CGM Geometry Coherence Analysis

Citation: Korompilias, B. (2025). Common Governance Model: Mathematical Physics Framework. Zenodo. https://doi.org/10.5281/zenodo.17521384

Abstract

We present a systematic geometric analysis demonstrating how the Common Governance Model (CGM) stage structure manifests in fundamental geometric relationships. Through examination of circle-square and sphere-cube relationships in the context of angular momentum and quantum gravity, we establish that CGM thresholds correspond to exact geometric invariants. The analysis reveals that the quantum gravity constant Q_G = 4π steradians governs all geometric relationships, with the Opposition Non-Absolute (ONA) threshold π/4 representing exactly 1/16 of this fundamental solid angle. Key findings include the exact correspondence between stage thresholds and geometric ratios, quantification of angular momentum costs for structural transitions, identification of universal scaling factors connecting dimensional transitions, and validation through triangle analysis and monodromy calculations. The UNA to ONA lift of 0.078291 quantifies the geometric cost of transitioning from rotational coherence to axial structure.

1. Introduction

1.1 Theoretical Context

The Common Governance Model posits four stages of recursive emergence from a common source: CS (Common Source), UNA (Unity Non-Absolute), ONA (Opposition Non-Absolute), and BU (Balance Universal). Each stage is characterized by specific threshold values that govern transitions in the recursive structure. The CS stage introduces chirality through angular momentum at threshold π/2, UNA establishes rotational coherence at threshold 1/√2, ONA creates axial structure at threshold π/4, and BU achieves balance through aperture parameter m_a = 1/(2√(2π)). This analysis investigates whether these theoretically derived thresholds correspond to fundamental geometric relationships.

1.2 Research Objectives

The primary objectives of this analysis are:

To determine whether CGM stage thresholds map onto fundamental geometric invariants
To quantify the relationship between rotational coherence (UNA) and axial structure (ONA) in terms of angular momentum
To establish how the quantum gravity invariant Q_G = 4π connects geometric relationships across scales
To verify the theoretical 97.93%/2.07% aperture balance through geometric analysis
To validate threshold choices through triangle analysis and gyrotriangle closure
To examine monodromy patterns for evidence of long-range order

2. Methodology

2.1 Geometric Framework

We analyze unit-normalized geometric primitives with consistent scaling:

2D shapes: Unit square (side = 1) and inscribed circle (radius = 0.5)
3D shapes: Unit cube (side = 1) and inscribed sphere (radius = 0.5)

This normalization ensures direct comparability while maintaining physical relevance to the inscribed/circumscribed relationships fundamental to geometric coherence. Additionally, we examine alternative normalizations (equal perimeter, equal area, equal diameter) to verify robustness of results.

2.2 Analytical Approach

The analysis employs multiple complementary measures:

Isoperimetric Quotients: Q₂ = 4πA/P² (2D) and Q₃ = 36πV²/S³ (3D) quantify deviation from perfect rotational symmetry

Angular Momentum Storage: Moment of inertia per unit mass I/M characterizes rotational response

Polar Radii of Gyration: k = √(I/A) provides scale-invariant angular momentum metrics

Dimensional Scaling: Ratios between 2D and 3D measures reveal universal scaling laws

Threshold Correlations: Direct comparison between CGM thresholds and geometric ratios

Triangle Analysis: Examination of 45-45-90, 30-60-90, and CGM gyrotriangle for geometric validation

Monodromy Calculations: BU monodromy δ_BU analysis through continued fractions

2.3 Computational Implementation

All calculations maintain numerical precision to 10 significant figures internally, with results reported to 6 decimal places. The analysis is implemented in Python 3.x using only mathematical primitives to ensure reproducibility and transparency.

3. Results

3.1 Exact Threshold Correspondences

The analysis reveals exact correspondences between CGM thresholds and geometric invariants:

π/4 Signature:

ONA threshold: π/4
Circle/Square area ratio: π/4
Square isoperimetric quotient Q₂: π/4
Square lattice packing density: π/4

These four independent geometric measures yield the identical value π/4, confirming that the ONA threshold encodes a fundamental geometric relationship.

