The Koide Relation as a Cyclotomic-12 Closed Form

Koide's 1983 relation Q ≡ (m_e + m_μ + m_τ)/(√(m_e) + √(m_μ) + √(m_τ))^2 = 2/3 has no accepted structural origin. We promote it from a numerical coincidence to a structural statement via three interlocking results. Geometric form. The relation is equivalent to v = (√(m_e), √(m_μ), √(m_τ)) lying on the 45^ cone about the democratic axis (1,1,1)/√(3); PDG values place v at 44.99974^ , just 0.95 arcseconds from exact. Closed form. Solving as a quadratic in √(m_τ) yields predicting m_τ = 1.776969~GeV against the PDG value 1.77686~GeV—a relative error of 61~ppm (0.91σ_ PDG). The surd coefficients satisfy cyclotomic-12 identities, placing the lepton mass spectrum in the atom family \π/12, (π/12), √(3)\. Structural origin. Within the UGP framework, the Koide phase θ = 2/9 is derived unconditionally from the QCD colour rank N_c = 3 via θ = (N_c^2 - 1)/(4N_c^2). The absolute lepton mass scale follows from first principles: y_τ = 1/98 arises as c_V/N_ mod2, where c_V = 1/49 is the canonical Z_7 potential coefficient and N_ mod2 = 2 the binary tape level, giving m_τ = v_H/(98√(2)) = 1776.57~MeV (-2.42σ from PDG~2024) with the SRRG Higgs VEV as the sole dimensional anchor. The closed form then predicts all three charged-lepton masses with zero free parameters. The combination N_c + θ = 29/9 also controls the neutrino mass-squared splitting ratio Δ m^2_21/Δ m^2_31 to 0.52\% from NuFIT~6.0. An S_3-equivariant Newton-step operator with the Koide null cone as its attractor provides a dynamical realisation of the 45^ condition. All identities are machine-checked in Lean~4 against Mathlib with zero .