Theory of a Single Fundamental Harmonic Field, Part I - ІІ - ІІІ: Foundations, proving the Yang–Mill
Here is a brief introduction to the theory, indicating the structures of the main parts, then a number of initial explanations with links to more detailed documents already published on scientific sites and simply on the Internet.

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• View the monograph in English on the scientific platform zenodo - in three parts in the most complete form

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• Download Part I of the monograph in English in PDF format

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• Appendix_ New and Refined Predictions of the G-Model for the First Part of the Study.pdf

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• Abstract to Parts II–III of the Monograph “Development of the G-Model and the Proof of the Great Theorem”.pdf

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• Key to proving the Yang–Mills _ mass gap theorem based on the theory of the harmonic field.pdf

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• New numbering of the structure of Part I.pdf

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• Preliminary numbering of the structure of Part I.pdf

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A brief description of the book's ideas - Part I of the Monograph
Introductory Explanations to the Monograph — Theory of the Single Fundamental Harmonic Field “EK = const”

0. Short characteristics

The study consists of three main parts. Each of them develops and refines what has been done before. Therefore, a number of notions and concepts receive their deeper interpretation and final explanations not immediately, but as if emerging from a certain initial mild indeterminacy.
Nevertheless, from the very beginning the research relied on clear and understandable meanings. A well-articulated philosophy is clearly transmitted by mathematics and physics. This means that the meaning embedded in the initial concepts, structures and postulates can be formally reproduced in mathematical language and in physical models.
Such correspondence is the main backbone of the modelling process: philosophical clarity
→ mathematical rigour → physical realizability.
Thus, within the G-field model a much wider space for further research opens up than the one presented in this monograph. However, precisely the first step — the structure given here — is the necessary foundation for unfolding a full-fledged theory of harmonic fundamental fields.
This monograph is the result of an authorial physico-mathematical and philosophical investigation of fundamental fields and their internal harmonic organization. It introduces and justifies for the first time the Lyashkevych harmonic postulate:
EK = const,

which is interpreted as a law of conservation of harmonicity in a stable self-governed system, where the energetic component (E) and the control component (K) remain harmoniously coordinated regardless of the state of the continuum.
This research forms a conceptual domain within which a strict physico-mathematical model of the birth, development and mutual adjustment of fundamental fields becomes possible, as well as the joint work of known physico-mathematical models.

1. Initial postulates

Fundamental fields exist as primary wholes that possess their own act of creation and intrinsic harmonic organization. Each such fundamental field (hereafter the G-field) gives rise to its own space–time
continuum, which is a secondary whole, derivative from the primary one.
The structures of the G-field form the continuum space; harmonic tasks correspond to times, and hence to different manifestations and phenomena, mass included.
Each G-field is expressed by its own complex harmonic constant Σ, which contains an invariant measure of energy and control. The conditional environment of G-fields we call an ensemble, in which each harmonic
spectrum is expressed by its own field. Therefore, in this environment there is no notion of hierarchy, ower, force, etc., although general criteria of energetic and other manifestations still operate.
General manifestations are built exclusively in the relations of all that exists, which is reflected in a reduced choral harmony of interaction.
The existence of a G-field is called forth by the general need precisely for its unique harmony. Thus such a field is the simplest and most effective solution to this need. It is postulated that the G-field is articulated from a first-point Ho, which is realized, both energetically and topologically, as the most accurate resolution of the missing component of general harmony and which represents the standard of its G-field.
We consider the energetic process of the emergence of harmony from its absence as a total articulation, a full-fledged filling along the entire spectrum of the need.
Such filling is of fractal character, where the basis is the first-point Ho and the harmonic invariant that describes it. Even though the energetic model of the G-field has one of its traits in the form of a harmonic phase and a zero, we consider the G-field itself as a proposal of pure potential.
Any “proposal” from the side of the G-field is strictly measured and rests on the multiplicity to its harmonic invariant, both in the integral energetic potential and in the topology of the harmonic spectrum.
All this can be called an ideal solution of beauty and goodness, which does not end merely as a project, because there is also a horizon of events which have not yet occurred at the moment of the field’s design.
The structuring of the G-field, based on fractal layers of Ho-points, which differ in topology and thus in inclusion into different chains of interrelations, is of project character.
However, reality also leaves its imprint, and the G-field possesses all necessary operators for this.
It is precisely the primary layers of Ho, in this or that structuring, that become the basis for the development of a multidimensional continuum — space–time, and also the electromagnetic field EM, containing superpositions of elementary particles (EP) — still without the massive fullness of their diversity.
The structures which unite Ho and acquire extended parametric continuum characteristics we distinguish as Lo and CH.
It is through them that the harmonic constant presents the ordering side, manifested as a tensor operator which unfolds the parameters of the field without violating the invariant. The continuum naturally arises as a response to the proposal of potential and topology from the structures of the G-field. Accordingly, from the side of the continuum there appears a demand for energy from secondary wholes. We consider the G-field itself as primary.
The project basis of the continuum consists both of superpositions of EP and of fragments of the G-field delegated to future secondary wholes and forming secondary fields. Secondary fields are constructed in the same way as the primary one, but not on the initial standard point Ho, rather on separate Ho–Lo chains, which have their own topological distinctions. Hence the quantization of the harmonic spectrum in the continuum. If the G-field remains one-dimensional and homogeneous in all its structural and topological
complexity, then each secondary whole of the delegated type (which initially receives its detached Ho–Lo chain, which becomes Ho_loc–Lo_loc) already exists in its own project regime H, which arises precisely from the topological separation of the secondary from the primary. And this defines all that follows.

2. Continuum, -deviations and project states

2.1. The continuum exists in two forms:
• project state — an ideal configuration, projectwise assigned by the G-field;
• realized state, which resolves the embedded harmonic deviations.
2.2. Harmonic deviations occur within the permissible -zone of secondary wholes — domains of delegated type and superdomains.
2.3. We use the notion of the continuum in the broadest sense of the multidimensionality of its manifestation, including discrete spaces and times, initially built by the G-field, and we mean that all this is combined into a whole by the field homogeneity and the universal properties of the G-field to structure phenomena and embodiments at all levels of manifestation of energy.
2.4. There is a constant further dispersion of this potential energy from the G-field within the kinetics of the continuum, as of that which has lost its ideal potential quality, passing into energetic processes where “-entropy” is present. However, in secondary wholes of the delegated type another layer acts as well: processes of refinement, growth of efficiency of local control (“+entropy”). How fully the dispersed energy loses local control over itself and whether it reaches complete loss of control is a subject for later study, but
the energetic balance in the whole field–continuum is always preserved, because harmonicity is about balance, stability and equilibrium, and the accompanying quality of effective self-governance.

3. Extended interpretation of zero

Within the model the notion of “zero” means not absence, but a topologically saturated state of potential fullness, from which harmonic structures unfold.
Clarification of zero:
• adds volume to mathematical language, translating it from planar descriptiveness into a topological space;
• creates a semantic basis for transitions between states of the field.

4. Extended understanding of entropy

Entropy, by its origin, is a notion that originally means “changes within oneself”. These changes can be directed in different ways. Some lead to dispersion and destruction of all that is unstable and not capable of
development; others — to the growth of quality, stability and conformity to a deeper harmonic structure.
In the author’s concept of the G-field theory, the basic image of entropy is divided into two, not directly but in a certain sense complementary components:
• “-entropy” — that part of self-changes which corresponds to the destruction of the unstable, the disintegration of obsolete configurations, the “cleaning away” of that which cannot be a carrier of development;
• “+entropy” — that part of self-changes which corresponds to the increase of quality, stability, equilibrium and harmonic efficiency of control, that is, to the development of wholes towards their maximal conformity to the G-field.

5. Lyashkevych formula

The formula EK = const = Σ is the basic invariant of the model and expresses:
• the unity of energy and control;
• the harmonic self-governance of the system;
• the law of conservation of harmonicity.


Extended description

1. Introductory clarifications.
To begin with, let us once again define, within the current understanding, what a fundamental harmonic field is. At first glance, it is an energetic self-governed harmonizer, a realizor of the missing harmonic constant, in which there is both a material basis and well-ordered internal structures — the beginning and
possible completion of one form of being and the onset of another.
There is also a multidimensional continuum present as a degree of fullness of manifestation of the G-field (the harmonized fundamental field), the forms and meanings of the field’s self-realization as a harmonizing (governing) potential precisely where the need for it arose — that is, in the region of the prior absence of harmony. We do not consider this absence as some alien continuum which should be replaced by something else, but strictly as the absence of the needed harmony, as a certain order.
When we use the notion of the continuum in the broadest sense of the multidimensionality of its manifestation, including discrete spaces and times, initially built by the G-field, we mean that all this is joined into a whole by the field homogeneity and the universal properties of the G-field to structure both phenomena and embodiments at all levels of manifestation of energy.
In this way, field harmonization is realized, first of all, by deepening its qualitative state in the region of boundless need for this harmony, as a result of self-governed processes and normalization of energetic states. We note that there is constant further dispersion of this potential energy within the kinetics of the continuum, as of that which has lost its ideal potential quality, passing into the kinetics of secondary whole self-governance, where “-entropy” is already present, while another layer is formed by processes of refinement and the growth of efficiency of local control (“+entropy”). How fully the dispersed energy loses
local control over itself and whether it reaches complete loss of control within the region of unfolding of the field is a subject for later study, but the energetic balance within the whole field–continuum is always preserved, because harmonicity is about balance, stability and equilibrium, and the accompanying quality of effective self-governance.

