Energy Scale Structure in the Common Governance Model: A Geometric Approach to Unification

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

Abstract

We present a systematic analysis of energy scales emerging from the Common Governance Model (CGM), a geometric framework that derives physical scales from fundamental structural principles. The framework establishes a hierarchy of energy scales through four stages (CS, UNA, ONA, BU) characterized by specific geometric thresholds. A central result is the optical conjugacy relation E_i^UV × E_i^IR = (E_CS × E_EW)/(4π)² that connects ultraviolet and infrared physics through a single invariant. We demonstrate how this geometric structure yields testable predictions including neutrino masses of approximately 0.06 eV through 48² quantization, proton lifetime of order 10^43 years, and a characteristic electromagnetic duality angle of 48°. The analysis employs standard renormalization group methods to verify consistency with gauge coupling evolution while maintaining clear distinction between dimensionless geometric ratios and dimensionful physical scales.

1. Introduction

The unification of fundamental forces remains a central challenge in theoretical physics. While the Standard Model successfully describes electromagnetic, weak, and strong interactions up to energies of order 10² GeV, the incorporation of gravity and the explanation of hierarchical structures require physics beyond current frameworks. Grand Unified Theories (GUTs) suggest unification at scales of approximately 10^16 GeV, while quantum gravity effects are expected near the Planck scale of 10^19 GeV.

The Common Governance Model (CGM) provides an alternative approach based on geometric principles rather than field-theoretic unification. The framework posits that energy scales emerge from a recursive geometric structure characterized by four stages, each with specific threshold values derived from fundamental angular relationships. This analysis examines the energy scale predictions of CGM and their consistency with observed physics.

2. Theoretical Framework

2.1 Stage Structure and Thresholds

The CGM framework identifies four stages of geometric evolution, each characterized by a threshold parameter:

CS (Common Source): s_p = π/2 [dimensionless]
UNA (Unity Non-Absolute): u_p = cos(π/4) = 1/√2 [dimensionless]
ONA (Opposition Non-Absolute): o_p = π/4 [dimensionless]
BU (Balance Universal): m_a = 1/(2√(2π)) ≈ 0.1995 [dimensionless]

The parameter m_a serves as the fundamental aperture parameter, governing the relationship between different stages.

2.2 Action Mapping

From these thresholds, we derive stage actions through the mapping:

S_CS = s_p /  m_a ≈ 7.875
S_UNA = u_p /  m_a ≈ 3.545
S_ONA = o_p /  m_a ≈ 3.937
S_BU =  m_a ≈ 0.199

All actions are dimensionless. The BU stage serves as a fixed point where S_BU = m_a, while other stages scale inversely with m_a.

2.3 GUT Action Construction

The GUT action emerges from treating UNA and ONA as parallel constraints with CS memory:

1/S_GUT = 1/S_UNA + 1/S_ONA + η/S_CS

where η represents the CS memory weight. For η = 1, we obtain S_GUT ≈ 1.508, yielding S_GUT/S_CS ≈ 0.192.

3. Energy Scale Derivation

3.1 Dimensionless Energy Ratios

Before introducing dimensional scales, the framework yields the following dimensionless energy ratios:

E_UNA/E_CS = 0.450158
E_ONA/E_CS = 0.500000
E_BU/E_CS = 0.025330
E_GUT/E_CS = 0.191518

These ratios are geometric invariants independent of the choice of units.

3.2 UV Energy Ladder

To obtain physical energy scales, we anchor the CS stage to the Planck scale:

E_CS = 1.22 × 10^19 GeV (Planck scale)

This yields:

E_UNA = 5.50 × 10^18 GeV
E_ONA = 6.10 × 10^18 GeV
E_GUT = 2.34 × 10^18 GeV
E_BU = 3.09 × 10^17 GeV

3.3 Optical Conjugacy and IR Scales

The framework's central result is the optical conjugacy relation:

E_i^UV × E_i^IR = (E_CS × E_BU)/(4π²)

where E_BU = 246.22 GeV is the Higgs vacuum expectation value (electroweak scale). This invariant holds for all stages i ∈ {CS, UNA, ONA, BU, GUT}, yielding:

Invariant K = 7.61 × 10^19 GeV²

The IR energy scales follow:

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

4. Physical Predictions

4.1 Neutrino Masses

Through 48² quantization of the GUT scale:

M_R = E_GUT / 48² = 1.01 × 10^15 GeV

The type-I seesaw mechanism yields:

m_ν = y² v² / M_R

For Yukawa coupling y ≈ 1, this gives m_ν ≈ 0.06 eV, consistent with observed neutrino oscillation data.

