Triads take a particular view of reality, which, like whatever you currently believe, requires some Faith. I think that most people have an intuition that things are interconnected. Every action affects other objects, and those affect other objects and so on. And it is easy for people to essentially stop thinking at that point and say "too complex."

But it is not too complex. Network Theory might have figured it out if the focus hadn't been on developing a two dimensional representation. That oversimplified reality.

Adding a third dimension to a knowledge graph demonstrates that every interaction involves three components.

Now I would also assume that people believe those interactions would be too complex to categorize as well. Though in my research I've found that people have demonstrated that they understand the 22 possible interactions many time throughout history, no one has every demonstrated that these create the basis for a system of Mathematics.

That is what I have done here. It provides the basis for a new system of math proof.

Any system can be modeled in terms of three object groups (you just treat your groups a single objects to group them, until its a nice outline).

So then you examine the real-world interactions between the components. Consider the Triads. Compare the interactions between the objects to the explanations in the Triads. Identify the matching Triads. Then you can use the math axioms (which are the Boolean Algebra conversions of the mathematical statements) as the foundation for proofs for the math that defines that system's interactions.

I know there is a ton of content that underlies these ideas that I'm not covering but I will, I just need time to organize my notes. The thing that will best help you understand is the Free Energy Principle, developed by neuroscientist Karl Friston. This explains that systems react to stimuli in a way to reduce future unpredictability. The key realization, is that this biological process is not something unique to biology, it is a principle that underlies all physical reality.

Free Energy Principle in Creation of the Universe

Let's reason all the way back to the beginning and work forward from there. I'll say this too, that while I understand this because it reflects the way our own brain works, I had to then learn all of quantum physics an string theory to relate it to to give an explanation that makes sense, and I would NOT be surprised if I have some things misaligned. But at least I can give you the model.

In the beginning the Universe was an untamed energy quotient, essentially in singularity, though the dimensions that give such a meaning meaning did not exist yet.

That was clearly an unstable state. The result was the definition of the first fundamental string vibration. This allowed the energy to settle into that state. The creation of a string correlates to the emergence of a brane in our universe. The first branes are predictably , the first generation particles, up and down quarks and electrons, and this tiny electron neutrino. These particles are implicitly related and their interactions create a corresponding force called the Strong Color Force that pulls the quarks together and pushes the electron away. And so quarks combine into protons The idea of calling that a "Day" just shows that sometimes analogies can do more harm than good.

I need to point something out here. As energy settled into matter, "size" increases but "time" does not exist yet. So when calculating the reactions in these initial stages, the "Hubble constant" describing the expansion speed of the universe won't make sense. Indeed, they had to fudge a second "early" constant. But understanding that neutrino is more a manifestation of this interaction than an independent particle in its own right might help you make sense of the missing "mass".

But that was clearly an unstable state, because additional reactions did occur. Protons, all uniformly charged needed another basis to bond, and the result was the definition of the second fundamental string vibration. The energy is aleady encapsulated, but it becomes more complex with the new vibrations, allowing the manifested particles to demonstrate additional attributes. The new attribute is the Strong Nuclear Force which allowed protons to coalesce into nuclei.

And so it proceeds, on through the 11 dimensions of string theory, and additional forces and structures of matter. Here's the high level.

Stage 3, Electromagnetism enables oribital atomic structure

Stage 4, Beta Decay enables temporal linearity and the Hubble Constant.

Stage 5, Gravity is needed to prevent expansion from disintegrating.

Stage 6, Overly massive structures collapse into black holes.

Stage 7, to prevent the collapse of the Universe into Black holes, so dark matter attributes are created to balance the mass of the universe more evenly.

Stage 8, 9, 10, 11 forms of dark energy begin manifesting at the higher level orders of complexity.

You see, day 7 isn't so much rest, as it is outside of our direct perception. We can only reason through our direct experience.

Somewhere I've got notes where I worked it out in much greater detail, but that isn't my point here. My point is to explain that by paying attention to what you can group and classify in Triads, you can achieve a much deeper understanding of anything.