II. What the G-field consists of at the base level.

It begins from the first manifestation point Ho. This is a field first-manifestation of a minimally sufficient point-like energetic potential, which arises as a standard, topologically calibrated realization precisely there, in the node where this specific harmony is absent (the reasons may differ, but we speak of a certain spectrum of absence of a concrete unique harmonic functional).
Thus this primary Ho carries, besides the point-like minimally sufficient energetic potential, the most important thing which actually reveals this point potential — a standardly precise topology (harmonic orientation), strictly corresponding to the general need for it.
It is this point topology (precise harmony), which itself is this realized Ho, that makes it possible for the minimally sufficient magnitude of energetic potential to manifest, which for our field corresponds, in its further manifestation in the form of action in the simplest continuum medium, to Planck’s constant.
In fact, Ho manifests in a complex form of a complementary pair E (minimally sufficient potential energy) and K (minimally sufficient, topologically precisely calibrated control), expressed by the formula of the basic invariant EminKmin = const. From this
standard point (standard measure of control of precisely this G-field) a rollout of harmonically governed potential over the entire region of need for itself unfolds. This happens layer by layer. The layering of new Ho on the field groundwork, topologically and controlwise complementary to the first point, has the same point sizes of potential, but the first layer of Ho above the basic one is created by the overall potential of need in the spectrum of this G-field with a somewhat different, more extended higher topology, reflected in this level’s Ho itself. Therefore the first Ho-layer is already a superstructure over the first point which manifests a unit complex potential.
This first Ho-layer is built as a fractal sequence and, as a result of microtopological discrepancies (since sequential placement occurs in the one-dimensionality of the spectrum of need), they together form what is called a state of potential angular motion with a unit standard of control at each point of the layer, but the sequence of positions of these micro-controls is capable of setting the frequency of the angular wave when such energy is released as a result of interaction with its user. The final design of the receiver that
will turn to Ho for energy remains.
In this way the first chain of Ho sequences is built over the standard first point of the field (the minimally sufficient measure of control and thus the minimal unit of control calibration 1gk). We note that the G-field expands, manifesting from the ideal first point with the whole spectrum of need in a one-dimensional way, but the continuum is also built discretely in the direction of project fullness of the G-field, which is the fullness of the set of degrees of freedom.
Thus the first degree of freedom in the continuum is manifested as the possibility of kinetics of the received energy and the implementation of angular motion. This forms the manifestation of the continuum — space–time–quality. For this potential of motion in the topology of the G-field is already a manifestation of the project span, which generates, when articulated, space, then time, and further quality. Therefore everything that is created along with it as a possibility of resolution — the continuum — arises from the
need to realize this potential.

III. Ho — Lo.

Hence we have the basic reason for building the continuum, without which the field is not able to articulate itself fully. And the basis of this articulation is the topologically complementary Ho, which are capable of structuring. That part of the whole field–continuum which, from the side of structuring Ho, is responsible for the features of the continuum, we call the leading structure Lo, basic for constructing the continuum.
Initially, Lo for us is a first cell with Ho, where there is a potential of motion of energy removal according to topological units of control. Since at this level there is no difference, in terms of calibration, between the field that proposes and the continuum that consumes, we postulate that the fundamental harmonic (G) field is precisely that within which we here and now exist in one way or another, and the confirmed theories and repeatable experiments carried out at the basic level of the field correspond to our theory as its components. We can then directly transfer our parameters to the already known scientific basis.
For example, if in the field at this level Kmin = 1gk, which is the standard calibrated harmonic state of the first-point and manifests a complex quantity (portion) of potential energy, then in the continuum this Kmin must correspond to the effective placement of action in space and time, here and now. Such placement is a mathematical problem that has a functional-formula solution. Therefore, we also speak about the projection of 1gk into the continuum.

All the more in the situation when, with the next Ho-layer of field expansion, the same governed potential two-dimensionality appears, which then finds its energetic resolution in the form of the basic electromagnetic field known to us. Through it, the field potentiality and standardness of the G-field find their realization. But in this way the energy is transferred into the continuum as kinetic energy. Thus, according to our logic (within the framework of a continuum whole being harmonized with the G-field), it must be effectively dispersed, with the growth of the structure of control of this process on site, harmonically coordinated with the G-field.
If we see the G-field as a conditional system of governance unique by its own harmony, then the continuum as a whole, in unity with the G-field, is naturally generalized as E — energy which in the end must be entirely dispersed, giving the G-field the possibility to reach fullness of quality and, in the state of acquired fullness, to metamorphose further.

So we may consider the whole G-field plus continuum also from another angle, where E (energy) in the continuum is kinetic energy of dispersion, which, while fulfilling project tasks with varying degree of quality, gradually loses further local control over itself. The process of self-governance in the continuum is the system of self-perfection of this or that secondary whole, which tries to make this dispersion as effective as possible. The governing functions in more complex, independent secondary wholes, where entropic phenomena are already present, are headed by local fields capable of growing in quality (and thus in harmonic relation with the G-field) and attaining fullness within the domain and above. Returning to the G-field, if there control is the fractality of state and topology calibrated in gk topological units, then in the continuum it is always the projection of the potentiality 1gk (realized or not) onto local metric correspondences with a project-permitted delta of harmony, which always sets the need for effective positioning of the quality of action in space and time. The higher the continuum structure, the more complex the task of attaining the maximum achievable quality of state that stands before it.
Thus even for the G-field itself, the continuum is a responsibility — the responsibility of the field to itself and to its project; this responsibility is a measure of the fullness of the entire spectrum of quality of the field. Hence the main principle of existence of the whole G-field–continuum: everything must occupy its own place. This is the avatar of the ideal (G-field) mathematics, which carries strict correspondences within itself, without gaps. Such “gaps” do not exist in the G-field itself. And even if at some level there is not yet
fullness, once it is resolved (found) mathematically, it is resolved in general. From here follows the abundance of mathematics in this research.
Whether there exists another mathematics that covers a larger space than that of
the G-field? Yes, but it does not exist within the G-space itself and does not reflect the directions and states of the G-field and its continuum.

4. The G-field from the context of our environment.

Thus at the level of the initial state of the G-field we have, in addition to the basic law of conservation of harmony EK = const, a defined standard first unit, which reveals in its state the complementarity of E and K in strict proportionality of the minimally sufficient portion of potential energy (equivalent to the action of Planck’s constant) and the precision of the “point-state” as a universal G-field measure of control — 1gk.
From this we may start an approach to the basic, known formats of the continuum state, where we immediately note the “burdens” in the form of m — mass and c — speed of propagation of electromagnetic interaction, quality of the state, etc.
Whether these burdens are present in the field itself? Apparently yes, but potentially. If the field were directly acting, it would fit into the general rules. But the field is a normalization for action and only the perfect state of its own harmony. Therefore it is somewhat detached from the general kinetic “burdens” for G-fields. To consider that each G-field is absolutely unique in all its manifestations and structures would be a mistake. Uniqueness is manifested precisely in its own governing quality, obtained and developed
from the uniqueness of EK = const, whereas generalized schemes and rules, burdens, exist also for G-fields, although they are not hierarchical restrictions but the result of combined harmonies from the interaction (chordal nature) of ensembles.
Approaching our realities, we understand that, for example, E = mc2 as a formula gains complete meaning beyond the threshold of the pure potential state of the G-field — where the continuum is already manifested as a medium needed for dispersion of energy.
But the G-field resolves problems of harmonicity of states and will not serve to multiply chaos in itself. Therefore in the continuum formula a control efficiency component must be present. Consequently, here we also have E = hν, that is, Planck’s constant as minimal standard action.
Thus we can project the standard position of the first point from the topology of the field (as a governing unit) into the most effective (qualitative) position in space–time (for the sake of harmonization) in the continuum — including G-mathematics.
It is clear that when we consider the appearance of the continuum, we are not speaking of a state of rest, but of motion and action. Hence we do not have a minimal real rest mass here, but only masses in motion, and at the boundary between field and continuum this motion is exclusively limited by c and the speed of EM-interactions.

5. Projection of 1gk.

If for massless particles 1gk means one-to-one reception, then the corresponding mass, as a manifestation and measure of the presence of a particle in the continuum structure — in the nested “matryoshka” of the domain secondary whole — is also a derivative of the difference between the standard control gk (from the side of the G-field — where there is full harmony and freedom in the complex) and the burden of self-governed processes in the -deviation of the local secondary whole, which together make up the domain fullness.
That is, an increase in mass indicates the degree of transition from the standard potential of the G-field to the kinetics of the secondary structure. Within it, there are its own nested structures in nested structures (project quotas from the most rigid allowed bounds at the bottom to freer ones; full H is realized only in the completeness of the corresponding domain level).

6. Measuring H.

Real calibration of H is possible also via the masses of elementary particles. We postulate that a certain series of elementary particles (EP) acquires mass by interacting within a domain whole at the lowest level of its secondary components, in correspondence with the H present there. At the lowest level of H, an EP attaches in the narrowest real H (in conditions of maximum rigidity of the state of that whole). These
are the primary building nodes (entry into the constructive of the whole). There are also nodes of purely energetic interactions, participation in entropic processes, with the understanding that such processes do not occur in the EP themselves. If an EP changes its state, it happens under the pressure of external forces.
We consider the density of the corresponding Ho–Lo structure of the second layer level responsible for the potential of a future EP: it gives the particle an appropriate integral potential and offers it through the EM-field. There is a local Lo_loc that receives this potential through the EM-field and thus materializes the EP. Thus an EP particle simultaneously resides in the EM-field and manifests kinetically in a corresponding region of the secondary whole, where the EP acquires the property of mass.
Knowing empirically the mass of such a particle and all the listed parameters — from the G-field and the EM parameters — we can determine local H as deviations from 1gk.
Ultimately, H as part of the project spectrum of the allowed is, in the continuum, the key not only to mass, but also to time, space and qualitative processes in our G-field theory.
The secondary field resembles the primary one in all respects, with the same operators and functions. It is only born not from the first-point Ho but from the Ho of the corresponding Ho–Lo structure of the G-field at its level. Hence a detached Ho (its chain) has its own H. In the G-field itself, topology is controlled by one-dimensionality and complete connectedness, whereas in the case of a secondary delegated field this onedimensional connection with the G-field is absent; therefore in the secondary field its own one-dimensionality is formed in a different H-interval — from which everything “on site” is taken.

