THE SPIRAL ENGINE OF ELECTROMECHANICAL TO ELECTROMAGNETIC ENERGY

The golden angle (360°/φ² = 137.508°) is coprime to 360 at integer resolution: advancing by 137° visits every position on a 360-point circle before repeating, completing exactly 137 revolutions. Each revolution constitutes one cycle of electromagnetic field energy. 137 revolutions constitute one proton. The energy scales as N² (inductor geometry), giving E = m(α⁻¹)² = mc². The fractional remainder (0.508°) never closes. It precipitates as mass. Hydrogen is the remainder. The spiral's first turn covers 38.2% of the circle (φ⁻²) and leaves a 61.8% gap (φ⁻¹). This partition generates a four-way decomposition of unity (φ⁻¹ + φ⁻³ + φ⁻⁵ + φ⁻⁶ = 1) that maps onto the cosmic energy budget: dark energy, dark matter, transition component, and baryonic matter. The periodic table terminates at Z = 137 because the bundle count (protons in the nucleus) meets the coil count (revolutions inside each proton). Maxwell's two vacuum constants (ε₀, μ₀) reduce to one: both are functions of α. His formula 1/√(ε₀μ₀) = c is the energy-to-mass ratio of the spiral, E = mc² in embryonic form, forty years before Einstein. Companion paper to Solving Alpha, Light Has No Speed, and The Constants Are Theorems. Part of the Primary Coherence Theory series. Zero free parameters throughout.