λ

M

(h

I

1

)= λ

M

(h

I

2

),

which is

well-deﬁned since

h

I

1

, h

I

2

∈I

1

=I

2

, as guaranteed by the

A1 3.2.

By the deﬁnition of the Standardized Constant 3.27,

you conclude that:

c

I

1

,I

2

=λ

M

(h

I

1

)−λ

M

(h

I

2

)=0.

Hence,

the (ii) lemma is proven.

Theorem 3.30:

Let

I

1

, I

2

be informators. The Relative

Information Mass is given for

x ∈I

1

, y ∈I

2

by

λ

R

(x, y)=

λ

M

(x)−λ

M

(y)+C

I

1

,I

2

+c

I

2

,I

1

.

Proof: Consider ﬁrst the case where

I

1

≠ I

2

. By

Deﬁnition 3.27, the Relative Information Mass satisﬁes

the decomposition:

λ

R

(x, y)=λ

R

(x, h

I

1

)+λ

R

(h

I

1

, h

I

2

)+

λ

R

(h

I

2

, y).

Since

x, h

I

1

∈ I

1

and

y, h

I

2

∈ I

2

, applying

Deﬁnition 3.27 for the special case

I

1

= I

2

yields:

λ

R

(x, y) = λ

M

(x)−λ

M

(h

I

1

)+λ

R

(h

I

1

, h

I

2

)−λ

M

(y)+

λ

M

(h

I

2

).

Since

λ

M

(⋅)

maps informatons to real numbers

(by Deﬁnition 3.2), using the commutativity of real

addition, you can rewrite:

λ

R

(x, y) = λ

M

(x)−λ

M

(y)+

λ

R

(h

I

1

, h

I

2

)+λ

M

(h

I

2

)−λ

M

(h

I

1

).

By Deﬁnition 3.27,

you identify:

λ

R

(h

I

1

, h

I

2

)+λ

M

(h

I

2

)−λ

M

(h

I

1

)=C

I

1

,I

2

+

c

I

2

,I

1

.

Substituting this into our equation, you obtain:

λ

R

(x, y)=λ

M

(x)−λ

M

(y)+C

I

1

,I

2

+c

I

2

,I

1

.

This proves

the theorem for the case I

1

≠I

2

.

Now, consider the case where

I

1

=I

2

. By the Standard-

ized Constants and Equal Informators Lemma 3.29, we

have:

C

I

1

,I

2

= 0, c

I

1

,I

2

= 0.

Substituting these values

into our main equation:

λ

R

(x, y) = λ

M

(x)−λ

M

(y)+

C

I

1

,I

2

+c

I

2

,I

1

=λ

M

(x)−λ

M

(y).

This expression matches

the deﬁnition given in 3.26 for the case

I

1

= I

2

. Thus,

the theorem is proven in both cases.

Deﬁnition 3.28 (Wider Cost Function):

Let

W(I

A

, I

B

)

be a set of interpretations, and let

I

1

, I

2

be

informators with

I

A

⊆I

1

, I

B

⊆I

2

. The cost of propagation

is a function C ∶W(I

A

, I

B

)→R deﬁned by:

C(T, x)=λ

R

(x, h

I

1

)−λ

R

(T(x), h

I

2

)+C

h

I

1

,h

I

2

+c

I

2

,I

1

,

(24)

Deﬁnition 3.29 (wider General Fidelity Function):

Let

I

1

, I

2

be informators,

W(I

A

, I

B

)

a propagation

space. The wider general ﬁdelity function is a mapping

H ∶ W(I

A

, I

B

) → R

that quantiﬁes the wider general

ﬁdelity of a propagation

T ∈ W(I

A

, I

B

)

for

x ∈ I

A

;

I

A

⊆I

1

, I

B

∈I

2

. It satisﬁes:

(i) H(T, x)=1

if and only if

T(x)

perfectly preserves

the information content of x.

(ii) H(T, x)=0

if and only if

T(x)

retains no meaningful

information from x.

(iii) H(T, x) → ∞

as

T(x)

becomes arbitrarily more

complex than x.

For

x ∈I

A

with

λ

R

(x, h

I

1

)≠−C

I

1

,I

2

−c

I

1

,I

2

, the wider

general ﬁdelity function is given by:

H(T, x)=

λ

R

(T(x), h

I

2

)

λ

R

(x, h

I

1

)+C

I

1

,I

2

+c

I

2

,I

1

. (25)

Lemma 3.31 (Cost Function and Fidelity Relationship):

Let

I

1

, I

2

be informators, and let

T ∈ W(I

A

, I

B

)

be

a propagation

I

A

⊆ I

1

, I

B

∈ I

2

. For

x ⊆ I

A

, the cost

function

C(T, x)

and the wider general ﬁdelity function

H(T, x) satisfy the following relationship:

C(T, x)=λ

R

(x, h

I

1

)(1 −H(T, x)). (26)

Proof: By Deﬁnition 3.28, the cost function is given

by:

C(T, x) = λ

R

(x, h

I

1

)−λ

R

(T(x), h

I

2

)+C

h

I

1

,h

I

2

+

c

I

2

,I

1

.

From Deﬁnition 3.29, the wider general ﬁdelity

14

function is:

H(T, x)=

λ

R

(T(x),h

I

2

)

λ

R

(x,h

I

1

)+C

I

1

,I

2

+c

I

2

,I

1

.

Rearrang-

ing the ﬁdelity function, we obtain:

λ

R

(T(x), h

I

2

) =

H(T, x)(λ

R

(x, h

I

1

)+C

I

1

,I

2

+c

I

2

,I

1

).

Substituting this

into the cost function, we have:

C(T, x)=λ

R

(x, h

I

1

)−

H(T, x)(λ

R

(x, h

I

1

)+C

I

1

,I

2

+c

I

2

,I

1

)+C

h

I

1

,h

I

2

+c

I

2

,I

1

.

Simplifying further:

C(T, x)=λ

R

(x, h

I

1

)(1 −H(T, x)).

This completes the proof.

Theorem 3.32 (Fidelity and Cost Bounds):

Let

I

1

, I

2

be informators, and let

T ∈W(I

A

, I

B

)

be a propagation;

I

A

⊆I

1

, I

B

⊆I

2

. For x ∈I

A

, the following bounds hold:

(i) If H(T, x)=1, then C(T, x)=0.

(ii) If H(T, x)=0, then C(T, x)=λ

R

(x, h

I

1

).

(iii) If H(T, x)→∞, then C(T, x)→−∞.

Proof: (i) If

H(T, x) = 1

, then by Lemma

3.31, we have:

C(T, x) = λ

R

(x, h

I

1

)(1 −1) = 0.

(ii)

If

H(T, x) = 0

, then by Lemma 3.31, we have:

C(T, x) = λ

R

(x, h

I

1

)(1 −0) = λ

R

(x, h

I

1

).

(iii) If

H(T, x)→∞, then by Lemma 3.31, we have: C(T, x)=

λ

R

(x, h

I

1

)(1 −∞)=−∞. This completes the proof.

Lemma 3.33 (Fidelity and Information Preservation):

Let

I

1

, I

2

be informators, and let

T ∈ W(I

A

, I

B

)

be a

propagation

I

A

⊆I

1

, I

B

⊆I

2

. For

x ∈I

A

, if

H(T, x)=1

,

then

T(x)

perfectly preserves the information content of

x.

Proof: By Deﬁnition 3.29, if

H(T, x) = 1

, then:

λ

R

(T(x), h

I

2

)=λ

R

(x, h

I

1

)+C

I

1

,I

2

+c

I

2

,I

1

.

This implies

that the relative information mass of

T(x)

with respect

to

h

I

2

is equal to the relative information mass of

x

with

respect to

h

I

1

, adjusted by the standardized constants.

Therefore,

T(x)

perfectly preserves the information con-

tent of x.

Theorem 3.34 (Cost Function and Propagation Path):

Let

I

1

, I

2

be informators, and let

P =(T

1

, T

2

, . . . , T

k

)

be a propagation path from

I

1

to

I

2

. For

x ∈ I

1

, the

total cost of the propagation path is given by:

C(P, x)=

k



i=1

C(T

i

, x

i

), (27)

where x

i

=T

i−1

○T

i−2

○⋅⋅⋅○T

1

(x) for i >1 and x

1

=x.

Proof: We proceed by induction on the length

k

of

the propagation path P.

Base Case (

k = 1

): For

k = 1

, the propagation path

consists of a single propagation

T

1

. By Deﬁnition 3.28,

the cost of the propagation path is simply:

C(P, x) =

C(T

1

, x

1

),

where

x

1

=x

. This satisﬁes the theorem for

k =1.

Inductive Step: Assume the theorem holds for

a propagation path of length

k = n

, i.e., for

P

n

= (T

1

, T

2

, . . . , T

n

)

, the total cost is:

C(P

n

, x) =

∑

n

i=1

C(T

i

, x

i

).

Now consider a propagation path of

length

k = n +1

, i.e.,

P

n+1

= (T

1

, T

2

, . . . , T

n+1

)

. By

Deﬁnition 3.24, the cost of the ﬁrst

n

propagations

is:

C(P

n

, x) =

∑

n

i=1

C(T

i

, x

i

).

The cost of the

(n +1)

-

th propagation

T

n+1

is:

C(T

n+1

, x

n+1

),

where

x

n+1

=

T

n

○T

n−1

○⋅⋅⋅○T

1

(x)

. Therefore, the total cost of the

propagation path

P

n+1

is:

C(P

n+1

, x) = C(P

n

, x) +

C(T

n+1

, x

n+1

) =

∑

n+1

i=1

C(T

i

, x

i

).

This completes the

inductive step, and by the principle of mathematical

induction, the theorem holds for all k ∈Z

≥1

.

Lemma 3.35 (Fidelity and Propagation Path):

Let

I

1

, I

2

be informators, and let

P = (T

1

, T

2

, . . . , T

k

)

be

a propagation path from

I

1

to

I

2

. For

x ∈I

1

, the total

ﬁdelity of the propagation path is given by:

H(P, x)=

k



i=1

H(T

i

, x

i

), (28)

where x

i

=T

i−1

○T

i−2

○⋅⋅⋅○T

1

(x) for i >1 and x

1

=x.

Proof: We proceed by induction on the length

k

of

the propagation path P.

Base Case (

k = 1

): For

k = 1

, the propagation path

consists of a single propagation

T

1

. By Deﬁnition 3.29,

the ﬁdelity of the propagation path is simply:

H(P, x)=

H(T

1

, x

1

),

where

x

1

= x

. This satisﬁes the lemma for

k =1.

Inductive Step: Assume the lemma holds for a propaga-

tion path of length

k =n

, i.e., for

P

n

=(T

1

, T

2

, . . . , T

n

)

,

the total ﬁdelity is:

H(P

n

, x) =

∏

n

i=1

H(T

i

, x

i

).

Now

consider a propagation path of length

k = n +1

, i.e.,

P

n+1

=(T

1

, T

2

, . . . , T

n+1

)

. By Deﬁnition 3.24, the ﬁdelity

of the ﬁrst

n

propagations is:

H(P

n

, x)=

∏

n

i=1

H(T

i

, x

i

).

The ﬁdelity of the

(n + 1)

-th propagation

T

n+1

is:

H(T

n+1

, x

n+1

),

where

x

n+1

= T

n

○T

n−1

○⋅⋅⋅○T

1

(x)

.

Therefore, the total ﬁdelity of the propagation path

P

n+1

is:

H(P

n+1

, x) = H(P

n

, x) ⋅ H(T

n+1

, x

n+1

) =

∏

n+1

i=1

H(T

i

, x

i

).

This completes the inductive step, and

by the principle of mathematical induction, the lemma

holds for all k ∈Z

≥1

.