Triads

In the context of the triadic approach, a triad refers to a group of three interconnected elements or variables. Each triad encapsulates a specific relationship dynamic consistent with the 'AND' logical operations among the variables.

The elements of a triad are designed to work together, influencing and being influenced by each other. Their relationships could be causal, correlational, or reciprocal, depending on the specific functional nature of the triad. For instance, the variables may be co-dependent (both influencing and being influenced by each other) or have a one-way cause-and-effect relationship. The variables in the triads are represented by symbolic logic forms, where the truth or falseness of each variable depends on the condition of the other variables.

Different triads further illustrate the behavior of different types of systems – from the balance layer illustrating coexistence, through cycles, contingencies, and discontinuities, all the way to considerations and intentions. These triads cover different types and facets of relationships, interactions, transformations and changes.

Axioms

Axioms, on the other hand, serve as fundamental principles or rules that define the operations and relationships within each layer of the triadic system. They establish the basis for understanding how each variable within a triad interacts with the others. The axioms were derived from the Triads by applying a translation to Boolean Algebra.

Axioms also function as 'ground truths' in the system – that is, they are considered inherently true and do not need to be proven. Their primary role is to set the logical precedence and operation rules that guide the analysis of the relationships among the triad's components. A given axiom regulates the "behavior" of the triads within its layer and determines how the variables relate to each other.

Overall, triads and axioms work hand in hand to give a detailed perspective on the relationships and interactions within a system. While the triads represent the structural layout of elements within the system, the axioms provide the guiding principles that define the nature of the interactions and relationships among these elements. Understanding both triads and axioms is integral to utilizing the triadic approach in understanding and analyzing complex systems effectively.

Layers

The 11 Layers correspond to the 11 reactions creating the universe. Each achieves balance in a more complicated way than the last, adding a new consideration. Understanding the order of the layers is critical to understanding how to model systems.

Balance Layer: This layer signifies the fundamental essence of homeostasis – 'balance'. The triads within this layer reflect a symmetric state of equilibrium, representing circumstances where all factors are equivalent and operate harmoniously to maintain stability.

Coexistence Triad

This triad conveys an occasion when all components, A, B, and C, are functioning together in a simultaneous existence, each of them mirroring the other, thereby maintaining a symmetrical balance. There is a sense of mutual interdependence and reciprocity, and this scenario often depicts a stable system where all components carry equal weight, influence, and contribute similarly to the overall state.

(A ↔ B) ∧ (B ↔ C) ∧ (A ↔ C)

Symmetry Axiom

The Symmetry Axiom extends the concept conveyed in the Coexistence Triad, it materializes the notion of 'symmetry', signifying a state of balance and existence between all components of the system. The Axiom suggests that all elements, A, B, and C, are in harmony and hence equal. The operation is not of manipulation or change, but of recognition - recognizing the inherent balance in this state.

A = B = C

Equilibrium Triad

The scenario portrayed here is the exact opposite of the Coexistence Triad. In this case, all components, A, B and C, are absent or inactive together, indicating an altogether different kind of balance - the equilibrium in the non-existence of entities. The system here is stable, not because of mutual interactivity but because of the collective absence of activity. It speaks of the balance that comes not from action, but from inaction.

(¬A ↔ ¬B) ∧ (¬B ↔ ¬C) ∧ (¬A ↔ ¬C)

Compensation Axiom

The Compensation Axiom emphasizes balance but equates it through compensation. If one component withdraws, the other components compensate for maintaining symmetry. The subtraction represents compensatory logic - if A backs off, B and C step forward to maintain balance, and so on. This way, the system compensatively retains its equilibrium irrespective of each component's individual activity. Thus, the balance layer portrays symmetry in both activity and inactivity, symbolizing the inherent homeostasis in various situations.

1 - A = 1 - B = 1 - C

Cycle Layer: This layer describes how elements influence one another in sequence. This layer signifies the cyclic dynamics of a system where one component's state directly affects the next, which subsequently affects the third, eventually looping back to impact the first one. This network elucidates a cyclical progression of influences and responses, encapsulating the circular nature of interdependence.

