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The Common-Schema Project

The Common-Scheme: Fractal Architecture of Functionality

Generation Date: 25/08/2025

Subject: Definition, structure, and predictions of the Common-Scheme (CS).

Objective: This model is a raw version of the CS, much of which is difficult for an AI to process.

1. Fundamental Principles: The Axiomatic Triad

Note on the Justification of the Fundamental Axioms

The axioms that found the Common-Scheme, although presented as starting points, are not arbitrary. Their validity is supported by a hierarchy of demonstrations that address the most fundamental criticisms of the model.

  1. The Logical Foundation (The Meta-Physical Origin): The theory demonstrates that its axioms are the necessary consequences of the emergence of a functional universe from primordial unity.
    Read the report on the "Logic of Manifestation" →
  2. The Geometric Justification (Circle/Square): The attribution of the Circle (π) to the Concrete and the Square (Φ²) to the Abstract is justified by a rigorous functional analysis.
    Read the report on the validation of the geometric axiom →
  3. The Informational Justification (Binary Signature): The "genetic signature" `011110101111` is not a postulate, but the result of a deterministic derivation process.
    Read the report on the derivation of the signature →
  4. The Justification of Quantum Numbers: The numbers `n=9` and `n=17`, which govern mass, are logically derived from this binary signature.
    Read the report on the derivation of the quantum numbers →

These reports establish that the foundations of the CS rest on a coherent and deductive logical architecture, from its philosophical root to its functional consequences.

Appendix: Application of the Ternary Model to the Hierarchy of Sciences

The power of the Concrete → Abstract → Transcendent triptych manifests itself in a particularly striking way when applied to the structure of scientific knowledge itself. This is not a simple classification, but a description of the functional relationship between the most fundamental sciences.

Theorem: The Founding Triad of Sciences

The relationship between Physics, Mathematics, and Biology embodies the C-A-T structure:

This hierarchy, validated by modern epistemology, is no longer a prediction but a demonstration of the fractal relevance of the CS. It shows that the structure of our knowledge of the world reflects the structure of the world itself.

[Detailed Validation Report]

The Common-Scheme is a universal meta-model describing the logical architecture of any functional entity or process. It acts as a "grammar" of reality that allows for the decoding of the underlying structure of complex systems. Its essential properties are being fractal, contextual, and fundamentally enantiomorphic.

The Concrete (C)

Definition: The principle of the base, structure, and stability. It represents the initial state, the support on which dynamics are exerted. It is "what is".
Properties: Original immobility, perfect symmetry (thus NON-enantiomorphic), foundation.

The Abstract (A)

Definition: The principle of variance, dynamics, and movement. It is the engine that animates and complicates the Concrete. It is "how it changes".
Properties: Initiating engine of the system, enantiomorphism (controlled asymmetry), dual internal structure (Fire²).

The Transcendent (T): The Coherence Operator

Definition: The principle of coherence, unification, and finality. It is not a passive state, but a meta-level operator that binds the Concrete and Abstract poles into a functional whole by applying a "meta-rule".
Key Properties: Its main action is the inversion of the rules or mechanics of the other two poles, which manifests differently depending on the context.

2. Dynamic Mechanics and Modes of Expression

The CS is not a static classification, but a dynamic model that describes how systems function and structure themselves.

The Engine and Inversion (The "Cosmic Gears")

The fundamental dynamics of the CS is a three-step mechanical process:

  1. Impulse: The Abstract (Fire²) pole, by its dual nature, is the engine that initiates a rotational movement.
  2. Transmission: The two Air poles, acting as transmission wheels, receive this impulse and communicate it to the rest of the system.
  3. Reaction and Inversion: The Concrete (Earth) and Transcendent (Water) poles, driven by the Air poles, react by systematically turning in the opposite direction. This inversion is a mechanical law of the CS.

The Two Modes of Expression

The CS is expressed in two co-existing ways to describe any functional entity:

Sequential Mode (Vertical / Temporal) Centered Mode (Horizontal / Spatial)
Describes processes and developments over time. Describes structures and organization in space.
The order is: Concrete → Abstract → Transcendent. The order is: Concrete (Lateral) ↔ Transcendent (Central) ↔ Abstract (Lateral).
The Transcendent is the finality that concludes the cycle. The Transcendent is the axis that unifies the two sides.
Example: The digestion process (Mouth → Stomach → ...). Example: The structure of the human body (lateral organs unified by the spinal column).