4.2 Proton Lifetime

Using the geometric GUT scale E_GUT = 2.34 × 10^18 GeV and coupling α_GUT^-1 ≈ 42, the estimated proton lifetime through dimension-6 operators is:

τ_p ≈ 8.6 × 10^43 years

This prediction greatly exceeds current experimental lower bounds (~10^34 years),
implying that proton decay is effectively unobservable in present or near-future experiments.
The result is therefore consistent with the absence of proton decay events to date.

4.3 Electromagnetic Duality Angle

The ratio of ONA to UNA actions yields:

θ = arctan(S_ONA/S_UNA) = 48.0°

This angle characterizes the electromagnetic duality rotation in the framework.

4.4 The Non-Observability of Sterile Neutrinos

4.4.1 Theoretical Basis

Within the CGM framework, sterile neutrinos with Majorana masses M_R ≈ 10^15 GeV belong fundamentally to the CS (Common Source) domain. CS itself is three-fold, as chirality requires a minimum of three reference points to establish non-commutativity. This three-fold structure at CS manifests as the three sterile neutrino families.

The CS domain is unobservable by principle. As the UV focus of the optical conjugacy relation, CS cannot manifest directly in observations, which occur at the BU (IR) focus. The sterile neutrinos therefore exist entirely within the unobservable CS domain, never appearing at UNA, ONA, or BU stages.

4.4.2 Direct versus Indirect Observability

The framework makes a precise distinction:

Direct observation impossible: The framework predicts sterile neutrinos cannot be detected as propagating particles at any observable stage. They remain forever confined to the unobservable CS focus.

Indirect effects observable: Their presence manifests only through:

Generation of light neutrino masses via the seesaw mechanism
Potential gravitational imprints from a primordial sterile neutrino background
The effective Weinberg operator (LH)(LH)/M_R appearing at low energies

The computed active-sterile mixing angles (θ₂ ≈ 10^-14, θ₃ ≈ 10^-13) confirm this extreme decoupling, preserving CS hiddenness.

4.4.3 Resolution of the Chirality Question

The apparent tension between CS left-bias and right-handed sterile neutrinos resolves through understanding the two-focus structure:

CS left-bias: Determines the order of stage evolution (CS→UNA→ONA→BU) and which focus becomes observable (BU)
Right-handed neutrinos: Are SU(2)_R states residing at the CS focus, consistent with CS hosting the unobservable complement to observable left-handed states
Three-fold CS: The three sterile families reflect the fundamental three-fold nature of CS required for chirality

The CS left-bias does not mean all particles at CS are left-handed. Rather, it ensures that right-handed states at CS remain unobservable while left-handed states at BU become observable.

4.4.4 Cosmological Implications

The framework hypothesizes a primordial sterile neutrino background residing at the CS focus. This background:

Predates the cosmic microwave background
Interacts with the observable sector only gravitationally
Cannot be directly detected by any experiment
May influence large-scale structure formation through gravitational effects alone

We hypothesise that no future cosmological probe (CMB-S4, 21cm, structure surveys) will ever see sterile features. Only indirect gravitational constraints (e.g. effective N_eff) are consistent.

4.4.5 Experimental Predictions and Falsification

This principle yields unambiguous predictions:

Null results guaranteed: All direct searches for sterile neutrinos must yield null results, regardless of energy scale or experimental technique.
Falsification criterion: Any direct observation of sterile neutrinos as propagating particles would immediately falsify the CGM framework.
Indirect signatures only: Sterile neutrino effects can only appear through intermediate mechanisms, never through direct detection.

The continuing null results from experiments worldwide support this geometric principle. These non-observations constitute positive evidence for the structural confinement of sterile neutrinos to the unobservable CS focus.

