The Neutrino Mixing Angles Are Geometric

Closed-form expressions for all four parameters of the PMNS neutrino mixing matrix, derived from three quantities established within the Pentagon Physics programme: the discriminant Δ = 5 of the self-referential axiom σ = 1/(1+σ), the generation count 3 (from Koide stability), and φ = (1+√5)/2. Predictions (zero free parameters): sin²θ₁₃ = 1/45 = 1/(5 × 3²) → 0.02222 (experiment: 0.02220 ± 0.00070, 0.05σ) sin²θ₁₂ = 2φ⁴/45 → 0.30463 (experiment: 0.304 ± 0.012, 0.16σ) sin²θ₂₃ = 3/(2φ²) → 0.57295 (experiment: 0.570 ± 0.008, 0.37σ) δ_PMNS = π + π/(5φ²) = π + δ_CKM/5 → 193.75° (experiment: 195° ± 40°, 0.03σ) The CKM–PMNS CP phase relationship δ_PMNS = π + δ_CKM/5 is the first quantitative link between quark and lepton CP violation in any framework. Ablation: 896 candidate expressions tested. The reactor and solar angle formulae are uniquely identified within the grammar. Monte Carlo false-positive rate: 1.7%. Five explicit kill conditions with named experiments: JUNO (reactor and solar angles), DUNE (CP phase and atmospheric octant), Hyper-Kamiokande (atmospheric angle). Falsification thresholds specified at 3σ. Part of the Pentagon Physics programme. ORCID: 0009-0009-6175-4408. Keywords: neutrino mixing, PMNS matrix, Pentagon Physics, CP violation, JUNO, DUNE