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The Fine-Structure Constant from Geometric First Principles

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

Abstract

We derive the fine-structure constant α ≈ 1/137.036 from the geometric structure of the Common Governance Model (CGM). The derivation uses the optical conjugacy relation between UV and IR foci, with CS as the unobservable UV focus and BU as the observable IR focus where electromagnetic interactions manifest. Starting from the base formula α₀ = δ_BU⁴/ m_a at the IR focus, we apply three systematic corrections that account for UV-IR transport, holonomy mapping between foci, and residual alignment. Each correction reduces error by orders of magnitude, yielding a final prediction accurate to 0.043 parts per billion. All parameters are measured geometric invariants from the CGM framework with no fitted values. This demonstrates that fundamental constants emerge from the geometric requirements of observation itself.

1. Introduction

The fine-structure constant α ≈ 1/137.036 governs electromagnetic interaction strength throughout quantum electrodynamics. Despite its fundamental importance, α has historically been treated as an empirical parameter rather than a derivable quantity. Previous attempts at derivation have not achieved experimental precision.

The Common Governance Model provides a new approach through its identification of Quantum Gravity as the geometric invariant Q_G = 4π, representing the complete solid angle required for coherent observation. Within CGM's four-stage structure (CS, UNA, ONA, BU), the optical conjugacy relation E^UV × E^IR = const connects high-energy physics at the CS focus with low-energy observable physics at the BU focus. We demonstrate that α emerges at the BU (IR) focus through geometric corrections accounting for transport from the UV focus.

2. Theoretical Foundation

2.1 UV-IR Foci Structure

The CGM framework establishes:

  • CS (Common Source): UV focus, unobservable, hosts high-energy physics
  • BU (Balance Universal): IR focus, observable, hosts electromagnetic phenomena
  • Optical Conjugacy: E_i^UV × E_i^IR = (E_CS × E_EW)/(4π²)

The fine-structure constant characterizes electromagnetic coupling at the observable BU focus.

2.2 Base Formula at IR Focus

The fundamental expression for α at the BU focus is:

α₀ = δ_BU⁴/ m_a (1)

where:

  • δ_BU = 0.195342176580 rad is the BU dual-pole monodromy (measured)
  • m_a = 1/(2√(2π)) = 0.199471140201 is the observational aperture parameter (exact)

The quartic scaling reflects electromagnetic interaction geometry, while normalization by m_a ensures observational coherence. This yields α₀ = 0.007299683322, differing from experiment by 319.398 ppm.

2.3 Aperture Structure

The system maintains 97.93% closure with 2.07% aperture:

Δ = 1 - δ_BU/ m_a = 0.020699553913 (2)

This aperture gap enables observation and serves as the expansion parameter for corrections.

3. Systematic Corrections via Foci Transport

3.1 UV-IR Curvature Correction

The first correction accounts for curvature between UV and IR foci:

α₁ = α₀ × [1 - (3/4)RΔ²] (3)

where:

  • 3/4 is the exact SU(2) Casimir invariant
  • R = 0.993434896272 is the measured Thomas-Wigner curvature ratio
  • Δ² represents quadratic aperture effects

The curvature R = (F̄/π)/ m_a with F̄ = 0.622543 measured at canonical thresholds. This correction captures how geometric transport from UV to IR focus modifies the coupling. Error reduces from 319.398 ppm to 0.052 ppm.

3.2 Holonomy Transport UV→IR

The second correction encodes holonomy mapping between foci:

α₂ = α₁ × [1 - (5/6)((φ_SU2/(3δ_BU)) - 1)(1 - Δ²h_ratio)Δ²/(4π√3)] (4)

where:

  • 5/6: Z₆ rotor with one leg open (aperture)
  • φ_SU2 = 2arccos((1 + 2√2)/4): exact SU(2) holonomy
  • h_ratio = 4.417034: measured 4-leg/8-leg holonomy ratio
  • 4π: complete solid angle (Q_G)
  • √3: 120° rotor geometry projection factor

This term captures how UV holonomy structure manifests at the IR focus through geometric projection. Error reduces to -0.000379 ppm.

3.3 IR Focus Alignment

The final correction aligns residual mismatch at the IR focus:

α₃ = α₂ × [1 + (1/ρ)diffΔ⁴] (5)

where:

  • ρ = δ_BU/ m_a = 0.979300: closure fraction
  • diff = φ_SU2 - 3δ_BU = 0.001874: monodromic residue
  • Δ⁴: fourth-order suppression

This ensures coherence at the observable focus after UV-IR transport. Final error: 0.043 ppb.

4. Complete Formula and Results

The complete formula incorporating all foci corrections:

α = (δ_BU⁴/m_a) × [1 - (3/4)RΔ²] × [1 - (5/6)((φ_SU2/(3δ_BU)) - 1)(1 - Δ²h_ratio)Δ²/(4π√3)] × [1 + (1/ρ)diffΔ⁴] (6)

Results:

  • CGM prediction: α = 0.007297352563
  • Experimental value: α = 0.007297352563
  • Error: 0.043 ppb (0.532 × experimental uncertainty)

Error reduction sequence:

  • Base (IR focus): 319.398 ppm
  • After UV-IR curvature: 0.052 ppm
  • After holonomy transport: -0.000379 ppm
  • After IR alignment: 0.043 ppb

5. Physical Interpretation

The derivation reveals α as emerging from the UV-IR foci structure:

  1. IR Focus Geometry: Base term δ_BU⁴/ m_a represents pure electromagnetic coupling at the observable BU focus.

  2. UV-IR Transport: Curvature correction accounts for geometric transport between unobservable UV (CS) and observable IR (BU) foci.

  3. Holonomy Mapping: Holographic factor encodes how UV holonomy structure projects onto IR observables through 4π solid angle.

  4. Focus Coherence: Final correction ensures geometric coherence at the IR focus after incorporating UV influences.

Within CGM, the value 1/137.036 thus emerges from the geometric requirements for electromagnetic phenomena to manifest at the observable focus while maintaining consistency with the unobservable UV origin.

6. Validation

The derivation's validity rests on:

  1. Measured parameters: All values are measured from CGM geometry, not fitted to α
  2. Systematic convergence: Each correction reduces error by orders of magnitude
  3. Foci consistency: Corrections follow UV→IR transport logic
  4. No free parameters: Complete determination from geometric structure

7. Conclusion

We have derived the fine-structure constant from the geometric requirements of observation in the CGM framework. The key insight is that α characterizes electromagnetic coupling at the observable IR focus (BU stage) with corrections accounting for transport from the unobservable UV focus (CS stage).

The optical conjugacy relation E^UV × E^IR = const provides the framework for understanding how high-energy geometric structure manifests in low-energy observables. The specific value α ≈ 1/137.036 emerges from:

  • 97.93% closure at the IR focus
  • Geometric transport between UV and IR foci
  • Holonomy projection through 4π steradians
  • Coherence requirements for observation

This demonstrates that within CGM, fundamental constants emerge from geometric requirements for observational coherence. The success suggests other constants may similarly arise from the UV-IR foci structure of the CGM framework.