7. Answers to some questions.

7.1. What is H-quantization in the G-model?
H-quantization does not refer to “quanta of energy by themselves” but to the fact that for each delegated whole (domain, secondary field) there exists a discrete set of allowed harmonic deviations from the project harmony of the G-field. That is, the scale is not continuous, with arbitrarily small perturbations, but consists of steps ΔHunit, 2ΔHunit, 3ΔHunit, . . .
Physically this means:
• each Ho_loc chain (local Ho-level in the structure of a secondary field) can be only in certain harmonic deviation modes (allowed H-intervals);
• the transition between these modes is precisely the “quantized” emission/absorption of energy;
• but energy does not “leave Ho_loc into emptiness”: it is always addressed to Lo_loc structures, which are receivers and carriers of kinetics in the continuum.
Thus, H-quantization is associated with the fact that Ho_loc can emit energy in quantized fashion, but more correctly: they change their H-state discretely, and this is reflected as quantized energy exchange in the Lo_loc continuum.

7.2. Spectrum of allowed states: what exactly is quantized?
In your picture the spectrum of allowed states is not an abstract “spectrum of the number” H but a spectrum of real states of Ho_loc layers, above which the following have already been built:
• continuum kinetics of EP-particles;
• local secondary fields;
• domain H-structures.
We can formulate it as follows:
• the Ho-level sets the potential: which deviations (H) are allowed in principle;
• Ho_loc chains specify this potential on site: for “this” domain, in “these” coordinates;
• Lo_loc gather it into a whole structure, which maintains a local one-dimensionality of time/quality in the domain and provides a stable configuration where EP-particles can “settle”, form bound states, possess mass, etc.

7.3. H-domains of delegated type and superdomains (3+1, 4+1, . . . ).
Here it is important to distinguish two levels.
Delegated H-domains are wholes in which:
• there is their own Ho_loc chain;
• their own Lo_loc structures;
• their own H-interval (allowed states of quality/deviation);
• their own local mass gap, local kinetics, local EP.
Superdomains 3+1, 4+1, . . . are dimensional continuum wholes: complete space–time “worlds” with a given structure of degrees of freedom; they are rigidly separated from each other in the continuum (3+1 does not topologically “mix” with 4+1); the connection between superdomains goes only through the G-field, and not through some “common continuum”.
Hence:
• H-domains of delegated type are “local rooms” inside a superdomain;
• a superdomain 3+1, 4+1, . . . is an entire “floor” of a building;
• H-quantization works inside each delegated domain;
• between superdomains other project constraints of the G-field operate.

7.4. Fluctuations between domains: what is allowed and what is not.
We postulate: fluctuations between domains (within given H-intervals) are in delegated, but not in superdomain-type wholes.
Physically, inside a single superdomain 3+1:
• delegated H-domains can fluctuate;
• can exchange energy via the EM-channel;
• can change their local H-states without leaving their allowed H-interval.
Between different superdomains (3+1 ↔ 4+1):
• there is no common continuum to easily transfer fluctuations;
• the connection is possible only through the G-field, and there quite different (stricter)
project conditions act.
So:
• H fluctuations between delegated domains are allowed,
• but they are always limited by:
– project H-intervals;
– local harmony requirements;
– quality factor Q (there cannot be infinitely “soft” oscillations).

7.5. How it all combines into one physical picture.
• The G-field:
– has a one-dimensional, fully connected topological Ho-line (in your sense of one-dimensionality);
– there is no H there — there is standard harmony and an invariant EK-level;
– the Ho-wave is a pure potential of an angular wave, still without frequency until a receiver appears.
• The secondary field:
– “copies” the operator structure of the G-field (the same types of operators, functionals, Lagrangians), but is born from the Ho–Lo structure of the G-field at its level, rather than from the primary Ho first-point;
– each detached Ho-chain receives its own H-interval — and this is already the beginning of a secondary domain.
• Ho_loc and Lo_loc:
– Ho_loc are local “oscillators of harmony” which can change their H-state discretely;
– Lo_loc are structural cells which gather Ho_loc into whole configurations,
maintain one-dimensionality and stability of the domain, and determine where in the continuum a given H-configuration may be realized.
• EM-field:
– the EM-field is not a domain and does not possess its own delegated fragment of the G-field;
– the EM-field is a channel through which a corresponding Ho-layer of the G-field in the given domain transfers energy into Lo_loc structures and secondary fields/EP of this domain, when Ho_loc–Lo_loc act as receivers at certain frequencies and in certain H-states.
• Mass gap and H-quantization:
– H-quantization at the domain level means: a minimal ΔHunit, a minimal excitation energy, a non-zero mass gap;
– the fact that ΔHunit and the Ho_loc/Lo_loc structure do not depend on the lattice spacing a makes the mass gap a physically stable phase parameter, rather than a numerical artefact of discretization.
In short:
• H-quantization is the discreteness of harmonic deviations, and not merely “quanta of energy into emptiness”;
• the spectrum of states is made of real states of Ho_loc-layers gathered by Lo_loc into domain wholes;
• delegated H-domains live in their H-intervals inside a single superdomain; superdomains 3+1, 4+1, . . . are more rigidly separated, only via the G-field;
• the EM-field and EP are already the kinetic “clothing” over this H-architecture.

7.6. Philosophical conclusion.
The monograph reveals the harmonic nature of creation, in which there is no external act of determination — creation is an internal property of the primary whole (the G-field).
This Annotation is a completed formula of the content core of the monograph. All subsequent sections are unfoldings of the above statements in geometric, physical, topological and simulation–algorithmic forms.

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Structure of the Pages of Part I of the Monograph

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0. General logic of Part I (conditional navigation)
Explanation. This document is the navigation map of Part I of the monograph: from the introductory ontological and axiomatic supports to the core of Step VI, its numerical modules, ΔH calibrations and applied blocks (gravity, elementary particles, appendices).
Numbering rule. The original position labels and document titles are preserved.
Page numbers are given conditionally (for later replacement by the actual ones in the assembled PDF).
A. Introduction, framework and basic documents of Part
I Pos. Document title Page

1 Structure of the Monograph (combined document) 2
2 Introductory Explanations to the Monograph — Theory of the Single Fundamental Harmonic Field “EK = const” 11
3 Dictionary-0: Minimal definitions and correspondences for the start of the monograph 22
4 Harmonic field constraints and the limiting speed of continuum interactions 26
5 Harmony — conceptual introduction 34
6 Introduction. Harmonic Field Theory. The Harmonic Structure of the Field as a Basis for the Reconciliation of Physical Theories 38
7 Introductory Clarifications on the G-Field 44
8 Why precisely the harmonic model of the existence of the fundamental field is chosen 51
9 Preamble to the blocks on ideal field mathematics and the G-field 55
10 Ideal field mathematics and G-mathematics: methodological core of the G-field theory 59
11 What is meant by “G-mathematics” 65
12 Axiomatic vision of the fundamental G-field 67
13 Chapter 1. Foundations of the active model of harmonic interaction 72
14 Appendix A. Formal analogies with the Maxwell and Schrцdinger equations 78
15 Appendix. Verification of the formula Eeff, Ho → 0 → (s) → CH 84
16 Chapter 2. Birth of Space and the Primary Angular Frequency of the Field 87
17 Chapter 3. Emergence and functioning of time(s) in the nonlinear continuum model 90
18 “Illumination” in the fundamental field 94
19 The Constant as a Tensor Operator of the Field 96
20 “Illumination” in the Fundamental Field. An Academic Clarification of the Term 100

Pos. Document title Page

21 Formal mathematical definition of “illumination” and of the channel of consciousness 102
22 Introductory appendix to Part V. Degrees of freedom, wholes and dimensionality in the G-field model 106
23 Clarification on the one-dimensional base o in block V 111
24 Part V-A. Philosophical basis of the active model of harmonic interaction 113
25 Part V-B. Mathematical basis of the active model of harmonic interaction 116
26 Part V-C. Applied methods and numerical approaches of the active harmony model 122
27 Part V-D. Pixel manifestation and resonance of the active harmonic system 128
28 Part V-E. Mathematical evolution of the harmonic invariant 132
29 Harmonic invariant base 136
30 “Signature of the Creator” and regions of chimerization 140
31 Questions and Resolutions. Self-Analysis Section 142
32 Types of wholes 145
33 Limits and mechanisms of harmonic completeness of the fundamental field (2) 149
34 Directedness of fundamental fields 151
35 Event horizon of the fundamental field 154
36 Complex harmonic constant CH 157
37 Axioms of the harmonious fundamental field (in the style of Yang–Mills and Hilbert–Einstein theories) 161
38 Definition of Zero and the initial interactions of the field before and at the moment of its appearance 166
39 Origin and Specificity of Lo (Basic consolidated version for further elaboration) 173
40 Bridge Microsection: From Primary Ontology to the Formal Core of Step VI 177
41 Origin and specificity of Lo 181
42 CH — Instant harmonicity and transfer of modes 185
43 Appendix. System connections and guidelines for Step VI 190
44 Appendix. Primary spectral norm Lo and projection of CH onto the modes of the field 194
45 CH, tensor operator and continuum Lo — mechanism of instantaneous coherence 198
46 Tabelle der Entsprechungen zwischen ontologischen Begriffen und mathematischen Symbolen 202
47 Appendix. Variational functional of Zero S0 207
48 System of Pre-Nuclear Sections and Their Relation to Step VI 210