5. Gauge Structure and Symmetry Breaking

5.1 Gauge Group

The framework suggests a left-right symmetric gauge structure:

G = SU(3)_c × SU(2)_L × SU(2)_R × U(1)_χ

5.2 Breaking Chain

Symmetry breaking proceeds through:

G → SU(3)_c × SU(2)_L × U(1)_Y → SU(3)_c × U(1)_em

with breaking scales v_R ≈ M_R ≈ 10^15 GeV and v_EW = 246.22 GeV (Higgs vev).

5.3 Charge Quantization

Electric charge emerges through:

Q = T_3L + T_3R + (B-L)/2

yielding the standard spectrum Q ∈ {-1, 0, -1/3, +2/3}.

6. Consistency Checks

6.1 Gauge Coupling Evolution

One-loop renormalization group analysis shows:

α_1 = α_2 at μ ≈ 10^13 GeV
α_2 = α_3 at μ ≈ 10^17 GeV

The geometric GUT scale E_GUT = 2.34 × 10^18 GeV lies above these crossing points, consistent with non-supersymmetric GUT expectations.

6.2 Anomaly Cancellation

All gauge anomalies cancel per generation:

[SU(3)]² U(1)_(B-L) = 0
[SU(2)L]² U(1)(B-L) = 0
[SU(2)R]² U(1)(B-L) = 0
[U(1)_(B-L)]³ = 0

6.3 Involution Property

The optical conjugacy satisfies an involution property—applying the transformation twice returns the original energy:

(E^UV → E^IR) → E^UV

with numerical verification showing errors < 10^-15.

7. Discussion

7.1 Interpretation of Results

The optical conjugacy relation E_i^UV × E_i^IR = const/(4π²) suggests a fundamental duality between UV and IR physics. The factor (4π²)^-1 ≈ 0.0253 provides a natural explanation for the apparent weakness of gravity at low energies through geometric dilution.

The 48² quantization factor appearing in neutrino mass generation suggests a deeper connection between the framework's geometric structure and particle physics. The exact factor 48 appears in multiple contexts, including the duality angle and state space dimensions.

7.2 Comparison with Standard Approaches

Unlike conventional GUTs that require fine-tuning or supersymmetry for gauge coupling unification, the CGM framework derives scales from geometric principles. The predicted neutrino masses and proton lifetime fall within experimentally viable ranges without additional parameter adjustment.

7.3 Limitations and Open Questions

Several aspects require further investigation:

The derivation of fermion mass hierarchies beyond neutrinos
The connection to cosmological observables

8. Conclusions

The CGM energy scale analysis demonstrates that a geometric framework based on four fundamental stages can yield a consistent hierarchy of physical scales. The optical conjugacy relation provides a unifying principle connecting UV and IR physics through a single invariant. Key predictions including neutrino masses of order 0.06 eV and proton lifetime of order 10^43 years fall within experimentally viable ranges.

The framework's strength lies in deriving energy ratios from geometric principles rather than empirical fitting. While questions remain regarding the fundamental origin of certain quantization factors, the internal consistency and testable predictions warrant further investigation of this geometric approach to unification.

Appendix: Assumptions and Methodology

A.1 Fundamental Assumptions

Geometric Origin: Energy scales emerge from geometric relationships rather than dynamical mechanisms
Four-Stage Structure: The CS, UNA, ONA, BU hierarchy is fundamental
Aperture Parameter: m_a = 1/(2√(2π)) is treated as a fundamental constant
Anchoring: The CS stage is identified with the Planck scale
Electroweak Scale: E_EW = 246.22 GeV is the Higgs vacuum expectation value v = (√2 G_F)^(-1/2)

A.2 Calculational Methods

Action Derivation: Actions computed as threshold/ m_a ratios (except BU)
Energy Scaling: Energies proportional to actions with single scale factor
Optical Projection: IR energies derived through E^IR = K/E^UV
Seesaw Mechanism: Standard type-I seesaw formula for neutrino masses
RG Evolution: One-loop beta functions for gauge coupling running

A.3 Numerical Precision

All calculations maintain at least 10 significant figures internally, with results reported to appropriate precision based on input uncertainties. The involution property verification achieves machine precision (errors < 10^-15).

