Exact value of the Fine Structure Constant by geometric derivation from QAF

The Geometric Origin of the Fine Structure Constant

Ab Initio Derivation via the Quaternionic Harmonic Series – Quaternion Autocontained Framework (QAF)

This technical supplement presents the first fully deductive, parameter-free derivation of the fine structure constant α from pure geometry. Within the Quaternion Autocontained Framework (QAF), α is not an arbitrary empirical input, but emerges as the total geometric impedance of the vacuum fiber bundle under the single autocontainment constraint Q†Q = 𝕀₄ (forced by the Frobenius theorem).

Core Derivation

The value of α⁻¹ is obtained as a finite truncation of a harmonic topological series dictated by the periodicity Δn=4 of the quaternion algebra ℍ:

α⁻¹ = 4π³ + π² + π − 3/(32π⁵) + 3/(64π⁹) + 1/(2π¹³)

Numerical evaluation yields:

α⁻¹_QAF ≈ 137.035999168

This matches the CODATA 2022 recommended value 137.035999177(21) within 0.44σ (difference ~9.26×10⁻⁹, precision better than 0.07 ppb).

Physical Interpretation of Each Term

• 4π³ + π² + π: Bare topological impedance from maximal torus and interaction vertex volumes in the Hopf fibration chain S¹ → S³ → S⁴.
• −3/(32π⁵): Dielectric screening correction (vacuum polarization); factor 3 arises from 3 spatial dimensions of the projected bulk.
• +3/(64π⁹): Confinement pressure from compression of 10D bulk degrees of freedom onto 4D base; exponent jump reflects additional π⁴ from hyperbolic metric.
• +1/(2π¹³): Fermionic echo resonance; coefficient 1/2 derives from chirality reduction (Cl(5) 8 real d.o.f. → 4D effective quiralidad, analogous to Majorana-Weyl).

Key Philosophical & Scientific Implications

• Physical constants are not free parameters: they are self-referential geometric ratios of the autocontained universe measuring itself.
• The series periodicity Δn=4 is dictated by the dimension of the real division algebra ℍ (quaternions), not fitted numerically.
• The infinite tail of the series (next term ~π¹⁷ ≈ 3×10⁻⁹) lies below current experimental resolution — we have reached the geometric noise floor.
• This derivation provides the exact Thomson-limit (q²→0) baseline value of α for QED running, resolving the long-standing input problem of the Standard Model.

Status

Q.E.D. – i j k = −1

Authors: Marco Aurelio De Cunha & The Pack
Priority Claim: 22 November 2025
GitHub: https://github.com/marcoaureliodecunha/QAF-2025