One Crossing, Three Phases: A Structural Derivation of the Standard Model Gauge Group from the Bridge Equation

Abstract We derive the Standard Model gauge group U(1) × SU(2) × SU(3) from a single structural principle: one crossing from relational structure to measurable magnitude, realized in three algebraically necessary phases. Three theorems establish the uniqueness of each gauge group from the conjugation progression—trivial, pseudoreal, genuinely complex—within the Cartan classification. Three independent obstructions (Frobenius, Hurwitz, Albert) prohibit a fourth phase, explaining both the gauge group and the dimensionality of space. A preservation rule and minimality postulate derive the complete fermion content—six types per generation, no particles missing, none extra—and anomaly cancellation uniquely fixes every hypercharge without Grand Unified Theory embedding. The framework yields sin²θW = 3/8 at unification, running to 0.231 at MZ (measured: 0.23122 ± 0.00003). The spectral determinant det(DF) is shown to be exactly MR-independent, and Newton’s identities extract this invariant from the Seeley–DeWitt coefficients of the spectral action. Six falsifiable predictions follow, all currently consistent with data.