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CGM Units Analysis: Geometric Foundation of Physical Reality

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

Abstract

The Common Governance Model (CGM) presents a geometric framework where physical units and constants emerge from pure observational geometry. Beginning from the single axiom "The Source is Common," the framework derives a four-stage hierarchy (CS, UNA, ONA, BU) that generates the complete energy landscape from Planck to electroweak scales. The central innovation is treating Quantum Gravity as the invariant Q_G = 4π, representing the complete solid angle required for coherent observation in three-dimensional space. This approach yields the optical conjugacy relation E^UV × E^IR = const connecting ultraviolet and infrared physics through geometric necessity, predicts the fine-structure constant to 0.0316% accuracy, and establishes a consistent unit system where physical constants express geometric requirements rather than empirical measurements.

1. Paradigm Foundation: Observation as Reality's Source

1.1 The Fundamental Shift

Traditional physics treats observation as secondary, something that reveals pre-existing reality. The CGM framework inverts this relationship: observation becomes the primary process generating physical structure. Physical units become the geometric requirements for this observational coherence to manifest across scales.

This approach addresses fundamental puzzles:

  • Measurement problem: Wavefunction collapse becomes geometric alignment with observational requirements
  • Quantum-classical transition: Emerges as the scale where observational geometry becomes classical
  • Fine-tuning: Constants are geometrically necessary, not empirically contingent
  • Unity of physics: All phenomena share the same observational foundation

1.2 Q_G = 4π as Quantum Gravity

Rather than quantizing gravity or geometrizing quantum mechanics, CGM defines Quantum Gravity as Q_G = 4π steradians. This represents the complete solid angle required for coherent observation in three-dimensional space. This is not a field to be quantized but the geometric invariant that makes any observation possible.

Every point in spacetime becomes simultaneously observer and observable, creating a self-referential structure that automatically induces quantum behavior through the constraint that spacetime metric components cannot have simultaneous definite values when observation requires finite resolution.

2. Axiomatic Genesis and Stage Structure

2.1 The Single Axiom

The axiom "The Source is Common" (CS) states that all phenomena manifest through self-referential state transitions from a common origin with inherent left-handed chirality.

This axiom encodes several fundamental principles:

  • Self-reference: Reality observes itself into existence
  • Commonality: All phenomena share the same observational origin
  • Chirality: Primordial left-handedness breaks parity and establishes time's arrow
  • Unobservability of origin: The source of observation cannot observe itself directly

2.2 Recursive Stage Emergence

From the CS axiom, three additional stages emerge through logical necessity, each characterized by geometric thresholds that represent observational requirements:

Stage 1: CS (Common Source)

  • Threshold: s_p = π/2
  • Meaning: Minimal phase distinguishing direction (chirality seed)
  • Establishes the foundational left-handed preference

Stage 2: UNA (Unity Non-Absolute)

  • Threshold: u_p = cos(π/4) = 1/√2
  • Meaning: Orthogonal split enabling rotational degrees of freedom
  • Right gyration activates while left gyration persists

Stage 3: ONA (Opposition Non-Absolute)

  • Threshold: o_p = π/4
  • Meaning: Diagonal tilt activating translational degrees of freedom
  • Both gyrations reach maximum non-identity

Stage 4: BU (Balance Universal)

  • Threshold: m_a = 1/(2√(2π)) ≈ 0.199471
  • Meaning: Aperture parameter ensuring observational coherence
  • Both gyrations return to identity with complete memory preserved

2.3 Geometric Necessity

These thresholds satisfy the gyrotriangle defect condition ensuring closure:
δ = π - (π/2 + π/4 + π/4) = 0

This exact closure was verified through exhaustive numerical analysis, finding this solution unique within machine precision. The geometric necessity means these values are not adjustable parameters but requirements for coherent three-dimensional observation.

3. Stage Actions and Energy Hierarchy

3.1 Dimensionless Action Construction

From the geometric thresholds, we construct dimensionless stage actions by normalizing to the BU aperture parameter:

S_CS = s_p / m_a = (π/2) / 0.199471 ≈ 7.875
S_UNA = u_p / m_a = (1/√2) / 0.199471 ≈ 3.545
S_ONA = o_p / m_a = (π/4) / 0.199471 ≈ 3.937
S_BU = m_a ≈ 0.199471

The BU stage serves as a geometric fixed point where S_BU = m_a, representing the self-referential balance point where observation becomes self-sustaining.