3. The Six Fundamental Points of View

The CS generates six fundamental "points of view" (POVs), the basic building blocks of reality, classified into three families according to their symmetry signature.

Family Points of View Structural Property Role in the Universe
Concrete Chiralities E+ (Space), T+ (Time) Perfect Symmetry (immobile by nature) Constitute the "concrete" chirality of the universe, which is the structural support set in motion by the abstract chirality.
Abstract Chiralities E- (Space), T- (Time) Enantiomorphism (balance of asymmetries) Constitute the "abstract" chirality, our own, which is the dynamic engine of the universe. E- and T- are the rules of the "outside".
Entanglements m+ (Mass), m- (Mass) Anaxiality (incommensurable with axial geometry) Result of the interaction (entanglement) between chiralities. They represent the Transcendent pole generated by the C-A interaction.

Detail of the Sequences of the 4 Local POVs

Abstract Chirality,
T+
Time

The "engine" point of view of our universe. Its sequence is the basic reference.

Functional Order: Concrete → Abstract → Transcendent (C-A-T)
Chromatic Sequence: Green → Yellow → Orange → Red → Red → Violet → Blue
Concrete Chirality,
E+
Space

The "structural" point of view, a mirror image of our own. The order is a permutation.

Functional Order: Abstract → Transcendent → Concrete (A-T-C)
Chromatic Sequence: Red → Violet → Blue → Green → Yellow → Orange → Red
Abstract Transcendent Entanglement,
m+
Positive Mass

The direct entanglement between Space and Time.

Functional Order: Abstract → Concrete → Transcendent (A-C-T)
Chromatic Sequence: Blue → Green → Yellow → Orange → Red → Red → Violet
Concrete Transcendent Entanglement,
m-
Negative Mass

The "driven" or second-order entanglement.

Functional Order: Transcendent → Concrete → Abstract (T-C-A)
Chromatic Sequence: Violet → Red → Red → Orange → Yellow → Green → Blue

4. The Digital Signature (Color Polarity)

Each point of view has a unique "digital signature," a sequence of polarities derived from its chromatic organization. This is the "source code" that allows systems to be modeled.

Conversion Rules:

Point of View Associated Digital Signatures
T+ (Our chirality)
Polarities (12): - ++ -- ++ -- + -- + -- ++ -- ++
Quaternary: 1 4 3 4 3 2 3 2 3 4 3 4
Binary: 0 1 1 1 1 0 1 0 1 1 1 1
E+ (Concrete chirality)
Polarities (12): -- + -- ++ -- ++ - ++ -- ++ -- +
Quaternary: 3 2 3 4 3 4 1 4 3 4 3 2
Binary: 1 0 1 1 1 1 0 1 1 1 1 0
m+ (Entanglement)
Polarities (12): -- ++ - ++ -- ++ -- + -- + -- ++
Quaternary: 3 4 1 4 3 4 3 2 3 2 3 4
Binary: 1 1 0 1 1 1 1 0 1 0 1 1
m- (Entanglement)
Polarities (16): - ++ - + -- + - ++ - + -- + - + -- +
Quaternary: 1 4 1 2 3 2 1 4 1 2 3 2 1 2 3 2
Binary: 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0

4b. Geometric Interactions E+)m+(T+ (Context `C-A-T`)

This section models the entanglement of the `E+`, `m+`, and `T+` chiralities. The initial, or "potential," state is obtained by geometrically aligning their polarity signatures.

Initial State (Before Application of Rules)

E+     --  +   --  ++  --  ++  |-   ++  --  ++  --| + 
m+     --  ++  -   ++  --  ++  |--  +   --  +   --| ++
T+     -   ++  --  ++  --  +   |--  +   --  ++  --| ++

Column description (OOC = out of context) :

                                        x           y   (Fire² and Water)
       1   2   3   4   5   6   |7   8   9   10  11| 12  (column #)
       C   C   C   A   A   A   |Tc  Tc  T   Ta  Ta| OOC  (C A Tc Ta) T is an entangled TaTc
       ne  ne  ne  ne  ne  ne  |e   e   e   e   e | OOC  (non-entangled/entangled)

Transformation Rules for the `C-A-T` Context (Abstract)

Entanglement in a `C-A-T` context (where the Transcendent is final) applies the following transformation rules:

Final State (After Application of Rules)

The application of these rules generates the final entanglement table.