B. Step VI — harmonic action functional and equations

Pos. Document title Page

49 Step VI — Academic version of the harmonic action functional 214
50 Analytical appendix. Critical questions of the foundations of field interaction 219

Pos. Document title Page

51 Comparative analytical document for Step VI 228
52 After the axioms. Appendix P_So. Variational functional of the zero state S0 and tuning–topological form Γ233
53 Towards numerical realization. Working program “Step VI —Harmonic action functional” 239
54 Formation of the harmonic Lagrangian and of the full field equation 244
55 Expanded equations and simulator of the harmonic field 249
56 Full Euler–Lagrange equations and simulator 257
57 Completion of the cycle of constructing the Lagrangian of harmonic action and the full Euler–Lagrange equations 265
58 Full equation for the wave field Ψ — general (non-simplified) case 270
59 EK_Harmonic_Simulator_New. One-dimensional simulator of the harmonic field (model CH–Ho– ˆ S) 275
60 Step VI. Full system of Euler–Lagrange equations for the harmonic field 285
61 Numerical models of the harmonic field 291
62 Harmonic_Simulator_Appendix_Recode.py 297
63 Dynamical–numerical module of the harmonic field theory 301
64 User Manual for the Harmonic Field Simulator 308
65 Appendix. Basic example of the variational principle for a One-Dimensional Field u(x, t) 314
66 Primary G-field, continuum and secondary wholes 317
67 Difference equations Σ, K, E — version with normalization and local content of K 320
68 Appendix. Convergence tests and implicit solver 325
69 Appendix. Implicit simulator of the one-dimensional harmonic module Σ, K, E 330
70 Implicit_Harmonic_Solver_1D.py 335
71 User guide to the implicit simulator 341
72 Updated axioms of the harmonic fundamental field. Section I 346
73 Complete system of harmonic field axioms (-model) 350
74 Explanatory bridge between the axioms of the field and the full Σ-model of the harmonic fundamental field 355
75 Mathematical deepening of the E-model of the harmonic G-field 361
76 Full completion of the axioms block (G-field) 366
77 Harmonic structure of the field as the basis for reconciling physical theories 370
78 Step VI. Harmonic action functional and Euler–Lagrange equations 377
79 Step VI. Full tensor formalization of the harmonic action functional 386
80 Step VI. Work-status table — updated version 392
81 Step VI. Dimensional analysis and dimensionless form 398
82 Step VI: Complete Table of Dimensions 403
83 Document No. 1r — Working clarification of the physical meaning of the multiplier 409
84 Document No. 2 — Axiomatic block of the EK multiplier, the invariant Σ0, time and “anti-temporality” 418
85 Document No. 3 — EK–Σ–K–e. Harmonized Core Fragment 422

Pos. Document title Page

86 “Step VI”: Dimensional Quick Reference 427

C. Step VIb — numerical realization and algorithmic
Modules
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87 VIb-A Theoretical foundations of numerical realization 431
88 VIb-C. Coordinated styled document of numerical realization (Step VI) 437
89 VIb-B. One-dimensional model HarmonicField1D (one practical case) 442
90 VIb-D. Final module HarmonicField1D 448
91 Bridge between “Step VI — Dimensional analysis and dimensionless form” and block VIb-A_B_C_D 453
92 Step VIb. Numerical realization — theoretical foundations and algorithms 457
93 Step VIb. Numerical realization of the one-dimensional harmonic field (dimensionless form) 462
94 Step VI. Dimensional analysis and dimensionless form 468
95 Meta-level comment to the axioms block of the G-field, taking into account the pair of tensors over Lo 474
96 How the self-illumination tensor Aμν “sees” and how the operator tensor Hμν “acts” 478
97 Appendix. Lagrangian of the G-field and the dependence L(ln(EK)) 480

D. Analytical blocks, Lo/Ho unfolding and preparation
of bridges
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98 Analysis of the Lyashkevych formula (EK––K–e, Step VI, node CH–S^–Ho–G–o) 485
99 Proof of the Lyashkevych formula (2) 494
100 Self-check of the model of the Lyashkevych formula. 500
101 One-dimensional mini-model of the expansion of the harmonic field from a single Lo (EK–Σ–K–e) 507
102 Complete Closure of the Axiom Block (G-field) 515
103 Chapter X. Σ0 as a fractal invariant shell of local and cluster invariants Σloc 521
104 Internal levels and modes of the harmonic field 528
105 Tensor of Illumination / Self-awareness of the Field 533
106 “One-dimensional harmonic model Lo” in the structure of the monograph 538
107 Table of physical dimensions of motion along the degrees of freedom 542

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108 Transition from the one-dimensional core Lo to multidimensionality 545
109 Refined symbols of the 1D-Lo block and the Harmonic-Field1DWithLo module 550
110 One-dimensional simulator 1D-Lo with measurement of the invariant E ・ K = Σ0 555
111 Paths to Representing the Multidimensionality of the Model, 2D code 563
112 The Triad “field – continuum – energy” 569
113 Real minimal working Python-file for the 1D-Lo model 572
114 Pseudocode and minimal Python-module 1D-Lo 581
115 Sketch of the 2D-Lo model: state, invariants and local rules 589
116 Attachment System of the Lo Layer to the Existing Harmonic-Field1D 595
117 One-dimensional model of harmonic unfolding of the field from a single Lo 599
118 Appendix. Explanation of the 1D-Lo demo code 608
119 Space, time and zero in the harmonic model of the fundamental field 613
120 Space, time, triad “field–continuum–energy” and Σ0 as fractal shell 618
121 Step VI. Block “ΔH → Action → Equation” 620
122 Schematic ontology — picture of the initial states of the G-field 626
123 Lo level — field or continuum, Ho levels of tensors 639
124 Physical interpretation of the main states of the G-field and its continuum 644
125 Critical questions to the physical interpretation of the ontological picture of the appearance of the G-field 657
126 Refinement of the physical processes in the foundation and unfolding of the G-field 662
127 Analysis of clarifications to the founding and unfolding of the G-field 668
128 Appendix to the axioms block (G-field). Clarification of the Lo level, time and fractal Σ0 673
129 Fractal faces of the project orbits of the G-field 678
130 Lo as an energetic factor in the G-field model 685
131 Time and “anti-temporality” in the G-field model 690
132 Tensor channels of connection between Hμν, Aμν and B(i) 696
133 Local conservation laws and topological invariants in the harmonic field 699
134 Chapter VI-A. Space, time and continuum in the block “ΔH →Action → Equation” 706
135 Bridge from Step VI to VIb: what is transferred into Harmonic-Field1D 712
136 The first three levels of unfolding: Ho0 → Lo1 (1D) → Ho-plane + Lo-lattice (2D germ of the continuum) 717
137 Reference note — G-field, wave frequency, speed of light and the role of “zero” 720
138 Use of the meanings of the notions “entropy” — “minus-entropy”, “plus-entropy” 726

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139 What the fundamental electromagnetic field is that combines K (control) and E, the active energy 730
140 Lagrangian Fragment of the G-Field with an EK-Term 736
141 Step VI — VI-B.2. Lagrangian fragment of the G-field with the EK term 740
142 EK-Lagrangian, pair of tensors (Hμν, Aμν) and Harmonic-Field1D1 744
143 Applied quality of the fundamental electromagnetic field in the G-field model 749
144 Formal introduction of the projection Π and functionals of “plusentropy” and “minus-entropy” 753
145 Lo positioning density, the continuum and G-integralities 758
146 Role of the tensor pair in the Lagrangian and the law EK = const (Step VI) 767

E. Formula bridges and VI-C (symmetries, currents, invariants)
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147 Formula bridges of G-mathematics with standard field theories and theories of elementary particles 772
148 Appendix 1. Mini-table “formula ↔ G-reading” and numerical example 781
149 Lagrangian fragment of the G-field with EK term 783
150 Appendix 2. Clarifications from the harmonization model as a basis for standard theories 787
151 Appendix 3. Small parameters and corrections in ε and ΔH 791
152 Appendix 4. Dictionary of correspondences of the G-model with QFT_GR 794
153 Appendix 5a. Mini computational examples for Appendix 5 803
154 Appendix 6. Methodological algorithm of transition from the standard Lagrangian to the G-description 807
155 Appendix 7. Test forms for EM waves in medium and gravitational effects 813
156 Appendix 8. Internal theorems of the G-model about the EK invariant and K(u) 819
157 Appendix 9. Generalized Noether block for the harmonic Lagrangian 824
158 Appendix 10. Compact map of predictions and zones where the G-model gives fundamentally new information 830
159 Triad “field–continuum–energy” in the Lagrangian scheme (Step VI) 835
160 Section VI-B. ΔH → Action → Equation. Harmonic action functional and Euler–Lagrange equations of the core of the model 840
161 Section VI-C — Detailing Item 1. Symmetries of the harmonic Lagrangian 846
162 Section VI-C — Detailing Item 2. Noether’s theorem and harmonized currents 850

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163 Section VI-C — Detailing Item 3. Local conservation laws of harmonic dynamics 855
164 Section VI-C — Detailing Item 4. Global evolution invariants and the connection with EK = const and Σ0, Lon 859
165 Preamble to Chapter VI-C. Symmetries, currents and invariants of harmonic dynamics 867
166 Meta-summary of Step VI. Generalization of the variational–Lagrangian structure of the model 870