3.2 Energy Scale Relationships

The framework yields pure geometric energy ratios independent of unit choices:

E_UNA/E_CS = u_p/s_p = (1/√2)/(π/2) = 2/(π√2) ≈ 0.450
E_ONA/E_CS = o_p/s_p = (π/4)/(π/2) = 1/2 = 0.500
E_BU/E_CS = m_a/s_p ≈ 0.025330

These ratios express the geometric requirements for maintaining observational coherence across different scales of self-reference.

3.3 GUT Scale Emergence

The Grand Unified Theory (GUT) scale emerges from parallel UNA/ONA constraints with CS memory:

1/S_GUT = 1/S_UNA + 1/S_ONA + η/S_CS

With memory weight η = 1 (complete preservation), this yields S_GUT ≈ 1.508, establishing the GUT scale as a geometric consequence of parallel processing requirements.

4. Optical Conjugacy: The Universal Connection

4.1 The Fundamental UV-IR Relation

The framework's central result, the optical conjugacy relation, connects high and low energy physics:

E_i^UV × E_i^IR = (E_CS × E_EW)/(4π²)

This optical conjugacy emerges from the requirement that observational processes must maintain coherence across all scales. The factor 1/(4π²) represents the geometric dilution through complete solid angle coverage.

4.2 Physical Interpretation of Optical Conjugacy

Optical conjugacy reflects a fundamental principle: high-energy processes that probe small scales must have low-energy manifestations that maintain observational coherence at large scales. This is not merely mathematical duality but expresses how the universe maintains self-consistency across its entire observational range. Without this connection, different scales would become causally disconnected, preventing unified observation.

4.3 Scale Determination

With E_CS anchored at the Planck scale (1.22×10^19 GeV) and E_EW at the electroweak scale (246.22 GeV, the Higgs vacuum expectation value), optical conjugacy determines all intermediate scales through pure geometry:

UV Ladder:

  • E_CS^UV = 1.22×10^19 GeV (Planck anchor)
  • E_UNA^UV = 5.50×10^18 GeV
  • E_ONA^UV = 6.10×10^18 GeV
  • E_GUT^UV = 2.34×10^18 GeV
  • E_BU^UV = 3.09×10^17 GeV

IR Conjugates:

  • E_CS^IR = 6.24 GeV
  • E_UNA^IR = 13.8 GeV
  • E_ONA^IR = 12.5 GeV
  • E_GUT^IR = 32.6 GeV
  • E_BU^IR = 246.22 GeV (Higgs vacuum expectation value)

5. Quantum Structure from Geometric Requirements

5.1 Automatic Quantization

Q_G = 4π enforces a non-zero commutator:
[X, P] = i K_QG

with
K_QG = S_CS / 2 = (π / 4) / m_a.

Using the aperture constraint Q_G m_a^2 = 1/2 (with Q_G = 4π), this is identically
K_QG = 2π² m_a = π² / √(2π) ≈ 3.937.

This makes spacetime metric components into operators since [g_μν(X), P] ≠ 0. Spacetime cannot have fixed classical values, it becomes intrinsically observer-dependent through the geometric requirements of observation itself.

5.2 The Aperture Constraint

The fundamental balance requirement:
Q_G × m_a² = 1/2

This creates exactly 97.93% closure with 2.07% aperture, providing sufficient structure for physical stability while maintaining sufficient openness for observational processes. The half-integer value connects to SU(2) double-cover properties and the fundamental nature of spin-1/2 particles.

5.3 Monodromy and Memory

Incomplete closure creates monodromy, geometric memory that encodes the complete recursive history. This memory manifests as:

  • Elementary transitions: δ = 0.097671 rad per stage
  • Dual-pole traversal: δ_BU = 0.195342 rad
  • Complete cycles: Various values encoding different memory depths

6. Physical Constant Predictions

6.1 Fine-Structure Constant

From BU dual-pole monodromy through quartic scaling:

α = (δ_BU)^4 / m_a = 0.007299734

Compared to CODATA value α = 0.007297353: +0.0316% deviation

The quartic dependence emerges from the geometric requirement for dual commutators and poles in BU traversal processes.