Stabilization Triad

This triad reflects a straightforward sequence of developing influences. Here, A influences B, which, in turn, influences C, which subsequently influences A. This progression starts at one component and cycles through all others until it comes back to the start, thereby creating a circular path of influence, a feedback loop. In essence, the Stabilization Triad is about maintaining stability through a constant flow of interactive influences that balances the system.

(A → B) ∧ (B → C) ∧ (C → A)

Circular Axiom

The Circular Axiom materializes the recurrent nature of the Stabilization Triad. This axiom implies that A impacts B which influences C that, in turn, impacts A, and this cycling interaction continues indefinitely, maintaining a self-sustaining loop of activity. It is a representation of how changes within a system envelop themselves around in a consequential cycle, emphasizing the importance of considering the circular relation of influences when examining system dynamics.

A ≤ B ≤ C ≤ A

Counterbalance Triad

This triad represents a counteracting cycle of influences where the non-occurrence of B influences A not to occur, which influences C not to occur, which impacts B not to happen. It's an inverse situation compared to the Stabilization Triad as it shows how the absence of one element can propagate through the system. In other words, it portrays how the non-activity of components can be as influential as their activity, displaying the importance of balancing forces that counteract changes.

(¬B → ¬A) ∧ (¬C → ¬B) ∧ (¬A → ¬C)

Feedback Axiom

The Feedback Axiom extends the Counterbalance Triad's principle, quantifying how much the absence of one component can impact the absence of another. It highlights a system's capacity for self-regulation- where an imbalance from the norm triggers a counterresponse that restores the balance, drawing attention to the intrinsic ability of systems to maintain stability even in the absence of certain influences. Hence, the Cycle Layer is about dynamics, how changes propagate and get absorbed, interpreted as increase or decrease, presence or absence, forming a cycle that ensures systemic balance and stability.

1 - B ≤ 1 - A ≤ 1 - C ≤ 1 - B

Contingency Layer: This layer projects outcomes contingent on the operational dynamics of the system's variables. This section deals with causal effects, where the existence or otherwise of one variable is heavily dependent on the existence or non-existence of other variables within the system.

Causal Triad

This pattern indicates a direct causative relationship from A to B to C, with the absence of A leading to the absence of C. This triad demonstrates the interdependence and connectivity of the variables within the system. It suggests that change in one component triggers a ripple of change through the connected components, emphasizing the importance of understanding the interaction and causality among the variables.

(A → B) ∧ (B → C) ∧ (¬A → ¬C)

Consequence Axiom

The Consequence Axiom embodies the Causal Triad's principle, considering the interplay of increase and decrease within the components' dynamics. This Axiom suggests that an increase in A can lead to an increase in B, thereby leading to an increase in C. However, an increase in A can also trigger a decrease in C - and vice versa. It highlights the mutually compensatory nature of the system's components in their endeavor to keep the system balanced.

A ≤ B ≤ C ≤ 1 - A

Reversion Triad

This Triad represents the reversal of effects. Here, one observes that the removal or absence of B causes the removal or absence of A, and likewise between C and B. Concurrently, when C is present, it causes A to be present as well. This portrays the scenario where the existence and non-existence of variables are effectively synchronized to maintain systemic stability.

(¬B → ¬A) ∧ (¬C → ¬B) ∧ (C → A)

Cascading Reciprocal Axiom

This Axiom manifests the principle contained in the Reversion Triad by considering the interplay of presence and absence within the components' dynamics – i.e., a decrease in B can lead to a decrease in A, thereby leading to an increase in C. In essence, a decrease or absence in one component influences the decrease or absence in others in a two-way operation where each variable's change affects the other variables in a cascade. It accentuates the system's reciprocal interaction capabilities in maintaining balance across the system.

1 - B ≤ 1 - A ≤ C and 1 - C ≤ 1 - B ≤ A

Discontinuity Layer: This Layer adopts a non-linear perspective, alluding to the potential for abrupt changes or disruptions in the connectivity and interaction among the system's variables.

