A Geometric Origin of the Koide Relation: Lepton Generations and Mass Hierarchy from a Degree-Three Monopole Bundle

Abstract

The charged-lepton masses satisfy Koide’s relation K = 2/3 to five decimal places — for four decades an unexplained regularity. We present a geometric origin. The three generations are the complete zero-mode kernel of the Dirac operator on a degree-three monopole bundle over an internal two-sphere, realized as the symmetric square of a single spinorial doublet whose antiperiodicity is forced, not chosen. A reality structure (Θ2 = +1) selects a coherent vacuum that locks the amplitude ratio to √ 2 — Koide’s relation — while transport around three symmetry-selected seats fixes the orientation angle to exactly −2/9, with stiffness given by the Fubini–Study metric and a unit winding source. The resulting mass ratios 1 : 206.770 : 3477.5 match measurement to four significant figures with only an overall scale free; the electron’s lightness becomes computed destructive interference, its seat 2.27◦ from an exact null. Six-dimensional consistency yields the Standard Model’s handedness pattern as an unaimed byproduct, and ties the generation count and the four-dimensional anomaly account to the same flux integer. Three postulates and one interpretive import are declared up front; the running-to-pole scheme question remains open, shared with all approaches to this observable.