F. Delegation, ΔH calibration and elementary particles
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167 Delegation of a Fraction of the G-Field into a Secondary Wholeness 875
168 Informational appendix. ΔH, frequency of addressing the G-field and domain integrities 881
169 Delegation and ΔH. Control and energy flows of the secondary whole (MATHPSI_I) 892
170 ΔH as a spectrum of eigenfrequencies of the domain 895
171 Elementary particles in the whole “G-field – continuum” 902
172 Informational appendix to joint calibration of the G-field and the continuum 911
173 Quantum of ΔH calibration via photon, ℏ and 1 gk 917
174 Formal steps in ΔH, Nn and verification of masses and frequencies ωn 923
175 Different degree of guiding participation of the G-field in processes in the continuum 928
176 State function of a building particle as carrier of λk and participation in structures 933
177 Criteria of confirmation of the G-model in the lepton sector 937
178 Building elementary particles and levels of delegation (MATHPSI_I) where “plus-” and “minus-entropy” act 940
179 Elementary particles in the whole “G-field – continuum” 944
180 Neutrino sector as an extreme ΔH test 957
181 Ho-linearization around ΔH = 0 (Step VI) 962
G. G-gravity, domain ΔH configurations and phenomenology

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182 Gravity. ΔH configuration of a single massive whole and domain curvature 969
183 Weak-field Newtonian limit of G-gravity (calibration of GN via H) 974
184 Connection of domain curvature Reff[ΔH] with the metric description of GR 982

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185 Domain-variational fixing of coefficients aD, bD, cD 990
186 PPN calibration of G-gravity. Parameters β, γ via aD, bD 995
187 ΔH analogue of a spherically symmetric “Schwarzschild” configuration 1001
188 Cosmological domain ΔHhom(t) and Friedmann-type equations 1005
189 Calibration of ΔHunit at the level of the photon and electron 1010
190 Neutrino sector as a “ΔH laboratory” for gravity 1015
191 Numerical experiments ΔH + Reff in a simple 1D/3D model 1020
192 Physical calibration of the 1D ΔH-gravity simulator 1025
193 Minimal 1D ΔH-gravity simulator 1033
194 Radial 1D ΔH-gravity simulator (spherical symmetry) 1039
195 Generalized Lagrangian of the gravitational sector of the G-model 1047
196 Part I. Meta-summary “G-gravity” 1052
197 Real calibration of ΔH in the context of masses of elementary particles 1057
198 Formal class of functions ΔH0(r) and f(Σ) capable of producing multiplicities Ni for leptons e, μ, τ1062
199 Practical way of calibrating ΔH via magnetic fields of the continuum 1068
200 Calibration of ΔH in continuum units via EM and magnetic fields 1073
201 Numerical estimates of ΔH for realistic magnetic fields 1078
202 Reminder note for the G-gravity block and the elementaryparticle block 1083
203 Extended numerical analysis of ΔH-gravity in 1D/3D models 1090
204 Scanning of parameters of G-gravity in PPN and cosmological regimes 1095
205 Construction of realistic ΔH profiles for astrophysical objects 1099
206 Predictive differences of G-gravity from GR in weak and strong fields 1104
207 Completion of the PPN block as a separate “sub-theory” 1110
208 Radial (Schwarzschild) ΔH block of a semi-analytical model 1115
209 Effective contribution of elementary particles to the cosmological ΔH domain 1121
210 Construction of classes of ΔH functions capable of qualitatively reproducing the lepton mass row 1126
211 Numerical experiments to test how Nn, λk and ΔHunit are related to realistic spectra 1132
212 Clarification of classes of ΔH functions and coordination with the neutrino sector 1138
213 Integration of the lepton ΔH block with the gravitational domain 1143
214 Fixing working ΔH profiles for lepton and domain scenarios 1148
215 Numerical experiments on the realization of ΔH and lepton profiles in 1D and radial simulators 1152
216 Phenomenological estimates of the lepton ΔH contribution to domain gravity 1157
217 Numerical tasks of the lepton ΔH contribution in gravitational domains 1164
218 Comparison of G-gravity with astronomical and cosmological data 1168
219 Relation of the ΔH block with the energy–momentum tensor Tμν and the standard QFT Lagrangian 1173

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220 Hard numerical calibration of the parameters ΔHunit, Lcorr, aD, bD, cD 1179
221 Full PPN analysis of the ΔH metric in G-gravity 1184
222 Radially symmetric ΔH block and TOV-like models with ΔH contribution 1189
223 Cosmological ΔH domain + neutrino block — simple FRW-like code 1194
224 Internal stability of the ΔH equations. Absence of bad dynamics 1198
225 Leptonic ΔH block. Explicit functional classes for Nn, Nν, k, λk 1202
226 Compact catalogue of “signals” where the G-model differs fundamentally from GR + SM 1208
227 Minimal “public” package of the G-model. Code and demonstration tasks (2) 1214
228 Benchmark Problem 1. “ΔH lump → gravitational well” for the 1D H-gravity simulator 1218
229 getting_started.md. Package of G-gravity and ΔH simulators 1224

H. Summary, examples and appendices
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230 Summary of Step VI for the reader 1227
231 On the Poincarй–Perelman theorem in the author’s harmonic Gfield model 1232
232 Riemann surfaces and harmonic wholeness: transformations without holes 1238

APPENDICES to the monograph

233 Extended proof of the Lyashkevych formula in the context of the G-model 1243
234 Axiomatic block EM1–EM3 for the EM channel in the G-field theory.tex 1250
235 G-field, EM mediator and the mass gap for elementary particles 1253
236 ΔH as the key to mass, time and space in the G-field theory.tex 1260
237 Appendix. New and clarified predictions of the G-model of the first part of the study 1264
238 PS. Polyvariance and parallel realities: detailed refutation in the G-model 1275
239 How the G-model theory can improve the work of Artificial Intelligence 1281

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- Download Part II of the monograph in English in PDF format

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Part II of the Monograph
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A Brief Annotation of G-Field Theory for the Block of Great Theorems

0. Purpose of the Annotation

Purpose. To briefly outline the ontological and mathematical foundations of G-field theory necessary for understanding the block of Great Theorems (the inner G-version and the outer Yang–Mills-formulation of the mass gap). This brief annotation is the introductory part of this block and is used as a brief scientific announcement.
1. The initial idea: the harmonic fundamental field The theory of the G-field (harmonic global field) is based on the assumption that in general, behind our fundamental reality – and all around it – there is a harmonic self-directed field, and not a bare continuum in empty space in the middle of nothing. And our continuum (space-time-quality) appears as a secondary tool for energy dissipation and the possibility of realizing its local, design degrees of freedom, but is not the primary basis of these processes.
The basic object of the G-field is the Ho-state-potential as the basis of the minimal potential wave of the sweep in its reference topology of this particular harmony, with minimally sufficient energy parameters Emin and control Kmin, which satisfy the invariant formula (Lyashkevych formula):
Emin · Kmin = const ≈ 1 gk,
This sets 1gk as the basic harmonic unit of the action potential. Further, above this initial point Ho (the reference energy and topological basis, including all subsequent calibration), the fractal sweep of the Ho layers is expressed. A similar fractal scaling is formed in the structural cells Lo, which are “pixels for the continuum” from the side of the harmonic field. The set of Ho–Lo structures is given by the initial design of the G-field as a complex harmonic spectrum, which by its expression solves the issue of the absence of this particular harmony. This is done through the multilayer Ho–Lo orbital framework, to which both the EM electromagnetic field and the continuum domains of secondary integers are bound.

2. ΔH-quantization and domain integrity

The key characteristic of purely secondary structures is the ΔH-quantization of states corresponding to the orbital framework of the G-field. The value ΔH is introduced as the positioning and measure of the obtained, design-defined, harmonic deviations not for the fundamental G-field itself, but for the states:
• secondary delegated fields,
• domain continuum integrity,
where, the fewer degrees of freedom, the narrower their ΔH.

Each secondary domain-type integrity has an admissible ΔH-interval of states, given by the design of the G-field itself. Within this interval, their local oscillations, evolution, and interactions are possible; outside it, the G-field does not support their energetics.
Important principle:
ΔH-intervals are specified at the level of the G-field project as the spectrum of the harmonic topological functional and determine which deviations of secondary structures are "allowed" in this model.
The concept of ΔH-quantization is based on this: there is a minimum step of the harmonic deviation ΔHunit, which fixes the "graininess" of the permissible states of the domains.

3. Secondary fields, Ho_loc–Lo_loc chains and EM channel

A secondary field in G-field theory, according to its level of existence, repeats the operator and functional structure of the primary G-field, but is born not from the reference initial point Ho, but from the corresponding Ho–Lo structure of the G-field of its level. Each Ho-chain torn off from that level upon the initiation of a secondary integrity of the delegated type (i.e., capable of secondary self-control) receives its own ΔH with respect to the reference one-dimensionality of the G-field; it receives on the basis of its basic topology, which, when breaking with the one-dimensionality of the G-field itself, forms this or that state ΔH. On this basis, the already characteristic secondary one-dimensionality of Ho_loc and Lo_loc in the corresponding domain is formed.
In these secondary structures:
• Ho_loc act as local energy receivers from the side of the G-field proposal and generators of harmonic energies;
• Lo_loc form structural cells of the continuum domain;
• continuum kinetics and potentials of the secondary field are realized through the Ho_loc layers and structural Lo_loc within their ΔH-intervals.
The EM-field in this picture is not a separate domain with its own, initially delegated fragment of the G-field. It is treated as a channel of interaction, transmission of a controlled potential, starting from the second Ho-layer through Lo – to the corresponding Lo_loc levels of secondary wholes.
The EM field "works" only where tuned receivers Ho_loc–Lo_loc exist in design-allowed ΔH states. Thus, the manifestation of elementary particles (EPs) in the continuum is interpreted as a consequence of the fact that there are corresponding receivers in a given secondary integrity, and the G field design-supports the channel of such interaction according to broader rules of interaction. That is, the G field is exceptional in terms of its unique harmony and topology of structures, but standard within the framework of more general (non-hierarchical) rules and principle schemes.

