The Koide Relation as a Cyclotomic-12 Closed Form

Koide's 1983 phenomenological identity Q (mₑ + 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 mₑ 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 S₃-equivariant Newton-step operator on R³ 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 Nc = 3 via the strand-count identity = (Nc² - 1) / (4Nc²), machine-checked in Lean 4 as koide\ₐngle\from\N\c\ₚure (zero sorry). . . .