Topological Mass Generation in a Prime-Partitioned Vacuum: The Lepton-Higgs Spectrum from First Principles

We present a parameter-free derivation of the lepton mass hierarchy and Higgs boson mass from the topology of a prime-indexed vacuum manifold. By treating the vacuum as a discrete substrate in logarithmic coordinate u = ln x (where x is the prime index), we demonstrate that stationary mass resonances occur at prime nodes where radial amplitude and angular phase achieve impedance-matched “Golden Lock” coupling. The Golden RatioΦ ≈ 1.618 emerges as the unique stability eigenvalue from the characteristic equation r2 −r −1 = 0. Dodecahedral symmetry at node Z3 (Prime 5) imposes a geometric correction γ = 5/4, and the topological winding number k = 3n is derived from phase-closure requirements in the 9D manifold via the Master Euler-Lagrange equation. The complete mass-scaling law: mnme= 5 4 · xn5 · Φ3npredicts particle masses with extraordinary agreement: muon (99.2%), tau (97.3%), and Higgs boson (99.9%)—with zero adjustable parameters. The Higgs emerges at Z432 (Prime 3037) as the n = 4 quadruple-winding structural resonance, dissolving the hierarchy problem and unifying electroweak symmetry breaking with lepton mass generation through a single geometric principle: topological phase-winding in a prime-partitioned manifold.