4. Elementary particles and mass calibration

In the G-field model, elementary particles do not have their own ΔH-processes, separate from the secondary and domain integrals to which they are attached, they do not have their own delegated fragment of the field φ or Σi. I are considered as lower ones, manifested at the places of interaction of the integrals, which:
• are determined by the G-field through the EM-channel in the composition of the corresponding secondary integral;
• participate in entropic ("plus-" and "minus-entropic") processes only as components of the domain type of integrals.
The mass of the EC in this concept depends on the ΔH-state of the level of the domain in which the particle is realized. The ΔH intervals and the Q-factor of the domain oscillations fix the minimum energy scales of the excitations. This provides a natural mechanism for mass calibration within the G-model:
• known experimental EC masses are used to calibrate the corresponding ΔH intervals and domain contour parameters;
• on the other hand, the ΔH architecture and Q-constraints imply a non-zero lower bound for the mass gap m0 > 0 in certain channels.
Thus, the mass scales are not arbitrary parameters, but are derived from the ΔH structure of the G-field and domain kinetics. This is an important bridge to the Great Mass Gap Theorems.

5. Ideal Field Mathematics and G-Mathematics

The theory introduces a meaningful distinction between:
• ideal field mathematics — a description of harmonic-reference states without Δ-deviations, which reflects the basic harmony of the G-field and its structure;
• G-mathematics — an extended formalism that describes not only the fundamental G-field, but also all admissible continuum processes, ΔH-deviations of secondary integrals, their entropic dynamics and critical states;
• abstract mathematics, which may not correlate with the environment of a specific G-field.
In this framework, the fundamental invariant of the type
E · K = Σ0 ≈ 1 gk
becomes the core of the mathematicians represented in the integrity of the G-field: through it, both the harmonic states and the allowed ΔH-deviations are calibrated. ΔH-intervals, domain structures, mass scales, and time scales of harmonization are considered as a single coherent mathematical object.

6. Connection with the Great Theorems (mass gap)

Within the G-model, an internal Great Theorem is formulated:
by axioms A1–A9 (existence of ΔH-quantization, domain integrals, 1gk-measure, etc.) and additional physical principles of mass gap stability (Volume II), in the corresponding domain channel there exists a local mass gap m0 > 0, which is RG-stable and does not vanish in the continuum limit.
A rigorous bridge to the Yang–Mills formulation is then built.

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Structure of Part II of the Monograph.

Refinement of the G-field and Proof of the Great Yang–Mills Theorem + mass gap (ΔH + 1gk + Yang–Mills + mass gap)
_____________________________________

0. General logic of the proof
The Great Theorem has two interrelated forms:
• Internal version in the G-model: existence and stability of a local mass gap m0 > 0
for H-quantization in the ΔH + 1gk model.
• External (Yang–Mills) version: existence of a mass gap in the standard 4D Yang–Mills model (over R4), obtained via a pushforward mapping from the G-model together with OS/Gibbs-type conditions.
The proof is arranged into major blocks (A–G), followed by a schematic chain (I).
A. Local polygonal mass-gap block (ΔH + U(1)/SU(N))
A.1. Minimal ΔH + U(1) model and a local mini-theorem of the mass gap №1 Short
Abstract to Parts II and III of the Monograph) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
№1_1 Yang–Mills, mass gap and the G-field — reference note) . . . . . . . . . . . . . . . . . . . . . . . . . 3
№1_2 Minimal model ΔH + U(1) + building elementary particle (mass-gap polygon) . . . 10
№2 Local mini-theorem of the mass gap for the ΔH + U(1) polygon + building elementary particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
№3 Lemma 1. Local mass gap for a scalar field with a positive lower bound of the mass potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Block function. Define an elementary polygonal ΔH-domain, a building elementary particle, and the minimal form of a mass gap in the simplest U(1) setting.
A.2. Transition to the SU(N) polygon and gauge-invariant operators
№4 Polygon SU(N) + ΔH-domains (preparation for the Yang–Mills mass gap) . . . . . . . . . 21
№5 Lemma 2. Gauge-invariant operator and the mass gap in a ΔH-domain (effective prototype for Yang–Mills) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Block function. Transfer the local U(1) picture to the SU(N) context and formulate the mass gap via gauge-invariant operators.
A.3. Proof structure for Yang–Mills + mass gap in the polygonal G-construction
№6 Yang–Mills, mass gap, and the G-field — proof structure . . . . . . . . . . . . . . . . . . . . . . . . . . 31
№7 Standard Yang–Mills theory, mass gap, and the ΔH-structure of the G-field (proof structure) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Block function. Assemble polygonal results into a framework for the transition to the standard Yang–Mills language.
A.4. Functional dependencies ΔH(F) and the mass operator
№8 ΔH(F) and M2(ΔH) in standard Yang–Mills theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
№9 Lemmas on ΔH(F), M2(ΔH), and the mass gap in Yang–Mills theory. . . . . . . . . . . . .50
№10 Lemmas on ΔH(F), M2(ΔH), and the mass gap in Yang–Mills theory (sharpened
functional-analytic version) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Block function. Establish strict relations between ΔH(F; x), the mass operator, and a local mass gap within the Yang–Mills framework.

A.5. Supplements and symbol glossary

№11 Ho–Lo and the mass gap — supplement to the ΔH and Yang–Mills block . . . . . . . . . 62
№12 Symbols: ΔH, Yang–Mills, and mass gap — reference list . . . . . . . . . . . . . . . . . . . . . . . 63
Block function. Fix the Ho–Lo picture and unify the notation used throughout the proof.
B. Beyond the polygon: 4D ΔH-field, Ho–Lo configurations, and the 1gk measure
B.1. 1gk as a measure on the Ho–Lo configuration space
№13 1gk as a measure on the Ho–Lo configuration space (polygonal version) . . . . . . . . . . . 70
№15 1gk as a functional measure of the ΔH field in 4D (beyond the polygon) . . . . . . . . . . . 76

Block function. Move from the polygonal scheme to a 4D functional measure μ1gk on the space of ΔH-configurations.
B.2. Mapping Ho–Lo → Yang–Mills field

№14 Ho–Lo → Yang–Mills field Aμ(x): mapping of configurations . . . . . . . . . . . . . . . . . . . . 82
№20 Step 2. Locality and gauge covariance Π(Ho–Lo,ΔH) → Aμ . . . . . . . . . . . . . . . . . . . . 89
Block function. Define the pushforward mapping Π from Ho–Lo/ΔHconfigurations to the
Yang–Mills field Aμ(x), with locality and gauge covariance.
B.3. 4D axiomatics of the ΔH-field and the action SG
№19 4D axiomatics of the ΔH field and the action SG with finiteness of 1gk and tightness of measures μ1gk(ΔH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Block function. Formulate 4D axioms for the ΔH-field and the action SG, ensuring finiteness and tightness properties needed for the Gibbs/Euclidean measure.

B.4. “Pure” Yang–Mills case and the first integration over ΔH, Ho, Lo
№17 OS conditions, correlation functions, and mass gap for gauge-invariant
operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
№18 “Pure” Yang–Mills case: integration over ΔH and Ho–Lo. . . . . . . . . . . . . . . . . . . . . . . 108

Block function. Show how OS conditions and correlation functions arise in the “pure” Yang–Mills regime from the ΔHand Ho–Lo constructions.

B.5. Pushforward measure μA and the effective action
№16 Pushforward measure Πμ and its relation to the YM action . . . . . . . . . . . . . . . . . . . . . . 113
№21 Pushforward measure μA = Π∗μ1gk and the effective action SYM + Scorr . . . . . . . . . 117
№22 OS conditions, correlation functions, and the mass gap in the ΔH +
Yang–Mills model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
№23 Pure Yang–Mills: integration over ΔH, Ho, Lo and stability of the mass gap . . . . . . 126
№39 Pushforward measure μA = Π∗μ1gk and the OS block (sharpened version) . . . . . . . . 131
№43 Pushforward measure μA = Π∗μ1gk and the OS mass-gap block (lemma level). . . . .136
№50 Pushforward measure μA = Π∗μ1gk and the OS block (strict version) . . . . . . . . . . . . . 142

Block function. Prove that the pushforward measure μA satisfies OS properties and yields an effective action SYM + Scorr with a stable mass-gap channel.
C. Axiomatics ΔH + 1gk + Π + Yang–Mills (A1–A12) and lemma blocks
C.1. Core axiomatics A1–A10 and full A1–A12

№27 Axiomatics ΔH + 1gk + Π + Yang–Mills (4D version). . . . . . . . . . . . . . . . . . . . . . . . . .148
№31 Axiomatics ΔH + 1gk + Π + Yang–Mills (4D version, A1–A12). . . . . . . . . . . . . . . . .153
№55 Axioms A10–A12: continuum limit, mass-gap stability, and correspondence to standard Yang–Mills theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
№63 Program for removing axioms A10–A12 within the G-model (technical Volumes I–III) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

Block function. Fix the complete set of axioms A1–A12, including the continuum limit and stability of the mass gap, and define the roadmap toward strict analytic volumes.
C.2. Lemma structure and lemma blocks for A1–A12

№30 Lemma structure for axioms A1–A10 (ΔH, 1gk, Π, Yang–Mills) . . . . . . . . . . . . . . . . 170
№25 Appendix. Clarifications to A3–A4 (Π and ΔH(F; x)) . . . . . . . . . . . . . . . . . . . . . . . . . . 176
№33 Lemma block for axioms A1–A3 (deepened version) . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
№34 Lemma block A4–A6: ΔH(F; x), quantization ΔHD, and the mass operator Hϕ . . 187
№35 Lemma block for axioms A7–A9 (OS properties, pushforward, and mass gap) . . . . . 192
№36 Lemma block for axioms A10–A12 (continuum, non-intersection, and ensemble
reality) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
№40 Technical lemma blocks for axioms A1–A12 (working outline) . . . . . . . . . . . . . . . . . . 200
№41 Lemma block A1–A2: ΔH-field, action SG, and measure μ1gk
(sharpened version) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
№42 Lemma block A3–A4: mapping Π and ΔH(F; x) (sharpened version) . . . . . . . . . . . . 209
№48 Lemma block A1–A2: ΔH-field, action SG, and measure μ1gk (basic version) . . . . . 214
№49 Lemma block A3–A4: mapping Π and the function ΔH(F; x) . . . . . . . . . . . . . . . . . . . 219
№51 Lemma block A5–A6: ΔH-quantization, operator H, and local mass gap . . . . . . . . . 224
№52 OS properties of μA, correlation functions, and reconstruction bH
(block A7–A9) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