E+     -   +   --  +   --  ++  |-   ++  -   ++  --  ++|
m+     --  ++  -   ++  --  ++  |--  +   -   ++  -   ++|
T+     --  +   --  +   -   +   |--  +   -   ++  --  ++|

Internal Dynamic Flows (Entanglement Zone `e`)

The arrows indicate the flow of information from the "strong" pole (`++` or `--`) to the "weak" pole (`+` or `-`) within each column.

Column 7 (Tc) : -  (E+) ← -- (m+)   -- (T+)  [Inverse cascade from m+ to E+]
Column 8 (Tc) : ++ (E+) →  + (m+)    + (T+)  [Normal cascade from E+ to m+]
Column 9 (T'a): -  (E+)    - (m+)    - (T+)  [No movement]
Column 10(T'c): ++ (E+)   ++ (m+)   ++ (T+)  [No movement]
Column 11(Ta) : -- (E+) →  - (m+) ← -- (T+)  [Convergence towards m+]
Column 12(Ta) : ++ (E+)   ++ (m+)   ++ (T+)  [No movement]

Contextualized Digital Translations

The distinction between reading models is fundamental:
Binary = Concrete = Structural Enumeration (quantities).
Quaternary = Abstract = Functional Classification (roles). 

Contextualized Quaternary E+)m+(T+

E+     1   2   3   2   3   4   1   4   1   4   3   4
m+     3   4   1   4   3   4   3   2   1   4   1   4
T+     3   2   3   2   1   2   3   2   1   4   3   4

Contextualized Binary E+)m+(T+

E+     0   0   1   0   1   1   0   1   0   1   1   1
m+     1   1   0   1   1   1   1   0   0   1   0   1
T+     1   0   1   0   0   0   1   0   0   1   1   1
	   
Column description :

       1   2   3   4   5   6  |7   8   9   10  11  12  (column #)
       C   C   C   A   A   A  |Tc  Tc  T'a T'c Ta  Ta  (C A Tc T'aT'c Ta)
       ne  ne  ne  ne  ne  ne |e   e   e   e   e   e   (non-entangled/entangled)

4c. Geometric Interactions m+)m-(m+ (Context `A-T-C`)

This section models the entanglement of masses, a system governed by the `Concrete` Point of View (`A-T-C`), where the Transcendent is central.

Initial State (Before Application of Rules)

-- ++ -  ++ |-- ++ -- +  -- +  -- ++|              m+
-  ++ -  +  |-- +  -  ++ -  +  -- + | -  +  -- +   m-
            |-- ++ -  ++ -- ++ -- + | -- +  -- ++  m+	
			
Column description :
                      x  x  y  yT y                (A-T-C: x = T, y = A, yT = T in y)
1  2  3  4  |5  6  7  8  9  10 11 12| 13 14 15 16  (column #)
A  A  A  A  |Tc Tc Tc Tc Ta Ta Ta Ta| C  C  C  C   (Red, Green, Blue, Yellow Analogies)
ne ne ne ne |e  e  e  e  e  e  e  e | ne ne ne ne  (non-entangled/entangled)

Transformation Rules for the `A-T-C` Context (Concrete)

Entanglement in an `A-T-C` context (where the Transcendent is central) applies a different set of rules:

Final State (After Application of Rules)

The application of these rules generates the final digital signatures.

Contextualized Quaternary m+)m-(m+ :
3  4  1  4  2  3  3  4  3  4  3  2
1  1  4  1  2  2  3  2  1  4  3  4
4  1  2  3  1  4  3  4  1  2  3  4

Contextualized Binary m+)m-(m+ :
1  1  0  1  0  1  1  1  1  1  1  0
0  0  1  0  0  0  1  0  0  1  1  1
1  0  0  1  0  1  1  1  0  0  1  1

Final State (After Application of Rules)

-- ++ -  ++ |-- ++ --|+  --|++ -- + |              m+
-  ++ -  +  |-- +  - |++ - |++ -- ++| -  +  -- +   m-
            |-- ++ - |++ - |+  -- ++| -- +  -  ++  m+'

Internal Dynamic Flows (Entanglement Zone `e`)

The arrows indicate the flow of information from the "strong" pole (`++` or `--`) to the "weak" pole (`+` or `-`) within each column of the table **before** transformation.