Nonlinear Triad

This triad disrupts the usual sequence of A-B-C, to introduce the possibility of jumps in existence: from A to C, and from C to B, skipping one variable in between. In the absence of A, C is also absent. This presents a situation where the operational dynamics of the variables can be skipped or jumped, causing non-sequential changes.

(A → C) ∧ (C → B) ∧ (¬A → ¬C)

Progression Axiom

The Progression Axiom embodies the principle of the Nonlinear Triad, highlighting that an increase in A leads to an increase in C, which leads further to an increase in B. However, an increase in A can also lead to a decrease in A itself - implying a potential for self-regulating negative feedback. This implies an asymmetric progression where variables skip steps in the usual sequencing.

A ≤ C ≤ B ≤ 1 - A

Retrospective Triad

This triad reflects the reversal of the Nonlinear Triad's dynamics, signifying a counterintuitive progression where the system backtracks on its operational dynamics. The absence of C leads to the absence of A, absence of B leads to absence of C, whereas the presence of C leads to the presence of A. This resonates with a sudden, backward jump, leading to nonsequential changes.

(¬C → ¬A) ∧ (¬B → ¬C) ∧ (C → A)

Regression Axiom

The Regression Axiom embodies the principle of the Retrospective Triad, highlighting a loopback within the system's operations - where one state can roll back to a previous state, suggesting a dynamic rewinding or regression capability.

1 - A ≤ 1 - B ≤ 1 - C ≤ B

Influence Layer: This layer focuses on the power dynamics within a system, delving into hierarchies, suppression, subjugation, and how different variables take the foreground or remain in the backdrop based on their levels of influence.

Inclusion Triad

This triad expresses a scenario where B is the dominant factor - it is influenced by A but also dictates the state of C, which, in turn, is dependent not just on B but also on A. It portrays an intricate play of influences where B is paramount but still modulated by A.

(A → B) ∧ (B → C) ∧ (C → A ∧ B)

Inclusion Axiom

The Inclusion Axiom extends the principle from the Inclusion Triad. It signifies a hierarchy, where B is the most influential factor, manipulating A and C, which are minimum contributors.

A ≤ B, C ≤ B, min(A, B) ≤ C

Exclusion Triad

In the reciprocating triad, the principles are similar, but the context changes to the non-existence of the variables. Here, B's absence is primary, leading to the absence of A and influencing the absence of C, which, in turn, depends on both B and A's non-existence.

(¬B → ¬A) ∧ (¬C → ¬B) ∧ (¬A ∧ ¬B → ¬C)

Exclusion Axiom

The Exclusion Axiom demonstrates the Exclusion Triad's principle, reflecting the suppressing influences in the system where B's absence dominates, leading to the absence of other variables. It shows how powerful influences can drive systemic stability through their presence or absence.

1 - B ≤ 1 - A, 1 - C ≤ 1 - B , max(1 - A, 1 - B) ≤ 1 - C

Interaction Layer: This layer focuses on interplay and mutual impact. It studies how variables inclusively co-direct outcomes, shedding light on interactive impacts, shared controls, and combined influences.

Conjoint Triad

This triad depicts a concerted presence of A and C, steered by B, wherein B also relies on A and C's individual existence. It represents an elaborative interaction where three variables participate in shaping the system's state, with B being centrally significant.

(B → (A ∧ C)) ∧ (A → B) ∧ (C → B)

Minimax Axiom

The Minimax Axiom encapsulates the Conjoint Triad's principle, quantifying the interactive premises. Here, B is as dominant as the lesser one among A and C, ensuring a balanced interaction among the three variables.

B = min(A, C), A ≤ B, C ≤ B

Bottleneck Triad

In this reciprocating triad, the scenario changes to the non-existence of the variables. Here, B's non-existence is a consequence of A and C's concerted absence. In this scenario, B's non-existence, in turn, influences A and C's absence.