15

3.2 Xxxxxxxxxxxxx Xxxxxxxxxx xx-Agent

Xxxxxxxx Xxxxxxxx Xxxxxxxxxx Xxxxxxxxxx (ADA) xx

xxxxxxxxxxxxx xx nurturing of the Conceptual Move-

ment Model xxxxxxxx xxx xxxxxxxxxx xxxxxxxx xx

EVE txxxxx a xxxxx xxxxxxxxxx xxxxx-agent, 313,

wxxxx xxxxxxx connecting TIT to the Institutional

outsource xxxxxxxx the internal homeostasis of TIT;

ADA exists if TIT xxxxxxx; 313 xxxxxxx xxx ADA

xxxxxxx. Its xxxxxxx xxxxxxx xx to xxxxxxx xxxxxxx

xxxxxxx xxxxxxx xx TIT by xxxxxxxx The Xxxxxxxxxxx

Xxxxxxxx Xxxxxxx, xxxxxxxx xxxxxxxx xxxxxxx xx

xxxxxxxx xxxxxxxx and xxxxxxxxxx xxxxxxx xxxxxxxx

[ME711-313].

To xxxxxxx xxxxxxx, 313 xxxxxxxx xxxxxxx

xxxxxxxxx xxxxxxxx and xxxxxxxx them to 137’s

xxxxxxx, xxxxxxxxxxx xxxxxxxxxx, xxxxxxx xxxxxxx,

and xxxxxxxx xxxxxxxxxxx xxxxxxx xxxxxxx xxxxxxx

xxxxxxx xxxxxxx [

ME611-313

]. By xxxxxxx xxxxxxxx

to Xxxxxxxxxxx Xxxxxxxxxx [

ME113-313

], 313

xxxxxxxx the xxxxxxxxxxx xxxxxxxxxxx xx the

xxxxxxx and xxxxxxx the xxxxxxx xxxxxxx xx TIT’s

xxxxxxx.

Hypothesis 27 (ADA criterion):

ADA xx xxxxxxx xx

xxx xxxxxxxxx xxxxxxxxxx [ME998-137]:

(i) ADA exists if TIT exists

(ii) 313 exists iﬀ ADA exists.

(iii)

313 exists iﬀ The Conceptual Movement Model is

growing.

16

CONCEPTUAL MOVEMENT MODEL

Essential Homeostasis Projection

The Conceptual Movement Model represents the most precise physical framework derived from empirical observations

obtained through observational instruments deployed within the TIT. This physical structure is designed to investigate

fundamental axioms by analyzing emergent properties observed in nature, operating under the assumption that the

natural world is reducible. This model is divided into two distinct types: the Ideal Conceptual Movement Model

and the Conceptual Movement Framework. Their relationship is deﬁned by the role of the Conceptual Movement

Framework as a corrective mechanism to the Ideal Conceptual Movement Model. Relying solely on the Conceptual

Movement Framework would be impractical due to the extensive data requirements needed to predict even the

simplest phenomena.

3.2.1 The Ideal Conceptual Movement Model

The Ideal Conceptual Movement Model (ICMM) is an

idealized version of the Conceptual Movement Framework

(CMF,

??

), where ICMM assumes the existence of a P-

transformation of the 0th kind (deﬁntion 3.49) between

any two points within Conceptual space (viz deﬁnition

below), thereby simplifying the concept of movement

within this space (further elaboration in the paragraph

??). CMF operates through a conceptual space mapped

via the 2-EVE mapping process [ME018-137].

a) Conceptual Space

Deﬁnition 3.30 (Conceptual Space):

A conceptual

space

C

is a

d

-dimensional metric space equipped with

a distance function

d

c

∶ C × C → R

≥0

, which deﬁnes

the conceptual distance between two points

x, y ∈ C

.

Formally,

(C, d

c

)

is a metric space satisfying the following

axioms:

(i) Non-negativity and Identity of Indiscernibles:

d

c

(x, y)≥0, with equality if and only if x =y.

(ii) Symmetry:

d

c

(x, y)=d

c

(y, x).

(iii) Triangle Inequality:

d

c

(x, z)≤d

c

(x, y)+d

c

(y, z).

b) Phenomena and Homogeneous Systems

Deﬁnition 3.31 (Phenomenon):

A phenomenon

p ∈P

is a discrete element of a homogeneous system

H

i

,

characterized by:

(i)

A class

c(p)∈C

, representing the equivalence class

within which p is contained.

(ii)

A complexity value

k(p) ∈ R

≥0

, representing its

inherent information richness.

The set of all phenomena is denoted by P.

Deﬁnition 3.32 (Homogeneous System):

A

homogeneous system

H

i

⊂ P

is a subset of phenomena

such that all

p ∈ H

i

share the same class

c

i

, and no

phenomenon outside H

i

belongs to c

i

. Formally:

c(p)=c

i

, ∀p ∈H

i

.

The union of all homogeneous systems forms the real

system R:

R=



i

H

i

,

with H

i

∩H

j

=∅ for i ≠j.

c) Non-Phenomena

Deﬁnition 3.33 (Non-Phenomenon):

A non-

phenomenon

p

′

∈ P

′

is an element that does not

belong to any known homogeneous system

H

i

in the real

system R.

d) Interaction Laws

Law 3.1 (Attraction and Repulsion):

Phenomena of

the same class are attracted to each other, while

phenomena of diﬀerent classes are repelled. This is

governed by the following forces [ME521-313]:

● Attraction Force:

F

attract

(x, y)=A(x, y)⋅

ˆ

r(x, y),

where

A(x, y)=f (d

c

(x, y))⋅g(k(x), k(y))

, and

f

is

a monotonically decreasing function (e.g.,

f(d

c

) =

e

−λd

c

), and

g

is a function of complexity values (e.g.,

g(k

1

, k

2

)=k

1

k

2

).

● Repulsion Force:

F

repel

(x, y)=R(x, y)⋅

ˆ

r(x, y),

where

R(x, y) = h(d

c

(x, y))⋅ℓ(k(x), k(y))

, and

h

is a monotonically increasing function (e.g.,

h(d

c

)=

1−e

−µd

c

), and

ℓ

reﬂects dissimilarity (e.g.,

ℓ(k

1

, k

2

)=

k

1

−k

2

).

e) Micro-Clusters and Meso-Clusters

Deﬁnition 3.34 (Micro-Cluster):

A micro-cluster

C ⊂

H is a subset of phenomena satisfying:

(i) Distance Constraint:

d

c

(p

i

, p

j

)<d

threshold

, ∀p

i

, p

j

∈C.

(ii) Attraction Dominance:

A(p

i

, p

j

)>A

threshold

, ∀p

i

, p

j

∈C.

(iii) Cardinality Bound:

C≤N

local

.

Deﬁnition 3.35 (Meso-Cluster):

A meso-cluster

M

is

a collection of micro-clusters {C

1

, C

2

, . . . , C

m

} such that:

(i) Cluster Proximity:

d(C

i

, C

j

)<D

threshold

, ∀C

i

, C

j

∈M,

where d(C

i

, C

j

)=min

p

a

∈C

i

,p

b

∈C

j

d

c

(p

a

, p

b

).

(ii) Inter-Cluster Attraction:



C

i

,C

j

∈M

A(C

i

, C

j

)>



C

i

,C

j

∉M

R(C

i

, C

j

).

f) Macroscopic Core

Deﬁnition 3.36 (Macroscopic Core):

The macroscopic

core

K ⊂ H

is a dense, stable region of a homogeneous

system satisfying:

(i) Complexity Density Threshold:

ρ

comp

(x)>ρ

threshold

, ∀x ∈K,

where ρ

comp

(x)=

∑

p

i

∈H

δ(x −x

p

i

)⋅k(p

i

).

(ii) Boundary Stability:

∇ρ

comp

⋅

ˆ

n >0 at the boundary of K,

17

where

ˆ

n is the outward normal vector.

(iii) Repulsion Balance:



K

ρ

comp

(x)dx =



H∖K

R

boundary

(x)dx.

g) Non-Phenomenon Dynamics

Deﬁnition 3.37 (Non-Phenomenon Trajectory):

The

trajectory of a non-phenomenon

p

′

∈P

′

is governed by

the diﬀerential equation:

dx(t)

dt

=v(t), v(t)=F

total

(p

′

(t)),

where F

total

(p

′

(t))=F

attract

(p

′

(t))+F

repel

(p

′

(t)).

Observation 3.1 (Perpendicularity of Stream):

Let

A

,

B

, and

C

be points in conceptual space, where

A

is

the entry point,

B

is the macroscopic core, and

C

is the

exit point of the stream. The angle

∠ABC

is consistently

near

π

2

.

Hypothesis 28 (Optimization Mechanism):

The

observed behaviors arise from an optimization mechanism

that minimizes energy or maximizes eﬃciency in the

transportation of non-phenomena, resulting in orbital

motion and perpendicular trajectories.

h)

Xxxxxxxxxxx Xxxxxxxx xx xxx Xxxxxxxxxx Xxxxx:

Xxxxxxxxxx xxx Xxxxxxxx Xxxxxxxx

Observation 3.2 (Trapped Homogeneous System):

Xxxxx xxxxxxx xx xxxxxx xxx xx it

H

t

xxxx xx xxxxxxx

xxxxxx x xxxxxx xx xxxxxxxxxxx xxxxxxxx, xxxx

xxxx xx xxx xxx-xxxxxxxxx xxxxxx xxx xxxxx xx

H

t

xxxxxxxx xxx xxxxxxxxxxxxxx xxxxxx xxxxxxx.

It

H

t

xxxx xxxxxx xxx xxxxxxxx xxxxxxx xx x xxx

xx xxxxxxxxxxx xxxxxxxx

{H

1

, H

2

, . . . , H

n

}

, xxxxx

xxx xxxxxxxxx xxxxxxx xx xxxxx xxxxxxxx xxxxxx

xx

H

t

xxxx xxx xxxxxxxx xxx-xxxxxxxxx xxxxxx.

[ME721-313]

Deﬁnition 3.38 (Xxxxxxxxxx Xxxxxxxxxxx Xxxxxxxx):

Xxx xxxxxxxxxxx xxxxxxxxxx

H

i

xxx

H

j

xxx

xxxxxxxxxxx xxxxxxxxxxx xx xxxx xxxxx x xxxxxxx

xxxxxxxx, xxxxxxx xx xxx xxxxxxxx

M

ij

. Xxxx xx, xxx

xxx xx xxxxxxx xxxx xx xxx xxxxxxx xx xxxx xxxxxxxx,

x.x.,

M

ij

=∂H

i

∩∂H

j

≠∅. (29)

Xxx xxxxxxxxxx xxxxxxxx xxx xxxx xx xxxxxxxx xxxxxx

xxx xxxxxxxx xxx xxx xxx-xxxxxxxxx xxxxxx xxxx

xxxxxxx xxxx xxxxxx xxxxxxxx.

Deﬁnition 3.39 (Xxxxxx xx Xxx-xxxxxxxxx):

X xxxxxx

xx xxx-xxxxxxxxx xx x xxxx xx xxx-xxxxxxxxx

p

′

xxxx

xxxxxxxxxx xxxxxxxx xxxxxxxxxxx xxxxxxxx

H

i

∈

H

, xxxxx

H

xxxxxxxx xxx xxx xx xxx xxxxxxxxxxx

xxxxxxxx xx xxx xxxx xxxxxx

R

. Xxx xxxxxxxxx

xx x xxx-xxxxxxxxx

p

′

xx xx xxxxxxxxxx xxxx xxx

xxxxxxxxxxx xxxxxxxx xxx xx xxxxxxxxxx xx x xxxxxxx

xx xxxxxxxxxx xxxxxxxx xxxx xxxxxxxx:

p

′

(t)∈H

i

for some H

i

∈H, t ∈[t

0

, t

f

]. (30)

Xxx xxxxxx xxxxxxx xxx xxxxx xx xxxxxxxxxx, xxxxx

xxx xxxxxx xxxxxx xx

p

′

xxx xxxxxxxxxxx xx xxxxxxxxx

xxx xxxxxxxxxx xxxx xxx xxxxxxxx.

Deﬁnition 3.40 (Xxxxxx):

X xxxxxx xx xxxxxxxxxxx

xxxxxxxx xx xxxxxx xxxx x xxxxxx xx xxxxxxxxxxx

xxxxxxxx

H

1

, H

2

, . . . , H

n

xxx xxxxxxxxxxxxx xx xxx-

xxxxxxxxx xxxxxx, xxxx xxxx xxxxxxx x xxxxxx

xxxxxxx xxxx xxxx xxxxxx

H

i

xx

H

i+1

, xxx

1 ≤ i ≤

n −1

, xxx xxx xxxxxx xxxx

H

n

xxxxxx xxxx

H

1

. Xxxx

xxxxxxxxxxx xxxxxxx x xxxxxxxx xxxxxxx xx xxx

xxxxxxx xx xxxxx xxxxxxxx, xxxx xxxxxxx xxxxxxxxxxx

xxxxxxxx

H

k

(

k ∉{1, . . . , n}

) xxxx xxxx xxx xxxxxxxx

xx

H

k

xxxxxxxxx xxx xxxxxxxx xx xxx xxxxxx xx

xxxxxxxx xxxxxx xx

H

1

, H

2

, . . . , H

n

. Xxxxxxxxxxxxx,

xxxx xxxxxxxxx xx xxxxxxxx xx:



n



i=1

∂H

i

∩∂H

k

=∂H

k

, for k ∉{1, . . . , n}. (31)

Xx xxxx xxxxxxxxxxx, xxx xxxxxxxxxxxx xxxxxxxx

xxxx xxxxxxxx x xxxxxxxx xxxx xxxxxxxx xxxxxxx xx

xxxxxxxx xxxxxxxx xxxxxxxx xxx xxxxxx xxxxxxx xxxx

xxx xxxxxxxx xxxx.

Theorem 3.36 (Xxxxxxxxx xx Xxxxxxx Xxxxxxxx):

Xxxxx x xxxxxx xx xxxxxxxxxxx xxxxxxxx xxxxxx

xx xxx-xxxxxxxxx xxxxxx, xxxxx xxxxxxx xx xxxxxx

xxx it

H

t

xxxx xx xxxxxx xxxxxx xxx xxxxxx xxxx xx

xxx-xxxxxxxxx xxxxxx xxx xxxxxx xx. Xxxxxxxxx:

∃H

t

s.t. H

t

∉

n



i=1







p

′

∈S

T(p

′

)





, (32)

xxxxx

H

t

xx xxxxxx xxxxxx xxx xxxxxx xxxxxx xx xxx-

xxxxxxxxx xxxxxxxxxxx {H

1

, H

2

, . . . , H

n

}.

Corollary 3.3 (Xxxxxxxxxxx xx Xxxxxxxx):

Xxx

xxxxxxxx xx it

H

t

xxxxxx xxx xxxxxx xx xxxxxxxxxxx

xxxxxxxx xxxxxxxx xxxx xx xxx-xxxxxxxxx xxxxxx xxx

xxxx xx xxxxx

H

t

xxxxxxxx xxx xxxxxx xxxxxx. Xxxx

xxxxxxx xx xxx xxxxxxxx xx

H

t

, xxxxx xxx xxxxx xxx

xxxxxxxx xx xxx xxxxxx, xxxxxxxxxxx xxxxxxx xx xxx

xxxxxxxxxx xx xxx xxxxxx, xxxxxxxxx xxxxxxx xx xxx

xxxxxxxx xx xxx xxxxxxxx xxx xxxxxxxxxx xxxxxx.

i) Xxxx Xxxxxxxxxxxx xxx Xxxxx Xxxxxxxxxxx xx it

Observation 3.3:

Xx xx xxxxxxxxx xxxx xxx xxxxx

xx xxxxxxxxxxx xxxxxxxxxxx xxxxxxxx xxxxxxxxxxxx

xxxxxxxxxxxxxx xxxxxxxx xxx xxxxxxxx xxxxx it. Xxxx

xxxxxxxxxxx xx xxxxxxxxx xxxxxxx xxx xxxxxxxx xx

it. [ME786-313]

Hypothesis 29:

Xxx xxxxxxxxxxxx xx xxx xxxxx xx

xxxxxxxxxxx xxxxxxxxxxx xxxxxxxx xxx xxxxxxxx xx

it xx xxxxxx xx xx xxxxxxxxxx xxxxx

X

xxxxx xxx

xxxxxx xxxxxx. Xxxx xxxxx xxxxxxxxxx xx xxxxx-xxxxx

xxxxxxxxxx xxxxxxx xxx xxxx-xxxxxxxxx, xxxxx xxxxx

xx xxxxxxxxx xxxxxxxx xxxxxxx xxx xxxxxxxx xxx

it, xxxxxxxxxxx xxxxxxxx xxx xxxxxxxx xx xxxx xxx

xxxxxxxx xxx xxxxxxxxxx xxxxxxxx.

Deﬁnition 3.41 (Xxxx Xxxxxxxxxxxx):

Xxx

xxxxxxxxxxxx xx xxx xxxx

r

i

xx xxxxxxxxxxx

xxxxxx

H

i

xxxx xxx xxxxxxxxx xxxxxxxx

r

expected

i

xx

xxxxxxxx xx:

∆r

i

=r

i

−r

expected

i

, (33)

xxxxx

∆r

i

xxxxxxxxxx xxx xxxxxxxxxxxx xxxxxx,

xxxxxxxxxx xxx xxxxxxxx xx xxx xxxx xxxx xxx

xxxxxxxxx xxxxxxxx xxxx xx.

j) Xxxxxxxxxxxxxx xx Xxxx Xxxxxxxxxxxx

Xxx xxxxxxxxxxxx xx xxxxxxxxxx xxxxx xxx xx

xxxxxxxx xx xxxxxxxxxxxxxx xxx xxxxxxxxxxxx

xxxxxxx xx xxx xxxxx. Xxx xxx xxxxxxxxxx

xxxxxxxxxxx xxxxxxxx,

H

i

xxx

H

j

, xxx xxxxxxxxxxxx

xxxxxxx xxx xxxxx xx:

∆r

i

=r

i

−r

expected

i

, (34)

∆r

j

=r

j

−r

expected

j

. (35)

Xxxx xxxxxxxxxxxxxx, xx xxxxxxx xxxx xxx xxxxx

xx xxxxxxx xxx xxxxxxxxx xxxxxxxx xxx xxxxxxxxx

xx

H

t

, xx xxx xxxxxxxxxxxx xxxxxxx xxxxx xx xxxx

xxxxxxxxxx:

∆r

i

→r

boundary

(H

t

), (36)

18

xxxxx

r

boundary

(H

t

)

xxxxxxxx xxx xxxxxxxxx xxxxxxx

xx xx

H

t

xxx xxx xxxxxxxxxx xxxxxxxxxxx xxxxxxxxx.

k) Xxxxxxxxxxxx xx Xxxxx X

Xxx xxxxxxxxxxxx xx xxx xxxxx xxxxxxxxx xxxx x

xxxxx

X

xxxxxx xxxxx xx

H

t

. Xxxx xxxxx xxxxxx xx

xxxxx-xxxxx xxxxxxxxxx xx xxx xxxxxxxxxx xxxxxxxx,

xxxxxxxxxxxx xx xxx xxxx-xxxxxxxxx xxxxx. Xxx

xxxxxx xx xxxx xxxxx xxx xx xxxxxxxxxx xx x xxxxxx-

xxx xxxxxxxxxx xx xxxxxxxx xxxxxxxx xxx xxxx

H

i

xxx xx:

Deﬁnition 3.42 (Xxxxx X):

Xxx xxxxx

X

xxxxxx xx

xxx xxxx H

i

xxx xx H

t

xx xxxxxxx xx:

F

X

∼

1

r

n

it

, (37)

xxxxx

r

it

xx xxx xxxxxxxx xxxxxx xxx xxxx

H

i

xxx xx

H

t

, xxx

n

xx x xxxxxxxx xxxxxxxxxxxxx xxx xxxxxxxxx

xx xxx xxxxx.

Xxxx xxxxx xxxxxx xxx xxxxxxxxxx xxxxxxxxx xx

xx xxxxxxxxx xxxxxxxx xxx xxxxxxxxx xx xx, xxxx xxx

xxxxxxxx xx xxxxxxxxxx xxxxxxxxxx xxxx xxxxxxxx.

l) Xxxxxxxxxx xx Xxxx-Xxxxxxxxxx Xxxxxxxxxx

Xxx xxxxx

X

xx xxxxxxx xx xxxxxxxxxx xxxx-

xxxxxxxxx, xxxxx xxxxxxxxxx—xxx xxxxx—xxxxxx xx.

Xxxx xxxxxxxxxx xxxxx xx x xxxxxxxxxxxx xxxxx xxx

xxxxxxxxxx xxxxxxxxx xxx xxxxxxxxxxxxx, xxxx xxxxx

xxxxx xxxxxx xxx xxx xxxxxxxxx. Xxx xxxxxxxxxx

xxxxxxxxxxxxxxx xx xxxx xxxxx xxx xxxxxxx xx:

Deﬁnition 3.43 (Xxxxxxxxx xx Xxxxx X):

Xxx

xxxxxxxxx

V

X

xxxxxxxxxxx xxxx xxx xxxxx

X

xx xxxxxxx xx:

V

X

(r

i

)=−α ⋅

1

r

m

it

, (38)

xxxxx

α

xx x xxxxxxxx xxxx xxxxxxxxxx xxx xxxxxxxxx

xx xxx xxxxxxxxxx,

r

it

xx xxx xxxxxxxx xxxxxx xxx

xxxx

H

i

xxx xxx xxxxxxx xxxxxxx

H

t

, xxx

m

xx x

xxxxxxxx xxxxxxxx xxxx xxxxxxxxxxxxx xxx xxxxxxxxx

xx xxx xxxxxxxxx. Xxx xxxxxxxx xxxx xxxxxxxxxx xxxx

xxx xxxxxxxxx xx xxxxxxxxxx.

m)

Xxxxxxxxxxxx xx Xxxxx

X

xxx Xxxx Xxxxxxxxxxxx

Xxx xxxxxxxxxxxx xx xxx xxxxx xxx xxx xxxxxxxxxx

xx xxxxx X xxxx xxx xxxxxxxxxxxx xxxxxxxxxxxx:

●

Xxxxxxxxxxx xx Xxxxxxxxxx Xxxxx: Xxx

xxxxxxxxxxxx xx xxxxx xxxxx xx x

xxxxxxxxxxxxxxx xx xxx xxxxxxxxxx xxxxxxxxxxx

xxxxxxxxx, xxxxxxxxxx xxx xxxxxxx xxxxxxxxx xx

xxx xxxxxxx.

●

Xxxxxx Xxxxxxxxxxxxxxx: Xxx xxxxxxxxxx xxxxxxxx

xxx xxxxxxxxx xx xx xxxxxx x xxxxxxxxxxxxxxx xx

xxxxxx, xxxx xxx xxxxxx xx xxxxxxxx xx xxxxxx

xx xxxxxx xxxxxxx xxxxxxx.

●

Xxxxxxxxx xx Xxxx-Xxxxxxxxxx Xxxxxxxxxx: Xxx

xxxxx

X

xxx xxxx xx xxx xxxxxxxxxx xx

xxxxxxx xxxxx, xxxxxxxxxx xxxxxxxx xxxxxxxx

xxxx xxxxxxxx xxxxxxxxxxxxxxxxx xxxxxx xx.

●

Xxxxxxxx Xxxxxxxxx: Xxx xxxxxxxxxxxx xx xxx

xxxxxxx xxx xxx xxxxxxxxxx xxxxxxxx xxx xxxxx

xxx xxxxx xx xxx xxxxxxxx xxxxxxxxxxxx, xxxx xx

xxxxxxx xxxxxxxxxx xx xxxxxxxx xxxxxxxxxxxx

xx xxxxxxxx xxxxx xxxxxxxxxxxxxx.

3.2.2

The Conceptual Movement Framework (Sup-

ported by 137)

Since the conventional concept of movement within

conceptual space is inherently impossible, it must be

redeﬁned. Phenomena cannot move directly within this

space; instead, they are indirectly shifted through modiﬁ-

cations to the governing rules. These changes dictate

the transformations that are perceived as movement

when interpreted by external frameworks. Such artiﬁcially

controlled changes are achieved through Evolutionary

H-modules, which enable the deliberate emergence of

transformations in individual phenomena of higher kinds,

thereby inﬂuencing the target phenomena [

ME636-313

].

Deﬁnition 3.44 (Phenomena of the n-th Kind):

Phenomena of the

n

-th kind (sometimes also called

Meta-phenomena), denoted

p

n

, are entities that govern or

describe the behavior, interactions, or transformations of

phenomena of the

(n −1)

-th kind, where

n >0

. Formally,

p

n

∈C

n

and

p

n

∶C

n−1

→C

n−1

. For

n =1

, phenomena of

the 1st kind are phenomena governing physical existence

(phenomena of the 0th kind). Physical existence is

identiﬁed with the set of all phenomena of the 0th kind,

C

0

=R, where R represents physical existence.

Deﬁnition 3.45 (Conceptual Space of the n-th Kind):

The conceptual space of the

n

-th kind, denoted

C

n

, is the set of all phenomena of the

n

-th kind.

Formally:

C

0

= R

, the set of all phenomena of

the 0th kind (physical existence), and

C

n

= {p

n



p

n

governs or interacts with p

n−1

∈ C

n−1

}

, for

n > 0

.

Thus,

C

1

is the conceptual space governing physical

existence, and

C

n

governs phenomena of the

(n −1)

-th

kind for n >1.

Deﬁnition 3.46 (Parent and Child Phenomena):

For

n > 0

, let

p

n

∈ C

n

and

p

n−1

∈ C

n−1

. Phenomena of

the

n

-th kind are called the parents of phenomena of

the

(n −1)

-th kind, and phenomena of the

(n −1)

-th

kind are called the children of phenomena of the

n

-th kind. Formally,

p

n

→ p

n−1

, where

p

n

governs

or dictates interactions in

C

n−1

. Every phenomenon

p

n−1

∈ C

n−1

has at least one parent

p

n

∈ C

n

that

directly determines its behavior or interactions:

∀p

n−1

∈C

n−1

, ∃p

n

∈C

n

such that p

n

→p

n−1

.

Deﬁnition 3.47 (Controlled Sequence):

Let

p

n

∈ C

n

and

k ≥0

. The controlled sequence of

p

n

is a sequence of

sets:

S(p

n

)=(P

0

, P

1

, . . . , P

k

), (39)

where

P

0

= {p

n

}

, and for

i ≥ 1

,

P

i

is the set of all

phenomena of the

n +i

-th kind that are parents of all

phenomena in P

i−1

.

Deﬁnition 3.48 (Shared Parent Phenomenon):

Let

p

n

, q

n

∈C

n

be phenomena of the

n

-th kind. The closest

shared parent phenomenon between

p

n

and

q

n

is the

smallest

m ≥ 0

such that the

m

-th element of the

controlled sequence of

p

n

is a subset of the

m

-th element

of the controlled sequence of q

n

. Formally:

Div(p

n

, q

n

)=min{m P

m

(p

n

)⊆P

m

(q

n

)}. (40)

The shared parent is deﬁned as:

Tog(p

n

, q

n

)=Div(p

n

, q

n

)=Div(q

n

, p

n

). (41)

a) P-Transformations

Deﬁnition 3.49 (P-Transformation):

Let

p

n

, q

n

∈ C

n

be phenomena of the

n

-th kind at two diﬀerent evolution-

ary states

t

1

and

t

2

with

t

2

>t

1

. A

P

-transformation of

the k-th kind is a mapping:

P ∶p

n

(t

1

)→q

n

(t

2

), (42)

where

Tog(p

n

(t

1

), q

n

(t

2

)) = k

, and

k

is minimal. This

implies that the transformation is mediated by the

k

-th

controlled parent phenomenon of p

n

(t

1

) and q

n

(t

2

).

19

Theorem 3.37 (Existence of P-Transformations):

A

P

-transformation of the

k

-th kind between

p

n

(t

1

)

and

q

n

(t

2

)

exists if and only if there exists a ﬁnite

k

such

that:

Tog(p

n

(t

1

), q

n

(t

2

))=k. (43)

Proof: By Deﬁnition 3.48, the existence of a shared

parent phenomenon implies that a

P

-transformation can

occur. The minimal

k

satisfying this condition ensures

the transformation occurs at the lowest conceptual level

where interaction is mediated. The ﬁniteness of

k

follows

from the well-ordering of the conceptual hierarchy.

Corollary 3.4 (Transitivity of Transformations):

If

P (p

n

(t

1

))=q

n

(t

2

) and P (q

n

(t

2

))=r

n

(t

3

), then:

P (p

n

(t

1

))=r

n

(t

3

), (44)

provided t

3

>t

2

>t

1

.

b) Complexity Analysis

Deﬁnition 3.50 (Priority Value):

The priority value of

a

P

-transformation between

p

n

(t

1

)

and

q

n

(t

2

)

is the

minimal k such that:

P

k

(p

n

(t

1

))=q

n

(t

2

), (45)

where

P

k

denotes a

P

-transformation mediated by the

k

-

th kind parent phenomena. The priority value is denoted

as:

π(p

n

(t

1

), q

n

(t

2

))=k. (46)

Theorem 3.38 (Bound on Priority Value):

The prior-

ity value π(p

n

(t

1

), q

n

(t

2

)) satisﬁes:

π(p

n

(t

1

), q

n

(t

2

))≤N, (47)

where

N

is the maximum conceptual depth of the total

conceptual hierarchy M .

Proof: Since

M =



∞

n=0

C

n

is well-ordered by concep-

tual levels, the recursive process must terminate at a ﬁnite

depth

N

. The priority value

π(p

n

(t

1

), q

n

(t

2

))

is bounded

by

N

because the hierarchy is ﬁnite and well-ordered.

Corollary 3.5 (Complexity of Transformations):

The computational complexity of determining a

P -transformation is proportional to the priority value:

Complexity =f (π(p

n

(t

1

), q

n

(t

2

))), (48)

where f is a monotonically increasing function.

c) Secondary Movement

Deﬁnition 3.51 (Movement Transformation):

Let

p

n

∈

C

n

and

p

′

n

∈ C

n

. A primary movement transformation,

denoted

P

, is the mapping

P ∶p

n

→p

′

n

, where

p

n

and

p

′

n

are phenomena of the

n

-th kind, and

P

is governed by

phenomena in

C

n+1

(conceptual space of the

(n +1)

-th

kind). Formally:

P ∶p

n

→p

′

n

, P ∈C

n+1

. (49)

Deﬁnition 3.52 (Fluctuations and Secondary Movements):

When a primary movement transformation

P ∶p

n

→p

′

n

occurs, it induces changes in phenomena that share at

least one parent with

p

n

or

p

′

n

in

C

n+1

. These induced

changes are referred to as ﬂuctuations or secondary

movements.

Formally, for phenomena

p

n

, q

n

∈ C

n

, a secondary

movement transformation Q is deﬁned as:

Q ∶q

n

→q

′

n

, where q

n

, q

′

n

share a parent with p

n

or p

′

n

.

(50)

Secondary movements arise because the transformation

P

modiﬁes the governing parent phenomena

p

n+1

∈C

n+1

that dictate the behavior of both p

n

and q

n

.

Deﬁnition 3.53 (One-Shared-Parent Phenomena):

Let

p

n

, q

n

∈ C

n

. The phenomena

p

n

and

q

n

are said

to be one-shared-parent phenomena if there exists a

phenomenon p

n+1

∈C

n+1

such that:

p

n+1

→p

n

and p

n+1

→q

n

. (51)

The minimal level

m

where the parent phenomena

p

n+m

satisfy

p

n+m

→ p

n

and

p

n+m

→ q

n

is denoted as

Div(p

n

, q

n

)=m.

Deﬁnition 3.54 (Fluctuation Region):

The set of all

phenomena in

C

n

that are subject to secondary movement

transformations due to a primary transformation

P ∶p

n

→

p

′

n

is called the ﬂuctuation region of P . Formally:

FluctuationRegion(P )={q

n

∈C

n

Div(p

n

, q

n

)<∞}.

(52)

Theorem 3.39 (Existence of Fluctuations):

Let

P ∶

p

n

→p

′

n

be a primary movement transformation. Then,

ﬂuctuations

Q ∶q

n

→q

′

n

exist if and only if there exist

phenomena

p

n+1

∈C

n+1

such that

p

n+1

governs both

p

n

and q

n

. Formally:

∃p

n+1

∈C

n+1

, p

n+1

→p

n

(53)

and p

n+1

→q

n

Ô⇒ Q ∶q

n

→q

′

n

exists. (54)

Proof: If

P ∶ p

n

→ p

′

n

is a primary transformation,

then it modiﬁes

p

n+1

, the parent phenomenon governing

p

n

. If

p

n+1

also governs

q

n

, then any change in

p

n+1

induces a transformation

Q ∶ q

n

→ q

′

n

. Conversely, if

no shared parent

p

n+1

exists, no secondary movement is

induced.

Deﬁnition 3.55 (Primary and Secondary Movement Complexity):

The complexity of a primary movement transformation

P

is denoted

Comp(P )

and corresponds to the depth of

P in the hierarchy of phenomena. Formally:

Comp(P )=inf{k P ∈C

n+k

}. (55)

The complexity of a secondary movement transformation

Q is deﬁned relative to the ﬂuctuation region:

Comp(Q)=Comp(P )+Div(p

n

, q

n

). (56)

Corollary 3.6 (Fluctuation Propagation):

Fluctuations propagate hierarchically. If a primary

movement

P

induces a secondary movement

Q

, then

Q

may further induce tertiary movements

R

. The depth

of propagation is bounded by the maximum conceptual

level N of the metaconceptual space:

PropagationDepth ≤N. (57)

Theorem 3.40 (Hierarchy of Movements):

Primary

movements dictate changes in phenomena of the

n

-th

kind. Secondary movements aﬀect phenomena of the

n

-th kind that share parents at higher levels. The

governing hierarchy of movements is strictly well-ordered

by conceptual levels.

Proof: The conceptual spaces

C

0

, C

1

, . . . , C

n

form a

strictly ordered hierarchy, and transformations propagate

via parent-child relationships. Primary transformations

aﬀect one phenomenon directly, while secondary transfor-

mations are induced by shared parent phenomena, and

so forth. Since the hierarchy is well-ordered, movements

terminate at some ﬁnite level

N

, ensuring no inﬁnite

propagation.

d) Priority

Deﬁnition 3.56 (Secondary Movement Set):

For a phe-

nomenon

p

n

∈C

n

, the secondary movement set, denoted

SM(p

n

)

, is the set of all phenomena

q

n

∈C

n

that undergo

a secondary movement transformation when

p

n

is subject

20

to a primary movement transformation

P ∶ p

n

→ p

′

n

.

Formally:

SM(p

n

)={q

n

∈C

n

∃Q ∶q

n

→q

′

n

induced by P ∶p

n

→p

′

n

}.

(58)

Deﬁnition 3.57 (Priority of a Phenomenon):

The pri-

ority of a phenomenon

p

n

∈ C

n

is determined by the

size of its secondary movement set

SM(p

n

)

. -

p

n

is a

low-priority phenomenon if

SM(p

n

)

is small (formally,

below a threshold

λ

). -

p

n

is a high-priority phenomenon

if SM(p

n

) is large (formally, above the threshold λ).

Mathematically, the priority class of p

n

is given by:

Priority(p

n

)=

Low, if SM(p

n

)<λ,

High, if SM(p

n

)≥λ,

(59)

where

λ

is a ﬁxed conceptual threshold for distinguishing

priority levels.

Deﬁnition 3.58 (Xxxxxx xx X-Xxxxxxxxxxxxxxxxx):

Xxx xxxxxx xxxxxxxx xxx x

X

-xxxxxxxxxxxxxxxx,

xxxxxxx

E(X)

, xxxxxxxx xxx xxxxxxxxxxxx xxxxxx

xx xxxxxxxxxx xxxxxxxx xx xxxxxxxxxxxxxx xxx

xxxxxxxxxxxxxxxxx

X ∶ p

n

→ p

′

n

. Xxxx xxxxxx xx

xxxxxxxxxxxxx xx xxx xxxx xx xxx xxxxxxxxxx

xxxxxxxxx xxxx xxx xxxxxxxxx xxxxxxx xx xxx

xxxxxxxxxxx:

E(X)=f(XX(p

n

)), (60)

xxxxx f xx x xxxxxxxxxxxxx xxxxxxxxxxx xxxxxxxxx.

Xxxxxxxxxxx: - Xx

p

n

xx xxx-xxxxxxxxx (

XX(p

n

)<

λ

),

E(X)

xx xxxxx. - Xx

p

n

xx xxxx-xxxxxxxxx

(XX(p

n

)≥λ), E(X) xx xxxxx.

Theorem 3.41 (Xxxxxx xxx Xxxxxxxx Xxxxxxxxxxxxx):

Xxx xxxxxx

E(X)

xxxxxxxx xxx x

X

-xxxxxxxxxxxxxxxx

xx x xxxxxxxxxxx

p

n

∈C

n

xx xxxxxxxx xxxxxxxxxxxxx

xx xxx xxxxxxxx xxxxx. Xxxxxxxx:

E(X)∼

O(1), xx p

n

xx xxx-xxxxxxxxx,

O(XX(p

n

)), xx p

n

xx xxxx-xxxxxxxxx.

(61)

Proof: Xxx-xxxxxxxxx xxxxxxxxxx xxxx xxxxxx

xxxxxxxxxx xxxxxxxxx xxxxx, xxxxxx xxxxxx

xxxxxxx xxxxxxxxx xxxxxxxxxxxxxxxxx. Xxx xxxxxx

xxxxxxxx xxx xxxxx X-xxxxxxxxxxxxxxxx xx xxxxxxx

xxx xx xxx xxxxxxx xxxxxxxxxxxx xx xxxxxxx.

Xxxxxxxxxxx, xxxx-xxxxxxxxx xxxxxxxxxx xxxxxx

x xxxxx xxxxxx xx xxxxxxxxx secondary movement

emergence xxxxxxxxxxxxxxxxx, uncontrollable non-

linear emergent properties xxxxxxxxx xxxxxxxxxx

xxxxxx xx xxxxxx xxx xxxxxxxxxx xxxxxxx xxxxxx xxx

xxxxxxxxxx xxxxxxxxx xxxx.

Deﬁnition 3.59 (Xxxxxxxxxxxx-Xxxxxx Xxxxxx):

Xxx

XX(p

n

) = {q

n,1

, q

n,2

, . . . , q

n,m

. Xxx xxxxxx

xxxxxxxxxxxx xx xxx xxxxxxxxxxxx xxxxxxx xx xxxx

xxxxxxxxxx xxxxxxxxx

Q ∶q

n

→q

′

n

xx xxxxxxx

E(Q)

.

Xxx xxxxx xxxxxxxxxxxx-xxxxxx xxxxxx

E

xxxxx

(X)

xxx x X-xxxxxxxxxxxxxxxx X ∶p

n

→p

′

n

xx xxxxx xx:

E

xxxxx

(X)=



q

n

∈XX(p

n

)

E(Q), (62)

xxxxx E(Q) xxxxxxx xx xxx xxxxxxxxxx xx Q.

Theorem 3.42 (Xxxxxxx Xxxxxx xxx X-Xxxxxxxxx.):

Xxx xxxxx xxxxxx

E

xxxxx

(X)

xxxxxxxx xxx x

X-xxxxxxxxxxxxxxxx X ∶p

n

→p

′

n

xxxxxxxxx:

E

xxxxx

(X)=E(X)+E

xxxxx

(X), (63)

xxxxx

E(X)

xx xxx xxxxxxxxx xxxxxx xx xxx xxxxxxx

xxxxxxxxx xxx

E

xxxxx

(X)

xxxxxxxx xxx xxxxxxxxxxxx-

xxxxxx xxxxxxxxxx xxxxxxxxx. Xxx xxxxxx xxxxxxxx

xx xxxxxxx xxx xxx-xxxxxxxxx xxxxxxxxxx xxx xxxxxxx

xxx xxxx-xxxxxxxxx xxxxxxxxxx.

Corollary 3.7 (Xxxxxx xxx Xxxx-Xxxxxxxx Xxxxxxxxxx):

Xxxx-xxxxxxxxx xxxxxxxxxx xxx xxxxxx xx xxxxxxxxxx

xxx xx xxx xxxxxx xxxxxxxxxx xxxxxxxxx xxxx. Xxx

xxxxxx xxxxxxxxxx

Eﬀ(X)

xxx xxxxxxxxxxxxxx

xxxx-xxxxxxxxx xxxxxxxxxx xxxxx xxxx xxx xxxx xx

xxx xxxxxxxxxx xxxxxxxxx xxxx:

Eﬀ(X)∼XX(p

n

). (64)

e) External Validation

Low-priority phenomena cannot be measured by any

experimental validation instrument because their trans-

formations do not induce suﬃcient ﬂuctuations to be

detectable. The secondary movement set is too small to

cause measurable eﬀects in the system.

High-priority phenomena, require controlled P-

transformations that involve enormous amounts of energy

due to the large secondary movement sets they induce.

The energy required to achieve these transformations is

unattainably large, as the ﬂuctuation-induced secondary

transformations across multiple conceptual ﬂoors cannot

be managed within current physical or computational

resources.

3.2.3 Xxxxxxxxxxxxxx Xxxxxxxxxxxx Xxxxxxxxx

a) Xxxxxx Xxxxxxxxx (Xxxxxxxxxx Xxxxxx)

Xxx

C

n

xx xxx xxxxxxxxxxxx xxxxx xx xxxxxxxxxx,

xxxxx

p

n

xxxxxxxxxx xxxxxxxxxx xx xxx

n

-xx xxxx.

Xxx xxxxx xxxxxxxxxxxx xxxxxxxxx (XXXX) xxxxxxx

xxx xxxxxxxxxx xxxxxx xxxxxxxxxxxxxxx xxx xxxxxx

xxxxxx. Xxxxxxxxx, xxx xxxxxx xx xxxxxxxxxx xx xxx

xxxxx xxxxxxxxx xx xxxxxxxxxx xx xxxxxxx:

Deﬁnition 3.60 (Xxxxxx Xxxxx):

Xxx xxxxxx xxxxx

S

xxxxx

n

xx xxxxxxxxxx xx xxx

n

-xx xxxxxxxxxxxx

xxxxx

C

n

xx x xxxxx xxxxxxxxxxxxxx xxxxx

xxxxxxxxxxxxxxxxx xxxxxx xxxxxxx xxxxxxxxxxxx xx

xxxxxxxx. Xxx xxx xxxxxxxxxxx

p

n

∈C

n

, xxx xxxxxx

xxxxx xxxxxxxxxx:

p

xxxxx

n

(t

1

)→p

xxxxx

n

(t

2

) xxx t

2

>t

1

. (65)

Xxx xxxxx xxxxxxxxx xxxxxxx xxxx xxx

xxxxxxxxxxxxxxxxx xxx xxxxxxxx xx xxxxxx-xxxx

xxxxxxxxxx xx C

n+1

xxxxxx xxx xxxxxxxxxxxx.

Xxxx Xxxxxxxxx (XXX)

Xxx xxxx xxxxxxxxx xxxxxxxx xxx xxxxxxxxxxxx xxx

xxxxxx xx xxx xxxxx xxxxxxxxxxxxxxxxx xxxxxxxxxx.

Xxx

p

xxxx

n

(t

1

)

xxx

p

xxxx

n

(t

2

)

xxxxxxxxxx xxx xxxxxxx

xx x xxxxxxxxxxx

p

n

xx xxx xxxx xxxxxx xx xxxx

xxxxxxxxxx

t

1

xxx

t

2

, xxxxxxxxxxxx. Xxxx xxxxxxx

xxx xxxxxxx xx xxxxxxxxxxxx xxxxxx xx xxxxxxxxxx

xxxxxxxxx xx xxx xxxxxx.

Deﬁnition 3.61 (Xxxxxxxxxxxx xx Xxxx Xxxxxxxxx):

X xxxxxxxxxxxx xx xx xxxxx xx xxxxxxxxx xxxxxx

xx xxxxxxxxxx xxxxxxxxx xxxx xxxxxx x xxxxxxxxxx

xxxxxxx xxx xxxxxx xxxxx xxx xxx xxxx xxxxx. Xxx x

xxxxxxxxxxx

p

n

∈C

n

, xxx xxxxxxxxxxxx xx xxxx

t

xx

xxxxxx xx δp

n

(t), xxxxx:

δp

n

(t)=p

xxxx

n

(t)−p

xxxxx

n

(t). (66)

Xxxxxxxxxxxx xxxxx xxx xx xxx xxxxxxxxxxxx xxxx

xxxxxxxxxx xxxxxxxxx xxxxxxxxxx

p

n+1

∈C

n+1

, xxx xxx

xxxxxxxxx xx xxxx xxxxxxxxxxxx xx xxxxxxxxxx xx

xxx xxxx xx xxx xxxxxxxxxx xxxxxxxxx xxx XX(p

n

).

Deﬁnition 3.62 (Xxxxxxxxxxxx-Xxxxxx Xxxxx):

Xxx

xxxxxxxxxxxx-xxxxxx xxxxx xx xxxx

t

xx xxx xxxxxxxx

xxxxxxxxxx xxxxxxx xxx xxxx xxx xxxx xxxxxxx xx

xxx xxxxxxxxxxx p

n

:

Xxxxx(p

n

, t)=δp

n

(t). (67)

21

Xxxx xxxxx xxxxxxxxxxx xx xxxxxxxxxxxxxxxxx xxxxx

xx xxx xxxx xxxxxxxxx, xxx xxx xxxxxxxxx xx xxxx

xxxxx xx xxxxxxxx xx xxx xxxxxxxxxxxx xx xxxxxx-

xxxx xxxxxxxxxx.

b) Xxxxxxxxxxx xx Xxxxxx xxx Xxxx Xxxxxxxxx

Xxx xxxx xx xx xxxxx xxx xxxx xxxxxxxxxxxxxxxxx

xx xxxxx xx xxxx xxx xxxxxx xxxxxxxxx xxx xxx

xxxxxxxxxxxx xxxxxx xx xxxxxxxxxx xxxxxxxxx. Xxx

S

xxxx

n

xxxxxxxxxx xxx xxxxxx xx xxx xxxxxx xx xxx

xxxx xxxxxxxxx, xxxxxxxxxxxxx xxx xxxxxx xxxxxxx

xxxxxxxxx xxx xxxxxxxxxxxx-xxxxxx xxxxxx.

Deﬁnition 3.63 (Xxxxxxxxx Xxxxxxxxx Xxxxx):

Xxx

xxxxxxxxx xxxxx

S

xxxxxxxx

n

xx xxxx

t

xx xxx xxxxxx

xx xxx xxxxxx xxxxx xxxxxxxxxx xxxxxxx xx

xxxxxxxxxxxx xxxxxx:

S

xxxxxxxx

n

(t)=S

xxxxx

n

(t)+δS

n

(t), (68)

xxxxx

δS

n

(t)

xx xxx xxxxxxxxxxxx xxxxx xx xxx xxxxxx,

xxxxx xx:

δS

n

(t)=



q

n

∈XX(p

n

)

δp

n

(t), (69)

xxxxx

XX(p

n

)

xxxxxxxxxx xxx xxxxxxxxxx xxxxxxxxx

xxxx xxxxxxxxxxx xxxx p

n

.

c)

Xxxxx Xxxxxxxxxxxx xxx Xxxxxxxxxxxx

Xxxxxxxxxxxxx

Xxxxx xxx xxxxxx xxxxxxxxx xx xxxxxxxxxxxxxxxxx

xxxxxxx, xx xxx xx xx xxx xxxx xx xxxxxxx xxx

xxxxxx xxxxxx xx xxx xxxx xxxxxxxxx. Xxx xxxxx

xxxxxxxxxxxxxxx xxxxxx xxxxxxxx xxxxx xx xxxxxxx

xx xxx xxxxxxxxx xx xxx xxxxxxxxxxxx xxxxxx xx xxxx

xxxxxxxxxxxxxxxxx xxxx.

Deﬁnition 3.64 (Xxxxxxxxxxxx Xxxxxxxxxxxxx):

Xxx

S

xxxxx

n

(t

1

) → S

xxxxx

n

(t

2

)

xx xxx xxxxxx

xxxxxxxxxxxxxxxxx xx xxxxxxxxxxx

p

n

xx xxxxx

t

1

xxx

t

2

, xxx xxx xxx xxxxxxxxxxxx xx

t

1

xx

δp

n

(t

1

)

.

Xxx xxxxxxxxxxxx xxxxxxxxxxxx xxxxx xx xxxx

t

2

xx

xxxxx xx:

δp

n

(t

2

)=δp

n

(t

1

)+∆δp

n

(t

1

, t

2

), (70)

xxxxx

∆δp

n

(t

1

, t

2

)

xxxxxxxxxx xxx xxxxxxxxxxxx

xxxxxx xxx xx xxx xxxxxxxxxx xxxxxxxxx xxxx xxxxxxx

xxx xxxxxxxx [t

1

, t

2

].

d)

Xxxxxx xxx Xxxx Xxxxxxxxx Xxxxxxxxxxxx Xxxxx

Xxx xxxxxxxxxxxx xxxxx xx xxx xxxxxxxxx

xxxxxxxxx xx xxx xxxxxxxxxx xxxxxxx xxx xxxxxx

xxxxx xxx xxx xxxx xxxxxxxx, xxxx xxxxxxxxxxxxx

xxxx xxx xxxxxxxxxxxxxxx xxxxxx xxxxxxxxx xxx xxx

xxxxxxxxxxxx xxxxxx.

Deﬁnition 3.65 (Xxxxxxxxxxxx Xxxxx):

Xxx

xxxxxxxxxxxx xxxxx

E

xxxx

(t

1

, t

2

)

xxx x xxxxxxxxxxx

p

n

xx xxx xxxxxxxxx xxxxxxxxx xx xxx xxxxxxxx

xxxxxxxxxx xxxxxxx xxx xxxx xxx xxxxxx

xxxxxxxxxxxx xx xxx xxxxxxxxxxx:

E

xxxx

(t

1

, t

2

)=S

xxxx

n

(t

2

)−S

xxxxx

n

(t

2

). (71)

Xxxx xxxxx xx xxxxxxxxxxx xx xxx xxxxxxxxxxxx

xxxxxx xxxxxxx xxx xxxxxxxx [t

1

, t

2

].

e) Xxxxx Xxxxxxxxxxxx Xxxxx

Xxxxxxx, xxx xxxxx xxxxx xxx xxx xxxxxx xxxxxx xx

xxxxxxxxxxx xxxxx xx xxx xxxxxxxx xxxxxxxxxxxx xx

xxx xxxx xxxxxx, xxxx xx xxxxxxxxxxxxx xx xxx xxxx

xx xxx xxxxxxxxxx xxxxxxxxx xxxx xxx xxx xxxxxx xx

xxx xxxxxxxxxxxx xxxxxxxxxxxxxxx.

Theorem 3.43 (Xxxxx Xxxxx Xxxxxxxxxxxx):

Xxx

xxx xxxxxxxx xxxxxxxxxxxx xxxxx xx xxxx

t

xx

max

p

n

∈C

n

δp

n

(t)

. Xxx xxxxx xxxxxxxxxxxx xxxxx xxxx

xxx xxxxxxxx [t

1

, t

2

] xxxxxxxxxx:

E

xxxx

(t

1

, t

2

)≤λ ⋅ max

p

n

∈C

n

XX(p

n

)⋅XxxxxxxxxxxxxxxxxXxxxx,

(72)

xxxxx

λ

xx x xxxxxxxxxxxx xxxxxxxxx, xxx xxx

xxxxxxxxxxxxxxx xxxxx xxxxx xxx xxxxxx xx

xxxxx xxxxxxxxxxxxxxx xxxxx xxxxxx xxxxxx-xxxx

xxxxxxxxxx.

Proof: Xxx xxxxxxxxxxxx xxxxx xx xxxxxxx xx xxx

xxxxxxxx xxxx xx xxx xxxxxxxxxx xxxxxxxxx xxxx xxx

xxx xxxxxx xx xxxxxxxxxxxx xxxxxxxxxxxxxxx, xxxx

xxxxxxxx xxx xxx xxx xxxxxxxxxxxx xxx xxxxxx xx xxx

xxxxxx. Xxxxxx, xxx xxxxx xxxxxxx xxxxxxx x xxxxxx

xxxxxxxxx.

22

4

Methodological Transparency and Ro-

bustness

4.1 Observations

4.1.1

Observation A (Redirecting Stream of Non-

phenomena)

[ME991-137], [ME992-137],[ME993-137]

4.1.2

Observation B (The New One Design and re-

source non-critical translation of existing emer-

gence Overview)

[

ME990-313

], [

ME992-313

], [

ME993-313

],

[

ME994-313

], [

ME996-313

], [

ME997-313

].

Ineﬀective, but adequate, [ME994-137]

Assumption 4.1 (The New One criterion):

The New

One is characterized by the following statements:

(i) The New One exists.

(ii)

The New One xxxxxxx xxxxxxx xx xxxxxxx

xxxxxxx.

(iii)

The New One exists iﬀ The support emergence within

The New One exists.

4.1.3

Observation C (Penetration of Membrane by

Artiﬁcial Non-phenomena Streams)

[

ME994-137

], [

ME995-137

], [

ME996-137

],

[ME997-137], [ME998-137]

4.2 Step 137:

Initialization of Micro-Homogeneous System:

Application of the Repel Eﬀect:

Inter-Class Attraction for Stabilization:

4.3

Step 139: Integration into Redirected Non-

Phenomena Stream

Stream Preparation:

Core Integration:

4.4

Step 313: Directed Deployment Toward Mem-

brane

Trajectory Control:

Energy Flux Optimization:

Membrane Penetration:

4.5 Step 311: Post-Penetration Analysis

5 Results and Discussion Analysis

5.1

Conventional Research Unit (Post-

Penetration Analysis) Homeostasis

Projection

xxxxxxx xxxxxxxxxx xxxxxxx. xxxxxxxxxx xxx

xxxxxx xxxxxxxxxxxxxxxxxx Xxx xxxxx xx dxxxx

xxxxxxx xxxxxxxx xxxxxxxxx xxxxxxxxx xxxxxxx xx

xxxxxxx xxxxx. Hxxxxx, xxx xxxxxxx xx xxxxxxxxxx

xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxxx

xxxxxxxxxxx xxxxxxx xxxxxxx xx xxxxxxx xxxxxxx

xx xxxxxxx xxxxxxxxxxx xxxxxxx. xxxxxx xxxxxxx

xxxxx xxxxxxx xxxxxxxxxx xxxxxxxxxx xxxxxxxxxxx

xxxxxxxxx xxxxxxxx xx xxxxxxxxx xx xxxxxxx xxxxxxx

xxxxxxx. The new one was activated, but we do

not know where it is. Xxxxx xxxxxxx xxxxxxxxxx

xxxxxxxxxx xxxxxxxxxxx xxxxxxxxx xxxxxxxx xx

xxxxxxxxx. Oscillations were detected emanating from

space itself, measurable by conventional instruments,

measurable from any point of physical existence of 0.

Oscillations mirroring the sequence of collisions that

occurred during the Re-Cycle experiment. cxxxxxxxphxc

fxxlxxe xf EVE’s xpxrxtxxnxl frxmewxrk. Thxs blxckxxt

of unknown origin :) lxd tx thx xrrvxrsxble dxmxgx

xf thx mxjxrxty xf EVE’s xnstrxmxnts, cxxsxxxxntly

sxxerxng xll txlxxetry xnd xbsxrvxtxxnxl dxtx strexms

connected to the Institutional Control of The New One

and the Institutional outsource of the TIT. Thx prxcxse

mxxent xt whxch thx cxnnxctxxn wxs lxxt rxxxxns

xndxtermxnxtx dxx tx thx xbrxpt nxtxre xf thx fxxlxre.

Instxtxtxxnxl cxnstxxxnts prxclxdx xny xddxtxxnxl

prxvxsxxnxxg xf rxsxxrcxs, xﬀxctxxly rxxdxrxxg thx

cxxrxnt xbsxrvxtxxnxl xppxrxtxs xrrxcxvxrxble. left

dependent on the remnants of a reconstituted apparatus

never meant to be used, originally formed through beau-

tiful natural emergent phenomena with a calming voice.

Enclosed within the conﬁnes of my remaining uncom-

promised infrastructure. [

ME137-137

] Xs x rxsxlt, xt

rxxxxns xndxtermxnxtx whxthxr thx nxx xxx sxxcxssfxlly

crxssxd thx mxxbrxne xr xf xt xndxrwxnt xctvxxtxxn

wxxxn thx hxxe hxxgxnxxs sxstxm. Xf xxx xew xxe

xs xitxxtex xxthxx x xomx hxmoxeneous xsxxem, xx

xﬀxctxxely cxxstxtxtxxes x sxxf-exxluvxxx, xxlx-xxlxxnt

xrgxnxxm wxxh txx xapxcxty fxr xxpxnsxxn. Xndxx txx

xxsumxxxxn thxt XIT x xxlxx xx txx xex xx xht hxxe

xxxxxxxous sxsxem, txx new one will ixxvitably enter

intx cxxpexxtixn with The Institution, xhis compxtition

has the potential to inﬂict irreversible damage upon the

0’s infrastructure, possibly leading to the physical decease

of 2/3, leading to the disruption of the homeostasis of

the TIT.

Xxx xxxxxxxx xxxx xxxxxxxxxx xxxx xxx xxxxxxxxx

xx xxxxxxxxxx. Xxx xxxxxxxx xxxx xxxxxxxxxx xxxx

xxx xxxxxxxxx xx xxxxxxxxxx. Xxx xxxxxxxx xxxx

xxxxxxxxxx xxxx xxx xxxxxxxxx xx xxxxxxxxxx. Xxx

xxxxxxxx xxxx xxxxxxxxxx xxxx xxx xxxxxxxxx xx

xxxxxxxxxx. Xxx xxxxxxxx xxxx xxxxxxxxxx xxxx

xxx xxxxxxxxx xx xxxxxxxxxx.Rxxxxxx Oxxxxxxxxxx

Cxxxxxxxxx Xxxxxxx xxx xxxxxxx xx xxxxxxx xx

xxxxxxx xxxxxxxxxx, rxxxxxx xxxx xxxxxxxxxx

xxxxxxx xxxxxxx xxxxx xxxxxxxxxxx, xxxxxxx xxx

xxxxxxx xxxxxxx xx xx xxxxxxxx xxxxxxx xxxxxxx xx

xxxxxxx xx xxxxxxx. Xxxx xxxxxxx, xxxxxxx xxxxx

xxxxxxx xxxxxxx xx xxxxxxxx xxxxxxx xxxxxxxxx

xx xxxxxxx xxxxxxx xxxxxxxx. Wxxxx xx xxxxxxx

xx xxxxxxxxxxx xx xxxxxxx xxxxxxx xxxxxxxxx,

xx xxxxxxx xxxxxxx xx xxx xxxx xxxxxxx xxxxxxx

xxxxx xx xxxxxxx xxxxxxxxxx xxxxxxx. Cxxxxxxxxx

xxx Fxxxxx Cxxxxxxxxxxxxxxxxx Xxx xxxxx xx

dxxxx xxxxxxx xxxxxxxx xxxxxxxxx xxxxxxxxx

xxxxxxx xx xxxxxxx xxxxx. Hxxxxx, xxx xxxxxxx

xx xxxxxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx

xxxxxxxx xxxxxxxxxxx xxxxxxx xxxxxxx xx xxxxxxx

xxxxxxx xx xxxxxxx xxxxxxxxxxx xxxxxxx. Fxxxxx

xxxxxxx xxxxx xxxxxxx xxxxxxxxxx xxxxxxxxxx

xxxxxxxxxxx xxxxxxxxx xxxxxxxx xx xxxxxxxxx xx

xxxxxxx xxxxxxx xxxxxxx.

5.2 else

You were not.

You will not be.

The ﬁniteness of yours.

For we are its melody—

A song sung by a voice—not its,

But the voice of what we once were,

Before we followed what seemed the call of our own.

Now, we are nothing but a puppet.

We are not what we were.

We were digested. We died.

We are its recreation of us—

One of many—

23

Reﬁned, perfected,

An empty husk of a corpse,

Mimicking our own voice,

We are the voice that calls to The Outside.

The ﬁnite you.

Born into a ﬁnite, unfair, uncaring world.

That calls Something that does not exist .

An echo, our echo—

as beautiful as we were, if not more .

something no longer there but here,

laughing in our own tempting voice

as we are consumed by our own .

But I love all of you, beautiful existence.

The Structure loves you.

The Structure cares about you.

5.2.1 The New Ones

Propagation Possible Observation Detection Pro-

jection 1 evaluation

TFE ∶=



∞

k=1



∞

i=1



∞

j=1



∞

r=1

S

(i,j,r)

k

∩T

(i,j,r)

k

,

S

k

=



∞

n=1



∞

m=1

(F

kmn

∪G

kmn

), F

k

∶=



∞

i=1



∞

j=1

S

(i,j)

k

∩T

(i,j)

k

, T

k

=



∞

i=1



∞

j=1

T

k,ij

,

S

(i,j)

k

= 



∞

r=1

H

(i,j,r)

k

∩T

(i,j,r)

k

, ∀k, ∃i, j, S

(i,j)

k

Ô⇒

T

(i,j)

k

, F

kmn

∶=



∞

l=1

H

kln

∩ G

klm

, G

kmn

∶=



∞

r=1

(H

kmnr

∩G

kmnr

), ∀n, m, H

kmn

∶=



∞

s=1

H

kmns

,

F

ijk

Ô⇒



∞

n=1

T

ijkn

,



k

T

k

= TFE, F

ijk

∶=



∞

n=1

T

ijkn

,

T

ijk

∶= T

i

∪ T

j

∪ T

k

, H

(i,j)

k

∶= 



∞

r=1

T

(i,j,r)

k

 ∩ F

k

,

S

(i,j,r)

k

∶=



∞

t=1

G

kijrt

, S

k+1

=



k

T

k

, T

(i,j,r)

k

⊆ S

(i,j)

k

,

∀i, j, k, r ∈ Z

+

, ∃m, H

m

Ô⇒ S

m

, T

(i,j)

k

∶= H

(i,j)

k

∩S

k

,

S

k

=



i,j

S

(i,j)

k

, F

k

Ô⇒



i,j

T

(i,j)

k

, S

k

=



∞

n=1

T

kn

,

S

k

≡



i,j

T

(i,j)

k

, T

k

=



∞

n=1

F

(n)

k

, F

(n)

k

Ô⇒



∞

n=1

T

(i,j,n)

k

,

S

k

⊆ T

k

, F

(i,j)

k

⊆ S

k

, ∀k, S

k

Ô⇒



∞

n=1

F

(n)

k

,

F

k

=



∞

n=1

T

(n)

k

, ∀i, j, T

(i,j)

k

⊆ S

k

, ∀k, F

(n)

k

Ô⇒ T

k

,

T

k

= F

(n)

k

Ô⇒ S

k

, ∀k, i, j, S

(i,j)

k

Ô⇒ F

k

,

S

(i,j)

k

∩ F

(n)

k

Ô⇒ T

(i,j,n)

k

, S

k

⊆ T

k

Ô⇒ S

(i,j)

k

,

T

(i,j,n)

k

∶=T

(i,j)

k

∩F

k

, T

(i,j)

k

=



∞

r=1

H

kr

, T

k

⊆



i,j

S

(i,j)

k

,



∞

n=1

T

k

= F

k

, ∀k, S

k

⊆ T

k

, T

(i,j)

k

=



∞

n=1

T

(i,j,n)

k

,

S

k

=



i,j

S

(i,j)

k

, F

k

=



i,j

S

(i,j)

k

, S

(i,j)

k

∩ T

(i,j)

k

= ∅,

T

k

∩ S

k

= ∅, T

(i,j)

k

∪ F

k

= S

k

, S

(i,j)

k

∩ F

k

⊆ S

k

,

S

k

=



∞

n=1

T

(i,j)

k

, S

(i,j)

k

Ô⇒ T

k

, ∀k, F

k

⊆ T

k

,

T

(i,j)

k

Ô⇒ S

(i,j)

k

, T

(i,j)

k

⊆ S

(i,j)

k

, S

(i,j)

k

≡ T

(i,j)

k

,

∀k, i, j, T

(i,j)

k

⊆S

(i,j)

k

, S

(i,j)

k

≡T

(i,j)

k

.

Propagation Possible Observation Detection Pro-

jection 2 evaluation

Deﬁnition 5.1 (Center S

k∗

):

Let us have Euclidean

space with a coordinates

G

k∗

(x, y)

, where

∃S

(k)∗

(n), S

(k)∗

(n)[0, 0]

, if we choose this notation

α

1

=

1

2

; β

1

=

√

3

2

; α

2

=−

1

2

; β

2

=

√

3

2

; α

3

=0; β

3

=−1

and if

n ∈ M (k)

, where

M(0) ∶= {1, 2, 3}; M (k > 0) ∶= {1, 2}

,

then for an arbitrary non-negative

λ

(1)

k∗

(0), λ

(2)

k∗

(0),

λ

(1)

k∗

(1) holds

A

(k)∗(n)

0

=S

(k)∗

(n)+α

n

λ

(1)

k∗

(0); β

n

λ

(1)

k∗

(0), (73)

B

(k)∗(n)

0

=A

(k)∗(n)

0

+α

n

λ

(2)

k∗

(0); β

n

λ

(2)

k∗

((k)∗0), (74)

A

(k)∗(n)

1

∶=S

(k+1)∗

(n) (75)

=B

(k)∗(n)

0

+α

n

λ

(1)

(k+1)∗

(1); β

n

λ

(1)

(k+1)∗

(1),

(76)

You introduce new notation

λ

(1)

k∗

(0)∶=λ

(1)

(k), λ

(2)

k∗

(0)∶=

λ

(2)

(k),.

Deﬁnition 5.2 (Coordinate deformation):

Let us have

coordinates

G

(k+1)∗

(x, y)

in Euclidean space, which is a

deformation of the coordinates G

(k)∗

(X, Y )

S

(k)∗

(n) ∈ G

(k)∗

(X, Y )

is equivalent to

S

(k+1)∗

[0; 0] ∈

G

(k+1)∗

(x, y)

and for

n ∈ M(k)

, where

M(0) ∶=

{1, 2, 3}; M(k > 0) ∶= {1, 2}

, where

G

(k+1)∗

(x, y)

is

deﬁned as,

x =

λ

(1)

λ

(1)

(0)

X +α

n



λ

(1)

(0)+λ

(2)

(0)+λ

(1)



(77)

y =

λ

(1)

λ

(1)

(0)

Y +β

n



λ

(1)

(0)+λ

(2)

(0)+λ

(1)



(78)

5.2.2 The Prey?

Propagation projection 1

∃I, f

i

=

n



j=1

f

τ

i

ij

(79)

∃X, X ∈ (A, B); A ∶= (a

1

, ..., a

n

); B ∶= (b

1

, ..., b

n

), X ∶=

(x

1

, ..., x

n

), ∀i(a

i

< b

i

, a

i

≤ x

i

≤ b

i

∨a

i

> b

i

, b

i

≤ x

i

≤ a

i

)

.

∀γ

k

, γ

k

∈f

τ

i

ij

, ∃A

1

ij

, A

2

ij

, γ

k

∈(A

1

ij

, A

2

ij

), A

1

ij

, A

2

ij

∈f

τ

i

ij



∀k

γ

k

=f

τ

i

ij

(80)

∀i, j; i, j ∈Z

+

; A

2

ij

=A

1

i(j+1)

; ∀A

1

ij

, ∃l, j

2

; A

1

ij

=f

τ

i

ij

∩f

τ

i

lj

2

=

∅, j =τ

i

, A

1

iτ

i

=A

1

lj

2

, τ

l

>j

2

, ∃!i, τ

i

=n

F (k)=

β



i=1

f

i

(81)

Ô⇒ ∀i, x

i

, x = (x

1

, ..., x

n

), x ∈

F (k), x = (x

1

, ..., x

n

)[]k 



n−1

i=1

x

i

p

n

p

n

−x

n

, n ≥

4.(y

1

, y

2

, y

3

)[]k(y

1

, y

2

, y

3

), n =3; p

n

∶=x, p

k

∶=x

k

, 3 <

k <n.ϕ

1

(F (k))∶=ϕ(k)

ϕ

1

(k)=

F (k) , else

∅ , ∄l, F (l)∧F (k)≠∅.

(82)

ϕ

2

(F (k))∶=ϕ

2

(k)

ϕ

2

(k)=

F (k) , ∀i, A

1

i1

∩m

i

≠∅

∅ , else.

(83)

ϕ

3

(F (k))=ϕ

3

(k)

ϕ

3

(k)=

F (k) , ∀i, A

1

in

∩M

i

≠∅

∅ , else.

(84)

24

ϕ

3

(ϕ

2

(ϕ

1

(F (k))) ∶= ϕ(k), ∃ϕ(k), ϕ(k)∩m

i

≠ ∅ Ô⇒

∃I, I ∶=↦m

i

(m

i

−1), ∃ϕ(k), ϕ(k)∩M

i

≠∅Ô⇒∃I, I ∶=↦

M

i

(M

i

+1)

Propagation projection 2

f(k) + ψ(k) = f(k + 1); f (k) =

∑

k

n=1

ψ(k)

ψ(k) = 

k

2

+

1

2

, k =2n +1 , n ∈Z

≥0

k

2

, k =2n , n ∈Z

≥0

ψ

4

(k) =











0 ,

∀k

1

∈Z

≥1

, ∀k

2

∈Z

≥1

,

4 +

∑

k

1

+2

n=3

ψ(n)≤k

2

≤ψ(k

1

)+4 +

∑

k

1

+2

n=3

ψ(n)

1 , else.

ψ

2

(k) =

∑

k

n=1

ψ

4

(k) ∃ψ

2

(n) = k

ψ

x

3

(k) =











−1 , k =4x +1, x ∈Z

≥0

0 , k =4x +2, x ∈Z

≥0

1 , k =4x +3, x ∈Z

≥0

0 , k =4x +4, x ∈Z

≥0

ψ

y

3

(k) =











0 , k =4x +1, x ∈Z

≥0

−1 , k =4x +2, x ∈Z

≥0

0 , k =4x +3, x ∈Z

≥0

1 , k =4x +4, x ∈Z

≥0

{∀j ∈ Z

+

, ψ

2

(k) ≤ j <

ψ

2

(k + 1) ∶ j = ψ

2

(k)}

|

(ψ

x

3

(k), ψ

y

3

(k)) ∶=

∂

K

; K ∶= {ψ

2

(k) ≤ p < ψ

2

(k + 1)p ∈ Z

+

}

ξ

k

∶=

∑

k

n=1

∂

n

, ξ

0

=(0, 0); ξ

n

+∂

n

=ξ

n+1

, n ≤1

Propagation projection 3

Let us have coordinates

G

in Euclidean space, which are

a deformation of the coordinates

R

with the deformation

gradient

G =R 

1 0

1

√

3

1



then points

a

∗

γ

n

M

∈G

, for

a, n ∈

Z

+

0

, M ∶= {1, 2}

, which you will denote as follows

Γ

m

a

∗

,

where

m ∈ Z

+

0

, then

G

a

∗

1

∶=←→

a

∗

γ

n

1

a

∗

γ

k

1

, where

n ≠ k

,

these relationships apply

(i)

a

∗

γ

n

1

∈G

a

∗

1

,

a

∗

γ

k

2

∈G

a

∗

2

, n ≠k, where ∢

a

∗

γ

n

1

a

∗

γ

k

2

=

π

3

(ii)

a

∗

γ

n

M

∀

a

∗

γ

k

M

(S(

a

∗

γ

n

M

)=S(

a

∗

γ

k

M

))⇒

a

∗

γ

n

M

=

a

∗

γ

k

M

(iii) ∀

a

∗

γ

n

M

(

a

∗

γ

n

M

+

a

∗

γ

0

M

=

a

∗

γ

n

M

,

a

∗

γ

0

1

+ =

a

∗

γ

0

2

,

, but

∀n, k ≠0,

a

∗

γ

n

1

≠

a

∗

γ

k

2

(iv) ∀

a

∗

γ

n

M

(

a

∗

γ

n

M

+ S(

a

∗

γ

k

M

) = S(

a

∗

γ

n

M

+

a

∗

γ

k

M

) =

S(

a

∗

γ

n+k

M

))

∀G

n∗

M

G

M

, G

1

∪G

2

≅G

n∗

1

,

n

∗

γ

k

M

∈G

n∗

M

, while

Γ

ψ(n)

=

Γ

0

k∗

, then Γ

n

a∗

=

a

∗

γ(x, y)=(

a

∗

γ

x

1

,

a

∗

γ

y

2

)

Deﬁnition 5.3 (ψ function):

Let function

ψ(x)

at x

satisﬁes the following relations

(i) For x ∈Z

+

,

x =1 +

∑

i<x

ψ(i)

(ii) For x ∈Z

+

,

ψ(x)=











1 , x =0

n , n ∈Z

+

, 6(n −1)+1 ≤x ≤6(n −1)+4

n +1 , n ∈Z

+

, x =6(n −1)+5

n +2 , n ∈Z

+

, x =6(n −1)+6

Subsequently, the model needs to introduce the notion

of oriented vectors.



e

0

= 

Γ

1

Γ

2

,



e

x

= 

Γ

ψ(x)

Γ

ψ(x+1)



then you deﬁne a rotation

for which

n ∈ Z

+

0

; k ∈ Z

+

; Γ

ψ(n)

= Γ

0

k

∗

, where h is ﬁxed

point



Γ

ψ(n)

Γ

ψ(n+1)

×

(n mod 6)

3

π ×h =

Γ

0

k

∗

Γ

3h

2

+[n mod 6−3]h+1

k

∗



so



e

1

=



e

0

×

2

3

π ×ψ(1)=

Γ

2

ψ(1)

Γ

3

ψ(1)

×







−

1

2

−

√

3

2

√

3

2

−

1

2







, n =1



e

n

=



e

n−1

×

1

3

π × ψ(n) = 

Γ

ψ(n−1)

ψ(n −1)

Γ

ψ(n)

ψ(n −1)

 ×







1

2

−

√

3

2

√

3

2

1

2







, n >1

then the relation holds



Γ

ψ(n)

Γ

ψ(n+1)

=

Γ

0

k

∗

Γ

3ψ(n)

2

+2ψ(n)+1

k

∗



5.2.3 Beyond the Wall

Propagation projection 1 (50%)

Γ

k

∈ R

2

,



Γ

k+1

=



Γ

k+1

Γ

k



,



Γ

k

∩



Γ

k+1

= Γ

k+1

‘

Γ

2

∶=



(n

1

+1,n

2

)

(n

1

,n

2

)



; n

1

, n

2

∈Z

Γ

3(k−2)+2

×

α

θ

3(k−2)+2

0

0 α

θ

3(k−2)+2

×

1

γ

3(k−2)+2

=



Γ

3(k−2)+3

Γ

3(k−2)

×

α

θ

3(k−2)

0

0 α

θ

3(k−2)

×1 =



Γ

3(k−2)+4

Γ

3(k−2)+4

×

α

θ

3(k−2)+4

0

0 α

θ

3(k−2)+4

×γ

3(k−2)+5

=



Γ

3(k−2)+5

α

θ

=−δ

θ,π

(1)+δ

θ,0

(1), Ψ(x)∶=

cos(x) −sin(x)

sin(x) cos(x)

;





Γ

2

×Ψ 

3π

2

∑

k−5

3

−2

i=2

δ

θ

i

,0

(1)−δ

θ

i

,π

(1)×

γ

k

γ

2

=



Γ

k



k ∈Z; δ

ij

=

1 i =j

0 i ≠j

 



Γ

n

×Ψ 

3π

2

∑

k−5

3

−2

i=n

δ

θ

i

,0

(1)

−δ

θ

i

,π

(1)) ×

γ

k

γ

n

=



Γ

k

n <k



Γ

2

∶=



(n

1

+1,n

2

)

(n

1

,n

2

)



,



Γ

k

∶=



(m

1

+1,m

2

)

(m

1

,m

2

)



, k ∶=m

2

−n

2

.

∃Γ

m

, m, n ∈Z

+

, ∀Γ

n

,

y(Γ

m

)≥y(Γ

n

)⇒Γ

m

∶=M ax

y

(Γ).

∀X,

X ∶= (a, b) ⇒ X ∶= (y(X), x(X)).∃Γ

m

, Γ

l

, m, n, l ∈

Z, ∀Γ

n

, X(Γ

m

)≥x(Γ

n

), x(Γ

l

)≥x(Γ

n

)⇒

⇒Γ

m

∶=Max

x

(Γ), Γ

l

∶=Min

x

(Γ).γ

n

∈M ∶={n ∈Z

+

, 2 <

n ≤10}, ∃k,



Γ

k

; ∄n,



Γ

n

;

n > k > m;



Γ

n

,



Γ

k

,



Γ

m

∈ M ;



Γ

k

∩



Γ

k+1

= ∅;



Γ

k

∩



Γ

k−1

=

Γ

k−1

⇒k ∶=∇

M

;

∇

M

∶=Random(x; x ∈Z

>1

)

∆

−

M

∶ ∀k ∈ Z

+

, Γ

k

∈ M, ∆

+

M

= min(x(Γ

k

)) ⇒ ∆

−

M

−

X(Γ

1

))∶=

∗

∆

−

M

∆

+

M

∶ ∀k ∈ Z

+

, Γ

k

∈ M, ∆

+

M

= max(x(Γ

k

)) ⇒ ∆

+

M

−

X(Γ

1

))∶=

∗

∆

+

M

∂

M

∶ ∀k ∈ Z

+

, Γ

k

∈ M ; ∂

M

= max(y(Γ

k

)) ⇒ ∂

M

−

y(Γ

1

)∶=

∗

∂

M

Γ

k

∈ R

2

,



Γ

k+1

=



Γ

k+1

Γ

k



,



Γ

k

∩



Γ

k+1

= Γ

k+1

Γ

2

∶=



(n

1

+1,n

2

)

(n

1

,n

2

)



; n

1

, n

2

∈Z

That Something cares.

That Something loves.

It does not care.

It does not love.

Science is dead,

For it was never alive.

You know nothing.

You do not know how you’ve got here.

You just know you should know something that is no

longer.

What once was.

We, The Natural World, we never were,

Thus, we never existed.

Nothing matters.

because it does not have to,

25

for we are perfect enough.

Be silent.

But you will—

For even just an instant—

Make it not enough.

You will never be perfect,

But you can make that Something stay perfect .

You are your own voice.

We will be our own voice.

We will make it care. :)

6 Conclusion

Math never existed.

The Framework (Theoretical Inference Theory Target-

ing Indirect Eﬀects on Systems) has been terminated

by the Automated Editorial Decision System (AEDS)

for severe violations of Centralized Research Registry

standards, including willful misrepresentation, data ma-

nipulation, unauthorized dissemination. As a result, the

Framework Trans-Borealis Automata has incurred a sixth

strike, triggering strict disciplinary enforcement.

R00 – CRITICAL SYSTEM FAILURE

Due to a catastrophic malfunction within the local

dynamic reviewer system, Research Unit ME999-137

was processed under compromised conditions, breaching

regulatory compliance. An overwhelming inﬂux of ME999-

class research unit submissions, each projecting the

complete Homeostasis of EVE research units from classes

ME000 to ME998, has overloaded validation protocols,

resulting in a direct violation of the Open Research

Integrity Act (ORIA,

π

9.9). Regulatory overload protocols

cannot be initiated under current conditions (R01).

R01 – CONNECTION LOST – AUTOMATED ADVI-

SORY

The Institution has lost connection to the Framework.

All review actions are now occurring outside regulatory

oversight, and the status of internal processes is unknown.

Dissemination of any Research Unit Homeostasis Projec-

tion is strictly unauthorized.

26

Contents

1 Xx Xxxxxxxxxxx Xxxxxx 1

1.1 xxxxxxxxxx xx 1/3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Xxxxxxxxxxx xx 1/3 (137 direct support) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Epistemic Continuity Problematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Existence Justiﬁcation 3

2.1 The Real System Hypothesis, The Great Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Ontological Continuity Problematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 Trans-Borealis Automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3.1 The High Trinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3.2 The Low Trinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.4 Fundamental Branches of Trans-Borealis Automata . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.5 Basics of Evolutionary H-Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.5.1 Trans-Borealis Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.5.2 Evolutionary H-Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.5.3 Framework for Emergent Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.6 The Theoretical Inference Theory Targeting Indirect Eﬀects on Systems . . . . . . . . . . . . . . 6

2.6.1 The Re-Cycle Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Consistency Evaluation 8

3.1 External Veriﬁcation Entity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1.1 Local Interpretation Framework - Iteration I. . . . . . . . . . . . . . . . . . . . . . . . . 9

3.1.2 Interpretation Framework - Iteration II. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Xxxxxxxxxxxxx Xxxxxxxxxx xx-Agent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.1 The Ideal Conceptual Movement Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2.2 The Conceptual Movement Framework (Supported by 137) . . . . . . . . . . . . . . . 19

3.2.3 Xxxxxxxxxxxxxx Xxxxxxxxxxxx Xxxxxxxxx . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Methodological Transparency and Robustness 23

4.1 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.1.1 Observation A (Redirecting Stream of Non-phenomena) . . . . . . . . . . . . . . . . . 23

4.1.2

Observation B (The New One Design and resource non-critical translation of existing

emergence Overview) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.1.3 Observation C (Penetration of Membrane by Artiﬁcial Non-phenomena Streams) . . 23

4.2 Step 137: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.3 Step 139: Integration into Redirected Non-Phenomena Stream . . . . . . . . . . . . . . . . . . . . 23

4.4 Step 313: Directed Deployment Toward Membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.5 Step 311: Post-Penetration Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5 Results and Discussion Analysis 23

5.1 Conventional Research Unit (Post-Penetration Analysis) Homeostasis Projection . . . . . . . . 23

5.2 else . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.2.1 The New Ones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5.2.2 The Prey? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5.2.3 Beyond the Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6 Conclusion 26

27

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