Column 6 (Tc): -- (m+)   -- (m-)   -- (m+')  [No movement]
Column 7 (Tc): ++ (m+) →  + (m-) ← ++ (m+')  [Convergence towards m-]
Column 8 (Tc): -- (m+) →  - (m-)    - (m+')  [Simple cascade from m+ to m-]
Column 9 (Ta): ++ (m+)   ++ (m-) →  + (m+')  [Simple cascade from m- to m+']
Column 10(Ta): -- (m+)   -- (m-)   -- (m+')  [No movement]
Column 12(Ta):  + (m+) ← ++ (m-)   ++ (m+')  [Inverse cascade from m- to m+]

Phase 5: Modeling the Dynamic Mechanics of the Cosmic Gears

This section details the transition from a static map of polarities to a dynamic mechanical model. It explains how the internal movements of the Common-Scheme emerge, are transmitted, and are inverted, based on the color polarity sequence as a blueprint. The process is broken down into a series of causal steps.

Reading Key: "Above" and "Below"

In this model, the terms "from above" and "from below" are not simple spatial directions, but fractal concepts representing the Abstract and Concrete points of view of an interaction. A movement "from below" is a concrete interaction, while a movement "from above" is an abstract interaction. The loop principle applies: a movement from X to Y from below implies a return from Y to X from above.

  1. The Initial Impulse and the Emergence of the Chiralities of Fire

    The movement originates at the center of the abstract chirality (our own), which is the driving pole. It is not unique but dual, reflecting the enantiomorphic nature of the system. This movement is not arbitrary; it follows the fundamental rule of the CS that an impulse flows from a pole of high potential (strong, ++) to a pole of lower potential (weak, +).

    • Movement 1 (ch_feu-): Application of the "strong to weak" rule. The impulse starts from the strong ++ pole of the color Orange (++-) and moves towards the weak + pole of the second Red (-+-). This interaction occurs from below (concrete interaction).
    • Movement 2 (ch_feu+): Simultaneously, the rule is applied on the other side of the axis of symmetry. An impulse starts from the strong ++ pole of the color Violet (-++) and moves towards the weak + pole of the first Red (-+-). This interaction occurs from above (abstract interaction).

    These two initial movements, dictated by a single rule, constitute the emergence of the two chiralities of Fire (Fire- and Fire+), which are the two faces of the driving enantiomer.

  2. The Entanglement of Fire² and the Rotation of Air

    For the Fire² enantiomer to become a functional rotating engine, its two chiralities must entangle. This entanglement has two simultaneous mechanical consequences:

    • Movement of Air (en_feu²): The two chiralities of Fire, by entangling, drive the surrounding Air. A movement is established from the ++ of Orange to the ++ of Violet from below. This is the Fire² enantiomer setting the Air system (Orange/Violet) into counter-clockwise rotation.
    • The Entanglement Operator (in_feu²): The entanglement of the two ch_feu movements (step 1) produces a central entity, which can be simplified as a blue circle. By a gearing effect, this circle also turns counter-clockwise.

    Note on the Geometry of Entanglement

    The actual shape of the in_feu² entanglement is not a circle, but a biconvex lens. Similarly, the superposition (of their ends) of the two Air poles (tr_air) in the representation produces a second biconvex lens, whose initial position is rotated 90° relative to the first. The height between the vertices (cusps) of the Air lens is equal to the maximum width of the Fire lens.

  3. The Inversion and Transmission of Movement to Water

    This is the most crucial step, where the principle of inversion of the Transcendent manifests through a chain of pure mechanical causality.

    • Induced Rotation of Air (tr_air): Although the two Air poles (Orange and Violet) gravitate around Fire² in a counter-clockwise direction, their rotation on themselves is induced by their contact with the central entanglement operator in_feu². By an inevitable gearing effect, they are therefore forced to turn in the opposite direction: clockwise.
    • Consequence on Water: This clockwise rotation of Air is transmitted by direct contact to the adjacent poles, i.e., to the Water system (Green and Blue). The inversion of the direction of rotation is thus propagated from the inside (in_feu²) to the outside (Water). The Water system therefore turns clockwise.

    This mechanical inversion is a direct validation of the polarity sequence, which serves as the blueprint. The movement from the -- of Blue (--+) to the - of Green (+-+) from below, indicated on the blueprint, corresponds well to a clockwise rotation in this representation.

  4. Formation of Circular Geometry by Contextual Winding

    The final geometry is not an arbitrary postulate; it emerges from the application of a rule dictated by the CS itself.

