The Koide Relation as a Cyclotomic-12 Closed Form

Koide's 1983 relation Q ≡ (mₑ + m_μ + m_τ) / (√ (mₑ) + √ (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 = (√ (mₑ), √ (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 Nc = 3 via θ = (Nc² - 1) / (4Nc²). The absolute lepton mass scale follows from first principles: y_τ = 1/98 arises as cV/N_ mod2, where cV = 1/49 is the canonical Z₇ potential coefficient and N_ mod2 = 2 the binary tape level, giving m_τ = vH/ (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 Nc + θ = 29/9 also controls the neutrino mass-squared splitting ratio Δ m²₂1/Δ m²₃1 to 0. 52\% from NuFIT~6. 0. An S₃-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.