The Proton-Electron Mass Ratio from Icosahedral Geometry: Two Paths Through 108

We derive the proton-to-electron mass ratio μ = m_p/m_e from first principles using only dimension D = 3 and icosahedral geometry. The primary formula μ = 6π⁵ + π/90 = 1836.153 matches the observed value to 0.19 ppm. Two independent integer paths yield μ = 1836: the proton path μ = V(V² + D²) and the electron path μ = (D!)² × D(F−D). Both coefficients are now fully derived. The proton formula uses V/D = 4, which counts vertices per orthogonal plane in the golden rectangle construction. The electron formula uses the identity E + F + 1 = D(F − D), which holds only for the icosahedron among Platonic solids. The number 108 = H(D) = 1¹ × 2² × 3³ serves as the bridge through which both particles form; its Euler totient φ(108) = 36 is the electron factor, while its divisor count d(108) = 12 equals the proton's vertex number V. Proton and electron are complementary aspects of a single geometric structure — inside (divisors) and outside (coprimes) of 108.