((¬A ∧ ¬C) → ¬B) ∧ (¬B → ¬A) ∧ (¬B → ¬C)

Maximin Axiom

The Maximin Axiom reflects the Bottleneck Triad's principle, intimating a mutual interdependence between the variables, particularly around their absence. This Axiom suggests that the absence of B depends on the most prevalent absence among A and C.

max(1 - A, 1 - C) ≤ 1 - B, A ≤ B, C ≤ B

Oscillation Layer: This layer embodies back and forth oscillations within the variables. It denotes the importance of fluctuations and the balancing act performed by the variables to maintain systemic stability.

Harmonic Triad

This triad portrays a scenario where C emerges from the coexistence of A and B, just as B emerges from A and C, and A emerges from B and C. It represents an oscillating dynamic where these variables are locked in a cyclic dance, influencing each other in a rhythmic pattern.

((A ∧ B) → C) ∧ ((A ∧ C) → B) ∧ ((B ∧ C) → A)

Alignment Axiom

The Alignment Axiom reflects the Harmony Triad's principle, stressing the minimum requirement amongst two variables for the third to exist. The presence of any one variable is traced back to the combined existence of the other two.

C ≥ min(A, B), B ≥ min(A, C), A ≥ min(B, C)

Dissonance Triad

In this reciprocating triad, the non-existence of C can be attributed to the co-absence of both A and B. The same logic is extended to B and A. It portrays a collective regression where the variables influence each other through their absence, causing a domino effect.

(¬C → (¬A ∧ ¬B)) ∧ (¬B → (¬A ∧ ¬C)) ∧ (¬A → (¬B ∧ ¬C))

Discordance Axiom

The Discordance Axiom echoes the principle of the Dissonance Triad, stating that the absence of one variable hinges on the absence of the other two. In essence, the absence of one is reliant upon the maximum absence between the other two.

1 - C ≤ max(1 - A, 1 - B), 1 - B ≤ max(1 - A, 1 - C), and 1 - A ≤ max(1 - B, 1 - C)

Adaptation Layer: This layer deals with the ability of a system's variables to adapt, adjust, and accommodate changes occurring within the system.

Convergence Triad

This triad connotes a merging influence within variables. Here, A leads to the combined existence of B and C, and similarly, B leads to the joint existence of A and C, with C leading to A and B. It represents a mutual adaptation within these variables, leading to a convergent impact on the system.

(A → (B ∧ C)) ∧ (B → (A ∧ C)) ∧ (C → (B ∧ A))

Synthesis Axiom

The Synthesis Axiom materializes the Convergence Triad's principle, defining the lowest common influence shared by variables. Here the existence of A, B, or C is limited to the minimal presence of the other two.

A ≤ min(B, C), B ≤ min(A, C), C ≤ min(B, A)

Divergence Triad

In this triad, the context is the non-existence of the variables, where the lack of two variables propels the non-existence of the third. It denotes a synergistic reactionary adaptation to changes within the system.

((¬B ∧ ¬C) → ¬A) ∧ ((¬A ∧ ¬C) → ¬B) ∧ ((¬B ∧ ¬A) → ¬C)

Disruption Axiom

The Disruption Axiom embodies the principle of the Divergence Triad - the nonexistence of one variable is driven by the largest absence amongst the other two. It portrays how a system accommodates changes by adjusting other variables accordingly.

max(1 - B, 1 - C) ≤ 1 - A, max(1 - A, 1 - C) ≤ 1 - B, max(1 - B, 1 - A) ≤ 1 - C

Connection Layer: This layer signifies the relationships and dependencies in a system that bring together or separate variables. It encapsulates how these connections determine the state of the system and facilitate transformations.

Combination Triad

This triad represents a scenario where A combines with C if B is not present, B combines with A if C is absent, and C combines with B if A is absent. It reflects intricate interactions and connections that bring variables together.