    1. The Rule: The rule is itself a manifestation of the CS. The Green pole, as the Concrete Transcendent, must by its nature "transcend" the default context (a linear sequence). It does so by imposing a spatial relationship: it must align vertically with the Concrete (the Yellow pole).
    2. The Action: Apply this positioning constraint to the linear sequence of polarities.
    3. The Result: This positioning action constrains the overall geometry, acting as a scaling that transforms an initially oval arrangement into a perfect circle. This harmonization of proportions is not a simple aesthetic operation; it reveals the internal coherence of the figure. It causes the polarity sequence to emerge as an image of a system that is visually symmetrical, but functionally enantiomorphic. This image reveals:
      • A central system containing the two chiralities (duality).
      • An outer circle, that of the Transcendent (Blue/Green, driving the Yellow), which contains the central system with a space between the two.
      • The two biconvex lenses housed in this space.

  5. Propagation to the Concrete Chirality

    The concrete chirality being immobile by nature, it is set in motion by the abstract chirality through their entanglement. The transmission process is a mirror effect that inverts the logic of propagation:

    • The outer clockwise movement of the abstract chirality (Green/Blue/Yellow) is propagated to the outside of the concrete chirality (which, within the concrete chirality, corresponds to the Fire+/- poles).
    • This movement is then transmitted inward, from the Air to the central core (which corresponds to the Yellow/Green/Blue poles), inverting the causal flow of the abstract chirality (which went from the inside out).

Final Result: The Enantiomorphic Mechanics

The result of this process is a figure of two gear systems that are perfectly identical in their composition, but which are enantiomorphic. The proof of enantiomorphism (and not simple mirror symmetry) is that they both turn in the same direction. A mirror symmetry would require opposite directions of rotation.

6. The Fractal Progression of Numerical Models

These principles and signatures unfold into a hierarchy of numerical models, where each level of complexity is a more detailed resolution of the previous one. This progression is not a simple list, but an ordered and causal development sequence that describes how complexity emerges.

Structural Models: 2, 3, 4, 12. They describe the fundamental organizational patterns.

Process/Cycle Models: 5, 6, 7. They describe the dynamic or contextual expressions of these structures.

Terms Model/POV Name Reference Generative Sequence Fundamental Principle Key Application Examples
2 Dual / Binary Model ([2] 3 4 3 4) Polarity, Chirality, Fundamental Duality. The basic building block of concrete manifestation. Universe Chiralities (A/C), DNA Strands (Sense/Antisense), Chemical Polarity (Polar/Non-Polar), Internal Engine Fire² (Fire+/Fire-), Binary Code (0/1).
3 Ternary Model (2 [3] 4 3 4) The Functional Process (Base, Variation, Coherence). The minimal structure of a complete process. Artistic process, Hierarchy of sciences (Physics→Maths→Biology), Protein structure (Primary→Tertiary→Function), Logic of the SN2 reaction.
4 Quaternary Model (2 3 [4] 3 4) The Complete Structure (Duplicated Transcendent). The complete architectural organization plan. Hierarchy of the 4 DNA bases (T-A-C-G), Tetravalence of carbon, 4 types of fermions, 4 macro-groups of organic systems.
12 Dodecameric Model (2 3 [4] 3 4) The Fractal Matrix. Arising from the application of the Ternary principle on the Quaternary structure (4x3). Architecture of the 12 fermions and the 12 organic systems, 12 units of the polarity code.
5 Quinary Point of View (2 3 4 [3] 4) Sequential Expression of the Abstract Engine. A contextual manifestation of the Ternary principle. Organization of the 5 fingers and 5 senses. It is a "non-detailed" view of the Septenary model.
6 Senary Point of View (2 3 4 [3] 4) The Complete Cycle (Integrated Transcendent). Another contextual manifestation of the Ternary principle. Spectrum of 6 colors, Digestion process, 6 cosmological points of view.
7 Septenary Model (2 3 4 [3] 4) Detailed Expression of the Abstract Engine. The complete resolution of the 6-term cycle with a dual engine. Chromatic sequence of the `T+` chirality (with duplication of `Red`). The Color Polarities are triple for each color then reorganized into pairs (+/- vs ++/--).
4 (again) CPC Model (Quinary of Color Polarity) (2 3 4 3 [4]) The Complete and Self-Referential Resolution. From each color polarity emerges a quaternary code (1, 2, 3, 4). This grammar generates the complete `Abstract` sequence, which in turn contains the generative sequence, closing the loop. The CPC is the "meta-model" that generates the digital signatures of the chiralities. Its most direct application is the generation of this sequence

