The Koide Relation as a Cyclotomic-12 Closed Form

Koide's 1983 phenomenological identity Q (m_e + m_ + m_ )/( + + )^2 = 2/3 remains one of the most striking numerical coincidences in the charged-lepton mass spectrum, with no accepted structural origin. We present three elementary observations that together promote the Koide relation from a numerical curiosity to a concrete structural statement about the lepton sector. (1) The relation is algebraically equivalent to the geometric condition that the vector v = (, , ) makes a 45^ angle with the democratic axis e = (1,1,1)/; the PDG values place v at 44.99974^ , i.e. 0.95 arcseconds from the exact /4 cone. (2) Solving the relation as a quadratic in with (, ) as inputs yields the closed form which, substituting PDG values for m_e and m_ , predicts m_ = 1.776969 GeV against the PDG value m_ = 1.77686 GeV — a relative error of 61 ppm or 0.91 _ PDG. (3) The surd coefficients of the closed form satisfy the cyclotomic-12 identities (2 + ) = 4 ^2( /12) and (1 + )^2 = 8 ^2( /12), placing the charged-lepton mass spectrum inside the cyclotomic-12 atom family /12, ( /12), \. All identities used are machine-checked in Lean 4 against Mathlib (standard axiom signature, zero sorry). An accompanying dynamical observation constructs an S_3-equivariant Newton-step operator on R^3 that has the Koide null cone as its attractor set, giving an explicit dynamical realisation of the 45^ condition; this result is presented but not required for the main prediction. None of the observations above depend on any particular physics framework, cosmological assumption, or new axiom; they are direct elementary facts about the charged-lepton mass spectrum. (4) The structural origin of the 45^ condition is identified : within the UGP framework, the Koide phase = 2/9 follows unconditionally from the QCD colour rank N_c = 3 via the strand-count identity = (N_c^2 - 1)/(4N_c^2), machine-checked in Lean 4 as koide\_angle\_from\_N\_c\_pure (zero sorry)....