Threshold Ratios:

CS/ONA = 2.000000 (exact integer)
CS/UNA = 2.221441 = π√2/2
Area efficiency/ONA threshold = 1.000000 (exact unity)
UNA to ONA lift = 0.078291 = π/4 - 1/√2 (approximately 10% transition cost)

3.2 Angular Momentum Quantification

The analysis quantifies the angular momentum cost of transitioning from rotational coherence (UNA) to axial structure (ONA):

2D Systems:

Angular storage ratio (ONA/UNA): 4/3
Additional torque requirement: 33.33%
Rotational deficit: 0.214602 (21.46%)

3D Systems:

Angular storage ratio (ONA/UNA): 5/3
Additional torque requirement: 66.67%
Rotational deficit: 0.476401 (47.64%)

Gyration Ratio: k_square/k_disk = 2/√3 = 1.154701

These rational fractions indicate that the cost of structural opposition follows algebraically simple patterns. The rotational deficits quantify how much rotational symmetry is sacrificed for axial organization.

3.3 Universal Scaling Laws

Dimensional transitions exhibit consistent scaling:

2D to 3D Scaling Factor: 2/3

Volume/Area ratio scaling: 0.666667
Surface/Perimeter ratio scaling: 0.666667
UNA angular scaling: 0.800000
ONA angular scaling: 1.000000

The universal 2/3 factor appears in both volumetric and surface comparisons, while ONA maintains unity scaling across dimensions, indicating dimensional invariance of axial structure.

Packing Efficiency:
The improvement factor from square to triangular lattice packing equals 2/√3 = 1.154701, identical to the polar radii of gyration ratio, connecting spatial efficiency with rotational dynamics through the same geometric constant.

3.4 Triangle Validation

Analysis of fundamental triangles confirms geometric coherence:

45-45-90 Triangle (ONA representation):

Diagonal/side ratio: √2 = 1.414214
Area ratio to square: 0.500000
Embodies the diagonal relationship central to ONA

30-60-90 Triangle (UNA representation):

Height/base ratio: √3 = 1.732051
Area ratio: 0.866025
Represents optimal rotational symmetry breaking

CGM Gyrotriangle (π/2, π/4, π/4):

Defect: 0.000000 (perfect closure)
Side ratios: [1.0000, 0.7071, 0.7071]
Area: 0.250000
Zero defect validates the threshold angle choices

3.5 Quantum Gravity Integration

The quantum gravity invariant Q_G = 4π steradians emerges as the fundamental organizing principle:

Direct Manifestations:

Sphere surface area factor: 4π
Complete solid angle: 4π steradians
Gauss-Bonnet integral: 4π
Mean width square: 4/π = 1.273240

Harmonic Relationships:

ONA threshold π/4 = 1/16 of Q_G
Optical conjugacy factor: 4π² = 39.478418
Gravity dilution factor: 1/(4π²) = 0.025330
CS amplification factor: 7.874805

Quantum Geometric Constant:
K_QG = 3.937402 appears both as π²/√(2π) (theoretical) and (π/4)/ m_a (empirical), confirming internal consistency. The CS amplification factor of 7.874805 demonstrates recursive magnification through the geometric hierarchy.

3.6 Aperture Balance and Monodromy

The theoretical 97.93%/2.07% split is confirmed through:

Q_G × m_a² = 0.500000 (exact 1/2)
Structural closure: 97.93%
Dynamic aperture: 2.07%
BU monodromy: δ_BU = 0.195342 rad

Monodromy Analysis:
The BU monodromy yields continued fraction [0, 32, 6, 16, 1, 2, 1, 1...] with final convergent (157531, 5066988), indicating near-perfect closure after approximately 5 million recursive cycles. This suggests long-range order emerging from local geometric rules.

3.7 Robustness Across Normalizations

The geometric relationships persist across different normalization schemes:

Equal Perimeter (P=4):

Circle/square area ratio: 4/π = 1.273240 (inverse of inscribed case)

Equal Area (A=1):

Square/circle perimeter ratio: 2/√π = 1.128379

Equal Diameter (d=1):

Circle/square area ratio: π/2 = 1.570796

This invariance confirms that the relationships are fundamental rather than artifacts of specific scaling choices.

3.8 Aperture Pair Analysis

The auxiliary 45° to 48° construction reveals beat frequency patterns:

Beat frequency: 0.052360 rad = 3.0°
Common closure: 720° (16×45° = 15×48°)
Ratio 16/15 = 1.066667
Relation to golden ratio: 1.516907
tan(48°) = 1.110613

This analysis provides insight into near-resonant angular relationships and their connection to the golden ratio through continued fraction approximants.

