The PMNS Neutrino Mixing Matrix from Foam Cell Geometry

Three of the four parameters of the PMNS neutrino mixing matrix, plus the neutrino mass-squared ratio, are derived from the geometry of the truncated octahedron. The solar angle is tan²θ₁₂ = √Δ/C_A² = √17/9 = 0.4581, where Δ = 17 is the discriminant of the master equation λ²−9λ+16 = 0 and C_A = 3 is the colour number. This matches the observed value 0.443 ± 0.027 to 3.4% (0.56σ). The atmospheric angle is θ₂₃ = π/4 (maximal mixing), enforced by the exact Z₂ exchange symmetry of the T₁u upper eigenspace. This gives sin²θ₂₃ = 0.500, compared to the observed 0.546 ± 0.021 (2.2σ). The reactor angle is sinθ₁₃ = √Δ/C_A³ = √17/27 = 0.1527, giving sin²θ₁₃ = 0.0233, compared to the observed 0.0220 ± 0.0006 (2.3σ). The mass-squared ratio is |Δm²₃₂|/Δm²₂₁ = 2Δ − 1 = 33, compared to the observed 32.6 ± 0.9 (0.5σ). All four results use only two foam integers: the discriminant Δ = 17 and the colour number C_A = 3. The CP-violating phase δ_CP remains open. Zero free parameters.