Eigenmode Ratios of the Minimal Quantum State Space and the Standard Model

The minimal quantum system — one binary degree of freedom — has pure state space S² = CP¹ and symmetry group SU (2) = S³, connected by the Hopf fibration. The eigenmode spectra of S² and S³ are completely determined by the spectral theorem, with zero adjustable parameters. We show that dimensionless ratios formed from the S² eigenmode degeneracies produce the Standard Model gauge group SU (3) × SU (2) × U (1), 12 gauge bosons, the SU (5) grand unification structure, the Standard Model hypercharge assignments, and the GUT-scale Weinberg angle sin²θW = 3/8 — all from eigenmode counting, with each step being a theorem of mathematics given the identification of eigenmode degeneracies with gauge representation dimensions. The Clifford algebra of the l = 2 eigenspace produces the chiral fermion content of one Standard Model generation (5̄⊕ 10 of SU (5), plus a right-handed neutrino), with chirality selected by the derived complex structure, anomaly cancellation forced by d₂ = 5, and three generations from the Hopf pullback dimension dim π*V₁ (S²) = 3. The Hopf pullback from S² to S³ induces a canonical splitting of the S³ eigenspace that derives the PMNS solar neutrino mixing angle sin²θ₁₂ = 4/13 via the same trace formula. The only inputs are the algebraic structure of quantum mechanics (empirical) and the completeness of binary observations. The gauge groups emerge from eigenmode geometry (theorem level for U (1), geometrically determined for SU (3) and SU (2) ), and Yang-Mills dynamics follow from topology, quantization, and derived physical constraints (strong physical argument, not theorem). The identification of canonical geometric structures with physical gauge fields involves interpretive steps that are explicitly assessed. The Born rule, Minkowski metric, and Koide lepton mass formula are shown to trace to the same algebraic inner product on the observable algebra. We report these results alongside explicit negative findings.