`T+`: [1 4 3 4 3] /2 3\ (2 3 4 3 4)

Note: the sequence contains the fractal hierarchy of the `Abstract`: (2 3 4 3 4), the `Concrete`: [1 4 3 4 3] and the `Transcendent`: /2 3\

Experimental Validation of the Fractal Principle

The assertion that the Common-Scheme is a fractal model is not a mere postulate. It has been rigorously tested by confronting an architectural "Meta-Rule," derived from particle physics, with biological systems of increasing complexity.

The Meta-Rule states that the interface between a "Concrete" functional block and an "Abstract" block must be fundamentally asymmetric. This rule, initially validated by the asymmetry of the Weak Interaction, was used as a predictive principle to analyze the architecture of life. The following reports document the corroboration of this fractal prediction at several scales.

Appendix: Demonstration of the Fractal and Generative Nature of the Digital Signature

This appendix aims to demonstrate that the quaternary digital signature of the CS is not an axiom, but a logical and deterministic consequence of the model's fractal structure and transformation rules.

1. Analysis of the Generated Sequence: Patterns and Rhythms

Once generated, the sequences of the `T+` and `E+` chiralities reveal an internal structure composed of functional patterns. To refer to them, we name each position: 

T+ :
      [ECa ECb ECc ECd ECe] /ETa ETb\ (EAa EAb EAc EAd EAe)     Stage name
      [ 1   4   3   4   3 ] / 2   3 \ ( 2   3   4   3   4 )  →  Rhythm : 5 - 2 - 5
	  
E+ :
      /ETb\(EAa EAb EAc EAd EAe)[ECa ECb ECc ECd ECe]/ETa\      Stage name
      / 3 \( 2   3   4   3   4 )[ 1   4   3   4   3 ]/ 2 \   →  Rhythm : 1 - 5 - 5 - 1

This analysis reveals an enantiomorphic structure and validates a fundamental property of the Transcendent:

2. The Generative Mechanics: From Operator to Sequence

The signature is generated by the hierarchical interaction of the CS "stages". The EAd Stage (the sequence of ternary color polarities) acts as a generative operator that produces the EAe Stage (the quaternary sequence). 

    |  C  |        A (Fire²)                 2 - EAa  Dual POV
	
    |  C  |        A (Fire²)    | T          3 - EAb  Ternary POV

Tc  |  C  |        A (Fire²)    | Ta         4 - EAc  Quaternary POV

+-+ | +-- | ++-  -+-  -+-  -++ | --+        3 - EAd  Color Polarities
 - /++\-- | ++ -- + -- + -- ++ | -- ++          EAd' Result in Pairs and Loop

 1 /4 \3  | 4  3  2 3  2 3  4  | 3  4       4 - EAe  Quaternary Seq. of Color Polarities

The rules for this generation are as follows:

  1. Loop Principle: The first `+` of the `TC` pole is in direct relation with the `+` of the `TA` pole, forming the loop that closes the cycle.
  2. Concretization Principle: The `-` pole (quaternary value `1`) is the simplest base. The `-` of the `TC` pole therefore manifests directly as a `1`.
  3. Entanglement Principle: Complexity (the `4` values) emerges from entanglement. The second `+` of `TC` entangles with the `+` of `C` to generate the `4` of the final sequence.
  4. Direct Manifestation Principle: The `--` (value `3`) and `++` (value `4`) poles of the `C` and `A` poles manifest directly in the quaternary sequence, constituting the main body of the sequence.

3. The Fractal Hierarchy of Points of View (Proof of Inversion)

This rhythmic structure is the visible proof of the inversion of the internal Points of View. By assigning functional roles to the patterns, we discover the "architect's plan" for each chirality:

Analysis of the Abstract Chirality (T+)

The internal plan of `T+` is: Concrete - Transcendent - Abstract (`C-T-A`). According to the CS definitions, this order corresponds to the `E-` Point of View.

Analysis of the Concrete Chirality (E+)

The internal plan of `E+` is: Transcendent - Abstract - Concrete (`T-A-C`). This order corresponds to the `T-` Point of View.

