Sommerfeld's Constant from Coherence Geometry: A Swarm Theory Derivation

Abstract

This paper derives the fine-structure constant (α) from coherence geometry within Swarm Theory. In conventional physics, α is treated as a dimensionless coupling constant with no known origin. Here, α emerges as a suppression factor from lattice geometry, coherence projection, and helical wave structure.

Key results:

• Geometric origin: α is revealed as a ratio arising from wave projection across lattice membranes, not a fundamental input.
• Numerical match: The derivation reproduces α = 1/137.035999… in exact agreement with CODATA, using the same calibrated parameters that yield Planck’s constant (h).
• Implication: Electromagnetic coupling strength is shown to be a geometric property of coherence, not an arbitrary constant.

This result places α alongside h, e, ε₀, μ₀, and Z₀ as a dependent consequence of lattice coherence, completing the electromagnetic scale in Swarm Theory.