The Koide Formula as a Resolution Geometry Theorem: Deriving the Lepton Mass Relation from Dimensional Projection

This companion paper demonstrates that the Koide formula—an unexplained empirical relationship between charged lepton masses discovered in 1981—is not numerological coincidence but geometric inevitability. It is the unique equilibrium condition for fermionic identity projected from a 3D fold onto a 2D scaffold. The mysterious factor of 2/3 in the Koide relation is derived as 〈sin²θ〉, the average projection factor for an isotropic 3D vector onto a 2D surface. This value emerges from pure geometry, requiring no fitted parameters. We show that: (1) the three lepton generations correspond to the three orthogonal pinning axes available in a 3D fold; (2) mass scales as the square of amplitude because the holographic scaffold charges for area while identity extends linearly; and (3) the Koide ratio is the fixed point of dimensional projection equilibrium—any deviation is unstable. The derivation matches observation to within 0.1% (predicted: 0.6667; observed: 0.6674), providing stronger precision than the framework's Dark Matter prediction (<1% error). This completes the Resolution Geometry derivation sequence, demonstrating that Black-Scholes (finance), the Heat Equation (thermodynamics), Schrödinger (quantum mechanics), and now Koide (particle masses) are expressions of the same constraint geometry—distinguished only by their ledger type and projection dimension.