Cyclotomic-12 Structure in the Charged-Fermion Mass Spectrum

We present two structural mass relations that reduce the nine charged-fermion masses to two empirical scale inputs, and identify the underlying cyclotomic-12 geometric and SU(5)/SO(10) representation-theoretic structure. (1) TT relation: (m_u_g/m_ _g) = ( /6) 2^g + /8 holds against PDG masses at $ 0.37 -free inter-generational identities, with null density 6 10^-6 under 10^6 random-coefficient trials. (2) VV relation: (m_d_g) = 9 (m_u_g) - 6 (m_ _g) - 14 holds at $ 0.17 10^-5 under random triple-permutation of log-mass inputs. Combined with the Koide closed form , all nine charged-fermion masses are determined from two scalar inputs. We identify the cyclotomic-12 geometric origin of the TT coefficient = /6: it is the half-opening angle of the A_2 Weyl chamber (proved in Lean 4 as , zero sorry). The full TT structure is realised by a two-flavon Froggatt-Nielsen UV completion whose flavon VEVs are global minima of a Z_6 Z_16-invariant Cartan-torus potential (Lean-certified), and the generation-doubling factor 2^g is proved via a binary cascade construction (Lean-certified). The three VV coefficient values (13/9,-7/6,-5/14) admit exact SU(5)/SO(10) group-theoretic identities: _d = 1 + rank(SU(5))/N_c^2, _d = -(1 + Y_Q_L), and _d = - (45_SU(5))/ (126_SO(10)), machine-checked in Lean as (zero sorry). These identities are exact group-theoretic facts [T]; however, each value also saturates the pre-registered algebraic basis (triple-target null rate $54.3 and the naive one-loop SM Yukawa RG realization of the EW-scale log-linear VV functional form fails (anomalous closure; see companion paper ). Accordingly the coefficient-value interpretations are classified [C] and the log-linear EW mechanism is algebraically identified ( ; vv\_mechanism\_algebraic, ). The Majorana Higgs dimension factors through the charged-lepton mirror offset as (126_SO(10)) = 2 N_c^2 , exhibiting a cross-sector bridge between the down-quark and neutrino sectors.... Supplementary information (supplementary_information.pdf) is bundled as a second file in this Zenodo record.