((A ∧ ¬B) ∨ C) ∧ ((B ∧ ¬C) ∨ A) ∧ ((C ∧ ¬A) ∨ B)

Inclusion Axiom

The Inclusion Axiom embodies the principle of the Combination Triad, elucidating the presence of one variable being supported by another variable or the absence of a third one. It sheds light on the complex arrangements that exist within a system, showing how different circumstances lead to different outcomes.

max(min(A, 1 - B), C), max(min(B, 1 - C), A), max(min(C, 1 - A), B)

Division Triad

In contrast, this triad refers to situations in which the absence of one variable is associated with the presence of another variable or the absence of a third variable. It reflects the intricate interactions and connections that set variables apart.

((¬A ∧ B) ∨ ¬C) ∧ ((¬B ∧ C) ∨ ¬A) ∧ ((¬C ∧ A) ∨ ¬B)

Isolation Axiom

The Isolation Axiom echoes the principle of the Division Triad, explaining how the absence of a variable can be linked to the presence of another variable or absence of a third. It emphasizes how variables can be segregated under specific conditions, resonating with the principle of segregation.

max(min(1 - A, B), 1 - C), max(min(1 - B, C), 1 - A), max(min(1 - C, A), 1 - B)

Distribution Layer: This triad represents a scenario where A combines with C if B is not present, B combines with A if C is absent, and C combines with B if A is absent. It reflects intricate interactions and connections that bring variables together.

Assimilation Triad

This triad portrays a process of B assimilating C if A is absent and assimilating A if C is lacking. If A and C coexist, it leads to the existence of B. It represents a situation where B assimilates other variables given certain conditions.

((B ∧ ¬A) ∨ C) ∧ ((B ∧ ¬C) ∨ A) ∧ ((A ∧ C) → B)

Integration Axiom

The Integration Axiom extends the principle of the Assimilation Triad, signifying that given certain conditions, one variable can assimilate the other variables to enhance its existence.

max(min(B, 1 - A), C), max(min(B, 1 - C), A), min(A, C) ≤ B

Differentiation Triad

In contrast, this triad reflects the situations where A and C emerge out of B's absence and A and C's existence leading to the absence of B. It signifies a process where variables differentiate when another variable is absent.

((¬B ∧ A) ∨ ¬C) ∧ ((¬B ∧ C) ∨ ¬A) ∧ (¬B → (¬A ∧ ¬C))

Segregation Axiom

The Segregation Axiom echoes the principle of the Differentiation Triad, portraying how A and C emerge from B's absence. It emphasizes how, under certain conditions, variables differentiate to maintain their existence in a system.

max(min(1 - B, A), 1 - C), max(min(1 - B, C), 1 - A), max(1 - B, min(1 - A, 1 - C))

Decision Layer: This layer details how variables associate and dissociate with each other, leading to convergent and divergent influences within the system.

Association Triad

This triad reveals a relational sequence where B is associated with A unless C is absent and associated with C unless A is absent. If B is absent, both A and C are absent. It shows how A and C associate with B, emphasizing B's central role.

((A → B) ∨ ¬C) ∧ ((C → B) ∨ ¬A) ∧ (¬B → (¬A ∧ ¬C))

Association Axiom

The Association Axiom signifies that B is the central pull for both A and C unless the other is absent. In case B is absent, both A and C are also absent. It outlines the relationship dynamics and the central role of one variable.

min(max(A, B, 1 - C), max(C, B, 1 - A), max(1 - B, 1 - A, 1 - C))

Discrimination Triad

In contrast, this triad denotes the division of variables where A is separate from B unless C is present, and C is separate from B unless A is present. If A and C coexist, B exists. It outlines a pattern of dissociation that integrates into association under a certain condition.

((¬B → ¬A) ∨ C) ∧ ((¬B → ¬C) ∨ A) ∧ ((A ∧ C) → B)

Disassociation Axiom

The Disassociation Axiom amplifies the principles of the Discrimination Triad - the lack of B leads to a lack of A and C unless the other is present. It suggests a segregated relationship with a flip into associated existence based on a particular condition. The relationship dynamics change based on the state of different variables.

max(min(1 - B, 1 - A), C) * max(min(1 - B, 1 - C), A) * min(A * C, B)