Block function. Provide a step-by-step lemma-level proof spine for each group of axioms A1–A12.
C.3. Tables of dimensions and symbols

№29 Large tables of dimensions and symbols (ΔH, Yang–Mills, G-field) . . . . . . . . . . . . . .234
№84 Full tables of dimensions and symbols (ΔH, Yang–Mills, G-field) . . . . . . . . . . . . . . . 242

Block function. Unify dimensions and notation across the entire axiomatics and lemma system.
C.4. Physical and technical links inside the axiomatics

№38 Mass gap as a consequence of ΔH-quantization (a separate item of
the Great Theorem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
№44 Relation of the pushforward-measure block μA + OS reconstruction to axioms A7–A9 and the Great Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
№45 Physical interpretation of the mass gap in standard Yang–Mills frameworks . . . . . . . 259
№54 Spectral representation SMG(t, x) and the role of minimal ΔHD . . . . . . . . . . . . . . . . . 265

Block function. Connect the formal blocks (quantization, OS reconstruction, pushforward) with the physical interpretation (spectrum, operators, mass-gap channel).
D. Internal Great Theorem of the G-model (H-quantization and local mass gap)
D.1. ΔH-quantization and the secondary field

№71_1 ExplanationoftheessenceofΔH − quantizationintheG − model . . . . . . . . . . . . 270
№85 ΔH-quantization and the secondary field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .275
№90 ΔH as a key to mass, time, and space in the G-field theory . . . . . . . . . . . . . . . . . . . 279

Block function. Describe ΔH-quantization as the mechanism of the emergence of mass, time, and space in the G-model, and its link to the secondary field.
D.2. Entropic trajectory of the domain and EM/UM levels

№57 ΔH, time, and the entropic trajectory of the domain: from perturbation to the EM level and a coherent gk-field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
№58 EM level, UM level, and YM vacuum — consistent definitions . . . . . . . . . . . . . . . . 288
№59 Polygonal modelsΔH(t, x), S+(t), S−(t) in 1D/2D,EMandUMlevels, and the transition
to the 4D limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
№60 Life cycle of a ΔH-domain and life cycle of a Yang–Mills configuration: formal correspondence (updated version with EM/UM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

Block function. Formulate the dynamic picture of a ΔH-domain and its formal correspondence to Yang–Mills configurations, including EM/UM levels.
D.3. Internal Great Theorem of H-quantization

№87_1 Internal Great Theorem of the G-model: H-quantization and
local mass gap m0 > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

Block function. State and prove the internal Great Theorem in terms of H-quantization and a local mass gap m0 > 0.
D.4. EM channel as a global instrument of the G-field

№89 Axiomatic block EM1–EM3 for the EM channel in the G-field theory . . . . . . . . . 307
№88 Extended proof of the Lyashkevych formula in the context of the G-model . . . . 310

Block function. Fix the EM-channel axiomatics and embed the Lyashkevych formula into the global ΔH+1gk picture.
E. External Great Theorem: Yang–Mills + mass gap via the G-model
E.1. Main formulations of the Great Theorem

№26 Great Theorem ΔH + 1gk + Yang–Mills + mass gap . . . . . . . . . . . . . . . . . . . . . . . . . 317
№28 Great Theorem ΔH + Yang–Mills + mass gap (version A1–A10) . . . . . . . . . . . . . . 321
№32 Great Theorem on the YM measure with a mass gap based on axiomatics A1–A12
(refined version) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
№37 Great Theorem: Yang–Mills + mass gap in the G-model ΔH, 1gk, and Π . . . . . 331
№56 Theorem (Yang–Mills + mass gap via the G-model): statement
and proof structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
№61 Theorem (Yang–Mills + mass gap via the G-model). Direct proof via A1–A12 . 340
№62 Great Theorem ΔH + 1gk + Yang–Mills + mass gap (conditional
version under A1–A12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
№87_7 External Great Theorem (YM formulation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

Block function. Collect the external Great Theorem variants and their equivalence under the axiom system A1–A12.
E.2. Confinement, Wilson loops, glueball spectrum, RG calibration

№46 Confinement, Wilson loops, and glueballs in the ΔH + Yang–Mills picture . . . . 352
№53 Gauge-invariant operators of the mass-gap channel: Wilson loops and
trace invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
№721 Note : ΔH − domains, confinement, andtherelationbetweenσG and m0 . . . . . 361
№731 Note : σG and Σ0 in the G-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

Block function. Detail the physical consequences of the mass-gap channel: confinement,gauge-invariant observables, spectral behavior, and scaling.
F. Tom I–II–IIβ–III: strict analytic framework (μ1gk, Scorr, RN, bridge G ↔ YM)
F.1. Tom I: construction of μ1gk and the continuum limit

№64 Quality of G-contours (Q) and stability of the Gibbs measure μ1gk . . . . . . . . . . . . 372
№65 Inductive contours Lo–Loloc and reflection positivity: induction kernel K
and OS form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
№66 Tom I. Construction of μ1gk, induction geometry of the G-field, and the continuum limit (A10) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
№87_5 Construction of the measure μ1gk for a given μYM . . . . . . . . . . . . . . . . . . . . . . . . . 387

Block function. Construct a stable Gibbs measure μ1gk with OS properties and formalize the induction geometry and the continuum limit.
F.2. Tome II and Tome IIβ: Scorr, RN derivatives, and mass-gap stability

№67 Physical principles of mass-gap stability in the G-model . . . . . . . . . . . . . . . . . . . . . . 391
№69 Tome II. Lemma block L1–L3, C1–C2, S1 and the mass-gap channel . . . . . . . . . 396
№70 Conditions on Scorr and the Ho–Lo structure (Volume II, axiom block S1). . . . .403
№722 Localactions, kernels, andstrictestimatesforScorr in Volume II . . . . . . . . . . . . . 409
№732 StrictestimatesofRadon−−NikodymderivativesandtheinfluenceofScorr on the
mass gap (Tom II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .415
№74 Tome II. Mathematically Rigorous block of mass-gap stability
(alternative version) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
№75 Tome II. Refined version: L1–L3, C2, S1 as lemmas and theorems from the RN Scorr construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
№75_1 Refined Volume II — mass-gap stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
№82 Volume IIβ. RN–Scorr block as a strict analytic instrument . . . . . . . . . . . . . . . . . . . 445
№87_3 Volume IIβ. A rigorous functional-analytic block for RN Scorr . . . . . . . . . . . . . .448

Block function. Provide strict functional-analytic control of Scorr and RN derivatives and prove persistence of the mass gap under corrective terms.
F.3. Volume III: bridge between the standard Yang–Mills model and the G-model

№76 Volume III. What we do next . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452
№77 Tom III— construction framework and main objectives . . . . . . . . . . . . . . . . . . . . . 455
№78 Tom III.1. From the G-model to the standard Yang–Mills language . . . . . . . . . . . 460
№80 Tom III.1. Physico-mathematical refinement of the bridge: G-model — standard
YM language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
№81 Volume III. Integrated version (III.1–III.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
№79 Volume III.2–III.4. Confinement, glueball spectrum, and RG calibration
of the mass gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
№71_2 Volume III.2. Confinement andWilson loops in
the G-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483
№71_3 A concrete class of Ho–Lo configurations for constructing the functionals Scorr
(Volume II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489
№87_2 Volume III. Bridge from the standard 4D Yang–Mills model to the G-model (ΔH
+ 1gk) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496
№87_4 Volume III. Strict bridge between the standard Yang–Mills model and the Gmodel
(ΔH + 1gk) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501

Block function. Formalize a two-way bridge between the standard YM model and the
G-model (ΔH + 1gk), including confinement and spectral consequences.
G. Integrative and meta-level documents of the Great Theorem
G.1. Master documents and summaries

№83 Master document of the Great Theorem of the G-model. . . . . . . . . . . . . . . . . . . . . .508
№86 Information-analytic summary of resolving critical issues in the proof of the Great Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514
№87_8 Final document of the Great Theorem and the G-field theory . . . . . . . . . . . . . . 520
№87_9 Concise annotation of the G-field theory for the Great Theorems block . . . . . 526

Block function. Consolidate the internal and external Great Theorems and the worldview/technical context of the G-model.
G.2. Auxiliary and control documents

№47 Is the Great Theorem proved at this stage. . . (control considerations) . . . . . . . . . 530

Block function. Track control questions and interpretative checkpoints.
I. Generalized “proof structure of the Great Theorem” (schematic chain)
1. Local polygonal mass gap (0_1, 2, 3, 4, 5)
2. Transition to SU(N) and the standard Yang–Mills mass-gap structure (6, 7, 8, 9,10, 11, 12)
3. Expansion to the 4D ΔH-field and the measure μ1gk on Ho–Lo configurations (13,
15, 19)
4. Mapping Π: Ho–Lo/ΔH→ Aμ and construction μA = Π∗μ1gk (14, 16, 20, 21, 23, 39,
43, 50)
5. OS properties, correlation functions, and the mass-gap channel (17, 18, 22, 52, 65)
6. Axiomatics A1–A12 and the corresponding lemma blocks (27, 31, 30, 33–36, 40–42,48–52, 55, 63, 25)
7. Internal Great Theorem of the G-model (H-quantization, ΔH-quantization, EM1–
EM3) (57–60, 71, 85, 88–90, 87_1)
8. External Great Theorem: YM + mass gap in standard language (26, 28, 32, 37, 38,45, 46, 53, 54, 56, 61, 62, 87_7, 72, 73, 79)
9. Volumes I–II–IIβ–III as a strict analytic framework (μ1gk, Scorr, RN, bridge) (64–66,67, 69–70, 72–75_1, 82, 87_2–87_5, 76–81)
10. Master documents, analytic summaries, and annotations (83, 84, 86, 87_8, 87_9) In this organization, each document has a defined place both in the global proof logic and in concrete technical steps (axioms, lemmas