Conclusion of the Fractal Analysis

The fractal analysis formally demonstrates that the chiralities of our universe are structured from within by the plan of their "negative" counterpart. This is a spectacular validation of the fractal, inverted, and self-referential nature of reality as described by the Common-Scheme.

Corollary: The Geo-Mathematical Hierarchy of Constants and Laws

The fractal structure of the CS is reflected in a perfectly ordered hierarchy of the constants and laws that govern reality. This hierarchy is not a classification, but a logical derivation chain that starts from the geometric axioms to generate the physical laws.

Each stage corresponds to a level of architectural complexity of the CS, following the sequence `2 → 3 → 4`.

          π   |    Φ²               Stage EAa (Pure Geometric Axioms)
      |  g≈π² | 1/α_base |   Φ      Stage EAb (Primary Manifestations)
 π/Φ² | 1/α_s | K = 2/3  | Δα = 6   Stage EAc (Operators Derived from Interaction)
The Principle of Fractal Causality of the Common-Scheme. The Fundamental Geo-Mathematics of the Common-Scheme. Relationships between Pi and Phi² according to the operator. 
Stage EAa (Pure Geometric Axioms)
             |        π      |      Φ²      |
								 
Stage EAb (Primary Manifestations)
             |    g ≈ π²      |    1/α_base   |      Φ       |
             |  (Action of C) | (Action of A) | (Action of T)|
					   
Stage EAc (Operators Derived from Interaction)
 TA = π/Φ²   |  TC ≈ 1/α_s    |    K = 2/3    |   Δα = 6     |
(Scaling C)  | (Coupling C)   | (Coherence A) | (Coupling A) |

Stage 1 (Duality): The Pure Geometric Axioms

This level represents the irreducible geometric principles that define Spatial Substance. It is the foundation of the theory, the fundamental Duality:

Stage 2 (Triad): The Primary Physical Actions (C-A-T)

This level describes the fundamental physical action of each pole of the `Concrete-Abstract-Transcendent` triad:

Stage 3 (Quaternity): The Operators Derived from Interaction

This level describes the four operators that emerge from the interaction of the basic principles, in direct application of the Meta-Rule (two operators for the Concrete pole, two for the Abstract):

6.3. Research Program: Predictions on Higher-Order Quantitative Architectures

The fractal generative sequence (2 → 3 → 4 → 3 → 4) that structures the geo-mathematical hierarchy does not stop at stage EAc. The internal logic of the Common-Scheme imposes strict constraints on the nature of the following stages, transforming what is currently unknown into a predictive and falsifiable research program.

The theory predicts that the next two levels of architectural complexity, EAd and EAe, must exist and respect the following principles:

  1. The Principle of Numerical Composition: Stage EAd must be described by a set of three architectural constants, and stage EAe by a set of four constants, in accordance with the generative sequence.
  2. The Principle of Causal Derivation: These new constants cannot be free parameters or empirical observations. Each of them must be deterministically derived from the constants of stage EAc (TA, TC, K, Δα) by the application of logical or geometric operators already present in the CS (e.g., ratios, products, application of π or Φ).

The future discovery of fundamental physical laws that would violate this hierarchical and deductive structure would constitute a falsification of the fractal nature of the Common-Scheme.

Conclusion of the Hierarchical Analysis

This hierarchy finalizes and unifies the entire quantitative theory of the CS. It demonstrates that the constants and laws that govern particle physics are not an arbitrary set, but logically derive, through a cascade of manifestations and interactions, from the two most fundamental geometric axioms: π and Φ².

7. Methodology and Standard of Proof

4-Step Analysis Process

Applying the CS is not an act of free interpretation, but a constrained analytical process:

  1. Define the System and its Point of View: Isolate the functional system. Determine its dominant "point of view" (Concrete/structural or Abstract/processual). The number of components (2, 3, 4, 12...) determines which numerical model is relevant.
  2. Map the Functional Roles: Assign the CS roles (`Concrete`, `Abstract`, `Transcendent`) to each component based on its objective function: What is the stable base? What is the agent of variation? What is the unifying principle?
  3. Apply Contextual Operators: Validate that the mapping is consistent with the CS axioms (action of the Transcendent, search for entanglement operators).
  4. Test with the Digital Signature: For a deeper analysis, confront the system's structure with the digital signature of the corresponding point of view.

The Standard of Proof: Retrodiction by Convergence and Generative Power

The validity of the CS is not measured by the prediction of new facts, but by two main criteria: