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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

Verification: experiments/cgm_alpha_analysis.py (base and correction chain). Companion operator evaluation: experiments/hqvm_corrections_analysis_1.py.

Abstract

We derive the fine-structure constant α 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, with δ_BU = 4 · arctan(k(π/4) · k(m_a)), we apply three systematic corrections that account for UV-IR transport, commutator mapping between foci, and residual alignment. The curvature correction reduces the base residual from about 319.43 ppm to about 0.086 ppm relative to CODATA 2018; the two higher-order terms leave a final residual of about 33.7 ppb. All parameters are geometric invariants from the CGM framework with no fitted values.

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 = 4 · arctan(k(π/4) · k(m_a)) with k(β) = β/(1 + √(1 − β²)), the BU dual-pole loop angle
  • m_a = 1/(2√(2π)) ≈ 0.199471140201, the observational aperture parameter (exact)

Evaluated at working precision, δ_BU ≈ 0.195342178258 and α₀ ≈ 0.007299683573. Relative to CODATA 2018 (α = 1/137.035999084 ≈ 0.007297352569), the base residual is about +319.43 ppm.

2.3 Aperture Structure

With ρ = δ_BU/m_a ≈ 0.979300454497 and

Δ = 1 − ρ = 1 − δ_BU/m_a ≈ 0.020699545503 (2)

the system maintains about 97.93% closure with 2.07% aperture. 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)R Δ²] (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. The residual falls from about 319.43 ppm to about 0.086 ppm versus CODATA 2018 (α₁ ≈ 0.007297353195).

3.2 Commutator Transport UV→IR

The second correction encodes commutator mapping between foci:

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

where:

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

This term captures how UV commutator structure manifests at the IR focus through geometric projection. The residual moves to about 0.033 ppm (α₂ ≈ 0.007297352813).

3.3 IR Focus Alignment

The final correction aligns residual mismatch at the IR focus:

α₃ = α₂ × [1 + (1/ρ) diff Δ⁴] (5)

where:

  • ρ = δ_BU/m_a ≈ 0.979300454497: closure fraction
  • diff = φ_SU2 − 3 δ_BU ≈ 0.001874227881: monodromic residue
  • Δ⁴: fourth-order suppression

This ensures coherence at the observable focus after UV-IR transport. The final value is α ≈ 0.007297352815, about 33.7 ppb from CODATA 2018.

4. Complete Formula and Results

The complete formula incorporating all foci corrections:

α = (δ_BU⁴/m_a) × [1 − (3/4)R Δ²] × [1 − (5/6)((φ_SU2/(3δ_BU)) − 1) Δ²/(4π√3)] × [1 + (1/ρ) diff Δ⁴] (6)

with R = 0.993434896272 and diff = φ_SU2 − 3 δ_BU, all evaluated from the closed-form δ_BU of Section 2.2.

Results versus CODATA 2018 (α = 1/137.035999084 ≈ 0.007297352569):

  • CGM prediction: α ≈ 0.007297352815
  • Residual: about 33.7 ppb

Error reduction sequence:

  • Base (IR focus): about 319.43 ppm
  • After UV-IR curvature: about 0.086 ppm
  • After commutator transport: about 0.033 ppm
  • After IR alignment: about 33.7 ppb

The dominant reduction is the curvature term. The commutator and IR-alignment factors are higher-order adjustments in Δ; under the closed-form loop angle they leave a residual of tens of ppb rather than a sub-ppb identity with a single experimental synthesis.

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. Commutator Mapping: The holographic factor encodes how UV commutator 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 near 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. Geometric parameters: δ_BU, m_a, ρ, Δ, φ_SU2, and R are fixed by CGM geometry, not fitted to α
  2. Systematic structure: Corrections follow UV→IR transport logic as an expansion in the aperture gap Δ
  3. Foci consistency: The base sits at the IR focus; corrections encode transport from the UV focus
  4. No free parameters tuned to α: The formula is completely determined once the geometric inputs are fixed

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 near α ≈ 1/137.036 emerges from:

  • 97.93% closure at the IR focus
  • Geometric transport between UV and IR foci
  • Commutator 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.