A parameter-free relation between the solar and Cabibbo mixing angles

The Standard Model of particle physics contains two large mixing angles that relate the quantum states of neutrinos: the solar angle (θ₁₂ ≈ 33.9°) and — across the quark–lepton divide — the Cabibbo angle (θ_C ≈ 13.0°). No existing theoretical framework predicts either angle from first principles without continuous adjustable parameters. This letter shows that the index-two subfactor — a minimal algebraic structure from which the discrete gauge architecture of the Standard Model is derived in companion papers I–IV — carries a two-sided closure structure whose forced amplitudes are 9/20 and 1/20. A single declared identification (that this graded pair reads as one lepton and one quark mixing angle) locks three consequences with no adjustable parameter: tan²θ₁₂ = 9/20, sin²θ_C = 1/20, and the exact covariant cross-sector relation tan²θ₁₂ = 9 sin²θ_C. Present data satisfy all three (0.3σ, 0.9σ, and 0.4σ respectively). The distribution of the pair to the correct sectors is structurally motivated by charge conservation under the modular conjugation; the functional forms (tan² for the Majorana sector, sin² for the Dirac) follow the native invariants of the respective state-space geometries. Both supporting arguments are stated as motivated rather than derived. The claim is falsifiable three independent ways: (F1) a factor 2–3 improvement in |V_us| precision discriminates the prediction; (F2) the identification requires neutrinos to be Majorana; (F3) it requires a hierarchical neutrino mass spectrum. Companion paper IV (the exhaustion theorem) proves that this pair is the complete forced flavour content of the minimal structure — no further mixing angles are derivable from first principles in this framework.