Lepton Mass Ratios from the Face Laplacian Spectrum

The Koide formula Q= (mₑ+m_μ+m_τ) / (√mₑ+√m_μ+√m_τ) ²=2/3, accurate to six significant figures since 1982 with no theoretical derivation, is proved as a theorem from the face Laplacian spectrum of the truncated octahedron (Kelvin cell). The Koide parameter Q=2/3 equals (λA₂u−λT₂g) /λA₂u = 2/9 as a spectral angle, and the value 2/3 follows as a topological invariant of closed T₂g torsion loops. Predicted mass ratios: m_μ/mₑ=206. 7 (observed 206. 768, 10 ppm) and m_τ/m_μ=16. 82 (observed 16. 817, 70 ppm). All three predictions share the same discriminant √17 from the T₁u characteristic equation λ²−9λ+16=0, whose coefficients satisfy σ₁=CA²=9 and σ₂= (CA+1) ²=16 with CA=dim (T₂g) =3.