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/CA² = √17/9 = 0. 4581, where Δ = 17 is the discriminant of the master equation λ²−9λ+16 = 0 and CA = 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/CA³ = √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 CA = 3. The CP-violating phase δCP remains open. Zero free parameters.