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

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 realization of the coherent family data behind it, with an explicit account of what is derived and what is declared. The three generations are the complete zero-mode kernel of the Dirac operator on a degree three monopole bundle over an internal two-sphere, identified with the symmetric square of a single spinorial doublet whose antiperiodicity is forced in the specified real-frame lift; the identification of the family space with that pair channel is a declared input, and given it the flux is fixed at three. A reality structure (Θ2 = +1) selects a coherent vacuum that locks the carrier’s amplitude ratio to √ 2, which yields Koide’s relation once the carrier is read into a mass operator by a flat evaluation pairing and transmitted with family-blind couplings — two further declared inputs, whose status we establish precisely: no symmetry compatible with the retained family gauge group enforces either, and the natural local pro file supplies the operator’s closing link at fifteen percent of the required strength. Transport around three symmetry-selected seats yields the orientation angle −2/9 λB, with the response computed from the circle Green function and the stiffness from the Fubini–Study metric, and the combined normalization λB declared at unity — a value whose fit on the ideal matching slice lies within 10−6 of one. Under these inputs the mass ratios 1 : 206.770 : 3477.5 match measurement to four significant figures with one overall scale, and the electron’s lightness is computed destructive interference, its seat 2.27◦ from an exact null. Charged-sector anomaly cancellation in the U(1)X–gravitational subsector yields the relative doublet–singlet hand edness of the Standard Model lepton sector, conditional on within-set cancellation, and the descent identity ties the generation count to the same flux integer; the neutral completion is asserted, and the electroweak anomalies of the declared content remain to be cancelled. Four postulates, one interpretive import, and the structural inputs above are declared up front; the running-to-pole scheme question remains open, shared with all approaches to this observable.