A.4 Unit Conventions

Energy scales reported in GeV
Dimensionless quantities explicitly noted
Natural units (ℏ = c = 1) implicit in particle physics calculations
Steradians specified where relevant for solid angle factors

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CGM Duality Parametrisation
1. Topology
  - BASE: the apex scale (CS)
  - UNION: the “middle” (CS–UNA–ONA)
  - SHELL: the observable shell (BU)
2. Perspective
  - BTM (UV-facing, CS focus)
  - TOP (IR-facing, BU focus)
3. Stages Structure:
  1. CS (Common Source): s_p = π/2 [dimensionless]
  2. UNA (Unity Non-Absolute): u_p = cos(π/4) = 1/√2 [dimensionless]
  3. ONA (Opposition Non-Absolute): o_p = π/4 [dimensionless]
  4. BU (Balance Universal): m_a = 1/(2√(2π)) ≈ 0.1995 [dimensionless]
  5. Geometric constant
    - m_a² = 1/(8π)
    - s_p/m_a² = 4π²
4. Action Mapping:
  1. S_CS = s_p / m_a ≈ 7.875
  2. S_UNA = u_p / m_a ≈ 3.545
  3. S_ONA = o_p / m_a ≈ 3.937
  4. S_BU = m_a ≈ 0.199 [identity]
5. Union Formation:
  1. 1/S_UNI = η/S_CS + 1/S_UNA + 1/S_ONA
  2. For η = 1: S_UNI ≈ 1.508, hence S_UNI/S_CS ≈ 0.192
6. Energy Ratios (dimensionless, single-perspective staging):
  1. E_UNA/E_CS = 0.450158
  2. E_ONA/E_CS = 0.500000
  3. E_BU/E_CS = 0.025330
  4. E_UNI/E_CS = 0.191518
7. UV Focus Scales (anchor CS at the Planck scale; BTM ≡ UV):
  1. E_CS^BTM = 1.22 × 10^19 GeV
    - E_UNA^BTM = 5.50 × 10^18 GeV
    - E_ONA^BTM = 6.10 × 10^18 GeV
    - E_BU^BTM = 3.09 × 10^17 GeV
  2. E_UNI^BTM = 2.34 × 10^18 GeV
8. Optical Conjugacy
  1. Invariant (for internal stages i ∈ {CS, UNA, ONA, UNI}; BU is the shell anchor):
    
    E_i^BTM × E_i^TOP = (E_CS^BTM × E_BU^TOP) / (4π²)
  2. Anchors: E_CS^BTM = 1.22 × 10^19 GeV, E_BU^TOP = 246.22 GeV (Higgs vev)
    - Invariant K = 7.61 × 10^19 GeV²
9. IR Focus Scales (TOP ≡ IR; from E_i^TOP = K / E_i^BTM):
  1. E_CS^IR = 6.24 GeV
    - E_UNA^IR = 13.8 GeV
    - E_ONA^IR = 12.5 GeV
    - E_BU^IR = 246.22 GeV (Higgs vacuum expectation value)
  2. E_UNI^IR = 32.6 GeV
10. Union Derivation
  1. Inversion map: E_i^BTM = K / E_i^TOP with K = (E_CS^BTM × E_BU^TOP) / (4π²)
  2. Fixed point (conjugacy centre): E_MID = √K ≈ 8.6 × 10^9 GeV
11. Invariant (equivalent forms)
  1. Stage-independent product: E_i^BTM × E_i^TOP = (E_BASE^BTM × E_SHELL^TOP) / (s_p/m_a²), with BASE = CS, SHELL = BU
  2. Normalised (cross-anchors): (E_i^BTM / E_CS^BTM) × (E_i^TOP / E_BU^TOP) = 1 / (4π²)
  3. Centred (via fixed point): (E_i^BTM / E_MID) × (E_i^TOP / E_MID) = 1
Centre Parametrisation

Define a dimensionless placement for each stage i relative to the fixed point:

ρ_i = E_i^TOP / E_MID

Then automatically:

E_i^BTM = E_MID / ρ_i

The product E_i^TOP × E_i^BTM = E_MID² is independent of i. Moving a stage up at TOP by a factor ρ_i moves it down at BTM by the reciprocal factor.