The Geometry of the Koide Relation: Null Cone, Spinor Square, and a Ring of Three

The charged-lepton masses obey Koide's relation K = 2/3, and in Brannen's parametrization their remaining freedom is a single phase theta, measured close to -2/9. We give both numbers exact geometric meanings. The relation K = 2/3 says that the vector of square-root masses lies on the null cone of a quadratic form of signature (1,2), a light cone in the mathematical sense, whose axis is the direction of equal masses. This cone lies exactly between the two natural cones of the positive octant, and it is the same 45-degree cone as Descartes' circle theorem, one dimension lower. Positive masses occupy three arcs of 30 degrees on it, and the electron is light because the leptons sit 2.27 degrees from the end of one arc. The cone is the set of squares of real two-component spinors: the radius sqrt(2) of Brannen's form is the null condition, and the phase is twice the spinor angle. Mixtures and quantum spreads lie strictly inside the cone, and in the hyperbolic plane each Koide triple is a horocycle. The phase 2/9 is the effective resistance between neighbouring seats of a ring of three, and the aspect of a squared rectangle. A bridge theorem joins the two halves: the resistance equals a difference of squared spinor path lengths, 2/9 = (1/2)^2 - (1/6)^2. The number three is special in three independent ways, one of which explains why the phase equals K/3. With current data, K - 2/3 = (-2.2 +/- 5.1) x 10^-6 and theta + 2/9 = (-2.5 +/- 6.3) x 10^-6. The paper is purely geometric; a physical mechanism is developed in a companion paper.