Deriving Koide's Ratio from a Self-Generating Quarter-Turn Relay

Koide's ratio $Q=2/3$, an unexplained numerical coincidence since 1983, is derived as an exact spectral identity of a four-axis relay. The relay follows from results derived in the companion paper~\cite{FoundationPaper} from the Bekenstein bound, Shannon--Nyquist sampling, and the empirical locality of interaction: a bandlimited continuum is information-theoretically identical to a lattice of cells; the state of the cells is a fixed point of their mutual renewal with no external sequencer; each cell carries the complete data from which every field is interpolated; no cell or axis is privileged; and the structure closes with no redundant part. From these results follows a corollary chain: quarter-turn renewal; non-abelian mutual action; the quaternion group $Q_8$; the quaternionic carrier $\mathbb{H}=\mathbb{R}^4$; three frames; and a unique relay coin. The coin's two-hop turn statistics---$1/2$ to stay in a frame, $1/4$ to each other---fix the turn amplitude to $|b|/a=1/\sqrt2$. Frame covariance then forces the mass-root operator on the three generations to be the Hermitian circulant $M=aI+bC+b^*C^\dagger$, and its spectral ratio is\[Q=\frac{\operatorname{tr}M^2}{(\operatorname{tr}M)^2}=\frac23\]exactly, with no free parameter. The relation among the three charged-lepton masses is the spectral ratio of the mass-root operator. The mass-root operator's entries are the turn amplitudes of the relay coin; its eigenvalues are the mass roots. The identity $Q=2/3$ follows from the turn statistics alone. The frame anticommutation additionally supplies the fermionic lift: the Jordan--Wigner second quantization of the three frames gives sector traces $(0,\tau,\tau+s,s)$ and shell factor $s=3a^2$. No measured value enters any derivation.