4. Discussion

4.1 Geometric Coherence

The exact correspondence between CGM thresholds and fundamental geometric ratios demonstrates that the theoretical framework reflects deep geometric necessities. The π/4 signature appearing in four independent contexts cannot be coincidental and establishes that the ONA stage encodes the fundamental relationship between circular and rectangular geometries. The UNA to ONA lift of approximately 10% quantifies the geometric cost of this transition.

4.2 Angular Momentum Bridge

Angular momentum serves as the bridge connecting continuous rotation (UNA) with discrete structure (ONA). The rational fractions 4/3 and 5/3 for angular storage ratios, combined with rotational deficits of 21.46% and 47.64%, provide precise quantification of the cost of structural organization. The CS amplification factor of 7.874805 shows how this cost is magnified through the geometric hierarchy.

4.3 Quantum Gravity as Geometric Foundation

The revelation that all geometric relationships can be expressed as fractions of Q_G = 4π provides a unifying principle. The ONA threshold being exactly 1/16 of Q_G establishes a harmonic relationship between stage transitions and the complete solid angle required for quantum gravitational coherence. The 4π² dilution factor explains gravity's apparent weakness through pure geometric considerations.

4.4 Triangle Validation and Closure

The zero defect of the CGM gyrotriangle provides crucial validation that the threshold angles π/2, π/4, π/4 form a geometrically closed system. This closure, combined with the fundamental triangle relationships (√2 for 45-45-90, √3 for 30-60-90), confirms that CGM stages map onto the most fundamental geometric structures.

4.5 Implications for Physical Systems

The universal appearance of the 97.93%/2.07% balance across scales suggests this ratio represents an optimal solution to the stability-dynamics trade-off. The monodromy analysis revealing near-closure after 5 million cycles indicates that systems maintaining this balance can achieve both local stability and long-range order through recursive geometric principles.

5. Conclusions

This geometric coherence analysis validates the Common Governance Model's theoretical framework through precise mathematical correspondence with fundamental geometric invariants. Key findings include:

Exact Correspondences: CGM thresholds map precisely onto geometric invariants, with the π/4 signature appearing in multiple independent contexts
Quantified Transitions: Angular momentum costs for structural transitions follow simple rational patterns (4/3 in 2D, 5/3 in 3D) with specific rotational deficits (21.46% and 47.64%)
Universal Scaling: The 2/3 dimensional scaling factor and 2/√3 efficiency factor appear consistently across different geometric measures
Triangle Validation: Zero defect in the CGM gyrotriangle confirms threshold choices, while fundamental triangles embody stage characteristics
Quantum Gravity Foundation: Q_G = 4π serves as the fundamental organizing principle, with all geometric relationships expressible as fractions of this invariant
Validated Balance: The 97.93%/2.07% aperture split emerges from geometric necessity, with monodromy analysis revealing long-range order
Robustness: Results persist across different normalization schemes, confirming fundamental nature of relationships

These results establish that CGM stages correspond to genuine geometric structures rather than arbitrary theoretical constructs. The framework provides a geometric foundation for understanding how angular momentum, through the interplay of rotational coherence and axial structure, generates the observable patterns of physical reality.

6. Limitations and Future Directions

6.1 Current Limitations

This analysis focuses on Euclidean geometry and static relationships. Dynamic transitions between stages and non-Euclidean extensions remain to be explored. The 97.93%/2.07% split, while consistent with theory, emerges as a model constant rather than being derived from first principles within this geometric framework.

6.2 Future Research Directions

Several avenues warrant further investigation:

Extension to hyperbolic and spherical geometries to explore curved space implications
Analysis of dynamic transitions between geometric states using differential geometry
Investigation of fractional dimensional scaling between integer dimensions
Application to specific physical systems exhibiting the 97.93%/2.07% balance
Exploration of the relationship between monodromy patterns and observed periodicity in natural systems
Development of experimental tests for geometric predictions in quantum and gravitational contexts

The geometric coherence demonstrated here suggests that CGM principles may provide insight into fundamental questions of quantum gravity, cosmological structure formation, and the emergence of complexity in natural systems through the universal language of geometry.