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Abstract to Parts II–III of the Monograph «Development of the G-Model and the Proof of the Great Theorem»

Parts II–III of the monograph refine the initial investigation against the background of a new task: to construct a completed, axiomatically rigorous proof of a Great Theorem of the type “ΔH + 1gk + Yang–Mills + mass gap” within the framework of the author’s theory of the harmonic fundamental field (the G-field). In this theory the fundamental field is treated as a harmonic self-controlled harmonizer, arising from an Ho-seed with the invariant
Emin ・ Kmin ≈ 1 gk,

and unfolded, according to the initial project, as a topological multilayer Ho–Lo structure.
The ΔH-structure, based on Ho–Lo components delegated into the continuum Holoc −
Loloc, specifies the admissible harmonic deviations of secondary wholes and domains in the continuum generated by this field. The continuum (space–time–quality) is interpreted as a secondary instrument for dissipating energy, whereas the mass gap appears not as an accidental characteristic of a particular Yang–Mills model, but as a projective property of the ΔH-architecture of the G-field.
In Part II, the internal and external forms of the Great Theorem are systematized. The internal version is formulated as the existence and stability of a local mass gap m0 > 0 for H-quantization in a ΔH-model with a 1gk-measure on the space of Ho–Lo configurations.
The external version is formulated for the standard 4D Yang–Mills model over R4, where the mass gap in a gauge-invariant channel arises via a pushforward mapping Π from the ΔH-field to the gauge field Aμ(x) and the fulfilment of OS/Gibbs-type conditions. An axiomatics A1–A12 is constructed for the coupled system “ΔH-field + measure μ1gk + mapping Π + Yang–Mills measure μYM”and a complete family of lemma-blocks is provided, which step by step derive the properties of the ΔH-field, the functional measure, OS-properties, the pushforward measure μA = Π∗μ1gk and the spectral mass gap. Separate sections are devoted to the internal Great Theorem (ΔH-quantization, secondary field, EM-channel EM1–EM3) and to the
reconciliation of all variants of the external Great Theorem in the standard Yang–Mills language.

Part III plays the role of an integrated proof spine. It is organized as a single controlled contour

RN/Scorr −→ OS −→ a → 0 −→ scale/renormalization −→ SU(N) −→ spectrum −→ gap.

Here the RN/Scorr block (Radon–Nikodym derivatives, sufficient conditions for formboundedness of Scorr and certification of transitions) is collected and harmonized with the OS-module (Euclidean measure, correlation functions, OS-reconstruction and compatibility with RN/Scorr), the controlled continuum limit a → 0, the scale–renormalization analysis in the ΔH-interval of the minimal mass gap, the SU(N)-lifting from prototype models, the description of the physical sector without gauge fixing, as well as the formulation of the Main Theorem, the proof-spine graph and local truth criteria.
The final modules contain an audit of the notation, “legal fixations” of the objects ΔH(f; x), the measure on the A-space and the renormalization condition, together with complete tables of correspondences between proof-spine nodes, individual documents, topologies and constants.
Taken together, Parts II–III constitute a completed axiomatically rigorous proof of a Great Theorem of the type “ΔH + 1gk + Yang–Mills + mass gap” within the G-model:
under axioms A1–A12, EM1–EM3 and technical conditions on μ1gk, μA and Scorr, the existence of a gauge-invariant mass gap m0 > 0 is proved for the 4D Yang–Mills measure.
The work may be of interest to specialists in mathematical physics, quantum field theory, probability theory and philosophy of physics who are looking for a structured, internally consistent route to solving the mass gap problem within a single harmonic model of the fundamental field. The study as a whole addresses the questions of the origin of space, time, mass, elementary particles and other related entities and phenomena. An extended interpretation of entropy as “minus-entropy” and “plus-entropy” is proposed, which makes it possible to distinguish more clearly and to answer current questions not only of mathematics and physics, but also of philosophy. Separately, the possibility is outlined of reestablishing present-day artificial-intelligence models from fragile, chimeric multilingual platforms onto a fundamental scientific basis.

μ1gk
Π −−−→
a→0
μA = Π∗μ1gk = μYM, m0 > 0, Aμ(x) R4

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- Download Part III of the monograph in Ukrainian in PDF format

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Part III of the monograph
__________________________________________________________
Structure of the pages of Part III of the monograph study
__________________________________________________________

0. Overall logic of Part III (aligned “proof line”)
Explanation. Part III is assembled as a single controlled contour (without logical cycles)
in the sense of the Proof Spine:
RN/Scorr → OS → a → 0 → scale/renormalization → SU(N) → spectrum → gap.
Numbering rule. The original document numbers are preserved below.
A. Introductory supports of Part III (framework and context)

No. 1 Integrated explanatory note on G-gravity, the Ech-block and the ΔH-simulator (1) 1
No. 6 Reference note. The place of CH in the G-field theory . . . . . . . . . . . . . . . . . . . . …… . . . . . 8
No. 8 Levels of mathematics in the G-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
B. RN/Scorr node (Radon–Nikodym, sufficient conditions, certification of transitions)
No. 2 Radon–Nikodym node_ strict bridge Scorr → stability A → 0 and spectral gap ……………18
No. 3 Rn_scorr_ technical subnode of sufficient conditions for Lp and form-bound estimates
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
No. 5 Binding the Rn_scorr node to Ch_l Corr and mass gap ↔ ΔHmin . . . . . . . . . . . . . ………….28
No. 7 Summary informational document: Rn_scorr node in Part II and its correspondences
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
No. 9 Certification lemma (am→gm) and application to Rn_scorr . . . . . . . . . . . . . . . . . . ……….. 36

C. OS-module (Euclidean measure, correlators, OS-reconstruction) + critical lemma compatible with RN/Scorr

No. 10 Os-module_ Euclidean measure → correlators → Os-reconstruction → spectral
mass gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
No. 18 Os–rn lemma_ reflection positivity is preserved under Rn_scorr (ch_lo microlocality)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
No. 25 Transition Os → spectrum_ standard reconstruction and mass gap as a spectral fact
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .49
D. Continuum limit a → 0 as a controlled transition (uniqueness or physical equivalence)
No. 11 Universality of the A − 0 limit in the ΔHmin channel. . . . . . . . . . . . . . . . . . . . . . . . . …..53
No. 19 The limit A → 0 as a controlled transition_ uniqueness or physical equivalence in the ΔHmin channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .57
No. 23 Continuum limit A → 0 as a single physical object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

E. Scale/renormalization in the ΔHmin channel (gap in physical units)

No. 16 Scale and renormalization in the ΔHmin channel_ control of A − 0 and physical mass-gap scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
No. 20 Scale/renormalization in the ΔHmin channel_ gap in physical units and inclusion in the Proof Spine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
No. 20a Scale/renormalization in the ΔHmin channel_ explicit renormalization condition and physical gap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74
F. SU(N) lifting (transfer theorem from prototype models to the non-Abelian setting)
No. 12 Su(n) lifting_ transfer of locality, invariance and estimates to the non-Abelian setting
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .78
No. 24 Su(n) lifting as a transfer theorem_ what does not depend on the group and where exactly Su(n) is needed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . …… 82
G. “What is physical” without gauge fixing (observables, A/G, separation of the physical sector)
No. 14 Physical observables and A_g without gauge fixing . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 87
No. 15 Physical observables and A_g without gauge fixing_ completed theorem . . . . . . . . . . .92
H. Main Theorem + Proof Spine (acyclic dependency graph) + truth criteria
No. 21 Main Theorem_ legally precise formulation (Part III) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
No. 22 Proof Spine_ acyclic dependency graph for the Main Theorem . . . . . . . . . . . . . . . . . . . .100
No. 22_1 Second step_ Proof Spine as an acyclic dependency graph (canonical N-node version) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105
No. 13 Main Theorem and Proof Spine for proving the mass gap in Su(n) via Rn_scorr, Os, A − 0, ΔHmin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
No. 17 Main Theorem and truth criteria (local tests). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

I. Notation alignment, “legal fixations”, proof metadocument and complete correspondence tables

No. 26 Audit-pass of notations and object statuses_ ΔH(f; x), Ch_lo locality, measure on A and physical Σ-algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .118
No. 27 Additions to minimize formal remarks_ uniformity of constants, topology of A → 0, node table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .124
No. 28 Three legal fixations_ ΔH(f; x), measure on A, renormalization condition . . . . . . . . .129
No. 29 Proof metadocument of the Great Theorem_ - Proof-Spine graph and nodes (goal_explanation_conclusion). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .131
No. 30 Complete correspondence tables of the Proof Spine (nodes –documents–objects
–topologies –constants) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
J. Final thematic node (linking mass gap with ΔH in the model formulations)
No. 4 Mass gap of Yang–Mills in the ΔH G-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
No. 31 Appendix: other predictions of the G-model from Part I
of the monograph research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .152
No. 32 PS Multivariantness, parallel realities_ - detailed refutation by the G-model . . . . . . . .163
No. 33 How the G-Model Theory Can Improve the Performance of Artificial Intelligence . . . . 169