Finite Information and the Fixed Point of Self-Renewal: Deriving Koide's Ratio

A universe is the totality of what exists relative to an internal observer. The paper's substantive claim is that this operational definition, when formalized algebraically alongside a single structural postulate (a self-generating phase advance), establishes the framework's algebraic spine. We derive the quaternion algebra $\mathbb H$ and the quaternion group $Q_8$, the carrier rigidity $\mathcal{H}(X_o) \cong \mathbb R^4$, the three frames, and a unique balanced relay coin. The quarter-turn is derived twice: once spectrally from the self-generation postulate, and once by a reductio from the structural zero-diagonal of the coin. The two derivations are not of equal strength: the spectral one is universal over all renewals and is the engine of the algebraic spine, while the interference one is specific to the balanced coin and serves as a structural check. Their convergence on the same angle from disjoint premise sets---one dynamic, one structural---is the quarter-turn's overdetermination. The carrier, the quarter-turn, and the frames are all $B_4$ data: the stencil is the $B_4$ root system, and the coin's axis part is a Clifford unit. We then introduce the physical correspondence principle identifying mass with the coherent intensity of the frame-renewal amplitude. From this correspondence, frame covariance forces the mass-root operator to be a Hermitian circulant. The coin's two-hop turn statistics---\(1/2\) to stay in a frame, \(1/4\) to each other---fix the off-diagonal modulus to \(1/\sqrt2\), yielding Koide's ratio $Q=2/3$ exactly.