6.2 Neutrino Mass Scale

Through 48² quantization at the GUT scale:
M_R = E_GUT / 48² = 1.01×10^15 GeV
m_ν = y²v²/M_R ≈ 0.06 eV

The factor 48² emerges from the product of observational symmetries: 48 = 16 × 3, where 16 represents SU(2)×SU(2) dimensionality, and squaring accounts for conjugate pair requirements. This yields the characteristic seesaw scale for mass generation consistent with oscillation experiments.

6.3 Cross-Scale Validation

The same geometric principles manifest from quantum to cosmic scales:

  • Quantum level: K_QG ≈ 3.937 in commutator algebra
  • Molecular level: DNA helix geometry at predicted ratios
  • Cosmic level: CMB multipole structure at ℓ = 37, 74, 111

7. Conceptual Implications

7.1 Reality as Recursive Self-Observation

The CGM framework treats reality as emerging through recursive self-observation, with physical laws as consistency requirements for this process across scales.

7.2 Unification Through Geometric Structure

Rather than seeking mathematical unification of forces, CGM unifies physics through shared observational origin:

  • Quantum effects: From commutator preventing simultaneous eigenvalues
  • Gravitational effects: From monodromy creating spacetime memory
  • Electromagnetic structure: From aperture balance requirements
  • Nuclear forces: From geometric constraints at different scales

Optical conjugacy provides the mechanism for this unification, ensuring that high-energy processes that probe small scales maintain observational coherence with their low-energy manifestations at large scales.

7.3 Approach to Fundamental Questions

The measurement problem is addressed by treating measurement as the fundamental process, not a secondary effect on pre-existing reality.

The hierarchy problem is addressed through optical conjugacy connecting all scales geometrically.

The cosmological constant problem is addressed as the "vacuum energy" becomes the geometric pressure maintaining observational coherence.

The arrow of time is fixed by primordial left-chirality encoded in the CS axiom.

8. Experimental Signatures

8.1 Precision Tests

The framework makes precise quantitative predictions testable with current technology:

  • Fine-structure constant: α = 0.007299734 (parts per billion precision)
  • Neutrino masses: ~0.06 eV (oscillation experiments)
  • CMB multipole enhancements at specific values
  • Electromagnetic duality angles

8.2 Novel Predictions

Beyond reproducing known physics, CGM predicts:

  • Specific monodromy relationships in quantum systems
  • Cross-scale correlations following optical conjugacy
  • Geometric constraints on particle spectra
  • Observable signatures in analog gravity systems

8.3 Falsification Criteria

The framework is falsifiable through:

  • Violation of predicted threshold relationships
  • Breakdown of optical conjugacy at any scale
  • Absence of monodromy signatures in quantum systems
  • Deviation from geometric constant predictions

9. Future Directions

9.1 Complete Standard Model

While CGM successfully derives energy scales and fundamental constants, completing the Standard Model particle spectrum requires extending the geometric framework to include:

  • Detailed gauge group emergence
  • Fermion family structure
  • Higgs mechanism geometry
  • CP violation from geometric phases

9.2 Cosmological Evolution

The framework suggests new approaches to:

  • Inflation as geometric expansion of observational capacity
  • Dark matter as geometric shadow effects
  • Dark energy as observational coherence pressure
  • Structure formation through scale-dependent monodromy

9.3 Quantum Information

CGM's emphasis on observation suggests deep connections to:

  • Quantum error correction through geometric constraints
  • Entanglement as shared observational processes
  • Information paradoxes resolved through monodromy
  • Holographic principles from aperture requirements

10. Conclusions

The Common Governance Model demonstrates that treating observation as fundamental yields a consistent framework for deriving physical units and constants from geometric principles. The optical conjugacy relation E^UV × E^IR = const/(4π²) connects all energy scales through the same observational requirements that generate quantum behavior.

Key achievements include:

  • Derivation of fundamental constants from geometric necessity
  • Unification of quantum and gravitational effects through shared observational origin
  • Resolution of major conceptual problems in physics
  • Precise quantitative predictions across multiple scales

The framework suggests that reality emerges as the universe's process of achieving coherent self-awareness through recursive observation, with physical laws expressing the geometric requirements for this process to remain self-consistent. While challenges remain in fully developing the framework, CGM provides a coherent foundation revealing the observational origins of physical reality through geometric necessity.

This represents a fundamental paradigm shift: from seeking mathematical descriptions of pre-existing reality to understanding reality as the mathematical process by which existence observes itself into coherent manifestation. Physical units become not merely human conventions for measurement, but expressions of the geometric requirements for existence to achieve self-aware coherence across all scales of observation.